Construction of noniterated boolean functions in the basis {&;, ∨, −} and estimation of their number

In this paper we consider noniterated Boolean functions in the basis {&;, ∨, ?}. We obtain the canonical form of the formula for a noniterated function in this basis. We construct the set of such formulas with respect to variables x ₁, …, x _n and calculate the number of its elements. Based on these results, we obtain the upper and lower bounds for the number of noniterated Boolean functions of n variables in the basis under consideration.     

Some test length bounds for nonrepeating functions in the {&;, ∨} basis

Testing relative to a nonrepeating alternative in a conjunction-disjunction basis is considered. A lower bound on the test length is established for all nonrepeating functions in this basis. A subsequence of easily testable functions is constructed and the corresponding tests are described. Individual lower test length bounds are proved for functions of a special form; minimality of the tests is established for the functions of the constructed subsequence.     

Synthesis of easy tested diagrams in the basis {&;, ∨, <Superscript>−</Superscript>} for systems of functions from some classes

A new method for synthesis of easy tested diagrams of functional elements in the basis {&;, ∨, ^?} is proposed for systems of m Boolean functions differing from constants and representable as disjunctive normal forms containing l variables x ₁,...,x _l , l > 0, without negations and the other variables x _l+1,...,x _n with negations only. All faults are assumed to be stack-at-1 faults at the outputs of elements. It is proved that the length of a complete verification test is not greater than min {m, l} for such diagrams.     

Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of ℂ n

We give a Schwarz-Pick estimate for bounded holomorphic functions on unit ball in Cn, and generalize some early work of Schwarz-Pick estimates for bounded holomorphic functions on unit disk in C.     

Approximation of functions of several variables by trigonometric polynomials with given number of harmonics,and estimates of ε-entropy

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Some estimates for the error in Fourier–Legendre expansions of functions of one variable

Some issues concerning expansions of functions in Fourier–Legendre series is considered in L₂[?1, 1]. In particular, the rate of their convergence in the classes of functions characterized by the generalized modulus of continuity are estimated, and estimates of the remainder terms are obtained.     

Asymptotics of the basis functions of generalized Taylor series for the class H ρ,2

We study the basis functions φ _n,k and ψ _n,p of generalized Taylor series for the classH _ρ,2 and obtain asymptotic expansions of the functions φ _n,0 ^(l) and $\psi _{n,2 \cdot 4^n - 1}^{(l)} $ . We prove the existence of an asymptotics for the functions φ _n,k and ψ _n,p for k ≠ 0 and p ≠ 2·4 ⁿ ? 1. The first term of the asymptotic expansions of these functions is obtained.     

Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the p−1 problem

We utilize the method of Bellman functions to derive new $L^p$-estimates of Littlewood–Paley type involving $p?1$. Among the applications to singular integrals we improve the $2(p?1)$ bounds for the Ahlfors–Beurling operator on $L^p(\mathbb{C})$ when $p\to \infty$. In addition, dimensionless estimates of Riesz transforms in the classical as well as in the Ornstein–Uhlenbeck setting are attained. To cite this article: O. Dragi?evi?, A. Volberg, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A note on “Reducing the number of binary variables in cutting stock problems”

This study proposes a deterministic model to solve the two-dimensional cutting stock problem (2DCSP) using a much smaller number of binary variables and thereby reducing the complexity of 2DCSP. Expressing a 2DCSP with $m$ stocks and $n$ cutting rectangles requires $2n^{2}+n(m+1)$ binary variables in the traditional model. In contrast, the proposed model uses $n^{2}+n\lceil {\log _2 m}\rceil $ binary variables to express the 2DCSP. Experimental results showed that the proposed model is more efficient than the existing model.     

Asymptotically optimally reliable circuits in the basis {x 1 & x 2 & x 3, x 1 ∨ x 2 ∨ x 3, \bar x_1 } for inverse faults at the inputs of elements

We prove that, in the basis {x ₁ & x ₂ & x ₃, x ₁ ∨ x ₂ ∨ x ₃, $ \bar x_1 $ }, for inverse faults at the inputs of functional elements, all Boolean functions f(x ₁, x ₂, ..., x _n ) can be realized by asymptotically optimally reliable circuits operating with unreliability asymptotically (as ? → 0) equal to: ? ₃ for the constants 0 and 1, ? for the functions $ \bar x_i $ , and 3? for f(x ₁, x ₂, ..., x _n ) ≠ 0, 1, $ \bar x_i $ , x _i , where ? is the error probability at each input of the functional element and i = 1, ..., n. The functions xi, i = 1, ..., n, can be realized absolutely reliably. The complexity of asymptotically optimally reliable circuits is equal in order to the complexity of minimal circuits constructed only from reliable elements.     

A distribution of the number of ℓ-steps in a random binary sequence subject to some constraints

There are investigated the joint distribution of random variables _kn(₁),..., _kn(_s), and distributions of some functionals of _kn(), for n. Here _kn(), 1ln–1 is the number of -steps in a binary sequence (b.s.), selected randomly and equiprobably from the totality of all n-dimensional b.s. that have a prescribed number of ones and k 1-steps. By an -step of a b.s. we understand a configuration of the form 1...0, where the ellipsis stands for an ( –1)-dimensional b.s.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1186–1193, September, 1991.     

Some new estimates of the Fourier transform in \mathbb{L}_2 (ℝ)

Given a function $\mathbb{L}_2 $ (?), its Fourier transform $g(x) = \hat f(x) = F[f](x) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {f(x)e^{ - ixt} dt} ,f(t) = F^{ - 1} [g](t) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {g(x)e^{ - ixt} dx} $ and the inverse Fourier transform are considered in the space f ε $\mathbb{L}_2 $ (?). New estimates are presented for the integral $\int\limits_{|t| \geqslant N} {|g(t)|^2 dt} = \int\limits_{|t| \geqslant N} {|\hat f(t)|^2 dt} ,N \geqslant 1,$ in the vase of f ε $\mathbb{L}_2 $ (?) characterized by the generalized modulus of continuity of the kth order constructed with the help of the Steklov function. Some other estimates associated with this integral are proved.     

Series ∑F(m)qm,where F(m) is the number of odd classes of binary quadratic forms of determinant −m

Consideration of the analytic continuation of the Eisenstein series of weight 3/2 for the group ₀(4) leads to a new proof of Mordell's formula connecting the values X()= _m=1 F(m)e^im, Im > 0, and (–1/). The behavior of the function () for ₀(4) is examined by the same method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 69–79, 1976.     

Sharp estimates for derivatives of functions in the Sobolev classes $$ \mathop {W_2^r }\limits^ \circ $$(−1,1)

Explicit formulas are obtained for the maximum possible values of the derivatives f ^(k)(x), x ∈ (−1, 1), k ∈ {0, 1, ..., r − 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r − 1 at the points ±1 and are such that $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 . As a corollary, it is shown that the first eigenvalue λ _1,r of the operator (−D ²) ^r with these boundary conditions is $ \sqrt 2 $ \sqrt 2 (2r)! (1 + O(1/r)), r → ∞.     

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.     

On some extremal problems in the theory of approximation of functions in the spaces S p , 1 ≤ p < ∞

We consider and study properties of the smoothness characteristics , of functions f(x) that belong to the space S ^p , 1 ≤ p < ∞, introduced by Stepanets. Exact inequalities of the Jackson type are obtained, and the exact values of the widths of the classes of functions defined by using Ω_m(f,t)_S ^p are calculated. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 303–316, March, 2006.     

On exact estimates of the convergence rate of fourier series for functions of one variable in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript>[–π, π]

The work is devoted to exact estimates of the convergence rate of Fourier series in the trigonometric system in the space of square summable 2π-periodic functions with the Euclidean norm on certain classes of functions characterized by the generalized modulus of continuity. Some N-widths of these classes are calculated, and the residual term of one quadrature formula over equally spaced nodes for a definite integral connected with the issues under consideration is found.     

Asymptotics and weighted estimates of meixner polynomials orthogonal on the gird {0, δ, 2δ, ...}

Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e ^−δ)^αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained: . The remainderv _n,N ^α (z) forn≤λN satisfies the estimate

Asymptotic normality of <Emphasis Type=Italic>b</Emphasis>-additive functions on polynomial sequences in number systems

We consider b-additive functions f where b is an algebraic integer over ℤ. In particular, let p be a polynomial, then we show that the asymptotic distribution of f(⌊ p(z)⌋), where ⌊⋅⌋ denotes the integer part with respect to basis b, when z runs through the elements of the ring ℤ[b] is the normal law. This is a generalization of results of Bassily and Kátai (for the integer case) and of Gittenberger and Thuswaldner (for the Gaussian integers).