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The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the question discussed in this paper. Estimate the degree $d$ of a polynomial in $n$ real variables, assumed to have non-negative coefficients and to be constant on a hyperplane, in terms of the number $N$ of its terms. No such estimate is possible when $n=1$. The sharp bound $d\le 2N-3$ is known when $n=2$. This paper includes two main results. The first provides a bound, not sharp for $n\ge 3$, for all $n\ge 2$. This bound implies the more easily stated bound $d\le {4(2N-3)\over 3(2n-3)}$ for $n\ge 3$. The second result is a stabilization theorem; if $n$ is sufficiently large given $d$, then the sharp bound $d \le {N-1 \over n-1}$ holds. In this situation we determine all polynomials for which the bound is sharp., 20 pages, minor corrections, accepted to Michigan Math. J

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their "regulator constants", and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations., 50 pages; minor corrections; final version, to appear in Invent. Math

An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues nu_j of the generator are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates nu_j. We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper's final section we present extensions of our results to more general chains., To appear in Journal of Theoretical Probability. Main change: addition of final section

Let $p$ be a prime, and let $n>0$ and $r$ be integers. In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\lfloor(n-1)/(p-1)\rfloor} \sum_{k=r(mod p)}\binom {n}{k}(-1)^k\in Z.$$ We determine $F_p(n,r)$ mod $p$ completely by certain number-theoretic and combinatorial methods; consequently, if $2\le n\le p$ then $$\sum_{k=1}^n(-1)^{pk-1}\binom{pn-1}{pk-1} \equiv(n-1)!B_{p-n}p^n (mod p^{n+1}),$$ where $B_0,B_1,...$ are Bernoulli numbers. We also establish the Kummer-type congruence $F_p(n+p^a(p-1),r)\equiv F_p(n,r) (mod p^a)$ for $a=1,2,3,...$, and reveal some connections between Fleck's quotients and class numbers of the quadratic fields $\Q(\sqrt{\pm p})$ and the $p$-th cyclotomic field $\Q(\zeta_p)$. In addition, generalized Fleck quotients are also studied in this paper., Comment: 28 pages

This paper surveys six notions of dynamical transitivity and mixing, in the context of group actions on topological spaces. We discuss the relations between these notions, and the manner in which they are inherited by subgroups, by taking products, and when passing to the induced action on hyperspace, i.e., the space of compact subsets. The focus of the paper is on the fact that certain standard notions, which are equivalent in the classical theory of the dynamics of flows and the iteration of single maps, are distinct for general group actions. The paper examines how the notions coalesce (a) for actions of abelian groups and (b) for chaotic actions. 0. Introduction. Consider an action of an infinite group G on a Hausdorff topological space M . This paper surveys six notions of dynamical transitivity and mixing for the action of G. We don’t assume any particular topology on G, but we assume that the action is “continuous” in the sense that, for each group element g, the corresponding map g : M → M is a homeomorphism. The fundamental transitivity and mixing notions are: Definition 1. The action of G on M is: (a) topologically transitive if, for every pair of nonempty open subsets U and V of M , there is an element g ∈ G such that gU ∩ V = ∅. (b) strongly topologically mixing if for any pair of nonempty open subsets U and V of M , the set {g ∈ G; gU ∩ V = ∅} is finite. (c) topologically k-transitive for k ∈ N, if the induced action of G on the k-fold Cartesian productM is topologically transitive. Topological 2-transitivity is also called weak topological mixing.

The aim of this paper is to prove that a pair of analytic sets X, Y $\subset$ Czm × Cwn is locally regularly separated with a uniform exponent α in the fibres taken over a proper projection π(z,w) = z of X ∩ Y (under the assumption that X ∩ Y has pure dimension): for all z $\in$ π (X ∩ Y) ∩ U, dist(w,Yz) ≥ const.dist(w,(X ∩ Y)z)α when w $\in$ Xz ∩ V, where U × V is a neighbourhood of a point a $\in$ X ∩ Y such that π(a) is regular in π(X ∩ Y). As an application of this we obtain a parameter version of the Łojasiewicz inequality for c-holomorphic mappings. Both results are a complex counterpart of the main result of [ŁW] from the subanalytic case, extended in this paper by a bound on the uniform exponent.

