Search

Zhang and Zheng (1990) conjectured on the structure of a solution for a two-dimensional Riemann problem for Euler equation. To resolve this illuminating conjecture, many researchers have studied the simplified 2 × 2 systems. In this paper, 3-pieces Riemann problem for two-dimensional 2 × 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave. We classify twelve topologically distinct solutions and construct analytical and numerical solutions. The computed numerical solutions clearly confirm the constructed analytic solutions.

We describe the limit surfaces of some equisymmetric $1$-complex dimensional families of Riemann surfaces in the boundary of the Deligne-Mumford compactification of moduli space. We provide a description of such nodal Riemann surfaces in terms of the group of automorphisms defining the family. We apply our method to some known examples.

The original proof-argument of Riemann in 1851 for the Riemann mapping theorem, one of the most central theorems in Complex analysis, was found faulty and essentially buried underneath the proof by $Carath{\acute{e}}odory$ of 1929, now accepted as the "textbook" proof. On the other hand,...

We present an example, different from Volterra's, of a bounded derivative that is not Riemann integrable. The existence of such functions was one of the motivations for Lebesgue to devise a stronge...

We solve the Riemann boundary value problem for holomorphic functions on a periodic non-rectifiable curve. Probably, this is the first study of the Riemann problem on infinite curve, which is not rectifiable. The solution is obtained in terms of generalized integrals over non-rectifiable path.

In this article, a new canonical function has been established to deal with Riemann boundary-value problem of periodic analytic functions discussed in [16]. In comparison with the corresponding result in [16], the expression of solution obtained here is much simpler. Then, we demonstrate the equivalence of solutions for the homogeneous Riemann problem. What's more, we obtain the precise rank of matrix of coefficients for the system of linear algebraic equations (4.35) in [16]. Those results can simplify the discussion of Riemann problem of single-periodic polyanalytic functions in [16].

We consider single-valued meromorphic functions and differentials on finite Riemann surfaces. We construct elementary q-differentials of all types. We find the dimension of two important quotient spaces of Abelian q-differentials, in particular, the first holomorphic de Rham cohomology group of 1-differentials. For these quotient spaces we construct bases that locally holomorphically depend on the moduli of the Riemann surfaces. Bibliography: 6 titles.

Our purpose is to define composition operators acting upon Hardy spaces of Riemann surfaces. In terms of counting functions related to analytic self-map on Riemann surfaces, the boundedness and compactness are characterized.

Natural Riemann extensions are pseudo-Riemannian metrics (introduced by Sekizawa and studied then by Kowalski–Sekizawa), which generalize the classical Riemann extension defined by Patterson–Walker. Let \(M\) be a manifold with an affine connection and let \(T^{*}M\) be the total space of its cotangent bundle. On \(T^{*}M\) endowed with a natural Riemann extension, we study here the Laplacian and give necessary and sufficient conditions for the harmonicity of a certain family of (local) functions. We also prove a gradient formula for natural Riemann extensions.

Thus, the Riemann hypothesis is an assertion concerning the infinite sequence of coefficients γ0, γ1, . . . known as Stieltjes constants (γ0 = γ = 0.577215 . . . is the Euler constant). As is well known, to prove the Riemann hypothesis, it would be sufficient to establish the validity of an appropriate infinite system of polynomial inequalities P1(γ0, . . . , γm1) > 0, . . . , Pn(γ0, . . . , γmn) > 0, . . . , (2) each of which contains only finitely many Stieltjes constants (and possibly also some classical constants) and hence admits a numerical verification. Polynomials Pn giving this reformulation of the Riemann hypothesis can be chosen in many ways. In one of the most well-known ways described in [1] (see also [2]), one has

The Riemann problem for a special Keyfitz-Kranzer system is investigated and then seven different Riemann solutions are constructed. When the initial data are chosen as three piecewise constant states under suitable assumptions, the global solutions to the perturbed Riemann problem are constructed explicitly by studying all occurring wave interactions in detail. Furthermore, the stabilities of solutions are obtained under the specific small perturbations of Riemann initial data.

This chapter completes the task of establishing the fundamentals of Calculus, this time studying the operation of integration on functions. As we shall see in the next section, in its most simple form this reduces to the computation of areas under the graphs of nonnegative continuous functions \(f: [a,b] \rightarrow \mathbb{R}\), suggesting that geometric intuition will play a strong role throughout. Among other byproducts of the coming discussions, we prove the irrationality of π and study the properties of two of the most ubiquitous functions of Mathematics, the exponential and logarithmic functions with base e.

