Commuting dual Toeplitz operators on the harmonic Dirichlet space

In this paper,we characterize the symbols for(semi-)commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space.We show that for φ,ψ∈W~(1,∞),S_φS_ψ=S_ψ Sφ on(D_h)~⊥ if and only if φ and ψ satisfy one of the following conditions:(1) Both φ and ψ are harmonic functions;(2) There exist complex constants α and β,not both 0,such that φ=αψ +β.     

Generalized CRF-structures

A generalized F-structure is a complex, isotropic subbundle E of \({T_cM \oplus T^*_cM}\) (\(T_cM = TM \otimes_{{\mathbb{R}}} {\mathbb{C}}\) and the metric is defined by pairing) such that \(E \cap \bar{E}^{\perp} = 0\). If E is also closed by the Courant bracket, E is a generalized CRF-structure. We show that a generalized F-structure is equivalent with a skew-symmetric endomorphism Φ of \(TM \oplus T^*M\) that satisfies the condition Φ³ +  Φ =  0 and we express the CRF-condition by means of the Courant-Nijenhuis torsion of Φ. The structures that we consider are generalizations of the F-structures defined by Yano and of the CR (Cauchy-Riemann) structures. We construct generalized CRF-structures from: a classical F-structure, a pair \(({\mathcal{V}}, \sigma)\) where \({\mathcal{V}}\) is an integrable subbundle of TM and σ is a 2-form on M, a generalized, normal, almost contact structure of codimension h. We show that a generalized complex structure on a manifold M? induces generalized CRF-structures into some submanifolds \(M \subseteq \tilde{M}\) . Finally, we consider compatible, generalized, Riemannian metrics and we define generalized CRFK-structures that extend the generalized Kähler structures and are equivalent with quadruples (γ, F ₊, F _?, ψ), where (γ, F _±) are classical, metric CRF-structures, ψ is a 2-form and some conditions expressible in terms of the exterior differential d ψ and the γ-Levi-Civita covariant derivatives ? F _± hold. If d ψ =  0, the conditions reduce to the existence of two partially Kähler reductions of the metric γ. The paper ends by an Appendix where we define and characterize generalized Sasakian structures.     

Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic

It was proved that the complexity of square root computation in the Galois field GF(3^s), s = 2^kr, is equal to O(M(2^k)M(r)k + M(r) log₂r) + 2^kkr^1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3^s) is equal to O(M(2^k)M(r)) and O(M(2^k)M(r)) + r^1+o(1), respectively. If the basis in the field GF(3^r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2^kkr^1+o(1) and r^1+o(1) can be omitted. For M(n) one may take n log₂nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O_r (M (s) log ₂s) = s (log ₂s)²ψ(s).     

Drift perturbation of subordinate Brownian motions with Gaussian component

Let d ≥ 1 and Z be a subordinate Brownian motion on R~d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L~b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p~b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L~b, C_c~∞(R~d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.     

Linear independence of values of Lerch functions

The number of linearly independent numbers among 1, Φ₁ (z, p/q), ...,Φ _a (z, p/q) is estimated depending on a natural number a, where Φ _s (z, p/q), s = 1, 2, ..., are Lerch functions.     

A four-point criterion for the Möbius property of a homeomorphism of plane domains

An ordered quadruple of pairwise distinct points T = {z ₁, z ₂, z ₃, z ₄} ? C is called regular whenever z ₂ and z ₄ lie at the opposite sides of the line through z ₁ and z ₃. Consider Φ(T) = ∠z ₁ z ₂ z ₃ + ∠z ₁ z ₄ z ₃ (the angles are undirected) as some geometric characteristic of a regular tetrad. We prove the following theorem: For every fixed α ∈ (0, 2π) the Möbius property of a homeomorphism f: D → D* of domains in C is equivalent to the requirement that each regular tetrad T ? D with Φ(T) = α whose image fT is also a regular tetrad satisfies Φ(fT) = α. In 1994 Haruki and Rassias established this criterion for the Möbius property only in the class of univalent analytic functions f(z).     

