Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I

We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces B _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) and L _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) of Nikol’skii-Besov and Lizorkin-Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system $ \mathcal{W}_m^\mathbb{I} $ \mathcal{W}_m^\mathbb{I} of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in B _pq ^sm ($ \mathbb{I} $ \mathbb{I} ) and L _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) by special partial sums of these series in the metric of L _r ($ \mathbb{I} $ \mathbb{I} ^k ) for a number of relations between the parameters s, p, q, r, and m (s = (s ₁, ..., s _n ) ∈ ℝ₊ ⁿ , 1 ≤ p, q, r ≤ ∞, m = (m ₁, ..., m _n ) ∈ ℕ ⁿ , k = m ₁ +... + m _n , and $ \mathbb{I} $ \mathbb{I} = ℝ or $ \mathbb{T} $ \mathbb{T} ). In the periodic case, we study the Fourier widths of these function classes.     

Linear maps that strongly preserve regular matrices over the Boolean algebra

The set of all m × n Boolean matrices is denoted by $ \mathbb{M} $ \mathbb{M} _m,n . We call a matrix A ∈ $ \mathbb{M} $ \mathbb{M} _m,n regular if there is a matrix G ∈ $ \mathbb{M} $ \mathbb{M} _n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $ \mathbb{M} $ \mathbb{M} _m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $ \mathbb{M} $ \mathbb{M} _m,n , or m = n and T(X) = UX ^T V for all X ∈ $ \mathbb{M} $ \mathbb{M} _n .     

On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.     

Completely nonmeasurable unions

Assume that no cardinal κ < 2 ^ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal $ \mathbb{I} $ \mathbb{I} of X contains uncountably many pairwise disjoint subfamilies $ \mathbb{I} $ \mathbb{I} -Bernstein unions ∪ $ \mathbb{I} $ \mathbb{I} -Bernstein if A and X \ A meet each Borel $ \mathbb{I} $ \mathbb{I} -positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.     

On the sharp Jackson-Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

The best constant C _n,m in the Jackson-Nikol’skii inequality between the uniform and integral norms of algebraic polynomials of a given total degree n ≥ 0 on the unit sphere $ \mathbb{S}^{m - 1} $ \mathbb{S}^{m - 1} of the Euclidean space ℝ ^m is studied. Two-sided estimates for the constant C _n,m are obtained, which, in particular, give the order n ^m−1 of its behavior with respect to n as n → +∞ for a fixed m.     

Trace class Toeplitz operators with unbounded symbols on weighted Bergman spaces

In this note we construct a function φ in L2（Bn,dμ） which is unbounded on any neighborhood of each boundary point of Bn such that Tφ is a trace class operator on weighted Bergman space Lα2（Bn,dμ） for several complex variables.     

Distortion theorem for Bloch mappings on the unit ball <Emphasis Type="Italic">ℬ</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H（B^n） satisfying ｜｜f｜｜0 = 1 and det f＇（0） = α ∈ （0, 1], where｜｜f｜｜0 = sup{（1 - ｜z｜^2 ）n＋1/2n det（f＇（z））[1/n ： z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f（0）. When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.     

The Bergman kernel function

In this note, we point out that a large family of n×n matrix valued kernel functions defined on the unit disc $ \mathbb{D} \subseteq \mathbb{C} $ \mathbb{D} \subseteq \mathbb{C} , which were constructed recently in [9], behave like the familiar Bergman kernel function on $ \mathbb{D} $ \mathbb{D} in several different ways. We show that a number of questions involving the multiplication operator on the corresponding Hilbert space of holomorphic functions on $ \mathbb{D} $ \mathbb{D} can be answered using this likeness.     

Integral-type operators from a mixed norm space to a bloch-type space on the unit ball

Let $ \mathbb{B} $ \mathbb{B} be the unit ball in ℂ ⁿ and let H($ \mathbb{B} $ \mathbb{B} ) be the space of all holomorphic functions on $ \mathbb{B} $ \mathbb{B} . We introduce the following integral-type operator on H($ \mathbb{B} $ \mathbb{B} ):

Weak convergence in the dual of weak L<Superscript>p</Superscript>

We consider a Dedekind σ-complete Banach lattice E whose dual is weakly sequentially complete. Suppose that E has a positive element u and a family of positive operators $ \mathcal{G} $ \mathcal{G} such that

