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.     

Isotopes of the alternative monster and the Skosyrsky algebra

We prove that the isotopes of the alternative monster and the Skosyrsky algebra satisfy the identity П_i=1⁴ [x_i, y_i] = 0. Hence, the algebras themselves satisfy the identity П_i=1⁴ (c, x_i, y_i) = 0. We also show that none of the identities П_i=1ⁿ(c, x_i, y_i) = 0 holds in all commutative alternative nil-algebras of index 3. Thus, we refute the Grishkov–Shestakov hypothesis about the structure of the free finitely generated commutative alternative nil-algebras of index 3.     

Minimal and maximal unconditional bases with respect to framings

This paper studies framings in Banach spaces, a concept raised by Casazza, Han and Larson, which is a natural generalization of traditional frames in Hilbert spaces and unconditional bases in Banach spaces. The minimal unconditional bases and the maximal unconditional bases with respect to framings are introduced. Our main result states that, if （xi, fi） is a framing of a Banach space X, and （eimin） and （eimax） are the minimal unconditional basis and the maximal unconditional basis with respect to （xi, fi）, respectively, then for any unconditional basis （ei） associated with （xi, fi）, there are A,B 〉 0 such that A｜｜i=1∑∞aieimin｜｜≤｜｜i=1∑∞aiei｜｜≤B｜｜i=1∑∞aieimax｜｜ for all （ai） ∈ c00.
It means that for any framing, the corresponding associated unconditional bases have common upper and lower bounds.     

Homotopical Morita theory for corings

A coring (A,C) consists of an algebra A in a symmetric monoidal category and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf–Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings (A,C) and (B,D) in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules V _A ^C and V _B ^D are Quillen equivalent. As an illustration of the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring.     

Estimation of the depth of reversible circuits consisting of NOT,CNOT and 2-CNOT gates

The paper discusses the asymptotic depth of a reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The reversible circuit depth function D(n, q) is introduced for a circuit implementing a mapping f: Z₂ⁿ → Z₂ⁿ as a function of n and the number q of additional inputs. It is proved that for the case of implementation of a permutation from A(Z₂ⁿ) with a reversible circuit having no additional inputs the depth is bounded as D(n, 0) ? 2ⁿ/(3log₂n). It is also proved that for the case of transformation f: Z₂ⁿ → Z₂ⁿ with a reversible circuit having q₀ ~ 2ⁿ additional inputs the depth is bounded as D(n,q₀) ? 3n.     

Isomorphisms,definable relations,and Scott families for integral domains and commutative semigroups

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ _α ⁰ -categorical integral domain (commutative semigroup) which is not relatively Δ _α ⁰ -categorical (i.e., no formally Σ _α ⁰ Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ _α ⁰ -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ _α ⁰ -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ _α ⁰ -dimension is equal to n. (4) For every computable successor ordinal α, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets X such that Δ _α ⁰ (X) is not Δ _α ⁰ . In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.     

Renormalized coupling constants for the three-dimensional scalar <Emphasis Type="Italic">λ</Emphasis><Emphasis Type="Italic">ϕ</Emphasis><Superscript>4</Superscript> field theory and pseudo-<Emphasis Type="Italic">ε</Emphasis>-expansion

The renormalized coupling constants g _2k that enter the equation of state and determine nonlinear susceptibilities of the system have universal values g _2k ^* at the Curie point. We use the pseudo-ε-expansion approach to calculate them together with the ratios R _2k = g _2k /g ₄ ^k-1 for the three-dimensional scalar λ ? ⁴ field theory. We derive pseudo-ε-expansions for g ₆ ^* , g ₈ ^* , R ₆ ^* , and R ₈ ^* in the five-loop approximation and present numerical estimates for R ₆ ^* and R ₈ ^* . The higher-order coefficients of the pseudo-ε-expansions for g ₆ ^* and R ₆ ^* are so small that simple Padé approximants turn out to suffice for very good numerical results. Using them gives R ₆ ^* = 1.650, while the recent lattice calculation gave R ₆ ^* = 1.649(2). The pseudo-ε-expansions of g ₈ ^* and R ₈ ^* are less favorable from the numerical standpoint. Nevertheless, Padé–Borel summation of the series for R ₈ ^* gives the estimate R ₈ ^* = 0.890, differing only slightly from the values R ₈ ^* = 0.871 and R ₈ ^* = 0.857 extracted from the results of lattice and field theory calculations.     