On the semiproportional character conjecture in groups Sp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Previously, the author made the following conjecture: if a finite group has two semiproportional irreducible characters φ and ψ, then φ(1) = ψ(1). In the present paper, a new confirmation of the conjecture is obtained. Namely, the conjecture is verified for symplectic groups Sp₄(q) and PSp₄(q).     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

Ergodic properties of skew products in infinite measure

Let (Ω, µ) be a shift of finite type with a Markov probability, and (Y, ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a non-singular automorphism of (Y, ν). We study skew products of the form (ω, y) ? (σω, Φ_ω0 (y)), where σ is the shift map on (Ω, µ). We prove that, when the skew product is recurrent, it is ergodic if and only if the Φ_i’s have no common non-trivial invariant set.     

On <Emphasis Type="Italic">s</Emphasis>-Elementary Super Frame Wavelets and Their Path-Connectedness

A super wavelet of length n is an n-tuple (ψ ₁,ψ ₂,…,ψ _n ) in the product space \(\prod_{j=1}^{n} L^{2}(\mathbb{R})\), such that the coordinated dilates of all its coordinated translates form an orthonormal basis for \(\prod_{j=1}^{n} L^{2} (\mathbb{R})\). This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (η ₁,η ₂,…,η _n ) in \(\prod_{j=1}^{n}L^{2} (\mathbb{R})\) such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for \(\prod_{j=1}^{n} L^{2}(\mathbb{R})\). In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E ₁,E ₂,…,E _m ) is said to be τ-disjoint if the E _j ’s are pair-wise disjoint under the 2π-translations. We prove that a τ-disjoint m-tuple (E ₁,E ₂,…,E _m ) of frame sets (i.e., η _j defined by \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\) is a frame wavelet for L ²(?) for each j) lead to a super frame wavelet (η ₁,η ₂,…,η _m ) for \(\prod_{j=1}^{m} L^{2} (\mathbb{R})\) where \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\). In the case of super tight frame wavelets, we prove that (η ₁,η ₂,…,η _m ), defined by \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\), is a super tight frame wavelet for ∏_1≤j≤m L ²(?) with frame bound k ₀ if and only if each η _j is a tight frame wavelet for L ²(?) with frame bound k ₀ and that (E ₁,E ₂,…,E _m ) is τ-disjoint. Denote the set of all τ-disjoint s-elementary super frame wavelets for ∏_1≤j≤m L ²(?) by \(\mathfrak{S}(m)\) and the set of all s-elementary super tight frame wavelets (with the same frame bound k ₀) for ∏_1≤j≤m L ²(?) by \(\mathfrak{S}^{k_{0}}(m)\). We further prove that \(\mathfrak{S}(m)\) and \(\mathfrak{S}^{k_{0}}(m)\) are both path-connected under the ∏_1≤j≤m L ²(?) norm, for any given positive integers m and k ₀.     

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R~n) for ? ∈ H~1(~(Sn-1)) and ? ∈ Llog~+L(S~(n-1)) ∪_1q∞(B~((0,0))_q(S~(n-1))), respectively.     

Small prime solutions to cubic equations

Let a_1,..., a_9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a_1,..., a_9 are pairwise coprime and a_1 + + a_9 ≡ b(mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a_1,..., a_9, n) such that the cubic equation a_1p_1~3+ + a9p_9~3= b is solvable with p_j 《 P for all 1 ≤ j ≤ 9. It is proved that one can take P = max{|a_1|,..., |a_9|}~c+ |b|~(1/3) with c = 2. This improves upon the earlier result with c = 14 due to Liu(2013).     

Weakly nonlinear Schrödinger equation with random initial data

It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ _t (x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x∈? ^d and t∈?. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ _t (x) has a limit as λ→0 for t=λ ^?2 τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.     

Quantization causes waves: Smooth finitely computable functions are affine

Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are F _p = {0, 1,..., p - 1} produces a 1-Lipschitz map f _A from the space Z _p of p-adic integers to Z _p . The map fA can naturally be plotted in a unit real square I² ? R²: To an m-letter non-empty word v = γ _m-1γ _m-2... γ0 there corresponds a number 0.v ∈ R with base-p expansion 0.γ _m-1γ _m-2... γ0; so to every m-letter input word w = α _m-1α _m-2 ··· α0 of A and to the respective m-letter output word a(w) = β _m-1β _m-2 ··· β0 of A there corresponds a point (0.w; 0.a(w)) ∈ R². Denote P(A) a closure of the point set (0.w; 0.a(w)) where w ranges over all non-empty words.We prove that once some points of P(A) constitute a C ²-smooth curve in R², the curve is a segment of a straight line with a rational slope. Moreover, when identifying P(A) with a subset of a 2-dimensional torus T² ∈ R³, the smooth curves from P(A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ(x, t) = e ^i(Ax-2πBt) (x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

