A continuous analogue of Sperner's theorem

One of the best-known results of extremal combinatorics is Sperner's theorem, which asserts that the maximum size of an antichain of subsets of an n-element set equals the binomial coefficient (n/(n/2)), that is, the maximum of the binomial coefficients. In the last twenty years, Sperner's theorem has been generalized to wide classes of partially ordered sets. It is the purpose of the present paper to propose yet another generalization that strikes in a different direction. We consider the lattice Mod(n) of linear subspaces (through the origin) of the vector space Rⁿ. Because this lattice is infinite, the usual methods of extremal set theory do not apply to it. It turns out, however, that the set of elements of rank k of the lattice Mod(n), that is, the set of all subspaces of dimension k of Rⁿ, or Grassmannian, possesses an invariant measure that is unique up to a multiplicative constant. Can this multiplicative constant be chosen in such a way that an analogue of Sperner's theorem holds for Mod(n), with measures on Grassmannians replacing binomial coefficients? We show that there is a way of choosing such constants for each level of the lattice Mod(n) that is natural and unique in the sense defined below and for which an analogue of Sperner's theorem can be proven. The methods of the present note indicate that other results of extremal set theory may be generalized to the lattice Mod(n) by similar reasoning. © 1997 John Wiley & Sons, Inc.     

Morse-Sard定理在无限维时的一个反例

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＭｏｒｓｅ Ｓａｒｄｔｈｅｏｒｅｍｉｓａｆｕｎｄａｍｅｎｔａｌｔｈｅｏｒｅｍｉｎａｎａｌｙｓｉｓ ,ｅｓｐｅｃｉａｌｌｙｉｎｔｈｅｂａｓｉｓｏｆｔｒａｎｓｖｅｒｓａｌｉｔｙｔｈｅｏｒｙａｎｄｄｉｆｆｅｒｅｎｔｉａｌｔｏｐｏｌｏｇｙ .ＴｈｅｃｌａｓｓｉｃａｌＭｏｒｓｅ Ｓａｒｄｔｈｅｏｒｅｍｓｔａｔｅｓｔｈａｔｔｈｅｉｍａｇｅｏｆｔｈｅｓｅｔｏｆｃｒｉｔｉｃａｌｐｏｉｎｔｓｏｆａｆｕｎｃｔｉｏｎ ｆ :Ｒｍ→ＲｌｏｆｃｌａｓｓＣｍ -ｌ+1ｈａｓｚｅｒｏＬｅｂｅｓｇｕｅｍ…     

A constructive proof of a permutation-based generalization of Sperner's lemma

In a recent paper, Gale has given an interesting generalization of the KKM lemma in combinatorial topology. We present a similar generalization of Sperner's well-known lemma and give a constructive proof. The argument uses the familiar idea of following simplicial paths in a triangulation. To demonstrate that the algorithm must work, orientation considerations are necessary. Gale's generalized KKM lemma is derived from the main result. A permutation-based generalization of Brouwer's fixed point theorem is also given.     

A counterexample to the closed graph theorem for bilinear maps

The closed graph theorem allows one to assert that if a bounded transformation has an inverse, the inverse is bounded. The corresponding statement for bilinear maps is shown to be false, at least in the category of general Banach spaces. This problem arose in connection with the problem of writing every function in H¹ of the polydisc as a product of functions in H². The method of proof involves a covering of the unit sphere by neighborhoods which are products but in which the bounds involved in these products tend to infinity.     

A counterexample to a theorem of S. Piccard

A pair of homometric sets having all distinct elements in the set of element differences has been found, in contradiction to a theorem of S. Piccard which disallowed their existence.     

A counterexample to the Bartle-Graves selection theorem for multilinear maps

We present an example showing that the multilinear version of the Bartle-Graves Selection Theorem is false, even on finite dimensional spaces.

     

A counterexample to the theorem of Beppo Levi in three dimensions

Supported by the Society of Fellows and NSF Grant DMS-86-06160     

A generalization of the Riesz-Fischer theorem

An analog is established, in a certain sense, of the Riesz-Fischer theorem for the space L^P, p1, and a corollary derived.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 365–372, October, 1972.     

A generalization of the Looman-Menchoff theorem

In this paper we give a generalization of the classical Looman-Menchoff theorem:If f is a complex-valued continuous function of a complex variable in a domain G, f has partial derivatives f _x and f _y everywhere in G and the Cauchy Riemann equations f _x +if _y = 0are satisfied almost everywhere, then f is holomorphic in G. From our generalization of this theorem, we deduce a theroem of Sindalovskii [9] as a corollary and also answer some of the questions raised in [9]. We note in this context that, as far as we know, Sindalovskii’s result is the best published to date in this area.     

A generalization of the rogosinski-rogosinski theorem

We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f _j ⊂ K(f ₁) ∀j≥2 and yield The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α _j (x) ≡ α_j, and the space L is the m-dimensional Euclidean space.     

A generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

A generalization of the Razmyslov-Procesi theorem

It is proved that all relations between the invariants of several n x n-matrices over an infinite field of arbitrary characteristic follow from σ_n+1,σ_n+2,... where σ_i is the ith coefficient of a characteristic polynomial extended to matrices of any order ≥i. Similarly, all relations between the concomitants are implied by X_n+1, X_n+2, …, where X_i is a characteristic polynomial in the general n x n-matrix, also extended to matrices of any order. Supported by RFFR grant No. 95-01-00513. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 433–457, July–August, 1996.     

A generalization of the ray-chaudhuri-wilson theorem

Let K = {k₁,…,k_r} and L = {l₁,…,l_s} be two sets of non-negative integers and assume k_i > l_j for every i,j. Let F be an L-intersecting family of subsets of a set of n elements. Assume the size of every set in F is a number from K. We conjecture that |F| ? (ⁿ_s). We prove that our conjecturer is true for any K. (with min k_i ? s) when L = {0,1,…,s ? 1}. We also show that for any K and any L, (with min k_i > max l_j) CALLING STATEMENT : © 1995 John Wiley & Sons, Inc.     

A generalization of the Motzkin-Taussky theorem

In this paper we generalize the Motzkin-Taussky theorem to matrices with polynomial entries.     

A generalization of the circumcircle theorem

The theorem relating the bisectors of the edges of a triangle and the corresponding circumscribing circle is established as a special case of a theorem for triangles with weighted vertices where the edges are partitioned with circular arcs in the proportions of the weights. The circular arcs are established as being uniquely determined by the weights and the triangle, and are given by three circles with collinear centres. These circles either intersect in zero, one or two real points, these latter points being the triple points.     

A generalization of the Gauss-Lucas theorem

Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.     

A generalization of the remainder theorem and factor theorem

We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division method. As a consequence we obtain an extension of the classical Factor Theorem that provides a general divisibility criterion for polynomials. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students.     

A generalization of the Stein-Rosenberg theorem to Banach spaces

Summary In the first part of this note we prove a generalization of the Stein-Rosenberg theorem; the context is that of real Banach spaces with a normal reproducing cone and the operators involved are positive and completely continuous. Our generalization of the Stein-Rosenberg theorem improves the modern version of it as stated by F. Robert in [5, §2]. In the second part, we discuss briefly how our results are related to other versions of the Stein-Rosenberg theorem. In the last section we describe a situation to which the results in the first part can be applied.