Uniform Morse lemma and isotopy criterion for Morse functions on surfaces

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let $ \mathcal{D}_0 $ \mathcal{D}_0 ⊂ Diff(M) be the group of diffeomorphisms homotopic to id _M . Two smooth functions f, g: M → ℝ are called isotopic if f = h ₂ ℴ g ℴ h ₁ for some diffeomorphisms h ₁ ∈ $ \mathcal{D}_0 $ \mathcal{D}_0 and h ₂ ∈ Diff⁺(ℝ). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from F to be isotopic is proved. For each Morse function f ∈ F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which continuously and Diff(M)-equivariantly depends on f in C ^∞-topology on F (“uniform Morse lemma”). Applications of these results to the problem of describing the homotopy type of the space F are formulated.     

A characteristic property of the Euclidean disc

In this paper the following is proved: Let K ⊂ $ \mathbb{E}^2 $ \mathbb{E}^2 be a smooth strictly convex body, and let L ⊂ $ \mathbb{E}^2 $ \mathbb{E}^2 be a line. Assume that for every point x ∈ L/K the two tangent segments from x to K have the same length, and the line joining the two contact points passes through a fixed point in the plane. Then K is an Euclidean disc.     

Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup

Let $ \mathfrak{S} $ \mathfrak{S} be a locally compact semigroup, ω be a weight function on $ \mathfrak{S} $ \mathfrak{S} , and M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) be the weighted semigroup algebra of $ \mathfrak{S} $ \mathfrak{S} . Let L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) be the C*-algebra of all M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)-measurable functions g on $ \mathfrak{S} $ \mathfrak{S} such that g/ω vanishes at infinity. We introduce and study a strict topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) and show that the Banach space L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) can be identified with the dual of M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) endowed with β ¹($ \mathfrak{S} $ \mathfrak{S} , ω). We finally investigate some properties of the locally convex topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω).     

Local limit theorem for the first crossing time of a fixed level by a random walk

Let X,X(1),X(2),... be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + ... + X(n), n = 1, 2,..., and define the Markov stopping time η _y = inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,.... In the case $ \mathbb{E} $ \mathbb{E} |X|³ < ∞, the following relation was obtained in [8]: $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) as n → ∞, where the constant R and the bounded sequence ν _n were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0 under the condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ only; In [1], an explicit form of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) was found under the same condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that this corrected version was formulated in [8] as a conjecture.     

Solution blow-up for a new stationary Sobolev-type equation

A new nonlinear stationary Sobolev-type equation with a parameter η ∈ ℝ¹ is derived. For η > 0, global solvability in the weak generalized sense is proved in the entire waveguide $ \mathbb{S} $ \mathbb{S} ⊗ ℝ₊¹. For η < 0, the strong generalized solution is shown to blow up in a certain waveguide cross section z = R ₀ > 0. An upper bound for R ₀ in terms of the original parameters of the problem is obtained.     

A construction of maximal linearly independent array

Let M be a subset of r-dimensional vector space Vτ （F2） over a finite field F2, consisting of n nonzero vectors, such that every t vectors of M are linearly independent over F2. Then M is called （n, t）-linearly independent array of length n over Vτ（F2）. The （n, t）-linearly independent array M that has the maximal number of elements is called the maximal （r, t）-linearly independent array, and the maximal number is denoted by M（r, t）. It is an interesting combinatorial structure, which has many applications in cryptography and coding theory. It can be used to construct orthogonal arrays, strong partial balanced designs. It can also be used to design good linear codes, In this paper, we construct a class of maximal （r, t）-linearly independent arrays of length r ＋ 2, and provide some enumerator theorems.     

Phase portraits for quadratic homogeneous polynomial vector fields on $$ \mathbb{S}^2 $$

Let X be a homogeneous polynomial vector field of degree 2 on $ \mathbb{S}^2 $ \mathbb{S}^2 . We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on $ \mathbb{S}^2 $ \mathbb{S}^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 ^th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on $ \mathbb{S}^2 $ \mathbb{S}^2 of degree n.     

Isometric immersions into warped product spaces

This paper concerns the submanifold geometry in the ambient space of warped productmanifolds F^n×σ R, this is a large family of manifolds including the usual space forms R^m, S^m and H^m. We give the fundamental theorem for isometric immersions of hypersurfaces into warped product space R^n×σ R, which extends this kind of results from the space forms and several spaces recently considered by Daniel to the cases of infinitely many ambient spaces.