Finiteness Theorem Research Articles

Subgroups of bounded rank in hyperbolic 3‐manifold groups

AbstractWe prove a finiteness theorem for subgroups of bounded rank in hyperbolic 3‐manifold groups. As a consequence, we show that every bounded rank covering tower of closed hyperbolic 3‐manifolds is a tower of finite covers associated to a fibration over a 1‐orbifold.

Equilibria in abstract economies with a continuum of agents with discontinuous and non-ordered preferences

This paper focus on the problem of the existence of an equilibrium in abstract economies and exchange economies. Spanning over the literature we have managed to extend and generalize some previous results. In particular, we generalize the main theorem of Yannelis (1987) on the existence of an equilibrium in an abstract economy with a continuum of agents, by allowing for discontinuous preferences. As a corollary of this result, we extend the finite agent Cournot–Nash equilibrium existence theorems with discontinuous preferences (e.g., Reny, 1999; Bareli and Meneghel, 2013; He and Yannelis, 2016; among others), to a continuum of agents. We also obtain an existence theorem for an abstract economy which allows for a convexifying effect on aggregation and nonconvex strategy and constraint sets. Furthermore, our new main theorem is used to prove the existence of a Walrasian equilibrium with a continuum of agents with discontinuous, non-ordered, interdependent and price-dependent preferences and thus extending the results of Aumman (1966) and Schmeidler (1969).

Homotopy Quotients and Comodules of Supercommutative Hopf Algebras

We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient A→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\rightarrow B$$\\end{document} satisfying some finiteness conditions, the Frobenius tensor category D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}$$\\end{document} of graded B-comodules with its stable model structure induces a monoidal model structure on C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}. We consider the corresponding homotopy quotient γ:C→HoC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma : {\\mathcal {C}} \\rightarrow Ho {\\mathcal {C}}$$\\end{document} and the induced quotient T→HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}} \\rightarrow Ho {\\mathcal {T}}$$\\end{document} for the tensor category T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document} of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho {\\mathcal {T}}$$\\end{document}. We apply these results in the Rep(GL(m|n))-case and study its homotopy category HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho {\\mathcal {T}}$$\\end{document} associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho{\\mathcal {T}}$$\\end{document} by the negligible morphisms is again the representation category of a supergroup scheme.

Finiteness theorems and counting conjectures for the flux landscape

In this paper, we explore the string theory landscape obtained from type IIB and F-theory flux compactifications. We first give a comprehensive introduction to a number of mathematical finiteness theorems, indicate how they have been obtained, and clarify their implications for the structure of the locus of flux vacua. Subsequently, in order to address finer details of the locus of flux vacua, we propose three mathematically precise conjectures on the expected number of connected components, geometric complexity, and dimensionality of the vacuum locus. With the recent breakthroughs on the tameness of Hodge theory, we believe that they are attainable to rigorous mathematical tools and can be successfully addressed in the near future. The remainder of the paper is concerned with more technical aspects of the finiteness theorems. In particular, we investigate their local implications and explain how infinite tails of disconnected vacua approaching the boundaries of the moduli space are forbidden. To make this precise, we present new results on asymptotic expansions of Hodge inner products near arbitrary boundaries of the complex structure moduli space.

Étale cohomology of algebraizable rigid analytic varieties via nearby cycles over general bases

We prove a finiteness theorem and a comparison theorem in the theory of étale cohomology of rigid analytic varieties. By a result of Huber, for a quasi-compact separated morphism of rigid analytic varieties with target being of dimension ≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le 1$$\\end{document}, the compactly supported higher direct image preserves quasi-constructibility. Though the analogous statement for morphisms with higher dimensional target fails in general, we prove that, in the algebraizable case, it holds after replacing the target with a modification. We deduce it from a known finiteness result in the theory of nearby cycles over general bases and a new comparison result, which gives an identification of the compactly supported higher direct image sheaves, up to modification of the target, in terms of nearby cycles over general bases.

KOBAYASHI-OCHIAI’S FINITENESS THEOREM FOR ORBIFOLD PAIRS OF GENERAL TYPE

AbstractKobayashi–Ochiai proved that the set of dominant maps from a fixed variety to a fixed variety of general type is finite. We prove the natural extension of their finiteness theorem to Campana’s orbifold pairs.

