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.     

Upper bound for the length of functions over a finite field in the class of pseudopolynomials

An exclusive-OR sum of pseudoproducts (ESPP), or a pseudopolynomial over a finite field is a sum of products of linear functions. The length of an ESPP is defined as the number of its pairwise distinct summands. The length of a function f over this field in the class of ESPPs is the minimum length of an ESPP representing this function. The Shannon length function L _k ^ESPP (n) on the set of functions over a finite field of k elements in the class of ESPPs is considered; it is defined as the maximum length of a function of n variables over this field in the class of ESPPs. It is proved that L _k ^ESPP (n) = O(k ⁿ /n ²).     

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.     

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.     

Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices

Let {c _j } _j=0 ⁿ be a sequence of matrix moments associated with a matrix of measures supported on the unit circle, and let {P _j } _j=0 ⁿ be its corresponding sequence of monic matrix orthogonal polynomials. In this contribution, we consider a perturbation on the moments and find an explicit relation for the perturbed orthogonal polynomials in terms of {P _j } _j=0 ⁿ . We also obtain an expression for the corresponding second kind polynomials.     

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.     

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 → ∞:

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

     

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.     

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.     

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.     

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.     

Generalized <Emphasis Type="Italic">p</Emphasis>-Adic Fourier Transform and Estimates of Integral Modulus of Continuity in Terms of This Transform

We consider a new class of functions on the p-adic linear space ? _p ⁿ for which a Fourier transform can be defined.We prove equalities of Parseval type, an inversion formula and a sufficient condition for a function to be represented as this Fourier transform. Also we give a sharp estimate of the L²(? _p ⁿ ) modulus of continuity in terms of Fourier transform generalizing the result of S. S. Platonov in the case n = 1. Finally we prove a generalization of this result and its converse for L^q(? _p ⁿ ) with appropriate q.     

On the Fekete and Szegö problem for starlike mappings of order <Emphasis Type="Italic">α</Emphasis>

Let S _α ^* be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szegö inequality for the class S _α ^* , and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in C ⁿ .     

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.     

Order of the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

An exclusive-OR sum of pseudoproducts (ESPP) is a modufo-2 sum of products of affine (linear) Boolean functions. The length of an ESPP is defined as the number of summands in this sum; the length of a Boolean function in the class of ESPPs is the minimum length of an ESPP representing this function. The Shannon length function L ^ESPP(n) on the set of Boolean functions in the class of ESPPs is considered; it is defined as the maximum length of a Boolean function of n variables in the class of ESPPs. It is proved that L ^ESPP(n) = ? (2 ⁿ /n ²). The quantity L ^ESPP(n) also equals the least number l such that any Boolean function of n variables can be represented as a modulo-2 sum of at most l multiaffine functions.     

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.     

On the Extremal Zagreb Indices of Graphs with Cut Edges

For a (molecular) graph, the first Zagreb index M ₁ is equal to the sum of squares of the vertex degrees, and the second Zagreb index M ₂ is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we show that all connected graphs with n vertices and k cut edges, the maximum (resp. minimum) M ₁- and M ₂-value are obtained, respectively, and uniquely, at K _n ^k (resp. P _n ^k ), where K _n ^k is a graph obtained by joining k independent vertices to one vertex of K _n?k and P _n ^k is a graph obtained by connecting a pendent path P _k+1 to one vertex of C _n?k.     

Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {A ^k : k ∈ ?}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ| > 1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let φ: ? ⁿ ×[0,∞) → [0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type H _A ^φ,∞ (? ⁿ ) and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to H _A ^φ,∞ (? ⁿ ) and the boundedness of the anisotropic Calderón-Zygmund operator from H _A ^φ,∞ (? ⁿ ) to L _A ^φ,∞ (? ⁿ ). It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of H _A ^φ,∞ (? ⁿ ) and the interpolation theorem.     

Non-generic unramified representations in metaplectic covering groups

Let G^(r) denote the metaplectic covering group of the linear algebraic group G. In this paper we study conditions on unramified representations of the group G^(r) not to have a nonzero Whittaker function. We state a general Conjecture about the possible unramified characters χ such that the unramified subrepresentation of \(Ind_{{B^{\left( r \right)}}}^{{G^{\left( r \right)}}}{X^{\delta _B^{1/2}}}\) will have no nonzero Whittaker function. We prove this Conjecture for the groups GL _n ^{( r)} with r ≥ n ? 1, and for the exceptional groups G ₂ ^{( r)} when r ≠ 2.     

Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system  = (0, 1, 2, . . .) of L2H(D), the space of holomorphic and absolutely square integrable functions in the bounded domain D of Cn, we construct a holomorphic imbedding ι : D →■(n, ∞), the complex infinite dimensional Grassmann manifold of rank n. It is known that in ■(n, ∞) there are n closed (μ, μ)-forms τμ (μ = 1, . . . , n) which are invariant under the holomorphic isometric automorphism of ■(n, ∞) and generate algebraically all the harmonic differential forms of ...