Problems in algebra inspired by universal algebraic geometry

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant K _Θ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H ₁ and H ₂ have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) K _Θ(H ₁) and K _Θ(H ₂) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ⁰ be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ⁰) for different varieties Θ and also the groups of autoequivalences of Θ⁰. The problem is to describe these groups for different Θ.     

Upper functions for positive random functionals. I. General setting and Gaussian random functions

In this paper we are interested in finding upper functions for a collection of real-valued random variables {Ψ(χ _θ ), θ ∈ Θ}. Here {χ _θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a nonrandom function U: Θ → ?₊ such that sup _θ∈Θ{Ψ(χ _θ ) ? U(θ)}₊ is “small” with prescribed probability. We apply the results obtained in the general setting to the variety of problems related to Gaussian random functions and empirical processes.     

Repeated games with public uncertain duration process

We consider repeated games where the number of repetitions θ is unknown. The information about the uncertain duration can change during the play of the game. This is described by an uncertain duration process Θ that defines the probability law of the signals that players receive at each stage about the duration. To each repeated game Γ and uncertain duration process Θ is associated the Θ-repeated game Γ_Θ. A public uncertain duration process is one where the uncertainty about the duration is the same for all players. We establish a recursive formula for the value V _Θ of a repeated two-person zero-sum game Γ_Θ with a public uncertain duration process Θ. We study asymptotic properties of the normalized value v _Θ = V _Θ/E(θ) as the expected duration E (θ) goes to infinity. We extend and unify several asymptotic results on the existence of lim v _n and lim v _λ and their equality to lim v _Θ. This analysis applies in particular to stochastic games and repeated games of incomplete information.     

Admissibility criteria for model subspaces with fast growth of the argument of the generating inner function

Let Θ be an inner function in the upper half-plane and let K_Θ = H² ⊖ ΘH² be the associated model subspace of the Hardy space H². We call a nonnegative function ω Θ-admissible if in the space K_Θ there exists a nonzero function f ∈ K_Θ such that |f| ≤ ω a.e. on ℝ. We give some sufficient conditions of Θ-admissibility for the case where Θ is a meromorphic function and arg Θ grows fast ((argΘ)′ tends to infinity). Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 55–84.     

Geométrie des suites de Kronecker

Let Θ be a element of the d-dimensional torus $\mathbb{T}$ ^d andτ the translationτ(x)=x + Θ. When d=1 there existe some partitions of $\mathbb{T}$ ¹ which are associated withτ. We prove the existence of partitions of $\mathbb{T}$ ^d which enjoyed the same kind of properties and whose elements (A _i ) _i≤n are convex polytopes. We also give a lower bound for the isotropic discrepancy of the sequence (nΘ) _nε?.     

Geometry and entropy of generalized rotation sets

For a continuous map f on a compact metric space we study the geometry and entropy of the generalized rotation set Rot(Φ). Here Φ = (?₁, ..., ? _m ) is a m-dimensional continuous potential and Rot(Φ) is the set of all µ-integrals of Φ and µ runs over all f-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of ? ^m . We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set K in ? ^m a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits Rot _Pt (Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = Rot _Pt (Φ). Finally, we study the entropy function w ? H(w),w ∈ Rot(Φ). We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems H(w) is determined by the growth rate of those hyperbolic periodic orbits whose Φ-integrals are close to w. We also show that for systems with strong thermodynamic properties (sub-shifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function w ? H(w) is real-analytic in the interior of the rotation set.     

Formulae concerning the computation of the Clausen integral Cl2(Θ)

The main problem under study concerns the expression of the Clausen integral Cl₂(Θ) in closed form in terms of known constants and special functions when Θ is equal to a rational multiple of π belonging to [0, 2π]. A general formula giving $\text{Cl}{}_2\text{(}\text{pπ}\text{q}\text{)}$ in terms of the derivative of the di-gamma function and the sine function is deduced from an appropriate Fourier series expansion. Some variants of this formula are obtained. In further sections, the formulae expressing Cl₂(2Θ) and, more generally, Cl₂(mΘ)(m=2,3,4,…) as linear combinations of terms of the form Cl₂(Θ+α) (α: const.) are established. The various results are illustrated by means of typical examples of practical application. The last section contains two simple approximations enabling the computation of Cl(Θ) for any Θ in [0,π] with a relative error smaller than 0.63% and 0.003%, resp.. The paper ends with an appendix in which, among other things, a peculiar trigonometric identity is established as a by-product.     

