Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space Topological convolution algebras Guy Salomon Joint work with Daniel Alpay Department of Mathematics Ben-Gurion University Beer-Sheva, Israel May 21 st , 2011
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Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Topological convolution algebras
Guy Salomon
Joint work with Daniel Alpay
Department of MathematicsBen-Gurion University
Beer-Sheva, Israel
May 21st, 2011
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Outline
Strong algebras: A new family of dF algebras A =⋃
Φ′pwhich satisfy ‖ab‖p ≤ Ap,q‖a‖q‖b‖p.Strong convolution algebras: A special case where A is aconvolution algebra.
Which convolution algebra is a strong one?Examples:
The algebra of germs of holomorphic functions in zero: Thelast theorem implies it is a strong algebra. In particular we willsee that ‖fg‖p ≤ (1− 4−(p−q))−
12 ‖f‖q‖g‖p.
Relation to Dirichlet series: e.g.∫∞1
(∫ x1
√byc⌊xy
⌋dyy
)2
dxxt+1 ≤ infr∈(1,t)
(ζ(t)ζ(r)t(t−r)r
).
A non commutative version of the Kondratiev space ofstochastic distributions: Together with the definition ofVoiculescu of the non commutative WNS, it will form anon-commutative counterpart to the Gelfand triplet of(S1,W,S−1).
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
L2(G, µ) is usually not an algebra
L2(G, µ) is usually not an algebra
Let G be a locally compact topological group with a Haar measureµ. The convolution of two measurable functions f, g is defined by
(f ∗ g)(x) =
∫Gf(y)g(y−1x)dµ(y).
It is well known that L1(G,µ) is a Banach algebra with theconvolution product, while L2(G,µ) is usually not closed underconvolution. More precisely,
Theorem (N.W. Rickert (1969))
For any locally compact group G, L2(G,µ) is closed underconvolution if and only if G is compact.
In case G is compact it holds that, ‖f ∗ g‖ ≤√µ(G)‖f‖‖g‖.
Thus, L2(G,µ) is a Banach algebra.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
So what can be done if G is not compact, but we still want an ”Hilbert environment”?
So what can be done if G is not compact, but we stillwant an ”Hilbert environment”?
We will try to answer this question in the sequel.
In the meanwhile...
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
So what can be done if G is not compact, but we still want an ”Hilbert environment”?
So what can be done if G is not compact, but we stillwant an ”Hilbert environment”?
We will try to answer this question in the sequel.
In the meanwhile...
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definition
Strong algebras
Definition (D. Alpay & S (2012))
Let A =⋃p Φ′p be an algebra which is also a dual of a countably
normed space⋂p Φp. We call A a strong algebra if it satisfies
the property that there exists a constant d such that for any q andfor any p > q + d there exists a positive constant Ap,q such thatfor any a ∈ Φ′q and b ∈ Φ′p,
‖ab‖p ≤ Ap,q‖a‖q‖b‖p.
Let A be a strong algebra equipped with its strong topology, thatis, a neighborhood of zero is defined by means of any bounded setB ⊆
⋂p Φp and any number ε > 0, as the set of all a ∈ A for
whichsupb∈B|a(b)| < ε.
With this topology A is locally convex.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definition
Strong algebras
In these settings, for any a ∈ Φ′q the multiplication operator
Ma : b 7→ ab
is bounded operator on Φ′p (p > q + d).
Moreover, we may easily consider power series. If∑∞
n=0 fnzn
converges in the open disk with radius R, then for any a ∈ Awith ‖a‖q < R
Ap,q(p > q + d), we obtain
∞∑n=0
|fn|‖an‖p ≤∞∑n=0
|fn|(Ap,q‖a‖q)n <∞,
and hence∑∞
n=0 fnan ∈ Φ′p ⊆ A.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
They are topological algebras with continuous inverse
Strong algebras
Theorem (D. Alpay & S (2012))
Let A be a strong algebra.
(a) The multiplication is a continuous function A×A → A.Hence (A,+, ·) is a topological algebra.
(b) GL(A) is open, and the function a 7→ a−1 is continuous (thus,GL(A) is a topological group).
In view of the above, we can apply to unital strong algebras theresults of Naimark on (locally convex) topological algebraswith continuous inverse. In particular,
Theorem (M.A. Naimark)
The spectrum of any element in a topological algebra withcontinuous inverse is non-empty and closed.
But is it bounded?
