A cover (with triples) of λKn is an ordered pair (V,B) where V is an n-element set and B is a set of 3-element subsets of V called blocks such that each 2-element subset of V appears in at least λ blocks. Define E(B) = {{x, y}, {x, z}, {y, z} | {x, y, z} ∈ B}. The excess (sometimes called the padding) of a cover is defined to be E(B)−E(λKn), although it is often convenient to think of the exces...

In 1991, one of the authors showed the existence of quadratic transformations between the Painlevé VI equations with local monodromy differences (1/2, a, b,±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformations. They are related to the quadratic transformations obtained by Manin and RamaniGrammaticos-Tamizhmani via Okamoto transformations. To avoid cumbersom...

A full-rank lattice L in R is a discrete subgroup of R which is the set of all integer linear combinations of n-linearly independent vectors, say b1, · · · ,bn i.e., L = { ∑n i=1 zibi | z1, · · · , zn ∈ Z}. The matrix B = [b1, · · · ,bn] is called the basis of the lattice and the matrix Q = B′B is called a Gram matrix of the lattice. A lattice is integral if its Gram matrix has only integer ent...

There are many good references for this material including [EKM], [L], [Pf] and [S]. 1 Quadratic forms Let k be a field with char k = 2. Definition 1.1. A quadratic form q : V → k on a finite-dimensional vector space V over k is a map satisfying: 1. q(λv) = λ 2 q(v) for v ∈ V , λ ∈ k. 2. The map b q : V × V → k, defined by b q (v, w) = 1 2 [q(v + w) − q(v) − q(w)] is bilinear. We denote a quadr...

We characterize the involutiveness of linear combinations form $a{\mathbf{A}} + b{\mathbf{B}}$ when $a,b$ are nonzero complex numbers, ${\mathbf{A}}$ is a quadratic $n \times n$ matrix and ${\mathbf{B}}$ an arbitrary matrix, under certain properties imposed on $\mathbf{A}$ $\mathbf{B}$.

This paper introduces the emph{interval unilateral quadratic matrix equation}, $IUQe$ and attempts to find various analytical results on its AE-solution sets in which $A,B$ and $CCC$ are known real interval matrices, while $X$ is an unknown matrix. These results are derived from a generalization of some results of Shary. We also give sufficient conditions for non-emptiness of some quasi-solutio...

Given a finite set T of maps on a finite ring R, we look at the finite simple graph G = (V,E) with vertex set V = R and edge set E = {(a, b) | ∃t ∈ T, b = t(a), and b 6= a}. An example is when R = Zn and T consists of a finite set of quadratic maps Ti(x) = x 2 + ai. Graphs defined like that have a surprisingly rich structure. This holds especially in an algebraic setup when T is generated by po...

An important question in designing cryptographic functions including substitution boxes (S-boxes) is the relationships among the various nonlinearity criteria each of which indicates the strength or weakness of a cryptographic function against a particular type of cryptanalytic attacks. In this paper we reveal, for the rst time, interesting connections among the strict avalanche characteristics...

The T-palindromic quadratic eigenvalue problem (λB + λC + A)x = 0, with A,B,C ∈ Cn×n, C = C and B = A, governs the vibration behavior of trains. One way to solve the problem is to apply a structure-preserving doubling algorithm (SDA) to the nonlinear matrix equation (NME) X + BX−1A = C and “square-root” the matrix quadratic involved. In this paper, we generalize the SDA for the solution of (odd...

This paper is a sequel to [6], in which we introduced quadratic forms on a module over a supertropical semiring R and analyzed the set of bilinear companions of a single quadratic form V → R in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering, which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic...