The topological center of the spectrum of some distal algebras

The topological center of the spectrum of the Weyl algebra W, i.e. the norm closure of the algebra generated by the set of functions , is characterized in a recent paper by Jabbari and Namioka (Ellis group and the topological center of the flow generated by the map , to appear in Milan J. Math.). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra W _k , the norm closure of the algebra generated by the set of functions , will be characterized, for all k∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.     

The number of all transformation groups on an infinite set

Let X be an infinite set, T(X)={f∈X~X:f is a bijection}, C={G:G is a transformation group on X}={G:G is a subgroup of T(X)}.Then |C|=2~(2~(|x|)). Inproof of this result, the AC is used.     

The finite basis problem for infinite involution semigroups of triangular 2 $$\times $$ 2 matrices

Let \(T_n(\mathbb {F})\) and \(UT_n(\mathbb {F})\) be the semigroups of all upper triangular \(n\times n\) matrices and all upper triangular \(n\times n\) matrices with 0s and/or 1s on the main diagonal over a field \(\mathbb {F}\) with \(\mathsf {char}(\mathbb {F})=0\), respectively. In this paper, we address the finite basis problem for \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) as involution semigroups under the skew transposition. By giving a sufficient condition under which an involution semigroup is nonfinitely based, we show that both \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) are nonfinitely based, and that there is a continuum of nonfinitely based involution monoid varieties between the involution monoid variety \(\mathsf {var} UT_2(\mathbb {F})\) generated by \(UT_2(\mathbb {F})\) and the involution monoid variety \(\mathsf {var} T_2(\mathbb {F})\) generated by \(T_2(\mathbb {F})\). Moreover, \(\mathsf {var} UT_2(\mathbb {F})\) cannot be defined within \(\mathsf {var} T_2(\mathbb {F})\) by any finite set of identities.     

Moore–Penrose inverse of an invertible infinite matrix

In this article we show that, contrary to finite matrices (with real or complex entries) an invertible infinite matrix V could have a Moore–Penrose inverse that is not a classical inverse of V. This also answers a recent open problem on infinite matrices.     

What is the limit of an infinite sequence？

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Sufficient Set of Integrability Conditions of an Orthonomic System

Every orthonomic system of partial differential equations is known to possess a finite number of integrability conditions sufficient to ensure the validity of them all. Here we show that a redundancy-free sufficient set of integrability conditions can be constructed in a time proportional to the number of equations cubed.     

Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree

In this paper,we study the strong law of large numbers and Shannon-McMillan (S-M) theorem for Markov chains indexed by an infinite tree with uniformly bounded degree.The results generalize the analogous results on a homogeneous tree.     

The topological types of some irregular Kähler surfaces

The topological type of generalized Kummer surfaces is described in terms of sphere bundles over Riemann surfaces and the complex projective plane. Explicit examples of sets of pairwise non-diffeomorphic K?hler surfaces of the same topological type are given. Received: 5 January 2000     

Fluid model criterion for instability of re-entrant line with infinite supply of work

Using a fluid model approach, we obtain a sufficient condition for re-entrant lines with infinite supply of work to be unstable, which generalizes the results for re-entrant line of Dai (Ann. Appl. Probab. 6: 751–757, 1996). We apply the result to two special re-entrant lines with infinite supply of work as follows. In addition, we get necessary conditions for the corresponding fluid model to be weakly stable.     

The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions

The BS Hermite spline quasi-interpolation scheme is presented. It is related to the continuous extension of the BS linear multistep methods, a class of Boundary Value Methods for the solution of Ordinary Differential Equations. In the ODE context, using the numerical solution and the associated numerical derivative produced by the BS methods, it is possible to compute, with a local approach, a suitable spline with knots at the mesh points collocating the differential equation at the knots and having the same convergence order as the numerical solution. Starting from this spline, here we derive a new quasi-interpolation scheme having the function and the derivative values at the knots as input data. When the knot distribution is uniform or the degree is low, explicit formulas can be given for the coefficients of the new quasi-interpolant in the B-spline basis. In the general case these coefficients are obtained as solution of suitable local linear systems of size 2d×2d, where d is the degree of the spline. The approximation order of the presented scheme is optimal and the numerical results prove that its performances can be very good, in particular when suitable knot distributions are used.     

The Intrinsic Diameter of the Surface of a Parallelepiped

In the paper we obtain an explicit formula for the intrinsic diameter of the surface of a rectangular parallelepiped in 3-dimensional Euclidean space. As a consequence, we prove that an parallelepiped with relation for its edge lengths has maximal surface area among all rectangular parallelepipeds with given intrinsic diameter.     

The Study for Some Subclasses of Biholomorphic Mappings by an Unified Method

LetΩ∈Cn be a bounded starlike circular domain with 0∈Ω. In this paper, we introduce a class of holomorphic mappings Mg onΩ. Let f(z) be a normalized locally biholomorphic mapping onΩsuch that Jf-1(z)f(z)∈Mg and z=0 is the zero of order k + 1 of f(z)-z. We obtain the growth and covering theorems for f(z). Especially, as corollaries, we unify and generalize many known results. Moreover, in view of proofs of corollaries, the essential relations among the subclasses of starlike mappings are shown.     

The unbounded solution of a periodic mixed Sturm–Liouville problem in an infinite strip for the Laplacian

The unbounded solution, at the points where the boundary conditions change, for a mixed Sturm–Liouville problem of the Dirichlet–Neumann type can be obtained using the method of the integral equation formulation. Since this formulation is usually reduced to an infinite algebraic system in which the unknowns are the Fourier coefficients of the unknown unbounded entity, a study of ?_p-solutions imposes itself concerning the influence of the truncation on such systems. This study is achieved and the well-known theorem on the ?₂-solutions of the infinite algebraic systems is generalized.     

The product formula for the topological degree of strict γ-contractions

Leray and Schauder [8] have defined a topological degree for mappings in Banach spaces of the form I-f, where f is compact. A product formula for this degree, i.e. a formula for the degree of the composed map (I-f)(I-g), has been given by Nagumo [10]. In recent years, this degree has been extended to mappings I-f, where f is a strict -contraction or -condensing with respect to a rather general measure of noncompactness . It has been shown that all the properties of the Leray-Schauder degree are still valid (see [11] and [12]), while the validity of the product formula remained an open question. Nussbaum [11] has announced this formula for the special case of ₁ -spaces and mappings f that are strict - and ß-contractions, simultaneously. Fenske [5] has shown that this formula holds in every Banach space provided f is a differentiable strict -contraction. In this note we shall establish this formula for an arbitrary Banach space and an arbitrary strict -contraction.     

The topological type of the α-sections of convex sets

An α=(α₁,…,αk)(0?αi?1) section of a family {K₁,…,Kk} of convex bodies in Rd is a transversal halfspace H⁺ for which Vold(Ki∩H⁺)=αi⋅Vold(Ki) for every 1?i?k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sd−k.     

The Contact Problems of the Mathematical Theory of Elasticity for Plates with an Elastic Inclusion

The contacts problem of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic inclusion of variable rigidity are considered. The problems are reduced to integral differential equation or to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, to get exact or approximate solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion.      

A characterization of arithmetical invariants by the monoid of relations

The investigation and classification of non-unique factorization phenomena have attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P. García-Sánchez, together with several co-authors, derived a method to calculate the catenary and tame degree from the monoid of relations, and they applied this method successfully in the case of numerical monoids. In this paper, we investigate the algebraic structure of this approach. Thereby, we dispense with the restriction to finitely generated monoids and give applications to other invariants of non-unique factorizations, such as the elasticity and the set of distances.