Additive Group Of Real Numbers Research Articles

Decidability bounds for Presburger arithmetic extended by sine

We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic (sin-PA), and systematically study decision problems for sets of sentences in sin-PA. In particular, we detail a decision algorithm for existential sin-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of sin-PA sentences. To do so, we explicitly interpret the weak monadic second-order theory of the grid, which is undecidable, in sin-PA.

Implementation of asynchronous mode in a linear periodic system with the nontrivial lower left block of averaging of the coefficient matrix

A linear control system with a periodic matrix of coefficients and program control is considered. The matrix under control is constant, rectangular (the number of columns does not exceed the number of rows) and its rank is not maximum, i. e. less than the number of columns. It is assumed that the control is periodic, and the module of its frequencies, i. e. the smallest additive group of real numbers, including all Fourier exponents of this control, is contained in the frequency module of the coefficient matrix. The following task is posed: to construct such a control from an admissible set that switches the system to asynchronous mode, i. e. the system must have periodic solutions such that the intersection of the frequency moduli of the solution and the coefficient matrix is trivial. The problem posed is called the problem of synthesis of asynchronous mode. The solution to the formulated problem significantly depends on the structure of the average value of the coefficient matrix. In particular, its solution is known for systems with zero average. In addition, solvability conditions were obtained in the case when the matrix under control has zero rows, the averaging of the coefficient matrix is reduced to the form with upper left diagonal block and with zero remaining blocks. In this paper we consider a more general case with a nontrivial left lower block. Assuming an incomplete column rank of the matrix function composed from the rows of oscillation path of the coefficient matrix, we construct the control explicitly. This control switches the system to asynchronous mode.

Solvability of the control problem of the asynchronous spectrum of linear almost periodic systems with a lower triangular representation of the averaging of coefficient matrix

A linear control system with an almost periodic matrix of coefficients and control in the form of the feedback linear in phase variables is considered. It is assumed that the feedback coefficient is almost periodic and its frequency module, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix. The system under consideration is studied in the case of a triangular average value of the matrix of coefficients. For the described class of systems, the control problem of the asynchronous spectrum with a target set of frequencies is solved. This task is to construct such a control from an admissible set that the system closed by this control has almost periodic solutions, a set of the Fourier exponents of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for the solvability of this problem are obtained.

A sufficient condition for the unsolvability of the control problem of the asynchronous spectrum of linear almost periodic systems with the diagonal averaging of the coefficient matrix

We consider a linear control system with an almost periodic matrix of the coefficients. The control has the form of feedback that is linear on the phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency modulus of the coefficient matrix. The following problem is formulated: choose a control from an admissible set for which the system closed by this control has almost periodic solutions with the frequency spectrum (a set of Fourier exponents) containing a predetermined subset, and the intersection of the frequency modules of solution and the coefficient matrix is trivial. The problem is called as the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with the target set of frequencies. At present, this problem has been studied only in a very special case, when the average value of the almost periodic coefficients matrix of the system is zero. In the case of nontrivial averaging, the question remains open. In the paper, a sufficient condition is obtained under which the control problem of the asynchronous spectrum of linear almost periodic systems with diagonal averaging of the coefficient matrix has no solution.

ON SOME METRIZABLE TOPOLOGIES FOR WEAKLY ALMOST PERIODIC FUNCTIONS

The article deals with studying metrizable topologies on the additive group of real numbers that compactify this group. The introduced metrizable topology is weaker than the original natural topology on the real axis. It is a modification of the Marchenko topology. In the introduced topology a countable system of neighborhoods is selected based on the spectrum of a given function. An invariant metric is constructed which defines an equivalent topology. The completed metric space is proved to be compact. A pseudometric using only the spectrum of a given scalar almost periodic function is considered. To obtain the Hausdorff space we pass to a factor space. On the factor space the pseudometric is a metric and it is shown that the values of a scalar almost periodic function on the original space coincide with those on the factor space. It is also proved that the set of scalar almost periodic functions on the axis coincides with the set of functions defined on a metric space, which are scalar uniformly continuous in this topology.