Gromov–Hausdorff convergence of metric pairs and metric tuples

We study the Gromov–Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov–Hausdorff equivariant convergence and a version of Grove–Petersen–Wu's finiteness theorem for stratified spaces.

A Finite Topological Type Theorem for Open Manifolds with Non-negative Ricci Curvature and Almost Maximal Local Rewinding Volume

Abstract In this paper, we present finite topological type theorems for open manifolds with non-negative Ricci curvature, under almost maximal local rewinding volume. Unlike previous related research, our theorems remove the constraints of sectional curvature or conjugate radius, which were crucial additional assumptions on metric regularity in prior results. Notably, our settings do not necessarily satisfy a triangle comparison of Toponogov type. In fact, the method we adopt also extends to many previous related studies.

Implementation of summation theorems of Andrews and Gessel-Stanton

Generalized hypergeometric functions and their natural generalizations in one and several variables appear in many mathematical problems and their applications. Solving partial differential equations encountered in many applied problems of mathematics physics is expressed in terms of such generalized hypergeometric functions. In particular, the Srivastava-Daoust double hypergeometric function (S-D function) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics. In this paper, we introduce two general double-series identities involving bounded sequences of arbitrary complex numbers employing the finite summation theorems of Gessel-Stanton and Andrews for terminating 3F2 hypergeometric series with arguments 3/4 and 4/3, respectively. Using these double-series identities, we establish two reduction formulas for the (S-D function) with arguments z, 3z/4 and z, −4z/3 expressed in terms of two generalized hypergeometric function of arguments proportional to z3 and −z3 respectively. All the results mentioned in the paper are verified numerically using Mathematica Program.

A finiteness theorem of block-transitive point-imprimitive 3-designs

In 2003, Tuan showed a finiteness theorem for block-transitive point-imprimitive 3-(k(k−1)2+1,k,λ) designs. As a generalization of this result, we consider the block-transitive point-imprimitive 3-(v,k,λ) designs with v<k(k−1)2+1. Let D=(P,B) be a 3-(v,k,λ) design admitting G as a block-transitive point-imprimitive automorphism group, and G preserve a partition C of the points into d imprimitivity classes of size c, here v=wk(k−1)s+1, s,w are two positive integers with gcd(s,w)=1 and 2w<s. We prove that, for a given positive integer s, there are only finitely many numbers v such that there exist nontrivial block-transitive point-imprimitive 3-(v,k,λ) designs with c,d≥3. Moreover, we obtain the classification for this type of designs when s≤10.

Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture

AbstractWe offer some theorems, mainly finiteness results, for certain patterns in elliptical billiards, related to periodic trajectories; these seem to be the first finiteness results in this context. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in , there are only finitely many directions for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results (which are shown not to hold for general billiards) have their origin in ‘relative’ cases of the Manin–Mumford conjecture and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell–Lang conjecture. In turn, this variant embraces cases, which, somewhat surprisingly, sometimes can be treated (only) by completely different methods compared to the former ones; here we shall offer an explicit example related to diophantine equations in algebraic tori.

Can we recover an integral quadratic form by representing all its subforms?

Let o be the ring of integers of a number field. If f is a quadratic form over o and g is another quadratic form over o which represents all proper subforms of f, does g represent f? We show that if g is indefinite, then g indeed represents f. However, when f is positive definite and indecomposable, then there exists a g which represents all proper subforms of f but not f itself. Along the way we give a new characterization of positive definite decomposable quadratic forms over o and a number-field generalization of the finiteness theorem of representations of quadratic forms by quadratic forms over Z[13, Theorem 3.3] which asserts that given any infinite set S of classes of positive definite integral quadratic forms over o of a fixed rank, there exists a finite subset S0 of S with the property that a positive definite quadratic form over o represents all classes in S if and only if it represents all classes in S0.

