The Nikodym property and cardinal characteristics of the continuum

We present a general method of constructing Boolean algebras with the Nikodym property and of some given cardinalities. The construction is dependent on the values of some classical cardinal characteristics of the continuum. As a result we obtain a consistent example of an infinite Boolean algebra with the Nikodym property and of cardinality strictly less than the continuum $\mathfrak{c}$. It follows that the existence of such an algebra is undecidable by the usual axioms of set theory. Besides, our results shed some new light on the Efimov problem and cofinalities of Boolean algebras.     

A non-treelike continuum that is not the 2-to-1 image of any continuum

Some thirteen years ago S. B. Nadler, Jr. and L. E. Ward, Jr., asked if any treelike continuum could be the 2-to-1 image of a continuum. In fact, it has been conjectured that the property of being treelike characterizes those continua that are not the 2-to-1 image of any continuum. But the characterization must be something else; this paper shows that many pseudo-solenoids are not the 2-to-1 image of any continuum.

     

Perfect cliques and colorings of Polish spaces

A coloring of a set is any subset of , where 1$"> is a natural number. We give some sufficient conditions for the existence of a perfect -homogeneous set, in the case where is and is a Polish space. In particular, we show that it is sufficient that there exist -homogeneous sets of arbitrarily large countable Cantor-Bendixson rank. We apply our methods to show that an analytic subset of the plane contains a perfect -clique if it contains any uncountable -clique, where is a natural number or (a set is a -clique in if the convex hull of any of its -element subsets is not contained in ).

     

Maximizing and minimizing the number of generalized colorings of trees

We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on $q$ vertices.     

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An interval$t$-coloring of a multigraph $G$ is a proper edge coloring with colors $1,\dots ,t$ such that the colors of the edges incident with every vertex of $G$ are colored by consecutive colors. A cyclic interval$t$-coloring of a multigraph $G$ is a proper edge coloring with colors $1,\dots ,t$ such that the colors of the edges incident with every vertex of $G$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. Denote by $w\left(G\right)$ ($w_c\left(G\right)$) and $W\left(G\right)$ ($W_c\left(G\right)$) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph $G$, respectively. We present some new sharp bounds on $w\left(G\right)$ and $W\left(G\right)$ for multigraphs $G$ satisfying various conditions. In particular, we show that if $G$ is a $2$-connected multigraph with an interval coloring, then $W\left(G\right)\le 1+?\frac{\left|V\left(G\right)\right|}{2}?(\Delta \left(G\right)?1)$. We also give several results towards the general conjecture that $W_c\left(G\right)\le \left|V\left(G\right)\right|$ for any triangle-free graph $G$ with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most $4$.     

On the connectivity of proper colorings of random graphs and hypergraphs

Let Ω_q=Ω_q(H) denote the set of proper [q]‐colorings of the hypergraph H. Let Γ_q be the graph with vertex set Ω_q where two colorings σ,τ are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_n,m;k, k ≥ 2, the random k‐uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Γ_q is connected if d is sufficiently large and . This is optimal up to the first order in d. Furthermore, with a few more colors, we find that the diameter of Γ_q is O(n) w.h.p., where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber dynamics Markov Chain on Ω_q is ergodic w.h.p.     

LF拓扑空间之间投影映射是开映射的一个充要条件

王国俊先生曾提出这样一个问题：设｛（Ｌ＾Ｘ，δｔ）｝ｔ∈Ｔ是一族ＬＦ拓扑空间，（Ｌ＾Ｘ，δ）是其乘积空间，Ｐｔ：Ｌ＾Ｘ→Ｌ＾Ｘｔ（ｔ∈Ｔ）是投影映射，问Ｐｔ是开映射的充要条件是什么？本文在较广范围－形如（Ｌ＾Ｘ，η）的拓扑分子格－内讨论类似问题，给出了上述问题的一个解答。     

Joint measurability and the one-way Fubini property for a continuum of independent random variables

As is well known, a continuous parameter process with mutually independent random variables is not jointly measurable in the usual sense. This paper proposes an extension of the usual product measure-theoretic framework, using a natural ``one-way Fubini' property. When the random variables are independent even in a very weak sense, this property guarantees joint measurability and defines a unique measure on a suitable minimal -algebra. However, a further extension to satisfy the usual (two-way) Fubini property, as in the case of Loeb product measures, may not be possible in general. Some applications are also given.

     

An irregular extension of the oscillation theory of the Sturm-Liouville spectral problem

Oscillation spectral properties (such as the number of zeros, the alternation of zeros for eigenfunctions, the simplicity of the spectrum, etc.) for the Sturm-Liouville problem with generalized coefficients are described.     

