We consider the complex interpolation method in the context of protoquantum spaces. We apply it to construct an ℒp-protoquantization of an arbitrary normed space for 1 < p < ∞.

We describe the location of the essential spectrum of one-dimensional Dirac operators with singular potentials supported on discrete sets in ℝ in terms of limit operators.

We give a new formula for the zeta determinant of a Sturm-Liouville operator on a line segment.

The paper considers the classical Bernoulli scheme, that is, a sequence of independent random variables identically distributed with respect to the Lebesgue measure m on the interval [0,1]. The space of realizations of this scheme is the infinite-dimensional cube \({\mathcal{X}} = ({[0,1]^{\mathbb{N}}},\mu )\) with Lebesgue measure μ = mℕ. It is proved that there exists a function k(·): (0, 1) → ℝ

Let V1, …, Vn be finite-dimensional spaces of smooth functions on a smooth n-manifold X. We determine a relationship between the average number of solutions to systems of equations {fi = ai ∣ fi ∈ Vi, ai ∈ ℝ, i =1, …, n} and mixed volumes of convex bodies. To this end, assuming the spaces Vi to be normed, we construct (1) measures on the spaces of systems of equations and (2) Banach convex bodies in

We consider two examples of a fully decodable combinatorial encoding of Bernoulli schemes: the encoding via Weyl simplices and the much more complicated encoding via the RSK (Robinson-Schensted-Knuth) correspondence. In the first case, decodability is quite a simple fact, while in the second case, this is a nontrivial result obtained by D. Romik and P. Sniady and based on [2], [12], and other papers

We show that the spectrum of the Schrödinger operator H = −Δ + V in a smooth cylinder with Robin boundary condition ∂vu = σu is purely absolutely continuous, assuming that the coefficients V and σ are periodic in the axial directions.

The paper deals with a three-dimensional family of diffusion processes on an infinite-dimensional simplex. These processes were constructed by Borodin and Olshanski in 2009 and 2010, and they include, as limit objects, the infinitely-many-neutral-allels diffusion model constructed by Ethier and Kurtz in 1981 and its extension found by Petrov in 2009. Each process X in our family possesses a unique

It is proved that any finite-dimensional Lie algebra over an algebraically closed field K of characteristic 0 can be embedded (realized) as a transitive Lie subalgebra of the Lie algebra of polynomial vector fields on the space Kn. The same is also proved for arbitrary real Lie algebras and in some other cases.

Let Ω be a C1,s bounded domain (s > 1/2) in ℝd, and let \({{\cal A}^\varepsilon } = - {\rm{div}}\;A(x,x/\varepsilon )\nabla \) be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of \({{\cal A}^\varepsilon }\) are locally periodic. For μ in the resolvent

A self-adjoint strongly elliptic second-order differential operator Aε on L2(ℝd;ℂn) is considered. It is assumed that the coefficients of Aε are periodic and depend on x/ε, where ε > 0 is a small parameter. Approximations for the operators cos(A 1/2ε τ) and A 1/2ε sin(A 1/2ε τ) in the norm of operators from the Sobolev space Hs(ℝd;ℂn) to L2(ℝd;ℂn) (for appropriate s) are obtained. Approximation with

The paper considers the Schrödinger equation with an additional linear potential on the entire real line. A transformation operator with a condition at −∞ is constructed. The Gelfand-Levitan integral equation is obtained on the half-line (−∞, x).

Geremia and Sullivan [Ann. Math Pure Appl. 127 (1981), 231–251] gave a necessary and sufficient condition for an ℓp-product of spaces to be 2-uniformly rotund. We extend this result to k-uniform rotundity for any integer k > 1. The nonreflexive, uniformly nonoctahedral Banach space \(\widetilde{X}\) constructed by James [Israel J. Math. 18 (1974), 145–155] does not contain arbitrarily close copies

We prove that the connected sum of two links is quasipositive if and only if each summand is quasipositive. The proof is based on the filling disk technique.

Conditions under which the system of dilations and translations of a function f in a symmetric space X is a representing system in X are found. Previously a similar result was known only for the spaces Lp, 1 ⩽ p < ∞. In particular, each function f with \(\int_0^1 {f(t)dt \ne 0} \) in a Lorentz space Λϕ generates an absolutely representing system of dilations and translations in this space if and only

We show that the extremal extrapolation spaces for an operator with any function ξ characterizing the growth of the operator norms are the sum and the intersection of spaces whose norms depend on ξ and are written out explicitly.

