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Let \(X\), \(Y\) and \(Z\) be Banach spaces and let \(U\) be a subspace of \(\mathcal{L}(X^*,Y)\), the Banach space of all operators from \(X^*\) to \(Y\). An operator \(S\colon U \to Z\) is said to be \((\ell^s_p,\ell_p)\)-summing (where \(1\leq p

For certain sequences $A$ of positive integers with missing $g$-adic digits, the Dirichlet series $F_A(s) = \sum_{a\in A} a^{-s}$ has abscissa of convergence $\sigma_c < 1$. The number $\sigma_c$ is computed. This generalizes and strengthens a classical theorem of Kempner on the convergence of the sum of the reciprocals of a sequence of integers with missing decimal digits., Comment: Minor improvements and corrected typos; 7 pages

A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with convergent harmonic series., Minor changes, 8 pages

Let $\Gamma$ be a countable group. A classical theorem of Thorisson states that if $X$ is a standard Borel $\Gamma$-space and $\mu$ and $\nu$ are Borel probability measures on $X$ which agree on every $\Gamma$-invariant subset, then $\mu$ and $\nu$ are equidecomposable, i.e. there are Borel measures $(\mu_\gamma)_{\gamma\in\Gamma}$ on $X$ such that $\mu = \sum_\gamma \mu_\gamma$ and $\nu = \sum_\gamma \gamma\mu_\gamma$. We establish a generalization of this result to cardinal algebras.

We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov. The main result is deduced from the classical theorem due to Marcel Berger saying that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the main result for billiards in spaces of constant curvature: space forms., Comment: 21 pages. The main result on commuting billiards and Berger's result on caustics are extended to billiards in spaces of constant curvature

We provide necessary and sufficient conditions for a pair $S,T$ of Hilbert space operators in order that they satisfy $S^*=T$ and $T^*=S$. As a main result we establish an improvement of von Neumann's classical theorem on the positive self-adjointness of $S^*S$ for two variables. We also give some new characterizations of self-adjointness and skew-adjointness of operators, not requiring their symmetry or skew-symmetry, respectively., 10 pages

[EN] With the help of C-contractions having a fixed point, we obtain a characterization of complete fuzzy metric spaces, in the sense of Kramosil and Michalek, that extends the classical theorem of H. Hu (see "Am. Math. Month. 1967, 74, 436-437") that a metric space is complete if and only if any Banach contraction on any of its closed subsets has a fixed point. We apply our main result to deduce that a well-known fixed point theorem due to D. Mihet (see "Fixed Point Theory 2005, 6, 71-78") also allows us to characterize the fuzzy metric completeness., This research was partially funded by Ministerio de Ciencia, Innovacion y Universidades, under grant PGC2018-095709-B-C21 and AEI/FEDER, UE funds.

International audience; The classical theorem of Erdős & Wintner furnishes a criterion for the existence of a limiting distribution for a real, additive arithmetical function. This work is devoted to providing an effective estimate for the remainder term under the assumption that the conditions in the criterion are fulfilled. We also investigate the case of a conditional distribution.

A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits., Comment: 18 pages; version 3: we specified that the contact manifolds are required to be connected. To appear in Commentarii Mathematici Helvetici

This paper is devoted to the investigation of a kind of generalized Caputo semilinear fractional differential inclusions with deviated-advanced nonlocal conditions. Solvability of the problem is established by means of the Leray-Schauder’s alternative approach with the help of the Lagrange mean-value classical theorem. Finally, some examples are given to delineate the efficient of theoretical results.

A classical theorem of W.Sierpinski, S. Mazurkiewicz and S.Kempisty says that the class of all differences of lower semicontinuous functions is uniformly dense in the space of all Baire-one functions. We show a generalization of this result to the case when finely continuous functions of either density topologies or both linear and nonlinear potential theory are involved. Moreover, we examine which topological properties play a crucial role when deriving approximation theorems in more general situations.

