On the rational equivalence of full decomposable forms

Let k be any finite normal extension of the rational field Q and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. Consider the set F of the full decomposable forms which correspond to the invertible fractional ideals of D. In a recent paper the author has given arithmetic criteria to determine which classes of improperly equivalent forms in F integrally represent a given positive rational integer m. These criteria are formulated in terms of certain integer sequences which satisfy a linear recursion and need only be considered modulo the primes dividing m. Here, for the most part, we consider partitioning F under rational equivalence. It is a found that the set of rationally equivalent classes in F is a group under composition of forms analogous to Gauss' and Dirichlet's classical results for binary quadratic forms. This leads us to given criteria as before to determine which classes of rationally equivalent forms in F rationally represent m. Moreover, by applying the genus theory of number fields, we find arithmetic criteria to determine when everywhere local norms are global norms if the Hasse norm principle fails to hold in $\text{k}\text{Q}$.     

Some results on sperner families

Let F be a Sperner family of subsets of {1,…,m}. Bollobás showed that if $\text{A ∈ F ?}\text{A}\text{= {1,…,m}βA ∈ F}$, and if the parameters of F are p₀,…,p_m then

$\text{∑}\text{i=0}\text{[}\text{m}\text{2}\text{Pi}\text{m?1}\text{i?1}\text{+}\text{∑}\text{i=[}\text{m}\text{2}\text{]+1}\text{m}\text{Pi}\text{m?1}\text{m?i?1}\text{? 2}$

Here we generalize this result and prove some analogues of it. A corollary of Bollobás' result is that $\text{|F| ? 2(}{}_{\mathrm{<mtext>[</mtext><mtext>m</mtext><mtext>2</mtext><mtext>]?1</mtext>}}{}^{\mathrm{m?1}}\text{)}$. Purdy proved that if $\text{A ∈ F ?}\text{A}\text{? F}$ then $\text{|F| ? (}{}_{\mathrm{<mtext>[</mtext><mtext>m</mtext><mtext>2</mtext><mtext>]+1</mtext>}}{}^{\mathrm{m}}\text{)}$, which implies Bollobás' corollary. We also show that Purdy's result may be deduced from Bollobás' by a short argument. Finally, we give a canonical form for Sperner families which are also pairwise intersecting.     

Nonexistence of maximal asymptotic union nonbases

Un sous-ensemble $\text{B}$ de Pensemble $\text{F}$ de toutes les parties finies de $\text{N}$ est une U-base asymptotique d'ordre h si chaque élément de $\text{F}$, à un nombre fini d'exceptions près, est l'union de h, pas nécessairement distincts, éléments de $\text{B}$. On démontre qu'une partie $\text{B}$ de $\text{F}$ qui n'est pas une U-base asymptotique d'ordre h ne peut pas vérifier le critère de maximalité ci-dessous: pour tout A appartenant à $\text{F}$|$\text{B}$, $\text{B}$∪{A} soit U-base asymptotique d'ore h.A subcollection $\text{B}$ of the collection $\text{F}$ of all finite subsets of $\text{N}$ is an asymptotic union basis of order h, if all but finitely many sets in $\text{F}$ are unions of h, not necessarily distinct, elements of $\text{B}$. Otherwise, $\text{B}$ is an asymptotic union nonbasis of order h. We prove that no asymptotic union nonbasis $\text{B}$ of order h fulfills the following criterion of maximality: for every A belonging to $\text{F}$ but not to $\text{B}$, $\text{B}$ ∪{A}; is an asymptotic union basis of order h.     

Enumeration of self-complementary structures

For functions f : D → R_k where D is a finite set and R_k = {0,1,… k} we define complementary and self-complementary functions. De Bruijn's generalization of Polya's theorem gives a formula for the number of non-isomorphic self-complementary functions $\text{f ∈ R}{}_{\mathrm{k}}{}^{\mathrm{D}}$. We consider the special cases of generalized graphs and m-placed relations. Among other results we prove that the number of non-isomorphic self-complementary relations over 2n elements is equal to the number of non-isomorphic self-complementary graphs with 4n + 1 points.     