Basic results for nonlinear operators are given. These results include nonlinear versions of the classical uniform boundedness theorem and Hahn-Banach theorem. Further, mappings of a metrizable space into another normed space can belong to some normed spaces if one defines suitable norms.

Given one-dimensional determinantal point processes induced by orthogonal projections with integrable kernels satisfying a certain growth condition, it is proved that their conditional measures with respect to the configuration in the complement of a compact interval are orthogonal polynomial ensembles with explicitly found weights. Examples include the sine-process and the process with Bessel kernel

We prove that the sigma functions of Weierstrass (g = 1) and Klein (g = 2) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of 2g linear differential heat equations in a nonholonomic frame (for a function of 3g variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic

For any small ε > 0, a two-dimensional elastic waveguide is constructed such that λe = ε4 is the only eigenvalue in the vicinity of the lower bound λ† = 0 of the continuous spectrum. This result is rather unexpected, because an acoustic waveguide (the Neumann problem for the Laplace operator) with an arbitrary small localized perturbation cannot support a trapped mode at a low frequency.

Compact pseudodifferential operators whose symbol fails to be smooth with respect to x on a given set are considered. Conditions under which Weyl’s law of spectral asymptotics remains valid for such operators are obtained. The results are applied to operators with symbols such that their order of decay as |ξ| → ∞ is a nonsmooth function of x.

We prove the attainability of the best constant in the fractional Hardy-Sobolev inequality with a boundary singularity for the spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin. A similar result has been proved earlier for the conventional Hardy-Sobolev inequality.

We study the center of the universal enveloping algebra of the strange Lie superalgebra q(N). We obtain an analogue of the well-known Perelomov-Popov formula [6] for the central elements of this algebra—an expression of the central characters through the highest weight parameters.

It is proved that any Bethe subalgebra corresponding to a regular semisimple element in a Yangian is a maximal commutative subalgebra and, moreover, the centralizer of its quadratic part. As a consequence, a description of such subalgebras as the traces of the R-matrix over all finite-dimensional representations of the Yangian is obtained.

Rules for calculating the densities of Borel measures which are absolutely continuous with respect to a positive nonatomic Radon measure are considered. The Borel measures are generated by composite functions which depend on continuous functions of bounded variation defined on an interval. Questions related to the absolute continuity of Borel measures generated by composite functions with respect to

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, r loops, one-dimensional framing, and dim V = 2. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank r. Hence it possesses a symplectic resolution with 2r fixed points with respect to a Hamiltonian torus action. We also identify its flavor deformation with

This paper describes a class of spectral curves and gives explicit formulas for the Darboux coordinates of the Hitchin systems of types Al, Bl, and Cl on hyperelliptic curves. The current state of the problem in the case of the systems of type Dl is described.

S. V. Chmutov, M. E. Kazarian, and S. K. Lando have recently introduced a class of graph invariants, which they called shadow invariants (these invariants are graded homomorphisms from the Hopf algebra of graphs to the Hopf algebra of polynomials in infinitely many variables). They proved that, after an appropriate rescaling of the variables, the result of the averaging of almost every such invariant

The variational problem on the equilibrium of a two-phase elastic medium is studied for conditions of the Signorini type. The strong convergence of its solutions to single-phase states as the temperature unboundedly increases is proved. A sufficient condition for the existence of phase transition temperatures for one-sided problems is given. A one-dimensional example illustrating the results is presented

We consider generalized Hausdorff operators and introduce the notion of the symbol of such an operator. Using this notion, we describe, under some natural conditions, the structure and investigate important properties (such as invertibility, spectrum, and norm) of normal generalized Hausdorff operators on Lebesgue spaces over ℝn. As an example we consider Cesàro operators.

Relative projection constants and strong unicity constants for a certain class of projection operators on the space l 2 n∞ are found. The maximum values of strong unicity constants are calculated for the projection operators with unit norm on certain codimension-2 subspaces formed by using hyperplanes in l 2 n∞ .

In this paper we strengthen the result of Fomin and Zelevinsky (2002) on the Laurent phenomenon for Somos-4 and Somos-5 sequences.

