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1.
A criterion of implicit completeness in a k-valued logic P k for arbitrary k ≥ 2 is established.  相似文献   

2.
We reveal a relation between the operations of α-completion and closure for the systems of functions of a k-valued logic. For k = 3, 4 we construct the α-bases consisting of two binary operations. We prove that the complete system T of functions of a 4-valued logic containing all permutations of the set E 4 = {0, 1, 2, 3} and the operation of addition modulo 4 is not α-complete, whereas its α-completion [T α] will be an α-complete system.  相似文献   

3.
We consider a problem of the realization of k-valued logics functions (k ≥ 3) by circuits in two bases: in the Rosser–Turkett basis and in its dual basis. We assume that the basis gates are exposed to faults at outputs: only of type 0 or only of type k ? 1, and they pass into faulty states independently of each other. We describe a constructive method for the synthesis of an asymptotically optimal reliable circuit for almost any function of k-valued logic, we found the upper and lower bounds of circuits unreliability and the class of functions for which the lower bounds are true.  相似文献   

4.
Let μ be a measurewith a k-concave density W on an open convex set V in Rm, that is, W is an integrable weight satisfying the condition
$$W(ax + (1 - a)y) \geqslant {(a{W^K}(x) + (1 - a){W^K}(y))^{1/k}},k \in ( - 1/m,\infty ]$$
for all xV, yV, and α ∈ [0, 1]. In this paper, we first show that the Fradelizi μ-distributional inequalities for polynomials P of m variables are sharp for each m and k ∈ (?1/m,∞]. Classes of extremal sets V, weights W, and polynomials P for these inequalities are presented. Sharpness of the Bobkov-Nazarov-Fradelizi dilation-type inequalities is established as well. Second, we find efficient conditions for k-concavity of a weight W and obtain new sharp polynomial inequalities.
  相似文献   

5.
Let \({\{ {f_{\lambda ;j}}\} _{\lambda \in V;1 \leqslant j \leqslant k}}\) be families of holomorphic functions in the open unit disk \({\text{D}} \subset {\Bbb C}\) ? ? depending holomorphically on a parameter λV ? ? n . We establish a Rolle type theorem for the generalized multiplicity (called cyclicity) of zeros of the family of univariate holomorphic functions \({\left\{ {\sum\nolimits_{j = 1}^k {{f_{\lambda ;j}}} } \right\}_{\lambda \in V}}\) at 0 ∈ D. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form \(\sum\nolimits_{k = 1}^m {{P_k}(z){e^{{Q_k}(z)}}} \), z ∈ ?, where P k and Q k are holomorphic polynomials of degrees p and q, respectively, parameterized by vectors of coefficients of P k and Q k .  相似文献   

6.
The article addresses the operator of positive closure on the set P k of functions of k-valued logic. For each k ? 3, k ≠ 4, the set H k of all homogeneous functions from P k is proved to form an atom in the lattice of the positively closed classes from P k . Also, we find all 17 positively closed classes from P 3 containing the class H 3 (i.e., discriminator positively closed classes). Positively generating systems of these classes are defined.  相似文献   

7.
The main result of this paper asserts that the distribution density of any non-constant polynomial f12,...) of degree d in independent standard Gaussian random variables ξ1 (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B1/d (R1) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on Rk to possess a density in the Nikol’skii–Besov class Bα(R)k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on Rk via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.  相似文献   

8.
Let fK{y} be an element of the ring of differential polynomials in one differential variable y with one differential operator δ. For any variable y k , the polynomial g = δ n (f) can be represented in the form g = A k y k + go, where go does not depend on y k . If y k is the leader of g, then A k is a separant of the polynomial f. A formula for A k is obtained for sufficiently large numbers n and k and some applications of this formula are presented.  相似文献   