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
The spectrum
Strong algebras
Theorem (D. Alpay & S (2012))
For any a ∈ A,
σ(a) ⊆ {z ∈ C : |z| ≤ inf{(p,q):p>q+d}
Ap,q‖a‖q}.
Thus, spectral theory can be easily done on strong algebras.
So what to strong algebras and our previous question?
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
The spectrum
Strong algebras
Theorem (D. Alpay & S (2012))
For any a ∈ A,
σ(a) ⊆ {z ∈ C : |z| ≤ inf{(p,q):p>q+d}
Ap,q‖a‖q}.
Thus, spectral theory can be easily done on strong algebras.
So what to strong algebras and our previous question?
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definition
Strong convolution algebras
Let G be a locally compact topological group with a Haar measureµ and let S ⊆ G be a Borel semi-group. Let (µp) be a sequence ofmeasures on G such that µ� µ1 � µ2 � · · · .
Definition⋃p L2(S, µp) is called a strong convolution algebra if there exists
a constant d such that for any p > q + d there exists a positiveconstant Ap,q such that for any f ∈ L2(µq) and g ∈ L2(µp),
‖f ∗ g‖L2(µp) ≤ Ap,q‖f‖L2(µq)‖g‖L2(µp)
(where (f ∗ g)(x) =∫S∩xS−1 f(x)g(y−1x)dµ(y)).
Due to the following theorem we can define a wide family of strongconvolution algebras.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
How to construct examples
Strong convolution algebras
Theorem (D. Alpay & S (2012))
If for any x, y ∈ S and for any p ∈ N,
dµpdµ (xy) ≤ dµp
dµ (x)dµpdµ (y),
then for any f ∈ L2(S, µq) and g ∈ L2(S, µp) such that p > q,
‖f ∗ g‖p ≤(∫
Sdµpdµq
dµ) 1
2 ‖f‖q‖g‖p.
In particular, if there exists d such that∫Sdµpdµq
dµ <∞ for any
p > q + d, then⋃p L2(S, µp) is a strong convolution algebra.
The simplest example: If G is compact, and we take µp = µ forall p, we obtain that ‖f ∗ g‖ ≤
√µ(G)‖f‖‖g‖. In particular,
L2(G,µ) is a SCA iff G is compact.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
An inverse theorem and a counter example
Strong convolution algebras
Proposition (The converse in a special case)
If dµp/dµ = (dµ1/dµ)p for any p ∈ N, then
‖f ∗ g‖p ≤ Ap,q‖f‖q‖g‖p.
implies
dµpdµ (xy) ≤ dµp
dµ (x)dµpdµ (y).
Example:Let s be the space of rapidly decreasing sequences (that is, thespace of all complex sequences (an) with supn |an(n+ 1)p| <∞for any p ∈ N). It can be shown that
s =⋂`2(n+1)2p
where `2(n+1)2p = {a ∈ CN0 :∑|an|2(n+ 1)2p <∞}.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
An inverse theorem and a counter example
Strong convolution algebras: a counter example
Therefore, its dual s′ can be viewed as
s′ =⋃p∈N
`2(n+1)−2p .
Thus s′ is of the form s′ =⋃p∈N L2(S, µp), where S = N0 and
µp(n) = (n+ 1)−2p.The convolution then becomes the standard one sided convolutionof sequences:
(a ∗ b)(n) =
n∑k=0
a(k)b(n− k).
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
An inverse theorem and a counter example
Strong convolution algebras: a counter example
One may ask whether s′ is a strong convolution algebra, i.e. ifthere exists a constant d such that for any p > q + d there existsAp,q such that for any a ∈ s′q and b ∈ s′p,
‖a ∗ b‖p ≤ Ap,q‖a‖q‖b‖p.
However since,
dµpdµ
(n) = (n+ 1)−2p =
(dµ1dµ
(n)
)pfor any p ∈ N,
and
dµpdµ
(n+m) = (n+m+1)2p 6≤ (n+1)2p(m+1)2p =dµpdµ
(n)dµpdµ
(m),
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
An inverse theorem and a counter example
Strong convolution algebras: a counter example
we conclude in view of the last theorem that the answer isnegative, that is,
Theorem
s′ is not a strong convolution algebra.
In a previous paper (2011, IDAQP), we replace the measures(n+ 1)−2p by 2−np, and obtain a strong convolution algebra thatcontains s′, and which can be identified as the dual of a space ofentire holomorphic functions that is included in the Schwartz spaceof complex-valued rapidly decreasing smooth functions.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
An inverse theorem and a counter example
Strong convolution algebras: the discrete case
Theorem (D. Alpay & S, IDAQP)
In the discrete case it holds that
A SCA is nuclear.