Sur une loi de probabilite conditionelle de kleitman

Nous donnons une généralisation et une démonstration très courte d'un théorème de Kleitman qui dit: Pour toute paire d'idéaux $\text{F}$, (β) d'éléments dans le produit cartésien de k ensembles totalement ordonnés P = [1, 2, … n₁] ? … ? [1, 2, … n_k], nous avons ($\text{|}\text{F}\text{|}\text{|P|}$). ($\text{|(β)|}\text{|P|}\text{) ?}\text{|}\text{F}\text{∩ (β)|}\text{|P|}$ ou en langage probabiliste $\text{Pr(}\text{F}\text{? Pr (}\text{F}\text{|(β))}$.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

On the existence of frames

This paper deals with two topics, namely, frames and pairwise balanced designs (PBD's). Frames, which were introduced by W.D. Wallis for the construction of (skew) Room squares, are shown to exist for most orders congruent to 1 (mod 4). This result relies heavily on the existence of PBD's since the set F = {v | there is a frame of order v] is shown to be PBD-closed. By employing a generalization of the usual recursive construction for PBD's, it is shown that $\text{B}${5, 9, 13, 17}?$\text{B}${5, 9, 13}∪{69, 77, 97, 137, 237, 277, 317, 377, 569}?{n | n  1 (mod 4), n>0}?{29, 33, 49, 57, 93, 129, 133}, where $\text{B}$(K) denotes the set of orders of PBD's of index one having block-sizes from the set K. Frames of orders 5, 9, 13 and 17 are exhibited which immediately implies that F?$\text{B}${5, 9, 13, 17}. D.R. Stinson and W.D. Wallis have shown that {29, 49}?F. Thus there is a frame of order υ for every positive integer υ congruent to 1 (mod 4) with the possible exceptions of υ ? {33, 57, 93, 133}.     

Further characterizations and properties of exactly covering congruences

If every nonnegative integer occurs in exactly one of the m integer sequences $\text{a}{}_{\mathrm{i}}\text{n\#\&62; + b}{}_{\mathrm{i}}\text{(n ? 0.1 ? i ? m)}$, then the system a_in + b_i is called an exactly covering congruence (ECC). Three characterizations of ECC's in terms of exponential functions, Bernoulli polynomials and Evler polynomials are given, from which several properties of ECC's are deduced, including a method of obtaining from an ECC with m moduli a_i several ECC's with ? m moduli.     

The 1-width of (0,1)-matrices having constant row sum 3

In this note we establish upper bounds for the 1-width of an m × n matrix of 0's and 1's having three 1's in every row and having a constant number, c, of 1's in every column. When c = 3, this upper bound is $\text{n}\text{2}$ and when c = 4 this estimate is $\text{5n}\text{9}$. In these cases the upper bound is best possible, in the sense that for every possible size there exist matrices with this maximal 1-width. The technique of proof is also used to improve the known bound for the 1-width of (0, 1)-matrices with constant line sum 4.     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

On effectively computable realizations of choice functions: Dedicated to Professors Kenneth J. Arrow and Anil Nerode

Let X be a compact, convex subset of R_n, and let $\text{〈R(X),}\text{F}{}_{\mathrm{R}}\text{〉}$ be a recursive space of alternatives, where R(X) is the image of X in a recursive metric space, and $\text{F}{}_{\mathrm{R}}$ is the family of all recursive subsets of R(X). If $\text{C:}\text{F}{}_{\mathrm{R}}\text{→}\text{F}{}_{\mathrm{R}}$ is a non-trivial recursively representable choice function that is rational in the sense of Richter, we prove that C has no recursive realization within Church's Thesis. Our proof is not a diagonalization argument and uses no paradoxical statements from formal systems. Instead, the proof is a Kleene-Post reduction style argument and uses the Turing equivalence between mechanical devices of computation and the recursive functions of Gödel and Kleene.     

The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions

Hecke's correspondence between modular forms and Dirichlet series is put into a quantitative form giving expansions of the Dirichlet series in series of incomplete gamma functions in two special cases. The expansion is applied to show, for example, the positivity of Epstein's zeta function at $\text{s =}\text{n}\text{4}$ when the n-ary positive real quadratic form involved has a “small” minimum over the integer lattice. Hecke's integral formula is used to consider consequences for the Dedekind zeta function of a number field.     

Generalizations of theorems of Wilson,Fermat and Euler

By using Pólya's and de Bruijn's theorems of enumeration, we prove some generalizations of Wilson's, Fermat's and Euler's theorems in number theory. We also present an algorithm for Pólya's and de Bruijn's theorems. By using the properties of the algorithm, we interpret the meanings of the integers $\text{(}\text{1}\text{p}\text{)((p ? 1)! + 1)}$ and $\text{(}\text{1}\text{p}\text{)(a}{}^{\mathrm{p}}\text{? a)}$ where p is a prime and a is a positive integer.     