In this paper we study the flow-property of graphs containing a spanning triangle-tree. Our main results provide a structure characterization of graphs with a spanning triangle-tree admitting a nowhere-zero $3$-flow. All these graphs without nowhere-zero $3$-flows are constructed from $K_4$ by a so-called bull-growing operation. This generalizes a result of Fan et al. in 2008 on triangularly-connected graphs and particularly shows that every $4$-edge-connected graph with a spanning triangle-tree has a nowhere-zero $3$-flow. A well-known classical theorem of Jaeger in 1979 shows that every graph with two edge-disjoint spanning trees admits a nowhere-zero $4$-flow. We prove that every graph with two edge-disjoint spanning triangle-trees has a flow strictly less than $3$., 16 pages, 8 figures

In 2003, Alladi, Andrews and Berkovich proved an identity for partitions where parts occur in eleven colors: four primary colors, six secondary colors, and one quaternary color. Their work answered a longstanding question of how to go beyond a classical theorem of G\"ollnitz, which uses three primary and three secondary colors. Their main tool was a deep and difficult four parameter $q$-series identity. In this paper we take a different approach. Instead of adding an eleventh quaternary color, we introduce forbidden patterns and give a bijective proof of a ten-colored partition identity lying beyond G\"ollnitz' theorem. Using a second bijection, we show that our identity is equivalent to the identity of Alladi, Andrews, and Berkovich. From a combinatorial viewpoint, the use of forbidden patterns is more natural and leads to a simpler formulation. In fact, in Part II of this series we will show how our method can be used to go beyond G\"ollnitz' theorem to any number of primary colors., Comment: 22 pages

Given an ideal $\mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $\mathcal{I}$-statistically convergent to $\ell$ provided that $$ \textstyle \left\{n \in \mathbf{N}: \frac{1}{n}|\{k \le n: x_k \notin U\}| \ge \varepsilon\right\} \in \mathcal{I} $$ for all neighborhoods $U$ of $\ell$ and all $\varepsilon>0$. First, we show that $\mathcal{I}$-statistical convergence coincides with $\mathcal{J}$-convergence, for some unique ideal $\mathcal{J}=\mathcal{J}(\mathcal{I})$. In addition, $\mathcal{J}$ is Borel [analytic, coanalytic, respectively] whenever $\mathcal{I}$ is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if $\mathcal{I}$ is the summable ideal $\{A\subseteq \mathbf{N}: \sum_{a \in A}1/a, Comment: 15 pages, comments are welcome

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables over a field [Formula: see text] of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations [Formula: see text] (known as Weitzenböck derivations), the algebra of constants [Formula: see text] is finitely generated. When the Weitzenböck derivation [Formula: see text] acts on the polynomial algebra [Formula: see text] in [Formula: see text] variables by [Formula: see text], [Formula: see text], [Formula: see text], Nowicki conjectured that [Formula: see text] is generated by [Formula: see text] and [Formula: see text] for all [Formula: see text]. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free [Formula: see text]-generated metabelian Lie algebra [Formula: see text], with few trivial exceptions, the algebra [Formula: see text] is not finitely generated. However, the vector subspace [Formula: see text] of the commutator ideal [Formula: see text] of [Formula: see text] is finitely generated as a [Formula: see text]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the [Formula: see text]-module [Formula: see text].

An undirected graph $G$ is $d$-degenerate if every subgraph of $G$ has a vertex of degree at most $d$. By the classical theorem of Erdös and Gallai from 1959, every graph of degeneracy $d>1$ contains a cycle of length at least $d+1$. The proof of Erdös and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least $d+1$. But can we decide in polynomial time whether a graph contains a cycle of length at least $d+2$? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: Deciding whether a graph has a cycle of length at least $d+2$ is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether $G$ contains a cycle of length at least $d+k$ can be done in time $2^{\mathcal{O}(k)}\cdot|V(G)|^{\mathcal{O}(1)}$. In other words, deciding whether a 2-connected $n$-vertex $G$ contains a cycle of length at least $d+\log{n}$ can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least $d+1$ is NP-complete. However, we prove that if graph $G$ is connected, then deciding whether $G$ contains a path of length at least $d+k$ can be done in time $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$. We complement these results by showing that the choice of degeneracy as the “above guarantee parameterization” is optimal in the following sense: For any $\varepsilon>0$ it is NP-complete to decide whether a connected (2-connected) graph of degeneracy $d$ has a path (cycle) of length at least $(1+\varepsilon)d$. publishedVersion

