Diophantine problems in variables restricted to the values 0 and 1

Let $\text{F}$x₁,…,x_s be a form of degree d with integer coefficients. How large must s be to ensure that the congruence $\text{F}$(x₁,…,x_s) ≡ 0 (mod m) has a nontrivial solution in integers 0 or 1? More generally, if $\text{F}$ has coefficients in a finite additive group G, how large must s be in order that the equation $\text{F}$(x₁,…,x_s) = 0 has a solution of this type? We deal with these questions as well as related problems in the group of integers modulo 1 and in the group of reals.     

The automorphisms of convolution algebras of C∞ functions

We find the automorphisms and the spectra of several different topological convolution algebras of C^∞-functions on the real line. Starting with the convolution algebra of compactly supported C^∞-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C^∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C^∞-functions of infinitly fast exponential decay at ∞, $\text{H J}$, on the one hand, and the algebra of C^∞-functions of only a finite degree $\text{e}{}^{\mathrm{?r¦x¦}}$ decay at ∞, $\text{J}$_r⁰, on the other hand, have quite different automorphisms, although $\text{H J}$ = ∩_r$\text{J}$_r⁰. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of $\text{J}$_r⁰, and that this representation does not extend to a representation on the Banach algebra L¹($\text{R}$).     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

Comparability graphs and intersection graphs

A function diagram (f-diagram) D consists of the family of curves {$\text{1}\text{?}$ñ} obtained from n continuous functions $\text{f}{}_{\mathrm{i}}\text{:[0,1]→}\text{R}\text{(1?i?n)}$. We call the intersection graph of D a function graph (f-graph). It is shown that a graph G is an f-graph if and only if its complement ? is a comparability graph. An f-diagram generalizes the notion of a permulation diagram where the f_i are linear functions. It is also shown that G is the intersection graph of the concatenation of ?k permutation diagrams if and only if the partial order dimension of $\text{G}\text{?}\text{is ?k+1}$. Computational complexity results are obtained for recognizing such graphs.     

Reflexive algebras with finite width lattices: Tensor products,cohomology, compact perturbations

Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of ¹-algebras. Tomita's tensor product commutation theorem can be restated in the form Alg($\text{L}$₁ ? $\text{L}$₂) = Alg $\text{L}$₁ ? Alg $\text{L}$₂, where each $\text{L}$_i is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each $\text{L}$_i is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if $\text{L}$ is a finite-width commutative subspace lattice and $\text{K}$ is the set of compact operators then the quasitriangular algebra Alg $\text{L}$ + $\text{K}$ is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.     

A new fuzzy compactness defined by fuzzy nets

Let (Ω, $\text{A}$, μ) be a probability space and let $\text{B}$ be a subsigma algebra of $\text{A}$. Let A= L_∞Ω, $\text{A}$, μ , let A= L_∞Ω, $\text{B}$, μ, and let f?A. It is shown that best L_∞-approximations of f by elements of B comprise an interval in B; that is, there exists $\text{f}\text{,}\text{f}\text{?B}$ such that a function g?B is a best L_∞-approximation to f if and only if $\text{f}\text{? g ?}\text{f}$ a.e. on Ω. The difference, $\text{f}\text{? f}$, of $\text{f}$ and f is completely characterized in terms of special sets that have been developed in [2]. Then it is established that the best best L_∞-approximation, f_{$\text{B}$,∞}, to f by elements of B is the average of $\text{f}$ and $\text{f}$, where the function f_{$\text{B}$,∞} is defined by f_{$\text{B}$,∞}(ω) lim_{p → ∞}f_$\text{B}$,P(ξ) and f_$\text{B}$,P denotes the best L_p-approximation to f elements of L_p(Ω, $\text{B}$, μ).     