The tensor product (with respect to the π or ε topology) oftwo SCA’s is again a SCA.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The space of germs of holomorphic functions in zero
Let H(C) be the space of entire holomorphic functions equippedwith the usual topology of convergence on compact sets. It can beshown that
H(C) =⋂p∈N
H2(2pD),
where H2(2pD) is the Hardy space of the disk 2pD. Thus,
H ′(C) =⋃p∈N
H2(2−pD),
and
‖f‖p = sup0<r<1
1
2π
∫|z|=r·2−p
|f(z)|2 dz =
∞∑n=0
|an|22−2np.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The space of germs of holomorphic functions in zero
For any two germs in H ′(C), f(z) =∑∞
n=0 anzn and
g(z) =∑∞
n=0 bnzn, we define
(fg)(z) =
∞∑n=0
(n∑
m=0
ambn−m
)zn.
Since for any n,m ∈ N0 and for any p ∈ N,dµpdµ (n+m) = 2−2(n+m)p = 2−2np2−2mp =
dµpdµ (n)
dµpdµ (m), and
since for any p > q∫N0
dµpdµq
dµ =
∞∑n=0
2−2np
2−2nq= (1− 4−(p−q))−1,
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
The algebra of germs of holomorphic functions in zero is strong
The space of germs of holomorphic functions in zero
In particular we obtain that
Theorem (The algebra of germs of holomorphic functions in zero)
H ′(C) is a strong (convolution) algebra. More precisely, for anyf ∈ H2(2−qD) and g ∈ H2(2−pD) (p > q) fg belongs toH2(2−pD) and
‖fg‖p ≤ (1− 4−(p−q))−1/2‖f‖q‖g‖p.
We note that since
σ(f − f(0)) ⊆ {z ∈ C : |z| ≤ inf{(p,q):p>q+d}
Ap,q‖f − f(0)‖q}
⊆ {z ∈ C : |z| ≤ infq
(1− 4−1)−1/2∑
06=n∈N|an|22−2nq} = {0},
f ∈ H ′(C) is invertible if and only if f(0) 6= 0.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The space P
Let P be the space of all functions f : [1,∞)→ R such that thereexists p ∈ N with ∫ ∞
1|f(x)|2 dx
xp+1<∞.
In particular, any restriction of a polynomial function into [1,∞)belongs to P. Thinking of [1,∞) as a semi-group with respect tothe multiplication of the real numbers, and since the Haar measureof ((0,∞), ·) is dx
x = d(ln(x)), we obtain the following convolution
(f ∗ g)(x) =
∫ x
1f(y)g
(x
y
)dy
y, ∀x ∈ [1,∞).
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The space P
We also note that for any p > q,∫ ∞1
dµpdµq
dµ =
∫ ∞1
x−(p+1)
x−(q+1)
dx
x=
1
p− q.
Thus, we obtain that
Theorem (The space P)
The space P is a strong convolution algebra. More preciesly, forany f ∈ L2([1,∞), dx
xq+1 ) and g ∈ L2([1,∞), dxxp+1 ), f ∗ g belongs
to L2([1,∞), dxxp+1 ) and
‖f ∗ g‖p ≤ (p− q)−1/2‖f‖q‖g‖p.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Mellin transform
The space P : Mellin transform
The last theorem can be expressed in terms of Mellin transform.
Definition
The one-sided Mellin transform is defined by
(Mf)(s) =
∫ ∞1
xsf(x)dx
x.
Thus, the inequality in the last theorem can be rewritten as
(M(f ∗ g)2)(−t) ≤ 1
t− r(Mf2)(−r)(Mg2)(−t) for any t > r.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Relations to Dirichlet series
The space P : relations to Dirichlet series
Now if we consider the Dirichlet series∞∑n=1
ann−s and
∞∑n=1
bnn−s.
Since for any s in the half-plane of absolute convergence,
∞∑n=1
ann−s = s
M∑n≤y
an
(−s),
we obtain
Corollary
For any r < t it holds that:
∫ ∞1
∫ x
1
√∑n≤y
an∑n≤ x
y
bndy
y
2
dx
xt+1≤ 1
t(t− r)r
∞∑n=1
ann−r
∞∑n=1
bnn−t.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Relations to Dirichlet series
The space P : Dirichlet series
For example, taking the zeta function of Riemann,ζ(s) =
∑∞n=1 n
−s, which converges for Re s > 1, one obtains∫ ∞1
(∫ x
1
√byc⌊x
y
⌋dy
y
)2
dx
xt+1≤ ζ(t)ζ(r)
t(t− r)rfor any 1 < r < t.
