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).     

The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Let φ be an N-function. Then the normal structure coefficients N and the weakly convergent sequence coefficients WCS of the Orlicz function spaces L ^φ[0, 1] generated by φ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If F _φ(t) = t _?(t)/φ(t) is decreasing and 1 < C _φ < 2 (where \(C_\Phi = \lim _{t \to + \infty } t\varphi (t)/\Phi (t)\)), then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1?1/Cφ. (ii) If F _φ(t) is increasing and C _φ > 2, then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1/Cφ.     

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.     

On the approximation of continuous functions of two variables

It is shown that if P _m ^α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ _mn ^α,β (x, y)} _m,n=0 ^r = {P _m ^α,β (x)P _n ^α,β (y)} _{m, n=0} ^r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω _N×N = ?ub;(x _i , y _i ) _i,j=0 ^N , where x _i and y _i are the zeros of the Jacobi polynomial P _n ^α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]², we construct the discrete partial Fourier-Jacobi sums of the rectangular type S _{m, n, N} ^α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S _{m, n, N} ^α,β ∥ of the discrete sums S _{m, n, N} ^α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) ^{q + 1/2}), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S _{m, n, N} ^α,β (f; x, y) are considered.     

On J. C. C. Nitsche type inequality for annuli on Riemann surfaces

Assume that (N, ?) and (M, S) are two Riemann surfaces with conformal metrics ? and S. We prove that if there is a harmonic homeomorphism between an annulus A ? N with a conformal modulus Mod(A) and a geodesic annulus A _S (p, ρ₁, ρ₂)?M, then we have ρ₂/ρ₁ ≥ Ψ _S Mod(A)²+ 1, where Ψ _S is a certain positive constant depending on the upper bound of Gaussian curvature of the metric S. An application for the minimal surfaces is given.     

Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x ? y) = Σ _j=1 ⁿ α _j (x)β _j (y) as well as of the form f₁(x + z)f₂(y + z)f₃(x + y ? z) = Σ _j=1 ^m φ _j (x, y)ψ _j (z) are solved for unknown entire functions f, g,α _j , β _j : ? → ? and f₁, f₂, f₃, ψ _j : ? → ?, φ _j : ?² → ? in the cases of n = 3 and m = 4.     

Closable Multipliers

Let (X, μ) and (Y, ν) be standard measure spaces. A function \({\varphi\in L^\infty(X\times Y,\mu\times\nu)}\) is called a (measurable) Schur multiplier if the map S _φ , defined on the space of Hilbert-Schmidt operators from L ₂(X, μ) to L ₂(Y, ν) by multiplying their integral kernels by φ, is bounded in the operator norm. The paper studies measurable functions φ for which S _φ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if φ is of Toeplitz type, that is, if φ(x, y) = f(x ? y), \({x,y\in G}\), where G is a locally compact abelian group, then the closability of φ is related to the local inclusion of f in the Fourier algebra A(G) of G. If φ is a divided difference, that is, a function of the form (f(x) ? f(y))/(x ? y), then its closability is related to the “operator smoothness” of the function f. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.     

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRODINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schr(o)dinger equation posed on finite interval α≤ x ≤β:{iyt yxx |y|2y = 0,y(α,t) = h1(t),y(β,t) = h2(t) for t ＞ 0 (S)is considered. It is shown that by choosing appropriate control inputs (hj), (j = 1,2) one can always guide the system (S) from a given initial state ψ∈ Hs(α,β),(s ∈ R) to a terminal state ψ∈ Hs(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schr(o)dinger equation posed on the whole line R. The discovered smoothing properties of Schr(o)dinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schr(o)dinger equation.     

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.     

Uniform extensions of partial geometries

A geometry of rank 2 is an incidence system (P, \(\mathcal{B}\)), where P is a set of points and \(\mathcal{B}\) is a set of subsets from P, called blocks. Two points are called collinear if they lie in a common block. A pair (a, B) from (P, \(\mathcal{B}\)) is called a flag if its point belongs to the block, and an antiflag otherwise. A geometry is called φ-uniform (φ is a natural number) if for any antiflag (a, B) the number of points in the block B collinear to the point a equals 0 or φ, and strongly φ-uniform if this number equals φ. In this paper, we study φ-uniform extensions of partial geometries pG _α (s, t) with φ = s and strongly φ-uniform geometries with φ = s ? 1. In particular, the results on extensions of generalized quadrangles, obtained earlier by Cameron and Fisher, are extended to the case of partial geometries.     

A numerical algorithm for solving a nonstationary problem of optimal control

Let {φ _n ^(α,β) (z)} _n=0 ^∞ be a system of Jacobi polynomials orthonormal on the circle |z| = 1 with respect to the weight (1 ? cos τ)^α+1/2(1 + cos τ)^β+1/2 (α, β > ?1), and let \(\psi _n^{\left( {\alpha ,\beta } \right)*} \left( z \right): = z^n \overline {\psi _n^{\left( {\alpha ,\beta } \right)} \left( {{1 \mathord{\left/ {\vphantom {1 {\bar z}}} \right. \kern-\nulldelimiterspace} {\bar z}}} \right)}\)). We establish relations between the polynomial φ _n ^(α,?1/2) (z) and the nth (C, α ? 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?3/2 and also between the polynomial φ _n ^(α,?1/2)* (z) and the nth (C, α + 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?1/2. We use these relations to derive an asymptotic formula for φ _n ^(α,?1/2) (z); the formula is uniform inside the disk |z| < 1. It follows that φ _n ^(α,?1/2) (z) ≠ 0 in the disk |z| ≤ ρ for fixed φ ∈ (0, 1) and α > ?1 if n is sufficiently large.     