A class of extension properties that are not simply generated

Let $\text{P}$ be a closed-hereditary topological property preserved by products. Call a space $\text{P}$-regular if it is homeomorphic to a subspace of a product of spaces with $\text{P}$. Suppose that each $\text{P}$-regular space possesses a $\text{P}$-regular compactification. It is well-known that each $\text{P}$-regular space X is densely embedded in a unique space γ_scPX with $\text{P}$ such that if f: X → Y is continuous and Y has $\text{P}$, then f extends continuously to γ_scPX. Call $\text{P}$-pseudocompact if γ_scPX is compact.Associated with $\text{P}$ is another topological property $\text{P}$^#, possessing all the properties hypothesized for $\text{P}$ above, defined as follows: a $\text{P}$-regular space X has $\text{P}$^# if each $\text{P}$-pseudocompact closed subspace of X is compact. It is known that the $\text{P}$-pseudocompact spaces coincide with the $\text{P}$^#-pseudocompact spaces, and that $\text{P}$^# is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if $\text{P}$ is not the property of being compact and $\text{P}$-regular, then $\text{P}$^# is not simply generated; in other words, there does not exist a space E such that the spaces with $\text{P}$^# are precisely those spaces homeomorphic to closed subspaces of powers of E.     

Analyticity and flows in von Neumann algebras

Let $\text{B}$ be a von Neumann algebra, let {α_t}_tεR be an ultraweakly continuous one-parameter group of ¹-automorphisms of $\text{B}$, and let $\text{U}$ be the set of all A such that for each ? in $\text{B}$₁, the function t → ?(α_t(A)) lies in H^∞($\text{R}$. Then $\text{U}$ is an ultraweakly closed subalgebra of $\text{B}$ containing the identity which is proper and non-self-adjoint if {α_t}_tεR is not trivial. In this paper, a systematic investigation into the structure theory of $\text{U}$ is begun. Two of the more note-worthy developments are these. First of all, conditions under which $\text{U}$ is a subdiagonal algebra in $\text{B}$, in the sense of Arveson, are determined. The analysis provides a common perspective from which to view a large number of hitherto unrelated algebras. Second, the invariant subspace structure of $\text{U}$ is determined and conditions under which $\text{U}$ is a reductive subalgebra of $\text{B}$ are found. These results are then used to produce examples where $\text{U}$ is a proper, non-self-adjoint, reductive subalgebra of $\text{B}$. The examples do not answer the reductive algebra question, however, because although ultraweakly closed, the subalgebras are weakly dense in $\text{B}$.     

Rectangular arrays

In this paper we studied m×n arrays with row sums $\text{nr}\text{(n,m)}$ and column sums $\text{mr}\text{(n,m)}$ where (n,m) denotes the greatest common divisor of m and n. We were able to show that the function H_m,n(r), which enumerates m×n arrays with row sums and column sums $\text{nr}\text{(m,n)}$ and $\text{mr}\text{(n,m)}$ respectively, is a polynomial in r of degree (m?1)(n?1). We found simple formulas to evaluate these polynomials for negative values, ?r, and we show that certain small negative integers are roots of these polynomials. When we considered the generating function G_m,n(y) = Σ_r?0H_m,n(r)y^r, it was found to be rational of degree less than zero. The denominator of G_m,n(y) is of the form (1?y)^(m?1)(n?1)+3, and the coefficients of the numerator are non-negative integers which enjoy a certain symmetric relation.     

The construction of SL(2, 3)-polynomials

Octic polynomials over $\text{Z}$ with Galois group SL(2, 3) are constructed. This is done via suited quartic totally real polynomials with group A₄ over $\text{Q}$. A table of the cycle patterns of the imprimitive transitive permutation groups of degree 8 is included.     

Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If $\text{D}$ is an atomic maximal Abelian subalgebra of $\text{B}$($\text{H}$), $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto $\text{D}$ and h is a complex homomorphism on $\text{D}$, then h ° $\text{P}$ is a pure state on $\text{B}$($\text{H}$). (2) If {P_n} is a sequence of mutually orthogonal projections with rank(P_n) = n and ∑ P_n = I, $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto {P_n}″ given by P(T)=∑trace_n(T)P_n and h is a homomorphism on {P_n}″ such that h(P_n) = 0 for all n then h ° $\text{P}$ induces a type II_∞ factor representation of the Calkin algebra. (3) If $\text{M}$ is a nonatomic maximal Abelian subalgebra of $\text{B}$($\text{H}$) then there is an atomic maximal Abelian subalgebra $\text{D}$ of $\text{B}$($\text{H}$) and a large family {Φ_α} of ¹-homomorphisms from $\text{D}$ onto $\text{M}$ such that for each α, Φ_α ° $\text{P}$ is an extreme point in the set of projections from $\text{B}$($\text{H}$) onto $\text{M}$. (Here $\text{P}$ denotes the projection of $\text{B}$($\text{H}$) onto $\text{D}$.)     