Hence,∫ ∞1
(∫ x
1
√byc⌊x
y
⌋dy
y
)2
dx
xt+1≤ inf
r∈(1,t)
(ζ(t)ζ(r)
t(t− r)r
)for any t > 1.
In a similar way, if we take ϕ(n) to be the phi Euler function, weobtain∫ ∞1
∫ x
1
√∑n≤y
ϕ(n)∑n≤x
y
ϕ(n)dy
y
2
dx
xt+1≤ inf
r∈(1,t)
(ζ(t− 1)ζ(r − 1)
tζ(t)(t− r)rζ(r)
).
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The Kondratiev space of stochastic distributions
Let H be an Hilbert space and let H◦n be the closed subspace ofH⊗n generated by all the vectors of the form
u1 ◦ · · · ◦ un =1
n!
∑σ∈Sn
uσ(1) ⊗ · · · ⊗ uσ(n).
Definition
The full Fock space associated to H is defined by
Γ(H) := ⊕∞n=0H⊗n = C⊕H⊕ (H⊗H)⊕ · · · .
The symmetric Fock space associated to H is defined by
Γ◦(H) := ⊕∞n=0H◦n = C⊕H⊕ (H ◦H)⊕ · · · .
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The Kondratiev space of stochastic distributions
The (commutative) white noise space is defined by
W = Γ◦(H0),
where it is conventional to fix H0 = L2(R). Letting (en) be anorthonormal basis of L2(R) (for example, the Hermite functions),the associated orthogonal basis of W can be viewed as follows.Denote by ` is the free commutative monoid generated by N, thatis
` = N(N)0 =
{α ∈ NN
0 : supp(α) is finite}
= ⊕n∈NN0en,
Then for α = (α1, α2, α3, . . .) ∈ `
eα := e◦α11 ◦ e◦α2
2 ◦ e◦α33 ◦ · · ·
is an orthogonal basis of W with ‖eα‖ = α! = α1!α2!α3! · · · .Thus, W is isometrically isomorphic to L2(`, ν), where ν(α) = α!.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The Kondratiev space of stochastic distributions
The Wick product is defined by (f, g) 7→ f ◦ g. In terms of thesecond representation we obtain that
f ◦ g =
(∑α∈`
fαeα
)◦
(∑α∈`
gαeα
)=∑α∈`
∑β≤α
fβgα−β
eα.
Since ` is not compact, in view of Rickert theorem W = L2(`, ν)cannot be closed under the Wick product. Kondratiev was lookingfor an algebra which includes W, he defined
Definition (Y. Kondratiev, 1978)
S−1 = {∑
α∈` fαeα :∑
α∈` |fα|2(2N)−αp <∞ for some p ∈ N}is called the Kondratiev space of stochastic distributions, where(2N)α = 2α1 · 4α2 · 6α3 · · · .
He also showed that this space is closed under the Wick product.We note that S−1 =
⋃p∈N L2(`, µp), where µp(α) = (2N)−αp.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The Kondratiev space of stochastic distributions
In 1992, Zhang showed that
Lemma (Zhang, 1992)∑α∈`(2N)−αd <∞ iff d > 1.
So actually, he showed that∫`dµpdµq
dµ <∞ iff p > q + 1.In 1996, Vage showed that
Theorem (Vage, 1996)
In the space S−1 it holds that,
‖f ◦ g‖p ≤ Ap−q‖f‖q‖g‖p
for any p > q + 1, and for any f ∈ L2(`, µq), g ∈ L2(`, µp), whereA2p−q =
∑α∈`(2N)−α(p−q) is finite due to Zhang.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Definitions
The Kondratiev space of stochastic distributions
So actually he showed that S−1 is a strong algebra (this could have
been done directly from our theorem, sincedµpdµ are multiplicative).
The fact that S−1 is a strong algebra is a key tool:
- In the theory of stochastic partial differential equations, see:
H. Holden, B. Øksendal, J. Ubøe, and T. Zhang. Stochastic partialdifferential equations. Probability and its Applications, 1996.
