In this exercise n is a large integer. a) Suppose that there is an integer F = ∏k i=1 p ei i with known primes pi and ei > 0 such that F |(n − 1) and F > √ N . Let R = (n − 1)/F and assume that for each i there is an integer ai such that the following conditions hold: a RF i ≡ 1 mod n and (a R(F/pi) i − 1, n) = 1. Show that Z/(n ·Z)× contains a primitive F−th root of unity and conclude that n i...

We prove an explicit formula for the number of n × n upper triangular matrices, over GF (q), whose square is the zero matrix. Theorem. The number of n × n upper-triangular matrices over GF (q) (the finite field with q elements), whose square is the zero matrix, is given by the polynomial C n (q), where, C 2n (q) = j 2n n − 3j − 2n n − 3j − 1 · q n 2 −3j 2 −j , C 2n+1 (q) = j 2n + 1 n − 3j − 2n ...

We consider the Schrödinger equation { −∆u+ V (x)u = λK(x)f(u) in R ; u ∈ H(R ), (Pλ) where N ≥ 2, λ ≥ 0 is a parameter, V,K : R → R are radially symmetric functions, and f : R → R is a continuous function with sublinear growth at infinity. We first prove that for λ small enough no non-zero solution exists for (Pλ), while for λ large enough at least two distinct non-zero radially symmetric solu...

Denote points in Euclidean n-space, Rn, by x = (x1, . . . ,xn) and let E and ∂E denote the closure and boundary of E ⊆Rn, respectively. Let 〈x, y〉 denote the standard inner product in Rn, |x| = 〈x,x〉1/2, and set B(x,r)= {y ∈Rn : |y− x| < r} whenever x ∈Rn, r > 0. Define k-dimensional Hausdorffmeasure, 1≤ k ≤ n, in Rn as follows: for fixed δ > 0 and E ⊆Rn, let L(δ)= {B(xi,ri)} be such that E ⊆ ⋃...

Let Pk n be the maximum value achieved by the permanent over k n, the set of (0, 1)-matrices of order n with exactly k ones in each row and column. Brègman proved that Pk n k!n/k . It is shown here that Pk n k!t r! where n = tk + r and 0 r < k. From this simple bound we derive that (P k n ) 1/n ∼ k!1/k whenever k = o(n) and deduce a number of structural results about matrices which achieve Pk n...

No problem is ever permanently closed. The editor is always pleased to consider for publication new solutions or new insights o n p a s t p r oblems. (1) Determine the set of all c > 0 such that whenever (1) holds, then we have x + y + z xy + yz+ zx. The positive r eals c asked for are those satisfying c 4. From (1), we have xy < c and z = c ; xy x + y + xy. Substituting this value of z into th...

INFINITE LOCALLY FINITE GROUPS OF TYPE PSL(2, K) OR Sz(K) ARE NOT MINIMAL UNDER CERTAIN CONDITIONS JAVIER OTAL AND JUAN MANUEL PENA In classifying certain infinite groups under minimal conditions it is needed to find non-simplicity criteria for the groups under consideration . We obtain some of such criteria as a consequence of the main result of the paper and the classification of finite simpl...

A central problem in coding theory is to determine Aq(n, 2e + 1), the maximal cardinality of a q-ary code of length n correcting up to e errors. When e is fixed and n is large, the best upper bound for A(n, 2e+1) (the binary case) is the well-known Johnson bound from 1962. This however simply reduces to the sphere-packing bound if a Steiner system S(e + 1, 2e + 1, n) exists. Despite the fact th...