9.
In this paper, we show that the truncated binomial polynomials defined by \(P_{n,k}(x)={\sum }_{j=0}^{k} {n \choose j} x^{j}\) are irreducible for each k≤6 and every nk+2. Under the same assumption nk+2, we also show that the polynomial P n,k cannot be expressed as a composition P n,k (x) = g(h(x)) with \(g \in \mathbb {Q}[x]\) of degree at least 2 and a quadratic polynomial \(h \in \mathbb {Q}[x]\). Finally, we show that for k≥2 and m,nk+1 the roots of the polynomial P m,k cannot be obtained from the roots of P n,k , where mn, by a linear map.  相似文献   

10.
A real polynomial in one variable is hyperbolic if it has only real roots. A function f is a primitive of order k of a function g if f (k) = g. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. In the paper, we prove a property of the domain of very hyperbolic polynomials and describe this domain in the case of degree 4.  相似文献   

11.
We define a probability measure on the space of polynomials over ? n in order to address questions regarding the attainment of the norm at given points and the validity of polynomial inequalities.Using this measure, we prove that for all degrees k ≥ 3, the probability that a k-homogeneous polynomial attains a local extremum at a vertex of the unit ball of ? 1 n tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.  相似文献   

12.
The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.  相似文献   

13.
An iterative procedure is proposed for calculating the number of k-valued functions of n variables such that each one has an endomorphism different from any constant and permutation. Based on this procedure, formulas are found for the number of three-valued functions of n variables such that each one has nontrivial endomorphisms. For any arbitrary semigroup of endomorphisms, the power is found of the set of all three-valued functions of n variables such that each one has endomorphisms from a specified semigroup.  相似文献   

14.
The 6j-symbols for representations of the q-deformed algebra of polynomials on \(\mathrm {SU}(2)\) are given by Jackson’s third q-Bessel functions. This interpretation leads to several summation identities for the q-Bessel functions. Multivariate q-Bessel functions are defined, which are shown to be limit cases of multivariate Askey–Wilson polynomials. The multivariate q-Bessel functions occur as 3nj-symbols.  相似文献   

15.
Withers and Nadarajah (Stat Pap 51:247–257; 2010) gave simple Edgeworth-type expansions for log densities of univariate estimates whose cumulants satisfy standard expansions. Here, we extend the Edgeworth-type expansions for multivariate estimates with coefficients expressed in terms of Bell polynomials. Their advantage over the usual Edgeworth expansion for the density is that the kth term is a polynomial of degree only k + 2, not 3k. Their advantage over those in Takemura and Takeuchi [Sankhyā, A, 50, 1998, 111-136] is computational efficiency  相似文献   

16.
We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λk(f), k ∈ N, defined by averaging the kth-order finite differences of functions fL2. On the basis of this, for differentiable functions in the classes L2r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ωk. For classes of functions defined by their smoothness characteristics Λk(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.  相似文献   

17.
We give a classification of second-order polynomial solutions for the homogeneous k-Hessian equation σ_k[u] = 0. There are only two classes of polynomial solutions: One is convex polynomial; another one must not be(k + 1)-convex, and in the second case, the k-Hessian equations are uniformly elliptic with respect to that solution. Based on this classification, we obtain the existence of C∞local solution for nonhomogeneous term f without sign assumptions.  相似文献   

18.
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.
  相似文献   

19.
An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andAB with μ(A)>0, then for some ø≠α one has μ(? i=1 k S i({α})A>0). We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the formS(α)=T(p(n(α))), wherep:Z t Z d is a polynomial vanishing at zero,T is a measure preservingZ d action andn is an IP system intoZ t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).  相似文献   

20.
In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials \({\Phi_{n}^{(\alpha)}(x,\nu)}\) of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers E n , Euler polynomials E n (x), generalized Euler numbers E n (a, b), generalized Euler polynomials E n (x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials \({{_HE}_n(x,y)}\) of Dattoli et al. and \({{_HE}_n^{(\alpha)} (x,y)}\) of Pathan are generalized to the one \({ {_HE}_n^{(\alpha)}(x,y,a,b,c)}\) which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between E n , E n (x), E n (a, b), E n (x; a, b, c) and \({{}_HE_n^{(\alpha)}(x,y;a,b,c)}\) are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions.  相似文献   

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