A new Caristi-type inequality is considered and Caristi’s fixed point theorem for mappings of complete metric spaces is developed (in both the single- and set-valued cases). On the basis of this development mappings of complete metric spaces which are contractions with respect to a function of two vector arguments are studied. This function is not required to be a metric or even a continuous function

Let T be a self-adjoint operator on a Hilbert space H with domain \(\mathscr{D}(T)\). Assume that the spectrum of T is contained in the union of disjoint intervals Δk = [α2k−1,α2k], k ∈ ℤ, the lengths of the gaps between which satisfy the inequalities$${\alpha _{2k + 1}} - {\alpha _{2k}}\geqslant b{\rm{|}}{\alpha _{2k + 1}} + {\alpha _{2k}}{{\rm{|}}^p}\;\;\;\;{\rm{for}}\;{\rm{some}}\;\;{\rm{b}} > 0

We give a complete solution of the inverse problem for finite Jacobi operators with matrix-valued coefficients.

We consider arbitrary noncompact hyperbolic Riemann surfaces of finite area. For such surfaces, we obtain identities relating the discrete spectrum of the Laplace operator to the resonance spectrum (formed by the poles of the scattering matrix). These identities depend on the choice of a test function. We indicate a class of admissible test functions and consider two examples corresponding to specific

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the nonsingular analytic curve W = {u =(u1, u3) ∈ ℂ2: σ(u)

The asymptotic behavior at large times of the solution of the Cauchy problem for the Airy equation—a third-order evolutionary equation—is established. We assume that the initial function is locally Lebesgue integrable and has a power-law asymptotics at infinity. For the solution in the form of a convolution integral with the Airy function, we use the auxiliary parameter method and the regularization

We consider foliations of compact complex manifolds by analytic curves. We suppose that the line bundle tangent to the foliation is negative. We show that, in the generic case, there exists a finitely smooth homomorphism holomorphic on the fibers and mapping fiberwise the manifold of universal coverings over the leaves passing through a given transversal B onto some domain with continuous boundary

We obtain a qualitative characterization of the convergence rate of the averages (with respect to the Margulis measure) of C2 functions over the iterations of domains in unstable manifolds of a topologically mixing C3 Anosov diffeomorphism with oriented invariant foliations. For this purpose, we extend the constructions of Margulis and Bufetov and introduce holonomy invariant families of finitely additive

Let Ω be a metric space. By At we denote the metric neighborhood of radius t of a set A ⊂ Ω and by \(\mathfrak{D}\), the lattice of open sets in Ω with partial order ⊆ and order convergence. The lattice of \(\mathfrak{D}\)-valued functions of t ∈ (0, ∞) with pointwise partial order and convergence contains the family I\(\mathfrak{D}\) = {A(·)| A(t) = At, A ∈ \(\mathfrak{D}\)}. Let ̃Ω be the set of

We suggest a combinatorial method for encoding continuous symbolic dynamical systems. We transform a continuous phase space, the infinite-dimensional cube, into the path space of a tree, and the shift corresponds to a transformation which we called “transfer.” The central problem is that of distinguishability: does the encoding distinguishes between almost all points of the space? The main result says

Several necessary conditions for the topological flatness of Banach modules are given. The main result is as follows: a Banach module over a relatively amenable Banach algebra which is topologically flat as a Banach space is topologically flat as a Banach module. Finally examples of topologically flat modules among classical modules of analysis are given.

Let M be a subharmonic function in a domain D ⊂ ℂn with Riesz measure νM, and let Z ⊂ D. As was shown in the first of the preceding papers, if there exists a holomorphic function f ≠ 0 in D such that f(Z) = 0 and |f| ⩽ exp M on D, then one has a scale of integral uniform upper bounds for the distribution of the set Z via νM. The present paper shows that for n = 1 this result "almost has a converse

Realizations of five-dimensional Lie algebras as algebras of holomorphic vector fields on homogeneous real hypersurfaces of a three-dimensional complex space are studied. In view of already known results, in the problem of describing such varieties only Levy nondegenerate hypersurfaces with exactly five-dimensional Lie algebras are of interest. It is shown that only two of the nine existing distinct

The present note deals with operator Hilbert systems, which are quantizations of unital cones in Hilbert spaces. One central result of the note is that the Pisier operator Hilbert space is an operator system whose quantum cone of positive elements is described in terms of the quantum ball of the relevant conjugate Hilbert space. Finally, we obtain a solution to the problem of Paulsen, Todorov and Tomforde

