Geometric Function Theory and Applications

BOOK OF ABSTRACTS

Generalized Functions 2004Topics in PDE, Harmonic analysis and Mathematical

physics

Novi Sad, September 22-28, 2004

Organized by:

• Department of MathematicsFaculty of ScienceUniversity of Novi Sad

• SANU

General Sponsors:

• Ministarstvo nauke i zastite zivotne okoline Srbije

• Pokrajinski sekretarijat za nauku Vojvodine

• Izvrsni odbor skupstine grada Novog Sad

• International Mathematical Union

Sponsors:

• NEXT, Subotica

• Elektrovojvodina, Novi Sad

• Mlekara, Novi Sad

• Pokrajinsko vece SSS Vojvodine

• Gradsko vece SSS Novog Sada

Program Committee:

B. Stankovic Honorary President, J.F. Colombeau, S-Y. Chung, V. Danilov,I. Dimovski, Yu. Drohzhinov, R. Estrada, T. Gramchev, J. A. Marti, M. Ober-guggenberger, S. Pilipovic, H. Render, L. Rodino, M. W. Wong

Organizing Committee:

S. Pilipovic (President), M. Veskovic (Vice-President, Vice- Rector of Novi SadUniversity), M. Nedeljkov (Vice-President), N. Teofanov (Secretary), D. Selesi(Vice-Secretary), N. Aleksic (Department Secretary), J. Aleksic, A. Eida, D.Herceg (Department Director), M. Jovanov, Z. Lozanov- Crvenkovic, S. Konjik,V. Maric (Secretary of the Novi Sad Branch of the Serbian Academy), M. Mija-tovic, Lj. Oparnica, D. Perisic, D. Rakic, M. Stojanovic, A. Takaci, Dj. Takaci,P. Tomic (Dean of the Faculty of Sciences), M. Zigic.

Associative algebras of the p-adic distributions

S. AlbeverioInstitut fur Angewandte Mathematik, Universitat Bonn, Germany SFB 611,Bonn, BiBoS, Bielefeld - Bonn, CERFIM, Locarno and Acc. Arch. USI

(Switzerland)

A. Yu. KhrennikovInternational Center for Mathematical Modeling in Physics and Cognitive

Sciences MSI, Vaxjo University, Swedene-mail: [email protected]

V. M. ShelkovichSt.-Petersburg State Architecture and Civil Engineering University, Russia

e-mail: [email protected]

The theory of p-adic distributions (generalized functions) plays an impor-tant part in various mathematical and physical problems, in particular it isintensively used in theory of p-adic strings and p-adic quantum mechanics. Inorder to deal with linear and nonlinear singular problems of the p-adic anal-ysis, we present some singular algebraic constructions of the p-adic theory ofdistributions [1], [2].

The p-adic Colombeau-Egorov algebra of generalized functions is constructed.This algebra is also an associative and commutative convolution algebra. Un-like the standard Colombeau theory, in the p-adic theory generalized functionsare uniquely determined by their pointvalues. The operations of (fractional)partial differentiation and (fractional) partial integration are introduced by theVladimirov’s pseudo-differential operator.

Using results of [3], we construct an associative and commutative algebra ofasymptotic distributions generated by the linear span of p-adic associated ho-mogeneous distributions. This algebra is also an associative and commutativeconvolution algebra. Any element of this algebra is represented as a weak asymp-totic series whose coefficients are associated homogeneous distributions. Usingideas from [4], one can identify the algebra of asymptotic distributions with thespace of p-adic Bruhat–Schwartz vector-valued distributions. This algebra canbe embedded as a subalgebra into p-adic Colombeau-Egorov algebra.

In the framework of these algebras the p-adic analog of the Schrodingerequation with delta-shaped potentials is studied.

REFERENCES

[1] S. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, p-adic Colombeau-Egorov typetheory of generalizef functions, Report 03008, Feb 2003, MSI, Vaxjo University,Sweden, 2003, pp. 16.

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[2] S. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, Nonlinear problems in p-adicanalysis: Algebras of p-adic distributions, Preprint no. 93, Bonn University, SFB611, Germany, 2003, pp. 45.

[3] S. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, Associated homogeneous p-adic generalized functions, Dokl. Ross. Akad. Nauk, 393, no. 3, (2003) 1–4. En-glish transl. in Russian Doklady Mathematics., 68.6, no. 1, (2003).

[4] A. Yu. Khrennikov, V. M. Shelkovich, O. G. Smolyanov, Locally convex spacesof vector-valued distributions with multiplicative structures, Infinite-DimensionalAnalysis, Quantum Probability and Related Topics, 5, no. 4, (2002), 1–20.

The sharp topology on the full Colombeau algebra

Jorge AragonaDepartamento de MatematikaUniversidade de Sao Paolo

Brasile-mail: [email protected]

We introduce a topology T (W ) on the full Colombeau algebra GW ). LetT be the topology induced by T (W ) on K(:= ring of the (full) generalizednumbers). Then, the topology induced by T (W ) (resp. T ) on Gs(W ) (resp.Ks) is the Scarpalezos sharp topology on Gs(W ) (resp. Ks).

Compactly Supported Generalized Integral KernelOperators

Severine Bernard

Joint work with J.-F. Colombeau and A. Delcroix

Equipe Analyse Algebrique Non LineaireLaboratoire Analyse, Optimisation, ControleFaculte des Sciences Exactes et NaturellesUniversite des Antilles et de la Guyane97159 Pointe-a-Pitre cedex Guadeloupee-mail: [email protected]

After the first approach on integral kernel operators, done by D. Scarpalezosin [3] and in parallel with the work of C. Garetto and alii [2], but with differ-ent goals, we propose a general study of integral operators in the frameworkof Colombeau generalized functions. Indeed, the theory of distributional kerneloperators has some limits since, for example, they can not be composed unre-strictedly. This is a drawback for some applications in theoretical physics. The

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theory of non linear generalized functions, which appears as a natural extensionof the distributional one, seems to be the suitable framework to overcome theselimitations.

In order to simply the computations and proofs, we will just present here thecase of compactly supported generalized integral kernel operators, but a moregeneral study has been done in [1].

We begin by defining such operators and showing that they are characterizedby their kernel. Indeed, if X (respectively Y ) is an open subset of IRm (respec-tively IRn) and H is an element of GC(X × Y ) (that is a compactly supportedgeneralized function) then the following map

H : G(Y ) → G(X)f 7→ H(f) =

∫H(·, y)f(y)dy

is a linear operator called generalized integral operator of kernel H. Moreover,the application H → H is injective from GC(X × Y ) to L (G(Y ),G(X)). Then,we make the link with the classical theory since the study of classical casesas H ∈ D(X × Y ) or H ∈ E ′(X × Y ) shows that our theory exactly extendsthe theory of distributional kernel operators. Furthermore, we show that suchoperators can be composed unrestrictedly. We take advantage of this to definethe exponential of a subclass of compactly supported generalized integral kerneloperators, that is whose kernel has good growth properties with respect to theparameter epsilon. And finally this exponential satisfies the expected functionalproperties of the exponential.

In the above mentioned paper [1], we show that these results are also avail-able for generalized integral operators whose kernel belongs to a special class ofgeneralized Sobolev algebras.

REFERENCES

[1] S. Bernard, J.-F. Colombeau, A. Delcroix. Generalized integral operators,Preprint AOC, 2004.

[2] C. Garetto, T. Gramchev, M. Oberguggenberger. Pseudo-differential operatorsand regularity theory, Preprint, 2003.

[3] D. Scarpalezos “Colombeau’s Generalized Functions: Topological Structure, Mi-crolocal Properties, a simplified point of view”, Prepublications mathematiquesde l’U.R.A.212 ”Theories geometriques”, Universite Paris 7, 1993.

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Hugoniot-Maslov chains of a shock wave in ConservationLaws

Panters Rodrıguez BermudezUniversidad de la Habana,Cubae-mail: [email protected]

Baldomero Valino AlonsoUniversidad de la Habana,Cuba


We give a theoretical foundation to the asymptotical development proposedby V.P.Maslov for shock type singular solutions of conservations laws, in theframework of Colombeau theory of generalized functions. Indeed, operatingwith Colombeau differential algebra of simplified generalized functions,we proofthat Maslov-Hugoniot chains are necessary conditions for the existence of shockwaves in conservation laws with polinomial flows. As a particular case, theseequations include the Hugoniot-Maslov chains for shock waves in Hopf-Burgersequation.

Approximate solution of the generalized Riemann problemfor a system of conservation laws with quadratic flow

Panters Rodrıguez BermudezUniversidad de la Habana,Cubae-mail: [email protected]

Baldomero Valino AlonsoUniversidad de la Habana,Cuba


In this work we solve the generalized Riemann problem for systems of con-servation laws with quadratic flow. First, we obtain the Hugoniot-Maslov chainapplying the theorem of the theoretic foundation in Colombeau language ofMaslov asymptotical method. Truncation of the chain allows to obtain an ap-proximation of the asymptotical development of the shock type singular solu-tion. By means of numerical simulations we ilustrate the efficacy of the methodsugested by Maslov.

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Hypoelliptic Systems Connected with Newton’sPolyhedron

Chikh BouzarDepartment of Mathematics. Oran Essenia University. Algeria


Leonid VolevichKeldysh Institute of Applied Mathematics. Russian Academy of Sciences.

Miusskaya Square 4. 125047 Moscow, Russia.e-mail: [email protected]

A class of hypoelliptic partial differential operators connected with the New-ton polyhedron of the symbol was introduced independently by different authors.These operators are a far reaching generalization of elliptic scalar differential op-erators, as well as q quasielliptic and their products with different q. They arecalled N quasielliptic or multi-quasielliptic differential operators , see [3] or [1].

This work is an attempt to include scalar N quasi-elliptic operators in a classof systems which also will be called N quasi-elliptic. We prove that positively Nquasi-elliptic systems with variable coefficients, in the case of two independentvariables, are hypoelliptic.

REFERENCES

[1] P. Boggiato, E. Buzano, L. Rodino, Global hypoellipticity and spectral theory,Academic Verlag,1996.

[2] C. Bouzar, R. Chaili, Gevrey vectors of multi-quasi-elliptic systems. Proc. Amer.Math. Soc., Vol. 131, N 5, P. 1565-1572, 2003.

[3] S.G. Gindikin,L.R. Volevich, The Method of Newton’s polyhedron in the theoryof partial differential equations. Kluwer Academic, 1992.

Complex powers of gobally hypoelliptic operators andapplications

Ernesto BuzanoUniversita’ di Torino and Fabio Nicola,

Politecnico di Torino, Italy

We define complex powers for globally hypoelliptic operators whose spectrummay have zero as an accumulation point and whose Weyl symbol is allowed totend both to zero and to infinity in different directions.

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Our analysis is based on the definition of complex powers of a non-negativeoperator due to Balakrishnan. As an application we also study the semigroupgenerated by a non-negative pseudodifferential operator, that is its heat kernel.

Multianisotropic Gevrey regularity and iterates ofoperators

Daniela CalvoDipartimento di Matematica

Via Carlo Alberto 10, 10123 Torino, Italye-mail: [email protected]

Gagik H. HakobyanChair of High Math., Department of Physics, Yerevan State University,

375025, Al Manoogian Str. 1, Armeniae-mail: [email protected]

We consider the iterates of a linear partial differential operator with constantcoefficients P (D) =

∑|α|≤m γαDα and the related Gevrey classes:

Let Ω be an open subset of Rn and d > 0. We say that f belongs to GdP (Ω) if

f ∈ C∞(Ω) and for any compact subset K of Ω there is a constant C > 0 suchthat ‖P j(D)f‖L2(K) ≤ Cj+1(j!)d, ∀j = 1, 2, . . . .It is known that the inclusion Gs ⊂ Gsm

P is always satisfied by any operator oforder m, while for any s > 1 the condition Gsm

P (Ω) = Gs(Ω) is equivalent tothe ellipticity of P (D) (cf. Bolley-Camus and Metivier).Considering the more general case of hypoelliptic operators, we study the rela-tion between the inclusion of Gd

P in some generalized Gevrey classes and growthconditions on the symbol P (ξ) =

∑|α|≤m γαξα of P (D). At this aim, we intro-

duce the multianisotropic Gevrey classes, associated to a complete polyhedronN in Rn (a convex polyhedron included in Rn

+ having vertices with rationalcoordinates and such that the outer normals of the faces of N have strictlypositive components). Namely:A function f ∈ C∞(Ω) belongs to the multianisotropic Gevrey class GN (Ω) iffor any compact subset K of Ω there is a constant C > 0 such that ‖Dαf‖K ≤Cj+1j!, ∀α ∈ N (j) ∩ Nn, j = 0, 1, . . . , where N (j) = ν ∈ Rn

+ : νj ∈ N.

Then we will prove our main result:Let P (D) be an operator, N be a completely regular polyhedron. If there is aconstant d > 0 such that Gd

P (Ω) ⊂ GN (Ω), then for a constant C > 0 we have

hN (ξ) ≤ C(|P (ξ)| 1d + 1

), ∀ξ ∈ Rn, (1)

where hN (ξ) =∑

α∈N 0 |ξα|, the sum ranging over N 0, the set of the verticesof N . Conversely, the condition (1) implies, under the hypothesis that P ishypoelliptic, the inclusion Gd

P (Ω) ⊂ GN (Ω).

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We also show the connection of our results with an important feature of hypoel-liptic operators, represented by the regular weight of hypoellipticity introducedby Kazharyan.In the particular case that N is the Newton polyhedron of P , we recapture foroperators with constant coefficients the results of Bouzar-Chaili for multi-quasi-elliptic operators, of Zanghirati for quasi-elliptic operators, and the standardresults for elliptic operators.

Analytic Functions Associated With Ultradistributions

Richard Carmichael

A survey of results concerning analytic functions in tubes which have ul-tradistribution boundary values is given. Recovery of the analytic functions interms of integral transforms associated with the boundary value is obtained.The ultradistributions of particular interest are those of type Lp and the tem-pered ultradistributions.

The Stability of some Boussinesq Equations

Daduan ChenDept. of Math, Shanghai University

99 Shangda Rd, Shanghai, P. R. of China, 200436e-mail: [email protected]

Youhua HeDept. of Math, Shanghai University

99 Shangda Rd, Shanghai, P. R. of China, 200436e-mail: [email protected]

Boussinesq equation is a simplified model of the atmospheric movement equa-tion . It is applicable to mesoscale, non-static equilibrium, quasi-incompressiblefluid movement. Ordinarily, it takes the following form:

A

d

V

dt = − 1ρ∇p− 2

Ω×

V − g +

F

∇ ·

V = 0dθdt = Q + ∆θdqdt = Qq + ∆q

If the influence of moisture is ruled out for consideration, the moisture equa-tion can be taken away. In movement equations, the viscosity of turbulent flow

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is mainly considered. The coefficients of viscosity and thermo-dissipation inhorizontal and vertical direction are regarded as different. The temperaturechange caused by the vertical motion is also taken into account. Under suchcircumstances, the equation goes like this:

(B)

d

V

dt = − 1ρ∇p− 2

Ω×

V − g +

(KH

M (∂xx + ∂yy) + KVM∂zz

)

V

∇ ·

V = 0dθdt = c(z)w + KH

H

(∂2θ∂x2 + ∂2θ

∂y2

)+ KV

H∂2θ∂z2

If the viscosity of turbulent flow is replaced by Reyleigh friction and thethermo-dissipation replaced by Newton cooling, we come to the model below:

(C)

d

V

dt = − 1ρ∇p− 2

Ω×

V − g − k

V

∇ ·

V = 0dθdt = c(z)w − k1θ

The main purpose of this paper is to study the stability of the above threemodels. The well posedness of the initial (boundary) value problem of thosemodels are also discussed based on stratification theory [1], [2].

First, we prove that the system of equations (A) is stable. For some suitableinitial (boundary) data, the initial (boundary) problem is well-posed. But if theinitial values are given on t = t0, the related problem is ill-posed.

Then, we prove that the system of equations (B) is unstable. For any initialvalue given on any hyper-surface, the problem is always ill-posed.

Finally, similar to (A), the system of equations (C) is stable.

REFERENCES

[1] Shih Weihui, Solutions analytiques de quelques equations aux derivees partiellesen mecanique des fluides. Paris: Hermann, 1992.

[2] Chen Daduan, Shi Weihui, On the Formal Solution of Initial Value Problem ofNavier-Stokes Equation, Applied Mathematics and Mechanics , Vol.21(12), 2000,1432-1439.

[3] Chen Daduan, He Youhua, Stabilities of Boussinesq Approximate Equations forNon-Static Rotating Fluid, Chineses Journal of Atmospheric Sciences, Vol.26,No.3, 2002, 293-298.

Identification of Conductivity in the Electrical Networks

Soon-Yeong ChungDepartment of Mathematics, Sogang University, Seoul 121-742, Korea

In this talk, we discuss the inverse problem of identifying the connectivityand the conductivity of the links between adjacent pair of nodes in a network,in terms of an input-output map. To do this we deal with an elliptic operator∆ω and an ω-harmonic function on the graph, with its physical interpretation

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as a diffusion equation on the graph, which models an electric network. Afterderiving the basic properties of ω-harmonic functions, we prove the solvability of(direct) problems such as the Dirichlet and Neumann boundary value problems.Our main result is the global uniqueness of the inverse conductivity problem fora network under a suitable monotonicity condition.

Generalized functions with spectral gaps

Yun-Sung ChungDepartment of Mathematics, Sogang University, Seoul 121-742, Korea

Jong-Ho KimDepartment of Mathematics, Sogang University, Seoul 121-742, Korea


We show that for a given ultradifferentiable function f , an ultradistributionT supported in the set x ∈ Rn : |f(x)| ≤ µ , µ > 0, can be characterized bythe behavior of the sequence of the successive multiplication powers of f actingon T . We also deal with the ‘dual’ characterization - an ultradistribution Tsupported in x ∈ Rn : |f(x)| ≥ µ can be characterized by the behavior ofa sequence Tkk∈N0

, where T0 = T and fTk+1 = Tk, k = 0, 1, 2, · · · . As anapplication, we generalize the results due to Gabardo on characterizing the sinefunctions.

