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Renormdynamic equations of motion and their solutions are given. New equation for negative binomial distribution and Riemann zeta function invented.
After short introduction, we consider different aspects of the renormdynamics. Then scaling functions of the multiparticle production processes and corresponding stochastic dynamics are considered. Nonperturbative quasi-particle dynamics is considered on the base of the toy QCD-O(N)-sigma model. Last section concerns to the NBD-Riemann zeta function connection.
Using the technology of harmonic analysis, we derive a crossing equation that acts only on the scalar primary operators of any two-dimensional conformal field theory with U(1) c symmetry. From this crossing equation, we derive bounds on the scalar gap of all such theories. Rather remarkably, our crossing equation contains information about all nontrivial zeros of the Riemann zeta function. As a result, we rephrase the Riemann hypothesis purely as a statement about the asymptotic density of scalar operators in certain two-dimensional conformal field theories. We discuss generalizations to theories with only Virasoro symmetry.
In this paper, we prove the existence of a large set of Gram points t{sub n} such that the values ζ(0.5+it{sub n}) are 'anomalously' close to zero. A lower bound for the negative 'discrete' moment of the Riemann zeta-function on the critical line is also given. Bibliography: 13 titles.
It was found that, in addition to trivial zeros in points (z = − 2N, N = 1, 2…, natural numbers), the Riemann’s zeta function ζ(z) has zeros only on the line {z=(1/2)+it_0, t_0 is real}. All zeros are numerated, and for each number, N, the positions of the non-overlap intervals with one zero inside are found. The simple equation for the determination of centers of intervals is obtained. The analytical function η(z), leading to the possibility fix the zeros of the zeta function ζ(z), was estimated. To perform the analysis, the well-known phenomenon, phase-slip events, is used. This phenomenon is the key ingredient for the investigation of dynamical processes in solid-state physics, for example, if we are trying to solve the TDGLE (time-dependent Ginzburg-Landau equation).
The function Ψ(x,y,s)=e{sup iy}Φ(−e{sup iy},s,x), where Φ(z,s,v) is Lerch's transcendent, satisfies the following two-dimensional formally self-adjoint second-order hyperbolic differential equation, where s=1/2+iλ. The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space L{sub 2}(Π), where Π=(0,1)×(0,2π). We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of Ψ(x,y,s). We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.
A summation method is described for computing the vocabulary size for given parameter values in the 1- and 2-parameter rank distributions. Two methods of determining the asymptotes for the family of 2-parameter rank-distribution curves are also described. Tables are computed and graphs are drawn relating paris of parameter values to the vocabulary size. The partial product formula for the Riemann zeta function is investigated as an approximation to the partial sum formula for the Riemann zeta function. An error bound is established that indicates that the partial product should not be used to approximate the partial sum in calculating the vocabulary size for the 2-parameter rank distribution.
A new effective approach to calculating the direct and inverse conformal mapping of rectangular polygons onto a half-plane is put forward; it is based on the use of Riemann theta functions. Bibliography: 14 titles.
In the paper, we obtain a new proof of Selberg formulae related to Gram's law in the theory of the Riemann zeta function. Bibliography: 12 titles.
The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS 5 × S 5 space. If the Wilson loop is Euclidean and confined to a plane (t, x) then the dual surface is Euclidean and lives in Lorentzian AdS 3 c AdS 5 . In this paper we study such minimal area surfaces generalizing previous results obtained in the Euclidean case. Since the surfaces we consider have the topology of a disk, the holonomy of the flat current vanishes which is equivalent to the condition that a certain boundary Schrödinger equation has all its solutions anti-periodic. If the potential for that Schrödinger equation is found then reconstructing the surface and finding the area become simpler. In particular we write a formula for the Area in terms of the Schwarzian derivative of the contour. Finally an infinite parameter family of analytical solutions using Riemann Theta functions is described. In this case, both the area and the shape of the surface are given analytically and used to check the previous results.
We prove known identities for the Khinchin constant and develop new identities for the more general Hoelder mean limits of continued fractions. Any of these constants can be developed as a rapidly converging series involving values of the Riemann zeta function and rational coefficients. Such identities allow for efficient numerical evaluation of the relevant constants. We present free-parameter, optimizable versions of the identities, and report numerical results.
A simple yet robust and accurate approach for capturing shock waves using a high-order discontinuous Galerkin (DG) method is presented. The method uses the physical viscous terms of the Navier-Stokes equations as suggested by others; however, the proposed formulation of the numerical viscosity is continuous and compact by construction, and does not require the solution of an auxiliary diffusion equation. This work also presents two analyses that guided the formulation of the numerical viscosity and certain aspects of the DG implementation. A local eigenvalue analysis of the DG discretization applied to a shock containing element is used to evaluate the robustness of several Riemann flux functions, and to evaluate algorithm choices that exist within the underlying DG discretization. A second analysis examines exact solutions to the DG discretization in a shock containing element, and identifies a "model" instability that will inevitably arise when solving the Euler equations using the DG method. This analysis identifies the minimum viscosity required for stability. The shock capturing method is demonstrated for high-speed flow over an inviscid cylinder and for an unsteady disturbance in a hypersonic boundary layer. Numerical tests are presented that evaluate several aspects of the shock detection terms. The sensitivity of the results to model parameters is examined with grid and order refinement studies.
At present, there are numerous multidimensional generalizations of holomorphic vectors. The most general of these is the four-dimensional generalization of the Cauchy–Riemann system. In the present work, by introducing two quaternion functions and the notion of quaternion differentiation, we obtain, for the first time, a five-dimensional generalization of holomorphic vectors. By using the representation of holomorphic vectors via the quaternion harmonic function and its derivatives, we consider the Riemann–Hilbert problem and one problem in a layer. A new solution of the Riemann–Hilbert problem in the five-dimensional half space is obtained.
We show that a generalized version of the 4-point string theory KLT double-copy map is the most general solution to the minimal-rank double-copy bootstrap in effective field theory. This follows from significant restrictions of the 4-point map resulting from the 6-point bootstrap analysis. The generalized 4-point double-copy map is defined by a function with only two parameters times a simple function that is symmetric in s,t,u. The two parameters can be interpreted as independent choices of α', one for each of the two sets of amplitudes double-copied with the map. Specifically, each of those two sets of amplitudes must obey either the string monodromy relations or the field theory KK & BCJ relations; there are no other options. We propose a closed form of the new double-copy map that interpolates between the original KLT string double-copy and the open & closed string period integrals. The construction clarifies the "single-valued projection" property of the Riemann zeta-function values for the 4-point string theory double copy.
The results of this work are referred to the well-known trend of the geometric theory of functions of complex variable, namely, to the extreme problems on the classes of nonoverlapping domains. It was started by Lavrent’ev’s classical work [1], where, in particular, the problem of the product of conformal radii of two nonoverlapping domains was first solved. Now, this trend is intensively developed. The main results can be found in [2–8] and [9–13]. Our results present a generalization of some results in [7].
Covered space, Riemann geometry extension, and partition function research on stochastic models for multivalued, multidimensional relations