We explore some properties of the so-called Zeckendorf representations of integers, where we write an integer as a sum of distinct, nonconsecutive Fibonacci numbers. We examine the combinatorics arising from the arithmetic of these representations, with a particular emphasis on understanding the Zeckendorf tree that encodes them. We introduce some possibly new results related to the tree, allowing us to develop a partial analog to Kummer’s classical theorem about counting the number of “carries” involved in arithmetic. Finally, we finish with some conjectures and possible future projects related to the combinatorics of these representations.

Answering a question posed by S.Maillot in MathOverFlow, for every $n\in\mathbb N$ we construct a locally connected subgroup $G\subset\mathbb R^{n+1}$ of dimension $dim(G)=n$, which is not locally compact., Comment: 2 pages

In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group $\vec{F}$. In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an oriented knot/link $\vec{L}$, there exists an element $g$ in $\vec{F}$ whose closure $\vec{\mathcal{L}}(g)$ is $\vec L$., To appear in Algebraic & Geometric Topology. Corrected definition

International audience; Let S ⊆ N be a numerical semigroup with multiplicity m = min(S \ {0}), conductor c = max(N \ S) + 1 and minimally generated by e elements. Let L be the set of elements of S which are smaller than c. Wilf conjectured in 1978 that |L| is bounded below by c/e. We show here that if c ≤ 3m, then S satisfies Wilf's conjecture. Combined with a recent result of Zhai, this implies that the conjecture is asymptotically true as the genus g(S) = |N \ S| goes to infinity. One main tool in this paper is a classical theorem of Macaulay on the growth of Hilbert functions of standard graded algebras.

In this paper, we extend and refine previous Tur\'an-type results on graphs with a given circumference. Let $W_{n,k,c}$ be the graph obtained from a clique $K_{c-k+1}$ by adding $n-(c-k+1)$ isolated vertices each joined to the same $k$ vertices of the clique, and let $f(n,k,c)=e(W_{n,k,c})$. Improving a celebrated theorem of Erd\H{o}s and Gallai, Kopylov proved that for $c\max{f(n,3,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)}$, then either $G$ is a subgraph of $W_{n,2,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, or $c$ is odd and $G$ is a subgraph of a member of two well-characterized families which we define as $\mathcal{X}_{n,c}$ and $\mathcal{Y}_{n,c}$. We prove that if $G$ is a 2-connected graph on $n$ vertices with minimum degree at least $k$ and circumference $c$ such that $10\leq c\max{f(n,k+1,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)}$, then one of the following holds: (i) $G$ is a subgraph of $W_{n,k,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, (ii) $k=2$, $c$ is odd, and $G$ is a subgraph of a member of $\mathcal{X}_{n,c}\cup \mathcal{Y}_{n,c}$, or (iii) $k\geq 3$ and $G$ is a subgraph of the union of a clique $K_{c-k+1}$ and some cliques $K_{k+1}$'s, where any two cliques share the same two vertices. This provides a unified generalization of the above result of F\"uredi et al. as well as a recent result of Li et al. and independently, of F\"uredi et al. on non-Hamiltonian graphs. Moreover, we prove a stability result on a classical theorem of Bondy on the circumference., Comment: 31 pages, to appear in Combinatorica

International audience; The existence of Macbeath regions is a classical theorem in convex geometry [13], with recent applications in discrete and computational geometry. In this paper, we initiate the study of Macbeath regions in a combinatorial setting—and not only for the Lebesgue measure as is the case in the classical theorem—and establish near-optimal bounds for several basic geometric set systems.

Let $(\lambda_n)$ be a nonnegative sequence, increasing to $+\infty$, $\tau=\limsup\limits_{n\to\infty}\frac{\ln n}{\lambda_n}$, and $\rho$ be a positive number. It follows from a classical theorem of G. Valiron that for every Dirichlet series of the form $F(s)=\sum a_ne^{s\lambda_n}$ we have $$\limsup_{\sigma\to+\infty}\frac{\ln \sup\{|F(s)|:\,\text{Re}\, s=\sigma\}}{e^{\rho\sigma}}\le e^{\rho\tau} \limsup_{n\to\infty}\frac{\lambda_n}{e\rho}|a_n|^\frac{\rho}{\lambda_n}.$$ The exactness of this estimation is proved in the paper.

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge coloring with $d+O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that $d+1$ colors are always enough.

A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation \(\gamma \) in \(\text {PGL}(2,\mathbb {Z})\) there exist s and t for which the complete quotients \(x_s\) and \(y_t\) coincide. In this paper we give an upper bound in terms of \(\gamma \) for the smallest indices s and t.

Given two systems $P=(P_j(D))_{j=1}^N$ and $Q=(Q_j(D))_{j=1}^M$ of linear partial differential operators with constant coefficients, we consider the spaces ${\mathcal E}_\omega^P$ and ${\mathcal E}_\omega^Q$ of $\omega$-ultradifferentiable functions with respect to the iterates of the systems $P$ and $Q$ respectively. We find necessary and sufficient conditions, on the systems and on the weights $\omega(t)$ and $\sigma(t)$, for the inclusion ${\mathcal E}_\omega^P\subseteq{\mathcal E}_\sigma^Q$. As a consequence we have a generalization of the classical Theorem of the Iterates.

We lift the classical theorem of Arnol'd on homological stability for configurations spaces of the plane to the motivic world. More precisely, we prove that the schemes of unordered configurations of points in the affine line satisfy stability with respect to the motivic t-structure on mixed Tate motives., 14 pages. I made a few small changes following the referee's suggestion

This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. In addition, we require rather general assumptions on the local operator. As far as we know, this result is new and represent a fractional version of a classical theorem obtained working with Laplacian equations.

For a real reductive group G, the center $\mathfrak{z}(\mathcal{U}(\mathfrak{g}))$ of the universal enveloping algebra of the Lie algebra $\mathfrak{g}$ of G acts on the space of distributions on G. This action proved to be very useful (see e.g. [HC63, HC65, Sha74, Bar03]). Over non-Archimedean local fields, one can replace this action by the action of the Bernstein center z of G, i.e. the center of the category of smooth representations. However, this action is not well studied. In this paper we provide some tools to work with this action and prove the following results. 1) The wave-front set of any z-finite distribution on G over any point $g\in G$ lies inside the nilpotent cone of $T_g^*G \cong \mathfrak{g}$. 2) Let $H_1,H_2 \subset G$ be symmetric subgroups. Consider the space J of $H_1\times H_2$-invariant distributions on G. We prove that the z-finite distributions in J form a dense subspace. In fact we prove this result in wider generality, where the groups $H_i$ are spherical groups of certain type and the invariance condition is replaced by equivariance. Further we apply those results to density and regularity of spherical characters. The first result can be viewed as a version of Howe's expansion of characters. The second result can be viewed as a spherical space analog of a classical theorem on density of characters of admissible representations. It can also be viewed as a spectral version of Bernstein's localization principle. In the Archimedean case, the first result is well-known and the second remains open., 27 pages. v2:version to appear in Advances in Mathematics

Graph Theory, Let G be a finite connected graph. We give an asymptotically tight upper bound on the size of G in terms of order, radius and minimum degree. Our result is a strengthening of an old classical theorem of Vizing (1967) if minimum degree is prescribed.

We give sufficient conditions for the convergence almost everywhere of the expansion with respect to an unconditional basis for functions in Lp p > 2. This result extends the classical theorem of Menchoff and Rademacher for orthogonal series in L².