Solution of fuzzy equations with extended operations

Assuming that 1 is any operation defined on a product set X × Y and taking values on a set Z, it can be extended to fuzzy sets by means of Zadeh's extension principle. Given a fuzzy subset C of Z, it is here shown how to solve the equation $\text{A 1 B = C}$ (or $\text{A 1 B ? C}$) when a fuzzy subset A of X (or a fuzzy subset B of Y) is given. The methodology we provide includes, as a special case, the resolution of fuzzy arithmetical operations, i.e. when 1 stands for +, ?, × or ÷, extended to fuzzy numbers (fuzzy subsets of the real line). The paper is illustrated with several examples in fuzzy arithmetic.     

Existence and regularity of extrema

Consider the minimization of a possibly noncoercive Gâteaux differentiable functional $\text{F:X→}\text{R}$. A modified notion of coercivity is introduced which may be usable to show existence of a minimum. Alternatively, $\text{F}\text{?}\text{:D→}\text{R}$ has a minimum at $\text{y}\text{ε}$$\text{D}$ ($\text{F}\text{?}$ not differentiable but the restriction F of $\text{F}\text{?}$ to $\text{X}$?$\text{D}$ differentiable), one may be able to show $\text{y}\text{?}$ is actually in $\text{X}$. The latter case is related to justification of formally calculated “necessary conditions” for optimal controls. The arguments are applications of Ekeland's “approximate variational principle” (J. Math. Anal. Appl.47 (1974), 324–353).     

The existence and convergence of subsequences of Padé approximants

We prove the existence of an infinite number of Padé approximants, and thereby remedy a defect in Nuttall's theorem. It is proved that the sequences of Padé approximants shown by Perron, Gammel, and Wallin to be everywhere divergent contain subsequences which are everywhere convergent. It is further proved that there always exist, for entire functions, everywhere convergent [1, N] and [2, N] subsequences of Padé approximants. There must exist subsequences of [m, N] Padé approximants (N → ∞) which converge almost everywhere in $\text{¦z¦ ? ? < R}$ to functions f(z) which are regular except for a finite number (n ? m) of poles in $\text{¦z¦ < R}$. We prove convergence of the [N, N + j] Padé approximants in the mean on the Riemann sphere for meromorphic functions.     

Some improvements in Neuberger's iteration procedure for solving partial differential equations

If m and n are positive integers then let S(m, n) denote the linear space over R whose elements are the real-valued symmetric n-linear functions on E_m with operations defined in the usual way. If $\text{U}$ is a function from some sphere S in E_m to R then let $\text{U}$⁽ⁱ⁾(x) denote the ith Frechet derivative of $\text{U}$ at x. In general $\text{U}$⁽ⁱ⁾(x)∈S(m,i). In the paper “An Iterative Method for Solving Nonlinear Partial Differential Equations” [Advances in Math. 19 (1976), 245–265] Neuberger presents an iterative procedure for solving a partial differential equation of the form

$\text{AU}{}^{\mathrm{n}}\text{(x)=F(x, U(x), U′(x),…,U}{}^{\mathrm{k}}\text{(x))}$

, where k > n, $\text{U}$ is the unknown from some sphere in E_m to R, A is a linear functional on S(m, n), and F is analytic. The defect in the theory presented there was that in order to prove that the iterates converged to a solution $\text{U}$ the condition $\text{k ?}\text{n}\text{2}$ was needed. In this paper an iteration procedure that is a slight variation on Neuberger's procedure is considered. Although the condition $\text{k ?}\text{n}\text{2}$ cannot as yet be eliminated, it is shown that in a broad class of cases depending upon the nature of the functional A the restriction $\text{k ?}\text{n}\text{2}$ may be replaced by the restriction $\text{k ?}\text{3n}\text{4}$.     

Fuzzy processes

In this paper, contributions to fuzzy probability and to differential equations with fuzzy parameters are made.After an introductory section, a review of fuzzy sets and fuzzy algebra is given in Section 2. The main new results of the investigation are contained in Section 3.In Section 3, Zadeh's definition of the probability of a ‘fuzzy event’ the average value of a fuzzy function are extended into the time domain. It is then shown that not only grades of membership, but also probabilistic processes with notions of fuzziness contained, can be defined which obey ordinary, matric, or integro-differential equations. Applications are also given in Section 3.