Solvability criterion of the control problem of an asynchronous spectrum of linear almost periodic systems with the trivial averaging of the coefficient matrix

A linear control system with an almost periodic matrix of coefficients and the control in the form of feedback linear in phase variables is considered. It is assumed that the feedback coefficient is almost periodic and its frequency module, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix. The system under consideration is studied in the case of a zero average value of the matrix of coefficients. For the described class of systems, the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies is solved. This task is as follows: to construct such a control from an admissible set so that the system closed by this control has almost periodic solutions, the set of Fourier exponents (frequency spectrum) that are contained in a predetermined subset; the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for solvability of the control problem of the asynchronous spectrum are obtained.

Necessary condition for solvability of the control problem of an asynchronous spectrum of linear almost periodic systems with trivial averaging of the coefficient matrix

We consider a linear control system with an almost periodic matrix of coefficients. The control has a form of feedback and is linear in phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i.e. the smallest additive group of real numbers, including all Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix.The following problem is formulated: choose such a control from an admissible set so that the closed system has almost periodic solutions, the frequency spectrum (a set of Fourier exponents) of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The problem is called the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies.The aim of the work aws to obtain a necessary solvability condition for the control problem of the asynchronous spectrum of linear almost periodic systems with trivial averaging of coefficient matrix The estimate of the power of the asynchronous spectrum was found in the case of trivial averaging of the coefficient matrix.

Discontinuous homomorphisms, selectors, and automorphisms of the complex field

We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$. This shows that a nonprincipal ultrafilter on the integers is not sufficient to construct a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas. This is an improved version of our paper in the Proceedings of the American Mathematical Society, which used a weak version of the Axiom of Choice for the same result.

Indicable groups and $p_c<1$

A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of $\mathbb{Z} $ has a non-trivial percolation phase. Our main results prove this conjecture for certain groups, and in particular prove that any group with a non-trivial homomorphism into the additive group of real numbers satisfies the conjecture. We use this to reduce the conjecture to the case of hereditary just-infinite groups. The novelty here is mainly in the methods used, combining the methods of EIT and evolving sets, and using the algebraic properties of the group to apply these methods.

Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation-related property and by having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.

The Bochner–Schoenberg–Eberlein Property for Totally Ordered Semigroup Algebras

The concepts of BSE property and BSE algebras were introduced and studied by Takahasi and Hatori in 1990 and later by Kaniuth and Ulger. This abbreviation refers to a famous theorem proved by Bochner and Schoenberg for \(L^1({\mathbb {R}})\), where \({\mathbb {R}}\) is the additive group of real numbers, and by Eberlein for \(L^1(G)\) of a locally compact abelian group G. In this paper we investigate this property for the Banach algebra \(L^p(S,\mu )\;(1\le p<\infty )\) where S is a compact totally ordered semigroup with multiplication \(xy=\max \{x,y\}\) and \(\mu \) is a regular bounded continuous measure on S. As an application, we have shown that \(L^1(S,\mu )\) is not an ideal in its second dual.

Completeness properties of group topologies for [formula omitted

We study the completeness properties of several different group topologies for the additive group of real numbers, and we also compute the corresponding dual groups. We first present two metrizable connected group topologies on R with topologically isomorphic dual groups, one of which is noncomplete and arcwise connected and the other one is compact (therefore complete), but not arcwise connected. Using a theorem about T-sequences and adapting a result about weakened analytic groups, we then describe a method for obtaining Hausdorff group topologies R that are strictly weaker than the usual topology and are complete. They are not Baire, and consequently not metrizable.

The topological center of the spectrum of the algebra generated by the maps $$\exp q(t)$$ exp q ( t )

In this paper, after giving a characterization of the spectrum \(M(W(R))\) of the normed closed invariant algebra \(W(R)\) (which we call the Weyl algebra of the discrete additive group of real numbers \(R\)) generated by the family of functions \(\{\exp (q(t));~q(t)\) is a real polynomial\(\}\), the topological center of \(M(W(R))\) will be characterized explicitly.