Bounded projectors and duality in the spaces of functions holomorphic in the unit ball

The paper studies the Banach spaces h _∞(φ), h ₀(φ), and h ¹(η) of harmonic functions over the unit ball in R ⁿ . These spaces depend on a weight function φ and a weight measure η. For a given function φ from a sufficiently broad class of functions, we solve the duality problem. that is, we construct measures η such that h ¹(η)* ～ h _∞(φ) and h ₀(φ)* ～ h ¹(η).     

The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S ₃ onto a genus 2 Riemann surface S ₂ by Hol(S ₃, S ₂). Call two mappings f and g in Hol(S ₃, S ₂) equivalent whenever there exist conformal automorphisms α and β of S ₃ and S ₂ respectively with f ? α = β ? g. It is known that Hol(S ₃, S ₂) always consists of at most two equivalence classes.We obtain the following results: If Hol(S ₃, S ₂) consists of two equivalence classes then both S ₃ and S ₂ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S ₃, S ₂) there exist anticonformal automorphisms α? and β? with f ? α? = β? ? g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S ₃, S ₂) such that Hol(S ₃, S ₂) consists of two equivalence classes.     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

On the rank of odd hyper-quasi-polynomials

Given any nonzero entire function g: ? → ?, the complex linear space F(g) consists of all entire functions f decomposable as f(z + w)g(z - w)=φ₁(z)ψ₁(w)+???+ φ_n(z)ψ_n(w) for some φ₁, ψ₁, …, φ_n, ψ_n: ? → ?. The rank of f with respect to g is defined as the minimum integer n for which such a decomposition is possible. It is proved that if g is an odd function, then the rank any function in F(g) is even.     

Differences of weighted composition operations on the unit polydisk

Let φ ₁ and φ ₂ be holomorphic self-maps of the unit polydisk \(\mathbb{D}^N\), and let u ₁ and u ₂ be holomorphic functions on \(\mathbb{D}^N\). We characterize the boundedness and compactness of the difference of weighted composition operators W _{φ1, u1} and W _{φ2, u2} from the weighted Bergman space \(A_{\vec \alpha }^p\), 0 < p < ∞, \(\vec \alpha = \left( {\alpha _1 , \ldots ,\alpha _{\rm N} } \right)\), α _j > ?1, j = 1,..., N, to the weighted-type space H _υ ^∞ of holomorphic functions on the unit polydisk \(\mathbb{D}^N\) in terms of inducing symbols φ ₁, φ ₂, u ₁, and u ₂.     

Dual submanifolds in rational homology spheres

Let Σ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M_+, M_-? Σ are called dual to each other if the complement Σ-M_+ strongly homotopy retracts onto M_- or vice-versa. In this paper, we are concerned with the basic problem of which integral triples(n; m_+, m-) ∈ N~3 can appear, where n = dimΣ-1 and m_± = codim M_±-1. The problem is motivated by several fundamental aspects in differential geometry.(i) The theory of isoparametric/Dupin hypersurfaces in the unit sphere S~(n+1) initiated by′Elie Cartan, where M_± are the focal manifolds of the isoparametric/Dupin hypersurface M ? S~(n+1), and m± coincide with the multiplicities of principal curvatures of M.(ii) The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres Σ,i.e., total spaces of smooth S~3-bundles over S~4 homeomorphic but not diffeomorphic to S~7, where M_± =P_±×_(SO(4))S~3, P → S~4 the principal SO(4)-bundle of Σ and P_± the singular orbits of a cohomogeneity one SO(4) × SO(3)-action on P which are both of codimension 2.Based on the important result of Grove-Halperin, we provide a surprisingly simple answer, namely, if and only if one of the following holds true:· m_+ = m_-= n;· m_+ = m_-=1/3n ∈ {1, 2, 4, 8};· m_+ = m_-=1/4n ∈ {1, 2};· m_+ = m_-=1/6n ∈ {1, 2};·n/(m_++m_-)= 1 or 2, and for the latter case, m_+ + m_-is odd if min(m_+, m_-)≥2.In addition, if Σ is a homotopy sphere and the ratio n/(m_++m_-)= 2(for simplicity let us assume 2 m_- m_+),we observe that the work of Stolz on the multiplicities of isoparametric hypersurfaces applies almost identically to conclude that, the pair can be realized if and only if, either(m_+, m_-) =(5, 4) or m_+ + m_-+ 1 is divisible by the integer δ(m_-)(see the table on Page 1551), which is equivalent to the existence of(m_--1) linearly independent vector fields on the sphere S~(m_++m_-)by Adams' celebrated work. In contrast, infinitely many counterexamples are given if Σ is a rational homology sphere.     

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p ₀, p ₁ ∈ (1,∞] be such that (1 ? θ)/p ₀ + θ/p ₁ = 1/p, and let φ, φ₀, φ₁ be some admissible functions such that φ, φ₀ ^p/p0 and φ₁ ^p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L _φ0 ^p0),λ (X), L _φ1 ^{p1), λ} (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L _φ ^p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.     

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.     

Characterizations of symmetrized polydisc

Let Γ_n, n ≥ 2, denote the symmetrized polydisc in ?ⁿ, and Γ₁ be the closed unit disc in ?. We provide some characterizations of elements in Γ_n. In particular, an element (s₁,..., s_n?1, p) ∈ ?ⁿ is in Γ_n if and only if \({s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p\), j = 1,..., n ? 1, for some (β₁,..., β_n?1) ∈ Γ_n?1, and |p| ≤ 1.