Euler's problem,Euler's method,and the standard map; or,the discrete charm of buckling

Summary We explore the relation between the classical continuum model of Euler buckling and an iterated mapping which is not only a mathematical discretization of the former but also has an exact, discrete mechanical analogue. We show that the latter possesses great numbers of “parasitic” solutions in addition to the natural discretizations of classical buckling modes. We investigate this rich bifurcational structure using both mechanical analysis of the boundary value problem and dynamical studies of the initial value problem, which is the familiar standard map. We use this example to explore the links between discrete initial and boundary value problems and, more generally, to illustrate the complex relations among physical systems, continuum and discrete models and the analytical and numerical methods for their study.     

A unifying formulation of the discrete and continuum approximations for embedded discontinuities

A methodology for the numerical implementation of embedded discontinuities into the finite element method is developed. This is applicable for the discrete and continuum approximations of discontinuities. The variational formulation of the problem of a solid with discontinuities is established for both approximations, yielding the equations used in this methodology. Three sets of equations are obtained by applying this methodology; all are suitable to be numerically implemented. To show the application potential of this method, the numerical simulation of the formation and propagation of a discontinuity in a concrete specimen is carried out and the results are compared with those from the physical experiment, demonstrating the adequacy of the methodology and its corresponding implementations to model discontinuities. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Consistent values and the core in continuum market games with two types of players

The consistent value is an extension of the Shapley value to the class of games with non-transferable utility.? In this paper, the consistent value will be characterized for market games with a continuum of players of two types. We will show that for such games the consistent value need not belong to the core, and provide conditions under which there is equivalence between the two concepts. Received: October 1998 RID="*" ID="*"  This thesis was completed under the supervision of Professor Sergiu Hart, The Center for Rationality and Interactive Decision Theory, Department of Mathematics, Department of Economics, The Hebrew University of Jerusalem. I would like to thank Professor Hart for introducing me to this area of research, for his help and guidance, and, especially, for all his patience.? I would also like to thank Michael Borns for improving the style, and an anonymous referee for helpful comments.     

二维渗流最低水平开穿透的宽度

在二维渗流中,设rn是矩形[O,n]2的最低水平开穿透,令ξn=max｛y：（xy）∈rn｝．本文证明了在临界状态p＝pc时,存在常数0＜C1＜C2＜1,当n充分大时有存在,而在上临界状态p＞pc时,有存在］＝0.     

小Excess与开流形的拓扑

本文中，我们应用比较几何的方法研究开流形的Excess与其拓扑之间的关系，我们证明了对于一个曲率下有界的开流形，当它的Excess被临界半径的某个函数所界定时，它就有有限拓扑型或微分同胚于n维z欧氏空间。     

The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients

The present paper discusses relations between regularity, Dirichlet, and Neumann problems. We investigate the boundary problems for block operators and prove, in particular, that the solvability of the regularity problem does not imply the solvability of the dual Dirichlet problem for general elliptic operators with complex bounded measurable coefficients. This is strikingly different from the case of real operators, for which such an implication was established in 1993 by C. Kenig, J. Pipher [Invent. Math. 113 (3) (1993) 447-509] and since then has served as an integral part of many results.     

A branch and bound algorithm for the maximum clique problem

We present a branch and bound algorithm for the maximum clique problem in arbitrary graphs. The main part of the algorithm consists in the determination of upper bounds by graph colorings. Using a modification of a known graph coloring method called DSATUR we simultaneously derive lower and upper bounds for the clique number.

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Zusammenfassung Wir stellen einen Branch and Bound Algorithmus für das Maximum Clique Problem in einem beliebigen Graphen vor. Das Hauptaugenmerk richtet sich dabei auf die Bestimmung oberer Schranken mit Hilfe von Färbungen von Graphen. Es wird eine Modifikation einer bekannten Färbungsmethode, genannt DSATUR, verwendet, mit der sich gleichzeitig obere und untere Schranken für die Cliquezahl erstellen lassen.

     

Finite volume method for the variational inequalities of first and second kinds

We propose and analyze the finite volume method for solving the variational inequalities of first and second kinds. The stability and convergence analysis are given for this method. For the elliptic obstacle problem, we derive the optimal error estimate in the H¹‐norm. For the simplified friction problem, we establish an abstract H¹‐error estimate, which implies the convergence if the exact solution u∈H¹(Ω) and the optimal error estimate if u∈H^{1 + α}(Ω),0 < α≤2. Copyright © 2015 John Wiley & Sons, Ltd.     

Periodic and boundary value problems for second order differential equations

In this paper we study second order scalar differential equations with Sturm-Liouville and periodic boundary conditions. The vector fieldf(t,x,y) is Caratheodory and in some instances the continuity condition onx ory is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well as truncation and penalization techniques, we show the existence of solutions and extremal solutions in the order interval determined by the upper and lower solutions. Also we establish some properties of the solutions and of the set they form.