We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann’s predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler.

We introduce a family of deformations of the Riemann xi-function endowed with two continuous parameters. We show that it has rich analytic structure and that its conjectural (mild) zero-free region for some fixed parameter is a sufficient condition for the Riemann hypothesis to hold for the Riemann zeta function.

The global solutions of the perturbed Riemann problem for the Leroux system are constructed explicitly under the suitable assumptions when the initial data are taken to be three piecewise constant states. The wave interaction problems are widely investigated during the process of constructing global solutions with the help of the geometrical structures of the shock and rarefaction curves in the phase plane. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.

In the present paper, we study the Riemann problem for degenerate equations in which the pressure is constant over some range of stains. By the Liu-entropy condition, the Riemann solutions of twelve regions in the v-u plane are completely constructed. Moreover, we obtain the invariant regions for the Riemann solutions. (C) 2012 Elsevier Inc. All rights reserved.

It is proved that the generalized Riemann problem for the Aw–Rascle vehicular traffic flow with phase transitions admits a unique local solution in the class of piecewise C 1 functions and in a neighborhood of the origin this solution possesses a structure similar to the solution of the corresponding Riemann problem. Especially, when the vacuum state is involved, the Riemann solution is also well-posed, which allows us to fit experimental data well.

In this paper, we give both lower and upper bound estimates of the Bergman kernel for a degeneration of Riemann surfaces with constant curvature −1. As a result, we give a geometric proof of the Riemann–Rock theorem for a singular Riemann surface.

Vortex motion of an incompressible inviscid flow on a infinitely thin curved shell specified as a Riemann surface is studied. The intrinsic dynamical equations are derived and compared on two conformally equivalent Riemann surfaces. In particular, the relations for the streamfunction, velocity, and vorticity are found by using explicit computations. Then, the result is applied to the problem of motion of point vortex dipoles on Riemann surfaces whose trajectories in time turn out to be a geodesic. Also, the case of more complicated surfaces with a boundary, is discussed with an example.

A natural Riemann extension is a natural lift of a manifold with a symmetric affine connection to its cotangent bundle. The corresponding structure on the cotangent bundle is a pseudo-Riemannian metric. The classical Riemann extension has been studied by many authors. The broader (two-parameter) family of all natural Riemann extensions was found by the second author in 1987. We prove the equivariance property for the natural Riemann extensions. We also prove some theorems for Ricci curvature and scalar curvature.

We discuss some recent results on algebraic differential properties of the Riemann ζ-function and the Euler Γ-function and prove two results in this direction. Especially, we will define a new class of meromorphic functions in terms of zeros of the Riemann ζ-function on the critical line and prove that algebraic equations with coefficients in this class cannot be satisfied by the Riemann ζ-function and the Euler Γ-function.

We give a complete classification up to equivalence of the holomorphic mappings of a Riemann surface of genus 3 onto a Riemann surface of genus 2. Also we establish that every Riemann surface of genus 3 has at most three holomorphic images of genus 2.

The first part contains a few facts about Riemann taken from correspondence (of Riemann and of Dedekind), and then describes Riemann’s letter to Weierstrass concerning his paper on the distribution of primes. It is followed by a brief outline of the paper itself.

The structure of Riemann solutions for certain systems of conservation laws can be so complicated that the classical constructions are unable to establish global existence and stability. For systems of two conservation laws, classically the local solution is found by intersecting two wave curves specified by the Riemann data. The intersection point 2000 Mathematics Subject Classification. 35L65.

We prove that the Riemann zeta-function is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in Γ, Γ' and Γ" over the field of complex numbers.

A compact Riemann surface [Formula: see text] of genus [Formula: see text] is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real [Formula: see text]-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order [Formula: see text] such that the quotient spaces have genus [Formula: see text].

Once-holed tori are the most primitive noncompact Riemann surfaces of positive genus, and can be used to measure the sizes of handles of Riemann surfaces of positive genus. We study some families of once-holed tori that are conformally embedded in target Riemann surfaces of conformal mappings of a given noncompact Riemann surface of genus one, and correct some results given in Masumoto (Math. Z. 257:453–464, 2007).

