Riemann's Theorem Research Articles

The complex Hamiltonian systems and quasi‐periodic solutions in the derivative nonlinear Schrödinger equations

AbstractThe complex Hamiltonian systems with real‐valued Hamiltonians are generalized to deduce quasi‐periodic solutions for a hierarchy of derivative nonlinear Schrödinger (DNLS) equations. The DNLS hierarchy is decomposed into a family of complex finite‐dimensional Hamiltonian systems by separating the temporal and spatial variables, and the complex Hamiltonian systems are then proved to be integrable in the Liouville sense. Due to the commutability of complex Hamiltonian flows, the relationship between the DNLS equations and the complex Hamiltonian systems is specified via the Bargmann map. The Abel‐Jacobi variable is elaborated to straighten out the DNLS flows as linear superpositions on the Jacobi variety of an invariant Riemann surface. Finally, by using the technique of Riemann‐Jacobi inversion, some quasi‐periodic solutions are obtained for the DNLS equations in view of the Riemann theorem and the trace formulas.

Quantum Electrodynamics (QED) Renormalization is a Logical Paradox, Zeta Function Regularization is Logically Invalid, and Both are Mathematically Invalid

Quantum Electrodynamics (QED) renormalizaion is a paradox. It uses the Euler-Mascheroni constant, which is defined by a conditionally convergent series. But Riemann's series theorem proves that any conditionally convergent series can be rearranged to be divergent. This contradiction (a series that is both convergent and divergent) is a paradox in classical logic, intuitionistic logic, and Zermelo-Fraenkel set theory, and also contradicts the commutative and associative properties of addition. Therefore QED is mathematically invalid. Zeta function regularization equates two definitions of the Zeta function at domain values where they contradict (where the Dirichlet series definition is divergent and Riemann's definition is convergent). Doing so either creates a paradox (if Riemann's definition is true), or is logically invalid (if Riemann's definition is false). We show that Riemann's definition is false, because the derivation of Riemann's definition includes a contradiction: the use of both the Hankel contour and Cauchy's integral theorem. Also, a third definition of the Zeta function is proven to be false. The Zeta function has no zeros, so the Riemann hypothesis is a paradox, due to material implication and vacuous subjects.

A Riemann-type theorem for a Riemann-type integral

For functions which are Henstock–Kurzweil integrable but not Lebesgue-integrable we prove a theorem which resembles the Riemann theorem on the rearrangement of conditionally convergent series.

Barycentric coordinates of Poisson — Riemann

The article deals with the problem of finding barycentric coordinates for arbitrary, simply connected, closed, discrete regions that are defined in and . Barycentric coordinates are given by a set of scalar parameters that unambiguously define a point of the affine space inside a simply connected, closed, discrete region through a specified point basis, which is given by the vertices of the region. Barycentriс coordinates being defined for the simply connected, closed, discrete region are harmonic and satisfy the properties of affine invariance, positive definiteness and equality to unit. The solution is based on the Riemann theorem on the uniqueness of conformal mapping and the Poisson integral formula for the ball. The paper shows the examples of approximation of the potential inside arbitrary, simply connected, closed, discrete regions using the proposed method, compared with the approximation using the finite element method.

A Direct Proof for Riemann Hypothesis Based on Jacobi Functional Equation and Schwarz Reflection Principle

Using the properties of theta-series and Schwarz reflection principle, a proof for Riemann hypothesis (RH) is directly presented and the first ten nontrivial zeros are easily obtained. From now on RH becomes Riemann Theorem (RT) and all its equivalent results and the consequences assuming RH are true.

Dispersionless 2D Toda hierarchy, Hurwitz numbers and Riemann theorem

We describe all formal symmetric solutions of dispersionless 2D Toda hierarchy. This classification we use for solving of two classical problems: 1) The calculation of conformal mapping of an arbitrary simply connected domain to the standard disk; 2) Calculation of 2- Hurwitz numbers of genus 0.

Tau Function Approach to Theta Functions

We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We describe the initial term of the expansion by the Schur function corresponding to the partition determined by the gap sequence of a certain flat line bundle. The other is the expansion of the theta function and its certain derivatives in one of the variables on the Abel-Jacobi images of k points on a Riemann surface with k less than or equal to g. We determine the initial term of the expansion as certain derivatives of the theta function successively. As byproducts, firstly we obtain a refinement of Riemann's singularity theorem. Secondly we determine normalization constants of higher genus sigma functions of a Riemann surface, defined by Korotkin and Shramchenko, such that they become modular invariant.

