Finiteness Conjecture Research Articles

Joint spectral radius and Hölder regularity of wavelets

We give preliminary results on the Hölder exponent of wavelets of compact support. In particular, we give a nearly complete map of this exponent for the family of four-coefficient multiresolution analyses and determine the smoothest one. This will resolve two conjectures by Colella and Heil. Wavelets of compact support can be generated via infinite products of certain matrices. The rate of growth of these products determines the regularity of the wavelet. This rate can be determined via joint, generalized, or common spectral radius of the given set of matrices. We outline a method for calculating this radius for a given set of matrices. The method relies on guessing the particular finite optimal product which satisfies the finiteness conjecture and exhibits the fastest growth. Then, we generate an optimal unit ball by taking the convex hull of the action of the semigroup of matrices, scaled by their joint radius, on the invariant ball of the scaled optimal product. If this process terminates in a finite number of steps and the convex hull does not grow, then the guessed optimal product is confirmed and the joint radius is determined.

Remarks on Methods of Fontaine and Faltings

In this note we prove the non existence of certain irreducible two dimensional representations of the the absolute Galois group. Such results are predicted by Serre's conjecture and we use Fontaine's methods to verify these predictions in a small number of cases. We also use a methods of Faltings to study certain finiteness conjectures for Galois representations. Key words: Serre's conjecture, Fontaine-Mazur conjectures, Galois representations.

Upper bound theorems for homology manifolds

In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that β k (Δ)⩽Σ{β i (Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where β i (Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.)

AUTOMORPHIC FORMS AND LORENTZIAN KAC–MOODY ALGEBRAS PART I

Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector. We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T. We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac–Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac–Moody algebras and formulate finiteness Conjecture for these forms. Detailed study of automorphic correction and Lie reflective automorphic forms for generalized Cartan matrices mentioned above will be given in Part II.

Neighborly cubical spheres and a cubical lower bound conjecture

Using mirrors and cyclic polytopes, we construct cubicald-spheres which are the analogs of cyclic polytopes in the sense that they have the ⌉d−1/2⌈-skeleta of cubes. The existence of these neighborly cubical spheres leads to a special case of an upper bound conjecture for cubical spheres, suggested by Kalai. We extend the same construction to show that the closed convex hull off-vectors of cubical spheres contains a cone described by Adin, as an analog to the generalized lower bound theorem for simplicial polytopes.

Torsion in rank 1 Drinfeld modules and the uniform boundedness conjecture

It is conjectured that for fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on the size of the torsion submodule of a Drinfeld $A$-module of rank $r$ over a degree $d$ extension $L$ of the fraction field $K$ of $A$. We verify the conjecture for $r=1$, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of $K$. Moreover, we show that within an $\Lbar$-isomorphism class, there are only finitely many Drinfeld modules up to isomorphism over $L$ which have nonzero torsion. For the case $A=\Fq[T]$, $r=1$, and $L=\Fq(T)$, we give an explicit description of the possible torsion submodules. We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic $p$ function fields proves the uniform boundedness for the $\pp$-primary part of the torsion for rank~2 Drinfeld $\Fq[T]$-modules over a fixed function field.

A Proof of KÜhnel’s Conjecture for n ⩾ k2 + 3k Eric Sparla

In this note we show that an Upper Bound Conjecture made by Kühnel for combinatorial 2k-manifolds holds for fixed k if its number of vertices is at least n ⩾ k2 + 3k. Together with known results this provides a simple proof of the conjecture for k = 1 and k = 2.

Uniformity of stably integral points on elliptic curves

A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. Following this practice, we demonstrate an analogue of a result of L. Caporaso, J. Harris and B. Mazur, showing that the Lang - Vojta conjecture implies a uniform bound on the number of stably integral points on an elliptic curve over a number field, as well as the uniform boundedness conjecture (Merel's theorem).

The finiteness conjecture for the generalized spectral radius of a set of matrices

The generalized spectral radius \\ ̄ g9(∑) of a set ∑ of n × n matrices is \\ ̄ g9(∑) = lim sup k→∞ \\ ̄ g9 k(∑) 1 k , where \\ ̄ g9 k(∑) = sup{ϱ( A 1 A 2… A k): each A i ∈ ∑} . The joint spectral radius \\ ̂ g9(∑) is \\ ̂ g9(∑) = lim sup k→∞ \\ ̂ g9 k(∑) 1 k , where \\ ̂ g9 k(∑) = sup{∥ A 1 … A k∥: each A i ∈ ∑} . It is known that \\ ̂ g9(∑) = \\ ̄ g9(∑) holds for any finite set ∑ of n × n matrices. The finiteness conjecture asserts that for any finite set ∑ of real n × n matrices there exists a finite k such that \\ ̂ g9(∑) = \\ ̄ g9(∑) = \\ ̄ g9 k(∑) 1 k . The normed finiteness conjecture for a given operator norm asserts that for any finite set ∑ = {A 1,…, A m } having all ∥A i ∥ op ≤ 1, either \\ ̂ g9(∑) < 1 or \\ ̂ g9(∑) = \\ ̄ g9(∑) = \\ ̄ g9 k(∑) 1 k = 1 for some finite k. It is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms. The normed finiteness conjecture is proved for a large class of operator norms, extending results of Gurvits. In particular, for polytope norms and for the Euclidean norm, explicit upper bounds are given for the least k having \\ ̄ g9(∑) = \\ ̄ g9 k(∑) 1 k . These results imply upper bounds for generalized critical exponents for these norms.

Boundedness conjecture for an output error adaptive algorithm

AbstractThis paper compares conditions for instability with conditions for global asymptotic stability of a discretetime output error adaptive estimation algorithm. This comparison leads to a boundedness conjecture which states that all signals within this adaptive system as well as the output error and parameter estimates remain globally bounded for all time despite any unstable behaviour. The investigation into the stability properties of this algorithm begins with a global stability criterion applied to a transfer function which is not strictly positive real. This criterion states that for large enough adaptation gain and/or input signal magnitude the system will be asymptotically stable. Then a local result is applied which states that for small enough adaptation gain and/or input signal magnitude the same system is unstable. The result is a bifurcation due to the decrease of the adaptation gain and/or input signal magnitude as the system goes from asymptotic stability to instability which remains bounded. The results suggested by the computer simulations are verified by an exact analysis of a linearized periodic version of the adaptive system.

Global and local stability properties of a non-SPR output error estimator

Conditions for local instability are compared with conditions for global asymptotic stability of an output error adaptive estimation algorithm for continuous-time systems. A boundedness conjecture is made that all signals within this adaptive system including the output error and parameter estimates remain globally bounded for all time despite any locally unstable behaviour. A global stability criterion applied to a transfer function, which is not strictly positive real, states that for a large enough adaptation gain the system will be asymptotically stable. However, a local result states that for a small enough adaptation gain the same system is unstable. The results suggested by the computer simulations are verified by an exact analysis of a linearized periodic version of the adaptive system.

On k-stacked polytopes

It is proved that equality in the Generalized Simplicial Lower Bound Conjecture can always be obtained by k-stacked polytopes.

A dual proof of the upper bound conjecture for convex polytopes.

A dual proof of the upper bound conjecture for convex polytopes.

Abelian varieties in characteristic p

In this paper Tate's finiteness conjecture for isogenies of polarized Abelian varieties in characteristic p>2 is proved. From this conjecture it is deduced that Tate modules are semisimple and that Tate's conjecture on the homomorphisms of Abelian varieties is valid.

The Upper Bound Conjecture and Cohen‐Macaulay Rings

Let Δ be a triangulation of a (d − 1)‐dimensional sphere with n vertices. The Upper Bound Conjecture states that the number of i‐dimensional faces of Δ is less than or equal to a certain explicit number ci(n, d). A proof is given of a more general result. The proof uses the result, proved by G. Reisner, that a certain commutative ring associated with Δ is a Cohen‐Macaulay ring.

ENDOMORPHISMS OF ABELIAN VARIETIES OVER FIELDS OF FINITE CHARACTERISTIC

The semisimplicity of -adic representations corresponding to one-dimensional etale cohomology, and a proof of the Tate's conjecture about homomorphisms of abelian varieties, are derived from the Tate's finiteness conjecture on isogenies of polarized abelian varieties.Bibliography: 5 items.

A REMARK ON ENDOMORPHISMS OF ABELIAN VARIETIES OVER FUNCTION FIELDS OF FINITE CHARACTERISTIC

Assuming Tate's finiteness conjecture, we prove some consequences of Tate's conjecture on homomorphisms of abelian varieties.

ISOGENIES OF ABELIAN VARIETIES OVER FIELDS OF FINITE CHARACTERISTIC

It is proved that Tate's finiteness conjecture for isogenies of Abelian varieties over fields of characteristic other than 2 can be formally deduced from the conjecture on the resolution of singularities. Bibliography: 4 items.

Some Results on the Upper Bound Conjecture for Convex Polytopes

Some Results on the Upper Bound Conjecture for Convex Polytopes

On the uniform Rasmussen–Tamagawa conjecture in the CM case

We prove a uniform version of a finiteness conjecture due to Rasmussen and Tamagawa in the case of CM abelian varieties. This extends recent results concerning CM elliptic curves to CM abelian varieties of arbitrary dimension.