Existence of linear Pfaffian systems whose lower characteristic set has positive measure in <Emphasis Type="Italic">R</Emphasis><Superscript>3</Superscript>

We prove the existence of a completely integrable Pfaffian system ?x/?t i = A _i (t)x, x ∈ R ⁿ , t = (t ₁, t ₂, t ₃) ∈ R ₊ ³ , i = 1, 2, 3, with infinitely differentiable bounded coefficients and with lower characteristic set of positive three-dimensional Lebesgue measure.     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

Categorical resolutions of a class of derived categories

We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D~b(A)and the subcategory K~b(P) of perfect complexes in D~b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K~b(P), and finding an example such that D_(hf)~b(A)≠K~b(P). We realize the bounded derived category D~b(A) as a Verdier quotient of the relative derived category D_C~b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT ∞ such that ~⊥T is finite, then D~b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.     

On necessary and sufficient conditions for the self-normalized central limit theorem

Let X₁, X₂, … be a sequence of independent random variables and S_n = Σ _i=1 ⁿ X_i and V _n ² = Σ _i=1 ⁿ X _i ² . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n=V_n converges to a standard normal distribution if and only if max_1?i?n|X_i|/V_n→0 in probability and the mean of X₁ is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max_1?i?n|X_i|/V_n→0 in probability, then these sufficient conditions are necessary.     

A partition theorem for a large dense linear order

Let κ be a cardinal which is measurable after generically adding ?_κ+ω many Cohen subsets to κ, and let ?_κ = (Q, ≤ _Q ) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value t _m ⁺ such that the set [Q] ^m of m-tuples from Q can be partitioned into classes 〈C _i : i < t _m ⁺ }〉 such that for any coloring a class C _i in fewer than κ colors, there is a copy ?* of ?_κ such that [?*] ^m ? C _i is monochromatic. It follows that \(\mathbb{Q}_\kappa \to (\mathbb{Q}_\kappa )_{ < \kappa /t_m^ + }^m \), that is, for any coloring of [?_κ] ^m with fewer than κ colors there is a copy Q′ ? Q of ?_κ such that [Q′] ^m has at most t _m ⁺ colors. On the other hand, we show that there are colorings of ?_κ such that if Q′ ? Q is any copy of ?_κ then C _i ? [Q′] ≠ ø; for all i < t _m ⁺ , and hence \(\mathbb{Q}_\kappa \nrightarrow [\mathbb{Q}_\kappa ]_{t_m^ + }^m \).We characterize t _m ⁺ as the cardinality of a certain finite set of ordered trees and obtain an upper and a lower bound on its value. In particular, t ₂ ⁺ = 2 and for m > 2 we have t _m ⁺ > t _m , the m-th tangent number.The stated condition on κ is the hypothesis for a result of Shelah on which our work relies. A model in which this condition holds simultaneously for all m can be obtained by forcing from a model with a κ^+ω-strong cardinal, but it follows from earlier results of Hajnal and Komjáth that our result, and hence Shelah’s theorem, is not directly implied by any large cardinal assumption.     

On the homotopy type of (<Emphasis Type="Italic">n</Emphasis>-1)-connected (3<Emphasis Type="Italic">n</Emphasis>+1)-dimensional free chain Lie algebra

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ?Q, then p=∞. Denote by DGL _n ^np , n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL _n ^np . In this work we intend to answer the following two questions: Given an object (L(V), ?) in DGL _n ³ⁿ⁺² and denote by S(L(V), ?) the class of objects homotopy equivalent to (L(V), ?). How we can characterize a free dgl to belong to S(L(V), ?)? Fix an object (L(V), ?) in DGL _n ³ⁿ⁺² . How many homotopy equivalence classes of objects (L(W), δ) in DGL _n ³ⁿ⁺² such that H _* (W, d′)?H _* (V, d) are there? Note that DGL _n ³ⁿ⁺² is a subcategory of DGL _n ^np when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.     