Theta divisors with curve summands and the Schottky problem

We prove the following converse of Riemann's Theorem: let (A,\Theta) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of W_{g-2}(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.

Logos, Pathos and Ethos in Ackroyd's Plato's Papers via an Interdisciplinary Psycho-philosophical Approach

The present moral crisis of humanity will be closely considered in Ackroyd's Plato's Papers via Riemann's Theorem of Earth Mapping, Kelly's Personal Construct theory and Plato's Idea of the Third Man with a view to showing that only the generally assumed relationship between logos and ethos could rescue the mentality of the entire world and retrieve temporality and morality. For this approach to be successfully turned to good account the Aristotelian concepts of Logos (the thinking part of the intellect) mediating with Pathos (the feeling self) through the actions of his acting self, the personal Ethos, will be employed in relation to Ackroyd's character, Plato, the philosopher-orator. Within Ackroyd's novel, Plato appears as a credible detector of the dark mysterious evolution of mankind, being at the same time, an important narrative device in the author's aesthetic attempt at retrieving the far-off history of London. This is accomplished through the technique of fast rewind, Ackroyd moving backwards from the year 3700 up to 3500 B.C. For the objective of the paper to be properly covered (the present moral crisis and the philosophical solution offered to it, which is also the main argument of the paper), logos will be dealt with in relation to Kelly's alternative perspectives where two concepts are equally significant: determinism and free will. The results and conclusions are related to Ethos based on choice, the choice of reconstructing humanity by recuperating light which equates with rescuing spiritual and cultural values.

On a High Dimensional Riemann's Removability Theorem

Let $M$ be a (connected) complex manifold and $E$ be a closed capacity zero set. Let $X$ be a (connected) complex compact Kobayashi hyperbolic space whose universal covering space is Stein and let $f$ be a holomorphic map of $M - E$ to $X$. Then $f$ can be extended holomorphically to a map of $M$ to $X$.

New integrable models with <italic>N</italic>-peakons and algebro-geometric solutions of soliton equations

In the soliton theory, it is a basic and challenging problem to search for integrable dynamical models with peaked soliton solutions and obtain the explicit solutions to soliton equations. This paper can be mainly divided into two parts. First, the derivation of two integrable dynamical systems with N -peakon, which provides new models for the family of nonlinear evolution equations with N -peakon; on the other hand, based on the knowledge of hyperelliptic curve, the method of nite-order expansion of Lax pairs is improved to obtain the solution of KdV6 equations. Moreover, to avoid the limit of using the Riemann theorem, the meromorphic function and the Baker-Akhiezer vector are introduced by which quasi-periodic solutions to the mKdV type equations and mixed AKNS equations are constructed according to their asymptotic properties and algebro-geometric characters.

Some Properties of Solutions to Weakly Hypoelliptic Equations

A linear different operatorLis called weakly hypoelliptic if any local solutionuofLu=0is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and anyLp-solution must vanish.

On the Lefschetz and Hodge–Riemann theorems

We give an abstract version of the hard Lefschetz theorem, the Lefschetz decomposition and the Hodge–Riemann theorem for compact Kähler manifolds. Some examples are studied for compact symplectic Kähler manifolds.

The Application of Neumann Type Systems for Solving Integrable Nonlinear Evolution Equations

An algorithm to obtain finite‐gap solutions of integrable nonlinear evolution equations (INLEEs) is provided by using the Neumann type systems in the framework of algebraic geometry. From the nonlinearization of Lax pairs, some INLEEs in 1+1 and 2+1 dimensions are reduced into a class of new Neumann type systems separating the spatial and temporal variables of INLEEs over a symplectic submanifold (M, ω2). Based on the Lax representations of INLEEs, we deduce the Lax–Moser matrix for those Neumann type systems that yield the integrals of motion, elliptic variables, and a hyperelliptic curve of Riemann surface. Then, we attain the Liouville integrability for a hierarchy of Neumann type systems in view of a Lax equation on (M, ω2) and a set of quasi‐Abel–Jacobi variables. We also specify the relationship between Neumann type systems and INLEEs, where the involutive solutions of Neumann type systems give rise to the finite parametric solutions of INLEEs and the Neumann map cuts out a finite dimensional invariant subspace for INLEEs. Under the Abel–Jacobi variables, the Neumann type flows, the 1+1, and 2+1 dimensional flows are integrated with Abel–Jacobi solutions; as a result, the finite‐gap solutions expressed by Riemann theta functions for some 1+1 and 2+1 dimensional INLEEs are achieved through the Jacobi inversion with the aid of the Riemann theorem.