In the recent paper [Adv. Applied Math., 38 (2007), 210--226] it is proved that the special matchings of permutations generate a Coxeter group. In this paper we generalize this result to a class of Coxeter groups which includes many Weyl and affine Weyl groups. Our proofs are simpler, and shorter, than those in [loc. cit.]

In this paper the constrained vector optimization problem mic C f(x), g(x) ∃ − K, is considered, where $$f:\mathbb{R}^n \to \mathbb{R}^m $$ and $$g:\mathbb{R}^n \to \mathbb{R}^p $$ are locally Lipschitz functions and $$C \subset \mathbb{R}^m $$ and $$K \subset \mathbb{R}^p $$ are closed convex cones. Several solution concepts are recalled, among them the concept of a properly efficient point (p-minimizer) and an isolated minimizer (i-minimizer). On the base of certain first-order optimalitty conditions it is shown that there is a close relation between the solutions of the constrained problem and some unconstrained problem. This consideration allows to “double” the solution concepts of the given constrained problem, calling sense II optimality concepts for the constrained problem the respective solutions of the related unconstrained problem, retaining the name of sense I concepts for the originally defined optimality solutions. The paper investigates the stability properties of thep-minimizers andi-minimizers. It is shown, that thep-minimizers are stable under perturbations of the cones, while thei-minimizers are stable under perturbations both of the cones and the functions in the data set. Further, it is shown, that sense I concepts are stable under perturbations of the objective data, while sense II concepts are stable under perturbations both of the objective and the constraints. Finally, the so called structural stability is discused.

We estimate the cardinalities of antipodal sets in oriented real Grassmann manifolds of low ranks. The author reduced the classification of antipodal sets in oriented real Grassmann manifolds to a certain combinatorial problem in a previous paper. So we can reduce estimates of the antipodal sets to those of certain combinatorial objects. The sequences of antipodal sets we obtained in previous papers show that the estimates we obtained in this paper are the best.

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a self-contained and elementary proof of the analog for the groups G(r,p,n), with r>1, of Gordon's theorem (previously Haiman's conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p, Comment: 20 pages; in the 3rd version we have omitted some well-known material to make the paper shorter and included a proof of the analog of Gordon's theorem on the diagonal coinvariant ring for the group G(r,p,n) that avoids the KZ functor

This paper studies $t$-interleaving on two-dimensional tori. Interleaving has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. A $t$-interleaving of a graph is defined as a vertex coloring in which any connected subgraph of $t$ or fewer vertices has a distinct color at every vertex. We say that a torus can be perfectly $t$-interleaved if its $t$-interleaving number (the minimum number of colors needed for a $t$-interleaving) meets the sphere-packing lower bound, $\lceil t^2/2 \rceil$. We show that a torus is perfectly $t$-interleavable if and only if its dimensions are both multiples of $\frac{t^2+1}{2}$ (if $t$ is odd) or $t$ (if $t$ is even). The next natural question is how much bigger the $t$-interleaving number is for those tori that are not perfectly $t$-interleavable, and the most important contribution of this paper is to find an optimal interleaving for all sufficiently large tori, proving that when a torus is large enough in both dimensions, its $t$-interleaving number is at most just one more than the sphere-packing lower bound. We also obtain bounds on $t$-interleaving numbers for the cases where one or both dimensions are not large, thus completing a general characterization of $t$-interleaving numbers for two-dimensional tori. Each of our upper bounds is accompanied by an efficient $t$-interleaving scheme that constructively achieves the bound.

This paper has been withdrawn by the author due to its publication, This paper has been withdrawn

In this paper, we study various strongly convex hyper S-subposets of hyper S-posets in detail. To begin with, we consider the decomposition of hyper S-posets. A unique decomposition theorem for hyper S-posets is given based on strongly convex indecomposable hyper S-subposets. Furthermore, we discuss the properties of minimal and maximal strongly convex hyper S-subposets of hyper S-posets. In the sequel, the concept of hyper C-subposets of a hyper S-poset is introduced, and several related properties are investigated. In particular, we discuss the relationship between greatest strongly convex hyper S-subposets and hyper C-subposets of hyper S-posets. Moreover, we introduce the concept of bases of a hyper S-poset and give out the sufficient and necessary conditions of the existence of the greatest hyper C-subposets of a hyper S-poset by the properties of bases.

