Varieties of quasigroups arising from 2-perfect m-cycle systems

For m = 6 and for all odd composite integers m, as well as for all even integers m 10 that satisfy certain conditions, 2-perfect m-cycle systems are constructed whose quasigroups have a homomorphism onto quasigroups which do not correspond to a 2-perfect m-cycle systems. Thus it is shown that for these values of m the class of quasigroups arising from all 2-perfect m-cycle systems does not form a variety.     

On Integrable roots in split Lie triple systems

We focus on the notion of an integrable root in the framework of split Lie triple systems T with a coherent 0-root space. As a main result, it is shown that if T has all its nonzero roots integrable, then its standard embedding is a split Lie algebra having all its nonzero roots integrable. As a consequence, a local finiteness theorem for split Lie triple systems, saying that whenever all nonzero roots of T are integrable then T is locally finite, is stated. Finally, a classification theorem for split simple Lie triple systems having all its nonzero roots integrable is given.     

Classification of Integrable Divergent N-Component Evolution Systems

We use a symmetry approach to solve the classification problem for integrable N-component evolution systems having the form of conservation laws. We obtain complete lists of both isotropic and anisotropic systems of this type and find auto-Bäcklund transformations with a spectral parameter for all systems.     

Further 6-sparse Steiner Triple Systems

We give a construction that produces 6-sparse Steiner triple systems of order v for all sufficiently large v of the form 3p, p prime and p ≡ 3 (mod 4). We also give a complete list of all 429 6-sparse systems with v < 10000 produced by this construction.     

On Approximate ℓ 1 Systems in Banach Spaces

Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (x_i)^∞_i=1 in X which satisfy ∑_iA±x_i|A|−f(|A|), for all finite A and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [x_i] is H.I., and we exhibit systems in all isomorphs of ℓ₁ which are not equivalent to the unit vector basis of ℓ₁. We also prove that certain lacunary Haar systems in L₁ are quasi-greedy basic sequences.     

A hierarchy of axiom systemZF

In this paper we develop a sequenceZ ₀, ...,Z _α,... of axiom systems for set theory, such that (1) the consistency of any system within the sequence is provable in its succeeding systems, (2) the first system in the sequence is Zermelo's system Z and the union of all systems in the sequence is justZF. And we prove that for ordinal number α>1, there exists a sequence of ℵ_a+1 axiom systems between systemsZ _α andZ _α+1 such that these systems satisfy the above condition (1).     

Pseudozero Set of Real Multivariate Polynomials

The pseudozero set of a system P of polynomials in n variables is the subset of C ⁿ consisting of the union of the zeros of all polynomial systems Q that are near to P in a suitable sense. This concept arises naturally in Scientific Computing where data often have a limited accuracy. When the polynomials of the system are polynomials with complex coefficients, the pseudozero set has already been studied. In this paper, we focus on the case where the polynomials of the system have real coefficients and such that all the polynomials in all the perturbed polynomial systems have real coefficients as well. We provide an explicit definition to compute this pseudozero set. At last, we analyze different methods to visualize this set.      

The random product homotopy and deficient polynomial systems

Summary Most systems ofn polynomial equations inn unknowns arising in applications aredeficient, in the sense that they have fewer solutions than that predicted by the total degree of the system. We introduce the random product homotopy, an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems. In many cases, the amount of computation required to find all solutions can be made roughly proportional to the number of solutions.     

Phase portraits of the quadratic vector fields with a polynomial first integral

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral. IfH(x, y) is a polynomial of degreen+1 then the differential systemx′=−∂H/∂y,y′=∂H/∂x is called a Hamiltonian system of degreen. We also prove that all the phase portraits that we obtain in this paper are realizable by Hamiltonian systems of degree 2.     

The entropies of renewal systems

Renewal systems are symbolic dynamical systems originally introduced by Adler. IfW is a finite set of words over a finite alphabetA, then the renewal system generated byW is the subshiftX _W ⊂A ^Z formed by bi-infinite concatenations of words fromW. Motivated by Adler’s question of whether every irreducible shift of finite type is conjugate to a renewal system, we prove that for every shift of finite type there is a renewal system having the same entropy. We also show that every shift of finite type can be approximated from above by renewal systems, and that by placing finite-type constraints on possible concatenations, we obtain all sofic systems. The authors were supported in part by NFS grants DMS-8706284, DMS-8814159 and DMS-8820716.     