Finite-gap solutions of 2+1 dimensional integrable nonlinear evolution equations generated by the Neumann systems

Each soliton equation in the Korteweg–de Vries (KdV) hierarchy, the 2+1 dimensional breaking soliton equation, and the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation are reduced to two or three Neumann systems on the tangent bundle TSN−1 of the unit sphere SN−1. The Lax–Moser matrix for the Neumann systems of degree N−1 is deduced in view of the Mckean–Trubowitz identity and a bilinear generating function, whose favorite characteristic accounts for the problem of the genus of Riemann surface matching to the number of elliptic variables. From the Lax–Moser matrix, the constrained Hamiltonians in the sense of Dirac–Poisson bracket for all the Neumann systems are written down in a uniform recursively determined by integrals of motion. The involution of integrals of motion and constrained Hamiltonians is completed on TSN−1 by using a Lax equation and their functional independence is displayed over a dense open subset of TSN−1 by a direct calculation, which contribute to the Liouville integrability of a family of Neumann systems in a new systematical way. We also construct the hyperelliptic curve of Riemann surface and the Abel map straightening out the restricted Neumann flows that naturally leads to the Jacobi inversion problem on the Jacobian with the aid of the holomorphic differentials, from which some finite-gap solutions expressed by Riemann theta functions for the 2+1 dimensional breaking soliton equation, the 2+1 dimensional CDGKS equation, the KdV, and the fifth-order KdV equations are presented by means of the Riemann theorem.

On Riemann's Rearrangement Theorem for the Alternating Harmonic Series

Let pn be the number of consecutive positive summands and let qn be the number of consecutive negative summands that appear in the classical Riemann’s rearrangement of the alternating harmonic series to sum a prescribed real number s. Assume that s > log2 and let x = (1=4)e 2s . It is shown that the sequence qn is constant equal to 1, and that the values of pn become stabilized: Eventually pn = bxc or pn =bxc + 1. Moreover, it is shown that x is rational if and only if the sequence pn is eventually periodic. The sequence pn is eventually constant if and only if x is integer, in which case pn =bxc for n big enough. Similar results are also true for s < log2.

Riemann's localization theorem. An estimate for the rate of convergence

An estimate for the Riemann localization principle for trigonometric series is established. At the same time, there is an error in treating this problem in the paper [3] which is now being corrected.

Towards an Effectivisation of the Riemann Theorem

Let Q be a connected and simply connected domain on the Riemann sphere, not coinciding with the Riemann sphere and with the whole complex plane ℂ. Then, according to the Riemann Theorem, there exists a conformal bijection between Q and the exterior of the unit disk. In this paper, we find an explicit form of this map for a broad class of domains with analytic boundaries.

A necessary and sufficient condition for Riemann's singularity theorem to hold on a Prym theta divisor

Let $(P,\Xi)$ be the naturally polarized model of the Prym variety associated to the etale double cover $\pi : \tilde C\rightarrow C$ of smooth connected curves defined over an algebraically closed field k of characteristic $\ne 2$ , where genus( C ) = $g \ge 3$ , Pic $^{(2g-2)}(\tilde C) \supset P = \{\mathcal L \in {\rm Pic}^{(2g-2)}(\tilde C) : {\rm Nm}(\mathcal L) = \omega_C$ and $h^0(\tilde C,\mathcal L)$ is even\} is the Prym variety, and $P \supset \Xi = \{\mathcal L \in P: h^0(\tilde C,\mathcal L) >0 \}$ is the Prym theta divisor with its reduced scheme structure. If $\mathcal L$ is any point on $\Xi$ , we prove that ‘Riemann's singularity theorem holds at $\mathcal L$ ’, i.e. mult $_{\mathcal L}(\Xi) = (1/2)h^0(\tilde C,\mathcal L)$ , if and only if $\mathcal L$ cannot be expressed as $\pi^*(\mathcal M)(B)$ where $B \ge 0$ is an effective divisor on $\tilde C$ , and $\mathcal M$ is a line bundle on C with $h^0(C,\mathcal M) >(1/2)h^0(\tilde C,\mathcal L)$ . This completely characterizes points of $\Xi$ where the tangent cone is the set theoretic restriction of the tangent cone of $\tilde {\Theta}$ , hence also those points on $\Xi$ where Mumford's Pfaffian equation defines the tangent cone to $\Xi$ .

Effectivisation of a String Solution of the 2D Toda Hierarchy and the Riemann Theorem About Complex Domains

Effectivisation of a String Solution of the 2D Toda Hierarchy and the Riemann Theorem About Complex Domains