In this paper multioperations defined on a two-element set and their closure operator based on composition operator and the equality predicate branching operator is considered. The composition operator is based on union of sets. The classification of multioperations based on their membership in precomplete sets has been obtained. It is shown that the number of equivalence classes is 129. All types of bases are described and it is proved that the maximum cardinality of a basis is 4.

For any given finitely presented group G there exists a closed oriented 4-manifold with fundamental group G. What pairs of integers can occur as the signature and Euler characteristic of such a 4-manifold? The first part of this paper develops general theory related to this question and applies this theory to resolve the question for a large collection of groups, including surface groups and 3-manifold groups. The second part of the paper offers an extended analysis in the case of free abelian groups, G = Z^n. The paper concludes with a list of open problems related to the general question., Minor expository changes. 33 pages, 6 figures

In this paper, we decorate leaves and edges of planar rooted forests simultaneously and use a part of them to construct free nonunitary Rota-Baxter family algebras. As a corollary, we obtain the construction of free nonunitary Rota-Baxter algebras.

Let \(G\) be a graph and \(f:V (G)\rightarrow P(\{1,2\})\) be a function where for every vertex \(v\in V(G)\), with \(f(v)=\emptyset\) we have \(\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}\). Then \(f\) is a \(2\)-rainbow dominating function or a \(2RDF\) of \(G\). The weight of \(f\) is \(\omega(f)=\sum_{v\in V(G)} |f(v)|\). The minimum weight of all \(2\)-rainbow dominating functions is \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{r2}(G)\). Let \(G_1\) and \(G_2\) be two copies of a graph G with disjoint vertex sets \(V(G_1)\) and \(V(G_2)\), and let \(\sigma\) be a function from \(V(G_1)\) to \(V(G_2)\). We define the functigraph \(C(G,\sigma)\) to be the graph that has the vertex set \(V(C(G,\sigma)) = V(G_1)\cup V(G_2)\), and the edge set \(E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}\). In this paper, \(2\)-rainbow domination number of the functigraph of \(C(G,\sigma)\) and its complement are investigated. We obtain a general bound for \(\gamma_{r2}(C(G,\sigma))\) and we show that this bound is sharp.

The block number $Bl(M)$ introduced in our previous paper is a new topological invariant of a closed orientable 3-manifold $M$ which estimates a combinatorial complexity of $M$ just like the Heegaard genus $HG(M)$. In our previous paper, we have shown an inequality $HG(M) \leq Bl(M)$ for any $M \ne S^2 \times S^1$. In this paper, we will show that $Bl(M)=HG(M)$ for any $M$ with $HG(M)=2$ and moreover that $Bl(M) \leq 4$ for any $M$ with $HG(M)=3$.

Let ${\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, smooth curves. For a family $C\to S$ of such curves over a connected base and a geometric point $\xi$ on $S$, the associated monodromy representation is the induced homomorphism $\pi_1(S,\xi)\to\pi_1({\cal M}_{g,n},[C_\xi])$ on algebraic fundamental groups. It is well known that, if $S$ is irreducible, reduced and locally of finite type over a field $k$ of characteristic zero, the fibre $C_\xi$ and the corresponding monodromy representation determine the relative isomorphism class of the family. In the first part of the paper, it is shown that suitable quotients of this representation suffice. These results are then applied to show that the monodromy representation associated to a family $C\to S$ of $n$-pointed, genus $g$, stable curves of compact type, i.e. the induced homomorphism $\pi_1(S,\xi)\to\pi_1(\widetilde{\cal M}_{g,n},[C_\xi])$ (where, $\widetilde{\cal M}_{g,n}$ denotes the moduli stack of $n$-pointed, genus $g$, stable curves of compact type), characterizes trivial and isotrivial families. Let $U$ be an open subscheme of a normal, irreducible, locally noetherian scheme $S$ over a field $k$ of characteristic zero and let $C\ra U$ be a family of stable curves of compact type. In the second part of the paper, a monodromy criterion is given for extending $C\to U$ to a family of stable curves of compact type over $S$., Comment: 13 pages; revised version (substantial changes in Section 1); published on IMRN

This paper concerns the coefficients of the chromatic polynomial of a graph. We first report on a computational verification of the strict log-concavity conjecture for chromatic polynomials for all graphs on at most $11$ vertices, as well as for certain cubic graphs. ¶ In the second part of the paper we give a number of conjectures and theorems regarding the behavior of the coefficients of the chromatic polynomial, in part motivated by our computations. Here our focus is on $\varepsilon(G)$, the average size of a broken-cycle-free subgraph of the graph $G$, whose behavior under edge deletion and contraction is studied.