New formulas of physics from a nonlinear theory ofgeneralized functions

Jean-Francois Colombeaue-mail: [email protected]

We explain how one can obtain easily new formulas on physical phenomena ofgreat interest and that can be checked experimentally. The point is to state thelaws of physics in a way which is deeper than permitted by classical mathematicsand distributions. The method appears to be general and so of wide interest.We explain the method with elastoplasticity and present a work under study onlinear acoustics in a fast moving medium.

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Delta-shock solutions of hyperbolic systems ofconservation laws. The theory of interaction.

V. G. DanilovMoscow Technical University of Communication and Informatics, Russia


V. M. ShelkovichSt.-Petersburg State Architecture and Civil Engineering University, Russia,


In the framework of the weak asymptotics method [1]–[4] the new definitionsof a δ-shock wave type solution for the system

ut +(F (u, v)

)x

= 0, vt +(G(u, v)

)x

= 0,

and zero-pressure gas dynamics system

vt +(vu

)x

= 0, (vu)t +(vu2

)x

= 0

are introduced, where F (u, v) and G(u, v) are smooth functions and linear withrespect to v, x ∈ R. These definitions are close to the standard definition ofL∞ type solutions and relevant to the notion of δ-shocks.

We study the dynamics of propagation and interaction of two δ-shocks forthe above systems, i.e., we solve the Cauchy problems with the initial data ofthe form

u0(x) = u00(x) +

∑2k=1 u0

k(x)H(−x + x0k),

v0(x) = v00(x) +

∑2k=1

(v0

k(x)H(−x + x0k) + e0

kδ(−x + x0k)

),

where u00(x), u0

k(x), v00(x), v0

k(x) are smooth functions, e0k are constants, k = 1, 2,

x01 < x0

2. The process of merging two δ-shock waves into one is analyticallydescribed.

REFERENCES

[1] V. G. Danilov, V. M. Shelkovich, Propagation and interaction of delta-shockwaves, The Ninth International Conference on Hyperbolic prorblems. Theory,Numerics, and Applications. Abstracts. California Institute of Technology, Cali-fornia USA, March 25–29, 2002, 106–110.

[2] V. G. Danilov, V. M. Shelkovich, Propagation and Interaction of Delta-ShockWaves of a Hyperbolic System of Conservation Laws, Hyperbolic Problems: The-ory, Numerics, Applications Proceedings of the Ninth International Conference onHyperbolic Problems held in CalTech, Pasadena, March 25-29, 2002 Hou, ThomasY.; Tadmor, Eitan (Eds.), Springer Verlag, 2003, 483–492.

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[3] V. G. Danilov, G. A. Omel′yanov, V. M. Shelkovich, Weak Asymptotics Methodand Interaction of Nonlinear Waves, in Mikhail Karasev (ed.), “AsymptoticMethods for Wave and Quantum Problems”, Amer. Math. Soc. Transl. Ser. 2,208, 2003, 33–165.

[4] V. G. Danilov, V. M. Shelkovich, Propagation and interaction of δ-shock wavesto hyperbolic systems of coservation laws, (To appear in Dokl. Ross. Akad. Nauk,394, no. 1, (2004)). English transl. in Russian Doklady Mathematics., 69.1, no. 1,(2004).

Generalized Integral Operators and Schwartz KernelTheorem

Antoine DelcroixEquipe Analyse Algebrique Non Lineaire

Laboratoire Analyse, Optimisation, ControleFaculte des sciences - Universite des Antilles et de la Guyane

97159 Pointe-a-Pitre Cedex Guadeloupee-mail: [email protected]

We continue the investigations in the field of generalized integral opera-tors initiated by a pioneering work of D. Scarpalezos [3], carried on by J.-F.Colombeau (personal communication) and S. Bernard et alii [1] in view of appli-cations to physics, by C. Garetto et al [2] with applications to pseudo differentialoperators theory and questions of regularity.

The following results holds: Every H belonging to G (Rm ×Rn) defines alinear operator from GC (Rn) to G (Rm) by the formula

H : GC (Rn) → G (Rm) , f 7→ H(f) with H(f)(x) =[∫

Hε(x, y)fε(y) dy

]where (Hε)ε (resp. (fε)ε) is any representative of H (resp. f) and [ · ] is theclass of an element in G

(Rd

). (G

(Rd

)denotes the usual quotient space of

Colombeau simplified generalized functions, while GC

(Rd

)is the subspace of

elements of G(Rd

)compactly supported.)

Conversely, in the distributional case, the well known Schwartz kernel theo-rem asserts that each linear map Λ from D (Rn) to D′ (Rm) continuous for thestrong topology of D′ can be represented by a kernel K ∈ D′ (Rm ×Rn) that is

∀f ∈ D (Rn) , ∀ϕ ∈ D (Rm) , (Λ (f) , ϕ) = (K, ϕ⊗ f) .

In the spirit of Schwartz theorem, we prove that in the framework of Colombeaugeneralized functions any net of linear maps (Lε : D (Rn) → C∞ (Rm))ε satisfy-ing some growth property with respect to the parameter ε (the strongly moderatenets) gives rise to a linear map L : GC (Rn) → G (Rm) which can be represented

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as an integral operator. This means that there exists a generalized functionHL ∈ G (Rm ×Rn) such that L(f) =

∫HL(·, y)f(y) dy for any f belonging to

a convenient subspace of GC (Rn).Moreover, this result is strongly related to Schwartz Kernel theorem in the

following sense. We can associate to each linear operator Λ : D (Rn) → D′ (Rm)satisfying the hypothesis above mentioned a strongly moderate map LΛ andconsequently a kernel HLΛ ∈ G (Rm ×Rn) such that, for all f in D (Rn), Λ (f)and HLΛ (f) are equal in the generalized distribution sense.

REFERENCES

[1] S. Bernard, J.-F. Colombeau, A. Delcroix. “Generalized Integral Opera-tors”. Preprint AOC, 2004.

[2] S. Garetto, T. Gramchev, M. Oberguggenberger. “Pseudo-Differentialoperators and regularity theory”. Preprint, 2003.

[3] D. Scarpalezos. Colombeau’s generalized functions: Topological structures; Mi-crolocal properties. A simplified point of view. Prepublication Mathematiques deParis 7/CNRS, URA212, 1993.

OPERATIONAL CALCULI FOR BOUNDARY VALUEPROBLEMS

Ivan DimovskiInstitute of Mathematics, Bulgarian Academy of Sciences

Sofia 1090, Bulgaria

For most of mathematicians and engineers there is only one operationalcalculus developed either by the Laplace transform or directly by Mikusinski’sapproach. Nevertheless, there exist infinitely many nonclassical operationalcalculi intended for BVPs. In principle, for each linear BVP for a PDE allowingseparation of the variables, it is possible to develop a special operational calculus.The main problem is to find a suitable convolution. This problem had beensolved by the author and his collaborators in the last 30 years (for example, see:I. Dimovski, Convolutional Calculus, Kluwer Acad. Publ. (Ser. Math. and ItsAppl-s, East European Ser. 43), Dordrecht-Boston-London, 1990).

The ultimate goal of such an operational calculus in solving a BVP is tofind explicit representation of the solution. This representation is given as anextension of the classical Duhamel principle, but for the space variables. In sucha way, an effectivization of the Fourier method could be achieved.

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On the spectra of operator matrices

Dragan S. DjordjevicDepartment of Mathematics and Informatics, Faculty of Science and

Mathematics,P.O. Box 224, 18000 Nis, Serbia

e-mail: [email protected], [email protected]

Some new results concerning the perturbation of the spectra of 2×2 operatormatrices are presented.

Generalized Functions on Adeles and their Applications inAdelic Quantum Mechanics

Branko DragovichInstitute of Physics, P.O.Box 57,

11001 Belgrade Serbia and Montenegro

Yauhen M. Radyna , Yakov V. RadynoMechanics and Mathematics Faculty, Belarusian State University,

F. Skaryna av. 4, 220050, Minsk, Belarus.

The following aspects will be discussed in this contribution:1. Adeles and ideles.2. Schwartz-Bruhat functions and distributions on adeles.3. Fourier transform of distributions on adeles. Tate formula and

connections to Riemann zeta-function.4. Mnemofunctions on p-adic numbers and adeles.5. Multiplication of distributions.6. Adelic Hilbert space.7. Adelic quantum mechanics.8. Evolution operator in adelic quantum mechanics.9. Generalized functions on adeles in some other adelic quantum models.

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Multidimensional Tauberian theorems for Banach-spacevalued generalized functions

Yu. N. DrozhzhinovSteklov Mathematical Institute, RAS

Multidimensional Tauberian theorems for the standard averages of temperedBanach-space valued distributions are discussed in the present talk. These re-sults enable one to judge by the asymptotic behaviour of the averages aboutthe asymptotic behaviour of the generalized function itself. The role of theasymptotic scale in these results is performed by the class of regularly varyingfunctions. Special attention is paid to averaging kernels such that several theirmoments or linear combinations of moments vanish. Important in these resultsis the structure of the zero set of the Fourier transformations of the kernels inquestions.

The results so established are applied to the study of the asymptotic prop-erties of solutions of the Cauchy problem for the heat equation in the classof tempered distributions, to the problem of the diffusion of many-componentgas, and to the problem of the absense of the phenomenon of compensation ofsingularities for holomorphic functions in tube domains over acute cones.

This research was carried out with the financial support of the RussianFoundation for Basic Research (grant no. 02-01-00076) and th Program of theSupport of Leading Scientific Schools (grant no. -1542.2003.1).

REFERENCES

[1] Yu.N. Drozhzhinov and B.I. Zavialov, ”Tauberian theorems for generalized func-tions with values in Banach spaces”, Izv. Ross. Akad. Nauk Ser. Mat. 66:4 (2002),47–119; English transl. in Izv. Math. 66 (2002).

[2] Yu.N. Drozhzhinov and B.I. Zavialov, ”On a theorem of the Tauberian type forgeneralized functions with values in Banach spaces”, Dokl. Ross. Akad. Nauk 391(2003), 158–161; English transl. in Docl. Math. 68 (2003).

[3] Yu.N. Drozhzhinov and B.I. Zavialov, ”Multidimensional Tauberian theorems forBanach-space valued generalized functions”, Mathem. Sb. 194:11, (2003), 17–64;English transl. in Sbornic: Mathematics 194:11 1599–1646.

Solvability on ultradistribution spaces

Atsuhiko EidaSchool of Computer Science, Tokyo University of Technology


We recall the space of tempered ultradistributions which is denoted by S∗′

and which is the strong dual of the projective or the inductive limit with respect

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to h > 0 of the spaces of the smooth functions ϕ on Rn which satisfy

ph(ϕ) = sup

h|α|

M|α||ϕ(α)(x)|eM(h|x|) ; α ∈ N,x ∈ Rn

< ∞,

where ∗ = (Mp) or Mp and M(ρ) = sup

logρp

Mp; p ∈ N

. Let P (D) be

a non-zero linear partial differential operator with constant coefficients. ThenP (D) : S∗′ −→ S∗′ is surjective.

An Inverse Function Theorem for Generalized Functions

Evelina ErlacherDepartment of Mathematics, University of Vienna

Nordbergstraße 15, A-1090 Vienna, Austria

Michael GrosserDepartment of Mathematics, University of Vienna

Nordbergstraße 15, A-1090 Vienna, Austria

The question of inversion of Colombeau functions in Rn has not been ad-dressed so far. However, in certain applications “discontinuous coordinate trans-forms” have been employed ([1],[2]). We discuss several notions of invertibilityof Colombeau functions and present a theorem providing sufficient conditionsto guarantee the latter.

REFERENCES

[1] Kunzinger, M., Steinbauer R., A note on the Penrose junction conditions, ClassicalQuantum Gravity, 16:1255–1264, 1999

[2] Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R., GeometricTheory of Generalized Functions, Math. Appl. 537, Kluwer Acad. Publ., Dordrecht2001

15

Radius of curvature and distributions

Ricardo EstradaDepartment of MathematicsLouisiana State University

Baton Rouge, Louisiana 70803U.S.A.

We show that any continuous plane path that turns to the left has a well-defined distribution, that corresponds to the usual radius of curvature of smoothpaths. We show that the distributional radius of curvature determines the pathuniquely except for a translation. We show that Dirac delta contributions in theradius of curvature correspond to facets, that is, flat sections of the path, andshow how a path can be deformed into a facet by letting the radius of curvatureapproach a delta function, and conversely. We also give applications to crystals,where the energy satisfies a differential equation whose source term is a multipleof the radius of curvature.

Duality theory and pseudodifferential techniques forColombeau algebras: generalized kernels and microlocal

analysis ( Parts I and II )

Claudia GarettoDipartimento di Matematica Universita‘ di Torino, Italia

Guenther HoermannFakultaet fuer Mathematik, Universitaet Wien, Austria

Starting from abstract topological modules over Colombeau generalized com-plex numbers we discuss duality of Colombeau algebras. In particular, we focuson generalized delta functionals and operator kernels as elements of dual spaces.A large class of examples is provided by pseudodifferential operators acting onColombeau algebras. By a refinement of symbol calculus we give a new char-acterization of the wave front set for generalized functions with applications tomicrolocal analysis.

16

The Cauchy problem and regularity of two - dimensionalwave maps

Vladimir GeorgievDipartimento di Matematica Universita’ di Pisa

via F.Buonarroti 2 56127 Pisa, Italye-mail: [email protected]

2000 AMS Subject Classification: 35L05, 35J10, 35P25, 35B40.Key words and Phrases: wave maps, global solution, hyperbolic target ; theauthors are partially supported by Research Training Network (RTN) HYKE,financed by the European Union, contract number : HPRN-CT-2002-00282.

We investigate the continuity properties of the solution operator to the wavemap system from R×Rn to a general nonflat target of arbitrary dimension, andwe prove by an explicit class of counterexamples that this map is not uniformlycontinuous in the critical norms on any neighbourhood of 0. This work generalizethe result of [3].

Further, we study the regularity properties of the wave maps, when thetarget i 2 - dimensional manifold of constant negative curvature. The result isclosely related to the particular cases of wave maps studied in [1] and [2].

REFERENCES

[1] Christodoulou D., Shadi Tahvildar - Zadeh A., On the regularity of spher-ically symmetric wave maps, Comm. Pure Appl. Math., 46, (1993) p. 1041 –1091

[2] Shatah, J.; Tahvildar-Zadeh, A. Sh., On the Cauchy problem for equivariantwave maps. Comm. Pure Appl. Math. 47 (1994), no. 5, 719–754.

[3] Tao, T., Ill-posedness for one-dimensional wave maps at the critical regularity.Amer. J. Math. 122 (2000), no. 3, 451–463.

Stability Properties for Nonlinear Evolution PDE withMultiple Characteristics

Todor GramchevDipartimento di Matematica e Informatica, Universita di Cagliari

via Ospedale 72, 09124 Cagliari, Italiae-mail: [email protected]

We propose iterative methods in scales of Banach spaces of both finitelysmooth and analytic-Gevrey functions for the study of Cauchy problems for

17

large classes of systems of evolution PDEs with multiple characteristics. Weinvestigate the dynamics for large t of the solutions to the Cauchy problem fornonlinear perturbations of the zero solutions. One of the crucial ingredients ofour approach is the use of ideas from normal form theory in Dynamical systemsand precise nonlinear estimates in scales of Sobolev and Gevrey type spaces.

Non-smooth differential geometry in the Colombeausetting

Michael GrosserInstitut fur Mathematik

Nordbergstraße 15A-1090 Wien

Austria

This talk is intended to provide an overview of the development of a “non-smooth differential geometry” which has recently taken place within the theoryof nonlinear generalized functions (cf. [1], [3], [2]).

A non-smooth differential geometry has to rest on two cornerstones: First,it has to be built upon (and, of course, to generalize) classical smooth geometry;secondly, it has to use a suitable theory of generalized functions permitting allthe relevant operations. If one tries to employ the well-known (linear) theory ofdistributions for this purpose (compare, e.g., [7]), one quickly arrives at severelimitations, caused by the fact that all constructions of differential geometryinvolving non-linear operations (e.g., multiplication) cannot be carried out dueto the very nature of distributions. Therefore, the theory of Colombeau algebrasprovides a promising way to generalize classical differential geometry to a non-smooth setting.

In the first part of this talk, we will present different types of Colombeaualgebras on smooth manifolds, outlining, in particular, their relative merits withrespect to different aims. Then, after a brief review of the limitations of apurely linear approach to differential geometry based on distributions, we willshow how to arrive at a satisfying non-linear theory by using concepts basedon Colombeau algebras. A topic of particular importance is that of manifold-valued generalized functions ([6]) which are indispensable for even formulatingthe notions of a flow of a given generalized vector field or of a geodesic in amanifold carrying a generalized (pseudo-)Riemannian structure (for the latter,see [4]). After mentioning some applications (e.g., in General Relativity) we willgive an outlook on some of the many promising routes of research in the fieldof non-smooth differential geometry which offer themselves for the near future(e.g., [5]).

REFERENCES

[1] Grosser, M., Farkas, E., Kunzinger, M., Steinbauer, R. On the foundations ofnonlinear generalized functions I, II. Mem. Amer. Math. Soc., 153(729), 2001.

18

[2] Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R. GeometricTheory of Generalized Functions, volume 537 of Mathematics and its Applications537. Kluwer Academic Publishers, Dordrecht, 2001.

[3] Grosser, M., Kunzinger, M., Steinbauer, R., Vickers, J. A global theory of algebrasof generalized functions. Adv. Math., 166:179–206, 2002.

[4] Kunzinger, M., Steinbauer, R. Generalized pseudo-Riemannian geometry. Trans.Amer. Math. Soc., 354(10):4179–4199, 2002.

[5] Kunzinger, M., Steinbauer, R., Vickers, J. Generalized connections and curvature.Preprint, 2003.

[6] Kunzinger, M., Steinbauer, R., Vickers, J. Intrinsic characterization of manifold-valued generalized functions. Proc. London Math. Soc., 87(2):451–470, 2003.

[7] Marsden, J. E. Generalized Hamiltonian mechanics. Arch. Rat. Mech. Anal.,28(4):323–361, 1968.

Time-Frequency Methods for Pseudodifferential Operators

Karlheinz GrochenigInstitute of Biomathematics and Biometry

GSF - National Research Center for Environment and HealthIngolstadter Landstrae 1

85764 Neuherberg, Germanye-mail: [email protected]

We will present a number of new results for pseudodifferential operators thatcan be obtained with methods from time-frequency analysis. The emphasis willbe on the action of pseudodifferential operators on modulation spaces and theuse of Banach algebra techniques.

