边加权有限图的Weil-Riemann-Roch定理

Riemann-Roch定理是数学中的一个重要结论,并有了广泛的应用。在有限图和边加权有限图等图中也有对应的Riemann-Roch定理以及应用,但所有这些工作都有一个共同点,那就是它们都聚焦于在除子或和除子线性等价的线丛的情况下,也就是秩为1的情况。为了得到高维秩的情形,可以借助多重除子的术语来描述。本文利用还原群GLn的root datum的概念给出了边加权有限图上主GLn-丛——向量丛的定义,并用多重除子的术语来描述向量丛,进而给出了边加权有限图的Weil-Riemann-Roch定理以及证明,推广了GROSS A.ULIRSCH M.和ZAKHAROV D的结果。

Proof of Riemann Conjecture Based on Contradiction between Xi-Function and Its Product Expression

Riemann proved three results: analytically continue ζ(s) over the whole complex plane s =σ + it with a pole s =1;(Theorem A) functional equation ξ(t) = G(s0)ζ (s0), s0 =1/2 + it and (Theorem B) product expression ξ1(t) by all roots of ξ(t). He stated Riemann conjecture (RC): All roots of ξ (t) are real. We find a mistake of Riemann: he used the same notation ξ(t) in two theorems. Theorem B must contain complex roots;it conflicts with RC. Thus theorem B can only be used by contradiction. Our research can be completed on s0 =1/2 + it. Using all real roots rk and (true) complex roots zj = tj + iaj of ξ (z), define product expressions w(t), w(0) =ξ(0) and Q(t) > 0, Q(0) =1 respectively, so ξ1(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s0)| =ω} for small ω > 0. If ξ(t) has complex roots, then ω =ωQ(t) on L(ω). Finally in a large interval of the first module |z1|>>1, we can find many points t ∈ L(ω) to make Q(t) . This contraction proves RC. In addition, Riemann hypothesis (RH) ζ for also holds, but it cannot be proved by ζ.

Stability Analysis of a Class of Nonlinear Fractional Differential Systems With Riemann-Liouville Derivative

Dear Editor,This letter investigates the stability of n-dimensional nonlinear fractional differential systems with Riemann-Liouville derivative.By using the Mittag-Leffler function,Laplace transform and the Gronwall-Bellman lemma,one sufficient condition is attained for the asymptotical stability of a class of nonlinear fractional differential systems whose order lies in(0,2).According to this theory,if the nonlinear term satisfies some conditions,then the stability condition for nonlinear fractional differential systems is the same as the ones for corresponding linear systems.Two examples are provided to illustrate the applications of our result.

How to Prove Riemann Conjecture by Riemann’s Four Theorems

Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

AGGREGATE SPECIAL FUNCTIONS TO APPROXIMATE PERMUTING TRI-HOMOMORPHISMS AND PERMUTING TRI-DERIVATIONS ASSOCIATED WITH A TRI-ADDITIVEψ-FUNCTIONAL INEQUALITY IN BANACH ALGEBRAS

In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.

Definite Answer for Riemann Hypothesis Zeta 3/2 Function Provided by New Material Yb2Si2O7 in Quantum Mechanics

This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x^,p^]=1/2in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.

Machine Learning Approaches for the Solution of the Riemann Problem in Fluid Dynamics:a Case Study

We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube.The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin methods.Therefore,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance.The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation.Prior to delving into the complexities of ML for the Riemann problem,we consider a much simpler formulation,yet very informative,problem of learning roots of quadratic equations based on their coefficients.We compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)approximations.Among these approaches,NNs prove to be more robust and efficient,although GP can be appreciably more accurate(about 30\%).We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state(EOS).We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.

耦合Aw-Rascle-Zhang模型的Riemann解及其稳定性

该文主要研究在单连通道路上具有不同压力项的耦合Aw-Rascle-Zhang(CARZ)交通流模型的黎曼问题,利用特征分析法和相变的相关理论,构造了文献[5]缺失的情形v−+η(ρ−)^(γ)=v+的稳定显式解,并修正了情形v++η(ρ−)γ<v+的黎曼解,完善了Herty[5]等人的工作.当CARZ模型中压力项前面的参数μ→η时,证明了该模型黎曼解的唯一性和稳定性.

协同拟凸函数的Riemann-Liouville分数阶积分不等式

基于Riemann-Liouville分数阶积分,对协同拟凸函数的Hermite-Hadamard分数阶积分不等式进行了研究。剖析Riemann-Liouville分数阶积分的运算特点,深入探究SAR?KAYA给出的Riemann-Liouville分数阶积分恒等式。在该Riemann-Liouville分数阶积分恒等式的基础上,利用二元函数的单调性和协同拟凸性,巧妙应用三角不等式和H?lder不等式等经典不等式,建立了若干个协同拟凸函数的Hermite-Hadamard分数阶积分不等式。

基于Riemann-Liouville分数阶积分Bernstein-Kantorovich算子的逼近性质

文章构造一种基于Riemann-Liouville分数阶积分的Bernstein-Kantorovich算子。利用一阶、二阶光滑模、Peetre’-K泛函和Lipschitz型极大函数等工具研究该算子的近似性质。然后利用一阶、二阶和四阶中心矩对该算子建立Vorononskaja型渐进公式。该算子的构建使得在曲线曲面造型方面对于给定的函数将会有更小的近似误差。

一类新型自适应反扩散近似Riemann求解器及其应用

对于包含激波、剪切层等复杂结构的流动问题,为了精确模拟剪切层等精细结构,且保证激波计算的稳定性,必须采用低耗散且强鲁棒的数值通量方法。传统的HLL近似Riemann求解器的耗散性较大,Roe、HLLEM和HLLC等近似Riemann求解器在计算某些含有强激波的物理问题时会出现非物理解,容易导致不稳定。针对这一问题,本文在Riemann求解器中通过合理设计反扩散矩阵,发展了一类具有自适应反扩散的新型Riemann求解器,并将其应用到高阶加权紧致格式,实现了高阶精度求解。通过典型数值算例验证了新型方法的计算精度和稳定性,结果表明本文提出的新型自适应反扩散Riemann求解器克服了传统Riemann求解器的缺陷,既能准确识别剪切层等精细结构,又能保证激波解的稳定性。

On the Absence of Zeros of Riemann Zeta-Function Out of ℜ(z) = 1/2

This work shows, after a brief introduction to Riemann zeta function , the demonstration that all non-trivial zeros of this function lies on the so-called “critical line”,, the one Hardy demonstrated in his famous work that infinite countable zeros of the above function can be found on it. Thus, out of this strip, the only remaining zeros of this function are the so-called “trivial ones” . After an analytical introduction reminding the existence of a germ from a generic zero lying in , we show through a Weierstrass-Hadamard representation approach of the above germ that non-trivial zeros out of cannot be found.

Riemann-Liouville型分数阶导数的非线性估计

本文主要研究任意有界连续信号的Riemann-Liouville分数阶导数估计问题.当分数阶α属于0到1时,首先利用滑模技术提出一种有界连续信号分数阶导数的非线性估计方法;然后将其结果推广至分数阶α∈R+的情况,并给出相应的非线性估计方案.借助Riemann-Liouville分数阶微积分频率分布模型,本文详细分析讨论了所给分数阶导数非线性估计的收敛性问题,并得到相应闭环系统是渐近稳定的结论.文中所提方法的主要优点是在事先未知给定信号分数阶导数上界的情况下,不仅能自适应地估计其Riemann-Liouville分数阶导数,而且当信号中含有随机噪声和不确定扰动时依然能正常工作.数值仿真实例验证了本文所给估计方法的可行性和有效性.