Berkovich, Chillag and Herzog characterized all finite groups $G$ in which all the nonlinear irreducible characters of $G$ have distinct degrees. In this paper we extend this result showing that a similar characterization holds for all finite solvable groups $G$ that contain a normal subgroup $N$, such that all the irreducible characters of $G$ that do not contain $N$ in their kernel have distinct degrees.

Let $r\ge 1$ be an integer and ${\bf U}:=\{U_n\}_{n\ge 0}$ be the Lucas sequence given by $U_0=0,~U_1=1$, and $U_{n+2}=rU_{n+1}+U_n$ for $n\ge 0$. In this paper, we explain how to find all the solutions of the Diophantine equation, $AU_{n}+BU_{m}=CU_{n_1}+DU_{m_1}$, in integers $r\ge 1$, $0\le m, Comment: 8 pages, Accepted Manuscript

Let $\Omega \subset {\mathbb{R}}^d $, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to{\mathbb{R}}$ be a non-negative subharmonic function. In this paper, we prove the inequality $$\begin{equation*} \frac{1}{|\Omega|}\int_{\Omega} f(x)\, \textrm{d}x \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\, \textrm{d}\sigma(x)\,. \end{equation*}$$Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset{\mathbb{R}}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then $$\begin{equation*} \|\nabla u\|_{L^\infty(\Omega)} < d\frac{|\Omega|}{|\partial\Omega|}\,. \end{equation*}$$Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al. [2].

Let $k$ be a positive integer. In this paper, we prove that if $\{k,k+1,c,d\}$ is a $D(-k)$-quadruple with $c>1$, then $d=1$., 15 pages

In this paper, we study variational solutions to parabolic equations of the type $$\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0$$ ∂ t u - div x ( D ξ f ( D u ) ) + D u g ( x , u ) = 0 , where u attains time-independent boundary values $$u_0$$ u 0 on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values $$u_0$$ u 0 admit a modulus of continuity $$\omega $$ ω and the estimate $$|u(x,t)-u_0(\gamma )| \le \omega (|x-\gamma |)$$ | u ( x , t ) - u 0 ( γ ) | ≤ ω ( | x - γ | ) holds, then u admits the same modulus of continuity in the spatial variable.

A classical theorem of Reinhold Baer shows that any group which is finite over its k-th centre has a finite (k + 1)-th term of its lower central series. Although the converse statement is false in general, Philip Hall proved that for any group G the finiteness of γk + 1(G) implies that the index $|G:\zeta_{2k}(G)|$ is finite. Similar situations have been investigated when finiteness is replaced by suitable weaker conditions. Moreover, it was proved by Yuri${\text{\u{\i}}}$ Merzljakov that Baer’s theorem and its potential converse do hold for linear groups. The aim of this paper is to obtain results of the latter type for several other finiteness conditions. Finally, although a result of Baer type does not hold for the class of soluble-by-finite reduced minimax groups, we prove that for this class a theorem of Hall type is true in arbitrary groups.

Given a finite order ideal $\mathcal O$ in the polynomial ring $K[x_1,\dots, x_n]$ over a field $K$, let $\partial \mathcal O$ be the border of $\mathcal O$ and $\mathcal P_{\mathcal O}$ the Pommaret basis of the ideal generated by the terms outside $\mathcal O$. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $\partial\mathcal O$-marked sets (resp. bases) and $\mathcal P_{\mathcal O}$-marked sets (resp. bases). We prove that a $\partial\mathcal O$-marked set $B$ is a marked basis if and only if the $\mathcal P_{\mathcal O}$-marked set $P$ contained in $B$ is a marked basis and generates the same ideal as $B$. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $\partial\mathcal O$-marked bases and $\mathcal P_{\mathcal O}$-marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gr\"obner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in smaller affine spaces. Furthermore, we observe that Pommaret marked schemes give an open covering of punctual Hilbert schemes. Several examples are given along all the paper., Comment: 17 pages; presentation improved, some references added

