On the Rosenthal Compacta and Analytic Sets

Totally Lindelöf and totally ω-narrow paratopological groups

Let {mathbb {K}} be a non-trivially valued non-Archimedean complete field. Let ell _{infty }({mathbb {N}}, {mathbb {K}}) [ell _c({mathbb {N}}, {mathbb {K}});c_0({mathbb {N}}, {mathbb {K}})] be the space of all sequences in {mathbb {K}} that are bounded [relatively compact; convergent to 0] with the topology of pointwise convergence (i.e. with the topology induced from {mathbb {K}}^{{mathbb {N}}}). Let X be an infinite ultraregular space and let C_p(X,{mathbb {K}}) be the space of all continuous functions from X to {mathbb {K}} endowed with the topology of pointwise convergence. It is easy to see that C_p(X,{mathbb {K}}) is metrizable if and only if X is countable. We show that for any X [with an infinite compact subset] the space C_p(X,{mathbb {K}}) has an infinite-dimensional [closed] metrizable subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). Next we prove that C_p(X,{mathbb {K}}) has a quotient isomorphic to c_0({mathbb {N}}, {mathbb {K}}) if and only if it has a complemented subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). It follows that for any extremally disconnected compact space X the space C_p(X,{mathbb {K}}) has no quotient isomorphic to the space c_0({mathbb {N}}, {mathbb {K}}); in particular, for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has no quotient isomorphic c_0({mathbb {N}}, {mathbb {K}}). Finally we investigate the question for which X the space C_p(X,{mathbb {K}}) has an infinite-dimensional metrizable quotient. We show that for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has an infinite-dimensional metrizable quotient isomorphic to some subspace ell _c^0({mathbb {N}}, {mathbb {K}}) of {mathbb {K}}^{{mathbb {N}}}. If {mathbb {K}} is locally compact then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _{infty }({mathbb {N}}, {mathbb {K}}). If |n1_{{mathbb {K}}}|ne 1 for some nin {mathbb {N}}, then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _c ({mathbb {N}}, {mathbb {K}}). In particular, C_p(beta D, {mathbb {Q}}_q) has a quotient isomorphic to ell _{infty }({mathbb {N}}, {mathbb {Q}}_q) and C_p(beta D, {mathbb {C}}_q) has a quotient isomorphic to ell _c({mathbb {N}}, {mathbb {C}}_q) for any prime number q.

We present results of a mathematical model for the gas flow in an internal combustion engine consisting of a single cylinder with an inlet and outlet pipe. In order to achieve optimal performance of the engine the dependence of the gas flow on physical parameters such as pipe dimensions and valve geometry need to be understood. A system of ordinary differential equations (in time t) with discontinuous right‐hand side describes the gas properties in the cylinder, whereas the gas flow in each pipe is modelled by the Euler equations, a system of hyperbolic partial differential equations. The explicit method of Euler and a TVD scheme are used for solving these equations. However, since the coupling of the pipe equations with the o.d.e. system in the cylinder on one side and atmospheric gas properties on the other appeared to be a main problem, we concentrate on appropriate coupling conditions. The numerical techniques involve discretization in space and time, and we present different methods of discrete coupling. As a main result we show that the various coupling methods lead to quite different numerical solutions. Therefore, a careful treatment of the coupling conditions is crucial.

The set of continuous maps of a space X to a space Y equipped with the topology of pointwise convergence will be denoted by Cp(X, Y). For Y = R the notation is abbreviated to Cp(X). We will be interested in connections between the topologies of spaces X and Cp(X). These connections have not been studied extensively. However, some progress has been made recently -interesting results have been obtained in problems of normality and final compactness of Cp(X) (e.g., see [i, 2]). We will study some problems associated with cardinalvalued invariants. All spaces considered are assumed to be completely regular. I. Separability Problems. The problem of separability of Cp(X) was first considered in [3~. The following result was obtained: TIIEOREM i. Cp(X) is separable if and only if X is condensed to a space of countable weight. The main result of the present note is: THEOREM 2. Cp(X) is hereditarily separable if and only if xn is hereditarily finally compact, n~-Ni==_ Proof. Necessity. Let Cp(X) be hereditarily separable. Statement i. Suppose that ~C~Cp(X) is hereditarily separable and separates points from closed sets (i.e., if x~F and F is closed, then there is f ~ ~ such that f(x) = 1 and f(F) ~ 0). Then X is hereditarily finally compact~ Indeed, let ? = {H} be a family of sets open in Xo For each point x ~ (here, ~ = [J {H: H~---_?} is the body of y), we choose H~:? and ]~__~ such that fx(x) = 1 and f~(x \ H~)~0 There exists a countable set B~=-~ such that {/2: x=_B} is dense in {/~: x~}. Then %% = U {H~: x~-B} Indeed, if x:~_~, then there exists g~B subject to condition [/x (x) --f~ (x) [ <: 1/2. Then fy (x) ~ 0 and x ~ I f v.

