Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

There is considerable evidence that the distribution of the length of time that a patient occupies a bed in a hospital department is best described by a sum of two or three exponential terms, because of the presence of acute care, rehabilitation, and possibly long term care patients in the department. The patient flow models implied by these mixed exponential distributions are presented and fitting them to observed data when the admission rate fluctuates is discussed. Unlike single exponential distributions, mixed exponential distributions imply that the average length of stay of patients currently resident in the department is much longer than the average length of stay of a group of patients discharged over a period of time, so that the latter way of measuring will not correctly indicate what portion of the resources are being used by rehabilitation and long term care patients. Also, the expected additional length of stay increases dramatically with the time already spent in the department. Applications to predicting the effects of policy changes and to long term monitoring of hospital departments are presented. Two American hospitals are analyzed. The occupancy times in the government supported hospital follow a mixed exponential distribution similar to those found in the United Kingdom, but in the private hospital they fit a single exponential distribution, indicating markedly different management practices.

We consider typical area-preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affirmatively Rokhlin’s multiple mixing question in this context. The main tool is a variation of the Ratner property originally proved by Ratner for the horocycle flow, i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behavior of these flows and has implications for joining classification. The main result is formulated in the language of special flows over interval exchange transformations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayad and Kanigowski’s main results, by showing that Arnold’s flows are mixing of all orders for almost every location of the singularities.

The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson-Lie groups $K$ on a symplectic manifold equipped with an $AN$-valued moment map, where $AN$ is the dual Poisson-Lie group of $K$. Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of $K$ with $\mathfrak{k}^*$-valued moment map to a Hamiltonian action with an $AN$-valued moment map, via a deformation of symplectic structures. We obtain our main result by proving a ``delinearization commutes with symplectic quotients'' theorem which is also of independent interest, and then putting this together with the local normal form theorem for classical Hamiltonian actions wtih $\mathfrak{k}^*$-valued moment maps. A key ingredient for our main result is the delinearization $\mathcal{D}(\omega_{can})$ of the canonical symplectic structure on $T^*K$, so we additionally take some steps toward explicit computations of $\mathcal{D}(\omega_{can})$. In particular, in the case $K=SU(2)$, we obtain explicit formulas for the matrix coefficients of $\mathcal{D}(\omega_{can})$ with respect to a natural choice of coordinates on $T^*SU(2)$.

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

We obtain orbital counting results for the class of strongly hyperbolic metrics on hyperbolic groups. To achieve this we combine ergodic theoretic techniques involving the Mineyev topological flow and symbolic dynamics. Our results apply to the Green metric associated to an admissible, finitely supported, symmetric random walk, to the Mineyev hat metric and to Hilbert length functions associated to Anosov representations. We also describe the domain of analyticity for the Poincaré series associated to these metrics, prove mixing results for the Mineyev topological flow and obtain correlation asymptotics for pairs of metrics.

Some inverse scattering problems for the three-dimensional biharmonic operator are considered. The operator is perturbed by first and zero order perturbations, which may be complex-valued and singular. We show the existence of the scattering solutions in the Sobolev space \(W^1_{\infty }(R^3)\). One of the main result of this paper is the proof of analogue of Saito’s formula (in different form as known before), which can be used to prove a uniqueness theorem for the inverse scattering problem. Another main result is to obtain the estimates for the kernel of the resolvent of the direct operator in \(W^1_{\infty }\) and to prove the reconstruction formula for the unknown coefficients of this perturbation.

Explicit matrix representation for NURBS curves and surfaces

The scientific and technical problem of mathematical modeling and calculation of separation of hydrocarbon mixtures in industrial columns with chaotic and regular nozzles is solved. Multicomponent transfer is calculated using a matrix of volumetric mass transfer coefficients, and the reverse (longitudinal) mixing of steam and liquid flows by Pekle numbers with back-mixing coefficients. The task of developing options for upgrading the stabilization column of the oil and gas condensate mixture at an oil refinery is considered. Calculated profiles of concentrations of key components in the mixture are presented. The required height of the nozzle layer has been determined. After upgrading the column with a new nozzle, a stable mode of operation of the unit is provided in the specified range of loads and the composition of the mixture. We also solved the problem of designing two industrial columns in the technological scheme for the separation of benzene from stable condensate (gasoline fraction). As a result of calculations based on a mathematical model, the operating and structural characteristics of columns with the use of chaotic metal nozzles "Injekhim-2012"were established.