On infinite-rank singular perturbations of the Schrödinger operator

Schrödinger operators with infinite-rank singular potentials V=Σ _i,j=1 ^∞ b _ij〈φ_j,·〉φ_i are studied under the condition that the singular elements ψ _j are ξ _j(t)-invariant with respect to scaling transformationsin ?³.     

Boundary Crossing Probabilities for General Exponential Families

We consider parametric exponential families of dimension K on the real line. We study a variant of boundary crossing probabilities coming from the multi-armed bandit literature, in the case when the real-valued distributions form an exponential family of dimension K. Formally, our result is a concentration inequality that bounds the probability that B ^ψ (θ? _n , θ*) ≥ f(t/n)/n, where θ* is the parameter of an unknown target distribution, θ? _n is the empirical parameter estimate built from n observations, ψ is the log-partition function of the exponential family and B ^ψ is the corresponding Bregman divergence. From the perspective of stochastic multi-armed bandits, we pay special attention to the case when the boundary function f is logarithmic, as it is enables to analyze the regret of the state-of-the-art KL-ucb and KL-ucb+ strategies, whose analysis was left open in such generality. Indeed, previous results only hold for the case when K = 1, while we provide results for arbitrary finite dimension K, thus considerably extending the existing results. Perhaps surprisingly, we highlight that the proof techniques to achieve these strong results already existed three decades ago in the work of T. L. Lai, and were apparently forgotten in the bandit community. We provide a modern rewriting of these beautiful techniques that we believe are useful beyond the application to stochastic multi-armed bandits.     

Frattini and related subgroups of mapping class groups

Let Γ_g,b denote the orientation-preserving mapping class group of a closed orientable surface of genus g with b punctures. For a group G let Φ_f(G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φ_f(G) is nilpotent when G is a finitely generated subgroup of Γ_g,b, in this paper we compute Φ_f(G) for certain subgroups of Γ_g,b. In particular, we answer Ivanov’s question in the affirmative for these subgroups of Γg,b.     

Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

We study the sharp Nikol’skii inequality between the uniform norm and the L _q norm of algebraic polynomials of a given (total) degree n ≥ 1 on the unit sphere \(\mathbb{S}^{m - 1} \) of the Euclidean space ? ^m for 1 ≤ q < ∞. We prove that the polynomial ? _n in one variable with unit leading coefficient that deviates least from zero in the space L _q ^ψ (?1, 1) of functions f such that |f| ^q is summable over (?1, 1) with the Jacobi weight ψ(t) = (1 - t)^α(1 + t)^β, α = (m - 1)/2, β = (m - 3)/2 as a zonal polynomial in one variable t = ξ _m , where x = (ξ ₁, ξ ₂, …, ξ _m ) ∈ \(\mathbb{S}^{m - 1} \), is (in a certain sense, unique) extremal polynomial in the Nikol’skii inequality on the sphere \(\mathbb{S}^{m - 1} \). The corresponding one-dimensional inequalities for algebraic polynomials on a closed interval are discussed.     

On the rate of decay of concentration functions of <Emphasis Type="Italic">n</Emphasis>-fold convolutions of probability distributions

Concentration functions of n-fold convolutions of probability distributions is shown to exhibit the following behavior. Let φ(n) be an arbitrary sequence tending to infinity as n tends to infinity, and ψ(x) be an arbitrary function tending to infinity as x tends to infinity. Then there exists a probability distribution F of a random variable X such that the mathematical expectation E _ψ(|X|) is infinite and, moreover, the upper limit of the sequence \(\sqrt n \phi \left( n \right)Q_n\) is equal to infinity, where Q _n is the maximal atom of the n-fold convolution of distribution F. Thus, no infinity conditions imposed on the moments can force the concentration functions of n-fold convolutions decay essentially faster than o(n ^?1/2).