In the first part of this paper, we consider 3 × 3 × 3 arrays with complex entries, and provide a complete self-contained proof of Kruskal’s theorem that the maximum rank is 5. In the second part, we provide a complete classification of the canonical forms of 3× 3× 3 arrays over F2; in particular, we obtain explicit examples of such arrays with rank 6. In 1989, Kruskal [6, page 10] stated witho...

The notation and terminology used in this paper have been introduced in the following papers: [3], [6], [4], [5], [10], [11], [7], [2], [1], [9], and [8]. In this paper X, Y denote sets, G denotes a group, and n denotes a natural number. Let us consider X. Note that ∅X,∅ is onto. Let us observe that every set which is permutational is also functional. Let us consider X. The functor permutations...

The articles [11], [7], [14], [13], [15], [5], [6], [2], [12], [1], [10], [8], [9], [3], and [4] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: A, B denote ordinal numbers, K, M, N denote cardinal numbers, x, y, z, X , Y , Z, Z1, Z2 denote sets, n denotes a natural number, and f , g denote functions. Let I1 be a function. We say that I1 is car...

c W W L Chen, 1984, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and ...

We adopt the following convention: x, y are real numbers, i, j, k are natural numbers, and p, q are finite sequences of elements of R. The following propositions are true: (1) Let A be a closed-interval subset of R and D be an element of divsA. If vol(A) 6= 0, then there exists i such that i ∈ domD and vol(divset(D, i)) > 0. (2) Let A be a closed-interval subset of R, D be an element of divsA, ...

we extend Painlevé’s determinateness theorem from the theory of ordinary differential equations in the complex domain allowing more general ’multiple-valued’ Cauchy’s problems. We study C0−continuability (near singularities) of solutions. Foreword and preliminaries In this paper we slightly improve Painlevé’s determinateness theorem (see [HIL], th.3.3.1), investigating the C0−continuability of ...

Exercise 1. Show that distances between sets do not necessarily satisfy the triangle inequality. That is, it is possible that d(S1, S2) + d(S2, S3) > d(S1, S3) for some sets S1, S2 and S3. Exercise 2. Prove that d(x, y) ≥ d(S, x)− d(S, y) and thus d(x, y) ≥ |d(S, x)− d(S, y)|. Proof. Fix ε > 0. Let y′ ∈ S be such that d(y′, y) ≤ d(S, y) + ε (if S is a finite set, there is y′ ∈ S s.t. d(y, y′) =...