Θ-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H _p to L _p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L ₁ converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.     

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

Equazioni monodimensionali di un gas viscoso barotropico con una perturbazione poco regolare

The equations for a barotropic viscous gas in one space dimensiondν=(μ(?ν_ε)_ε?p _ε)dt+dG,? _t +?²ν_ε=0,p=?^γ with a perturbationdG are considered under the assumption thatG is only a function of bounded variation inL ²(Θ) orH ₀ ¹ (Θ) (Θ=]0, α[) and the esistence and the uniqueness of the global solution in a class of solutions of «strong type» as well as in a class of solutions of «weak type» are proved. This result constitutes a generalization of the result of Kazhikhov [8] and that of Shelukhin [10] and contains preliminary considerations for the corrisponding stochastic equations.     

The K-admissibility of SL(2, 5)

Let K be a field and let G be a finite group. G is K-admissible if there exists a Galois extension L of K with G=Gal(L/K) such that L is a maximal subfield of a central K-division algebra. We characterize those number fields K such that H is K-admissible where H is any subgroup of SL(2, 5) which contains a S ₂-group. The method also yields refinements and alternate proofs of some known results including the fact that A ₅ is K-admissible for every number field K.Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthdayThe first author was partly supported by NSF fellowship DMS-8601130; the second author was partly supported by NSF grant DMS-8806371.     

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators L _Θ,Ω in L ²(Ω; d ⁿ x) ^m , m, n ∈ ?, associated with divergence form elliptic partial differential expressions L with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains Ω ? ? ⁿ . In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions L which act as $$Lu = - \left( {\sum\limits_{j,k = 1}^n {\partial _j } \left( {\sum\limits_{\beta = 1}^m {a_{j,k}^{\alpha ,\beta } \partial _k u_\beta } } \right)} \right)_{1 \leqslant \alpha \leqslant m} , u = \left( {u_1 , \ldots ,u_m } \right).$$ The (nonlocal) Robin-type boundary conditions are then of the form $$v \cdot ADu + \Theta [u|_{\partial \Omega } ] = 0{\text{ on }}\partial \Omega ,$$ where Θ represents an appropriate operator acting on Sobolev spaces associated with the boundary ?Ω of Ω, ν denotes the outward pointing normal unit vector on ?Ω, and $Du: = \left( {\partial _j u_\alpha } \right)_{_{1 \leqslant j \leqslant n}^{1 \leqslant \alpha \leqslant m} } .$ Assuming Θ ≥ 0 in the scalar case m = 1, we prove Gaussian heat kernel bounds for L _Θ,Ω, by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on ?Ω. We also discuss additional zero-order potential coefficients V and hence operators corresponding to the form sum L _Θ,Ω + V.     

Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π₊ of G how does π₊ decompose on restriction to H. Here G is GL⁺(2,F), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL⁺(2,F) the subgroup of index 2 in GL(2,F) consisting of those matrices whose determinant is in NK/F(K^∗), π₊ is an irreducible, admissible supercuspidal representation of GL⁺(2,F) and H=K^∗ under an embedding of K^∗ into GL(2,F).     

On the restriction of cross characteristic representations of 2 F 4(q) to proper subgroups

We prove that the restriction of any nontrivial representation of the Ree groups ² F ₄(q), q = 2²ⁿ⁺¹ ≥ 8 in odd characteristic to any proper subgroup is reducible. We also determine all triples (K, V, H) such that ${K \in \{^2F_4(2), ^2F_4(2)'\} }We prove that the restriction of any nontrivial representation of the Ree groups ² F ₄(q), q = 2²ⁿ⁺¹ ≥ 8 in odd characteristic to any proper subgroup is reducible. We also determine all triples (K, V, H) such that K ? {²F₄(2), ²F₄(2)￠}{K \in \{^2F_4(2), ^2F_4(2)'\} } , H is a proper subgroup of K, and V is a representation of K in odd characteristic restricting absolutely irreducibly to H.     