The Riemann zeta-function ζ has the following well-known properties (M) It is meromorphic in ℂ with a simple pole at z = 1 with residue 1. (SR) The symmetry relation ζ(z) = $$ \overline {\zeta \left( {\bar z} \right)} $$ holds for z ≠ 1. (FE) The following functional equation holds: $$ \zeta \left( z \right)\Gamma \left( {{z \mathord{\left/ {\vphantom {z 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\pi ^{ - {z \mathord{\left/ {\vphantom {z 2}} \right. \kern-\nulldelimiterspace} 2}} = \zeta \left( {1 - z} \right)\Gamma \left( {{{\left( {1 - z} \right)} \mathord{\left/ {\vphantom {{\left( {1 - z} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\pi ^{ - {{\left( {1 - z} \right)} \mathord{\left/ {\vphantom {{\left( {1 - z} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} $$ . Moreover, ζ has a universality property due to Voronin (1975). We show that arbitrarily close approximations of the Riemann zeta-function that satisfy (M),(SR),(FE) may have a different universality property. Consequently, these approximations do not satisfy the Riemann hypothesis. Moreover, we investigate the set of all “Birkhoff-universal” functions satisfying (M),(SR),(FE).

In this article, we deal with boundary value problems on a symmetric Riemann surface in the sense of Klein, showing how symmetric conditions on the boundary translate into symmetric solutions, which generate solutions to similar problems for the non-orientable surface. We use the model of analysis on non-orientable Riemann surfaces introduced in Barza (Calculus on non-orientable Riemann surfaces, Libertas Math., Vol. XV (1995), 1–45); Barza (Integration on non-orientable Riemann surfaces. In: Almost Complex Structures. (Eds, K. Sekigawa and S. Dimiev). World Scientific, Singapore, New Jersey, London, Hong Kong, 1995, 63–97) and Barza and Ghisa (Boundary value problems on non-orientable surfaces. Rev. Roumaine Math. Pures Appl., 43(1998), pp. 67–79.) and the solutions given by Gakhov (Boundary Value Problems. Moscow, 1963 (Russian); English translation, Pergamon Press, Oxford, 1966) and Muskhelishvili (Singular Integral Equations. Moscow, 1946 (Russian); English translation, Noordhoff, Groningen, 1953).

We find the upper bound for the total number of ovals of k (5 ≤ k ≤ 8) non-conjugate symmetries of a Riemann surface of genus g, without an additional assumption concerning their separabilities, filling this way the gap existing in known literature of the subject. We also prove that our bound is attained for infinitely many values of g. Finally, as a byproduct of our considerations, we obtain the algebraic structure of the group generated by the extremal configurations of the symmetries, provided it is a 2-group.

In this paper, we study ovals of symmetries and the fixed points of their products on Riemann surfaces of genus g ≥ 2. We show how the number of these points affects the total number of ovals of symmetries. We give a generalisation of Bujalance, Costa and Singerman's theorems in which we show upper bounds for the total number of ovals of two symmetries in terms of g, the order n and the number m of the fixed points of their product, and we show their attainments for n holding some divisibility conditions. Finally, we give an upper bound for m in terms of n and g, and we study conditions under which it has given parity.

Involutions without fixed points on hyperbolic closed Riemann surface are discussed. For an orientable surface $X$ of even genus with an arbitrary Riemannian metric $d$ admitting an involution $\tau$, it is known that $\min_{p\in X}d(p,\tau(p))$ is bounded by a constant which depends on the genus of $X$. The equivalent result is proved to be false in odd genus, and the optimal constant for hyperbolic Riemann surfaces is calculated in genus 2., Comment: 12 pages, 8 figures

tion of the exact number of primes < x, which is denoted by n(x), from the estimate x/ log x that had been conjectured by Gauss, Legendre, and others. Riemann alluded to returning to this matter later by saying that he was "setting it aside for the time being." Apparently Riemann did not live long enough to do that. To this day, no one has been able to prove the Riemann hypothesis despite overwhelming numerical evidence in its favor. However, many generalizations and analogs of the Riemann zeta function have been formulated by, among others, Dirichlet, Dedekind, E. Artin, F. K. Schmidt, and Weil, and the Riemann hypothesis has been shown to be true in some of these cases. One such case is the Riemann hypothesis for elliptic curves, originally conjectured by E. Artin (see [1, pp. 1-94]) and proved by Hasse, and therefore also known as Hasse's theorem. We begin by laying out the statement of this result in Section 2 below. We then turn to the two main topics of this article: (i) a brief explanation of the fact that these two Riemann hypotheses are not only closely analogous, but indeed two examples of a single more general framework; and (ii) an elementary proof of the Riemann hypothesis for elliptic curves over finite fields. This is carried out in Sections 3 and 4 respectively, and these may be read independently of one another. Our proof is based on an idea of Manin. The presentation is self-contained except for an appeal to the "Basic Identity," which is a technical lemma stated in (19) be low. The proof of the Basic Identity, although somewhat complicated, is completely elementary (see [5]) and is the least illuminating part of our proof of this Riemann hypothesis.