Let M be an open, connected manifold. A classical theorem of McDuff and Segal states that the sequence of configuration spaces of n unordered, distinct points in M is homologically stable with coefficients in Z: in each degree, the integral homology is eventually independent of n. The purpose of this note is to prove that this phenomenon also holds for homology with twisted coefficients. We first define an appropriate notion of finite-degree twisted coefficient system for configuration spaces and then use a spectral sequence argument to deduce the result from the untwisted homological stability result of McDuff and Segal. The result and the methods are generalisations of those of Betley for the symmetric groups., v3: 25 pages

The classical theorem of Bishop-Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension of the theorem gives a quantitative description of just how dense the norm-achieving functionals have to be: if (x,f) is in X x X* with ||x||=||f||=1 and |1-f(x)|< h^2/4 then there are (x',f') in X x X* with ||x'||= ||f'||=1, ||x-x'||, ||f-f'||< h and f'(x')=1. This means that there are always "proximinal" hyperplanes H in X (a nonempty subset E of a metric space is said to be "proximinal" if, for x not in E, the distance d(x,E) is always achieved - there is always an e in E with d(x,E)=d(x,e)); for if H= ker f (f in X*) then it is easy to see that H is proximinal if and only if f is norm-achieving. Indeed the set of proximinal hyperplanes H is, in the appropriate sense, dense in the set of all closed hyperplanes H in X. Quite a long time ago [Problem 2.1 in his monograph "The Theory of Best approximation and Functional Analysis" Regional Conference series in Applied Mathematics, SIAM, 1974], Ivan Singer asked if this result generalized to closed subspaces of finite codimension - if every Banach space has a proximinal subspace of codimension 2, for example. In this paper I show that there is a Banach space X such that X has no proximinal subspace of finite codimension n>1. So we have a converse to Bishop-Phelps-Bollobas: a dense set of proximinal hyperplanes can always be found, but proximinal subspaces of larger, finite codimension need not be., The paper has been submitted for publication to the Israel Journal of Mathematics

Let $G$ be a finite group. We show that the order of the subgroup generated by coprime $\gamma_k$-commutators (respectively $\delta_k$-commutators) is bounded in terms of the size of the set of coprime $\gamma_k$-commutators (respectively $\delta_k$-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words $\gamma_k$ and $\delta_k$ are concise., Comment: 7 pages

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain nonarchimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage-Enriques-Petri on equations for the canonical embedding of a curve., Comment: 17 pages, 4 figures

Generalising a classical theorem by Ueno, we prove structure results for manifolds with nef or semiample cotangent bundle., 9 pages, changed metadata

A bipartite graph on 2n vertices is bipancyclic if it contains cycles of all even lengths from 4 to 2n. In this paper we prove that the random bipartite graph $G(n,n,p)$ with $p(n)\gg n^{-2/3}$ asymptotically almost surely has the following resilience property: Every Hamiltonian subgraph $G'$ of $G(n,n,p)$ with more than $(1/2+o(1))n^2p$ edges is bipancyclic. This result is tight in two ways. First, the range of $p$ is essentially best possible. Second, the proportion 1/2 of edges cannot be reduced. Our result extends a classical theorem of Mitchem and Schmeichel., 14 pages, 3 figures. arXiv admin note: text overlap with arXiv:1005.5716 by other authors

International audience; Let $K$ be a field of characteristic $p>0$ and let $f(t_1,\ldots ,t_d)$ be a power series in $d$ variables with coefficients in $K$ that is algebraic over the field of multivariate rational functions $K(t_1,\ldots ,t_d)$. We prove a generalization of both Derksen's recent analogue of the Skolem--Mahler--Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices $(n_1,\ldots,n_d)\in \mathbb{N}^d$ for which the coefficient of $t_1^{n_1}\cdots t_d^{n_d}$ in $f(t_1,\ldots ,t_d)$ is zero is a $p$-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to $S$-unit equations and more generally to the Mordell--Lang Theorem over fields of positive characteristic.