Circuits with delay elements are very popular and important in the simulation of very-large-scale integration (VLSI) systems. Neutral systems (NSs) with multiple constant delays (MCDs), for example, can be used to model the partial element equivalent circuits (PEECs), which are widely used in high-frequency electromagnetic (EM) analysis. In this paper, the model order reduction (MOR) problem for the NS with MCDs is addressed by moment matching method. The nonlinear exponential terms coming from the delayed states and the derivative of the delayed states in the transfer function of the original NS are first approximated by a Padé approximation or a Taylor series expansion. This has the consequence that the transfer function of the original NS is exponential-free and the standard moment matching method for reduction is readily applied. The Padé approximation of exponential terms gives an expanded delay-free system, which is further reduced to a delay-free reduced-order model (ROM). A Taylor series expansion of exponential terms lets the inverse in the original transfer function have only powers-of-s terms, whose coefficient matrices are of the same size as the original NS, which results in a ROM modeled by a lower-order NS. Numerical examples are included to show the effectiveness of the proposed algorithms and the comparison with existing MOR methods, such as the linear matrix inequality (LMI)-based method.

We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur--Merlin games where Arthur is limited to polynomial time and constant space. Dwork and Stockmeyer [SIAM J. Comput., 19 (1990), pp. 1011--1023] asked whether these npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or $\bar{L}$ is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the "characteristic matrix" of L, namely, the binary matrix with a row and column for each string of length $\le n$, where entry [x,y]=1 if and only if the string $xy \in L$. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in $2^{\Omega (\sqrt{\log n})}$ infinitely often. The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof extends techniques of Frobenius and Iohvidov developed for Hankel matrices [Sitzungsber. der Königl. Preuss. Akad. der Wiss., 1894, pp. 407--431], [Hankel and Toeplitz Matrices and Forms: Algebraic Theory, Birkhauser, Boston, 1982].

In this note we give generalisations of Livsic's result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of: (a) 0-transformations; (b) rational maps; and (c) planar maps with indifferent periodic points. Such examples are not immediately covered by Livsic's original approach either due to a lack of continuity or hyperbolicity. 0. INTRODUCTION In 1972 Livsic showed that given a mixing subshift of finite type T: X -* X and a H6lder continuous function c: X -* IR then any solution to the equation (1) c(x) u(Tx) u(x) with u: X ---1iR measurable and essentially bounded (with respect to an equilibrium measure for a H6lder continuous function) must have a continuous version i.e. eu': X -* R continuous such that u(x) = u'(x) a.e. [2], [3] This is an elementary type of rigidity result. Livsic also applied this result to Anosov systems. Another approach to this result for subshifts of finite type and using Perron-Frobenius type operators appeared in [6]. In this note we shall discuss some simple generalisations of Livsic's result to other examples of dynamical systems. We shall be interested in systems with discontinuities or which are not uniformly hyperbolic and thus are not covered by Livsic's original results. In this context the Perron-Frobenius type operator method adapts easily to prove the generalisations of Livsic's original results. To illustrate this approach we shall apply this method to 3-transformations (section 1), rational maps (section 2) and certain multi-dimensional maps (section 3). 1. }5-TRANSFORMATION We begin by recalling the definition of the well-known 3-transformation on the unit interval. Let / > 1 and define T: [0,1) -* [0,1) by T(x) = fx (mod 1). Let v denote Lebesgue measure on the interval [0,1] and Ll ([0, 1], 13, v) denotes the space of integrable functions on [0,1]. Received by the editors August 5, 1996. 1991 Mathematics Subject Classification. Primary 58Fxx. (?)1999 American Mathematical Society 559 This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms 560 M. POLLICOTT AND M. YURI Definition. We define a Perron-Frobenius operator Li : Ll ([O, 1], 6,B v) --Ll ([O, 1], 6,B v) by Ty= 13 where the summation is over the [,3] or [a1] + 1 pre-images of x. We denote by BBV the space of functions of bounded variation i.e. if f E BBV then we have that var(f) :=sup{ZEif(Xi+1)-f(xi)I: O xoX 0 for x E [0,1). We shall also need the following more general operator. Definition. Given a Lipschitz function c: [0,11 --? R. We define a c-weighted Perron-Frobenius operator L2: L' ([O, 1], B, v) -* L1 ([O, 1], 13, v) by L2 f (x) = E ecp f (y) TY=X (When c= 0 then L2 clearly reduces to Lp.) It is convenient to denote cn (x) = c(x) + c(Tx) +.. + c(Tn lx) for n > 1. The following result is important in proving Theorem 1. Lemma 2. (i) Vf E L1([0, 1], B, v) we have I2nf h(f fdv) in the L1 norm, (ii) Let c: [0,11 -? IR be a Lipschitz function. There exists w E BBV with w > 0 and A > 0 such that the sequence of functions An (T-y=x BBV. The convergence Ijf -* h(J f dv), as n -4 +oo is proved in [9, Theorem 10] when f is uniformly continuous. Since any function in L1 ([O,1], B, v) can be approximated from above and below by such functions, this is sufficient. This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms REGULARITY OF SOLUTIONS l'O THE MEASURABLE LIVSIC EQUATION 561 (ii) The operator L2: BBV -BBV has a simple maximal eigenvalue A, say, with positive eigenvector w E BBV ([9, Lemma 9]). Moreover, there exists a measure ,t on [0, 1] such that fn =~ --> w f dti, as n +oo [9, Theorem 10]. 1] Our main result on the measurable Livsic equation for fl-transformations is the following. Theorem 1. If c: [0, 1t -* IR is Lipschitz then for any measurable and essentially bounded solution u to (1') c(x) = u(Tx) u(x) there exists uo(x) = v(x) log h(x), where v(x) E BBV such that u == uo a.e. (nt). Proof. We can make the choice f 1 and then iterating the operator 12 introduced above we can write (L n1)(X)= ASn fn (X) Tny3/X In particular, substituting from the identity (1') we get that f Ix eCn(y) fn (x) An E , n