- In the theory of stochastic linear systems, see:
D. Alpay and D. Levanony. Linear stochastic systems: a white noiseapproach. Acta Applicandae Mathematicae, 2010.
D. Alpay, D. Levanony, and A. Pinhas. Linear stochastic state spacetheory in the white noise space setting. SIAM Journal of Control andOptimization, 2010.
- In the theory of interpolations, see:
D. Alpay and H. Attia. An interpolation problem for functions with valuesin a commutative ring. Operators Theory: Advances and Applications,2011.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
A non commutative version
The Kondratiev space: a non commutative version
The non-commutative white noise space is defined by
W = Γ(H0),
(where again, it’s conventional to fix H0 = L2(R)).
In the same manner as before, if we denote by ˜ the free(non-commutative) monoid generated by N, we obtain that W isisometrically isomorphic to L2(˜, ν), where ν is now the countingmeasure (the α! has now disappeared since we are no longer in thesymmetric case).
The Wick product is now defined by (f, g) 7→ f ⊗ g, and again canbe viewed as a convolution of functions over ˜. Again, since ˜ isnot compact we obtain in view of Rickert theorem that W is notclosed under the Wick product.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
A non commutative version
The Kondratiev space: a non commutative version
When we try to construct the non commutative version of theKondratiev space, we wish that a similar inequality will hold.
Definition (D. Alpay & S, 2012)
The non commutative Kondratiev space of stochastic distributionsis defined by
S1 =⋃p∈N
Γ(Hp),
where Hp = {∑fnen :
∑|fn|2(2n)−p <∞} ⊇ H0 and (en) is an
orthonormal basis of H0 = L2(R).
In the same manner as before, if for any α = zα1i1zα2i2· · · zαnin ∈ ˜ we
denote µp(α) = (2N)α =∏nk=1(2ik)
αk . We obtain, that
S−1 =⋃p∈N
L2(˜, µp).
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
Thus, for any 2 integers p > q + 1 we obtain∫˜dµpdµq
dµ <∞.
Moreover, we note thatdµpdµ are multiplicative. Thus, we conclude
Corollary
S−1 is a strong (convolution) algebra. More precisely,
‖f ⊗ g‖p ≤ Bp−q‖f‖q‖g‖p
for any p > q + 1, and for any f ∈ L2(`, µq), g ∈ L2(`, µp), whereB2p−q =
∑α∈˜(2N)−α(p−q) <∞.
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
A non commutative version
The Kondratiev space: a non commutative version
More generally, in case S is discrete, anddµpdµ = (dµ1dµ )p, it holds
that
Theorem (D. Alpay & S, 2012)⋃p∈N Γ(L2(S, µp)) with the multiplication ⊗ is a strong algebra if
and only if⋃p∈N L2(S, µp) is nuclear.
(In the case of S−1, S = N, µp(n) = (2n)p,⋃p∈N L2(S, µp) = s′
which is nuclear, and thus S−1 =⋃p∈N Γ(L2(S, µp)) is a SA).
The fact that the non commutative version of the Kondratievspace of stochastic distributions is a strong algebra, gives rise to atheory of non commutative stochastic linear systems, and to noncommutative versions of the other applications which weredescribed before.
But this will be done in a future work...
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
A non commutative version
The Kondratiev space: a non commutative version
More generally, in case S is discrete, anddµpdµ = (dµ1dµ )p, it holds
that
Theorem (D. Alpay & S, 2012)⋃p∈N Γ(L2(S, µp)) with the multiplication ⊗ is a strong algebra if
and only if⋃p∈N L2(S, µp) is nuclear.
(In the case of S−1, S = N, µp(n) = (2n)p,⋃p∈N L2(S, µp) = s′
which is nuclear, and thus S−1 =⋃p∈N Γ(L2(S, µp)) is a SA).
The fact that the non commutative version of the Kondratievspace of stochastic distributions is a strong algebra, gives rise to atheory of non commutative stochastic linear systems, and to noncommutative versions of the other applications which weredescribed before.
But this will be done in a future work...
Outline Introduction Strong algebras Strong convolution algebras Space of germs The space P The Kondratiev space
A non commutative version
Further reading
For further reading on strong algebras, see:
D. Alpay and G. Salomon. A new family of C-algebras withapplications in linear systems. Infinite Dimensional Analysis,Quantum Probability and Related Topics, 2012.
D. Alpay and G. Salomon. Topological convolution algebraspreprint on arXiv, 2012.