The Symmetry of Riemann <i>ξ</i>-Function

To prove RH, studying ζ and using pure analysis method likely are two kinds of the incorrect guide. Actually, a unique hope may study Riemann function by geometric analysis, which has the symmetry: v = 0 if β = 0, and Assume that |u| is single peak in each root-interval of u for any fixed β ∈ (0,1/2], using the slope ut of the single peak, we prove that v has opposite signs at two end-points of Ij, there surely is an inner point so that v = 0, so {|u|,|v|/β}form a local peak-valley structure, and have positive lower bound in Ij. Because each t must lie in some Ij , then ||ξ|| > 0 is valid for any t. In this way, the summation process of ξ is avoided. We have proved the main theorem: Assume that u (t, β) is single peak, then RH is valid for any . If using the equivalence of Lagarias (1999), the assumption of single peak can be canceled. Therefore our new thinking is that we have found the local peak-valley structure of ξ, which may be the geometry structure expected by Bombieri (2000), and proposed a basic framework of proving RH by geometric analysis.

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures,and the solutions admit the concentration of mass.It is found that under the requirement of satisfying the over-compressing entropy condition:(i)there is a unique delta shock solution,corresponding to the case that has two strong classical Lax shocks;(ii)for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave,or two shocks with one being weak,there are infinitely many solutions,each consists of a delta shock and a rarefaction wave;(iii)there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves.These solutions are self-similar.Furthermore,for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data,there always exists a unique delta shock for at least a short time.It could be prolonged to a global solution.Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass(particle).Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.We also discuss possible physical interpretations and applications of these new solutions.

指数函数曲线上的矩阵值Riemann边值问题

本文讨论的是指数函数曲线上的矩阵值Riemann边值问题,我们主要是对一类特殊的下三角矩阵值Riemann边值问题进行求解。首先使用双线性形式给出指数函数曲线上的伪正交多项式;其次给出特殊的下三角矩阵值边值问题并转化为边值问题;最后使用Liouville定理和伪正交多项式给出矩阵值Riemann边值问题的求解。

关于Riemann-Lebesgue引理的一个推广

通过“把Riemann-Lebesgue引理看做是在一个周期上可积的周期函数的一种性质”的这个角度,提出了亚周期函数的概念,并得到有界亚周期函数的刻画.

一类Riemann-Liouville分数阶时滞随机发展方程的Ulam-Hyers稳定性

用不动点定理研究Hilbert空间中一类含α∈(0,1)阶Riemann-Liouville分数阶导数的时滞随机发展方程温和解的存在唯一性,并证明该解的Ulam-Hyers稳定性.最后举例说明所得结论的适用性.

Comparative study of the adsorption performance of NH_(2)-functionalized metal organic frameworks with activated carbon composites for the treatment of phenolic wastewaters

To better understand the role of the-NH_(2)group in adsorption process of phenolic wastewaters,NH_(2)-functionalized MIL-53(Al)composites with activated carbon(NH_(2)-M(Al)@(B)AC)were prepared.The results showed that the-NH_(2)group could increase the mesopore volume for composites,which promotes mass transfer and full utilization of active sites,because hierarchical mesopore structure makes the adsorbent easier to enter the internal adsorption sites.Furthermore,the introduction of the-NH_(2)group can improve the adsorption capacity,decrease the activation energy,and enhance the interaction between the adsorbent and p-nitrophenol,demonstrating that the-NH_(2)group plays a crucial role in the adsorption of p-nitrophenol.The density functional theory calculation results show that the H-bond interaction between the-NH_(2)group in the adsorbent and the-NO_(2)in the p-nitrophenol(adsorption energy of -35.5 kJ·mol^(-1)),and base-acid interaction between the primary-NH_(2)group in the adsorbent and the acidic-OH group in the p-nitrophenol(adsorption energy of -27.3 kJ·mol^(-1))are predominant mechanisms for adsorption in terms of the NH_(2)-functionalized adsorbent.Both NH_(2)-functionalized M(Al)@AC and M(Al)@BAC composites exhibited higher p-nitrophenol adsorption capacity than corresponding nonfunctionalized composites.Among the composites,the NH_(2)-M(Al)@BAC had the highest p-nitrophenol adsorption capacity of 474 mg·g^(-1).