We present a new polynomially solvable case of the Quadratic Assignment Problem in Koopmans-Beckman form QAP(A,B), by showing that the identity permutation is optimal when A and B are respectively a Robinson similarity and dissimilarity matrix and one of A or B is a Toeplitz matrix. A Robinson (dis)similarity matrix is a symmetric matrix whose entries (increase) decrease monotonically along row...

1. If p is a prime other than 2, half of the numbers 1, 2, ..., p-l are quadratic residues (modp) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality ...

A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant ∆n(a, b) of the trinomial fn,a,b(t) = t n + at+ b, where n ≥ 5 is a fixed integer. In particular, it is shown that, under the abc-conjecture, for every n ≡ 1 (mod 4), the quadratic fields Q (√ ∆n(a, b) ) are pairwise distinct for a positive proportion of such discriminants with i...

We present a concise and self-contained definition of the ideal class group, which is useful for proving facts about zero sets of Diophantine equations, and discuss a few relevant key facts. We approach this by first assembling some preliminary definitions regarding algebraic integers, and subsequently delving into several useful results about lattices, including Minkowski’s lemma. Then, return...

Abstract We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set real numbers, and the differences consecutive elements are all distinct. It known that $|A+B|\geq c|A||B|^{1/2}$ for any finite numbers $B.$ The bound tight up to constant multiplier. give new proof this result using bounds on crossing geometric graphs. construct examples show...

We exhibit, for each n ≥ 5, infinitely many quadratic number fields admitting unramified degree n extensions with prescribed signature whose normal closures have Galois group An. This generalizes a result of Uchida and Yamamoto, which did not include the ability to restrict the signature, and a result of Yamamura, which was the case n = 5. It is a folk conjecture that for n ≥ 5, all but finitel...

in the present paper a solution of the generalizedquadratic functional equation$$f(kx+ y)+f(kx+sigma(y))=2k^{2}f(x)+2f(y),phantom{+} x,yin{e}$$ isgiven where $sigma$ is an involution of the normed space $e$ and$k$ is a fixed positive integer. furthermore we investigate thehyers-ulam-rassias stability of the functional equation. thehyers-ulam stability on unbounded domains is also studied.applic...

in this article, we characterize the involutiveness of the linear combination of the forma1a1 +a2a2 when a1, a2 are nonzero complex numbers, a1 is a quadratic or tripotent matrix,and a2 is arbitrary, under certain properties imposed on a1 and a2.

This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields Q(vu), for which d has at most thtee distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms. 1. Let hid) denote the class number of the quadratic field Qi\ß) and let h id) denote the number of classes of primitive bina...

The Garman–Klass unbiased estimator of the variance per unit time of a zero–drift Brownian Motion (B), based on the usual financial data that reports for time windows of equal length the open (OPEN), minimum (MIN), maximum (MAX) and close (CLOSE) values, is quadratic in the statistic S1 = (CLOSE − OPEN, OPEN − MIN,MAX − OPEN). This estimator, with efficiency 7.4 with respect to the classical es...