Codes from incidence matrices of graphs

We examine the p-ary codes, for any prime p, from the row span over ${\mathbb {F}_p}$ of |V| × |E| incidence matrices of connected graphs Γ = (V, E), showing that certain properties of the codes can be directly derived from the parameters and properties of the graphs. Using the edge-connectivity of Γ (defined as the minimum number of edges whose removal renders Γ disconnected) we show that, subject to various conditions, the codes from such matrices for a wide range of classes of connected graphs have the property of having dimension |V| or |V| ? 1, minimum weight the minimum degree δ(Γ), and the minimum words the scalar multiples of the rows of the incidence matrix of this weight. We also show that, in the k-regular case, there is a gap in the weight enumerator between k and 2k ? 2 of the binary code, and also for the p-ary code, for any prime p, if Γ is bipartite. We examine also the implications for the binary codes from adjacency matrices of line graphs. Finally we show that the codes of many of these classes of graphs can be used for permutation decoding for full error correction with any information set.     

Strong polynomiality of the Gass-Saaty shadow-vertex pivoting rule for controlled random walks

We consider the subclass of linear programs that formulate Markov Decision Processes (mdps). We show that the Simplex algorithm with the Gass-Saaty shadow-vertex pivoting rule is strongly polynomial for a subclass of mdps, called controlled random walks (CRWs); the running time is O(|S|³?|U|²), where |S| denotes the number of states and |U| denotes the number of actions per state. This result improves the running time of Zadorojniy et al. (Mathematics of Operations Research 34(4):992?C1007, 2009) algorithm by a factor of |S|. In particular, the number of iterations needed by the Simplex algorithm for CRWs is linear in the number of states and does not depend on the discount factor.     

Hermitian-einstein metrics on parabolic stable bundles

Let $\overline M $ be a compact complex manifold of complex dimension two with a smooth Kähler metric and D a smooth divisor on $\overline M $ . If E is a rank 2 holomorphic vector bundle on $\overline M $ with a stable parabolic structure along D, we prove the existence of a metric on $E'{\text{ = }}E|_{\overline M \backslash D} $ (compatible with the parabolic structure) which is Hermitian-Einstein with respect to the restriction of the Kähler metric to $\overline M $ ěD. A converse is also proved.     

Das Existenzproblem multivalenter Lösungen bei Differentialgleichungen

We consider linear differential equations $$w^{(n)} + \sum\limits_{i = 0}^{n - 1} {\sigma _i w^{(i)} = 0 in |z|}< 1.$$ If the coefficients σi inH ^∞ the solutions of theis equations have only a finite number of zeros and therefore these solutions are multivalent in |z|<1.     

Small Divisor Problem in Dynamical Systems and Analytic Solutions of a q-Difference Equation with a Singularity at the Origin

In this paper, we consider a q-difference equation $$\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(z)(y(q^jz))^{t}=G(z)$$ in the complex field ${\mathbb C,}$ where C _t,j (z) and G(z) have a h ₁ order pole and a h ₂ order pole at z = 0, respectively. Under the case 0 < |q| < 1 or |q| = 1, we give the existence of local analytic solutions for the above equation by using small divisor theory in dynamical systems.     

Critical behavior and the limit distribution for long-range oriented percolation. I

We consider oriented percolation on ${\mathbb{Z}}^d\times{\mathbb{Z}}_+$ whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on ${\mathbb{Z}}^d$ . Suppose that D(x) decays as |x|^?d?α for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension $d_c=2(\alpha\wedge2)$ . We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to $e^{-c|k|^{\alpha\wedge2}}$ for some c > 0.     

On polygonal measures with vanishing harmonic moments

A polygonal measure is the sum of finitely many real constant density measures supported on triangles in ?. Given a finite set S ? ?, we study the existence of polygonal measures spanned by triangles with vertices in S, all of whose harmonic moments vanish. We show that for generic S, the dimension of the linear space of such measures is \(\left( {_2^{|S| - 3} } \right)\) . We also investigate the situation in which the density for such measure takes on only values 0 or ±1. This corresponds to pairs of polygons of unit density having the same logarithmic potential at ∞. We show that such (signed) measures do not exist for |S| ≤ 5, but that for each n ≥ 6 one can construct an S, with |S| = n, giving rise to such a measure.     

Resonance-free region in scattering by a strictly convex obstacle

We prove the existence of a resonance-free region in scattering by a strictly convex obstacle \(\mathcal{O}\) with the Robin boundary condition \(\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0\) . More precisely, we show that the scattering resonances lie below a cubic curve ?ζ=?S|ζ|^1/3+C. The constant S is the same as in the case of the Neumann boundary condition γ=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.     

