We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is E and also when the membership relation is E', and in both cases the formulas are allowed to contain both E and E', then (M,E) and (M,E') are isomorphic, and the isomorphism is definable in (M,E,E'). This extends Zermelo's 1930 theorem about second order ZFC.

Jouko Väänänen

Let $E$ be a Banach space with the $c^1$-norm $\|\cdot\|$ in $ E \backslash \{0\}$ and $S(E)=\{e\in E: \|e\|=1\}.$ In this paper, a geometry characteristic for $E$ is presented by using a geometrical construct of $S(E).$ That is, the following theorem holds : the norm of $E$ is of $c^1$ in $ E \backslash \{0\}$ if and only if $S(E)$ is a $c^1$-submanifold of $E,$ with ${\rm codim}S(E)=1.$ The theorem is very clear, however, its proof is non-trivial, which shows an intrinsic connection between the continuous differentiability of the norm $\|\cdot\|$ in $ E \backslash \{0\}$ and differential structure of $S(E).$

Jipu Ma

We derive a generalization of forward fitting for X-ray spectroscopy to include linear polarization of X-ray sources, appropriate for the anticipated next generation of space-based photoelectric polarimeters. We show that the inclusion of polarization sensitivity requires joint fitting to three observed spectra, one for each of the Stoke's parameters, I(E), U(E), and Q(E). The equations for Stoke's I(E) (the total intensity spectrum) are identical to the familiar case with no polarization sensitivity, and for which the model-predicted spectrum is obtained by a convolution of the source spectrum, F(E'), with the familiar energy response function, e(E')*R(E', E), where e(E') and R(E', E) are the effective area and energy redistribution matrix, respectively. In addition to the energy spectrum, the two new relations for U(E) and Q(E) include the source polarization fraction and position angle versus energy, a(E'), and psi'_0(E'), respectively, and the model-predicted spectra for these relations are obtained by a convolution with the "modulated" energy response function, m(E')*e(E')R(E, E'), where m(E') is the energy-dependent modulation fraction that quantifies a polarimeter's angular response to 100% polarized radiation. We present results of simulations with response parameters appropriate for the proposed PRAXyS Small Explorer observatory to illustrate the procedures and methods, and we discuss some aspects of photoelectric polarimeters with relevance to understanding their calibration and operation.

Tod E. Strohmayer

Let $E$ be a module over a domain $A$, and $W(E)^{\#}=W(E)-ann(E)$ where $W(E)=\{a\in A:aE\neq E\}$. We define an equivalence relation $\sim$ on $W(E)^{\#}$ as follows: $a\sim b$ if and only if $aE=bE$ for any $a,b\in W(E)^{\#}$ and denote $EC(W(E)^{\#})$ to be the set of all equivalence classes $[a]$ of $W(E)^{\#}$. We first show that the family $\{U_a\}_{a\in W(E)^\#}$ generates a topology which we called the quasi divisor topology of $A$-module $E$ denoted by $qD_A(E)$ where $U_{a}=\{[b]\in EC(W(E)^{\#}):\ aE\subseteq bE\}$ for every $a\in W(E)^{\#}$. This paper examines the connections between topological properties of the quasi divisor topology $qD_{A}(E)$ and algebraic properties of $A$-module $E$. These include each separation axioms, compactness, connectedness and first and second countability. Also, we characterize some important class of rings/modules such as divisible modules and uniserial modules by means of $qD_{A}(E)$. Furthermore, we introduce quasi second modules and study its algebraic properties to decide when $qD_A(E)$ is a $T_1$-space.

Mesut Buğday Dilara Erdemir Ünsal Tekir Suat Koç

A graph $G=(V,E)$ is strongly antimagic, if there is a bijective mapping $f: E \to \{1,2,\ldots,|E|\}$ such that for any two vertices $u\neq v$, not only $\sum_{e \in E(u)}f(e) \ne \sum_{e\in E(v)}f(e)$ and also $\sum_{e \in E(u)}f(e) < \sum_{e\in E(v)}f(e)$ whenever $\deg(u)< \deg(v) $, where $E(u)$ is the set of edges incident to $u$. In this paper, we prove that double spiders, the trees contains exactly two vertices of degree at least 3, are strongly antimagic.