We choose some special unit vectors $\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$ in $\mathbb{R}^3$ and denote by $\mathscr{L}\subset\mathbb{R}^5$ the set of all points $(L_1,\dots,L_5)\in\mathbb{R}^5$ with the following property: there exists a compact convex polytope $P\subset\mathbb{R}^3$ such that the vectors $\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$ (and no other vector) are unit outward normals to the faces of $P$ and the perimeter of the face with the outward normal $\boldsymbol{n}_k$ is equal to $L_k$ for all $k=1,\dots,5$. Our main result reads that $\mathscr{L}$ is not a locally-analytic set, i.\,e., we prove that, for some point $(L_1,\dots,L_5)\in\mathscr{L}$, it is not possible to find a neighborhood $U\subset\mathbb{R}^5$ and an analytic set $A\subset\mathbb{R}^5$ such that $\mathscr{L}\cap U=A\cap U$. We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces.

A variational splitting of high-order linear multistep methods for heat transfer and advection–diffusion parabolic problems

The main result is Theorem 1. If is a holomorphic function on the polydisk in , and for each fixed in some nonpluripolar set the function can be continued holomorphically to the whole plane with the exception of some polar set of singularities, then can be continued holomorphically to , where is a closed pluripolar subset of . Some generalizations are also given, along with corollaries on extension of functions with analytic sets of singularities. Bibliography: 13 titles.

In this paper, we estimate the rate of pointwise convergence of the Stancu type Bernstein operators for functions defined on the interval. To prove our main result, we have used some methods and techniques from probability theory.

A recent theorem of Orey [12] (see also [1], [6], [7], [13]) asserts that if $T$ is an $L_1$ operator induced on a discrete measure space by an irreducible recurrent aperiodic Markov matrix, then the condition (C) holds: $f \epsilon L_1, \int f = 0$ implies that $T^n f$ converges to zero in $L_1$. In an attempt to determine when (C) holds for more general operators, we at first prove the following (Theorem 1.1): Let $T$ be a positive linear contraction operator on $L_1$; if $T^nf$ and $T^{n+1}f$ intersect slightly, but uniformly in $f$ in the unit sphere of $L_1$, then $T^nf - T^{n+1}f$ converges to zero in norm. (C) follows if $T$ is conservative and ergodic (Corollary 1.3). In Section 2 we derive from this a simple proof of Orey's theorem. The main result of the paper is in Section 3 and could be called a "zero-two" theorem: Let $P(x, A)$ be a Markov kernel, and assume that there is a $\sigma$-finite measure $m$ such that for each $A, m(A) = 0$ implies $P(x, A) = 0$ a.e. and $m(A) > 0$ implies $\sum^\infty_{n=0} P^{(n)}(x, A) = \infty$ a.e. Then the total variation of the measure $P^{(n)}(x, \cdot) - P^{(n+1)}(x, \cdot)$ is either a.e. 2 for all $n$ or it converges a.e. to 0 as $n \rightarrow \infty$. In Section 4 it is shown that a version of the zero-two theorem essentially contains the Jamison-Orey generalization of Orey's theorem to Harris processes. Section 1 and Section 2 of this paper do not assume any knowledge of either operator ergodic theory or probability. Some known results in ergodic theory are applied in Section 3, but the proof of the main theorem does not depend on them.

Robust μ-stability analysis of Markovian switching uncertain stochastic genetic regulatory networks with unbounded time-varying delays

In this paper we discuss a class of team problems with discrete action spaces. We introduce multimodularity into team theory as a natural alternative to convexity in continuous spaces. The main result relates coordinatewise-optimal (cw-optimal) points to the optimal team decision for a class of team problems. The method is based on a characterization of coordinatewise minima of multimodular functions.

MathematikaVolume 36, Issue 2 p. 290-300 Research Article Note on pointwise convergence of sequences of analytic sets Roman Pol, Roman Pol Wydziat Matematyki U.W., PKiN IXp., 00-901 Warszawa, PolandSearch for more papers by this author Roman Pol, Roman Pol Wydziat Matematyki U.W., PKiN IXp., 00-901 Warszawa, PolandSearch for more papers by this author First published: 01 December 1989 https://doi.org/10.1112/S0025579300013140Citations: 9AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume36, Issue2December 1989Pages 290-300 RelatedInformation

Banach limits and uniform almost summability