Optimal maps for the multidimensional Monge-Kantorovich problem

Let μ₁,…, μ_N be Borel probability measures on ℝ^d. Denote by Γ(μ₁,…, μ_N) the set of all N-tuples T=(T₁,…, T_N) such that T_i:ℝ^d ↔ ℝ^d (i = 1,…, N) are Borel-measurable and satisfy μ₁ = μ_i[V] for all Borel V ⊂ ℝ^d. The multidimensional Monge-Kantorovich problem investigated in this paper consists of finding S=(S₁,…, S_N) ∈ Γ(μ₁,…, μ_N) minimizing over the set Γ(μ₁, ···, μ_N). We study the case where the μ_i's have finite second moments and vanish on (d - 1)-rectifiable sets. We prove existence and uniqueness of optimal maps S when we impose that S₁( x ) ≡ x and give an explicit form of the maps S_i. The result is obtained by variational methods and to the best of our knowledge is the first available in the literature in this generality. As a consequence, we obtain uniqueness and characterization of an optimal measure for the multidimensional Kantorovich problem. © 1998 John Wiley & Sons, Inc.     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspaceĨ _d ^m of the noncommutative invariant algebraĨ _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(Ĩ _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition ofĨ _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Products of Toeplitz Operators on a Vector Valued Bergman Space

We give a necessary and a sufficient condition for the boundedness of the Toeplitz product T _F T _G* on the vector valued Bergman space L_a²(\mathbbCⁿ){L_a^2(\mathbb{C}^n)}, where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [13]. We also characterize boundedness and invertibility of Toeplitz products T _F T _G* in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space [17].     

Algebra Forms with $$d^{N} = 0$$ on Quantum Plane. Generalized Clifford Algebra Approach

We construct a q-analog of exterior calculus with a differential d satisfying d ^N = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.     

Noncommutative unitons

By Uhlenbeck’s results, every harmonic map from the Riemann sphere S² to the unitary group U(n) decomposes into a product of so-called unitons: special maps from S² to the Grassmannians Gr _k(ℂⁿ) ⊂ U(n) satisfying certain systems of first-order differential equations. We construct a noncommutative analogue of this factorization, applicable to those solutions of the noncommutative unitary sigma model that are finite-dimensional perturbations of zero-energy solutions. In particular, we prove that the energy of each such solution is an integer multiple of 8π, give examples of solutions that are not equivalent to Grassmannian solutions, and study the realization of non-Grassmannian zero modes of the Hessian of the energy functional by directions tangent to the moduli space of solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 220–239, February, 2008.     

On Transference of Multipliers on Matrix Weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

We consider a periodic matrix weight W defined on ℝ ^d and taking values in the N×N positive-definite matrices. For such weights, we prove transference results between multiplier operators on L _p (ℝ ^d ;W) and L_p(\mathbb T^d;W)L_{p}(\mathbb {T}^{d};W), 1<p<∞, respectively. As a specific application, we study transference results for homogeneous multipliers of degree zero.     

Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complex. 23:262–295, 2007. We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S _d given as the d-fold tensor product of a compact linear operator S ₁ for d=1,2,…, with ‖S ₁‖=1 and S ₁ having at least two positive singular values. Let n(ε,S _d ) be the minimal number of information evaluations needed to approximate S _d to within ε∈[0,1]. We study generalized tractability by verifying when n(ε,S _d ) can be bounded by a multiple of a power of T(ε ⁻¹,d) for all (ε ⁻¹,d)∈Ω⊆[1,∞)×ℕ. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T(ε ⁻¹,d) whose multiple bounds n(ε,S _d ). We also study weak tractability, i.e., when . In our previous paper, we studied generalized tractability for proper subsets Ω of [1,∞)×ℕ, whereas in this paper we take the unrestricted domain Ω ^unr=[1,∞)×ℕ. We consider the three cases for which we have only finitely many positive singular values of S ₁, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S ₁ decay slightly faster than logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S ₁ and mostly asymptotic properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T(x,y)=x ^1+ln y . We also study tractability functions T of product form, T(x,y)=f ₁(x)f ₂(x). Assume that a _i =lim inf  _x→∞(ln ln f _i (x))/(ln ln x) is finite for i=1,2. Then generalized tractability takes place iff

Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces

Summary.   Let X,X ₁,X ₂,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ ^d . Assume that E X=0 and that X is not concentrated in a proper subspace of ℝ ^d . Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S _N ] of the normalized sums S _N =N ^−1/2(X ₁+⋯+X _N ) and show that

On a generalized dimension of self‐affine fractals

For a d ×d expanding matrix A, we de.ne a pseudo‐norm w (x) in terms of A and use this pseudo‐norm (instead of the Euclidean norm) to define the Hausdorff measure and the Hausdorff dimension dim^w _H E for subsets E in R ^d . We show that this new approach gives convenient estimations to the classical Hausdorff dimension dim^w _H E, and in the case that the eigenvalues of A have the same modulus, then dim^w _H E and dim_H E coincide. This setup is particularly useful to study self‐affine sets T generated by ?j (x) = A^–1(x +dj), dj ∈ R ^d , j = 1, …, N. We use it to investigate the fractality of T for the case that {?j }^N _{j =1} satisfying the open set condition as well as the cases without the open set condition. We extend some well‐known results in the self‐similar sets to the self‐af.ne sets. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wavelets on the 2-Sphere: A Group-Theoretical Approach

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S², based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and ⁺_* for dilations on S², which are embedded into the Lorentz group SO₀(3, 1) via the Iwasawa decomposition, so that X SO₀(3, 1) M Y S O L N, where N . We select an appropriate unitary representation of SO₀(3, 1) acting in the space L²(S², d μ) of finite energy signals on S². This representation is square integrable over X; thus it yields immediately the wavelets on S² and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S² is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R → ∞. Then the parameter space goes into the similitude group of ² and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.     

