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Let a, b and n be non-negative integers such that 1 ≤ a ≤ b, and let G be a graph of order p with $\(p\geq\frac{(a+b-1)(a+b-2)+bn-2}{a}\)$ and f be an integer-valued function defined on V(G) such that a ≤ f(x) ≤ b for all x ∈ V(G). Let h: E(G) → [0, 1] be a function. If ∑e∋xh(e) = f(x) holds for any x ∈ V(G), then we call G[Fh] a fractional f-factor of G with indicator function h, where Fh = {e ∈ E(G): h(e) > 0}. A graph G is called a fractional (f, n)-critical graph if after deleting any n vertices of G the remaining graph of G has a fractional f-factor. In this paper, it is proved that G is a fractional (f, n)-critical graph if $\(|N_G(X)|>\frac{(b-1)p+|X|+bn-1}{a+b-1}\)$ for every non-empty independent subset X of V(G), and $\(\delta(G)>\frac{(b-1)p+a+b+bn-2}{a+b-1}\)$. Furthermore, it is shown that the result in this paper is best possible in some sense.

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.

SUMMARYThe workspace of a robotic manipulator is a very important issue and design criteria in the context of optimum design of robots, especially for parallel manipulators. Though, considerable research has been paid to the investigations of the three-dimensional (3D) constant orientation workspace or position workspace of parallel manipulators, very few works exist on the topic of the 3D orientation workspace, especially the nonsingular orientation workspace and practical orientation workspace. This paper addresses the orientation workspace analysis of a special class of the Stewart–Gough parallel manipulators in which the moving and base platforms are two similar semisymmetrical hexagons. Based on the half-angle transformation, a polynomial expression of 13 degree that represents the orientation singularity locus of this special class of the Stewart–Gough parallel manipulators at a fixed position is derived and graphical representations of the orientation singularity locus of this special class of the Stewart–Gough manipulators are illustrated with examples to demonstrate the result. Exploiting this half-angle transformation and the inverse kinematics solution of this special class of the Stewart–Gough parallel manipulators, a discretization method is proposed for computing the orientation workspace of this special class of the Stewart–Gough parallel manipulators taking limitations of active and passive joints and the link interference all into consideration. Based on this algorithm, this paper also presents a new discretization method for computing the nonsingular orientation workspace of this class of the manipulators, which not only can satisfy all the kinematics demand of this class of the manipulators but also can guarantee the manipulator is nonsingular in the whole orientation workspace, and the practical orientation workspace of this class of the manipulators, which not only can guarantee the manipulator is nonsingular and will never encounter any kinematic interference but also can satisfy the demand of the orientation workspace with a regular shape in practical application, respectively. Examples of a 6/6-SPS Stewart–Gough parallel manipulator of this special class are given to demonstrate these theoretical results.

In the classical theorems about lower and upper vector functions for systems of linear differential equations very heavy restrictions on the signs of coefficients are assumed. These restrictions in many cases become necessary if we wish to compare all the components of a solution vector. The formulas of the integral representation of the general solution explain that these theorems claim actually the positivity of all elements of the Green's matrix. In this paper we define a principle of partial monotonicity (comparison of only several components of the solution vector), which assumes only the positivity of elements in a corresponding row of the Green's matrix. The main theorem of the paper claims the equivalence of positivity of all elements in the nth row of the Green's matrices of the initial and two other problems, non-oscillation of the nth component of the solution vector and a corresponding assertion about differential inequality of the de La Vallee Poussin type. Necessary and sufficient conditions of the partial monotonicity are obtained. It is demonstrated that our sufficient tests of positivity of the elements in the nth row of the Cauchy matrix are exact in corresponding cases. The main idea in our approach is a construction of an equation for the nth component of the solution vector. In this sense we can say that an analog of the classical Gauss method for solving systems of functional differential equations is proposed in the paper.

This thesis is concerned with the study of stochastic epidemic models for infectious diseases in heterogeneous populations. All diseases treated are of SIR type, i.e. individuals are either Susceptible, Infectious or Recovered (and immune). The transitions between these states are according to S to I to R. The thesis consists of five papers. Papers I and II treat approximations for the distribution of the time to extinction. In Paper I, a sub-community version of the SIR model with demography is considered. The interest is in how the distribution of the time to extinction is affected by varying the degree of interaction between the sub-communities. Paper II is concerned with a two-type version of Bartlett's model. The distribution of the time to extinction is studied when the difference in susceptibility/infectivity between the types of individuals is varied. Papers III and IV treat random intersection graphs with tunable clustering. In Paper III a Reed-Frost epidemic is run on such a random intersection graph. The critical parameter R_0 and the probability of a large outbreak are derived and it is investigated how these quantities are affected by the clustering in the graph. In Paper IV the interest is in the component structure of such a graph, i.e. the size and the emergence of a giant component is studied. The last paper, Paper V, treats the situation when a simple epidemic is running in a varying environment. A varying environment is in this context any external factor that affects the contact rate in the population, but is itself unaffected by the population. The model treated is a term-time forced version of the stochastic general epidemic where the contact rate is modelled by an alternating renewal process. A threshold parameter R_* and the probability of a large outbreak are derived and studied.

SUMMARYA Jacobian-based algorithm that is useful for planning the motion of a floating rigid body operated using two input torques is addressed in this paper. The rigid body undergoes a four-rotation fully reversed (FR) sequence of rotations which consists of two initial rotations about the axes of a coordinate frame attached to the body and two subsequent rotations that undo the preceding rotations. Although a Jacobian-based algorithm has been useful in exploring the inverse kinematics of conventional robot manipulators, it is not apparent how a correct FR sequence for a desired orientation could be found because the Jacobian of FR sequences is singular as well as being a null matrix at the identity. To discover the FR sequences that can synthesize the desired orientation circumventing these difficulties, the Jacobian algorithm is reformulated and implemented from arbitrary orientations where the Jacobian is not singular. Due to the insufficient degrees-of-freedom of four-rotation FR sequences required to achieve all possible orientations, the rigid body cannot achieve certain orientations in the configuration space. To best approximate these infeasible orientations, the Jacobian-based algorithm is implemented in the sense of least squares. As some orientations can never be attained by a single four-rotation FR sequence, two different four-rotation FR sequences are exploited alternately to ensure the convergence of the proposed algorithm. Assuming the orientation is supposed to be manipulated using three input torques, the switching Jacobian algorithm proposed in this paper has significant practical importance in planning paths for aerospace and underwater vehicles which are maneuvered using only two input torques due to the failure of one of the torque-generation mechanisms.

Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.

Determining meaningful lower bounds on the supremal strict p-negative type of classes of finite metric spaces is a difficult nonlinear problem. In this paper we use an elementary approach to obtain the following result: given a finite metric space (X,d) there is a constant ζ>0, dependent only on n=∣X∣ and the scaled diameter 𝔇=(diamX)/min{d(x,y)∣x⁄=y} of X (which we may assume is >1), such that (X,d) has p-negative type for all p∈[0,ζ] and strict p-negative type for all p∈[0,ζ). In fact, we obtain A consideration of basic examples shows that our value of ζ is optimal provided that 𝔇≤2. In other words, for each 𝔇∈(1,2] and natural number n≥3, there exists an n-point metric space of scaled diameter 𝔇 whose supremal strict p-negative type is exactly ζ. The results of this paper hold more generally for all finite semi-metric spaces since the triangle inequality is not used in any of the proofs. Moreover, ζ is always optimal in the case of finite semi-metric spaces.

Jørgensen’s famous inequality gives a necessary condition for a subgroup of PSL(2,ℂ) to be discrete. It is also true that if Jørgensen’s inequality holds for every nonelementary two-generator subgroup, the group is discrete. The sufficient condition has been generalized to many settings. In this paper, we continue the work of Wang, Li and Cao (‘Discreteness criteria for Möbius groups acting on $\overline {\mathbb {R}}^n$’, Israel J. Math.150 (2005), 357–368) and find three more (infinite) discreteness criteria for groups acting on $\overline {\mathbb {R}}^n$; we also correct a linguistic ambiguity of their Theorem 3.3 where one of the necessary conditions might be vacuously fulfilled. The results of this paper are obtained by using known results regarding two-generator subgroups and a careful analysis of the relation among the fixed point sets of various elements of the group.

In this paper, we introduce an iterative scheme using an extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz-continuous mapping. We obtain a weak convergence theorem for three sequences generated by this process. Based on this result, we also obtain several interesting results. The results in this paper generalize and extend some well-known weak convergence theorems in the literature.

This paper deals with families of conformal iterated function systems (CIFSs). The space CIFS(X,I) of all CIFSs, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and the so-calledλ-topology. We show just how bad the topology of pointwise convergence is: although the Hausdorff dimension function is continuous on a dense Gδ-set, it is also discontinuous on a dense subset of CIFS(X,I). Moreover, all of the different types of systems (irregular, critically regular, etc.), have empty interior, have the whole space as boundary, and thus are dense in CIFS(X,I), which goes against intuition and conception of a natural topology on CIFS(X,I). We then prove how good the λ-topology is: Roy and Urbański [Regularity properties of Hausdorff dimension in infinite conformal IFSs. Ergod. Th. & Dynam. Sys.25(6) (2005), 1961–1983] have previously pointed out that the Hausdorff dimension function is then continuous everywhere on CIFS(X,I). We go further in this paper. We show that (almost) all of the different types of systems have natural topological properties. We also show that, despite not being metrizable (as it does not satisfy the first axiom of countability), the λ-topology makes the space CIFS(X,I) normal. Moreover, this space has no isolated points. We further prove that the conformal Gibbs measures and invariant Gibbs measures depend continuously on Φ∈CIFS(X,I) and on the parameter t of the potential and pressure functions. However, we demonstrate that the coding map and the closure of the limit set are discontinuous on an important subset of CIFS(X,I).

We investigate the setIof parametersκ∈ℂ for which the singular orbit (0,eκ,…) ofEκ(z):=exp (z+κ) converges to$\infty $. These parameters are organized in curves in parameter space calledparameter rays, together with endpoints of certain rays. Parameter rays are an important tool to understand the detailed structure of exponential parameter space. In this paper, we construct and investigate these parameter rays. Based on these results, a complete classification of the setIis given in the following paper [M. Förster, L. Rempe and D. Schleicher. Classification of escaping exponential maps.Proc. Amer. Math. Soc.136(2008), 651–663].

We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primespat which the group defining the Shimura variety ramifies. We describe ‘good’p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

Morozov’s discrepancy principle is one of the simplest and most widely used parameter choice strategies in the context of regularization of ill-posed operator equations. Although many authors have considered this principle under general source conditions for linear ill-posed problems, such study for nonlinear problems is restricted to only a few papers. The aim of this paper is to apply Morozov’s discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems under general source conditions.

SUMMARYThis paper describes a numerical algorithm to solve the inverse kinematics of parallel robots based on numerical integration. Inverse kinematics algorithms based on numerical integration involve the drift phenomena of the solution; as a consequence, errors are generated when the end-effector location differs from that desired. The proposed algorithm associates a novel method to describe the differential kinematics with a simple numerical integration method. The methodology is presented in this paper and its exponential stability is proved. A numerical example and a real application are presented to outline its advantages.

In this paper we present a simple formula that can often be used to assist in the approximation of the sum of an alternating seriesThe ideas presented here were motivated by translating Euler's papers [1] and [2]. Suppose we begin by summing a terms of this series exactly, then the approximation that we will obtain is given by

We give an explicit Krasnoselski–Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: where C is a closed convex subset of a Hilbert space H, G:C×C→ℝ is an equilibrium function, T:C→C is a nonexpansive mapping with Fix(T) its set of fixed points and f:C→C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, ‘Weak and strong convergence of Krasnosel’skiĭ–Mann iteration for hierarchical fixed point problems’, Inverse Problems24 (2008), 501–508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction f was considered. The paper follows the lines of research of corresponding results of Moudafi and Théra.