A large dam model is an object of study of this paper. The parameters $L^{lower}$ and $L^{upper}$ are its lower and upper levels, $L=L^{upper}-L^{lower}$ is large, and if a current level of water is between these bounds, then the dam is assumed to be in normal state. Passage one or other bound leads to damage. Let $J_1$ $(J_2)$ denote the damage cost of crossing the lower (upper) level. It is assumed that input stream of water is described by a Poisson process, while the output stream is state-dependent (the exact formulation of the problem is given in the paper). Let $L_t$ denote the dam level at time $t$, and let $p_1=\lim_{t\to\infty}\mathbf{P}\{L_t= L^{lower}\}$, $p_2=\lim_{t\to\infty}\mathbf{P}\{L_t> L^{upper}\}$ exist. The long-run average cost $J=p_1J_1+p_2J_2$ is a performance measure. The aim of the paper is to choose the parameter of output stream (exactly specified in the paper) minimizing $J$., To appear in "Journal of Applied Probability" 44 (2007), No.1

This paper proves that several interactive proof systems are zero-knowledge against quantum attacks. This includes a few well-known classical zero-knowledge proof systems as well as quantum interactive proof systems for the complexity class HVQSZK, which comprises all problems having "honest verifier" quantum statistical zero-knowledge proofs. It is also proved that zero-knowledge proofs for every language in NP exist that are secure against quantum attacks, assuming the existence of quantum computationally concealing commitment schemes. Previously no non-trivial proof systems were known to be zero-knowledge against quantum attacks, except in restricted settings such as the honest-verifier and common reference string models. This paper therefore establishes for the first time that true zero-knowledge is indeed possible in the presence of quantum information and computation., 21 pages

In this paper, we obtain a sufficient condition for the univalence of analyticfunctions in the open unit disk U. This condition involves two arbitrary functions g ( z )and h ( z ) analytic in U. Replacing g ( z ) and h ( z ) by some particular functions, we findthe well-known conditions for univalency established by Z. Nehari ( Bull. Amer. Math.Soc. 55 (1949)) and S. Ozaki and M. Nunokawa ( Proc. Amer. Math. Soc. 33 (1972)).Likewise we find other new sufficient conditions. Key words : univalent function, L¨owner chain, Nehari criterion, Ozaki criterion. 1. IntroductionWe denote by U r = {z ∈ C: |z| < r} the disk of z -plane, where r ∈ (0 , 1] , U 1 = Uand I = [0 , ∞ ). Let A be the class of functions f ( z ) whichare analytic in Uwith the normalizations f (0) = 0 and f 0 (0) = 1. Inthe present paper, we consider the following conditions for univalency offunctions f ( z ) belonging to the class A .Theorem 1.1 ([1]) Let f ( z ) ∈ A. If, for all z ∈ U , f ( z ) satisfies flfl {f ; z}

This paper is concerned with the tight closure of an ideal $I$ in a commutative Noetherian ring $R$ of prime characteristic $p$. The formal definition requires, on the face of things, an infinite number of checks to determine whether or not an element of $R$ belongs to the tight closure of $I$. The situation in this respect is much improved by Hochster's and Huneke's test elements for tight closure, which exist when $R$ is a reduced algebra of finite type over an excellent local ring of characteristic $p$. More recently, Hochster and Huneke have introduced the concept of test exponent for tight closure: existence of these test exponents would mean that one would have to perform just one single check to determine whether or not an element of $R$ belongs to the tight closure of $I$. However, to quote Hochster and Huneke, 'it is not at all clear whether to expect test exponents to exist; roughly speaking, test exponents exist if and only if tight closure commutes with localization'. The main purpose of this paper is to provide a short direct proof that test exponents exist for parameter ideals in a reduced excellent equidimensional local ring of characteristic $p$., This is to appear in the Michigan Mathematical Journal

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its proper subsystems does. We show that if $G$ is cyclic, or $G$ is finite and $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\geq m+f([G:\bigcap_{i=1}^kG_i])$, where $f(\prod_{t=1}^r p_t^{\alpha_t})=\sum_{t=1}^r\alpha_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $\alpha_1,...,\alpha_r$ are nonnegative integers. When all the $a_i$ are the identity element of $G$ and all the $G_i$ are subnormal in $G$, we prove that there is a composition series from $\bigcap_{i=1}^kG_i$ to $G$ whose factors are of prime orders. The paper also includes some other results and two challenging conjectures., Comment: 19 pages