The rate of convergence of the nearest neighbor (NN) rule is investigated when independent identically distributed samples take values in a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> -dimensional Euclidean space. The common distribution of the sample points need not be absolutely continuous. An upper bound consisting of two exponential terms is given for the probability of large deviations of error probability from the asymptotic error found by Cover and Hart. The asymptotically dominant first term of this bound is distribution-free, and its negative exponent goes to infinity approximately as fast as the square root of the number of preclassified samples. The second term depends on the underlying distributions, but its exponent is proportional to the sample size. The main term is explicitly given and depends very weakly on the dimension of the space.

An alternative derivation of Birkhoff’s formula for the contraction coefficient of a positive matrix

In this paper, the event‐triggered dynamic output feedback control problem for linear systems with actuator saturation is investigated. Event‐triggered scheme only transmits the corresponding signal when the event‐triggered condition is violated. Due to its advantage of saving communication resources, it is utilized to design the dynamic output feedback controller. A criterion is established to guarantee the stability of the closed‐loop system by introducing an exponential term for the Lyapunov function, which corresponds to the exponential term in the event‐triggered condition. The explicit design of the coefficient matrices of the controller is presented. Furthermore, a lower bound of the inter‐event time is calculated to avoid Zeno behavior. An optimization algorithm is then formulated to maximize the estimation of the domain of attraction. Finally, a numerical example is given to illustrate the effectiveness of our methods and to show the trade‐off between the size of the domain of attraction and communication resources saving.

: This report is divided into four independent sections. Section 1 contains a theorem giving sufficient conditions for the asymptotic distribution of a standardized location-scale random variable to have the distribution of a power of the random variable, and examples showing that the conditions are not necessary. Section 2 gives illustrations of the problem of deciding whether or not one random variable is a transform of another; and, in each case in which a transformation of the one random variable into the other is assured, the set of all such transformations is investigated. Section 3 consists of an example of the notion of robustness of a test as well as a tentative general definition of the concept of robustness of a test, and a brief study of the Kolmogorov metric on the space of location-parameter distributions. Section 4 presents simple iterative solutions of special systems of differential-difference equations, in which the constant coefficient matrices are triangular and satisfy conditions sufficient to insure that the solutions involve only exponential terms or terms that are products of linear factors and exponential factors. These methods are applied to the simple stochastic epidemic and to the general stochastic epidemic.