On Q-sobriety

The study of fixed-basis variety-based topology was initiated by S.A. Solovyov (in 2008), which, among other things, generalizes fuzzy topology. We extend within this framework, an earlier result due to Srivastava et al. (in 1998), which showed that the category of sober fuzzy topological spaces is the epireflective hull of the fuzzy Sierpinski space in the category of T₀-fuzzy topological spaces.     

On the curvature of algebraic hypersurfaces

Using algebraic residue theory, we try to generalize a theorem of Chasles about osculating circles of plane algebraic curves to algebraic hypersurfaces over algebraically closed fields of characteristic zero.     

Fixed point property and formal concept analysis

The purpose of this paper is to interpret, with the language of formal concept analysis, the fixed point free and order-preserving self-mappings of ordered sets as formal concepts of a context. With this interpretation one can derive a practicable algorithm for determining if a given finite ordered set has the fixed point property. As a side product it is proved that dismantlability of finite ordered sets can be tested in polynomial time.     

The Dunkl-Williams equality in pre-Hilbert C-modules

In this paper we consider the case of equality in some generalizations of the Dunkl-Williams inequality for elements of a pre-Hilbert C^∗-module.     

Testing the equality of linear single-index models

Comparison of nonparametric regression models has been extensively discussed in the literature for the one-dimensional covariate case. The comparison problem largely remains open for completely nonparametric models with multi-dimensional covariates. We address this issue under the assumption that models are single-index models (SIMs). We propose a test for checking the equality of the mean functions of two (or more) SIM’s. The asymptotic normality of the test statistic is established and an empirical study is conducted to evaluate the finite-sample performance of the proposed procedure.     

The dimension of cycle-free orders

The existence of a four-dimensional cycle-free order is proved. This answers a question of Ma and Spinrad. Two similar problems are also discussed.Research partially supported by Office of Naval Research grant N00014-90-J-1206Research partially supported by the National Science Foundation under grant DMS     

Finite-dimensional time-invariant linear dynamical systems: Algebraic theory

Willems' approach to dynamical systems without a priori distinguishing between inputs and outputs is accepted, and with this as a starting point, new linear dynamical systems are introduced and studied. It is proved in particular that (in the complex case) the set of isomorphism classes of completely observable (or completely reachable) linear systems with given input and output numbers and McMillan degree, has a natural structure of a compact algebraic variety. This variety is closely connected to the one constructed by Hazewinkel using the Rosenbrock linear systems % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]=Ax+Bu, v=Cx+D(·)u, where D is a polynomial matrix, and may be regarded as the most natural compactification of it. (The connection is very similar to that of Grass_m,mx+p() and Mat_m.p(). Input/output linear systems, i.e. linear systems equipped with an extra structure which distinguishes input and output signals, are also considered. It is shown that each of them may be represented by the equations K% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]+L% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeyDayaaca% aaaa!35D8!\[{\rm{\dot u}}\]=Fx+Gu, v=Hx+Ju (det(K–sF)0). Such systems clearly contain the so-called generalized linear systems. They also contain the Rosenbrock linear systems mentioned above and essentially coincide with them.     

On the convergence of solutions of some evolution differential equations

Two conjectures concerning the asymptotic behaviour of solutions of two types of evolution differential equations are presented.     

Linear systems of arbitrary genus

Kalman's theory is extended to a complete, smooth, and irreducible algebraic curve of arbitrary genus.     

On subtractive varieties,I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a classical ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.Presented by A. F. Pixley.     

On the parseval equality

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 5, pp. 178–181, Spetember–October, 1990.     

The critical exponent for continuous conventional powers of doubly nonnegative matrices

We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than n-2 and we conjecture that it is always n-2 (as it is with Hadamard powering). We prove this conjecture when n<6 and in certain other special cases. We establish a quadratic bound for the critical exponent in general.     

On Mckenzie's method

This is an expository account of R. McKenzie's recent refutation of the RS conjecture.Dedicated to László Fuchs on the occasion of his 70th birthday     

On the congruence modularity conjecture

A new condition of compatibility with projections, applicable to some Maltsev filters, is defined and shown to hold, among others, for the filter of congruence-modular varieties. As a consequence, it is shown that there exist no simple counterexamples (in a specified sense) to the modularity conjecture. This paper is dedicated to Walter Taylor. Received November 5, 2005; accepted in final form April 3, 2006.     

On algebraic properties of monotone clones

Let R ₈ denote the 8-element bounded tower. G. Tardos has shown that C(R ₈), the clone of all monotone functions on R ₈, is not finitely generated. In this paper we show that the clone of all nonsurjective functions is finitely generated.     

Stochastic matrices and a property of the infinite sequences of linear functionals

Our starting point is the proof of the following property of a particular class of matrices. Let T={T_i,j} be a n×m non-negative matrix such that ∑_jT_i,j=1 for each i. Suppose that for every pair of indices (i,j), there exists an index l such that T_i,l≠T_j,l. Then, there exists a real vector k=(k₁,k₂,…,k_m)^T,k_i≠k_j,i≠j;0<k_i?1, such that, if i≠j.Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals , corresponding to an infinite sequence of probability measures {μ_(·)(i)}_i∈N, on the Borel σ-algebra such that, . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that

     

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.