Ez: DYNAMICS AND BIFURCATIONS

In view of the central symmetry of the set of Riemann zeros, \(\rho \leftrightarrow 1 - \rho\) which crucially reflects the functional equation (3.26), there is no clear reason to pick either the ρ themselves (with \(\rho \equiv \frac{1}{2} \pm i\tau_{k}, {\rm Re} \tau_{k} > 0\)) or the \(\rho(1 - \rho) = \frac{1}{4} + \tau_{k}^{2}\) as the basic set upon which to build zeta functions over the Riemann zeros. This is a nonlinear remapping freedom as discussed in Sect. 1.1, which leads to generalized zeta functions of several kinds. This chapter gives an introductory overview of the various possibilities. In decreasing order of analytical tractability, and in our older notation to be soon discarded, we may define: Functions of first kind: \(\sum_{\rho}(x - \rho)^{-s}, \qquad {\rm Re} s > 1;\) Functions of second kind: \(\sum\limits_{k = 1}^{\infty}(\tau_{k}^{2} + v)^{-\sigma},\qquad \mathfrak{R}\sigma > \frac{1}{2};\) Functions of third kind: \(\sum\limits_{k = 1}^{\infty}(\tau_{k}^{2} + \tau)^{-s},\qquad {\rm Re} s > 1.\) In all three kinds the argument (in exponent) is the principal variable, the other one a shift parameter. Each parametric family starts as an analytic function of its argument in its indicated half-plane of convergence, due to (4.7) or (4.26), but it will extend to a meromorphic function in the entire complex plane (of its argument), with a computable singular part. The domain of each shift parameter is left as large as possible: simply, none of the quantities to be raised to a power should lie on the cut \(\mathbb{R}_{-}\); this will hardly matter with the fairly small parameter values that we will emphasize.KeywordsMeromorphic FunctionZeta FunctionCentral SymmetryShift ParameterRiemann HypothesisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A complex valued function defined and analytic in the entire complex plane is called an entire function or an integral function. In other words, in the extended complex plane, it can have a singularity only at infinity. If f is an entire function different from a polynomial and M(r, f) [or M(r) when there is no confusion] denotes the maximum of |f(z)| on the circle |z| = r, it follows from Liou-ville’s theorem that M(r, f) increases more rapidly than any power of r. Thus, to study the growth of the function we have to compare with exponentials.

In this section, we obtain theorems on the approximation of functions of the complex variable dened on closed contours bounding a simply connected domain in the complex plane; the approximations are constructed in the space of continuous functions and in Lebesgue spaces with the use of Lozinskii interpolation polynomials. These results are used to obtain a theoretical justication of the subdomain method for the approximate solution of singular integral equations dened on closed contours of various smoothness. The solvability of the computational scheme of this method is proved by the technique in [1]. Note that issues related to the approximate solution of singular integral equations have been considered in numerous papers (see [2{10] and the bibliography therein). However, the subdomain method for singular integral equations on closed contours other than the unit circle has not been studied. Let be a closed smooth contour bounding a connected domain F + containing the origin, and let F = CnfF + [ g ,w hereC is the entire complex plane. We say that belongs to the class (m) ,w herem is a positive integer, if the Riemann function t = (w), w2 0 =f : jj =1 g, (1 )= 1, 0 (1 )= a1 (a1 > 0), of this contour has continuous :

This paper presents the realization of complex impedance standard, which achieve the calibration of digital bridge on entire complex plane through generating voltage and current waveform. The proposed standard can simulate any modulus and angle of complex impedance, which is focused on some technologies, such as digital waveform synthesis, all digital phase locked loop, DDS. Based on the concept of complex impedance and the calibration object, this paper achieves the design and realization of complex impedance standard. The measurement data at 1kHz is provided in the end, which can establish the foundation for calibration.

The celebrated Liouville theorem in complex analysis which asserts that any bounded holomorphic function on the entire complex plane has to be constant, has been extended to various areas of analysis.

A general definition of ``cosine'' and ``sine'' operators, C and S, for harmonic oscillator phase is proposed and its consequences examined. An important feature of the spectral analysis is the ``chain sequence'' condition which ensures that C and S have unit norm. The (nonunitary) operator U = C + iS is shown to be an annihilation-type operator whose spectral properties bear a remarkable analogy to those of the standard annihilation operator, although its spectrum fills the unit circle rather than the entire complex plane. Statistical properties of the eigenstates of U are discussed briefly.

We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplication-differentiation operators acting on L^2 (\mathbb R^n) which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behaviorwith the range of the symbol and the long-time behavior with the eigenvalues of their generators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane.

