Absolutely Singular Dynamical Foliations

Let A ³ be the product of the automorphism of T ² and of the identity on T ¹. A small perturbation g of A ³ among volume preserving diffeomorphisms will have an invariant family of smooth circles Γ forming a continuous foliation of T ³. Corresponding to the vector bundle tangent to the circles Γ there is a “central” Lyapunov exponent of (g, volume), which is nonzero for an open set of ergodic g's. This surprising result of Shub and Wilkinson is complemented here by showing that the volume on T ³ has atomic conditional measures on the Γ's: there is a finite k such that almost every Γ carries $k$ atoms of mass 1/k. Received: 26 September 2000 / Accepted: 8 December 2000     

Determination of Partition Lines from Dynamical Foliations for the Hénon Map

The binary partition lines for the Hhnon map are numerically constructed from tangencies between the contracting and expanding foliations. The ordering of foliations acording to their symbolic sequences are examined.     

Zeta Function Resummation of Spectral Functions

This article continues an extensive analysis of spectral functions such as for both general and explicitly known spectra {λ_m}. In physical applications (which in quantum field theory are numerous) the spectral functions are mode sums. Our main analytic tool is the ζ-function resummation method which expresses f(s|x) in powers of x and perhaps other simple functions of x. Here the general spectrum will be replaced by its asymptotic form λ_m = (const) m^α with α > 0 (Weyl's theorem). This preserves certain global features of the general spectrum problem but enables one to work entirely in terms of known functions. This simplified problem will be fully analysed and certain aspects of it (in particular the continuum limit) studied for the first time. Several mode-sum calculations illustrate physical application of the method.     

Transfer Operators and Dynamical Zeta Functions for a Class of Lattice Spin Models

 We investigate the location of zeros and poles of a dynamical zeta function for a family of subshifts of finite type with an interaction function depending on the parameters . The system corresponds to the well known Kac-Baker lattice spin model in statistical mechanics. Its dynamical zeta function can be expressed in terms of the Fredholm determinants of two transfer operators and with the Ruelle operator acting in a Banach space of holomorphic functions, and an integral operator introduced originally by Kac, which acts in the space with a kernel which is symmetric and positive definite for positive β. By relating via the Segal-Bargmann transform to an operator closely related to the Kac operator we can prove equality of their spectra and hence reality, respectively positivity, for the eigenvalues of the operator for real, respectively positive, β. For a restricted range of parameters we can determine the asymptotic behavior of the eigenvalues of for large positive and negative values of β and deduce from this the existence of infinitely many non-trivial zeros and poles of the dynamical zeta functions on the real β line at least for generic . For the special choice , we find a family of eigenfunctions and eigenvalues of leading to an infinite sequence of equally spaced ``trivial' zeros and poles of the zeta function on a line parallel to the imaginary β-axis. Hence there seems to hold some generalized Riemann hypothesis also for this kind of dynamical zeta functions. Received: 14 March 2002 / Accepted: 24 June 2002 Published online: 14 November 2002     

Zeta Function on a Generalised Cone

The analytic properties of the -function for a Laplace operator on a generalised cone are studied in some detail using Cheeger's approach and explicit expressions are given. In the compact case, the -function of the Laplace operator turns out to be singular at the origin. As a result, strictly speaking, the -function regularisation does not regularise and a further subtraction is required for the related one-loop effective potential.     

Analytic Surgery of the Zeta Function

In this paper we study the asymptotic behavior (in the sense of meromorphic functions) of the zeta function of a Laplace-type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing hypersurface, separating the manifold into two manifolds with infinite cylindrical ends. We also study the related problem on a manifold with boundary as the manifold is stretched in the direction normal to its boundary, forming a manifold with an infinite cylindrical end. Such singular deformations fall under the category of “analytic surgery”, developed originally by Hassell (Comm Anal Geom 6:255–289, 1998), Hassell et al. (Comm Anal Geom 3:115–222, 1995) and Mazzeo and Melrose (Geom Funct Anal 5:14–75, 1995) in the context of eta invariants and determinants.     

The Selberg Zeta Function for Convex Co-Compact Schottky Groups

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ⁿ⁺¹: in strips parallel to the imaginary axis the zeta function is bounded by exp (C|s|) where is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp (C|s|ⁿ⁺¹) , and it gives new bounds on the number of resonances (scattering poles) of \ⁿ⁺¹ . The proof of this result is based on the application of holomorphic L²-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider \ⁿ⁺¹ as the simplest model of quantum chaotic scattering.     

Large Numbers Generated by the Riemann Zeta Function

Journal of Experimental and Theoretical Physics - Anomalously large numbers generated of the Riemann zeta function are analyzed. A set of Mersenne primes is investigated. The equations connecting...     