Lower bounds on the number of maximal subgroups in a finite group

For a finite group G, let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. When G is a cyclic group, an elementary calculation proves that |m(G)| = |π(G)|. In this paper, we prove lower bounds on |m(G)| when G is not cyclic. In general, ${|m(G)| \geq |\pi(G)|+p}$ | m ( G ) | ≥ | π ( G ) | + p , where ${p \in \pi(G)}$ p ∈ π ( G ) is the smallest prime that divides |G|. If G has a noncyclic Sylow subgroup and ${q \in \pi(G)}$ q ∈ π ( G ) is the smallest prime such that ${Q \in {\rm syl}_q(G)}$ Q ∈ syl q ( G ) is noncyclic, then ${|m(G)| \geq |\pi(G)|+q}$ | m ( G ) | ≥ | π ( G ) | + q . Both lower bounds are best possible.     

A remark on Fano 4-folds having (3,1)-type extremal contractions

Let π : X → Y be the blow-up of a four-dimensional complex manifold Y along a smooth curve C. Assume that X is a Fano manifold and has another (3,1)-type extremal contraction ${\varphi : X \to Z}$ whose exceptional divisor meet that of the blow-up π : X → Y. We show that if the exceptional divisor of ${\varphi}$ is smooth, then Y is isomorphic to four-dimensional projective space and C is an elliptic curve of degree 4.     

The Monotone Catenary Degree of Krull Monoids

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree c _mon (H) of H is the smallest integer m with the following property: for each ${a \in H}$ and each two factorizations z, z′ of a with length |z| ≤  |z′|, there exist factorizations z = z ₀, ... ,z _k  = z′ of a with increasing lengths—that is, |z ₀| ≤  ... ≤  |z _k |—such that, for each ${i \in [1,k]}$ , z _i arises from z _i-1 by replacing at most m atoms from z _i-1 by at most m new atoms. Up to now there was only an abstract finiteness result for c _mon (H), but the present paper offers the first explicit upper and lower bounds for c _mon (H) in terms of the group invariants of G.     

On the fastest moving off from a vertex in directed regular graphs

Let Γ be a directed regular locally finite graph, and let $\bar \Gamma $ be the undirected graph obtained by forgetting the orientation of Γ. Let x be a vertex of Γ and let n be a nonnegative integer. We study the length of the shortest directed path in Γ starting at x and ending outside of the ball of radius n centered at x in $\bar \Gamma $ .     

Exactn-widths of hardy-sobolev classes

Let $\tilde h^r _{\infty ,\beta } $ and $\tilde H^r _{\infty ,\beta } $ denote those 2π-periodic, real-valued functions onR that are analytic in the strip |Imz|<β and satisfy the restrictions |Ref ^(r)(z)| ≤ 1 and |f ^(r)(z)| ≤ 1, respectively. We determine the Kolmogorov, linear, and Gel’fand widths of $\tilde h^r _{\infty ,\beta } $ inL _q[0, 2π], 1 ≤q ≤ ∞, and $\tilde H^r _{\infty ,\beta } $ inL _∞[0, 2π].     

Minimum degree of bipartite graphs and the existence ofk-factors

LetG be a bipartite graph with bipartition (X, Y) andk a positive integer. If (i) $$\left| X \right| = \left| Y \right|,$$ (ii) $$\delta (G) \geqslant \left\lceil {\frac{{\left| X \right|}}{2}} \right\rceil \geqslant k,$$ \(\left| X \right| \geqslant 4k - 4\sqrt k + 1\) when |X| is odd and |X| ≥ 4k ? 2 when |X| is even, thenG has ak-factor.     

An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

Given ${\Gamma \subset \mathbb{Q}^*}$ a multiplicative subgroup and ${m \in \mathbb{N}^+}$ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which ind _p Γ = m, where ind _p Γ = (p ? 1)/|Γ _p | and Γ _p is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi–Pappalardi, Lenstra, Moree, Murata, Wagstaff, and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations.     

Strongly dependent theories

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ? ? and |I|≥?_|T|+(µ), then some J ? I of cardinality µ⁺ is an indiscernible sequence over A.     

On families of complex lines sufficient for holomorphic extension

It is shown that the set $ \mathfrak{L}_\Gamma $ of all complex lines passing through a germ of a generating manifold Γ is sufficient for any continuous function f defined on the boundary of a bounded domain D ? ? ⁿ with connected smooth boundary and having the holomorphic one-dimensional extension property along all lines from $ \mathfrak{L}_\Gamma $ to admit a holomorphic extension to D as a function of many complex variables.