Fei-Huang Chang Pinhui Chin Wei-Tian Li Zhishi Pan

Brylawski and Seymour independently proved that if $M$ is a connected matroid with a connected minor $N$, and $e \in E(M) - E(N)$, then $M \backslash e$ or $M / e$ is connected having $N$ as a minor. This paper proves an analogous but somewhat weaker result for $2$-polymatroids. Specifically, if $M$ is a connected $2$-polymatroid with a proper connected minor $N$, then there is an element $e$ of $E(M) - E(N)$ such that $M \backslash e$ or $M / e$ is connected having $N$ as a minor. We also consider what can be said about the uniqueness of the way in which the elements of $E(M) - E(N)$ can be removed so that connectedness is always maintained.

Zachary Gershkoff James Oxley

In this paper we consider the price of anarchy (PoA) in multi-commodity flows where the latency or delay function on an edge has a heterogeneous dependency on the flow commodities, i.e. when the delay on each link is dependent on the flow of individual commodities, rather than on the aggregate flow. An application of this study is the performance analysis of a network with differentiated traffic that may arise when traffic is prioritized according to some type classification. This study has implications in the debate on net-neutrality. We provide price of anarchy bounds for networks with $k$ (types of) commodities where each link is associated with heterogeneous polynomial delays, i.e. commodity $i$ on edge $e$ faces delay specified by $g_{i1}(e)f^{\theta}_1(e) + g_{i2}(e)f^{\theta}_2(e) + \ldots + g_{ik}(e)f^{\theta}_k(e) + c_i(e), $ where $f_i(e)$ is the flow of the $i$th commodity through edge $e$, $\theta \in {\cal N}$, $g_{i1}(e), g_{i2}(e), \ldots, g_{ik}(e)$ and $c_i(e)$ are nonnegative constants. We consider both atomic and non-atomic flows. For networks with decomposable delay functions where the delay induced by a particular commodity is the same, i.e. delays on edge $e$ are defined by $a_1(e)f_1^\theta(e) + a_2(e)f_2^\theta(e) + \ldots + c(e)$ where $\forall j , \forall e: g_{1j}(e) = g_{2j}(e) = \ldots = a_j(e)$, we show an improved bound on the price of anarchy. Further, we show bounds on the price of anarchy for uniform latency functions where each edge of the network has the same delay function.

Sanjiv Kapoor Junghwan Shin

In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity $h(n,G)$ is defined to be the maximum number of edges in an $n$-vertex graph $H$ such that there exists a mapping $f: E(H)\rightarrow E(H)$ with $f(e)\neq e$ for all $e\in E$ and further in all copies $G'$ of $G$ in $H$ there exists $e\in E(G')$ with $f(e)\in E(G')$. Among other results, we determine $h(n, G)$ when $G$ is a matching and $n$ is large enough. As a related concept, we say that $H$ is unavoidable for $G$ if for any mapping $f: E(H)\rightarrow E(H)$ with $f(e)\neq e$ there exists a copy $G'$ of $G$ in $H$ such that $f(e)\notin E(G')$ for all $e\in E(G)$. The set of minimal unavoidable graphs for $G$ is denoted by $\mathcal{M}(G)$. We prove that if $F$ is a forest, then $\mathcal{M}(F)$ is finite if and only if $F$ is a matching, and we conjecture that for all non-forest graphs $G$, the set $\mathcal{M}(G)$ is infinite. Several other parameters are defined with basic results proved. Lots of open problems remain.

Yair Caro Balázs Patkós Zsolt Tuza Máté Vizer

Given Banach spaces E and F, we denote by ${\mathcal P}(^k!E,F)$ the space of all k-homogeneous (continuous) polynomials from E into F, and by ${\mathcal P}_{wb}(^k!E,F)$ the subspace of polynomials which are weak-to-norm continuous on bounded sets. It is shown that if E has an unconditional finite dimensional expansion of the identity, the following assertions are equivalent: (a) ${\mathcal P}(^k!E,F)={\mathcal P}_{wb}(^k!E,F)$; (b) ${\mathcal P}_{wb}(^k!E,F)$ contains no copy of $c_0$; (c) ${\mathcal P}(^k!E,F)$ contains no copy of $\ell_\infty$; (d) ${\mathcal P}_{wb}(^k!E,F)$ is complemented in ${\mathcal P}(^k!E,F)$. This result was obtained by Kalton for linear operators. As an application, we show that if E has Pe\l czy\'nski's property (V) and satisfies ${\mathcal P}(^k!E) ={\mathcal P}_{wb}(^k!E)$ then, for all F, every unconditionally converging $P\in{\mathcal P}(^k!E,F)$ is weakly compact. If E has an unconditional finite dimensional expansion of the identity, then the converse is also true.

Manuel González Joaquín M. Gutiérrez

Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

Xavier Tolsa