A composite functional equation from algebraic aspect

In this paper we discuss the composite functional equation $$f(x+2f(y)) = f(x)+y+f(y)$$ on an Abelian group. This equation originates from Problem 10854 of the American Mathematical Monthly. We give an algebraic description of the solutions on uniquely 3-divisible Abelian groups, and then we construct all solutions f of this equation on finite Abelian groups without elements of order 3 and on divisible Abelian groups without elements of order 3 including the additive group of real numbers.

TRIVIALITY OF A TRACE ON THE SPACE OF COMMUTING TRACE-CLASS SELF-ADJOINT OPERATORS

Abstract. In the present article, we investigate the possibility of a real-valued map on the space of tuples of commuting trace-class self-adjointoperators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with con-tinuous group homomorphisms from the Milnor’s K -group of the realnumbers into the additive group of real numbers. Using this connection,it is shown that any such trace map must be trivial, but it is proposedthat the target group of a nontrivial trace should be a linearized ver-sion of Milnor’s K -theory as with the case of universal determinant forcommuting tuples of matrices rather than just the eld of constants. 1. IntroductionCommuting self-adjoint operators are constantly encountered in various ma-thematical and physical researches. As an example of physics, let H be aHilbert space whose elements represent physical states. In quantum mechanics,observables are represented by self-adjoint linear operators on

Defining additive subgroups of the reals from convex subsets

Let G be a subgroup of the additive group of real numbers and let C ⊆ G be infinite and convex in G. We show that G is definable in (R, +,.,C) and that Z is definable if G has finite rank. This has a number of consequences for expansions of certain o-minimal structures on the real field by multiplicative groups of complex numbers.

Lower Semimodular Inverse Semigroups

An inverse semigroup S is said to be lower semimodular if its lattice LF(S) has that property. Previous results of the second author have essentially reduced the study of such semigroups to the 0-simple ones and, as we show, of those only the simple ones that are not groups are of concern. We show that such an inverse semigroup S is lower semimodular if and only if it is combinatorial, its semilattice of idempotents is archimedean in S and the poset of idempotents of each D-class contains at most two mutually incomparable elements, each of which is maximal. As easy corollaries, we recover the descriptions of modular and distributive inverse semigroups. With each lattice ordered group G, Reilly associated an "l-bisimple" inverse semigroup. We show that the bisimple lower semimodular inverse semigroups with totally ordered idempotents are, up to isomorphism, precisely the l-bisimple inverse semigroups over subgroups of the additive group of real numbers. There is, up to isomorphism, only one bisimple, non-group, lower semimodular inverse semigroup that is not of this type.

Phylogenetic Diversity Over an Abelian Group

There is a natural way to associate to any tree T with leaf set X, and with edges weighted by elements from an abelian group G, a map from the power set of X into G—simply add the elements on the edges that connect the leaves in that subset. This map has been well-studied in the case where G has no elements of order 2 (particularly when G is the additive group of real numbers) and, for this setting, subsets of leaves of size two play a crucial role. However, the existence and uniqueness results in that setting do not extend to arbitrary abelian groups. We study this more general problem here, and by working instead with both, pairs and triples of leaves, we obtain analogous existence and uniqueness results. Some particular results for elementary abelian 2-groups are also described.

Equivariant Embeddings of the n-Dimensional Space into a Hilbert Space

The present paper is devoted to the study of equivariant embeddings of the n-dimensional space into a Hilbert space. We consider a representation of a group of similarities. The existence of a cocycle for this representation implies the existence of an isometric embedding of a metric group into the Hilbert space. Then we describe all cocycles of a representation of the additive group of real numbers and construct an embedding of the n-dimensional space with metric d(x,y)=|x-y|α into the Hilbert space. Bibliography: 5 titles.

Cardinalities of locally compact groups and their Stone-Čech compactifications

Dedicated to Edwin HewittIf G is any Hausdorff topological group and βG is the Stone-Čech compactification then where |G| denotes the cardinalty of G It is known that if G is a discrete group then and if G is the additive group of real numbers with the Euclidean topology, then |βG| = 2|G|. In this paper the cardinality and weight of βG, for a locally compact group G, is calculated in terms of the character and Lindelöf degree of G The results make it possible to give a reasonably complete description of locally compact groups G for which |βG| = 2|G| or even |βG| = |G|.