Some results for operators on a model space

We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space K _θ = H²ΘθH² is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H²-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.     

The scaling window of the 2‐SAT transition

We consider the random 2‐satisfiability (2‐SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨y, chosen uniformly at random from among all 2‐clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n→α, the problem is known to have a phase transition at α_c=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite‐size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α_?(n,δ), α₊(n,δ)) such that the probability of satisfiability is greater than 1?δ for α<α_? and is less than δ for α>α₊. We show that W(n,δ)=(1?Θ(n^?1/3), 1+Θ(n^?1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+ε)n, where ε may depend on n as long as |ε| is sufficiently small and |ε|n^1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(?Θ(nε³)) above the window, and goes to one like 1?Θ(n^?1|ε|^?3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2‐SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2‐SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001     

Light graphs in families of polyhedral graphs with prescribed minimum degree, face size, edge and dual edge weight

A graph H is defined to be light in a family H of graphs if there exists a finite number φ(H,H) such that each G∈H which contains H as a subgraph, contains also a subgraph K≅H such that the Δ_G(K)≤φ(H,H). We study light graphs in families of polyhedral graphs with prescribed minimum vertex degree δ, minimum face degree ρ, minimum edge weight w and dual edge weight w^∗. For those families, we show that there exists a variety of small light cycles; on the other hand, we also present particular constructions showing that, for certain families, the spectrum of short cycles contains irregularly scattered cycles that are not light.     

Resolvable balanced incomplete block designs with subdesigns of block size 4

In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly (v,4,1) RBIBDs. The problem we investigate is the existence of (v,4,1) RBIBDs containing a (w,4,1) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB(v,w). The necessary conditions for the existence of an IRB(v,w) are that v?4w and . We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever and whenever w?1852.We also give some results on pairwise balanced designs on v points containing (at least one) block of size w, i.e., a (v,{K,w^*},1)-PBD.If the list of permitted block sizes, K₅, contains all integers of size 5 or more, and v,w∈K₅, then a necessary condition on this PBD is v?4w+1. We show this condition is not sufficient for any w?5 and give the complete spectrum (in v) for 5?w?8, as well as showing the condition v?5w is sufficient with some definite exceptions for w=5 and 6, and some possible exceptions when w=15, namely 77?v?79. The existence of this PBD implies the existence of an IRB(12v+4,12w+4).If the list of permitted block sizes, K₁₍₄₎, contains all integers , and v,w∈K₁₍₄₎, then a necessary condition on this PBD is v?4w+1. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w?1037. The existence of this PBD implies the existence of an IRB(3v+1,3w+1).     

Effective constructions in well-partially- ordered free monoids

For strings υ and w define υ?w if and only if υ is a scattered substring of w. We give a general solution for a problem of Haines to effectively determine regular expressions for L?= {w:?_{υ ? L}υ ? w} and L?={υ:?_{w ? L} υ ? w} when L denotes an arbitrary context-free language. We show by an inductive argument that one can effectively determine L? and L? for each language L in the algebraic extension of some family K if and only if one can do so for each language in K.     

On the Coifman type inequality for the oscillation of one-sided averages

In this paper we study the Coifman type estimate for an oscillation operator related to the one-sided discrete square function S⁺. We prove that for any weight w, the Lp(w)-norm of this operator, and therefore the Lp(w)-norm of S⁺, is dominated by a constant times the Lp(w)-norm of the one-sided Hardy-Littlewood maximal function iterated two times. For the kth commutator with a BMO function we show that k+2 iterates of the one-sided Hardy-Littlewood maximal function are sufficient.