Complexity of realization of functions from a class of three-valued logic by formulas of special type

A set of functions of the three-valued logic is considered and upper estimates for the Shannon function in the class of formulas of special form are obtained for this set. Some examples of sequences of functions from this set are considered and exponential lower estimates of complexity are obtained. In this case the values of the Shannon function are obtained for the considered class with the accuracy up to an additive constant.     

Mixed id-expansion of multivalued logic functions

The conception of id-expansion of multivalued logic function is introduced. For any k ≥ 2, the degree of mixed id-expansion of the class of all k-valued logic functions and the degree of mixed id-expansion of the class of all Boolean functions over the class of self-dual monotonic functions are defined.     

Complexity of implementation of Boolean functions by real formulas

Upper complexity estimates are proved for implementation of Boolean functions by formulas in bases consisting of a finite number of continuous real functions and a continuum of constants. For some bases upper complexity estimates coincide with lower ones.     

Lower complexity estimates for some sequences of functions of multivalued logic

The problem of realization of functions of a multivalued logic by formulas is considered. Some sequences of functions whose realization complexity exceeds exponential one are presented.     

Complexity and depth of formulas for symmetric Boolean functions

A new approach for implementation of the counting function for a Boolean set is proposed. The approach is based on approximate calculation of sums. Using this approach, new upper bounds for the size and depth of symmetric functions over the basis B₂ of all dyadic functions and over the standard basis B₀ = {∧, ∨,^- } were non-constructively obtained. In particular, the depth of multiplication of n-bit binary numbers is asymptotically estimated from above by 4.02 log₂n relative to the basis B₂ and by 5.14log₂n relative to the basis B₀.     

Complexity of realization of Boolean functions from some classes related to finite grammars by formulas of alternation depth 3

The realization complexity of Boolean functions associated with finite grammars in the class of formulas of alternation depth 3 is studied. High accuracy asymptotic bounds are obtained for the corresponding Shannon function.     

Complexity of functions from some classes of three-valued logic

The problem of the realization complexity for functions of the three-valued logic taking values from the set {0, 1} by formulas over incomplete generating systems is considered. Upper and lower asymptotic estimates for the corresponding Shannon functions are obtained.     

On zeros of entire functions of special form

Sufficient conditions for an entire function of special form to have no zeros in the open lower half-plane are obtained.     

Orders of discriminator classes in multivalued logic

Mathematical Notes - For k ≥ 2, discriminator classes, that is, closed classes of functions of k-valued logic containing the ternary discriminator p, are considered. It is proved that any...     

Complexity of realization of Boolean functions with small number of unit values by selfcorrecting contact circuits

An asymptotics for the complexity of realization of Boolean functions taking the unit value on a comparatively small set of collections of variables by self-correcting contact circuits is obtained.     

Multiplication formulas for Kubert functions

The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of （generalized） Kubert functions. Second, we apply the special case of Carlitz＇s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz＇s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz （for sums） and Mordell （for integrals）, both of which are generalizations of preceding results by Frasnel, Landau, Mikolas, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck＇s lamma is the same as Carlitz＇s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.     

Structure formulas for univalent functions

With the aid of special integrating factors for the Loewner-Kufarev equation, one determines the general form of the structure formulas for functions of the class S and one obtains a series of new concrete structure formulas for certain subclasses of functions from S.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 128–134, 1983.     

Approximation of classes of analytic functions by a linear method of special form

We establish asymptotic equalities for the least upper bounds of deviations of trigonometric polynomials generated by a linear approximation method of a special form on classes of convolutions of analytic functions in the uniform and integral metrics.     

On consistency in probabilistic logic for logical formulas represented by B-hypertrees

One of the problems (mainly unsolved) in probabilistic logic is to consistently assign probabilities to logical formulas. In this paper we consider Horn formulas represented by B-hypertrees. We give a set of necessary conditions that any valid assignment of probabilities to the logical formulas should fulfill. If a certain condition is imposed on the B-hypertree, the necessary conditions are also sufficient, thus describing exactly which rules the assigned probabilities should obey to be consistent.     

Equilibrium and least element problems for multivalued functions

The principal aim of this paper is to extend some recent results which concern problems involving bifunctions to similar generalized problems for multivalued bifunctions. To this end, by using the appropriate notions of strict pseudomonotonicity we establish the relationships between generalized vector equilibrium problems and generalized minimal element problems of feasible sets. Moreover relationships between generalized least element problems of feasible sets and generalized vector equilibrium problems are studied by employing the concept of Z-multibifunctions.     

Complexity of approximate realization of functions,satisfying the lipschitz condition,by schemes in continuous bases

Translated from Matematicheskie Zametki, Vol. 43, No. 4, pp. 543–557, April, 1988.     

Bosonic formulas for affine branching functions

In this paper, we derive two bosonic (alternating sign) formulas for branching functions of affine Kac-Moody Lie algebras \(\mathfrak{g}\). Both formulas are expressed in terms of the Weyl group and string functions of \(\mathfrak{g}\).     

On approximation of functions regular in convex polygons by exponential polynomials of special form

We construct exponential polynomials of special form which sufficiently well approximate functions that are regular in an open convex polygon and continuous on its closure.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 424–426, March, 1992.