Specifically, we will discuss three important results of Sjostrand on a newsymbol class that coincides with the modulation space M∞,1. We will give a newproof of the Banach algebra property and the Wiener property of this symbolclass and draw conclusions about spectral invariance of such pseudodifferentialoperators on modulation spaces.

The second topic, joint work with Elena Cordero, concerns a symbolic cal-culus of time-frequency localization operators. As a consequence we obtainstatements about the Fredholm property of certain localization operators andnew isomorphisms between weighted modulation spaces.

19

Polaroid operators and Weyl’s theorem

Robin E. HarteTrinity College, Dublin 2, Irelande-mail: [email protected]

“Polaroid” operators satisfy the isoloid condition of Berberian, together withpart of the condition “Weyl’s theorem holds”.

Colombeau type generalized functions through sequencespaces

M. F. Hasler(with A. Delcroix, S. Pilipovic, V. Valmorin)


We describe various Colombeau type algebras of generalized functions in aunified way as completion of sequence spaces equipped with ultra-seminorms,defined by sequences of exponential weights. This way the sharp topology ispresent from the very beginning of the construction.

The new approach allowed us to give new and improve existing results onultradistributions and periodic hyperfunctions, to cope with functorial issues,and describe different notions of association.

It also makes a connection to other classical branches of analysis, thus open-ing new perspectives and directions of investigations.

The results presented here are based on joint work with A. Delcroix, S. Pilipovicand V. Valmorin, to whom I express my deep gratitude.

On the Properties of Atmospheric Circulation Equation

Youhua HeDept. of Mathematics, Shanghai University99 Shangda Rd., Shanghai, China, 200436


Daduan ChenDept. of Mathematics, Shanghai University

99 Shangda Rd., Shanghai, China, 200436 e-mail:[email protected]

In the stratification theory [1] any system of partial differential equationsis represented by a subset D of Jk(Rn, Rm), the k-th Ehresmann space from

20

Rn into Rm. In this paper, we use this framework to study some theoreticalproblems of the atmospheric circulation equation in p coordinate system [2]:

D :

dudt = −∂Φ

∂x + ∂∂x (ν1

∂u∂x ) + ∂

∂y (ν1∂u∂y ) + ∂

∂p (ν2∂u∂p ) + fv

dvdt = −∂Φ

∂y + ∂∂x (ν1

∂v∂x ) + ∂

∂y (ν1∂v∂y ) + ∂

∂p (ν2∂v∂p )− fu

dTdt = − ω

Cp

∂Φ∂p + ∂

∂x (µ1∂T∂x ) + ∂

∂y (µ1∂T∂y ) + ∂

∂p (µ2∂T∂p ) + ηω

∂u∂x + ∂v

∂y + ∂ω∂p = 0

∂Φ∂p + RT

p = 0

We discuss D’s topological property, stability and well-posedness of Cauchyproblem and mixed problem. For a well posed problem, we can provide themethod to obtain an analytic solution. More precisely, the following results willbe presented in detail in the paper:

1. Atmospheric circulation equation’s canonical stratification [1] [3] is:

W3,k−1(R4, R5) = St3,k−1(D) ∪ S2

3,k−1(D) ∪ S53,k−1(D) ∪ T3,k−1(D), k ≥ 2

Because St3,k−1(D) is nonempty, the atmospheric circulation equation is sta-

ble.2. Atmospheric circulation equation’s general boundary value problem

Du|p=h(x,y,t) = u0(x, y, t)∂pu|p=h(x,y,t) = u1(x, y, t)

is always well-posed, if the given initial condition satisfies a group of compatibleconditions (where u = (u, v, w,Φ, T )).

3. We obtained some sufficient and necessary conditions for the well-posednessof atmospheric circulation equation’s general initial value problem

Du|t=g(x,y,p) = u0(x, y, p)∂tu|t=g(x,y,p) = u1(x, y, p)

4. The following mixed problem of atmospheric circulation equation is ill-posed:

Du|t=0 = u0

ω|p=pS= 0

5. For some suitable initial data, the following mixed problem of atmosphericcirculation equation admits a unique solution:

Du|t=0 = u0

ω|p=pS= 0, Φ|p=pS

= 0

21

REFERENCES

[1] Shih Weihui, Solutions analytiques de quelques equations aux derivees partiellesen mecanique des fluides. Paris: Hermann, 1992.

[2] Norbury, J., Roulstone, I., Large-scale atmosphere-ocean dynamics, CambridgeUniversity Press, 2002.

[3] He Youhua, Shi Weihui, The Ck Instability of Navier-Stokes Equation Append-ing Polynomials of Unknown Functions. Applied Mathematics and Mechanics,Vol.21(12), 2000, 1440-1449.

On Mean-Square and Asymptotic Stability for NumericalApproximations of Stochastic Ordinary Differential

Equations

Rozsa Horvath-BokorDepartment of Mathematics and Computing University of Veszprem

Taketomo MitsuiGraduate School of Information Science, Nagoya University, Japan

This talk connects a stochastic mean-square stability [Saito and Mitsui,SIAM J. Numer.Anal., 33(1996), pp. 2254-2267] and asymptotic stability [D.J.Higham, SIAM J. Numer. Anal., 38(2000), pp. 753-769].

Saito and Mitsui generalizes the deterministic A-stability for a stochasticdifferential equation test problem with multiplicative noise.

For the test equation, we know that the asymptotic stability in the mean-square sense implies the stochastic asymptotic stability in large. We will provethe same property for numerical schemes.

On One Asymptotic Method for StochasticIntergrodifferential Equations

Dejan IlicUniversity od Nis, Faculty of Science and Mathematics

Visegradska 33, 18000 Nis, Serbia and Montenegro

Svetlana JankovicUniversity od Nis, Faculty of Science and Mathematics


In this paper we consider a general stochastic intgrodifferential equationof the Ito type, which is not linear with respect to one-dimensional and double

22

Lebesgue and Ito integrals. Since this equation is not effectively solvable in mostcases, it is important to find its approximate solution in an explicit form, or in aform suitable for applications of numerical methods. We linearize this equationby using the Taylor expansion of its coefficients, up to the first derivatives, ona partition of the time interval. The rate of the closeness of the original andapproximate solution is measured in the sense of the Lp norm, p ≥ 2, and withprobability one.

Some Approximate Method for Stochastic HereditaryIntegrodifferential Equations



Miljana JovanovicUniversity od Nis, Faculty of Science and Mathematics


In this paper we consider an analytic iterative method for solving stochastichereditary integrodifferential equations of the Ito type, so that the sequence ofiterations converges with probability one to the solution of the considered equa-tion. Some special iterative methods, including some types of linearizations ofthe original equations, are represented. We show that the iterative procedure,utilized to prove the existence and uniqueness of the solution of the stochastichereditary integrodifferential equation which coefficients have a bounded ran-dom integral contractor, is also a special case of this general iterative method.We also consider some types of perturbed hereditary stochastic differential equa-tions with discrete small parameters and we show that, under some sufficientconditions, they can be treated as a special iterative procedure of the presentediterative method, so that the sequence of iterations tends with probability oneto the solution of the unperturbed equation.

Equality of Two Diffeomorphism Invariant ColombeauAlgebras.

Jirı JelınekDepartment of Mathematics, Charles University,Sokolovska 83, 186 00 Praha 8, Czech Republic


Equality of two diffoumorphism invariant Colombeau algebras introducedin M. Grosser, E. Farkas, M. Kunzinger, R. Steinbauer: On the foundations

23

of nonlinear generalized functions I, II, Mem. Am. Math. Soc. 729 (2001)is verified. Available on http://www.karlin.mff.cuni.cz/ms-preprints/prep.php(Matematicke sekce 2001, 33–34).

On a Class of Nonlinear Differential Equation

Biljana Jolevska-TuneskaFaculty of Electrical Engineering, Karpos II bb,

Skopje, Republic of Macedoniae-mail: [email protected]

Arpad TakaciFaculty of Science and Mathematics , Trg Dositeja Obradovica 4,

21000 Novi Sad, Serbia and Montenegroe-mail: [email protected]

In this paper a class of nonlinear differential equation with coefficients gen-eralized functions is considered. Using the framework of Colombeau algebras ofgeneralized functions existence and uniqueness of the solution of this equationare given.

Some Analytic Approximations for Stochastic DifferentialEquations

Miljana JovanovicUniversity od Nis, Faculty of Science and Mathematics




In many fields of science and engineering there is a large number of problemswhich depend on parametric and random excitations of a Gaussian white noisetype. These phenomena are usually mathematically described by stochasticdifferential equations of the Ito type, which are essentially complex and noteffectively solvable.

In the present paper we consider some classes of perturbed stochastic dif-ferential and integrodifferential equations of the Ito type, as well as of some

24

perturbed hereditary differential and integrodifferential equations. We estimateintervals of the closeness of the solutions for the perturbed and unperturbedequations and also the rate of the closeness of these solutions, in the senseof the L2m norm. Precisely, we show that, under some sufficient conditions,these solutions are close on finite intervals or on intervals whose length tends toinfinity as small perturbations tend to zero.

An application of the Dafermos – DiPerna theory to thecompressible Navier – Stokes system

Vladimir JovanovicFaculty of Natural Sciences

Mladena Stojanovica 2Banjaluka, Bosnia and Herzegovina


The stability theory due to C. Dafermos and R. DiPerna, which originallyconsiders nonlinear systems of conservation laws, is applied here to the com-pressible Navier – Stokes system in order to get a stability result for its smoothsolutions.

Algebraic Theory of Colombeau Generalized Functions

Stanley Orlando JuriaansIME-USP-Brazil

In this talk we shall focus on the algebraic theory of Colombeau GenerelizedFunctions, which led to the definition of an intrinsic geometric theory. Wewill apply this to study prime and maximal ideals of the ring of ColombeauGeneralized Numbers and Colombeau Generalized Fucntions.

25

Exponetial Formula for One-time Integrated Semigroups

Senada KalabusicDepartment of Mathematics, University of Sarajevo,

Sarajevo, Bosnia and Herzegovinae-mail: [email protected]

Fikret VajzovicDepartment of Mathematics, University of Sarajevo,

Sarajevo, Bosnia and Herzegovina

In this paper we prove that

limn→∞

∫ T

0

(n + 1

t

)n+1

Rn+1

(n + 1

t;A

)dt = S(T ),

where S(T ) is one-time integrated exponentially bounded semigroup and thelimit is uniform in T > 0 on any bounded interval.

Linear continuous operators commuting with somegeneralized integration operators

Andrzej Kaminski , Svetlana Mincheva-KaminskaInstitute of Mathematics, University of Rzeszow

35-959 Rzeszow, Rejtana 16 A, Polande-mail: [email protected]

The generalized integration operator in C[0, 1], i.e. the right inverse operator

of the differentiationd

dtin C[0, 1], can be described as the linear continuous

operator L on C[0, 1] of the form:

(Lf)(t) :=∫ t

0

f(τ) dτ − Φ(f), f ∈ C[0, 1], (2)

where Φ is a linear continuous functional on C[0, 1] and so, by the Riesz theorem,

Φ(f) =∫ 1

0

f(x) dν(x), f ∈ C[0, 1],

for some function ν of bounded variation on [0, 1]. We are interested in char-acterizing all linear continuous operators A : C[0, 1] → C[0, 1] commuting withthe given generalized integration operator L of the form (2) with nonzero Φ(satisfying the normalizing condition Φ1 = 1).

26

By the Lebesgue decomposition theorem, it suffices to consider the two cases:a) ν is an absolutely continuous function, b) ν is a step function, i.e. thefunctional Φ is of the form:

Φ(f) :=n∑

k=0

pkf(ak), f ∈ C[0, 1], (3)

for some positive integer n, numbers a0, a1, . . . , an ∈ [0, 1] and real numbersp0, p1, . . . , pn such that

∑nk=0 pk = 1.

The first case was presented during the Symposium on Wavelets and IntegralTransforms in Novi Sad in 2002. Now we discuss the second case and presentas the main result the following characterization:Theorem. A linear continuous operator M : C[0, 1] → C[0, 1] commutes withthe given generalized integration operator L of the form (2) with Φ given by (3)if and only if the operator M is of the form (Mf)(t) = d

dt (m∗f)(t) for t ∈ [0, 1],where ∗ is the Dimovski convolution; equivalently,

(Mf)(t) = µf(t) +n∑

i=0

pi

∫ t

ai

f(t + ai − τ) dm(τ), t ∈ [0, 1],

where m := M1 is a continuous function of bounded variation in [0, 1] andµ :=

∑ni=0 pim(ai).

Stochastic models in the risk theory and theirapplications in insurance mathematics

Elena KarashtanovaDepartment of computer science, South-West University ”N. Rilski”

bul. Ivan Mihailov 66, Blagoevgrad 2700, Bulgariae-mail: [email protected]

Irena AtanassovaDepartment of computer science, South-West University ”N. Rilski”

bul. Ivan Mihailov 66, Blagoevgrad 2700, Bulgariae-mail: [email protected]

In this paper is presented an algorithm estimating compound distributionsbased on the fast Fourier transformation.

27

Fractional derivatives of power functions via Fouriertransforms

Siegmar KempfleHelmut Schmidt Universitat

Universitat der Bundeswehr HamburgD–22393 Hamburg, Germany

A fractional derivative of some convenient function x(t) can be establishedas PDO via

Dqx(t) := F−1(iω)qx(ω) , (4)

where F denotes the (unitary) Fourier transform in L2(R), x = Fx(t) andq ∈ R+ is the order of differentiation. To avoid calamities, (iω)q must bedefined as principal branch of the complex power function, i.e.

(iω)q := |ω|qeiπ sign(ω)/2 .

It is well understood and sketched in the talk, how the definition can beextended from L2 in distributional spaces like S ′,D′, E ′.

In view to get some characterization of the maximal subspace K ⊂ D′ suchthat (4) is well–defined we get difficulties by functions with Fourier transformswhich are singular in the origin. To demonstrate this we use the above machineryto get fractional derivatives of some power of t formally as

Dqtr = (−1)q Γ(q − r)Γ(−r)

tr−q. (5)

Again formally, this equation fulfils the semigroup property

Dp(Dqtr) = (−1)p+q Γ(p + q − r)Γ(q − r)Γ(q − r)Γ(−r)

tr−q−p = Dp+qtr (6)

To see that the common integer ordered differentiation is included (q = k ∈ N)one uses the functional equation of the Γ–function to get

Dktr =Γ(r + 1)

Γ(r + 1− k)tr−k , (k ∈ N) . (7)

But we run into some paradoxes, one is demonstrated along the simple exampleDt

12 .

1. Formula (7) yields Dt12 = 1

2 t−12

2. Obviously we can compose this result into (e.g.) Dt12 = D

13

(D

23 t

12

)3. But we cannot act via Dt

12 = D

12

(D

12 t

12

)using formula (5) for D

12 t

12

because of its singularity.

28

4. But using (5) only formally, i.e., dealing with the whole term on the right-hand side, using (6) and then (7), we get the correct result.

In the talk we will discuss this and further paradoxes along the existence ofproducts (iω)q Drδ(ω).

Periodic Solutions of Second Order Differential Equationsin Banach spaces

Valentin KeyantuoUniversity of Puerto Rico, Department of Mathematics,

Faculty of Natural Sciences, PO Box 23355, PR - 00931 U.S.A.e-mail: [email protected]

We consider the second order abstract differential equation:u′′(t)− aAu(t)− αAu′(t) = f(t), 0 ≤ t ≤ 2π

u(0) = u(2π),u′(0) = u′(2π),

(8)

in the spaces Lp2π(R;X), 1 < p < ∞ (resp. Cs

2π(R;X), 0 < s < 1) where X is aBanach space, aα ∈ R and A : D(A) ⊂ X −→ X is a closed linear operator.

We use operator-valued Fourier multiplier theorems to study (8). Solutionsin the X−valued Besov spaces Bs

pq are considered as well. We establish maximalregularity results in Lp and Cs for strong solutions of a complete second orderequation. This equation has as a model a linearized version of a quasi-linearwave equation with strong dissipation.

In the second part, we study mild (generalized) solutions for the secondorder problem. Two types of mild solutions are considered. When the operatorA involved is the generator of a strongly continuous cosine function, we givecharacterizations in terms of Fourier multipliers and spectral properties of thecosine function. The results obtained are applied to elliptic partial differentialoperators.

This is joint work with Carlos Lizama of the University of Santiago de Chile.

The author is supported in part by Convenio de Cooperacion Internacional(CONICYT-Chile) Grant # 7010675

2000 Mathematics Subject Classification Primary 47D06, 47D09; Secondary47F05, 35L70, 35G20

29

Zeros of generalized holomorphic functions

A. Khelif, D. ScarpalezosU.F.R. Mathematics, Case 7012, Universite Paris 7,

2 place Jussieu, Paris, 75005 Francee-mail: [email protected]

Holomorphic Colombeau generalized functions are just the solutions in Colombeaualgebra of Dirac equation ∂f = 0. In the case of Colombeau generalized func-tions it has been noted that a generalized function on Ω can take zero valueon all points of Ω without being necessarily zero. Thus point values do notcompletely determine a Colombeau generalized fonction.

This led M. Oberguggenberger and M. Kunzinger to introduce the notion ofcompactly supported generalized points and this time the values on compactlysupported generalized points completely determine the generalized function.

As noted by S.Pilipovic it was not known whether something similar wasnecessary for the holomorphic case i.e. it was not known whether a holomorphicColombeau generalized function is necessarily zero if all its values on ordinarypoints are zero.

In this paper we prove some stronger results. More precisely we prove thatif the zero set of a Colombeau generalized function f is of positive measure thenf is zero. Using the same ideas we prove a more precise result that implies forexample that if two holomorphic generalized functions take the same values ona line then they are equal.

The analogy with holomorphic fuctions cannot go much further because wegive a counterexample of a nonzero holomorphic generalized function whose zeroset is a dense Gδ subset of Ω. Since there exist many Colombeau holomorphicgeneralized functions whose values on ordinary points are invertible but whichare not invertible it is clear that it is interesting to study the generalized zeroset i.e. the set of generalized points on which it takes value zero.

We prove results analogous to our main theorems but concerning this timethe generalized zero set.

30

On the Matrix Convolution Product and SomeApplications

Adem KilicmanDepartment of Mathematics and Institute for Mathematical Research

University Putra Malaysia (UPM), Malaysia43400 UPM, Serdang, Selangor, Malaysia

Zeyad Al ZhourDepartment of Mathematics and Institute for Mathematical Research

University Putra Malaysia (UPM), Malaysia43400 UPM, Serdang, Selangor, Malaysia

In addition to the usual matrix multiplication; recently there has been re-newed interest in another kind matrix multiplication that is also known as con-volution product and it is very useful in applications of matrix equations andmatrix differential equations in matrix theory, engineering and many other sub-jects.

In this work, we study several properties (equalities and inequalities) of theconvolution product of matrices. We define the Dirac identity matrix and itsproperties which behaves like a group identity element under the convolutionmatrix operation. We also derive several elementary properties of the matrixconvolution product. The connections between the usual product and the con-volution product of matrices are established. Further some applications of theconvolution matrix product are also studied. These applications involve withrenewal matrix equation and non homogeneous matrix differential equations.

Keywords: Convolution product of functions, Convolution product of ma-trices, Kronecker product (sum) of matrices, Matrix norm, Vector-Operator,Laplace transform, Dirac identity matrix, Renewal matrix equation, Matrix dif-ferential equations, Exponential matrix, Correlation matrix.

Identification of the location of the dipole source in thebrain by MEG

Jung Eun KimDepartment of Mathematics, Seoul National University, Seoul, Korea


The problem of determining the neuronal current inside the brain from mea-surements of the induced magnetic field outside the head is discussed under the

31

assumption that the space occupied by the brain is approximately ellipsoidal.In this talk, we invert the Geselowitz equation to find the exact location of singledipole source using only the radial part of the induced magnetic field outside.Furthermore, we discuss the uniqueness in the identification of location of theelectrical source by a measurement outside the brain when the magnetic field isinduced by double dipole sources.

Characterization of Eigenfunctions of DifferentialOperators

Jong-Ho KimDepartment of Mathematics, Sogang University, Seoul 121-742, Korea

Yun-Sung ChungDepartment of Mathematics, Sogang University, Seoul 121-742, Korea


Suppose that P (D) is a linear differential operator with complex coefficientsand fkk∈Z is a two-way sequence of complex-valued functions defined on Rwith the properties :(i) fk+1 = P (D)fk

(ii)there exists a sequence Mnn∈N0 with the property that for any ε > 0, thesequence

Mn

(1+ε)n

n∈N0

has a bounded subsequence, for constants a ∈ [0, 1) and

N ≥ 0 such that |fk(x)| ≤ M|k| exp(N |x|a).

Then we show that f0 should be an entire function with almost exponentialgrowth. This generalizes the results of Roe, Burkill and Howard.

Multi-index Mittag-Leffler Functions, GeneralizedFractional Calculus and Laplace Type Transform

Virginia KiryakovaInstitute of Mathematics, Bulgarian Academy of Sciences

Sofia 1090, Bulgariae-mail: [email protected] , [email protected]

Recently, the interest in the Mittag-Leffler (M-L) functions has increasedin view of their important role and applications in fractional calculus and re-lated fractional order differential and integral equations (FODIE). We have in-troduced and studied analogues of these functions, Em

(1/ρ1,...,1/ρm),(µ1,...,µm)(z),

32

depending on two (m-tuple, m ≥ 2) sets of multi-indices. They happen togenerate operators of the generalized fractional calculus (V. Kiryakova, Gener-alized Fractional Calculus and Applications, Longman, Harlow and J. Wiley, N.York, 1994) and Laplace-type integral transforms involving the Fox H-function.These new special functions appear to be “fractional indices” analogues of thehyper-Bessel functions of Delerue, and the associated respective differential andintegral equations are fractional (multi-)order analogues of the Bessel type equa-tions arising so often in problems of mathematical physics and engineering.

Even the classical Mittag-Leffler functions, for a long time, have been almosttotally ignored in the common handbooks on special functions and existing ta-bles of Laplace transforms, although a description of their properties has ap-peared yet in the third volume of the Bateman Project (A. Erdelyi et al. (Ed-s),Higher Transcendental Functions. McGraw, N. York, 1953), in a chapter de-voted to “miscellaneous functions”. Recently, they have been widely recognizedas solutions of FODIE arising in many problems of physics, mechanics, con-trol theory etc. and as inverse Laplace transforms of rational (fractional-order)transfer functions (see e.g. I. Podlubny, Fractional Differential Equations, Acad.Press, San Diego etc., 1999 and the publications in the “FCAA” Journal: Frac-tional Calculus & Applied Analysis, vol. 1-vol. 7 (1998-2004), IMI-BAS, Sofia).

In the special case m = 2 falling in our scheme, such generalizations of theM-L functions were considered by M.M. Dzrbahjan in 1960, but remained not sopopular due to publication in Russian in Izv. AN Arm. SSSR. It is worthy andinteresting to mention about their close relation to the Wright (called sometimesalso Bessel-Maitland) functions, studied in earlier paper by L. Gajic and B.Stankovic in Publ. l”Institut Math. Beograd, Nouv. Ser. 20 (34), 91-98.

Generalized solutions to strongly singular initial-boundaryproblems for hyperbolic systems

Irina KmitDepartment of Numerical Mathematics & Programming,

State University “Lvivska Polytechnika”,Bandera St. 12, 290646 Lviv, Ukraine


We investigate existence and uniqueness of Colombeau solutions to mixedfirst-order hyperbolic problems where strong singularities are present both incoefficients and in initial and boundary conditions. The boundary conditionscan be given in classical as well as in nonclassical (nonseparable and integral)form. We explore regularity of Colombeau solutions using the framework of thedistribution theory and the notion of a delta-wave. For a particular problemarising in the population dynamics we succeed in constructing a distributionalsolution. These results develop a previous work joint with Gunther Hormann(J. for Analysis and its Applications 20(3):637–659, 2001).

33

Generalized Solutions for a Model in Kinetic Theory

Irina KmitDepartment of Numerical Mathematics & Programming,

State University “Lvivska Polytechnika”,Bandera St. 12, 290646 Lviv, Ukraine


Michael KunzingerDepartment of Mathematics, University of Vienna

Nordbergstr. 15, A-1090 Wien, Austriae-mail: [email protected]

Roland SteinbauerDepartment of Mathematics, University of Vienna


In kinetic theory one often considers large ensembles of collisionless (classi-cal) particles which interact only by fields they create collectively—a situationcommonly referred to as the mean field limit of a many-particle system. Thereis an extensive literature on this topic (cf. e.g. [1]) and we are going to focuson interaction by non-relativistic, gravitational or electrostatic fields, i.e., theVlasov-Poisson system of PDEs given by

∂tf + v · ∂xf + E · ∂vf = 0, E(t, x) = −γ

∫x− y

|x− y|3ρ(t, y)dy (γ = ±1).

Here f = f(t, x, v) ≥ 0 (x, v ∈ R3) is the phase space distribution functionof the particles, and ρ(t, x) =

∫f(t, x, v) dv, their density in space. The initial

value problem for this system is well understood, e.g. there exist unique smoothglobal-in-time solutions for general data. However, the Vlasov-Poisson systemhas as singular limit cases the Euler-Poisson system (with pressure zero) ([2])and the classical n-body problem ([3]), for which in general no global-in-timesolutions exist.

In this contribution we report on work in progress concentrating on gener-alized solutions (in the sense of J. F. Colombeaus special construction) of thespherically symmetric Vlasov-Poisson system with the principal aim of study-ing solutions with the initial data f(x, y) concentrated either in space or inmomentum-space, hence approaching the singular limit systems of the Vlasov-Poisson system.

REFERENCES

[1] R. Glassey, The Cauchy Problem in Kinetic Theory, (SIAM, Philadelphia, PA,1996).

34

[2] V. Sandor, The Euler-Poisson-System with Pressure Zero as Singular Limit ofthe Vlasov-Poisson System—the Spherically Symmetric Case, preprint, 1996.

[3] H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in C.Cercignani (Ed.) Kinetic theories and the Boltzmann equation, Lecture Notes inMath. 1048, 60–110, (Springer, Berlin, 1984).

A global approach to symmetry group analysis in algebrasof generalized functions

S. KonjikDepartment of Agricultural Engineering, University of Novi Sad

Trg D. Obradovica 8, 21000 Novi Sad, SCGe-mail: [email protected]

M. KunzingerDepartment of Mathematics, University of Vienna


Symmetry group analysis is a powerful tool in the study of differential equa-tions, in particular for determining invariance properties, constructing specialsolutions or deriving conservation laws in systems of physical interest. Over thepast years, the classical (smooth) theory of group analysis of PDEs has beenextended to include weak solution concepts, in particular distributional andColombeau generalized functions. So far this extension has been confined to alocal setting. Based on recent developments in global analysis in the Colombeausetting, in this talk we present a global approach to symmetries of generalizedsolutions to differential equations.

Convoluted semigroups and convoluted cosine functions

Marko Kosticsupported by Stevan Pilipovic

Department of Mathematics and Informatics, University of Novi Sad

Convoluted cosine functions in Banach spaces are studied and related to ultra-distribution and hyperfunction sines as well as to introduced analytic convolutedsemigroups. Examples of convoluted cosine functions are presented.

In this talk, we will present the basic results related to convoluted semi-groups, introduced by I. Cioranescu and G. Lumer in 1994, and convolutedcosine functions, introduced and investigated in [13] and [11]. The class of

35

convoluted cosine functions, resp., convoluted semigroups is essentially largerthan the class of integrated cosine functions, resp., integrated semigroups. Wemostly deal with global exponentially bounded convoluted cosine functions andsemigroups. Our investigations are based on [12] and [13], where the rela-tions of convoluted semigroups with ultradistribution semigroups and (Fourier)hyperfunction semigroups are analyzed. Cioranescu and Lumer related convo-luted semigroups to ultradistribution semigroups, see [5], [16] and [14]. But,we want to note that convoluted semigroups (cosine functions) can be used inanalysis of operators whose spectrum has non-empty intersection with any ray(ω,∞), ω ∈ R.

We study exponentially bounded convoluted cosine functions using the Laplacetransform. Komatsu introduced Laplace hyperfunctions through a very generalconcept of the Laplace transform and relate them to Laplace hyperfunctionsines. We do not follow this general approach as well as the aproach with theasymptotic Laplace transform [15]. The later approach is useful for the inves-tigations of the Laplace operator and the biharmonic operator ∆2 in L2[0, π].Our analysis it is done by the use of standard Laplace transform.

The obtained results are closely connected with the abstract Cauchy prob-lem:

(ACP2)Θ :

u ∈ C([0, τ) : D(A)) ∩ C2([0, τ) : E),

u′′(t) = Au(t) + Θ(t)x +t∫0

Θ(s)yds,

u(0) = 0, u′(0) = 0 (x, y ∈ E).

The existence of a unique solution is connected with the existence of the uniqueK-convoluted mild solution of the problem (ACP2) (see Section 6) and theexistence of a (local) K-convoluted cosine function generated by A. Similarly,convoluted semigroups are important in the study of the corresponding (Θ)-problem:

(Θ) :

u ∈ C([0, τ) : D(A)) ∩ C1([0, τ) : E),u′(t) = Au(t) + Θ(t)x,

u(0) = 0, (x ∈ E).

We also introduce and investigate analytic convoluted semigroups. The mainpart of the talk is related to the connections of convoluted cosine functionswith ultradistribution and hyperfunction sines as well as analytic convolutedsemigroups. The obtained results can be used in the analysis of Cauchy problemsquoted above in the framework of various vector valued generalized functionspaces.

REFERENCES

[1] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valuedLaplace Transforms and Cauchy Problems, Birkhauser Verlag, 2001.

[2] B. Baumer, Approximate solutions to the abstract Cauchy problem, in: Evolu-tion Equations and Their Applications in Physical and Life Sciences (Bad Her-renalb,1998), 33-41. Lecture Notes in Pure and Appl. Math., 215, Dekker, NewYork, 2001.

36

[3] R. Beals, On the abstract Cauchy problem, J. Funct. Analysis 10 (1972), 281–299.

[4] R. Beals, Semigroups and abstract Gevrey spaces, J. Funct. Anal. 10 (1972),300-308.

[5] J. Chazarain, Problemes de Cauchy abstraites et applications a quelquesproblemes mixtes, J. Funct. Anal. 7 (1971), 386-446.

[6] I. Cioranescu, Local convoluted semigroups, in: Evolution Equations (BatonRauge, LA, 1992), 107–122, Dekker, New York, 1995.

[7] I. Cioranescu, G. Lumer, Problemes d’evolution regularises par un noyangeneralK(t). Formule de Duhamel, prolongements, theoremes de generation, C.R.Acad. Sci. Paris Ser. I Math. 319 (1995), 1273–1278.

[8] I. Cioranescu, G. Lumer, On K(t)-convoluted semigroups, in: Recent Devel-opments in Evolution Equations (Glasgow, 1994), 86–93. Longman Sci. Tech.,Harlow, 1995.

[9] H. Komatsu, Ultradistributions, I. Structure theorems and a characterization,J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25–105.

[10] H. Komatsu, Ultradistributions, III. Vector valued ultradistributions the theoryof kernels, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 653–718.

[11] M. Kostic, Convoluted C-cosine functions and convoluted C-semigroups, Bull.Cl. Sci. Math. Nat. Sci. Math. 28 (2003), 75–92.

[12] M. Kostic, S. Pilipovic, Generalized semigroups, preprint.

[13] M. Kostic, S. Pilipovic, Convoluted cosine functions: relations to ultradistri-bution and hyperfunction sines, preprint.

[14] P.C. Kunstmann, Banach space valued ultradistributions and applications toabstract Cauchy problems, preprint.

[15] G. Lumer, F. Neubrander, The asymptotic Laplace transform: new results andrelation to Komatsu’s Laplace transform of hyperfunctions, in: Partial DifferentialEquations on Multistructures (Luminy, 1999), 147–162. Lecture Notes in Pure andAppl. Math., 219, Dekker, New York, 2001.

[16] I.V. Melnikova, A.I. Filinkov, ”Abstract Cauchy problems: Three Ap-proaches”, Chapman & Hall/CRC, 2001.

[17] S. Ouchi, Hyperfunction solutions of the abstract Cauchy problems, Proc. JapanAcad. 47 (1971), 541–544.

37

Some Remarks of the Infinitesimal Generators of anIntegrated C-semigroups of Unbounded Linear Operators

in Banach Spaces

Ratko KravarusicInstitute of Mathematics, University of Novi Sad,

Trg Dositeja Obradovica 4, 21000 Novi Sad, Yugoslavia

Milorad MijatovicInstitute of Mathematics, University of Novi Sad,

Trg Dositeja Obradovica 4, 21000 Novi Sad, Yugoslavia

AMS Mathematical Subject Classification (1991): 47D06.Key word and phrases: infinitesimal generator, integrated C-semigroups, familyof unbounded linear operators.

We introduced and investigated infinitesimal generator A of an integratedC-semigroups of unbounded linear operators (S(t))t≥0. A Banach spaces (Eω, ‖·‖ω) are used for a construction of a family of infinitesimal generators Aω, ω > 0which determine an infinitesimal generator A integrated C-semigroups of un-bounded linear operators in Banach spaces.

Generalized Flows and singular ODEs on differentiablemanifolds

M. KunzingerFakultaet fuer Mathematik,Universitaet Wien, Austria

In this talk we report on the recent development of a theory of singularordinary differential equations on smooth manifolds in the setting of Colombeaugeneralized functions ([1]). In particular, we compare this new setting with thepurely distributional approach and develop criteria securing that a sequence ofsmooth flows corresponding to a regularization of a given singular vector fieldconverges to a measurable limiting flow.

REFERENCES

[1] Kunzinger, M., Steinbauer, R., Oberguggenberger, M., Vickers, J. Generalizedflows and singular ODEs on differentiable manifolds. Acta Appl. Math. 80, 221-241, 2004.

38

Deconvolution of Multi-sensor Problems

Nam Kee LeeDepartment of Mathematics, Seoul National University, Seoul, Korea

Soon-Yeong ChungDepartment of Mathematics, Sogang University, Seoul, Korea

Meisters and Peterson gave an equivalence condition under which the decon-volution problem has solutions when convolvers are given as two characteristicfunctions on intervals. The solvability of this problem depends only on the ra-tio of the length of the intervals. In this talk we find the conditions that thedeconvolution of multisensor problems are solvable. And then we extend theresult to the space of Gevrey distributions and we prove that every linear op-erator from the space of Gevrey differentiable functions with compact supportonto itself, which commutes with translations can be represented as convolutionwith a unique Gevrey distribution of compact support. Finally we find explicitformula of deconvolvers, using the nonperiodic sampling method.

An extension of Yosida approximation

Ludovic Dan LemleUniversite Blaise Pascal Clermont2,

UFR de Recherche Scientifique et Technique,63177 Aubiere, France

andUniversity ”Politehnica” of Timisoara,

the Faculty of Engineering of Hunedoara,331128 Hunedoara, Romania


The aim of this note is to formulate a generalisation of Yosida approximationfor infinitesimal generator of an strongly continuous semigroup. Some propertysand applications of generalized Yosida approximation will also be discussed.The surprising fact is that we can obtain an characterisation of C0-semigroupsgenerated by an operator not neccesarily densely defined.

39

Distribution Analogue of the Tumarkin Result

Vesna Manova-ErakovicFaculty of Natural Sciences and Mathematics, Institute of Mathematics

P.O. Box 162, 1000 Skopje, MACEDONIAe-mail: [email protected]

Nikola PandeskiFaculty of Natural Sciences and Mathematics, Institute of Mathematics

P.O. Box 162, 1000 Skopje, MACEDONIA

Ljupco NastovskiFaculty of Natural Sciences and Mathematics, Institute of Mathematics

P.O. Box 162, 1000 Skopje, MACEDONIA

In this paper we give distribution analogue of the Tumarkin result thatconcerns approximation of some functions by sequence of rational functionswith given poles.

Dispersive Systems in Quantum Hydrodynamics forSemiconductors

Pierangelo MarcatiDipartimento di Matematica Pura ed Applicata Via Vetoio

I-67010 Coppito (L’Aquila) AQ - Italye-mail: [email protected]

This talk is devoted to illustrate recent results on the study of the existenceand the time-asymptotic of multi-dimensional quantum hydrodynamic equa-tions for the electron particle density, the current density and the electrostaticpotential in spatial periodic domain, in the irrotational case. The results arepublished in

H.Li , P.Marcati, Existence and Asymptotic Behavior of Multi-DimensionalQuantum Hydrodynamic Model for Semiconductors, Commun. Math. Phys.245 (2004), 215-247.