The program iteration method in a game problem of guidance

Let a sequence of d-dimensional vectors n _k = (n _k ¹ , n _k ² ,..., n _k ^d ) with positive integer coordinates satisfy the condition n _k ^j = α _j m _k +O(1), k ∈ ?, 1 ≤ j ≤ d, where α ₁ > 0,..., α _d > 0 and {m _k } _k=1 ^∞ is an increasing sequence of positive integers. Under some conditions on a function φ: [0,+∞) → [0,+∞), it is proved that, if the sequence of Fourier sums \({S_{{m_k}}}\) (g, x) converges almost everywhere for any function g ∈ φ(L)([0, 2π)), then, for any d ∈ ? and f ∈ φ(L)(ln⁺ L) ^d?1([0, 2π) ^d ), the sequence \({S_{{n_k}}}\) (f, x) of rectangular partial sums of the multiple trigonometric Fourier series of the function f and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.     

On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C _n ² for p≠ 3 and d(G′) ≤ C _n ² + C _n ³ for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C _n ² . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C _n ² + C _n ³ , then G″ is an elementary abelian group.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for Riesz transform and their commutators associated with Schrödinger type operator

Let H ₂ = (?Δ)² + V ² be the Schrödinger type operator, where V satisfies reverse Hölder inequality. In this paper, we establish the L ^p boundedness for V ² H ₂ ^?1 , H ₂ ^?1 V ², VH ₂ ^?1/2 and H ₂ ^?1 V ², and that of their commutators. We also prove that H ₂ ^?1 V ²,VH ₂ ^?1/2 are bounded from BMO _L to BMO _L .     

Smoothness of solutions of almost hypoelliptic equations

The paper studies a class of almost hypoelliptic equations P(D)U = ? in a strip. It is proved that for \(\mathcal{H}\) great enough and for δ > 0 small enough all solutions of this equation, which are square summable with the weight e ^?δ|x| and for which \(D_2^{\alpha _2 } U\), where α ₂ = 0, …, \(ord_{\alpha _2 } P\), are infinitely differentiable in x ₁ functions, provided D ₁ ^j ? ∈ L ₂(\(\Omega _\mathcal{H} \)) for any j.     

The SL(<Emphasis Type="Italic">V</Emphasis>)-ample cone of product of flag varieties

Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set ω = (\(\vec d\)(1), ..., \(\vec d\)(m)) of sequences of positive integers, denote by L _ω the ample line bundle corresponding to the polarization on the product X = Π _i=1 ^m Flag(V, \(\vec n\)(i)) of flag varieties of type \(\vec n\)(i) determined by ω. We study the SL(V)-linearization of the diagonal action of SL(V) on X with respect to L _ω. We give a sufficient and necessary condition on ω such that X ^ss (L _ω) ≠ \(\not 0\) (resp., X ^s (L _ω) ≠ \(\not 0\)). As a consequence, we characterize the SL(V)-ample cone (for the diagonal action of SL(V) on X), which turns out to be a polyhedral convex cone.     

Invariance of the generalized oscillator under a linear transformation of the related system of orthogonal polynomials

We consider the families of polynomials P = { P _n (x)} _n=0 ^∞ and Q = { Q _n (x)} _n=0 ^∞ orthogonal on the real line with respect to the respective probability measures μ and ν. We assume that { Q _n (x)} _n=0 ^∞ and {P _n (x)} _n=0 ^∞ are connected by linear relations. In the case k = 2, we describe all pairs (P,Q) for which the algebras A _P and A _Q of generalized oscillators generated by { Qn(x)} _n=0 ^∞ and { Pn(x)} _n=0 ^∞ coincide. We construct generalized oscillators corresponding to pairs (P,Q) for arbitrary k ≥ 1.     

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l _r,k ^α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$

, and generated by the classical orthogonal Laguerre polynomials L _k ^α (x) (k = 0, 1,...). The polynomials l _r,k ^α (x) are represented as expressions containing the Laguerre polynomials L _n ^α?r (x). An explicit form of the polynomials l _r,k+r ^α (x) is established as an expansion in the powers x ^r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l _r,k ^α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.