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2 + iλn. By constructing a continuous family of scaling-like operators involving the Gauss–Jacobi theta series and by invoking a novel [Formula: see text]-invariant Quantum Mechanics, involving a judicious charge conjugation [Formula: see text] and time reversal [Formula: see text] operation, we show why the Riemann Hypothesis is true. An infinite family of theta series and their Mellin transform leads to the same conclusions.

The Riemann hypothesis (RH) states that the non-trivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert–Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu–Sprung potential (that capture the average level density of zeros) by recurring to a weighted superposition of Weierstrass functions ∑p W(x, p, D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu–Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss–Jacobi theta series. An explicit completion relation ("trace formula") related to a superposition of eigenfunctions of these scaling-like operators is defined. If the completion relation is satisfied, this could be another test of the Riemann Hypothesis. In an appendix, we briefly describe our recent findings showing why the Riemann Hypothesis is a consequence of [Formula: see text]-invariant Quantum Mechanics, because [Formula: see text] where s are the complex eigenvalues of the scaling-like operators. We show why [Formula: see text] invariance requires that s(1 - s) = real , which implies that s is real and/or it lies in the critical Riemann line.

The Fundamental Theorem of Calculus is a statement about the inverse relationship between differentiation and integration. It comes in two parts, depending on whether we are differentiating an integral or integrating a derivative. Under suitable hypotheses on the functions f and F, the Fundamental Theorem of Calculus states that $$ \begin{array}{l} \left( i \right)\,\int_a^b {F'\left( x \right)} dx = F\left( b \right) - F\left( a \right)\;and\\ \left( {ii} \right)\,if\,G\left( x \right) = \int_a^x {f\left( t \right)} dt,\,then\,G'\left( x \right) = f\left( x \right). \end{array} $$

Let X be a 2-dimensional C 0 manifold in the sense of Chap. IX, n0 11, (ii). If (U, φ) is a chart for X, φ may be considered a homeomorphism from the open set U onto an open subset of \(\mathbb{C}\,:\,{\rm if }{\, \xi_1(x), \xi_2(x)}\) are the coordinates of \(\varphi(x)\,\in\,{{\mathbb{R}}^2}\,{\rm for}\,x \in U \), it suffices to agree that \(\varphi(x)\,=\,\xi_1(x)\,+\,i\xi_2(x) \).

We return to the conservation law (1.2), but now study the case of systems, i.e., $$\displaystyle u_{t}+f(u)_{x}=0,$$ (5.1) where \(u=u(x,t)=(u_{1},\dots,u_{n})\) and \(f=f(u)=(f_{1},\dots,f_{n})\in C^{2}\) are vectors in \(\mathbb{R}^{n}\). (We will not distinguish between row and column vectors, and use whatever is more convenient.) Furthermore, in this chapter we will consider only systems on the line; i.e., the dimension of the underlying physical space is still one. In Chapt. 2 we proved existence, uniqueness, and stability of the Cauchy problem for the scalar conservation law in one space dimension, i.e., well-posedness in the sense of Hadamard. However, this is a more subtle question in the case of systems of hyperbolic conservation laws. We will here first discuss the basic concepts for systems: fundamental properties of shock waves and rarefaction waves. In particular, we will discuss various entropy conditions to select the right solutions of the Rankine–Hugoniot relations.

We study a nonlinear hyperbolic system of balance laws that arises from an entropy-based moment closure of a kinetic equation. We show that the corresponding homogeneous Riemann problem can be solved without smallness assumption, and we exhibit invariant regions.

This article introduces one of the outstanding problems of today-a legacy of Bernhard Riemann

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Fredholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on \( \Lambda _{2,a}^1 (M) \)for Riemann surface M are defined and some properties of these operators are discussed.

A conjecture of Ehrenpreis states that any two compact Riemann surfaces of genus at least two have finite degree unbranched holomorphic covers that are arbitrarily close to each other in moduli space. Here we prove a weaker result where certain branched covers associated with arithmetic Riemann surfaces are allowed, and investigate the connection of our result with the original conjecture.