In this paper, a regularized trace formula is obtained for a higher-order differential operator on an interval under a perturbation of the lower-order term by a finite charge. Arbitrary regular boundary conditions and an arbitrary order n ⩾ 3 of the operator are considered. A new effect is discovered: for an even order of the operator, an additional term appears depending on the jump of the charge

We prove that (1) for any q ∈ (0, 1), all complex conjugate pairs of zeros of the partial theta function \(\theta (q,x): = \sum\nolimits_{j = 0}^\infty {{q^{j(j + 1)/2}}{x^j}}\) belong to the set {Re x ∈ (−5792.7,0), |Im x| < 132} ∪ {|x| < 18} and (2) for any q ∈ (−1,0), they belong to the rectangle {|Re x| < 364.2, |Im x| < 132}.

We show that on the sphere Sm, m ≠ 1, 3, 7, there exists an nm-valued multiplication with unit for some nm ∈ {2, 4, 8}. We also explicitly construct a 2k−1-fold branched covering of \(S^{m_1}\;\times\cdots\times\;S^{m_k}\) the product Sm1× ··· × Smk of k spheres over the sphere Sm, m = m1 + ··· + mk.

In the present paper we construct quantum Markov chains associated with open quantum random walks in the sense that the transition operator of a chain is determined by an open quantum random walk and the restriction of the chain to the commutative subalgebra coincides with the distribution ℙρ of the walk. This sheds new light on some properties of the measure ℙρ. For example, this measure can be considered

Questions related to “very weak” solutions of physical models of acoustic and shallow water wave propagation with singular dissipation are studied. The existence of a new type of solutions is proved. An existence theorem for a very weak solution of the problem is obtained. Finally it is shown that very weak solutions are consistent with classical ones in a certain sense, provided that the latter exist

This brief communication contains estimates for the zero divisors of holomorphic functions from weight classes in terms of subharmonic real test functions.

In a bounded domain \(\mathscr{O}\) ⊂ ℝ3 of class C1,1, the stationary Maxwell system with boundary conditions of perfect conductivity is considered. It is assumed that the dielectric permittivity and the magnetic permeability are given by η(x/ε) and μ(x/ε), where η and μ are symmetric bounded positive definite matrix-valued functions periodic with respect to some lattice in ℝ3. Here ε > 0 is a small

We study the Hölder exponents of self-affine similar functions on the interval [0; 1]. Closed-form expressions for the Hölder exponents via the self-similarity parameters are obtained.

We give an Abelian type result relating the quasiasymptotic boundedness of tempered distributions to the asymptotics of their directional short-time Fourier transform (DSTFT). We also prove several Abelian-Tauberian results characterizing the quasiasymptotic behavior of distributions in \(\mathscr{S}'\)(ℝn) in terms of their DSTFT with fixed direction.

The solvability of the equation A(x) = f(x) on the sphere of a Hilbert space and the dimension of its solution set are studied in the case where A is a closed surjective operator and f is an odd Lipschitz mapping. A kind of analogue of the infinite-dimensional version of the Borsuk-Ulam theorem is obtained.

The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrödinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Schrödinger equation at the zero energy level is a smooth M-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so

Let d > 0 be a square-free integer, and let Ld be the corresponding Hilbert lattice. Suppose given a finite-index subgroup Γ of O+(Ld) generated by reflections and containing -id and let A(Γ) be the algebra of Γ-automorphic forms. It is proved that if the algebra A(Γ) is free, then d ∈ {2, 3, 5, 6,13, 21}.

Hurwitz spaces are spaces of meromorphic functions with prescribed orders of poles on curves of a given genus. In this article, we derive new formulas for the degrees of the strata of Hurwitz spaces of genus 0 corresponding to functions that have two degenerate critical values with prescribed partitions of multiplicities of the preimages. More precisely, one of the critical values has an arbitrary

A weak stability bound for the symmetrization Θ = (f(·) − f(−·))/2 of a general ε-isometry f from a Banach space X to a Banach space Y is presented. As a corollary, the following somewhat surprising weak stability result is obtained: For every x* ∈ X*, there exists ϕ ∈ Y* with ‖ϕ‖ = ‖x*‖ ≔ r such that \(\mid\langle{x}^*,x\rangle-\langle\varphi,\Theta(x)\rangle\mid\;\leqslant\frac{3}{2}r\varepsilon