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Following a recent paper by Faltings, we study the integral points on $\bm{P}_2\setminus\mathcal{D}$ , where $\mathcal{D}$ is the branch locus of a projection from a surface $\mathcal{X}$ ; a crucial point in the analysis is that the pull-back of $\mathcal{D}$ in the Galois closure of the projection often splits into several components. As in the paper by Faltings, under certain assumptions we obtain finiteness of the integral points (Theorem 3.1); for instance, we shall find that it suffices if the projection is sufficiently general and if $\mathcal{X}$ has Kodaira number $\ge0$ (Corollary 4.1). We have borrowed freely from Faltings’s paper, for the whole geometrical setting. As to the arithmetic, our method is in part different, relying on the recent paper by Corvaja and Zannier and leading to apparently new conditions. We shall also use a more elementary approach to study a similar situation in arbitrary dimension, where the projection is taken from a hypersurface (Theorem 2.1). In concrete terms, these results deal with certain diophantine equations $F(x_0,\dots,x_n)=c$ . AMS 2000 Mathematics subject classification: Primary 11D72; 11G35; 11G99

The goal of this paper is to study two related problems. The first is a certain maximality property of the Shimura subgroup among multiplicative-type subgroups of the Jacobian J0(N ). The second is a conjecture of Stevens, which characterizes the unique curve of minimal Faltings-Parshin height in an isogeny class of elliptic curves over Q. As a consequence, we obtain a number of arithmetic applications to elliptic curves with positive μ-invariant, and to the existence of canonical parametrizations by modular curves. To explain the first object of our study, we let X1(N ) = 1(N )\H ∗ and X0(N ) = 0(N )\H ∗ denote the usual modular curves. We fix models for these curves over Q, for example, the canonical models of Shimura. Since 0(N ) ⊂ 1(N ), there is a finite map π : X1(N ) → X0(N ) given over the complex numbers by the natural projection 1(N )\H ∗ → 0(N )\H ∗ . In fact, π is defined over Q, and gives a finite map of the canonical Q-models. By Picard functoriality, we deduce a morphism of Jacobians π∗ : J0(N ) → J1(N ). The kernel of π∗ is a finite subgroup V of J0(N )(Q), known as the Shimura subgroup. Since π∗ is a Q-rational map, the subgroup V is stable under the action of Gal(Q/Q). It is known that V is of multiplicative type, meaning that the Cartier dual W = Hom(V ,Gm ) is trivial for the action of Galois. Furthermore, it is known that V is Eisenstein for the action of the Hecke algebra T, in the sense that it is annihilated by the operators Tp − p − 1, for p N . For this, and other details about the structure of V (including a formula for the order) we refer the reader to the paper [LO91].

Here, we obtain some bounds on the number of solutions of some congruences and some exponential sums. We present them in forms more general that we need in the paper, but which we believe to be of independent interest. The implied constants in symbols 'O' and ' 0). In particular, they do not depend on the auxiliary parameter i which occurs in our estimates. Received 25th October, 2004 During the preparation of this paper, the second author was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and the third author was supported in part by ARC grant DP0211459. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/05 SA2.00+0.00.

The purpose of the paper is to show that weak order convergence of a net of Dedekind complete Riesz space-valued σ-measures is uniform over uniformly bounded, uniformly equicontinuous classes of functions. The paper ends with generalizing Ulam's theorem for tightness of positive, finite Borel measures to Riesz space-valued σ-measures.

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

Let Mn be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere n+1, then Mn is associated with a so-called Möbius metric g, a Möbius second fundamental form B and a Möbius form Φ which are invariants of Mn under the Möbius transformation group of n+1. A classical theorem of Möbius geometry states that Mn (n ≥ 3) is in fact characterized by g and B up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hyper-surfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.In this paper, we prove that a Möbius isoparametric hypersurface in 4 is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of ℝP2 into 4. The classification of hypersurfaces in n+1 (n ≥ 2) with parallel Möbius second fundamental form has been accomplished in our previous paper [6]. The present result is a counterpart of Pinkall’s classification for Dupin hypersurfaces in 4 up to Lie equivalence.

In this paper we consider inequalities of the form, WhereMis a mean. The main results of the paper offer sufficient conditions onMso that the above inequality holds with a finite constantC. The results obtained extend Hardy's and Carleman's classical inequalities together with their various generalisations in a new dirction.

In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space $X$ admits a Lyapunov one-form $\omega$ lying in a prescribed \v{C}ech cohomology class $\xi\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions., Comment: 27 pages, 3 figures. This revised version incorporates a few minor improvements

The notion of an atomic operator between spaces of measurable functions was introduced in 2002 in a paper by Drakhlin, Ponosov and Stepanov in order to provide a reasonable generalization of local operators useful for applications. It has been shown that, roughly speaking, atomic operators amount to compositions of local operators with shifts. A natural problem is then when a continuous-in-measure atomic operator can be represented as a composition of a Nemytskiiˇ (composition) operator generated by a Carathéodory function, and a shift operator. In this paper we will show that the answer to this question is inherently related to the possibility of extending an atomic operator with continuity from a space of functions measurable with respect to some $\sigma$-algebra to a larger space of functions measurable with respect to a larger $\sigma$-algebra, as well as to the possibility of extending any $\sigma$-homomorphism from a smaller-measure algebra to a $\sigma$-homomorphism on a larger-measure algebra. We characterize precisely the condition on the respective $\sigma$-algebras which provides such possibilities and induces the positive answer to the above representation problem.AMS 2000 Mathematics subject classification: Primary 47B38; 47A67; 34K05

In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s.Questions about precisely when one shape fits into another attract wide attention. In 1993 Post [1] gave necessary and sufficient conditions on the six sides of two triangles for the first to fit into the second. Recently, the necessary and sufficient conditions for squares to fit in triangles [2], equilateral triangles in triangles [3], rectangles in triangles [4] and rectangles in rectangles [5] are given. In [2] Wetzel asked when a given triangle fits into a given square. In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s. For the sake of convenience, let α, β, γ denote the angles opposite sides a, b, c respectively, and we may assume without loss of generality that a ≥ b ≥ c, which implies that α ≥ β ≥ γ.

This paper studies a discrete-time single-server queue with two independent inputs and service interruptions. One of the inputs to the queue is an independent and identically distributed process. The other is a much more general process and it is not required to be Markov nor is it required to be stationary. The service interruption process is also general and it is not required to be Markov or to be stationary. This paper shows that a stochastic decomposition property for the virtual waiting-time process holds in the discrete-time single-server queue with service interruptions. To the best of the author's knowledge, no stochastic decomposition results for virtual waiting-time processes in non-work-conserving queues, such as queues with service interruptions, have been obtained before and only work-conserving queues have been studied in the literature.

For any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.

In this paper, we study the on-line parameter estimation problem for a partially observable system subject to deterioration and random failure. The state of the system evolves according to a continuous-time homogeneous Markov process with a finite state space. The state of the system is hidden except for the failure state. When the system is operating, only the information obtained by condition monitoring, which is related to the working state of the system, is available. The condition monitoring observations are assumed to be in continuous range, so that no discretization is required. A recursive maximum likelihood (RML) algorithm is proposed for the on-line parameter estimation of the model. The new RML algorithm proposed in the paper is superior to other RML algorithms in the literature in that no projection is needed and no calculation of the gradient on the surface of the constraint manifolds is required. A numerical example is provided to illustrate the algorithm.

In this paper we review some recent work on limit results on realised power variation, that is, sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, quadratic variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.

The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau(\a^t)$ associated to a given ideal $\a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper, which gives a characterization of the ideal $\tau(\a^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $\tau(R)$. Moreover, we prove an analog of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Brian\c{c}on--Skoda theorem.", Comment: 11 pages, AMS-LaTeX; v.2: minor changes, to appear in Nagoya Math. J

We correct two technical errors in the original paper. The main result in the original paper remains valid without any changes.

This paper is concerned with the well-known and long-standingk(G V)-problem: If the finite groupGacts faithfully and irreducibly on the finiteGF(p)-moduleVandpdoes not divide the order ofG, is the numberk(GV) of conjugacy classes of the semidirect productGVbounded above by the order ofV? Over the past two decades, through the work of numerous people, by using deep character theoretic arguments this question has been answered in the affirmative except for ρ = 5 for which it is still open. In this paper we suggest a new approach to thek(G V)-problem which is independent of most of the previous work on the problem and which is mainly group theoretical. To demonstrate the potential of the new line of attack we use it to solve thek(G V)-problem for solvableGand large ρ.

Let S be a semigroup contained in a locally compact Abelian group G. Let Ĝ denote the Bohr compactification of G. We say that a sequence contained in S is Hartman uniform distributed on G iffor any character χ in Ĝ. Suppose that (Tg)g∈s is a semigroup of measurable measure preserving transformations of a probability space (X, β, μ) and B is an element of the σ-algebra β of positive μ measure. For a map T: X → X and a set A ⊆ X let T−1A denote {x ∈ X: Tx ∈ A}. In an earlier paper, the author showed that if k is Hartman uniform distributed thenIn this paper we show that ≥ cannot be replaced by =. A more detailed discussion of this situation ensues.

This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing.In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility.In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied.Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary.A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility.In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion.Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.

Suppose that {X s , 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝ m → [0,∞) is a (weighted) l q (ℝ m )-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the L p (μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(X s ), 0 ≤ s ≤ T. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.

In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron–Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.

Let Un{f) = / fd(Fn — F) be the function-indexed Kaplan-Meier integral process constructed from the random censorship model. We study a uniform version of the law of large numbers of Glivenko-Cantelli type for {Un} under the bracketing entropy condition. The main result is that the almost sure convergence and convergence in the mean of the process Un holds uniformly in T. In proving the result we shall employ the bracketing method which is used in the proof of the uniform law of large numbers for the complete data of the independent and identically distributed model. In this paper, we obtain a uniform version of the law of large numbers of GlivenkoCantelli type for the function-indexed Kaplan-Meier integral process based on the incomplete data of the random censorship model. In obtaining the uniform law of large numbers, we observe the role played by bracketing of the indexed class of functions of the process and slightly modify the underlying metric. Then we employ the idea of DeHardt [1] of the bracketing method to the Kaplan-Meier integral process. The uniform law of large numbers of the present paper extends the one dimensional law of large numbers for random censoring that was established by Stute and Wang [4] and the DeHardt's uniform law of large numbers for independent and identically distributed random variables [1]. Among others our results are stated not only for almost sure convergence but also for convergence in the mean. The results may be used in nonparametric statistical inference in verifying uniform consistency. See Van de Geer [6] for applications. We begin by introducing the integral version of the usual empirical process based

For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.

In the past few years meshless methods for numerically solving partial differential equations have come into the focus of interest, especially in the engineering community. This class of methods was essentially stimulated by difficulties related to mesh generation. Mesh generation is delicate in many situations, for instance, when the domain has complicated geometry; when the mesh changes with time, as in crack propagation, and remeshing is required at each time step; when a Lagrangian formulation is employed, especially with nonlinear PDEs. In addition, the need for flexibility in the selection of approximating functions (e.g., the flexibility to use non-polynomial approximating functions), has played a significant role in the development of meshless methods. There are many recent papers, and two books, on meshless methods; most of them are of an engineering character, without any mathematical analysis.In this paper we address meshless methods and the closely related generalized finite element methods for solving linear elliptic equations, using variational principles. We give a unified mathematical theory with proofs, briefly address implementational aspects, present illustrative numerical examples, and provide a list of references to the current literature.The aim of the paper is to provide a survey of a part of this new field, with emphasis on mathematics. We present proofs of essential theorems because we feel these proofs are essential for the understanding of the mathematical aspects of meshless methods, which has approximation theory as a major ingredient. As always, any new field is stimulated by and related to older ideas. This will be visible in our paper.

Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

A continuous linear operator on a complex Banach space is said to be paranormal if ‖Tx‖2 ≤ ‖T2x‖ ‖x‖ for all x ∈ X. T is called totally paranormal if T–λ is paranormal for every λ ∈ C. In this paper we investigate the class of totally paranormal operators. We shall see that Weyl's theorem holds for operators in this class. We also show that for totally paranormal operators the Weyl spectrum satisfies the spectral mapping theorem. In Section 5 of this paper we investigate the operator equations eT = eS and eTeS = eSeT for totally paranormal operators T and S.

In a recent paper [3] Dales and Pandey have shown that the class Sp of Segal algebras is weakly amenable. In this paper, for various classes of Segal algebras, we characterize derivations and multipliers from a Segal algebra into itself and into its dual module. In particular, we prove that every Segal algebra on a locally compact abelian group is weakly amenable and an abstract Segal subalgebra of a commutative weakly amenable Banach algebra is weakly amenable. We also introduce the Lebesgue–Fourier algebra of a locally compact group G and study its Arens regularity when G is discrete or compact.

A number r > 0 is called a rendezvous number for a metric space (M, d) if for any n ∈ ℕ and any x1,…xn ∈ M, there exists x ∈ M such that . A rendezvous number for a normed space X is a rendezvous number for its unit sphere. A surprising theorem due to O. Gross states that every finite dimensional normed space has one and only one average number, denoted by r (X). In a recent paper, A. Hinrichs solves a conjecture raised by R. Wolf. He proves that for any n-dimensional real normed space. In this paper, we prove the analogous inequality in the complex case for n ≥ 3.

The main result of this paper is a sufficient condition for the minimax relation to hold for the canonical bilinear form on X × Y, where X is a nonempty convex subset of a real locally convex space and Y is a nonempty convex subset of its dual. Using the known “converse minimax theorem”, this result leads easily to a nonlinear generalisation of James's (“sup”) theorem. We give a brief discussion of the connections with the “sup-limsup theorem” and, in the appendix to the paper, we give a simple, direct proof (using Goldstine's theorem) of the converse minimax theorem referred to above, valid for the special case of a normed space.

In this paper, we establish the existence of infinitely many solutions to a Neumann problem involving the p-Laplacian and with discontinuous nonlinearities. The technical approach is mainly based on a very recent result on critical points for possibly non-smooth functionals in a Banach space due to Marano and Motreanu, namely Theorem 1.1 in a paper that is to appear in the journal J. Diff. Eqns (see Theorem 2.3 in the body of this paper). Some applications are presented. Throughout the sequel, Ω is a non-empty bounded open set of the real Euclidean space R n , n 3, with boundary of class C ∞ , a belongs to L ∞ (Ω), with ess inf Ω a> 0; α(x), β(x) lie in L 1 (Ω) with min{α(x) ,β (x)} 0 a.e. in Ω, and p ∈ )n, +∞(. Consider the

In this article, we prove the existence and uniqueness of solution for the Cauchy problem of the Landau- Lifshitz equation of ferromagnetism with external magnetic field. We also show that the solution is globally regular with the exception of at most finitely many blow-up points. An energy identity at blow-up points is presented. In this paper we discuss the Landau-Lifshitz equation (10) which models the fer- romagnetic spin chain on surfaces with the Gilbert damping term in the presence of an external magnetic field h.x; t/. This equation plays an important rin the understanding of non-equilibrium magnetism. Let and be closed oriented Riemannian surfaces with metric tensors D ./1;2 and g D.g ij /1i; j2 respectively. Let be isometrically embedded in 3 and denote its second fundamental form by A. We shall assume that the principal curvatures of and their derivatives are uniformly bounded. Throughout this paper, C will denote constants which depend only on and . The notation '' for the dot product of vectors will be suppressed. For u0 V ! and t > 0, the Landau-Lifshitz system is given by

In this paper, we consider denumerable-state continuous-time Markov decision processes with (possibly unbounded) transition and reward rates and general action space under the discounted criterion. We provide a set of conditions weaker than those previously known and then prove the existence of optimal stationary policies within the class of all possibly randomized Markov policies. Moreover, the results in this paper are illustrated by considering the birth-and-death processes with controlled immigration in which the conditions in this paper are satisfied, whereas the earlier conditions fail to hold.

The object of this paper is to find all the irreducible algebraic surfaces which (for special values of the parameters b, r, s) are invariant under the Lorenz systemx˙ = X(x, y, z) = s(y−x), y˙ = Y(x, y, z) = rx−y−xz, ż = Z(x, y, z) =−bz+xy. (1)It is customary in considering the Lorenz system to require the parameters b, r, s to be all strictly positive; however for this particular problem we shall follow previous practice in only imposing the condition s ≠ 0. (If s = 0 the equations are trivially integrable and x is constant on any trajectory; thus x should be regarded as a parameter and the question discussed in this paper ceases to be a natural one.)

This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.

Let G be a compact Lie group. A G-cell of dimension n is a space of the form G/H x D, where H is a closed subgroup of G and D is an n-cell. A G-CW complex X (or an equivariant CW complex in the terminology of [9]) is constructed by iterated attaching of G-cells. It is the union of G-spaces X^> such that X^ is a disjoint union of G-cells of dimension 0, that is, orbits G/H, and X' + 1 ) is obtained from X n ) by attaching G-cells of dimension n + 1 along equivariant attaching maps G/H x dD -* X^K The space X^"\ which is called the n-skeleton of X, is thus the union of all G-cells of dimension at most n (the topological dimension of X^ is in general greater than n). For basic facts about G-complexes see the original papers [5] and [3] or the exposition in [9]. For discrete groups G it is well known that every G-CW complex is also a CW complex with a cellular action of G (this follows for example from [9, Proposition 1.16, p. 102]). For non-discrete groups, Illman [4] gave an example showing that a G-CW complex X does not always admit a CW decomposition, compatible with the given GCW decomposition, and proved that there always exists a homotopy equivalent CW complex Y which is finite if X is a finite G-complex. In this paper we consider the following problem. Given a G-CW complex X, does there exist a G-space Y, G-homotopy equivalent to X, with a CW decomposition such that the action p: G xY -> Y is a. cellular map with respect to some decomposition of G. The existence of such a Y is interesting from the point of view of equivariant homology and cohomology. For example, Greenlees and May showed that for some groups G the generalised Tate cohomology defined in [1] can be calculated from the CW decomposition

This paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

In this paper we study the small-scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is, the coefficients take value $+1$ with probability p and $-1$ with probability $1-p$ . Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ( $p=0.5$ ) coin to retain equidistribution.

