scholarly journals The strong Suslin reciprocity law

2021 ◽  
Vol 157 (4) ◽  
pp. 649-676
Author(s):  
Daniil Rudenko
Keyword(s):  
Rational Functions ◽  
Projective Curve ◽  
Main Ingredient ◽  
Homotopy Invariance ◽  
Reciprocity Law ◽  
Hyperbolic Volume ◽  
Scissors Congruence ◽  
Dehn Invariant ◽  
Contracting Homotopy ◽  
Invariance Theorem

We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$ -theory. The Milnor $K$ -groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

scholarly journals -algebras associated with curves and rational functions on . I

2014 ◽  
Vol 150 (4) ◽  
pp. 621-667 ◽  
Author(s):  
Robert Fisette ◽  
Alexander Polishchuk
Keyword(s):  
Homotopy Class ◽  
Rational Functions ◽  
Coherent Sheaf ◽  
Linear Series ◽  
Projective Curve ◽  
Canonical Embedding ◽  
Rational Map ◽  
Projective Embedding ◽  
Smooth Projective Curve ◽  
Weierstrass Points

AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater than$1$. In the case$n=g$we prove that the$A_{\infty }$-structure is determined uniquely (up to homotopy) by the products$m_i$with$i\le 6$. Also, in this case we study the rational map$\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$associated with the homotopy class of$m_3$. We prove that for$g\ge 6$it is birational onto its image, while for$g\le 5$it is dominant. We also give an interpretation of this map in terms of tangents to$C$in the canonical embedding and in the projective embedding given by the linear series$|2(p_1+\cdots +p_g)|$.

scholarly journals On $$\mathsf {G} $$-isoshtukas over function fields

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Paul Hamacher ◽  
Wansu Kim
Keyword(s):  
Moduli Spaces ◽  
Function Fields ◽  
Rational Functions ◽  
Conjugacy Classes ◽  
Projective Curve ◽  
Smooth Projective Curve

AbstractIn this paper we classify isogeny classes of global $$\mathsf {G} $$ G -shtukas over a smooth projective curve $$C/{\mathbb {F}}_q$$ C / F q (or equivalently $$\sigma $$ σ -conjugacy classes in $$\mathsf {G} (\mathsf {F} \otimes _{{\mathbb {F}}_q} \overline{{\mathbb {F}}_q})$$ G ( F ⊗ F q F q ¯ ) where $$\mathsf {F} $$ F is the field of rational functions of C) by two invariants $${\bar{\kappa }},{\bar{\nu }}$$ κ ¯ , ν ¯ extending previous works of Kottwitz. This result can be applied to study points of moduli spaces of $$\mathsf {G} $$ G -shtukas and thus is helpful to calculate their cohomology.

scholarly journals A nonlinear population dynamics model of patient diagnosis and treatment involving in two level medical institutions and its qualitative analysis of positive singularity

2022 ◽  
Vol 19 (3) ◽  
pp. 2575-2591
Author(s):  
Xiaoxia Zhao ◽  
◽  
Lihong Jiang ◽  
Kaihong Zhao ◽  
◽  
...  
Keyword(s):  
Sufficient Conditions ◽  
Practical Application ◽  
Dynamics Model ◽  
Qualitative Properties ◽  
Homotopy Invariance ◽  
Patient Diagnosis ◽  
Population Dynamics Model ◽  
Medical Institutions ◽  
And Function ◽  
Invariance Theorem

<abstract><p>In this article, we firstly establish a nonlinear population dynamical model to describe the changes and interaction of the density of patient population of China's primary medical institutions (PHCIs) and hospitals in China's medical system. Next we get some sufficient conditions of existence of positive singularity by utilising homotopy invariance theorem of topological degree. Meanwhile, we study the qualitative properties of positive singularity based on Perron's first theorem. Furthermore, we briefly analyze the significance and function of the mathematical results obtained in this paper in practical application. As verifications, some numerical examples are ultimately exploited the correctness of our main results. Combined with the numerical simulation results and practical application, we give some corresponding suggestions. Our research can provide a certain theoretical basis for government departments to formulate relevant policies.</p></abstract>

Dehn Invariant and Scissors Congruence of Flexible Polyhedra

2018 ◽  
Vol 302 (1) ◽  
pp. 130-145 ◽  
Author(s):  
Alexander A. Gaifullin ◽  
Leonid S. Ignashchenko
Keyword(s):  
Scissors Congruence ◽  
Dehn Invariant

A Generalization of Global Class Field Theory

1969 ◽  
Vol 21 ◽  
pp. 609-614
Author(s):  
Tae Kun Seo ◽  
G. Whaples
Keyword(s):  
Field Theory ◽  
Finite Field ◽  
Class Group ◽  
Rational Functions ◽  
Class Field Theory ◽  
Usual Sense ◽  
Finite Degree ◽  
Class Field ◽  
Reciprocity Law

Let R be a field of rational functions of one variable over a field of constants R0. Dock Sang Rim (6) has proved that the global reciprocity law in exactly the usual sense holds whenever R0 is an absolutely algebraic quasi-fini te field of characteristic not equal to 0: this was known before only when R0 was a finite field. We shall give another proof of Rim's result by means of a noteworthy generalization of the usual global reciprocity law. Namely, let R0 be a finite field and let F be the set of all fields k contained in some fixed Ralg.clos. and of finite degree over R. The reciprocity law states that there exists a family {fk}, k ∈ F, of functions fk: Ck → G(kabel.clos./k) (where Ck is the idèle class group of k) enjoying certain properties such as the norm transfer law.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

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