Minimal codes in abelian group algebras

In this paper we show that two minimal codes $\text{M}$₁ and $\text{M}$₂ in the group algebra $\text{F}$₂[G] have the same (Hamming) weight distribution if and only if there exists an automorphism θ of G whose linear extension to $\text{F}$₂[G] maps $\text{M}$₁ onto $\text{M}$₂. If θ(M₁) = M₂, then $\text{M}$₁ and $\text{M}$₂ are called equivalent. We also show that there are exactly τ(l) inequivalent minimal codes in $\text{F}$₂[G], where ? is the exponent of G, and τ(?) is the number of divisors of ?.     

Transcendence order over Qp in Cp

Let (K, ∥ · ∥) be a valued transcendence degree 1 extension of $\text{Q}$_p. An element x ∈ K transcendental over $\text{Q}$_p is said to have order ≤a (a > 0) if there exists C_x > 0 such that every polynomial P(X) ∈ $\text{Q}$_p [X] satisfies

$\text{?}\text{log;}\text{(P(x))? ?}\text{log}\text{∥P∥+c}{}_{\mathrm{x}}\text{(}\text{deg}\text{P)}{}^{\mathrm{a}}$

when ∥ · ∥ is the Gauss norm on $\text{Q}$_p[X]. No x ∈ $\text{C}$_p can have order ≤α if α < 1 but we construct some x ∈ $\text{C}$_p with order ≤ 1. Furthermore, we prove order ≤α is stable by algebraic extension.     

On Artin's conjecture

Let $\text{F}$ be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of $\text{F}$. The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠K. Assuming the generalized Riemann hypothesis and some mild restrictions on $\text{F}$, we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if $\text{F}$ is the family of fields obtained by adjoining to $\text{Q}$ the q-division points of an elliptic curve E over $\text{Q}$, the Artin problem determines how often E($\text{F}$_p) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields.     

On graphs with equal edge connectivity and minimum degree

The note contains some conditions on a graph implying that the edge connectivity is equal to the minimum degree. One of these conditions is that if d₁?d₂???d_n is the degree sequence then ∑^l_l?1(d₁+d_n?1)>In?1 for $\text{1 ? l? min {}\text{n}\text{2?1}\text{, d}{}_{\mathrm{n}}\text{}}$.     

On the existence and exactness of the associated sheaf functor

Let $\text{C}$ be a category with inverse limits. A category $\text{x}$is called an $\text{A}$-topos if there is a site (?, τ), i.e. a small category ? together with a Grothendieck topology τ such that $\text{x}$is equivalent to the category Sh_τ[$\text{C}$⁰. $\text{A}$] of τ-sheaves on $\text{C}$ with values in $\text{C}$. If $\text{x}$is an $\text{C}$-topos, then so is Sh_τ'[?⁰, $\text{x}$] for any site (?', τ'). It is shown that if for every site (?,τ) the associated sheaf functor from presheaves to τ-sheaves with values in $\text{A}$ exists (and preserves finite inverse limits), then the same holds if $\text{A}$is replaced by any $\text{A}$-topos $\text{x}$. Roughly speaking, the main result is that for a site (?,τ) the associated sheaf functor [?⁰, $\text{A}$] → Sh_τ [?⁰, $\text{A}$] exists and preserves finite inverse limits, provided $\text{A}$ has filtered direct limits which commute with finite inverse limits, e.g. if $\text{A}$ is a Grothendieck category or a category of sheaves with values in a locally finitely presentable category [8. 7.1]. Analogous results hold in the additive case.     

The construction of stationary two-dimensional Markoff fields with an application to quantum field theory

It is shown that if φ(f)  ∝_R^dφ(y) f(y) dy is a Markoff random field and X_α are multiplicative functionals of φ (with E(X_α) = 1) which converge locally in L₁, then there exists a locally Markoff random field $\text{φ}{}_1$ such that $\text{E(exp(iφ}{}_1\text{(f))) =}\text{lim}{}_{\mathrm{\alpha }}\text{E(X}{}_{\mathrm{<mtext>\alpha </mtext>}}\text{exp}\text{(iφ(φ)))}$. We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take X_α proportional to exp(?λ∝_R² : P(φ(y)) : g_α(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of g_α → 1 and small λ, {X_α} does converge locally in L₁ and that the corresponding $\text{φ}{}_1$ is stationary.