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal.158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

In this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.

This paper investigates the ordering properties of largest claim amounts in heterogeneous insurance portfolios in the sense of some transform orders, including the convex transform order and the star order. It is shown that the largest claim amount from a set of independent and heterogeneous exponential claims is more skewed than that from a set of independent and homogeneous exponential claims in the sense of the convex transform order. As a result, a lower bound for the coefficient of variation of the largest claim amount is established without any restrictions on the parameters of the distributions of claim severities. Furthermore, sufficient conditions are presented to compare the skewness of the largest claim amounts from two sets of independent multiple-outlier scaled claims according to the star order. Some comparison results are also developed for the multiple-outlier proportional hazard rates claims. Numerical examples are presented to illustrate these theoretical results.

We set out some general criteria to prove the K-property, refining the assumptions used in an earlier paper for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda $ -decompositions, as well as multiple techniques for checking the pressure gap required to show the K-property. We apply our results to the family of Mañé diffeomorphisms and the Katok map. Our argument builds on the orbit decomposition theory of Climenhaga and Thompson.

For a $\psi $ -mixing process $\xi _0,\xi _1,\xi _2,\ldots $ we consider the number ${\mathcal N}_N$ of multiple returns $\{\xi _{q_{i,N}(n)}\in {\Gamma }_N,\, i=1,\ldots ,\ell \}$ to a set ${\Gamma }_N$ for n until either a fixed number N or until the moment $\tau _N$ when another multiple return $\{\xi _{q_{i,N}(n)}\in {\Delta }_N,\, i=1,\ldots ,\ell \}$ , takes place for the first time where ${\Gamma }_N\cap {\Delta }_N=\emptyset $ and $q_{i,N}$ , $i=1,\ldots ,\ell $ are certain functions of n taking on non-negative integer values when n runs from 0 to N. The dependence of $q_{i,N}(n)$ on both n and N is the main novelty of the paper. Under some restrictions on the functions $q_{i,N}$ we obtain Poisson distributions limits of ${\mathcal N}_N$ when counting is until N as $N\to \infty $ and geometric distributions limits when counting is until $\tau _N$ as $N\to \infty $ . We obtain also similar results in the dynamical systems setup considering a $\psi $ -mixing shift T on a sequence space ${\Omega }$ and studying the number of multiple returns $\{ T^{q_{i,N}(n)}{\omega }\in A^a_n,\, i=1,\ldots ,\ell \}$ until the first occurrence of another multiple return $\{ T^{q_{i,N}(n)}{\omega }\in A^b_m,\, i=1,\ldots ,\ell \}$ where $A^a_n,\, A_m^b$ are cylinder sets of length n and m constructed by sequences $a,b\in {\Omega }$ , respectively, and chosen so that their probabilities have the same order.

In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$. In this process, we establish the decomposition of Chow groups for the cases of Cayley's trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds., Comment: v2. Many grammatical problems fixed. References updated

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

In this paper we prove bounds for ergodic averages for nilflows on general higher-step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become equidistributed at polynomial speed. We analyze the rate of decay which is determined by the number of steps and structure of general nilpotent Lie algebras. Our main result follows from the technique of controlling scaling operators in irreducible representations and measure estimation on close return orbits on general nilmanifolds.

In this paper we focus on compacta$K \subseteq \mathbb {R}^3$which possess a neighbourhood basis that consists of nested solid tori$T_i$. We call these sets toroidal. Making use of the classical notion of the geometric index of a curve inside a torus, we introduce the self-geometric index of a toroidal setK, which roughly captures how each torus$T_{i+1}$winds inside the previous$T_i$as$i \rightarrow +\infty $. We then use this index to obtain some results about the realizability of toroidal sets as attractors for homeomorphisms of$\mathbb {R}^3$.

Martin Josefsson [1] has coined the term ‘bisect-diagonal quadrilateral’ for a quadrilateral with at least one diagonal bisected by the other diagonal, and extensively explored some of its properties. This quadrilateral has also been called a ‘bisecting quadrilateral’ [2], a ‘sloping-kite’ or ‘sliding-kite’ [3], or ‘slant kite’ [4]. The purpose of this paper is to explore some more properties of this quadrilateral.