Full measure of a set of singular continuous measures

On the space of structurally similar measures, we construct a nontrivial measure m such that the subclass of all purely singular continuous measures is a set of full m-measure. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 83–91, January, 2009.     

Basic theonean triangles

Properties of the set of triangles with integer sides whose area is the square of an integer are studied. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 561–569, April, 1997. Translated by V. N. Dubrovsky     

Subdifferentiability and superdifferentiability of distance functions

We obtain necessary and sufficient conditions for the subdifferentiability and superdifferentiability (in the Dem'yanov-Rubinov sense) of the distance in an arbitrary norm from a point to a set for the finitedimensional case. The geometric structure of the subdifferential and the superdifferential is described. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 530–542, April, 1997. Translated by N. K. Kulman     

Clone classification of dually discriminator algebras with finite support

Dually discriminator algebras are considered up to clones generated by the algebra operations. In terms of binary relations, all clones of the operators on a finite set that contain the Pixley dual discriminator are efficiently described. As a consequence, a similar clone classification of quasi-primal algebras with finite support is determined. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 359–366, March, 1997. Translated by A. I. Shtern     

Construction of exact solutions in two-field cosmological models

We construct a dark energy model with a phantom scalar field, a standard scalar field, and a polynomial potential inspired by string field theory. We find a two-parameter set of exact solutions of the Friedmann equations. We find a potential satisfying the conditions obtained from the string theory and such that at large times, some of the exact solutions correspond to the state parameter w_DE > −1 while the others correspond to w_DE < −1. We demonstrate that the superpotential method is very effective for seeking new exact solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 47–61, April, 2008.     

Adaptive learning and <Emphasis Type="Italic">p</Emphasis>-best response sets

A product set of strategies is a p-best response set if for each agent it contains all best responses to any distribution placing at least probability p on his opponents’ profiles belonging to the product set. A p-best response set is minimal if it does not properly contain another p-best response set. We study a perturbed joint fictitious play process with bounded memory and sample and a perturbed independent fictitious play process as in Young (Econometrica 61:57–84, 1993). We show that in n-person games only strategies contained in the unique minimal p-best response set can be selected in the long run by both types of processes provided that the rate of perturbations and p are sufficiently low. For each process, an explicit bound of p is given and we analyze how this critical value evolves when n increases. Our results are robust to the degree of incompleteness of sampling relative to memory.     

Rings over which each module possesses a maximal submodule

Right Bass rings are investigated, that is, rings over which any nonzero right module has a maximal submodule. In particular, it is proved that if any prime quotient ring of a ringA is algebraic over its center, thenA is a right perfect ring iffA is a right Bass ring that contains no infinite set of orthogonal idempotents. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 407–415, March, 1997. Translated by A. I. Shtern     

The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

We consider the Riemann–Hilbert problem in the following setting: find a function whose boundary values ϕ⁺(t) satisfy the condition a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, and K ^{p( · )}(D;ω) is the set of functions ϕ(z) representable in the form , where ω(z) is a weight function and (K _Γφ )(z) is a Cauchy type integral whose density φ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewise-Lyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-H?lder condition. The solvability conditions are established and solutions are constructed. These solutions largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points. Submitted: May 13, 2007. Revised: August 8, 2007 and August 28, 2007. Accepted: November 8, 2007.     

The element sets and value sets of two-dimensional continued fractions

For two-dimensional continued fractions we prove the existence and uniqueness of an optimal sequence of value sets corresponding to an arbitrarily given sequence of element sets. We compute the element set for a given sequence of disk value sets and as a corollary, give the element sets and value sets that are used in convergence criteria for two-dimensional continued fractions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 55–61.     

On the Cohen–Lusk theorem

Let G be a finite group and X be a G-space. For a map f: X → ℝ ^m , the partial coincidence set A(f, k), k ≤ |G|, is the set of points x ∈ X such that there exist k elements g ₁,…, g _k of the group G, for which f(g ₁ x) = ⋅⋅⋅ = f(g _k x) holds. We prove that the partial coincidence set is nonempty for G = ℤ _p ⁿ under some additional assumptions. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 61–67, 2007.     

Classes of biorthonormal systems

Summary The (A)-summability and mean-convergence of Fourier series in terms of certain biorthonormal systems have been investigated. The problem of completing a set of polynomials to a biothonormal system has been solved in the case of certain sets of type zero (Sheffer’s classification). The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States air Force, through its European Office, under Contract N₀ AF 61 (052)-324.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient , for certain external subspaces of the hyperfinite dimensional Banach space , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and are isometrically isomorphic as Banach algebras. Research of both authors supported by a grant by VEGA – Scientific Grant Agency of Slovak Republic.     

Relations for relative Grothendieck groups of rings of finite representation type

For a ring R of finite representation type the set of free generators is described for the group of relations of the relative Grothendieck group K₀(R,F) corresponding to an additive subfunctor F of the functorExt ¹ _R, which yields an extension to the absolute case of the corresponding results of Butler and Auslander. Bibliography:8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 61–65.     

Element number of the Platonic solids

Let Σ be a set of polyhedra. A set Ω of polyhedra is said to be an element set for Σ if each polyhedron in Σ is the union of a finite number of polyhedra in Ω. We call each polyhedron of the element set Ω an element for Σ. In this paper, we determine one element set for the set Π of the Platonic solids, and prove that this element set is, in fact, best possible; it achieves the minimum in terms of cardinality among all the element sets for Π. We also introduce the notion of indecomposability of a polyhedron and present a conjecture in Sect. 3.     

Schnorr trivial reals: a construction

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null ${\Sigma^0_1}A real is Martin-L?f (Schnorr) random if it does not belong to any effectively presented null (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if , then h contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees. This material is based upon work supported under a National Science Foundation Graduate Research Fellowship and appears in the author’s Ph.D. thesis. A preliminary version of this paper appeared in Electronic Notes in Theoretical Computer Science     

Topological Structure of Non-wandering Set of a Graph Map

Let G be a graph （i.e., a finite one-dimensional polyhedron） and f ： G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non-wandering point; every accumulation point of the set of non-wandering points of f with infinite orbit is a two-order accumulation point of the set of recurrent points of f; the derived set of an ω-limit set of f is equal to the derived set of an the set of recurrent points of f; and the two-order derived set of non-wandering set of f is equal to the two-order derived set of the set of recurrent points of f.     

Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for <Emphasis Type="Italic">Q</Emphasis>-mappings

We prove that an open discrete Q-mapping has a continuous extension to an isolated boundary point if the function Q(x) has finite mean oscillation or logarithmic singularities of order at most n – 1 at this point. Moreover, the extended mapping is open and discrete and is a Q-mapping. As a corollary, we obtain an analog of the well-known Sokhotskii–Weierstrass theorem on Q-mappings. In particular, we prove that an open discrete Q-mapping takes any value infinitely many times in the neighborhood of an essential singularity, except, possibly, for a certain set of capacity zero. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 116–126, January, 2009.     

Error estimates for the numerical approximation of Neumann control problems

We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193–219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L ²(Γ). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45–61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided. The authors were supported by Ministerio de Ciencia y Tecnología (Spain).     

Matrix representations of octonions and their applications

As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because is a non-associative algebra over ℝ. However since is an extension of ℍ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.     

Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.