This paper proves that two possible notions of Stein fillability for a contact structure are the same. A contact geometer typically says a contact manifold $(M,\xi)$(M,ξ) is Stein fillable if $M$M is the noncritical level set of a proper plurisubharmonic function $f\colon X\to [0,\infty)$f:X→[0,∞) and $\xi$ξ is the set of complex tangencies to the level set $M.$M. This is a very useful notion in contact geometry and implies strong geometric properties for the contact structure if $M$M is 3-dimensional. (In particular, the contact structure will be tight. Moreover, many tight contact structures are constructed by constructing a Stein filling of a 3-manifold.) There is another, "more intrinsic'', notion of Stein fillability. Suppose $X$X is a Stein manifold and $X$X is the interior of a closed manifold $\overline{X}$X−− such that $\overline{X}=M\cup X$X−−=M∪X and the complex structure on $X$X extends to one on $\overline{X}. $X−−. Then we can say the contact manifold $(M,\xi)$(M,ξ) is filled by $X$X, where $\xi$ξ is the set of complex tangencies to $M.$M. In this case the authors say that $M$M is the "intrinsic'' boundary of $X.$X. The main result of this paper shows the equivalence of these two notions. The first notion obviously implies the second, but the converse is non-obvious. Along the way the authors prove several results. In particular, they show that if $(M,\xi)$(M,ξ) is a 3-dimensional Stein fillable contact manifold then $M$M embeds in complex 4-space such that $\xi$ξ is the set of complex tangencies to the embedding. They also show that if $M$M is an intrinsic boundary of $X$X then $\overline{X}$X−− embeds as a domain in a larger open complex manifold with strictly pseudoconvex boundary $M.$M. However, the germ of this extension of $\overline{X}$X−− is not unique.

This paper deals with queues and insurance risk processes where a generic service time, resp. generic claim, has the form U ∧ K for some r.v. U with distribution B which is heavy-tailed, say Pareto or Weibull, and a typically large K, say much larger than $$\mathbb{E}U$$ . We study the compound Poisson ruin probability ψ(u) or, equivalently, the tail $$\mathbb{P}{\left( {W > u} \right)}$$ of the M/G/1 steady-state waiting time W. In the first part of the paper, we present numerical values of ψ(u) for different values of K by using the classical Siegmund algorithm as well as a more recent algorithm designed for heavy-tailed claims/service times, and compare the results to different approximations of ψ(u) in order to figure out the threshold between the light-tailed regime and the heavy-tailed regime. In the second part, we investigate the asymptotics as K → ∞ of the asymptotic exponential decay rate γ = γ (K) in a more general truncated Levy process setting, and give a discussion of some of the implications for the approximations.

Logarithmic signatures (LS) are a kind of factorization of finite groups which are used as a main component of cryptographic keys for secret key cryptosystems such as PGM and public key cryptosystems like MST1. As such, logarithmic signatures of short length are of special interest. In the present paper we deal with the fundamental question of the existence of logarithmic signatures of shortest length, called minimal logarithmic signatures (MLS), for finite groups. Studies of the problem can be found in several articles, especially in “On Minimal Length Factorizations of Finite Groups,” where González Vasco, Rö}tteler, and Steinwandt show that minimal logarithmic signatures exist for all groups of order less than 175,560 by direct computation using the method of factorization of a group into "disjoint'' subgroups. We introduce new approaches to deal with the question. The first method uses the double coset decomposition to construct minimal logarithmic signatures. This method allows one to prove, for instance, that if {\small $\gcd(n,q-1)\in \{1,4, p \ | \ p \ \mbox{prime} \}$}, then the projective special linear groups {\small $L_n(q)$} have an MLS. Another main goal of this paper is to construct MLS for all finite groups of order {\small $\leq 10^{10}$}. Surprisingly, the method of double coset decomposition turns out to be very effective, as we can construct MLS for all groups in the range except eight groups. We are also able to prove that if an MLS for any of these eight groups exists, then it cannot be constructed by the method of double coset decomposition. We further discuss a method of construction of MLS for groups of the form {\small $G=A.B$} with subgroups {\small $A$}, {\small $B$} and {\small $A\cap B \not= 1$}, by building suitable MLS for {\small $A$} and {\small $B$} and "gluing'' them together.

