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Subspaces that minimize the condition number of a matrix
Matrix the condition number subspace subspace
2015/8/10
We define the condition number of a nonsingular matrix on a subspace, and consider the problem of finding a subspace of given dimension that minimizes the condition number of a given matrix. We give a...
Solving the Odd Perfect Number Problem: Some New Approaches
Odd perfect number Euler factor inequalities OPN components non-injective and non-surjective mapping
2012/6/29
A conjecture predicting an injective and surjective mapping $X = \displaystyle\frac{\sigma(p^k)}{p^k}, Y = \displaystyle\frac{\sigma(m^2)}{m^2}$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$)...
The variance of the number of prime polynomials in short intervals and in residue classes
variance the number of prime polynomials short intervals residue classes Number Theory
2012/4/23
We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial i...
Counting rational points over number fields on a singular cubic surface
rational points over number fields singular cubic surface Number Theory
2012/4/18
A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successful...
The number of integer points in a family of anisotropically expanding domains
number of integer points family of anisotropically expanding domains Number Theory
2012/4/17
We investigate the remainder in the asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space, which remain unchanged along some linear subspace and exp...
On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields
Surjectivity Galois Representations Elliptic Curves over Number Fields Number Theory
2012/4/17
Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states t...
A universal first order formula defining the ring of integers in a number field
universal first order formula defining ring integers number field
2012/3/1
We show that the complement of the ring of integers in a number field K is Diophantine. This means the set of ring of integers in K can be written as {t in K | for all x_1, ..., x_N in K, f(t,x_1, ......
Every even number is equal to the difference of two prime number
sieve method two-dimension sieve method even number
2011/9/21
This paper have proved a conjecture about that every even number is equal to the difference of two prime numbers by using a two-dimension sieve method of using odd composites to the difference formula...
A Note on Terence Tao's Paper "On the Number of Solutions to 4/p=1/n_1+1/n_2+1/n_3"
Terence Tao's Paper Number of Solutions Number Theory
2011/9/21
Abstract: For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1,\,n_2,\,n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}....
The Skewes number for twin primes: counting sign changes of $π_2(x)-C_2 \Li_2(x)$
Primes twins Skewes number Number Theory
2011/9/6
Abstract: The results of the computer investigation of the sign changes of the difference between the number of twin primes $\pi_2(x)$ and the Hardy--Littlewood conjecture $C_2\Li_2(x)$ are reported. ...
An algorithm for list decoding number field codes
algorithm number field codes Number Theory
2011/9/5
Abstract: We present an algorithm for list decoding codewords of algebraic number field codes in polynomial time. This is the first explicit procedure for decoding number field codes whose constructio...
Counting the number of solutions to the Erdos-Straus equation on unit fractions
the Erdos-Straus equation unit fractions Number Theory
2011/8/25
Abstract: For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ with $x,y,z$ positive integers. Th...
Upper Bounds for the Number of Number Fields with Alternating Galois Group
Number Fields Alternating Galois Group Number Theory
2011/8/26
Abstract: We study the number $N(n, A_n, X)$ of number fields of degree $n$ whose Galois closure has Galois group $A_n$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect th...
Exact Covering Systems in Number Fields
Exact covering systems Lattice parallelotopes Chinese Remainder Theorem
2011/2/22
It is well known that in an exact covering system in Z, the biggest modulus must be repeated. Very recently, S. Kim proved an analogous result for certain quadratic fields. In this paper, we prove tha...
On the number of simple arrangements of five double pseudolines
number of simple arrangements five double pseudolines
2010/12/15
We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces...