Habiro–Le invariants dominate sl2 Witten–Reshetikhin–Turaev invariants of rational homology 3–spheres at roots of unity of order coprime with the torsion. In this paper we give a formula for the Habiro–Le invariant of a rational homology 3–sphere obtained by rational surgery on a link in S. As an application, we compute this invariant for Seifert fibered spaces and for Dehn surgeries on twist k...

For x real, let { } $ \lbrace \rbrace$ be the fractional part of (that is, = − ⌊ ⌋ $\lbrace x\rbrace - \lfloor \rfloor$ ). The lonely runner conjecture can stated as follows: for any n positive integers v 1 < 2 ⋯ v_1 v_2 \dots v_n$ , there exists a real number t such that / ( + ) ⩽ i 1/(n+1) \leqslant v_i t\rbrace n/(n+1)$ 1, n$ . In this paper, we prove if ε > 0 \epsilon >0$ and is sufficientl...

We state a number of important results which we owe to Tarlok Shorey. 1 Shorey’s Contributions to Linear Form Estimates and Some Applications One of the first results of Shorey concerns a sharpening of a theorem of Sylvester. Sylvester proved in 1892 that a product of k consecutive positive integers greater than k is divisible by a prime exceeding k. By combining a result of Jutila which depend...

The idea of the NFS is rooted in work by Coppersmith, Odlyzko, and Schroeppel [9], who used the Gaussian integers, Z[i], to compute discrete logarithms in GF (p) in subexponential time. They looked for coprime integers, a and b such that the integer a+bx is smooth for some fixed integer value of x and over some factor base, and such that a+ bi is smooth over some factor base of primes in Z[i]. ...

has only finitely many solutions in integers x and y. In the first part of this paper we shall establish upper bounds for the number of solutions of (1) in coprime integers x and y under the assumption that the discriminant D(F) of F is nonzero. For most integers h these bounds improve upon those obtained by Bombieri and Schmidt in [5]. In the course of proving these bounds we shall establish a...

Given integers d ≥ 3 and N 3, let G be a finite abelian group acting faithfully linearly on smooth hypersurface of degree in the complex projective space ℙN−1. Suppose ⊂ PGL(N, ℂ) can lifted to subgroup GL(N, ℂ). moreover that there exists an element g such G/❬g❭ has order coprime − 1. Then all possible are determined (Theorem 4.3). As application, we derive 4.8) orders linear automorphisms hyp...

in this paper we study the coprime graph of a group $g$. the coprime graph of a group $g$, is a graph whose vertices are elements of $g$ and two distinct vertices $x$ and $y$ are adjacent iff $(|x|,|y|)=1$. in this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. also we study the automorphism group of the coprime graph.

Coprime linear microphone arrays allow for narrower beams with fewer sensors. A coprime microphone array consists of two staggered uniform linear subarrays with M and N microphones, where M and N are coprime with each other. By applying spatial filtering to both subarrays and combining their outputs, M+N-1 microphones yield M⋅N directional bands. In this work, the coprime sampling theory is imp...

Let m>2 and r be integers, let y0 be an integer in the least residue system mod m, and let X be an integer coprime to m with X ^ ± 1 (mod m) and (X ~ l)y0 + r ^ 0 (mod m). A sequence y0,yl9...of integers in the least residue system mod m is generated by the recursion yn+ x = Xyn + r (mod m) for n = 0, 1, . . . . In the homogeneous case r = 0 (mod m), one chooses y0 to be coprime to m. The seque...