This paper studies the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n, together with 1/2. The subsemigroup of integers in this semigroup is called the wild integer semigroup, and the wild numbers are the irreducible elements in this subsemigroup. This paper presents evidence that the wild numbers consist of all the prime numbers p except 3. The subsemigroup of integers in the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n is called the Wooley integer semigroup and its irreducible elements are called Wooley numbers. This semigroup is shown to be recursive, and various open problems are formulated about Wooley integers., 11 pages, latex; revised with title change and some notation changes

This paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety We characterize a family of arcs which corresponds to a divisorial valuation by this map. We can see that both the Nash map and a certain McKay correspondence are the restrictions of this map. This paper also gives the affirmative answer to the Nash problem for a non-normal variety in a certain category. As a corollary, we get the affirmative answer for a non-normal toric variety., One wrong statement in Theorem 3.6 was deleted. This does not affect the rest of this paper

Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \fr {\p^2}{\p z^2_i}$ the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to the following what we call {\it vanishing conjecture}: for any homogeneous polynomial $P(z)$ of degree $d=4$, if $\Delta^m P^m(z)=0$ for all $m \geq 1$, then $\Delta^m P^{m+1}(z)=0$ when $m>>0$, or equivalently, $\Delta^m P^{m+1}(z)=0$ when $m> \fr 32 (3^{n-2}-1)$. It is also shown in this paper that the condition $\Delta^m P^m(z)=0$ ($m \geq 1$) above is equivalent to the condition that $P(z)$ is Hessian nilpotent, i.e. the Hessian matrix $\Hes P(z)=(\fr {\p^2 P}{\p z_i\p z_j})$ is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen \cite{BE1} and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived., Comment: Latex, 34 pages

A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that correspond to the product action of a wreath product. Previously we identified six classes of Cartesian decompositions that can be acted upon transitively by an innately transitive group with a non-abelian plinth. The inclusions studied in this paper correspond to three of the six classes. We find that in each case the isomorphism type of the acting group is restricted, and some interesting combinatorial structures are left invariant. We also show how to construct examples of inclusions for each type., v1 is replaced after minor alterations not concerning maths content

A \emph{queue layout} of a graph consists of a total order of the vertices, and a partition of the edges into \emph{queues}, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line grid) drawing} of a graph represents the vertices by points in $\mathbb{Z}^3$ and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph $G$ is closely related to the queue-number of $G$. In particular, if $G$ is an $n$-vertex member of a proper minor-closed family of graphs (such as a planar graph), then $G$ has a $O(1)\times O(1)\times O(n)$ drawing if and only if $G$ has O(1) queue-number. (2) It is proved that queue-number is bounded by tree-width, thus resolving an open problem due to Ganley and Heath (2001), and disproving a conjecture of Pemmaraju (1992). This result provides renewed hope for the positive resolution of a number of open problems in the theory of queue layouts. (3) It is proved that graphs of bounded tree-width have three-dimensional drawings with O(n) volume. This is the most general family of graphs known to admit three-dimensional drawings with O(n) volume. The proofs depend upon our results regarding \emph{track layouts} and \emph{tree-partitions} of graphs, which may be of independent interest., This is a revised version of a journal paper submitted in October 2002. This paper incorporates the following conference papers: (1) Dujmovic', Morin & Wood. Path-width and three-dimensional straight-line grid drawings of graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts, tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS 2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of $k$-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 2003

Let A be a finite-dimensional, power-associative algebra over a field F, either R or C, and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0 not equal a is an element of S, and f(alpha a) = |alpha| f(a) for all a is an element of S and alpha is an element of F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant sigma > 0 so that f(a(m)) less than or equal to sigma f(a)(m) for all a is an element of S and m = 1, 2, 3.... The purpose of this paper is to provide an updated account of our study of stable subnorms on subsets of finite-dimensional, power-associative algebras over F. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets.

Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method to study chains of minimal rational curves via varieties of minimal rational tangents and their higher secants. For many examples of Fano manifolds this method can be used to compute the minimal length of chains needed to join two general points. One consequence of our computation is a bound on the multiplicities of divisors at a general point of the moduli of stable bundles of rank two on a curve., This is the final version of the paper, to appear in Crelle's Journal. An optimized PDF file with additional graphics is available at http://www.mi.uni-koeln.de/~kebekus/publications-e.html

A collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behaviour of a square two–dimensional lattice of such resistors when both types coexist in fixed proportions in the lattice. The macroscopic behaviour is that of an anisotropic conductor at the continuum level and the goal of the paper is to describe the set of all possible such conductors. This is thus a problem of bounds, following in the footsteps of an abundant literature on the topic in the continuum case. The originality of the paper is that the investigation focuses on the interplay between homogenization and the passage from a discrete network to a continuum. A set of bounds is proposed and its optimality is shown when the proportion of each resistor on the discrete lattice is½. We conjecture that the derived bounds are optimal for all proportions.

Let ( X , T ) be a topological dynamical system and let $\mu$ be a T -invariant probability measure on X . In this paper, we study two properties of the notions of measure theoretical entropy for a measurable cover $\mathcal{U},\ h_\mu^+(\mathcal{U},T)$ and $h_\mu^-(\mathcal{U},T)$ introduced by P. P. Romagnoli ( Ergod. Th. & Dynam. Sys . 23 (2003), 1601–1610). The main result of the paper states that entropy pairs for the measure $\mu$ can be defined using either $h_\mu^+$ or $h_\mu^-$ . We also prove that both $h_\mu^+$ and $h_\mu^-$ have an ergodic decomposition and we use it to prove a local Abramov formula for $h_\mu^-$ .

Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a mutant of 4-genus n. A second result of this paper is a crossing change formula for the algebraic concordance class of a knot, which is then applied to prove the invariance of the algebraic concordance class under mutation. The paper concludes with an application of crossing change formulas to give a short new proof of Long's theorem that strongly positive amphicheiral knots are algebraically slice., 16 pages, 6 figures

Let X \subset Proj(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope \Delta(X). The polytope \Delta(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope \Delta(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n,C) is shown to hold for other classical groups., Comment: 17 pages, LaTex, uses the package xy

We investigate the worst-case behavior of the simplex algorithm on linear programs with three variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we consider, the ``random edge'' rule yields the best asymptotic behavior as well as the most complicated analysis. All other rules turn out to be much easier to study, but also produce worse results: Most of them show essentially worst-possible behavior; this includes both Kalai's ``random-facet'' rule, which without dimension restriction is known to be subexponential, as well as Zadeh's deterministic history dependent rule, for which no non-polynomial instances in general dimensions have been found so far., 24 pages, to appear in: SIAM J. Comp.; the paper comprises the contents of our paper 'The Random Edge Rule on Three-Dimensional Linear Programs' (math.CO/0301076)

Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that S^2 V^* \otimes L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r > 0. The main result of this paper is that if d > N-r, then the locus of points where the rank of the quadratic form is at most r-1 is nonempty. We give some applications to subschemes of matrices, and to degeneracy loci associated to embeddings in projective space. The paper concludes with an appendix on Gysin maps. The main result of the appendix identifies a Gysin map with the natural map from ordinary to relative cohomology., 31 pages

The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in $\r^3$ passing through one point. It turns out that to be rectifiable all circles need to pass through some other common point. The second main result is a complete description of geometries in $\r^3$ in which all the geodesics are circles. This is a consequence of an extension of Beltrami's theorem by replacing straight lines with circles.}, 18 pages

In this paper, we consider Meyer's function of hyperelliptic mapping class groups of orientable closed surfaces and give certain explicit formulae for it. Moreover we study geometric aspects of Meyer's function, and relate it to the h- invariant of the signature operator and Morita's homomorphism, which is the core of the Casson invariant of integral homology 3-spheres. where G n denotes the mapping class group of S 2 leaving n points invariant. As is known, Dg ¼ Gg if g ¼ 1;2 and Dg 0Gg for gb3. The group Dg is called the hyperelliptic mapping class group. Our main object in the present paper is Meyer's signature cocycle (Me), which is a group 2-cocycle of the Siegel modular group Spð2g;ZÞ. Topologi- cally, this presents the signature of total spaces of surface bundles over a surface. Since the group Dg is acyclic over Q (cf. (C), (K)), the restriction of the pull-back of the signature cocycle via the classical representation