Casimir Effect for a Perfectly Conducting Wedge in Terms of Local Zeta Function

The vacuum energy density of electromagnetic field inside a perfectly conducting wedge is calculated by making use of the local zeta function technique. This regularization completely eliminates divergent expressions in the course of calculations and gives rise to a finite expression for the energy density in question without any subtractions. Employment of the Hertz potentials for constructing the general solution to the Maxwell equations results in a considerable simplification of the calculations. Transition to the global zeta function is carried out by introducing a cutoff nearby the cusp at the origin. Proceeding from this the heat kernel coefficients are calculated and the high temperature asymptotics of the Helmholtz free energy and of the torque of the Casimir forces are found. The wedge singularity gives rise to a specific high temperature behavior ∼T2 of the quantities under consideration. The obtained results are directly applicable to the free energy of a scalar massless field and electromagnetic field on the background of a cosmic string.     

On The Spectral Zeta Function For The Noncommutative Harmonic Oscillator

The spectral zeta function for the so-called noncommutative harmonic oscillator is able to be meromorphically extended to the whole complex plane, having only one simple pole at the same point s = 1 where Riemann's zeta function ζ(s) has, and possesses a trivial zero at each nonpositive even integer. The essential part of its proof is sketched. A new result is also given on the lower and upper bounds of the eigenvalues of the noncommutative harmonic oscillator.     

A Riemannian geometry for thermodynamic state space

We consider a class of thermodynamic systems in which the dynamics of the spontaneous approach to equilibrium is governed by the gradient of negentropy, where the gradient is taken with respect to a Riemannian metric. In open systems (dissipative structures) this gradient field is superposed with a vector field of interactions with environment. We consider three characteristics of the economy of dissipative structures: negentropy inflow (income), negentropy consumption (i.e. entropy production), and negentropy surplus (reserves). We derive explicit formulas for these characteristics and for the relations between them.     

余弦BOC信号伪相关函数无模糊跟踪方法

二进制偏移载波 (binary offset carrier,BOC) 调制方式实现了频谱分离和频段共享,被应用于全球导航卫星系统中;但由于BOC信号自相关函数的主峰和副峰幅度差异较小,容易引起捕获和跟踪模糊,导致较大的测距误差;余弦相位BOC(Cosine-phased BOC,CosBOC)信号自相关函数形式相对复杂,给其无模糊处理带来了挑战;基于伪相关函数法(pseudo correlation function, PCF)思想,针对CosBOC信号设计本地参考信号的特殊码片波形,与接收信号相关后经过非线性组合可获得单峰无模糊相关函数,消除了跟踪模糊性;仿真结果表明:参数取值合适时该方法的跟踪和抗多径性能均优于同等条件下的BPSK-like方法。     

A Pseudo Arc-Length Method for Numerical Simulation of Shock Waves

A pseudo arc-length method is proposed for the numerical simulation of shock wave propagations. This method passes the discontinuities and establishes adaptive moving meshes in the physical space by introducing the arc-length parameter and transforming the computational domain. Numerical experiments of the Sod problem, double Mach reflection problem and explosion problem demonstrate that this approach is more efficient than traditional numerical methods in capturing and tracking discontinuous solutions of singular or nearly singular problems.     

Robustness and Flexibility of Neural Function through Dynamical Criticality

In theoretical biology, robustness refers to the ability of a biological system to function properly even under perturbation of basic parameters (e.g., temperature or pH), which in mathematical models is reflected in not needing to fine-tune basic parameter constants; flexibility refers to the ability of a system to switch functions or behaviors easily and effortlessly. While there are extensive explorations of the concept of robustness and what it requires mathematically, understanding flexibility has proven more elusive, as well as also elucidating the apparent opposition between what is required mathematically for models to implement either. In this paper we address a number of arguments in theoretical neuroscience showing that both robustness and flexibility can be attained by systems that poise themselves at the onset of a large number of dynamical bifurcations, or dynamical criticality, and how such poising can have a profound influence on integration of information processing and function. Finally, we examine critical map lattices, which are coupled map lattices where the coupling is dynamically critical in the sense of having purely imaginary eigenvalues. We show that these map lattices provide an explicit connection between dynamical criticality in the sense we have used and “edge of chaos” criticality.     

Geometric calculus: A new computational tool for Riemannian geometry

We compare geometric calculus applied to Riemannian geometry with Cartan's exterior calculus method. The correspondence between the two methods is clearly established. The results obtained by a package written in an algebraic language and doing general manipulations on multivectors are compared. We see that the geometric calculus is as powerful as exterior calculus.     

Cohomological tautness for Riemannian foliations

In this paper, we present new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold.     

Moments of the Derivative of Characteristic Polynomials with an Application to the Riemann Zeta Function

We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann ζ function on the critical line. We do the same for the analogue of Hardy’s Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum.     

The Status of the Wave Function in Dynamical Collapse Models

No Heading The idea that in dynamical wave function collapse models the wave function is superfluous is investigated. Evidence is presented for the conjecture that, in a model of a field theory on a 1+1 lightcone lattice, knowing the field configuration on the lattice back to some time in the past, allows the wave function or quantum state at the present moment to be calculated, to arbitrary accuracy so long as enough of the past field configuration is known.     

A Central Limit Theorem for Time-Dependent Dynamical Systems

The work by Ott et al. (Math. Res. Lett. 16:463–475, 2009) established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random examples are also found, which are not small perturbations of a given map. Our approach uses martingale approximation technique in the form of Sethuraman and Varadhan (Electron. J. Probab. 10:121–1235, 2005).