We obtain several new characterizations of ultrametric spaces in terms of roundness, generalized roundness, strict p-negative type, and p-polygonal equalities (p > 0). This allows new insight into the isometric embedding of ultrametric spaces into Euclidean spaces. We also consider roundness properties additive metric spaces which are not ultrametric., 12 pages

We reconsider a classical theorem by Bican and El Bashir, which guarantees the existence of non-trivial relatively pure submodules in a module category over a ring with unit. Our aim is to generalize the theorem to module categories over rings with several objects. As an application we then consider the special case of alpha-pure objects in such module categories., 11 pages, corrected several typos, some references and explanations have been added, two errors in the proof of the main theorem have been corrected, the results remain unchanged

Let $G$ be a $k$-connected graph with $k\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree of the vertices other than $x$ and $z$. Further, with this result, we prove that: If $G$ has $n$ vertices and $m$ edges, then it contains a cycle of length at least $2m/(n-1)$ passing through its any $k-1$ specified vertices. Our results generalize a theorem of Fan on the existence of long paths and a classical theorem of Erd\"os and Gallai on the existence of long cycles under the average degree condition.

Aiming at a generalization of a classical theorem of Moebius, we study maps that take line intervals to plane curves, and also maps that take line intervals to conics from certain linear systems., 9 pages

We consider quasi-martingales indexed by a linearly order set. We show that such processes are isomorphic to a given class of (finitely additive) measures. From this result we easily derive the classical theorem of Stricker as well as the decompositions of Riesz, Rao and the supermartingale decomposition of Doob and Meyer.

In a recent work, Coronel et al. initiated the study of the relation between the diversity-multiplexing tradeoff (DMT) performance of a multiuser multiple-input multiple-output (MU-MIMO) lattice code and the rate of the decay of the determinants of the code matrix as a function of the size of the signal constellation. In this note, we state a simple general upper bound on the decay function and study the promising code proposed by Badr and Belfiore in close detail. We derive a lower bound to its decay function based on a classical theorem due to Liouville. The resulting bound is applicable also to other codes with constructions based on algebraic number theory. Further, we study an example sequence of small determinants within the Badr-Belfiore code and derive a tighter upper bound to its decay function. The upper bound has certain conjectural asymptotic uncertainties, whence we also list the exact bound for several finite data rates., 5 pages, submitted to ITW 2010, Cairo, Egypt

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of R\'{e}nyi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton--Watson process conditioned on nonextinction. The primary tools are an adaptation of Stein's method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory., Comment: Published in at http://dx.doi.org/10.1214/10-AOP559 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

If F is a global function field of characteristic p>3, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F., 28 pages, final version, few minor changes

A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf's result to topological manifolds, replacing vector fields with path fields. In this note, we give an equivariant analog of Brown's theorem for locally smooth $G$-manifolds where $G$ is a finite group., Comment: 17 pages, 6 figures

The study of the local behavior of nonlinear systems in the neighborhood of a periodic orbit is a classical problem in nonlinear dynamics. Most of our knowledge stems from simulations or the numerical integration of the variational equation. Only in the case of planar oscillators, the solution of the variational equation can be found analytically, provided that an explicit expression for the periodic trajectory is available. The aim of this paper is to extend a classical theorem due to S. P. Diliberto to higher dimensional systems. In doing so, we show how the fundamental matrix solution to the variational equation of higher order differential equations can be obtained in a closed analytical form. To obtain this result, the knowledge of the periodic trajectory is not sufficient anymore, and a specific set of orthogonal vectors has to be determined. The analysis of some examples reveals that finding these vectors may be easier than solving the variational equations.

International audience; In this paper, we prove some noncommutative analogues of Yano's classical extrapolation theorem. Applying one of them to noncommutative martingales, we obtain a maximal inequality for noncommutative martingales from Llog2 L to L1. Moreover, the exponent 2 is optimal. We also obtain the noncommutative analogue of the classical theorem of Burkholder and Chow on the iterations of two conditional expectations.

In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna Slice Theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pairs (GL(n+k,F), GL(n,F) x GL(k,F)) and (GL(n,E), GL(n,F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proven earlier by Jacquet and Rallis and the second by Flicker. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-Archimedean F the latter is a classical theorem of Gelfand and Kazhdan., A merge of arXiv:0803.3395 and arXiv:0803.3397 (with no additional material). Appendix D by Avraham Aizenbud, Dmitry Gourevitch and Eitan Sayag. v2,v3: minor changes. v4: correction in the speciality criterion (7.3.7). v5: minor correction in the proof of Proposition 7.2.1