Discrete Arzelà-Ascoli Theorem on the Half Line and Its Application

The compactness criterion in l_◇={x(·)∈l~∞(0,∞))|lim t→∞ x(t)=x(∞)} is given and as an application,we study the existence of positive solutions of second order boundary value problems of difference equation on the half line by the fixedpoint theorem in cones.     

Coordinating Numeric and Linear Units: Elementary Students’ Strategies for Locating Whole Numbers on the Number Line

Two investigations of fifth graders’ strategies for locating whole numbers on number lines revealed patterns in students’ coordination of numeric and linear units. In Study 1, we investigated the effects of context on students’ placements of three numbers on an open number line. For one group (n?=?24), the line was presented in a thematic context as a “race course,” and, for a second group (n?=?24), the line was presented as a conventional number line. Most students in both groups placed consecutive whole numbers at appropriate linear distances, but the thematic context group was more likely to place nonconsecutive whole numbers at appropriate linear distances. In Study 2 (n?=?24), students placed numbers on lines marked with two numbers. Most students placed a third number appropriately when the marked numbers were consecutive whole numbers, but not when the labeled numbers were nonconsecutive whole numbers. The findings reveal fifth graders’ conceptual difficulties in coordinating numeric and linear units on the number line and a thematic context that can support this coordination.     

On Small Values of the Riemann Zeta‐Function on the Critical Line and Gaps Between Zeros

Small values of are investigated, using the value distribution results of A. Selberg. This gives an asymptotic formula for Some related problems involving values of and gaps between ordinates of consecutive zeros of (s) are also discussed.     

Discrete Zγ: Embedded Circle Patterns with the Square Grid Combinatorics and Discrete Painlevé Equations

We study a discrete analogue of the holomorphic map z . It is given by Schramm's circle pattern with the square grid combinatorics. We show that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlevé equations. We establish global properties of these solutions and of the discrete z .     

The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.     

The Relation Between the Petits Integral and the McShane Integral of Banach-valued Functions Defined on R~n

R.A.Gordon[1]generalizedthedefinitionoftheMcShaneintegralforrealvaluedfunctionstotheabstractfunctionsfromR1toBanachspaces.Inthispaper,wedefinetheMcShaneintegralforabstractfunctionsfromRntoBanachspacesandprovethatifeveryfunctionffromIoinRntoBanachspacesisMcShaneintegrablethenfisPettisintegrable,andiftheunitballB(X)inX*isrelativelyweaklycompact,fisPetitsintegrablethenfisMcShaneintegrableonIo.ThisgivesapositiveanswertoanopenproblemleftbyR.A.Gordonin[1].ThroughoutthispaperXwilldenoteare…     

The Monge Problem of “Piles and Holes” on the Torus and the Problem of Small Denominators

We discuss the problem of existence of a smooth endomorphism of a closed n-dimensional manifold carrying a differential n-form into a prescribed volume form. Of course, we assume that the integrals of these forms over the whole manifold are equal. The solution of this problem for the n-dimensional torus reduces to the problem of small denominators well known in analysis.     

The Non–symmetric Discrete Algebraic Riccati Equation and Canonical Factorization of Rational Matrix Functions on the Unit Circle

Canonical factorization of a rational matrix function on the unit circle is described explicitly in terms of a stabilizing solution of a discrete algebraic Riccati equation using a special state space representation of the symbol. The corresponding Riccati difference equation is also discussed.     

The “Circolo Matematico di Palermo” and the First World War: The crisis of scientific internationalism: a view through the unedited correspondence of De Franchis with Edmund Landau and other mathematicians

In this work the situation of the “Circolo Matematico di Palermo” between 1914 and 1928 is analyzed. It will be observed that during this time the Circolo was among the few European scientific associations with German as well as French associates. During the 1930s, the nationalist politics of Fascism and above all the racial laws dealt a deadly blow to the Circolo as an international scientific association. We will use the rich correspondence in the Circolo's archives to shed some light on this. In particular, the correspondence between M. De Franchis and E. Landau and other recently found documents will figure prominently.