We prove the local-in-time existence of the solutions, in the case of thegeneral, nonconvex pressure-density relation and large and regular initial data.Furthermore we propose a quantum correction for the subsonic type stabilitycondition of the classical hydrodynamics. When this condition is satisfied, thelocal-in-time solutions exist globally in-time and converge time exponentially to-ward the corresponding steady-state. Since for this problem classical methodslike, for instance, the Friedrichs theory for symmetric hyperbolic systems can-not be used, we investigate, via an iterative scheme, an extended system, which

40

incorporates the one under investigation as a special case. In particular thedispersive terms appear in the form of a fourth-order wave type equation. Re-cent improvements obtained by a more careful analysis of dispersive properties(Strichartz type bounds) will be also shown.

GL-Microlocal analysis of generalized functions

Jean-Andre MartiEquipe Analyse Algebrique Non Lineaire

Laboratoire Analyse, Optimisation, ControleFaculte des sciences - Universite des Antilles et de la Guyane

97159 Pointe-a-Pitre Cedex Guadeloupee-mail: [email protected]

The study of generalized functions has been developed for twenty years([1],[2],[6],[13]), and their theory appears as a natural continuation of the distri-butions one, specially efficient to pose and solve non linear differential problemswith irregular data ([10],[9]).

The algebras of generalized functions have a sheaf structure ([8]) and somesheaves of these algebras may contain some subsheaves with particular proper-ties. It is the case of the sheaf G of the simplified Colombeau algebra ([1],[12]):it contains the subsheaf G∞ of “regular sections” of G such that the embedding:G∞ → G is the natural extension of the classical one: C∞ → D′, in the sensegiven by the equality G∞ ∩ D′ = C∞, proved in ([13]).

In the paper we begin by constructing GL as a special subsheaf of G∞ andG, related to the sheaf of functions of CL class ([7]) (including analytic andGevrey ones) verifying some estimation involving a special increasing sequenceL = (Lk)k∈N. We prove that GL∩G∞ = GL∩D′ = CL, and then the embeddings:GL → G∞ → G are the natural extension of: CL → C∞ → D′.

Through the sheaf structure, we can define the GL-regularity (as well as theG∞ one) of a section u of G above some open set Ω ⊂ Rn which leads to thenotion of GL-singular support of u: sing suppL u.

But it is not easy to give a microlocal description of the GL-singularities ofu as it is done for the G∞-singularities ([12],[3]). We begin to give a charac-terization of local GL-regularity by means of some sequence uk of generalizedfunctions with compact support whose the Fourier transform uk verifies a spe-cial estimation involving the sequence (Lk)k∈N. Indeed, uk is constructed asproduct of u and a suitable cutoff sequence Xk whose derivatives are controledup to the order k ([7]). This leads to define the GL-wave front set of a gen-eralized function: WFL

g (u) ⊂ Ω × (Rn \ 0) and prove, by refining the cutoffsequence Xk, that its projection in Ω equal to sing suppL u. Then, WFL

g (u)gives a spectral decomposition of sing suppL u.

We finish the paper by the first steps in the study of the propagation ofGL-singularities of u: they don’t increase under the action of some linear partial

41

differential operator with coefficients in CL or GL. Further results followingthose in ([4],[5]) are expected for analytic (or GL) pseudo-differential operatorsin a sense generalized from the classical one ([11]).

REFERENCES

[1] Colombeau J.F. New Generalized Functions and Multiplication of Distributions.North-Holland, Amsterdam, Oxford, New-York, 1984.

[2] Colombeau J.F. Elementary introduction to New generalized Functions. North-Holland, Amsterdam, Oxford, New-York, 1985.

[3] Delcroix A. Fourier transform of rapidly decreasing generalized functions. Ap-plication to microlocal regularity. Preprint 2004.

[4] Garetto S., Gramchev T., Oberguggenberger M. Pseudo-Differential op-erators and regularity theory Preprint, 2003.

[5] Garetto C., Hormann G. Microlocal analysis of generalized func-tions: pseudodifferential techniques and propagation of singularities.arXiv:math.AP/0401397v1 28 Jan 2004.

[6] Grosser M., Kunzinger M., Oberguggenberger M., Steinbauer R. Geo-metric Theory of Generalized Functions with Applications to General Relativity.Kluwer Academic Press, 2001.

[7] Hormander L. The analysis of Linear Partial Differential Operators I,distribution theory and Fourier Analysis. Grundlehren der mathematischenWissenchaften 256. Springer Verlag, Berlin, Heidelberg, New York, 2nd edition1990.

[8] Marti J.-A. Fundamental structures and asymptotic microlocalization in sheavesof generalized functions. Integral Transforms and Special Functions 6(1-4):223-228, 1998.

[9] Marti J.-A. Non linear Algebraic analysis of delta shock wave to Burgers’ equa-tion. Pacific J. Math. 210(1):165-187, 2003.

[10] Marti J.-A., Nuiro P., Valmorin V. Algebres differentielles et problemes deGoursat non lineaires a donnees irregulieres. Ann. Fac. Sci. Toulouse, VI. Ser.Math. 7:135-159, 1998.

[11] Matsuzawa T., A calculus approach to hyperfunctions II Trans. A.M.S. 313(2):2,619-654, 1989.

[12] Nedeljkov M., Pilipovic S., Scarpalezos D. The linear theory of Colombeaugeneralized functions. Pitman Research Notes in Mathematics Series, 385. Long-man, 1998.

[13] Oberguggenberger M. Multiplication of Distributions and Applications to Par-tial Differential Equations. Longman Scientific & Technical, 1992.

42

Extremal problems

Miodrag MateljevicMatematicki fakultet, Studentski trg 16

Beograd, Serbia and Montenegro

Extremal problems for quasiconformal mappings, compactness propertiesand connection with Sobolev spaces are subject of our talk. Teichmuller ,following Grotzsch, showed that any homotopic equivalence class of quasicon-formal mappinngs from a compact Riemann surface M to a compact Riemannsurface N contains a unique mapping ( what we now call a Teichmuller map-pinng ) whose maximal dilatation K is minimal. Moreover, this unique mappingcan be described geometrically in terms of holomorphic differentials on M , whichgive a way of cutting up the surface M into euclideian rectangles. It was sug-gested by Teichmuller that this is true for Riemann surface (in general, whichare not compact). During the last several years, important progress has beendone in characterizing the conditions under which unique extremality occurs.

There are many examples of extremal Beltrami differentials with noncon-stant modulus, but all examples of uniquely extremal Beltrami differentialsknown up to resent author’s series of joint papers, were of the general Te-ichmuller type. Moreover, many results obtained studying the extremal prob-lems speak in favour of the conjecture that all uniquely extremal Beltrami differ-entials µ satisfy |µ(z)| = ||µ||∞, for almost all z. Surprisingly, we disprove thisconjecture and show that there are uniquely extremal Beltrami differentials withnonconstant modulus. We refer to the proof of this result as the construction ofuniquely extremal dilatation with nonconstant modulus (shortly the construc-tion). During Lectures on quasiconformal mappings, Scoala Normala SuperioaraBuchurest, (SNSB), 2003-2004, we have introduced notion which is a general-ization of uniquely extremal dilatation. Using this notion we generalize resultsrelated to uniquely extremal dilatation. This leads to better understanding ofunique extremality. Thanks to the characterization of unique extremality byRe-polynomials, Runge’s and Mergelyan’s theorems, we can make interestingconstructions of uniquely extremal dilatations. In particular, we present a newconstruction which is more visual and which gives uniquely extremal dilatationof Teichmuller type out of the set K of positive measure which has emptyinterior and has arbitrary values on K.

43

FBI-transformation and Non-isotropic GevreyHypoellipticity for Extended Grushin Class

Tadato MatsuzawaDepartment of Mathematics, Meijo University

Nagoya 468-8502, Japane-mail: [email protected]

We denote x = (x1, x2, ..., xn) ∈ Rn, and D = (D1, D2, ..., Dn), Dj =−i∂j , j = 1, ..., n, as usual.

Definition 1. Let Ω be an open set in Rn and ϕ ∈ C∞(Ω). We say thatϕ ∈ Gs1,s2,...,sn(Ω), 0 < s1, s2, ..., sn < ∞, if for any compact subset K of Ωthere are positive constants C0 and C1 such that

(1) supx∈K

|Dαϕ(x)| ≤ C0C|α|1 α1!s1α2!s2 · · ·αn!sn , α ∈ Zn

+.

We denote often Gs(Ω) = Gs1,s2,...,sn(Ω), s = (s1, s2, . . . , sn) and α!s =α1!s1α2!s2 · · ·αn!sn .

We suppose sj ≥ 1, j = 1, 2, . . . , n, and denote 〈ξ〉s = (1+ ξ21)

12s1 + · · ·+(1+

ξ2n)

12sn , ξ ∈ Rn. We recall now a slight version of a representation formula for

distributions, which seems originated by Sjostrand, [5 ].Lemma 1. (FBI-inversion formula) Let u ∈ E ′(Rn). Then for any x0 ∈ Rn

and ε > 0 we have(2) + wε(x),

u(x) = (2π)−n

∫∫|β−x0|<2ε

ξ∈Rn

uy(ei〈x−y,ξ〉−(β−y)2〈ξ〉s/2)(〈ξ〉s2π

)n/2

dβdξ

where wε(x) ∈ S ′(Rn) and wε(x) ∈ Gs(|x− x0| < ε).Definition 2. Let u ∈ E ′(Rn) and s = (s1, s2, . . . , sn), sj ≥ 1, j = 1, . . . , n.

We define

(3) Fsu = Fsu(x, β, ξ) = 〈uy, ei〈x−y,ξ〉−(β−y)2〈ξ〉s/2〉.

We shall give a characterization of the classes Gs.Theorem 1. Let u ∈ E ′(Rn) and x0 ∈ Rn. Then the following assertions

are mutually equivalent:(i) u ∈ Gs at x0.(ii) There exist positive constants C, δ and ε such that

|Fsu(x, β, ξ)) ≤ Ce−δ〈ξ〉s , |x− x0| < ε, |β − x0| < ε, ξ ∈ Rn.

Now we shall give the definition of a slightly extended class of Grushin operators.We separate the variables as (x, y) = (x1, . . . , xk, y1, . . . , yn) ∈ Rk+n. Let m be

44

an even positive integer and denote σ = (σ1, σ2, . . . , σk), q = (q1, q2, . . . , qk)whose elements are rational numbers such that

σ1, . . . , σp > 0, σp+1 = · · · = σk = 0, (0 ≤ p ≤ k),

q1 ≥ q2 ≥ · · · ≥ qk ≥ 0, q1 > 0.

Furthermore we assume mqj ∈ Z, j = 1, . . . , k;mqj

σj∈ Z, j = 1, . . . , p. We

pose the following major hypothesis:Hypothesis (G) Suppose 1 + qp > σ0 ≡ max(σ1, . . . , σp).Remark 1. Grushin’s original major hypothesis given in [2] was 1 + qk >

σ0 = max(σ1, . . . , σp). We shall see that this condition can be weakened asabove. The assumption on q1, q2, . . . , qk given in [2] is also slightly weakened asabove. When p = 0, we consider σ0 = 0 and σ = (0, . . . , 0).

We divide the variable x into two parts such that x = (x′, x′′) when 1 ≤ p <k, where x′ = (x1, . . . , xp) and x′′ = (xp+1, . . . , xk). We consider x = x′ whenp = k and x = x′′ when p = 0 and σ = (0, . . . , 0). Now we shall consider adifferential operator with polynomial coefficients under the hypothesis (G):

(4) P (x′, y,Dx, Dy) =∑

〈σ,ν〉+|γ|=〈q,α〉+|α+β|−m|α+β|≤m

aαβνγx′νyγDαx Dβ

y , aαβνγ ∈ C,

α, ν ∈ Zk+, β, γ ∈ Zn

+,

where aαβνγ can be non zero only when |γ| = 〈q, α〉+ |α + β| −m = 〈σ, ν〉 is anon negative integer and here we write such as |α + β| = |α|+ |β|. We may alsoconsider ν = (ν1, . . . , νp, 0, . . . , 0). We can see the symbol P (x′, y, ξ, η) satisfiesthe following condition.

Condition 1. (quasi-homogeneity) We have

P (λ−σx′, λ−1y, λ1+qξ, λη) = λmP (x′, y, ξ, η),

λ > 0, x, ξ,∈ Rk, y, η ∈ Rn,

where λ−σx′ = (λ−σ1x1, . . . , λ−σpxp) and λ1+qξ = (λ1+q1ξ1, . . . , λ

1+qkξk).We add the two more conditions on P .Condition 2. (ellipticity) The operator P is elliptic for |x′|+ |y| = 1.Condition 3. (non-zero eigenvalue) For all ω, |ω| = 1, the equation

P (x′, y, ω, Dy)v(y) = 0 in Rny

has no non-trivial solution in S(Rny ).

We set the Gevrey indices with respect to the operator P as follows.

θj = max(1 + qj

1 + qk,

1 + qk

1 + qp − σ0) for j = 1, . . . , p,

θj =1 + qj

1 + qkfor j = p + 1, . . . , k, d = max

1≤j≤kθj + qj

1 + qj.

45

We write alsod = max

1≤j≤kθj + qj

1 + qj · In = (d, . . . , d).

Theorem 2. Let Ω be an open neighborhood of (0, 0) ∈ Rk+nx,y and consider

the equation

(5) P (x′, y,Dx, Dy)u(x, y) = f(x, y) in Ω,

where u(x, y) ∈ D′(Ω) and f(x, y) ∈ Gθ,dx,y (Ω). Then we have u(x, y) ∈ G

θ,dx,y (Ω).

Remark 2. In the above theorem we can find that

(i) p = 0, θ1 = 1 ⇐⇒ (θ, d) = (1, 1, . . . , 1),

(ii) p = 0, θ1 > 1 =⇒ 1 < d =θ1 + q1

1 + q1< θ1.

Examples

(a) For the operator P1 = D2y + y2kD2

x, (k = 1, 2, . . .), we have

p = 0, q1 = k, σ0 = 0 and θ1 = 1, d = 1.

(b) For the operator P2 = D2y + (x2l + y2k)D2

x, (k, l = 1, 2, . . .), we have

p = 1, q1 = k, σ1 = σ0 =k

l;

1 < θ1 =l(1 + k)

l(1 + k)− k, 1 < d =

θ1 + k

1 + k< θ1.

(c) For the operator P3 = D2y + (x2l + y2k)(D2

x + D2z), (k, l = 1, 2, . . .), we have

p = 1, q1 = q2 = k, σ1 = σ0 =k

l, σ2 = 0, x′ = x, x′′ = z;

θ1 =l(1 + k)

l(1 + k)− k, θ2 = 1, 1 < d =

θ1 + k

1 + k< θ1.

(d) For the operator P4 = D2y + (x2l + y2k)D2

x + D2z , (k ≥ 1, l ≥ 0), we have

p = 1, q1 = k, q2 = 0, σ1 = σ0 =k

lif l ≥ 1, σ1 = 0 if l = 0, σ2 = 0;

θ1 = 1 + k, θ2 = 1, 1 < d =θ1 + k

1 + k=

1 + 2k

1 + k< 1 + k.

We remark that this operator P4 does not satisfy the original hypothesis (G) ofGrushin when 1 ≤ l ≤ k.(e) An example with 1 < d1 < d2 is given by

P5 = D2y + (x4 + y4)D2

x + (x2 + y2)D2z ,

46

where we havep = 2, q1 = 2, q2 = 1, σ1 = σ2 = 1;

θ1 = θ2 = 2, d1 =43

< d2 =32

= d.

Remark 3. By virtue of Theorem 1, (ii), our main purpose will be to provethat there exist a small neighborhood V of (0, 0) ∈ Rk+n

x,y and positive constantsC and δ such that

(6) |Fθ,du(x, y, β, ξ, η)| ≤ Ce−δ(∑|ξj |

1θj +|η|

1d ), (x, y, β, ξ, η) ∈ V × V ×Rk+n

ξ,η ,

where

Fθ,du(x, y, β, ξ, η) =∫

u(x, y)ei(〈x−x,ξ〉+〈y−y,η〉)−((βk−x)2+(βn−y)2)〈µ〉θ,d/2dxdy,

β = (βk, βn), µ = (ξ, η) ∈ Rk+n, 〈µ〉θ,d = 〈ξ〉θ + 〈η〉d ∼k∑

j=1

|ξj |1

θj + |η| 1d .

The estimation (6) will be proved in the paper [4] by the method suggested bythat one used in the paper [1].

REFERENCES

[1] Christ, M.: Intermediate optimal Gevrey exponents occur, Comm. in Partialdifferential Equations, 22(3-4), 1997(359-379).

[2] Grushin, V. V.: Hypoelliptic differential equations and pseudodifferential oper-ators with operator-valued symmbols, Math. USSR Sbornik, 17, No.2, 1972(497-514).

[3] Hashimoto, Y. T. Hoshino and T. Matsuzawa: Non-isotropic Gevrey hy-poellipticity for Grushin operators, Publ. RIMS, Kyoto Univ. 38, 2002(289-319).

[4] Matsuzawa, T.: FBI-transformation and non-isotropic Gevrey hypoellipticityfor extended Grushin class, Preprint.

[5] Sjostrand, F.: Propagation of analytic singularities for second order Dirichletproblems, Comm. in Partial Differential Equations, 5, No.1, 1980(41-94).

47

Distribution Methods as Regularization for Ill-PosedProblems

Irina MelnikovaUral State University, Ekaterinburg, RUSSIA


We consider the Cauchy problem

u′(t) = Au(t), t ∈ [0, T ), T ≤ ∞, u(0) = f, (CP)

with linear differential operator A = A(D) as a special case of the abstract Cauchyproblem in a Banach space, assuming (CP) to be not uniformly well-posed. We takeinto consideration differential specific features of

A(D) =∑|α|≤r

AαDα Dα = Dα1

1 Dα22 . . .Dαn

n , Dk = i∂/∂xk.