We continue the study of a mathematical model for a forest ecosystem which has been presented by Y. A. Kuznetsov, M. Y. Antonovsky, V. N. Biktashev and A. Aponina (A cross-diffusion model of forest boundary dynamics, J. Math. Biol. 32 (1994), 219–232). In the preceding two papers (L. H. Chuan and A. Yagi, Dynamical systemfor forest kinematic model, Adv. Math. Sci. Appl. 16 (2006), 393–409; L. H. Chuan, T. Tsujikawa and A. Yagi, Aysmptotic behavior of solutions for forest kinematic model, Funkcial. Ekvac. 49 (2006), 427–449), the present authors already constructed a dynamical system and investigated asymptotic behaviour of trajectories of the dynamical system. This paper is then devoted to studying not only the structure (including stability and instability) of homogeneous stationary solutions but also the existence of inhomogeneous stationary solutions. Especially it shall be shown that in some cases, one can construct an infinite number of discontinuous stationary solutions.

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.

This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$, then the number of equivalence classes is uncountable (Theorem A). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems B and C)., Comment: 38 pages, 5 figures. To appear in Ergodic Theory and Dynamical Systems

Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=\dim_{\mathbb{R}} K$. Assume $d(n-1) \geq 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $\Gamma$-space $(X,\mu_X)$, we assume that a measurable cocycle $\sigma:\Gamma \times X \rightarrow \text{G}(m)$ admits an essentially unique boundary map $\phi:\partial_\infty \mathbb{H}^n_K \times X \rightarrow \partial_\infty \mathbb{H}^m_K$ whose slices $\phi_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are atomless for almost every $x \in X$. Then, there exists a $\sigma$-equivariant measurable map $F: \mathbb{H}^n_K \times X \rightarrow \mathbb{H}^m_K$ whose slices $F_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are differentiable for almost every $x \in X$ and such that $\text{Jac}_a F_x \leq 1$ for every $a \in \mathbb{H}^n_K$ and almost every $x \in X$. The previous properties allow us to define the natural volume $\text{NV}(\sigma)$ of the cocycle $\sigma$. This number satisfies the inequality $\text{NV}(\sigma) \leq \text{Vol}(\Gamma \backslash \mathbb{H}^n_K)$. Additionally, the equality holds if and only if $\sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \text{G}(n) \leq \text{G}(m)$, modulo possibly a compact subgroup of $\text{G}(m)$ when $m>n$. Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context., Comment: 27 pages, to appear on J. Inst. Math. Jussieu

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

A mathematical model is proposed for a revolute joint mechanism with an n-degree of freedom (DOF). The matrix approach is used for finding the relation between two consecutive links to determine desired link parameters such as position, velocity and acceleration using the forward kinematic approach. The matrix approach was confirmed for a proposed 10 DOF revolute type (R-type) human upper limb model with servo motors at each joint. Two DOFs are considered each at shoulder, elbow and wrist joint, followed by four DOF for the fingers. Two DOFs were considered for metacarpophalangeal (mcp) and one DOF each for proximal interphalangeal (pip) and distal interphalangeal (dip) joints. MATLAB script function was used to evaluate the mathematical model for determining kinematic parameters for all the proposed human upper limb model joints. The simplified method for kinematic analysis proposed in this paper will further simplify the dynamic modeling of any mechanism for determining joint torques and hence, easy to design control system for joint movements.

In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].

In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.

Let f be a Hénon–Sibony map, also known as a regular polynomial automorphism of $\mathbb {C}^k$ , and let $\mu $ be the equilibrium measure of f. In this paper we prove that $\mu $ is exponentially mixing for plurisubharmonic observables.

In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations \begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*} where $0, $\Omega = \mathbb {R}^{N}$ or $\Omega$ is a smooth bounded domain, $\Gamma$ is a singular subset of $\Omega$ with fractional capacity zero, $f(t)$ is locally bounded and positive for $t\in [0,\,\infty )$, and $f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in $t$ for large $t$, rather than for every $t>0$. Our main result is that the solutions satisfy the estimate \begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*} This estimate is new even for $\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.

Using a method due to Rieger [‘Remark on a paper of Stux concerning squarefree numbers in non-linear sequences’, Pacific J. Math.78(1) (1978), 241–242], we prove that the Piatetski-Shapiro sequence defined by $\{\lfloor n^c \rfloor : n\in \mathbb {N}\}$ contains infinitely many consecutive square-free integers whenever $1.

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Sch\"utzenberger groups. The main result of this paper states that for a finitely generated commutative semigroup $S$, these three separability conditions coincide and are equivalent to every $\mathcal{H}$-class of $S$ being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal{H}$-classes, we investigate whether it has one of these properties if and only if all its Sch\"utzenberger groups have the property., Comment: 27 pages

A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux–Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive$\mathcal {S}$-adic representation where the morphisms in$\mathcal {S}$are positive tame automorphisms of the free group generated by the alphabet. In this paper, we investigate those$\mathcal {S}$-adic representations, heading towards an$\mathcal {S}$-adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with two vertices.

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

In this paper we consider a new generalized finite mixture model formed by dependent and identically distributed (d.i.d.) components. We then establish results for the comparisons of lifetimes of two such generalized finite mixture models in two different cases: (i) when the two mixture models are formed from two random vectors $\textbf{X}$ and $\textbf{Y}$ but with the same weights, and (ii) when the two mixture models are formed with the same random vectors but with different weights. Because the lifetimes of k-out-of-n systems and coherent systems are special cases of the mixture model considered, we used the established results to compare the lifetimes of k-out-of-n systems and coherent systems with respect to the reversed hazard rate and hazard rate orderings.

In this paper, we first prove an existence theorem for the integrodifferential equation(*)wheref,k,xare functions with values in a Banach spaceEand the integral is taken in the sense of Henstock–Kurzweil–Pettis. In the second part of the paper we show that the setSof all solutions of the problem (*) is compact and connected in (C(Id,E),ω), where$I_{d} \subset I_{a} $.

In this paper, we obtain algebraic equations for all genus 2 compact Riemann surfaces that admit a semi-regular (or uniform) covering of the Riemann sphere with more than two lifting symmetries. By a lifting symmetry, we mean an automorphism of the target surface which can be lifted to the covering. We restrict ourselves to the genus 2 surfaces in order to make computations easier and to make possible to find their algebraic equations as well. At the same time, the main ingredient (Main Proposition) depends neither on the genus, nor on the order of the group of lifting symmetries. Because of this, the paper can be thought as a generalisation for the non-normal case to the question of lifting automorphisms of a compact Riemann surface to a normal covering, treated, for instance, by E. Bujalance and M. Conder in a joint paper, or by P. Turbek solely.

In this paper, we study an upper bound of the fractal dimension of the exponential attractor for the chemotaxis–growth system in a two-dimensional domain. We apply the technique given by Eden, Foias, Nicolaenko and Temam. Our results show that the bound is estimated by polynomial order with respect to the chemotactic coefficient in the equation similar to our preceding papers.

Let Mm be a closed smooth manifold with an involution having fixed point set of the form Fn ∪ F2, where Fn and F2 are submanifolds with dimensions n and 2, respectively, where n ≥ 4 is even (n < m). Suppose that the normal bundle of F2 in Mm, μ → F2, does not bound, and denote by β the stable cobordism class of μ → F2. In this paper, we determine the upper bound for m in terms of the pair (n, β) for many such pairs. The similar question for n odd (n ≥ 3) was completely solved in a previous paper of the authors. The existence of these upper bounds is guaranteed by the famous 5/2-theorem of Boardman, which establishes that, under the above hypotheses, m ≤ 5/2n.