Joint state and fault estimation for nonlinear systems with missing measurements and random component faults under Round-Robin Protocol

This work studies the fault estimation problem for a class of nonlinear systems subjected to missing measurements and random component faults under the Round-Robin protocol. Considering that the communication bandwidth is not infinite in the actual projects, the Round-Robin protocol is used for data scheduling, in which sensor nodes cyclically transmit data to the controller/filter. Meanwhile, component faults are also considered which usually caused by actuator faults. It should be emphasized that the introduction of RR protocol will make the system output equation more complex, the MMs and component faults will generate more unknown parameters, which will bring new challenges during the design of the filter. In respect of the problems above, a recursive joint state and fault estimation (JSFE) algorithm is proposed to solve the difficulties caused by missing measurements, component faults, and nonlinearity. The performance of the designed JSFE filter is analyzed by using the boundedness theorem and the reasonable parameter condition is derived for this filter. Finally, an engineering example is given to verify the effectiveness and feasibility of the proposed joint state and fault estimation scheme. This study provides a new theoretical basis and reference for the JSFE of nonlinear systems with mixed faults under RR protocol.

Finiteness for self-dual classes in integral variations of Hodge structure

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.

Finiteness theorems for potentially equivalent Galois representations: Extension of Faltings’ finiteness criteria

Finiteness theorems for potentially equivalent Galois representations: Extension of Faltings’ finiteness criteria

Bubble-tree convergence and local diffeomorphism finiteness for gradient Ricci shrinkers

We prove bubble-tree convergence of sequences of gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds, refining the compactness theory of Haslhofer and Müller (Geom Funct Anal 21:1091–1116, 2011; Proc Am Math Soc 143(10):4433–4437, 2015). In particular, we show that no energy concentrates in neck regions, a result which implies a local energy identity for the sequence. Direct consequences of these results are an identity for the Euler characteristic and a local diffeomorphism finiteness theorem.

A geometric generalization of Kaplansky’s direct finiteness conjecture

Let G G be a group and let k k be a field. Kaplansky’s direct finiteness conjecture states that every one-sided unit of the group ring k [ G ] k[G] must be a two-sided unit. In this paper, we establish a geometric direct finiteness theorem for endomorphisms of symbolic algebraic varieties. Whenever G G is a sofic group or more generally a surjunctive group, our result implies a generalization of Kaplansky’s direct finiteness conjecture for the near ring R ( k , G ) R(k,G) which is k [ X g : g ∈ G ] k[X_g\colon g \in G] as a group and which contains naturally k [ G ] k[G] as the subring of homogeneous polynomials of degree one. We also prove that Kaplansky’s stable finiteness conjecture is a consequence of Gottschalk’s Surjunctivity Conjecture.

A "nonlinear duality" approach to $ W_0^{1, 1} $ solutions in elliptic systems related to the Keller-Segel model

&lt;abstract&gt;&lt;p&gt;In this paper, we prove existence of distributional solutions of a nonlinear elliptic system, related to the Keller-Segel model. Our starting point is the boundedness theorem (for solutions of elliptic equations) proved by Guido Stampacchia and Neil Trudinger.&lt;/p&gt;&lt;/abstract&gt;

Robust stability and boundedness of uncertain conformable fractional-order delay systems under input saturation

&lt;abstract&gt;&lt;p&gt;In this article, a class of uncertain conformable fractional-order delay systems under input saturation is considered. By establishing the Lyapunov boundedness theorem for conformable fractional-order delay systems, some sufficient conditions for robust stability and boundedness of the systems are obtained. Examples are given to illustrate the obtained theory.&lt;/p&gt;&lt;/abstract&gt;

Factorisation de la cohomologie étale p-adique de la tour de Drinfeld

Résumé For a finite extension F of ${\mathbf Q}_p$ , Drinfeld defined a tower of coverings of (the Drinfeld half-plane). For $F = {\mathbf Q}_p$ , we describe a decomposition of the p-adic geometric étale cohomology of this tower analogous to Emerton’s decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finiteness theorem for the arithmetic étale cohomology modulo p whose proof uses Scholze’s functor, global ingredients, and a computation of nearby cycles which makes it possible to prove that this cohomology has finite presentation. This last result holds for all F; for $F\neq {\mathbf Q}_p$ , it implies that the representations of $\mathrm{GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F = {\mathbf Q}_p$ .