Self-Affine Sets and Graph-Directed Systems

   Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ⊂ Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ≠  ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.     

Cohomology of arbitrary spin currents in AdS3

We study conserved currents of any integer or half-integer spin built from massless scalar and spinor fields inAdS ₃. We show that 2-forms dual to the conserved currents inAdS ₃ are exact in the class of infinite expansions in higher derivatives of the matter fields with the coefficients containing inverse powers of the cosmological constant. This property has no analogue in the flat space and may be related to the holography of the AdS spaces. “Improvements” to the physical currents are described as the trivial local current cohomology class. A complex (T^S, D) of spin-s currents is defined, and the cohomology group H¹(T^S, D)= ℂ^2s+1is found. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 3–25, April, 2000.     

Mixing of Cartesian squares of positive operators

LetT be a power bounded positive operator inL ₁(X, Σ, m)of a probability space, given by a transition measureP (x, A). The Cartesian squareS is the operator onL ₁ (X × X, Σ × Σ, m × m) induced by the transition measure Q((x, y), A × B)=P(x, A)P(y, B).T iscompletely mixing if ∝u e dm=0 impliesT ⁿ u→0 weakly (where 0≦e ∈L _∞ withT ^* e=e).Theorem. IfT has no fixed points, thenT is completely mixing if and only ifS is completely mixing. Part of this research was done at The Hebrew University of Jerusalem.     

ON NUMBER OF LEAVES ANDBANDWIDTH OF TREES

1.IntroductionFOragraphG=(V,E)oforderp,aonetoonemappingfromVinto{l,2,',p}iscalledanumberingofG.Definition1.1.SupposefisanumberingofG.LetBj(G)=(u57teif(u)--f(v)l.ThebandwidthofG,denotedbyB(G),isdefinedbyB(G)=min{Bf(G)IfisanumberingofG}.Thebandwidthproblemofgraphshasbecomeveryimportantsincethemid-sixties(see[21or[4]).Itisverydifficulttodeterminethebandwidthofagraph.GareyetallllshowedthatthebandwidthproblemisNP-completeevenifitisrestrictedtotreeswithmaximumdegree3.Soitisinterestingtoe…     

Talagrand＇s T2-transportation Inequality w.r.t, a Uniform Metric for Diffusions

Using the method of Girsanov transformation, we establish the Talagrand‘s T2-inequality for diffusion on the path space C([0, N], R^d) with respect to a uniform metric, with the constant independent of N. This improves the known results for the L2-metric.     

Unitary dilations and theC* algebra0 2

LetH be an infinite-dimensional separable Hilbert space, and letS=(S _ij)∈teB(H)⊗M ₂ be a unitary 2 × 2 matrix with operator entries. We study theC*-algebra generated by the operatorsS _ij, and show that the study of unitary dilations of isometriesT inH reduces to the special case whereS ₁₁ =T, andS ₂₁ = 0. We useC*-algebraic techniques to obtain detailed results about the set of all unitary dilations ofT. Work supported in part by NSF.     

Logarithmic mean oscillation on the polydisc,endpoint results for multi-parameter paraproducts,and commutators on BMO

We study boundedness properties of a class ofmultiparameter paraproducts on the dual space of the dyadic Hardy space H _d ¹ (T ^N ), the dyadic product BMO space BMO ^d (T ^N ). For this, we introduce a notion of logarithmic mean oscillation on the polydisc. We also obtain a result on the boundedness of iterated commutators on BMO [0, 1] ^N ).     

Rank-one cross commutators on backward shift invariant subspaces on the bidisk

For a backward shift invariant subspace N in H^2（Г^2）, the operators Sz and Sw on N are defined by Sz = PNTz｜N and Sw, = PNTw｜N, where PN is the orthogonal projection from L^2（Г^2） onto N. We give a characterization of N satisfying rank [Sz, Sw^＊] = 1.     

Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems

Using Grozman’s formalism of invariant differential operators we demonstrate the derivation of N=2 Camassa-Holm equation from the action of Vect(S ^1|2) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N=2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H ¹-metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N=1 super peakon type equations, known as N=1 super b-field equations, are derived from the action of Vect(S ^1|1) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S ^1|1) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N=1 super b-field equations.     

Twistor quantization of the space of half-differentiable vector functions on the circle revisited

We discuss the twistor quantization problem for the classical system (V _d ,A _d ), represented by the phase space V _d , identified with the Sobolev space H ₀^1/2 (S ¹,ℝ ^d ) of half-differentiable vector functions on the circle, and the algebra of observables A _d , identified with the semi-direct product of the Heisenberg algebra of V _d and the algebra Vect(S ₁) of tangent vector fields on the circle.