The independent set polynomial of a graph has one variable for each vertex and one monomial for each independent set, comprising the product of the corresponding variables. Given a graph G on n vertices and a vector p ∈ [0,1)ⁿ, a central problem in statistical mechanics is determining whether the independent set polynomial of G is non-vanishing in the polydisk of p, i.e., whether |Z_G(x)| > 0 for every x ∈ ℂⁿ such that |x_i| ≤ p_i. Remarkably, when this holds, Z_G(-p) is a lower bound for the avoidance probability when G is a dependency graph for n events whose probabilities form vector p. A local sufficient condition for |Z_G| > 0 in the polydisk of p is the Lovász Local Lemma (LLL). 
\nIn this work we derive several new results on the efficient evaluation and bounding of Z_G. Our starting point is a monotone mapping from subgraphs of G to truncations of the tree of self-avoiding walks of G. Using this mapping our first result is a local upper bound for Z(-p), similar in spirit to the local lower bound for Z(-p) provided by the LLL. Next, using this mapping, we show that when G is chordal, Z_G can be computed exactly and in linear time on the entire complex plane, implying perfect sampling for the hard-core model on chordal graphs. We also revisit the task of bounding Z(-p) from below, i.e., the LLL setting, and derive four new lower bounds of increasing sophistication. Already our simplest (and weakest) bound yields a strict improvement of the famous asymmetric LLL, i.e., a strict relaxation of the inequalities of the asymmetric LLL without any further assumptions. This new asymmetric local lemma is sharp enough to recover Shearer’s optimal bound in terms of the maximum degree Δ(G). We also apply our more sophisticated bounds to estimate the zero-free region of the hard-core model on the triangular lattice (hard hexagons model).

It is shown that the integral equation \[ \int_0^\infty {f(t)} \frac{{\sin (x - t)}} {{\pi (x - t)}}dt = \lambda f(x)\quad (x > 0)\] has a solution f for any (complex) $\lambda $, excluding the real numbers $\lambda \leq 0$, $\lambda > 1$. The closure of the set of eigenvalues, i.e., the spectrum of the integral operator (over all functions in its domain) is then the entire complex plane.

This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.

We show that, under certain additional assumptions, analytic solutions of sufficiently general nonlinear functional differential equations are representable by Dirichlet series of unique structure on the entire real axis ℝ and, in some cases, on the entire complex plane ℂ. We investigate the dependence of these solutions on the coefficients of the basic exponents of the expansion into a Dirichlet series. We obtain sufficient conditions for the representability of solutions of the main initial-value problem by series of exponents.

This work explores the Julia and Mandelbrot sets of the Gamma function by extending the function to the entire complex plane through analytic continuation and functional equations. Various Julia and Mandelbrot sets associated with the Gamma function are generated using the iterative function , with different parameter values. To produce an accurate result using the integral definition of the Gamma function, a large number of terms would have to be added during the numerical integration procedure; this makes computation of Gamma function a very difficult task. To overcome this challenge, the Lanczos approximation of the Gamma function which presents an efficient and easy way to compute algorithms for approximating the Gamma function to an arbitrary precision is used. The resulting images reveal that the fractal (chaotic) behaviour found in elementary functions is also found in the Gamma function. The chaotic nature of the Julia and Mandelbrot sets provides a way of understanding complexity in systems as well as just in shapes.

Throughout the echocardiographic acoustic quantification technique, a non-invasive estimate of left ventricular volume (VV) is obtained and acquired in real-time. An automated algorithm is applied to extract the beat-by-beat values of ventricular function parameters. Their beat-by-beat variability has been recently studied, together with heart period (RR) variability, by spectral techniques. In this study, we propose to investigate this variability by nonlinear techniques, such as Poincare plots (PPS) and symbolic dynamics (SyD) patterns. A group of 17 normal young subjects (NY, 25 /spl plusmn/ 1 years) and a group of 12 normal old subjects (NO, 64 /spl plusmn/ 2 years) were studied to observe if this approach could be able to detect modifications in VV function connected with the ageing process. Differences in RR, peak filling and peak ejection rate were evidenced by PPs indexes. SyD analysis evidenced different patterns: in RR a change from increasing/decreasing in NY to alternating pattern in NO was found, together with a change in SV pattern from random in NY to alternating in NO. Consistent alternating patterns were found in VV derivative parameters. In conclusion, PPs and SyD patterns seemed able to evidence differences between NY and NO ventricular function, otherwise not captured by linear analysis techniques.

In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetric and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false–periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third–order differential equation y'''(x) = λy(x) are described. The general form of degenerate boundary conditions for the fourth–order differentiation operator D4 is found. 12 classes of boundary value eigenvalue problems are described for the operator D4, the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): are there similar problems for odd–order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): can a spectral boundary–value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary–value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker’s tenth problem is resolved.