Quadrics on Riemannian spaces of constant curvature. Separation of variables and connection with the Gaudin magnet

Integrable systems associated with separation of the variables in real Riemannian spaces of constant curvature are considered. An isomorphism between all such systems and the hyperbolic Gaudin magnet is established. This isomorphism is used in a classification of all coordinate systems that admit separation of the variables, the basis of which is the classification of the correspondingL operators of the Gaudin magnet.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 83–111, April, 1992.     

Check Character Systems Using Quasigroups: II

In this part of the article we continue to research check character systems with one check character over quasigroups under check equations which have one permutation. These systems always detect all single errors (i.e. errors in only one component of a code word) and can detect some other errors arising during transmission of data. We study check character systems over T-quasigroups. These quasigroups are isotopic to abelian groups and generalize the well-known class of medial quasigroups. We establish some properties of a T-quasigroup so that the check character systems over it are able to detect transpositions, jump transpositions, twin errors and jump twin errors. We also give some models of T-quasigroups, which satisfy all of the required properties for detection of errors of each of the considered types. Communicated by: P. Wild     

Galois theory for sets of operations closed under permutation,cylindrification, and composition

A set of operations on A is shown to be the set of linear term operations of some algebra on A if and only if it is closed under permutation of variables, addition of inessential variables, and composition, and if it contains all projections. A Galois framework is introduced to describe the sets of operations that are closed under the operations mentioned above, not necessarily containing all projections. The dual objects of this Galois connection are systems of pointed multisets, and the Galois closed sets of dual objects are described accordingly. Moreover, the closure systems associated with this Galois connection are shown to be uncountable (even if the closed sets of operations are assumed to contain all projections).     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

Categorical abstract algebraic logic: The criterion for deductive equivalence

Equivalent deductive systems were introduced in [4] with the goal of treating 1‐deductive systems and algebraic 2‐deductive systems in a uniform way. Results of [3], appropriately translated and strengthened, show that two deductive systems over the same language type are equivalent if and only if their lattices of theories are isomorphic via an isomorphism that commutes with substitutions. Deductive equivalence of π‐institutions [14, 15] generalizes the notion of equivalence of deductive systems. In [15, Theorem 10.26] this criterion for the equivalence of deductive systems was generalized to a criterion for the deductive equivalence of term π‐institutions, forming a subclass of all π‐institutions that contains those π‐institutions directly corresponding to deductive systems. This criterion is generalized here to cover the case of arbitrary π‐institutions.     

Embedding directed and undirected partial cycle systems of index λ > 1

Recent results have found small embeddings for partial m-cycle systems of order n with λ= 1. However, if λ> 1 then the best known techniques produce embeddings that are often quadratic functions of both m and n and linear functions of λ. In this article we obtain embeddings for partial m-cycle systems of order n, and of partial directed m-cycle systems, for all values of m. These embeddings are independent of λ and linear in both n and m. © 1993 John Wiley & Sons, Inc.     

Existence of solutions to second-order BVPs on time scales

We consider a boundary value problem (BVP) for systems of second-order dynamic equations on time scales. Using methods involving dynamic inequalities, we formulate conditions under which all solutions to a certain family of systems of dynamic equations satisfy certain a priori bounds. These results are then applied to guarantee the existence of solutions to BVPs for systems of dynamic equations on time scales.     

Stability of Classes of Mappings and Holder Continuity of Higher Derivatives of Elliptic Solutions to Systems of Nonlinear Differential Equations

Nirenberg published the following well-known result in 1954: Let a function z be a twice continuously differentiable solution to a nonlinear second-order elliptic equation. Suppose that the function F defining the equation is continuous and has continuous first-order partial derivatives with respect to all of its arguments (i.e., independent together with z and the symbols of all first- and second-order partial derivatives of z). Then the partial derivatives of z are locally Holder continuous. Simultaneously with Nirenberg, Morrey obtained an analogous result for elliptic systems of second-order nonlinear equations. In this article, we get the same result for the higher derivatives of elliptic solutions to systems of nonlinear partial differential equations of arbitrary order and a rather general shape. The proof is based on the results of the author's recent research on the study of the stability phenomena in the C ^l-norm of classes of mappings.