Distribution, semigroup, Fourier transform methods, and ill-posed problem theorymethods are applied for constructing solutions to the problem.

The basic idea is the following: distribution, semigroup, and generalized Fouriertransform methods are methods of regularization in a broad sense. ”In a broad sense”means that we construct a corrected (smoothed) solution without a concern to anapproximation in contrast to regularization by ill-posed problems methods that supplycertain approximate solutions.

It is shown that regularization in distribution and generalized Fourier transformmethods is due to correction of solution operators by appropriate test functions andin semigroup methods due to correction of a resolvent of A (equal to the Laplacetransform of solution operators) by appropriate correcting functions or operators. Suchapproach to solutions constructed by methods mentioned above allows to constructsome new regularizing operators for ill-posed problems.

Special attention in the report will be given to construction of generalized solutions(in t) in spaces of distributions and ultradistributions in dependence on a behaviourof a resolvent of A and to construction of generalized solutions (in spatial variable xof D(A)) in spaces of Gelfand’s test functions in dependence on a behaviour of theFourier transform of solution operators.

The work was supported by grant RFBR No 03-01-00310.

REFERENCES

[1] Irina V. Melnikova and Alexei Filinkov The Cauchy problem.Three approaches. Monographs and Surveys in Pure and Applied Mathematics,120, London-New York: CRC, Chapman & Hall, 2001.

[2] Melnikova I.V. About regularization in a broad sense. Proceedings of the XIV-thCrimean Autumn Math. School-Symposium. Simpferopol, 2004.

48

Exact Solutions For a Generalized Weakly HyperbolicEquation Assigned to Dunkel Operrators

Assal MiloudDepartment de Mathematiques, Capmus universitaire, MRAZKA, IPEIN,

Nabeul, 8000. Tunisia.e-mail: [email protected]

Jelassi MouradDepartment de Mathematiques, Faculte des Sciences de Gabes, 6029 Tunisia.


In this paper we study a generalized weakly hyperbolic equation to obtain an exactrepresentation of the solutions for the Cauchy problem assigned to the hyperbolicoperator

2(a) = ∂2t − a(t)Lα, with a(t) > 0

On the half space IR× IR+, where Lα, (α > − 12) is the Dunkel operator given by

Lαu(x) :=d

dxu(x) +

α+ 1/2

x(u(x)− u(−x)).

Explicit solutions are given in the form of generalized functions and a global regularityresults are obtained in a generalized Sobolev-type spaces.

Exact Solutions For a Generalized Weakly HyperbolicEquation on the Dual of the Lauerre Hypergroup

Assal MiloudDepartment de Mathematiques, Capmus universitaire, MRAZKA, IPEIN,

Nabeul, 8000. Tunisia.e-mail: [email protected]

Using the harmonic analysis on the Dual of the Laguerre hypergroup we studya generalized weakly hyperbolic equation and we give an exact representation of thesolutions for the Cauchy problem assigned to the hyperbolic operator

2(a) = ∂2t − a(t)Λα, with a(t) > 0

On the half space (IR× IN)×IR+, where Λα, (α ≥ 0) is the derivation operator on thedual on the Laguerre hypergroup defined for an appropriate function Φ on (IR × IN)by

ΛαΦ(λ,m) =

(Λ2

1 − (2Λ2 + 2∂

∂λ)2)

Φ(λ,m),

where

• Λ1Ψ(λ,m) = 1|λ|

(m∆+∆−Ψ(λ,m) + (α+ 1)∆+Ψ(λ,m)

).

• Λ2Ψ(λ,m) = −12λ

((α+m+ 1)∆+Ψ(λ,m) +m∆−Ψ(λ,m)

).

49

• ∆+Ψ(λ,m) = Ψ(λ,m+ 1)−Ψ(λ,m).

• ∆−Ψ(λ,m) = Ψ(λ,m)−Ψ(λ,m− 1), if m ≥ 1 and ∆−Ψ(λ, 0) = Ψ(λ, 0).

Explicit solutions are given in the form of generalized functions and a global reg-ularity results are obtained.

Weak asymptotic of shock wave formation process

Darko MitrovicUniversity of Montenegro, SCG


We construct an asymptotic (in a weak sense) solution corresponding to the shockwave formation in a special situation. Passage from continuous to discontinuous partof the solution is described uniformly in t ∈ R+.

Continuity of the pseudodifferential operators on localizedBesov space

Madani MoussaiDepartment of Mathematics, M’Sila University,

P.O. Box 166, 28000 Algeriae-mail: [email protected]

We shall treat the continuity of pseudodifferential operators (ps.d.o.) of order m onLocalized Besov space

(Bs,q

p

)`r . The operators have a symbol in the class Sm

ρ,δ (ω,K, a)defined as ∣∣∣∂α

ξ ∂βxσ (x, ξ)

∣∣∣ ≤ c |ξ|a (1 + |ξ|)m−ρ|α|+δ|β| and∣∣∣∂αξ ∂

βxσ (x+ h, ξ)− ∂α

ξ ∂βxσ (x, ξ)

∣∣∣ ≤ c |ξ|a (1 + |ξ|)m−ρ|α|+δ|β| ω(|h| |ξ|δ

),

where m, δ, ρ ∈ R, a ≥ 0, α, β ∈ Nn, |β| ≤ K, ω a positive, nondecreasing, concavefunction on [0,∞) and(∫ 1

0

(ω (t)

ts−[s]+a

)qdt

t

)1/q

<∞, (1 ≤ q ≤ ∞, s > 0, a ≥ 0).

All ps.d.o., with σ ∈ S0ρ,δ (ω, s, 0) , is a bounded operator from

(Bs,q

p

)`r into itself, (G.

Bourdaud and M. Moussai 1988). We shall extend this result to the class Smρ,δ (ω, s, a)

for some ρ, δ and a.As(Bs,q

p

)`∞

, for 1 ≤ p ≤ q ≤ ∞ and s ≥ n/p, characterize the pointwise multi-

pliers space of Besov M(Bs,q

p

), (W. Sickel and I. Smirnov 1999), then every ps.d.o.

of order 0 is bounded on M(Bs,q

p

). We shall study the cases s ≥ n/p with p ≥ q and

0 ≤ s ≤ n/p for n = 1,(n ≥ 2 G. Bourdaud and M. Moussai 1988).

50

Singular shock wave and interaction phenomenons insystems of conservation laws

Marko NedeljkovDepartment for Mathematics and Informatics, Faculty of Science,

University of Novi Sad, Serbia and Montenegroe-mail: [email protected]

Some conservation law systems pemit solutions in the form of so called singularshock waves. The main property of such solutions is that they contain delta measure,like delta shock waves.

As a particular case one can use systems affine in some component of a solution,for example a 2× 2 system of the form:

(f1(u) + f2(u)v)t + (f3(u) + f4(u)v)x = 0

(g1(u) + g2(u)v)t + (g3(u) + g4(u)v)x = 0.

Existence of singular shock waves for such systems in evolution form are discussed inM. Nedeljkov, Delta and singular delta locus for one dimensional systems of con-

servation laws, Math. Meth. Appl. Sci. 27, 931-955 (2004).There are still a lot of open questions concerning their admissibility and uniqueness.The second part ot the talk consists of several examples concerning interaction

of singular shock wave with another one (singular shock, shock, rarefaction wave orcontact discontinuity).

Generalized uniformly continuous semigroups andsemilinear hyperbolic systems

Marko NedeljkovDepartment for Mathematics and Informatics, Faculty of Science,


Danijela Rajter-CiricDepartment for Mathematics and Informatics, Faculty of Science,


We introduce generalized uniformly continuous semigroups and use them in solvingsemilinear hyperbolic first-order systems with regularized derivatives. The existenceand uniqueness of a solution is proved. At the end we give some results on coherencewith the previously known results for the same class of systems.

51

Regularity theory in algebras of generalized functions: asurvey

Michael OberguggenbergerInstitut fur Technische Mathematik, Geometrie und Bauinformatik

Universitat Innsbruck, A - 6020 Innsbruck, Austria

The purpose of this talk is to study families of differential equations

Aε(x,D)uε(x) = fε(x)

where x varies in an open subset Ω of Rn and ε > 0. We wish to characterize theregularity of the solutions in terms of their asymptotic growth or decay properties withrespect to ε→ 0.

Equations of this type arise in semiclassical analysis, singular perturbations, oper-ators with regularized coefficients and/or data, asymptotic series solutions of analyticdifferential equations, and in Colombeau algebras. We present a number of instancesshowing how the regularity of a distribution w ∈ D′(Ω) can be expressed in termsof the asymptotics of its regularizations w ∗ ϕε, where ϕε is a mollifier of the formϕε(x) = ε−nϕ(x/ε).

The core of the talk is the study of regularity of solutions to differential equations asabove. In increasing generality, we first study linear partial differential equations withconstant, ε-dependent coefficients, then pseudodifferential operators with ε-dependentamplitudes, and finally we will arrive at some results about nonlinear equations.

The talk is a survey of ongoing joint work with C. Garetto, T. Gramchev, G.Hormann and S. Pilipovic.

On Laplace and Fourier Transforms of Mittag-LefflerFunctions and Applications to Fractional PDE’s

Zaid Odibat

Nabil Shawagfeh

The Mittag-Leffler functions have proved their efficiency as solutions of fractionaldifferential equations and thus have become important elements of fractional calculustheory and applications. In this paper we introduce the inverse Laplace transform ofthe Mittag-Leffler functions and their ordinary derivatives, the relationship betweenthe Multiple Mittag-Leffler functions and the inverse Fourier transform of a class ofMittag-Leffler functions and their ordinary derivatives. This work deals with solutionof two classes of fractional PDE’s. The solutions obtained are calculated in terms ofinverse Laplace and Fourier transforms of Mittag-Leffler functions.

52

Local Solvability of Partial Differential Operators inGevrey Classes

Alessandro OliaroDepartment of Mathematics, University of TorinoVia Carlo Alberto 10, I-10123 Torino (TO) - Italy

We present some results concerning Partial Differential Equations, in particularin Gevrey classes: we consider equations whose principal part is an elliptic operatorin a certain set of variables, and we prove local solvability in some Gevrey (or mixedGevrey-C∞) spaces. In some cases the results apply also to the semilinear equation.

Interaction of nonlinear waves for non integrable equations

Georgii Omel’yanovUniversidad de Sonora, Mexico

We consider non integrable versions of the KdV and sine-Gordon equations suchthat the equations have exact solitary wave solutions. The main result consists in ob-taining sufficient conditions, under which the interaction of solitary waves preserves theKdV scenario. This means that the interaction results without changing waves shapeand with shifts of trajectories. As the main tool here we use the weak asymptoticsmethod.

Another problem appears in the gas dynamics theory in the case of shock wavesinteraction. Scenarios of interactions are known, however the uniqueness of weaksolutions remains unproved. Using again the weak asymptotics method, we constructuniform in time asymptotic solutions for the isothermal case. The most interestingresult here is the description of a rarefaction wave appearance.

On the model of viscoelastic rod in unilateral contact witha rigid wall

Ljubica Oparnica , Teodor M. AtanackovicFaculty of Technical Sciences, University of Novi Sad

Stevan PilipovicDepartment of Mathematics and Mathematics, University of Novi Sad

We study translatory motion of a body to which a viscoelastic rod with the con-stitutive equation having fractional derivatives is attached. The body with a rodimpacts against a rigid wall. It is shown that the problem is described with coupledsystem of differential equations having integer and fractional derivatives having theform x(2) = −f ; f + af (α) = x + bx(α), x (0) = 0, x(1) (0) = 1. The unique solvability

53

in S′+ is proved and interpretation of solutions is given. Also some a priori estimatesof the solution are given. In particular we showed that restrictions on coefficientsthat follow from the Second Law of Thermodynamics imply that the velocity after theimpact is smaller than the velocity before the impact.Keywords: Viscoelasticity, fractional derivative, impact

Representation of Interpolating Function by Distribution

Nikola Pandeski

Vesna Manova-Erakovic

Ljupco Nastovski

We find distribution whose analytic representation is given interpolating function.

The Wavelet Transform of Gevrey Distributions

R. S. Pathak

The continuous wavelet transform is studied on certain Gevrey function and distri-bution spaces. Boundedness and continuity results are obtained in certain generalizedfunction spaces.

Bounded Solutions of Difference Equation withContinuous Time

Hajnalka PeicsUniversity of Novi Sad

Faculty of Civil Engineering, SuboticaSerbia and Montenegro


In this work we study the scalar difference equation with continuous time of theform

x(t) = (1− c(t))x(t− 1) + c(t)x(p(t)),

where t0 > 0, c, p : [t0,∞) → R are given real functions such that limt→∞ p(t) = ∞,and give some conditions for the existence of bounded solutions.

54

Laplace-Fourier Analysis when integrals and series diverge

Yves PeraireUniversite Blaise Pascal (Clermont II),

Departement de mathematiques,63177 Aubiere Cedex, France.


The underlying idea of my speach is that the discourse in mathematical languagemust aim the target of the facts directly, (the word ”fact” must be understood in abroad sense). To this purpose I will use a richer mathematical language, namely thatof the relative set theory, that is a conservative extension of ZFC. We are able thus tointroduce the words necessary to explore more finely the basic physical concepts. Mostof the physics concepts, ( point, equality, infinitesimality, infinity, delta function ...)involve partial indetermination; we chose to assert directly this indetermination withwords of this specific mathematical language while the traditional approach consistsin introducing suitably structured << small paradises >> in which indeterminationis hidden, such as distributions or ultra-distributions spaces, Hilbert spaces etc . . .My discourse thread is the presentation of various concepts for physical equality ( orindiscernability )

Dirac equality,D= : This equality allows us to speak directly of Dirac functions

and their derivatives in a semantical satisfactory way. All the products are autho-rized. No space of generalized function is needed.

Laplace equality,L= : It makes a general Heaviside formal calculus possible even if

no Laplace transform exists. It also permits to use divergent truncated series as gen-eralized Laplace image.

Fourier equality,F= : This equality neglects the inaudible signals. A kind of Fourier

analysis is developed, which is not encumbered by problems of integrability.

Generalized hyperfunctions and algebra of megafunctions

Stevan PilipovicDepartment of Mathematics and Informatics, University of Novi Sad,Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia and Montenegro


Sheaves of algebras of holomorphic generalized functions GH(Ω), where Ω is openin C, is introduced by Colombeau. We refer to papers of Colombeau and Aragona forthe properties of generalized holomorphic functions.

If f ∈ GH(Ω), then there exists a representative consisting of a net of holomorphicfunctions, global holomorphic representative (Pilipovic, Valmorin). Using this result,Khelif and Scarpalezos proved that it is equal to zero if its value, in the sense ofgeneralized complex numbers, at any point of Ω, is equal to zero.

55

Sheaves of spaces of generalized hyperfunctions BG and algebras of megafunctionsMG is introduced as follows. Let Ω be open in C and contain an open set ω ∈ R as aclosed subset. The space of generalized hyperfunctions is defined as BG(ω) = GH(Ω \ω)/GH(Ω) while the algebra of megafunctions is defined as MG(ω) = G(Ω \ ω)/G(Ω).

The first one is a flabby sheaf. Moreover, there exist injective sheaf homomor-phisms G → BG and BG → MG, where G is the algebra of Colombeau generalizedfunctions.

Nonlinear hyperbolic PDE-singularities and propagation

P. R. PopivanovBulg.Acad.of Sciences

This talk deals with the singularities of the solutions of several classes of nonlinearPDE and systems. In our investigations we have used two different approaches-theclassical method of the characteristics in the case of systems with one space variableand the machinery of the paradifferential operators in the multidimensional case. Atthe first part of the talk we discuss the propagation and interaction of singularitiesof one -dimensional semilinear hyperbolic systems. The solutions are allowed to havejump discontinuities in the initial data. We propose a detailed study of the behaviourof jump discontinuities ”in the future”. The singularities mentioned before propagatealong the corresponding characteristics till their interaction/as in the linear case/. Af-ter their collision new jump type singularities can appear which will propagate alongthe full set of characteristics starting from the collision point. The newly created sin-gularities are weaker than the initial ones. Similar results on creation/interaction/ ofsingularities hold true in the multi-dimensional case and in the scale of Sobolev spaces,the notion of characteristics being replaced by the notion of characteristic hypersur-faces. To prove the corresponding theorems/mainly due to Bony/the machinery ofparadiferrential operators is used. At the end of this talk we consider model examplesof first order semilinear non-strictly hyperbolic systems with two space variables andthe generalized Cauchy problem for them. We find out a necessary and sufficient con-dition for creation of logarithmic type singularity of the solution. The initial data areassumed to have finite jump type discontinuities along two transversal each to othercharacteristic surfaces. Different examples from geometry and physics are illustratingthe results here proposed.

Problems of the Navier-Stokes approximation to kineticequations

E. Radkevich

There is a well known fact of “ultra-violet catastrophe” which occurs in the lower-order truncations of Chapman-Enskog expansion [1]. It has been first demostratedby Bobylev [2] even in the simplest regime (one-dimensional linear deviations off theglobal equilibrium state):

56

post Navier-Stokes hydrodynamic equations violate basic physics behind the Boltz-mann equation.

Namely, sufficiently short acoustic waves are increasing with time instead of de-caying.

We consider a system of conservation laws with hard relaxation term, which in-cludes the moment approximations of kinetic equation:

∂tU +∇x · F (U) + ε−1 P(U) = 0, (9)

where U = U(x, t) is a Rn-valued function representing the density vector of basicphysical variables over the space variable x ∈ Rd. The homogeneous system is hyper-bolic, the relaxation term is endowed with an n × N matrix O of rank n < N suchthat

OP(U) = 0 ∀U.This yields n independent conserved quantities u = OU . In addition we assume eachsuch u determines uniquely a local equilibrium value U = E(u) such that

PE(u) = 0, OE(u) = u ∀u.

Variable ε is the relaxation time, which is small in many physical situations.The existence of the projection for (9) to the phase space of conserved variables

has been proved in the class of hyperbolic systems with relaxation. This allows one tosplit the dynamics (9) to the dynamics in the phase space of conserved variables andthe dynamics of nonequilibrium quantities that give an explanation of phenomena of“ultra-violet catastrophe”.

References1. Chapman S. C., Cowling T. C. The mathematical theory of non-uniform gases.

