Implicatively selector sets

Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an n-ary selector general recursive function f such that (∀x₁,...,x_n)(β(χ(x₁),...,χ(x_n))=1⟹f(x₁,...,x_n)∈A), where χ is the characteristic function of A. Let F^(m), m≥1, be the family of all d _m+1 ^* -IS sets, where , F⁽⁰⁾=N, and F^(∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one of F^(m), m≥0, or F^(∞), and, moreover, the inclusions F⁽⁰⁾⊂F⁽¹⁾⊂...⊂F^(∞) hold. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 145–153, March–April, 1996.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

A New View on Fuzzy Hypermodules

We describe the relationship between the fuzzy sets and the algebraic hyperstructures. In fact, this paper is a continuation of the ideas presented by Davvaz in （Fuzzy Sets Syst., 117： 477- 484, 2001） and Bhakat and Das in （Fuzzy Sets Syst., 80： 359-368, 1996）. The concept of the quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set is introduced and this is a natural generalization of the quasi-coincidence of a fuzzy point in fuzzy sets. By using this new idea, the concept of interval-valued （α,β）-fuzzy sub-hypermodules of a hypermodule is defined. This newly defined interval-valued （α,β）-fuzzy sub-hypermodule is a We shall study such fuzzy sub-hypermodules and sub-hypermodules of a hypermodule. generalization of the usual fuzzy sub-hypermodule. consider the implication-based interval-valued fuzzy     

Discrete spectrum in spectral gaps of a self-adjoint operator under unbounded perturbations

Let A be a self-adjoint operator, let (α,β) be a gap in the spectrum of A, and let B=A+V, where, in general, the perturbation operator V is unbounded. We establish some abstract conditions under which the spectrum of B in (α,β) is discrete; does not accumulate to β; is finite. An estimate of the number of eigenvalues of B in (α,β) is obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 237–241. Translated by S. V. Kislyakov.     

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

We introduce new potential type operators J^a_b = (E+(-D)^b/2)^-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with J^α_β. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).     

Irreducible characters with equal roots in the groups Sn and An

We show that treating of (non-trivial) pairs of irreducible characters of the group S_n sharing the same set of roots on one of the sets A_n and S_n \ A_n is divided into three parts. This, in particular, implies that any pair of such characters χ^α and χ^β (α and β are respective partitions of a number n) possesses the following property: lengths d(α) and d(β) of principal diagonals of Young diagrams for α and β differ by at most 1. Supported by RFBR grant No. 04-01-00463 and by RFBR-NSFC grant No. 05-01-39000. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 3–25, January–February, 2007.     

Mixed weak continuity on generalized topological spaces

We introduce the notion of mixed weak (μ,ν₁ν₂)-continuity between a generalized topology μ and two generalized topologies ν₁, ν₂. We characterize such continuity in terms of mixed generalized open sets: (ν₁,ν₂)′-semiopen sets, (ν₁,ν₂)′-preopen sets, (ν₁,ν₂)-preopen sets [2], (ν₁,ν₂)′-β′-open sets and θ(ν₁,ν₂)-open sets [3]. In particular, we show that for a given mixed weakly (μ,ν₁ν₂)-continuous function, if the codomain of the given function is mixed regular (=(ν₁,ν₂)-regular), then the function is also (μ,ν₁)-continuous.     

On wavelet decomposition of Hermite type splines

We consider wavelet decompositions of spaces of Hermite type splines of class C¹(α, β) that are defined by a 4-component vector-valued function ϕ(t) ∈ C¹ (α, β) by means of a grid X (not necessarily uniform) on (α, β) ∈ ℝ¹ (the special case ϕ(t)^def = (1, t, t²,t³)^T corresponds to cubic Hermite splines). The basis wavelets obtained are compactly supported. The decomposition and reconstruction formulas are given. Bibliography: 8 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 33–46     

Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity

So far there is no systematic attempt to construct Boolean functions with maximum annihilator immunity. In this paper we present a construction keeping in mind the basic theory of annihilator immunity. This construction provides functions with the maximum possible annihilator immunity and the weight, nonlinearity and algebraic degree of the functions can be properly calculated under certain cases. The basic construction is that of symmetric Boolean functions and applying linear transformation on the input variables of these functions, one can get a large class of non-symmetric functions too. Moreover, we also study several other modifications on the basic symmetric functions to identify interesting non-symmetric functions with maximum annihilator immunity. In the process we also present an algorithm to compute the Walsh spectra of a symmetric Boolean function with O(n²) time and O(n) space complexity. We use the term “Annihilator Immunity” instead of “Algebraic Immunity” referred in the recent papers [3–5, 9, 18, 19]. Please see Remark 1 for the details of this notational change     

A Cheeger–Müller theorem for delocalized analytic torsion

We study the delocalized L ²-analytic torsion introduced by John Lott, and define the delocalized L ²-combinatorial torsion. By using the method of Bismut–Zhang, under the conditions of positive Novikov–Shubin invariants and finite conjugacy class, we get the Cheeger–Müller type relation between the delocalized L ²-analytic torsion and the delocalized L ²-combinatorial torsion.      

