2. (Jordan) A transitive permutation group of degree n > 1 contains a derangement. In fact (Cameron and Cohen) the proportion of derangements in a transitive group G is at least 1/n. Equality holds if and only if G is sharply 2transitive, and hence is the affine group {x 7→ ax + b : a, b ∈ F, a 6= 0} over a nearfield F. The finite nearfields were determined by Zassenhaus. They all have prime po...

We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integers a and b, let G n denote the graph for which V {0, 1, . . . , n − 1} is the set of vertices and there is an edge between a and b if the congruence a ≡ b modn is solvable. Let n p1 1 p k2 2 · · · pr r be the prime power factorization of an integer n, where p1 < p2 < · · · < ...

We present the first direct measurement of proper motion pulsar J1124-5916 in young, oxygen-rich supernova remnant G292.0+1.8. Using deep Chandra ACIS-I observations from 2006 and 2016, we measure a positional change $0.^{\prime\prime}21$ $\pm$ $0.^{\prime\prime}05$ over $\sim$ 10 year baseline, or $0.^{\prime\prime}02$ yr$^{-1}$. At distance 6.2 0.9 kpc, this corresponds to kick velocity plane...

Assuming the well known conjecture that for any γ > 0 and x sufficiently large the interval [x, x+xγ ] always contains a prime number, we prove the following unexpected result: There exist numbers 0 < ρ < 1 arbitrarily close to 0, and arbitrarily large primes q, such that if S is any subset of Z/qZ of density at least ρ, having the least number of 3-term arithmetic progressions among all such s...

This paper studies a data recovery problem in compressed sensing (CS), given a measurement vector b with corruptions:     0 0 b Ax f   , can we recover   0 x and   0 f via the reweighted 1  minimization: , 1 1 min || || || , . || . x f x f s t Ax f b     ? Where the m n  measurement matrix A is a partial Fourier matrix,   0 x denotes the n dimensional ground true signal vector...

Section 2. Planar functions and semifields and other algebraic structures Suppose that f is a function over Fq where pn = q and p is an odd prime. We define about the behavior of the number of solutions of difference equations of f as the following. For a, b ∈ Fq, Nf (a, b) := {x ∈ Fq | f(x+ a)− f(x) = b}, Nf = Maxa,b∈Fq, a =0Nf (a, b). If Nf ≤ δ, a function f is called to be differentially δ-u...

Let p be an odd prime and a be a positive integer. We show that p−1 k=0 (−1) (a−1)k p − 1 k a ≡ 2 a(p−1) + a(a − 1)(3a − 4) 48 p 3 B p−3 (mod p 4), which is a generalization of a congruence due to Carlitz.

We prove the nonexistence of [gq(4, d), 4, d]q codes for (a) d = q /2−q− 2q+1 for q = 2, h ≥ 3, (b) d = 2q−3q−2q+1 for q ≥ 7, and (c) d = 2q−rq−q+1 for 3 ≤ r ≤ q − q/p, q = p with p prime, where gq(k, d) = ∑k−1 i=0 ⌈ d/q ⌉

Let p be an odd prime and a be a positive integer. We show that p−1 k=0 (−1) (a−1)k p − 1 k a ≡ 2 a(p−1) + a(a − 1)(3a − 4) 48 p 3 B p−3 (mod p 4), which is a generalization of a congruence due to Morley.

in this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. we prove that if $n$ is a weakly prime subtractive ternary subsemimodule of a ternary $r$-semimodule $m$, then either $n$ is a prime ternary subsemimodule or $(n : m)(n : m)n = 0$. if $n$ is a $q$-...