— Cambridge: Cambridge University Press, 19702. A. V. Bobylev, Sov. Phys. Dokl., 27, 29 (1982)3. Radkevich E. V. Well-posedness of mathematical models in continuum mechan-

ics and thermodynamics// Contemporary Mathematics. Fundamental Directions. —2003. — 3. — C. 5-32

4. Zkharchenko P. A., Radkevich E. V. Central manifold and problems of theChapman-Enskog projection for the Boltzmann-Peierls equation.//Dokl. RAN, to ap-pear

Multiplication of Distributions on the Group of Adeles

Yauhen Radyna

Distributions on the group of adeles (locally compact group built of the completionsof the rational numbers with respect to all possible valuations) are of great interestboth in Number Theory (A. Weil, Basic Number Theory) and Mathematical Physics(for examples of adelic distributions and formulation of adelic quantum mechanicssee papers of B. Dragovich). Adelic distributions inherit the main problem of theirreal analogs, there is no multiplication operation on the space of adelic distributionswhich would satisfy certain reasonable conditions. Thus no distributional solution ofa non-linear equation can be defined.

57

We handle the mentioned problem using apparatus of mnemofunctions (also callednon-linear generalized functions; for this theory in case of a real domain see papersof J.-F. Colombeau, M. Oberguggenberger, J.-A. Marti, Ya.V. Radyno, A.B. An-tonevich).

There is an interesting connections of mnemofunctions with trace formulas of Num-ber Theory. This happens because a cut-off used in trace formulas is essentially a reg-ularization procedure used to embed distributions into mnemofunction algebra. Thus,for example, Connes trace formula (at least in a local case) can be reformulated as astatement about a specific mnemofunction.

Subharmonic almost periodic functions and it’sFourier-Bohr coefficients

Adriy RakhninV.N. Karazin Kharkiv National University, Ukraine


A function f(z) subharmonic in open horizontal strip S ∈ C is called almostperiodic (in sense of distributions) if for any test function ϕ(z) in S the function∫ϕ(z − t)f(z)dxdy is a continuous, almost periodic function on the axis.

We prove that subharmonic function f(z) is almost periodic if and only if f(z) isalmost periodic in sense of Stepanov.

For every subharmonic almost periodic function there is Fourier-Bohr series

f(z) ∼∑

λ∈Λf

aλ(y) exp(iλx),

where Λf is spectrum of f and aλ(y) are Fourier-Bohr coefficients, which, as we prove,are continuous functions.

Idempotent operators on Banach and Hilbert spaces

Vladimir RakocevicDepartment of Mathematics and Informatics, Faculty of Science and Mathematics,

P.O. Box 224, 18000 Nis, Serbiae-mail: [email protected]

In this talk some new results concerning the idempotent operators on Banach andHilbert spaces are presented.

58

On the analytic representations of distributions

Nikola Rechoski, Vasko RechoskiUniversity of Ohrid, Macedonia

In theorem 1 we give another proof for the continuation of a distribution T ∈ D′ to adistribution of Oα

′, from the proof given by N. Bremerman in “Distributions, complexvariables, and Fourier transforms”.

In theorem 2 first we make some remarks for a lemma presented by R. Carmishaeland D. Mitrinovic in “Distributions and analytic functions”, and then we dive anotherproof for one part as a theorem.

PDE-methods in Multivariate Spline Analysis

Hermann RenderUniversity of Duisburg-Essen, Germanye-mail: [email protected]

andUniversidad de La Rioja, Logrono, Spain


In this talk we survey the theory of polysplines. Roughly speaking, polysplines oforder p are functions on the euclidean space Rn which are piecewise solutions of theequation ∆pu (x) = 0 (where ∆p is the p-th iterate of the Laplace operator ∆) andwhich obey certain matching conditions on given knot surfaces of codimension 1; fora precise definition and applications of polysplines see the monograph [2].

Special emphasis is given to cardinal polysplines u : Rn → C on strips: by defini-tion, u is then 2p − 2 times continuously differentiable, and ∆pu (x) = 0 for all x inthe open strips (j, j + 1)× Rn−1 for each j ∈ Z. We present an interpolation theoremfor polysplines in the flavor of the famous interpolation theorem of Schoenberg forunivariate polynomial splines. A major result is the fact that interpolation with poly-harmonic splines (introduced in [5] being a special case of interpolation with radialbasis functions) on lattices of the form Z× aZn−1 for a real number a > 0 leads in thelimit a→ 0 in a natural way to polysplines on strips.

In the last part we present applications of polysplines in Wavelet Analysis.The talk is based joint work with A. Bejancu (University of Leeds, UK) and O.

Kounchev (Bulgarian Academy of Sciences).

REFERENCES

[1] A. Bejancu, A. Kounchev, O., Render, H., Cardinal Interpolation with bihar-monic polysplines on strips, Proceedings of the Fifth International Conference on”Curves and Surfaces”, Saint-Malo, 27.6.–3.7.2002

[2] O. Kounchev, Multivariate Polysplines. Applications to Numerical and WaveletAnalysis, Academic Press, London–San Diego, 2001.

59

[3] Kounchev, O., Render, H., The Approximation order of Polysplines, Proc. Amer.Math. Soc. 132 (2004), 455-461.

[4] Kounchev, O., Render, H., Wavelet Analysis of cardinal L-splines and Construc-tion of multivariate Prewavelets. Approximation theory, X (St. Louis, MO, 2001),333–353, Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2002

[5] Madych, W.R. , Nelson, S.A., Polyharmonic Cardinal Splines, J. Approx. Theory60 (1990), 141–156.

Inhomogeneous Gevrey ultradistributions and Cauchyproblem

Luigi RodinoDipartimento di Matematica

Via Carlo Alberto, 10 10123 Torino, Italye-mail: [email protected]

Daniela CalvoDipartimento di Matematica

Via Carlo Alberto, 10 10123 Torino, Italye-mail: [email protected]

The inhomogeneous Gevrey ultradistributions D′s,λ are the topological duals ofthe inhomogeneous Gevrey classes, that represent a natural extension of the standardGevrey classes by means of Fourier transform, in terms of a fixed weight function λ,cf. Liess-Rodino [2] and Calvo-Morando-Rodino [1].

A relevant characterization of the inhomogeneous Gevrey ultradistributions withcompact support E ′s,λ(Rn), for any s > 1, is given by the following condition: for everyε > 0 there exists a constant Cε > 0 such that:

|u(ξ)| ≤ Cε exp(ελ(ξ)

1s

), ∀ξ ∈ Rn,

where u is the Fourier transform of u.Starting from this definition, we analyze the topological and algebraic properties

of the inhomogeneous Gevrey ultradistributions.In this frame we prove the well-posedness of the Cauchy problem in Rt × Rn

x :P (D)u(t, x) = Dm

t u(t, x) +∑

|ν|+j≤m, j 6=m

aνjDνxD

jtu(t, x) = 0

Dkt u(0, x) = fk(x), k = 0, . . . ,m− 1,

under the condition of (s, λ)-hyperbolicity of the operator P (D), i.e. we ask that itssymbol satisfies for a constant C > 0 the condition:

τm +∑

|ν|+j≤m, j 6=m

aνjξντ j = 0 for (τ, ξ) ∈ Cτ × Rn

ξ ⇒ =τ ≥ −Cλ(ξ)1s .

60

REFERENCES

[1] D. CALVO - A. MORANDO - L. RODINO, Inhomogeneous Gevrey classes andultradistributions, to appear in J. Math. Anal. Appl.

[2] O. LIESS - L. RODINO, Inhomogeneous Gevrey classes and related pseudo-differential operators, Anal. Funz. Appl., Suppl. Boll. U.M.I., Vol. 3, 1-C (1984),233-323.

Integral functionals and the compressible Euler equationson manifolds

Olga S. RozanovaMoscow State University, Moscow, Russia


It is shown that on the solutions to the compressible Euler equations given on asmooth manifold it is possible to consider a system of especially constructed integralfunctionals. This system is closed or not in dependence on the metrics of the manifold.My means of this system one can estimate the time of existence of smooth solutionsto Cauchy problem and construct special classes of exact solutions.

Cosine Operator Functions and Hilbert Transforms

Amina SahovicUniversity Dzemal Bijedic, Mostar, BiH


Fikret VajzovicUniversity of Sarajevo, BiH

In this paper we consider bounded cosine operator functions and their connectionto Hilbert transforms.

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Asymptotic Expansion of the Distributional WaveletTransform

Katerina SanevaInstitute of Mathematics and Physics, Faculty of Electrical Engineering

P.O.Box574, 1000 Skopje, Macedoniae-mail: [email protected]

Aneta BuckovskaInstitute of Mathematics and Physics, Faculty of Electrical Engineering

P.O.Box574, 1000 Skopje, Macedoniae-mail: [email protected]

In this paper we investigate the asymptotic behavior of the distributional wavelettransform at the origin 0. Using the notion of the quasiasymptotic behavior of a dis-tribution from S ′(R) at 0, we obtain results for the ordinary asymptotic behavior ofits wavelet transform at 0. We also give Abelian type results for the ordinary asymp-totic expansion of the wavelet transform of a tempered distribution with appropriatequasiasymptotic expansion.

On the Generalized Benjamin Type Equations

M. ScialomIMECC-Unicamp, CP 6065, Campinas, 13083-970, SP, Brazil


F. LinaresIMPA, Estrada Dona Castorina, 110; Rio de Janeiro, 22460-320, RJ, Brazil


We establish local and global well-posedness for the initial value problem associatedto the generalized Benjamin equation and generalizations of this in the energy space.We also studied the limit process of solutions when the surface tension is becomingsmall. To establish these results we make use of sharp theory developed to the studyof the generalized Korteweg-de Vries equation.

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Expansion Theorems for Generalized Random Processes,Wick Products and

Applications to Stochastic Differential Equations

Dora SelesiDepartment of Mathematics and Informatics, University of Novi Sad,Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia and Montenegro


Stevan PilipovicDepartment of Mathematics and Informatics, University of Novi Sad,Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia and Montenegro


Generalized stochastic processes can be regarded as a family of generalized ran-dom variables (generalized stochastic functions) with a time, space, time-space, or evenwith a multi-dimensional parameter system. Spaces of generalized stochastic functions,such as the Hida, the Kondratiev spaces etc. are infinite-dimensional analogues of theSchwartz distribution space S ′(Rd). Processes obtained in this way, as generalizedstochastic functions pointwisely defined with respect to the multi-dimensional param-eter t ∈ Rd, will be called generalized random processes of type (O), or we will justuse the acronym GRPs (O).

Another possibility is to consider generalized stochastic processes as linear con-tinuous mappings from a test-space A into a space of classical or generalized randomvariables. Processes defined in this way will be called GRPs (I). As in the deterministictheory of distributions, where we identify locally integrable functions and regular dis-tributions, we can identify elements f ∈ L1

loc(Rd, (S)−1) with the corresponding linearmappings f ∈ L(S(Rd), (S)−1). In other words, the class of locally Pettis-integrableGRPs (O) can be embedded into the class of GRPs (I).

The purpose of this paper is to develop series expansion theorems for the epony-mous GRPs (I) and hence, to pursue the idea of relating the approach to GRPs of T.Hida, Y. Kondratiev, B. Øksendal and the approach of K. Ito, H. Inaba, I. Gel’fand.In this genuine context processes are considered as generalized ones both with respectto the multi-dimensional parameter t ∈ Rd and by the random parameter ω ∈ Ω.

We construct a new Gel’fand triple exp(S)1 ⊆ (L)2 ⊆ exp(S)−1 as extension ofthe known Kondratiev one (S)1 ⊆ (L)2 ⊆ (S)−1. We also extend the Wick product ofGRPs (O) from the Kondratiev spaces to the spaces exp(S)−1. Expansion theoremsfor generalized stochastic processes considered as elements of the spaces L(A, (S)−1)and L(A, exp(S)−1) are derived. We prove that each GRP (I) can be representedin the form

∑∞j=1 fj ⊗ Hαj , where fj , j ∈ N, are deterministic generalized functions

from the Zemanian space of generalized functions A′. This series expansion is usedfor solving a class of evolution stochastic differential equations.

The second main result is the establishment of the Wick product for GRPs (I) viatheir chaos expansion. If we want to define the Wick multiplication of two GRPs (I) inthe same form as for GRPs (O), following question arises: How should one multiplicatethe coefficients fj , j ∈ N of the expansion? The answer is: We also develop a Wick-type multiplication for deterministic generalized functions in A′. The Wick product(we denote it by ) of GRPs (I) has a distinguished feature: It can be reduced to the

63

Wick product on the spaces of generalized stochastic functions (denoted by ♦), andalso to the Wick product acting on the Zemanian spaces (denoted by ).

Shock Structure in Hyperbolic Dissipative Systems

Srboljub S. SimicFaculty of Technical Sciences, Department of Mechanics

Trg Dositeja Obradovica 6, 21000 Novi Sad, Serbia

In hyperbolic systems of conservation laws shocks represent discontinuous solutionswith jumps located at the singular surface - shock front. When dissipative mechanisms,like viscosity, heat conduction or diffusion, are taken into account shocks are smearedand transformed into a shock structure - continuous solution with steep gradients. Inclassical approach they are obtained as perturbed solutions of the system of balancelaws endowed with vanishing viscosity terms, thus representing so-called viscous pro-files of k−shocks which bifurcate from the eigenvalues satisfying genuine nonlinearitycondition.

Rational extended thermodynamics provides a systematic way for derivation ofsystem balance laws retaining its hyperbolicity. Usual field variables, which enter inthe equilibrium subsystem of conservation laws, are adjoined with non-equilibriumvaribales (like stress tensor, heat flux or higher order moments) producing a first-order quasi-linear hyperbolic system of PDE’s. Eigenvalues of equilibrium subsystem,whose values are different from the eigenvalues of the extended system, satisfy so-calledsubcharacteristic conditions.

It is demonstrated, through a series of examples, that extended systems, whichobey certain dissipative properties, generate a shock structure solution for the k−shocksof equilibrium subsystem. These solutions are obtained for k−shocks which bifurcatefrom genuinely nonlinear eigenvalues of the equilibrium subsystem.

Laplace Transform of Distribution-Valued Functions andIts Applications

B. Stankovic

T. Atanackovic

For the distribution valued functions it is defined the Laplace transform and it isproved some properties of them. The space of distribution valued functions can beregarded as the smallest space of generalized functions which can be useful in manyproblems of Mechanics. In order to illustrate the usefulness of the elaborated results, itis proved the unicity of solution to a system of partial differential equations, containinginteger and real (fractional) order derivatives. The system is mathematical model ofan axially loaded rod, with rotary inertia, positioned on fractional derivative type ofviscoelastic foundation.

64

INEQUALITIES WHICH INCLUDE q-INTEGRALS

Miomir S. StankovicDepartment of Mathematics, Faculty of Occupational Safety


Predrag M. RajkovicDepartment of Mathematics, Faculty of Mechanical Engineering


Sladjana D. MarinkovicDepartment of Mathematics, Faculty of Electronic Engineering


University of Nis, Serbia and Montenegro

The main problem in analyzing inequalities which include q-integrals is the factthat q-integral of a function f(t) over an interval (a, b) is defined by difference

Iq(f, a, b) =

∫ b

a

f(t)dq(t) :=

∫ b

0

f(t)dq(t)−∫ a

0

f(t)dq(t),

where

Iq(f, 0, a) =

∫ a

0

f(t)dq(t) := a(1− q)

∞∑n=0

f(aqn)qn.

Here, discussion of the integral properties must include the points outside of intervalof integration.

In this paper, we will signify to some directions for solving this problem.

REFERENCES

[1] H. Gauchman, Integral Inequalities in q-Calculus, Computers and Mathematicswith Applications, vol. 47, (2004), 281–300.

[2] D.S. Mitrinovic, J.E. Pecaric and A.M. Fink, ”Classical and New Inequalitiesin Analysis”, Kluwer Academic Publishers, 1993.

65

p- frames and reconstruction series in separable Banachspaces

Diana T. StoevaDepartment of Mathematics, University of Architecture, Civil Engineering and

GeodesyBlvd. Christo Smirnenski 1, 1046 Sofia, Bulgaria

e-mail: stoeva [email protected]

Joint work with O. Christensen

The motivation behind the present work comes from frames in Hilbert spaces,which plays an increasing role in pure and applied mathematics. A countable sequencegii∈I in a Hilbert space (H, < ·, · >) is a frame for H if

∃B ≥ A > 0 : A‖f‖2 ≤∑i∈I

| < f, gi > |2 ≤ B‖f‖2, ∀f ∈ H.

One of the most useful properties of a frame gii∈I for H is that there exists a dualframe fii∈I for H, such that

f =∑i∈I

< f, gi > fi =∑i∈I

< f, fi > gi, ∀f ∈ H.

As a generalization of frames to Banach spaces, Aldroubi, Sun and Tang introducedp-frames and considered them in shift-invariant subspaces of Lp. A countable familygii∈I in the dual spaceX∗ of a Banach spaceX is called a p-frame forX (1 < p <∞)if there exist constants B ≥ A > 0 such that

A‖f‖X ≤

(∑i∈I

|gi(f)|p)1/p

≤ B‖f‖X , ∀f ∈ X.

In the present talk p-frames in general Banach spaces are considered. It is observed,that appropriate “dual” family for a p-frame, in connection to reconstruction series,is a q-frame ( 1

p+ 1

q= 1). It is proved, that a p-frame, without further assumptions,

allows every g ∈ X∗ to be represented as

g =∑i∈I

digi

for some coefficients dii∈I ∈ `q(I). However, expansions like

f =∑i∈I

gi(f)fi, ∀f ∈ X and g =∑i∈I

g(fi)gi, ∀g ∈ X∗ (10)

via a dual q-frame fii∈I ⊂ X are only possible under extra assumptions.

As a generalization of Riesz bases, q-Riesz bases are considered. A family gii∈I in aBanach space Y is a q-Riesz basis for Y (1 < q < ∞) if spangii∈I = Y and thereexist constants B ≥ A > 0 such that for all finite scalar sequences di,

A(∑

|di|q)1/q

≤∥∥∥∑ digi

∥∥∥Y≤ B

(∑|di|q

)1/q

.

66

A connection between q-Riesz bases and p-frames is determined ( 1p

+ 1q

= 1). For aq-Riesz basis gii∈I for X∗, reconstruction series like (10) are valid via a dual p-Rieszbasis fii∈I for X.

Regularization of fractional integral. Application tononlinear equations with singularities

Mirjana StojanovicDepartment of Mathematics and Informatics, University of Novi Sad,

Trg D.Obradovica 4, 21 000 Novi Sad, Serbia and Montenegro,e-mail: [email protected]

We study existence and uniqueness theorems for nonlinear equations with singu-larities in Colombeau algebras. To do so we use a method called regularization forfractional integral by delta sequence.

We recognize the fractional part of the kernel in integral form of some evolutionequations and regularize it by delta sequence. Since the fractional part of the kernelis a function with respect to t, the regularization is given for the time variable t.The aim of this method of regularization is to analyze the mathematical structure ofkernels by means of the fractional part which displays some of the main features of thegeneral problem. In addition, we regularize singular initial data, potential, and non-Lipschitz nonlinear term, with respect to the space variable. We illustrate the influenceof the procedure of regularization on the existence-uniqueness results established bythis procedure in corresponding Colombeau algebras. The selection of good mollifiersoccurring in kernels, initial data and potentials, depend on special information on thestructure and form of the equations and kernels, and the type of effects one wants toobserve. We find the link between the singularities appearing in the equation throughthe growths of mollifiers.

In particular, the fractional part of the heat kernel is regularized by delta sequencerelative to the time variable t in the case of nonlinear parabolic equations of ordi-nary type, equations with nonlinear conservative term, as well as nonlinear parabolicequations with Schrodinger’s type of operators, involving singular initial data and non-Lipschitz nonlinearities. The existence-uniqueness theorems in Colombeau vector typespaces GC1,(Lp,Lq)([0, T ), Rn), 1 ≤ p, q ≤ ∞, is established, for those choices of p, qand the dimension space n for which corresponding Colombeau space is an algebrawith multiplication. The same is done for linear Schrodinger equation with singularpotential and the initial data.

To show the generality of the method, we prove the existence-uniqueness theoremto the system of nonlinear Volterra’s type equations with δ-like free term, a polar kerneland non-Lipschitz nonlinearities. Since Volterra’s type of equations is governed byODEs, PDEs (especially of parabolic type) and they arise as model processes, in thatway, by our method, we cover a wide range of nonlinear equations with singularities.

67

The Bochner-Schwartz theorem for the Fourierultra-hyperfunctions

Masanori SuwaDepartement of English Language, Sophia Junior College,999 Kamiootuki, Sannoudai, Hatano-shi, Kanagawa, Japan

Kunio YoshinoDepartment of Mathematics, Sophia University,

7-1 Kioicho, Chiyoda-ku, Tokyo, Japan.

We shall treat the space of the positive definite Fourier ultra-hyperfunctions.Then we obtain that the elements of the positive Fourier ultra-hyperfunctions arethe elements of the distributions of exponential growth and the positive measure withsome exponential growth, and that the elements of the positive definite Fourier ultra-hyperfunctions are the elements of the tempered ultra-hyperfunctions.

The numerical methods in the field of Mikusinskioperators

Djurdjica TakaciDepartment of Mathematics and InformaticsFaculty of Science, University of Novi Sad

Trg Dositeja Obradovica 4, Novi Sad, Serbia and Montenegrotel/fax: (+381–21)–350–458


AMS Mathematics Subject Classification (2000): 44A40, 65J10.Key words and phrases: Mikusinski operators, integro–differential equations, characterof the approximate solutions.

We consider a class of nonlinear differential equations in the field of Mikusinskioperators corresponding to the partial integro–differential equations. We apply themethod of successive approximations in order to determine the approximate solutionand analyze the character of obtained solutions

REFERENCES

[1] Takaci, Dj., Takaci, A., The Approximate Solution of Nonlinear Equations, Pub-licationes Mathematicae Debrecen, 65(1-2), 2004 (in print).

[2] Takaci, Dj., Takaci, A., The Euler method in the field of operators, NumericalFunctional Analysis and Optimization (in print).

68

Multiresolution Expansions of Tempered Distributions

Nenad TeofanovDepartment of Mathematics and Informatics,Faculty of Science, University of Novi Sad

Trg Dositeja Obradovica 4, Novi Sad, Serbia and Montenegroe-mail: [email protected]

In this short communication we study properties of projections hj of a tempereddistribution h to the corresponding spaces Vj , in a regular multiresolution approxima-tion of L2(R).

We show that the derivatives of hj converge almost uniformly to the correspondingderivative of h as j tends to infinity, providing that h is smooth enough. Moreover,the convergence is in the norm of the space Sr, whose projective limit is the Schwartzspace S. Results related to tempered distributions are obtained by duality.

Furthermore, we prove Abelian and Tauberian type theorems concerning the quasi-asymptotic behavior of h at infinity via its multiresolution expansion.

Math Subject Classifications. 42C15, 42C40, 46F12.

Nontrivial Solutions of Boundary Value Problems forSemilinear Fourth and Sixth Order Differential Equations

Stepan Agop TersianCenter of Appled Mathematics and Informatics

University of Rousse8, Studentska, 7017, Rousse, Bulgaria

We study the existence of nontrivial solutions of the fourth and sixth order differ-ential equations equations, arising in phase transition models, as follows

uiv − pu′′ − a (x)u+ b (x)u3 = 0

anduvi +Auiv +Bu′′ + u− u3 = 0,

where p 6= 0 is a constant, a and b are continuous positive functions,A and B are con-stants. The boundary value problems (P1) and (P2) for these equations are consideredrespectively with the boundary conditions

u (0) = u (L) = u′′ (0) = u′′ (L) = 0.

andu (0) = u′′ (0) = uiv (0) = u (L) = u′′ (L) = uiv (L) = 0.

Existence of nontrivial solutions of problems (P1) and (P2) are studied using a min-imization theorem, Clark’s theorem and abstract theorems for critical points of thefunctionals in Hilbert spaces.

69

Positivity in twisted convolution algebra with applicationsto Weyl Calculus

Joachim ToftDepartment of Mathematics and Systems Engineering,

Vaxjo University, Vaxjo, Swedene-mail: [email protected]

We discuss positivity in the algebra of twisted convolution, and discuss applicationsto positivity in pseudo-differential calculus, especially the Weyl calculus.

If a ∈ D ′(R2n) and ϕ ∈ C∞0 (R2n), then the twisted convolution a ∗σ ϕ is definedby

(a ∗σ ϕ)(X) = (2/π)n/2

∫a(X − Y )ϕ(Y )e2iσ(X,Y ) dY

and we let S ′+(R2n) be the set of positive elements a in this algebra, i. e. (a∗σϕ,ϕ) ≥ 0

for every ϕ ∈ C∞0 (R2n). Here σ is the symplectic form.In many situations we have that if a ∈ S ′

+ satisfies a certain regularity or boundedproperty at the origin, then a and its Fourier transform a = Fa satisfy the sameregularity or bounded property everywhere. For example, the following are true forsome choice of Fourier transform Fσ:

1. (Growing properties at infinity.) If a ∈ S ′+, then a ∈ S ′.

2. (Trace-class properties.) If a ∈ S ′+, then the Weyl quantization for Fσa if and

only if a is continuous near the origin.

3. (Local boundedness with respect to Fourier Lp-spaces.) If a ∈ S ′+ and χa ∈ FLp

for some χ ∈ S such that χ(0) 6= 0, then ψa ∈ FLp and ψa ∈ FLp, for anyψ ∈ S .

4. (Classical regularity questions.) If a ∈ S ′+ is a C2N -function near the origin

for some integer N ≥ 0, then a and a belong to C2N ∩ H2N . Consequently,

S ′+ ∩ C∞ ⊂ S .

We also present some consequences in the theory of wave-front set and to modu-lation spaces theory.

Finally, these results have immediate applications to the Weyl calculus, since a ∈S ′

+(R2n) if and only if (Fσa)w(x,D) ≥ 0.

70

On the causal and anticausal n-dimensional convolutionequation related to the Diamond kernel of Marcel Riesz.

Susana Elena TrioneFacultad de Ciencias Exactas y Naturales - UBA, IAM-CONICET - Saavedra

15-3er.Piso-(1058),Buenos Aires. TE: 4954-6781/82 - FAX: 4954-6782


In this Note we extend the Theorem 4.1 ([3], p. 37) due to A. Kananthai whichsays that “Given the linear differential equation of the form

(eαt♦kδ) ∗ u(t) = Lku(t) = δ; (I, 1)

thenu(t) = eαt(−1)kS2k ∗R2k(t), (I, 2)

is an elementary solution of (I,1) or, equivalently, of the Diamond kernel of MarcelRiesz of (I,1), where S2k(t) and R2k(t) are defined, respectively by (2,1) and (2,3) of[1] with γ = 2k.

Our main result is the Theorem V.1, formula (V,3) which expresses that: “Giventhe linear partial differential equation of the form

(eαt♦kδ) ∗ (P ± i0)α−n

2 = Lk(P ± i0)α−n

2 = δ. (I, 3)

Here L is the partial differential operator of Diamond type defined by (IV,2). Then

(P ± i0)2k−n

2 = eαt(−1)k · S2k(P ′ ± i0) ∗R2k(P ± i0), (I, 4)

is an elementary solution of (I,3) where S2k(P ′ ± i0) and R2k(P ± i0) are defined by(II,10) and (II,7), respectively.

REFERENCES

[1] A. Kananthai, “On the Solution of the n-Dimensional Diamond Operator”, Ap-plied Mathematics and Computation, Elsevier, 88, (1997), 27-37.

[2] S. E. Trione, “On the solutions of the causal and anticausal n-dimensional Di-amond operators”, to appear in Trabajos de Matematica, IAM-CONICET yFCEyN-UBA.

[3] A. Kananthai, “On the convolution equation related to the Diamond kernel ofMarcel Riesz”. Journal of Computational and Applied Mathematics 100 (1998)33-39.

[4] I.M. Gelfand and G.E. Shilov, “Generalized Functions”, Vol.I, Academic Press,New York, 1964.

[5] S. E. Trione, “On the elementary (P ± i0)λ-ultrahyperbolic solution of the Klein-Gordon operator iterated k-times”, Integral Transforms and Special Functions,2000, Vol.9, Nr.2, 149-162.

[6] Y. Nozaki, “On the Riemann-Liouville integral of ultra-hyperbolic type”, KodaiMathematical Seminar Reports, 6(2) (1964), 69-87.

71

[7] M. Riesz, “L’integrale de Riemann-Liouville et le probleme de Cauchy”, ActaMathematica (81), 1949, 1-223.

[8] S. E. Trione, “Distributional Products”, Cursos de Matematica, 3, IAM-CONICET, Buenos Aires, 1980.

[9] W. F. Donoghue, “Distributions and Fourier transforms”, Academic Press, 1969.

[10] A. Gonzalez Domınguez and S. E. Trione, “On the Laplace transforms of retardedLorentz-invariant functions”, Trabajos de Matematica, preprint, Serie I, 13, IAM-CONICET, Buenos Aires, Argentina, 1977 and Advances in Mathematics, Volume30, number 2, November 1978, 51-62.

[11] S. E. Trione, “Sobre una formula de L. Schwartz”, Revista de la Union MatematicaArgentina, Vol. 26, 1973, 250-259.

Heaviside Generalized Functions and Shock Waves for aBurger Kind Equation

Francisco VillarrealDepartamento de Matematica, FEIS-UNESP15.385-000, Ilha Solteira, Sao Paulo, Brasil


Key words and phrases: Distributions, Generalized functions, Heaviside generalizedfunctions, Shock wave solutions, Burgers equation.

AMS (2000): 46F30, 35G20

The purpose of the work is to study the existence and nonexistence of shock wavesolutions ([3], [4]) for the equations ut + u divx u = 0 and ut + u divx u ≈ 0 whereu ∈ Gs(IR

n × IR) and the symbol “≈” denotes the weak equality used in generalizedfunctions theory (also called association relation) ([1], [3], [4]). The study is developedin the context of Colombeau’s theory of generalized functions ([1], [3]). The introduc-tion of the considered equations was suggested by the classical inviscid Burgers equa-tion ([2], [3]). The shock wave solutions are given in terms of generalized functions thathave the Heaviside function, in x ∈ IRn variable and in (x, t) ∈ IRn × IR variables, asmacroscopic aspect ([1]). Solutions of the form u(x, t) = (ur(t)−u`)H(x−y(t))+u` areinvestigated where y ∈ C∞(IR; IRn), ur ∈ C∞(IR) and H = H(x) ∈ Gs(IR

n) is a Heav-iside generalized function. Solutions of the form u(x, t) = (ur(t)−u`)H(x−y(t), t)+u`

are also investigated where H = H(x, t) ∈ Gs(IRn × IR). We show that a necessary

and sufficient condition so that the equation ut + u divx u ≈ 0 may have a shock wavesolution is that a jump condition in hydrodynamics holds. We also show that theequation ut + u divx u = 0 has no shock wave solutions.

REFERENCES

[1] Aragona, J. & Biagioni, H. An Intrinsic Definition of Colombeau Algebra ofGeneralized Functions, Anal. Math., T. 17, Fasc.2, (1991), pp. 75–132.

[2] Hop, E. The Partial Differential Equation ut + uux = µuxx, Comm. Pure Appl.Math. 3 (1950), 201–230.

72

[3] Oberguggenberger, M. Multiplication of Distributions and Applications toPartial Differential Equations. Pitman Research Notes in Math. V. 259. Harlow,Essex, England: Longman ST, 1993.

[4] Villarreal, F. Colombeau’s theory and shock wave solutions for systems of

PDEs, Electron. J. Diff. Eqns., V. 2000, N0

21, (2000), pp. 1–17.

From Weyl Transforms to Heat Kernelson Compact Lie Groups

M. W. WongDepartment of Mathematics and Statistics, York University

4700 Keele Street, Toronto, Ontario M3J 1P3, Canadae-mail: [email protected]

Pseudo-differential operators and their variants, first systematically developed byKohn and Nirenberg in 1965 and modified by Hormander and others for problemsin partial differential equations, have roots in quantization due to Hermann Weylmore than thirty years earlier. As such, they are known as Weyl transforms. Theproduct formula for these classic Weyl transforms entails a twisted convolution, whichis essentially a convolution on the Heisenberg group. This suggests a notion of Weyltransforms on locally compact and Hausdorff topological groups, which we present inthe context of compact groups in this talk. These Weyl transforms play a fundamentalrole in the formula for the heat kernels of the Laplacians on compact Lie groups.

On cohomologies of algebras of smooth functions

Victor ZharinovSteklov Mathematical Institute, Moscow, Russia


The linear space of all smooth functions over a smooth manifold, equipped withthe natural topology, gives rise to a number of algebras, widely used in applications.Important characteristics of these algebras are their cohomologies, in many cases thecohomological language is most appropriate. In my lecture I present several calculatedexamples.

73

Index

Albeverio, S., 1Alonso, B. V., 4Aragona, J., 2Atanackovic, T., 51, 61Atanassova, I., 26

Bermudez, P. R., 3, 4Bernard, S., 2Bouzar, C., 4Buckovska, A., 58Buzano, E., 5

Calvo, D., 6, 57Carmichael, R., 7Chen, D., 7, 19Chung, S.-Y., 8, 9, 30, 37Chung, Y.-S., 9, 30Colombeau, J.-F., 2, 9

Danilov, V. G. , 9Delcroix, A., 2, 11Dimovski, I., 12Djordjevic, D. S., 12Dragovich, B., 13Drozhzhinov, Yu. N., 13

Eida, A., 14Erlacher, E., 14Estrada, R., 15

Garetto, C., 15Georgiev, V., 16Grochenig, K., 18Gramchev, T., 17Grosser, M., 14, 17

Hakobyan, G. H., 6Harte, R. E., 19Hasler, M. F., 19He, Y., 7, 19Hoermann, G., 15Horvath-Bokor, R., 21

Ilic, D., 21

Jankovic, S., 21–23Jelinek, J., 22Jolevska-Tuneska, B., 23

Jovanovic, M., 22, 23Jovanovic, V., 24Juriaans, S. O., 24

Kalabisic, S., 24Kaminski, A., 25Karashtanova, E., 26Kempfle, S., 26Keyantuo, V., 27Khelif, A., 28Khrennikov, A. Yu., 1Kilicman, A., 29Kim, J. E., 30Kim, J.-H., 9, 30Kiryakova, V., 31Kmit, I., 32Konjik, S., 33Kostic, M., 34Kravarusic, R., 36Kunzinger, M., 32, 33, 36

Lee, N. K., 37Lemle, L. D., 37Linares, F., 59

Manova-Erakovic, V., 38, 51Marcati, P., 38Marinkovic, S. D., 61Marti, J.-A., 39Mateljevic, M., 41Matsuzawa, T., 41Melnikova, I., 45Mijatovic, M., 36Miloud, A., 46, 47Mincheva-Kaminska, S., 25Mitrovic, D., 47Mitsui, T., 21Mourad, J., 46Moussai, M., 47

Nastovski, Lj., 38, 51Nedeljkov, M., 48, 49

Oberguggenberger, M., 49Odibat, Z., 50Oliaro, A., 50Omel’yanov, G., 50

74

Oparnica, Lj., 51

Peics, H., 51Peraire, Y., 52Pandeski, N., 38, 51Pathak, R. S., 51Pilipovic, S., 34, 51, 53, 59Popivanov, P., 53

Radkevich, E., 54Radyna, Y., 54Radyna, Y. M., 13Radyno, Y. V., 13Rajkovic, P. M., 61Rajter-Ciric, D., 49Rakhnin, A., 55Rakocevic, V., 55Rechoski, N., 56Rechoski, V., 56Render, H., 56Rodino, L., 57Rozanova, O. S., 58

Sahovic, A., 58Scarpalezos, D., 28Scialom, M., 59Selesi, D., 59Shawagfeh, N., 50Shelkovich, V. M., 1, 9Simic, S. S., 60Stankovic, B., 61Stankovic, M. S., 61Steinbauer, R., 32Stoeva, D. T., 62Stojanovic, M., 63Suwa, M., 64

Takaci, A., 23Takaci, Dj., 64Teofanov, N., 65Tersian, S. A., 65Toft, J., 66Trione, S. E., 67

Vajzovic, F., 24, 58Villarreal, F., 68Volevich, L., 5

Wong, M. W., 69

Yoshino, K., 64

Zharinov, V., 69Zhour, Z. A., 29

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Geometric Function Theory and Applications

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