On Relative Difference Sets in Dihedral Groups

In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters of the form (uλ,u,uλ, λ) or (uλ+2,u, uλ+1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral groups and Menon–Hadamard difference sets. In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes.     

Regularities of the Distribution of β-adic van der Corput Sequences

For Pisot numbers β with irreducible β-polynomial, we prove that the discrepancy function D(N, [0,y)) of the β-adic van der Corput sequence is bounded if and only if the β-expansion of y is finite or its tail is the same as that of the expansion of 1. If β is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length y ? \Bbb Q(b) y \notin {\Bbb Q}(\beta) . We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse β-substitution.     

Irreducible characters of the group S n that are semiproportional on A n

Previously, we dubbed the conjecture that the alternating group A_n has no semiproportional irreducible characters for any natural n [1]. This conjecture was then shown to be equivalent to the following [3]. Let α and β be partitions of a number n such that their corresponding characters χ^α and χ^β in the group S_n are semiproportional on A_n. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis and the conclusion of the latter conjecture. Supported by RFBR (grant No. 07-01-00148) and by RFBR-NSFC (grant No. 05-01-39000). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 135–156, March–April, 2008.     

Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.     

On the embedding problem with non-Abelian kernel of orderp 4. IV

In the present paper, the embedding problem is considered for number fields with p-groups whose kernel is either of two groups with two generators α and β and with the following relations: (1) α^ρ=1, α^ρ=1, [α,β,β]=1, [α,β,α,α]=1, or (2) α^ρ=[α, β α], β^ρ=1, [α,β,β]=1. It is shown that for the solvability of the original embedding problem it is necessary and sufficient to have the solvability of the associated Abelian and local problems for all completions of the base fields. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 120–126. Translated by V. V. Ishkhanov.     

Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220.     

Coprimitive sets and inextendable codes

Complete (n,r)-arcs in PG(k−1,q) and projective (n,k,n−r) _q -codes that admit no projective extensions are equivalent objects. We show that projective codes of reasonable length admit only projective extensions. Thus, we are able to prove the maximality of many known linear codes. At the same time our results sharply limit the possibilities for constructing long non-linear codes. We also show that certain short linear codes are maximal. The methods here may be just as interesting as the results. They are based on the Bruen–Silverman model of linear codes (see Alderson TL (2002) PhD. Thesis, University of Western Ontario; Alderson TL (to appear) J Combin Theory Ser A; Bruen AA, Silverman R (1988) Geom Dedicata 28(1): 31–43; Silverman R (1960) Can J Math 12: 158–176) as well as the theory of Rédei blocking sets first introduced in Bruen AA, Levinger B (1973) Can J Math 25: 1060–1065.      

Condition for inducing topology in a vector lattice by the order completion

Properties of an order topology in vector lattices and Boolean algebras are studied. The main result is the following: in a vector lattice or a Boolean algebra with the condition of “closure by one step” (a generalization of the well-known “regularity” property of Boolean algebras and K-spaces) the order topology is induced by the topology of its Dedekind completion. Bibliography: 4 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16, 1997, pp. 204–207.     

Rational sets in finitely generated nilpotent groups

We deal with a class of rational subsets of a group, that is, the least class of its subsets which contains all finite subsets and is closed under taking union. a product of two sets, and under generating of a submonoid by a set. It is proved that the class of rational subsets of a finitely generated nilpotent group G is a Boolean algebra iff G is Abelian-by-finite. We also study the question asking under which conditions the set of solutions for equations in groups will be rational. It is shown that the set of solutions for an arbitrary equation in one variable in a finitely generated nilpotent group of class 2 is rational. And we give an example of an equation in one variable in a free nilpotent group of nilpotency class 3 and rank 2 whose set of solutions is not rational. Supported by RFFR grant No. 98-01-00932. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 379–394, July–August, 2000.     

Subalgebras of Orthomodular Lattices

Sachs (Can J Math 14:451–460, 1962) showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham (Int J Theor Phys 37(11):2669–2733, 1998, Int J Theor Phys 38(3):827–859, 1999), at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.