Results 1 - 10 of 175

Counting Modulo Quantifiers on Finite Structures

by Juha Nurmonen - Information and Computation , 1996
"... We give a combinatorial method for proving elementary equivalence in firstorder logic FO with counting modulo n quantifiers D n . Inexpressibility results for FO(D n ) with built-in linear order are also considered. For instance, the class of linear orders of length divisible by n + 1 cannot be exp ..."
Abstract - Cited by 16 (2 self) - Add to MetaCart
We give a combinatorial method for proving elementary equivalence in firstorder logic FO with counting modulo n quantifiers D n . Inexpressibility results for FO(D n ) with built-in linear order are also considered. For instance, the class of linear orders of length divisible by n + 1 cannot

Counting subgraphs via homomorphisms

by Omid Amini, Fedor V. Fomin, Saket Saurabh - In Automata, Languages and Programming: Thirty-Sixth International Colloquium (ICALP , 2009
"... We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Björklund, ..."
Abstract - Cited by 10 (3 self) - Add to MetaCart
We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Björklund

Quartic and octic characters modulo n

by Steven Finch
"... Abstract. The average number of primitive quadratic Dirichlet characters of modulus n tends to a constant as n → ∞. The same is true for primitive cubic characters. It is therefore surprising that, as n → ∞, the average number of primitive quartic characters of modulus n grows with ln(n), and that t ..."
Abstract - Cited by 3 (3 self) - Add to MetaCart
) of nonzero complex numbers. We wish to count homomorphisms χ: Z ∗ n → C ∗ satisfying certain requirements. A Dirichlet character χ is quadratic if χ(k) 2 = 1 for every k in Z ∗ n. It is well-known that # quadratic Dirichlet characters of modulus ≤ N as N → ∞, where a(n) is multiplicative with a(2r ⎧ ⎨ 1 if r

Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes

by Sam Buss, Dima Grigoriev, Russell Impagliazzo, Toniann Pitassi - JOURNAL OF COMPUTER AND SYSTEM SCIENCES , 1999
"... This paper gives nearly optimal lower bounds on the minimum degree of polynomial calculus refutations of Tseitin's graph tautologies and the mod p counting principles, p 2. The lower bounds apply to the polynomial calculus over fields or rings. These are the first linear lower bounds for ..."
Abstract - Cited by 36 (9 self) - Add to MetaCart
This paper gives nearly optimal lower bounds on the minimum degree of polynomial calculus refutations of Tseitin's graph tautologies and the mod p counting principles, p 2. The lower bounds apply to the polynomial calculus over fields or rings. These are the first linear lower bounds

COUNTING PRIMES IN RESIDUE CLASSES

by Marc Deléglise, Pierre Dusart, Xavier-françois Roblot , 2004
"... We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x) can be used for computing efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1lessthanx for several values of x ..."
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We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x) can be used for computing efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1lessthanx for several values

Time-Space Tradeoffs for Counting NP Solutions Modulo Integers

by R. Ryan Williams - In Proceedings of the 22nd IEEE Conference on Computational Complexity , 2007
"... We prove the first time-space tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known time-space tradeoffs for Sat. Let m> 0 be an integer, and define MODm-Sat to be the problem of determining if a given Boolean formula has exactly km sat ..."
Abstract - Cited by 15 (5 self) - Add to MetaCart
the tradeoff. We prove that the same limitation holds for Sat and MOD6-Sat, as well as MODm-Sat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo

On The Norm Residue Homomorphism For Fields

by A. S. Merkurjev , 1996
"... . It is proved that the triviality of the Bockstein map of degree n for all fields of characteristic different from 2 implies the bijectivity of the norm residue homomorphism of degree n modulo 2. In particular, the bijectivity of the norm residue homomorphism follows from its surjectivity. 1991 Mat ..."
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Mathematics Subject Classification: 19E20 Keywords and Phrases: Bockstein map, norm residue homomorphism. Let F be a field and m be a natural number prime to the characteristic of F . The interaction between Milnor K-theory and Galois cohomology of the field F is presented by the norm residue homomorphism

THE FIBONACCI SEQUENCE MODULO N

by Andrew Vince
"... Let n be a positive integer. The Fibonacci sequence, when considered modulo n, must repeat. In this note we investigate the period of repetition and the related unsolved problem of finding the smallest Fibonacci number divisible by n. The results given here are similar to those of the simple problem ..."
Abstract - Cited by 2 (0 self) - Add to MetaCart
problem of determining the period of repetition of the decimal representation of 1/p. If p is a prime other than-2 or 5, it is an easy matter to verify that the period of repetition is the order of the element 10 in the multiplicative group Zp of residues modulo p. Analogously, the period of repetition

Counting points modulo p for some finitely generated subgroups of algebraic group

by C. R. Matthews - Bull. London Math. Soc , 1982
"... We begin by explaining the basic idea of this paper in a simple case. We write np for the order of 2 modulo the prime p, so that np is the number of powers of 2 which are distinct mod p. We have the elementary bounds logp < £ np ^ p-1. ..."
Abstract - Cited by 6 (0 self) - Add to MetaCart
We begin by explaining the basic idea of this paper in a simple case. We write np for the order of 2 modulo the prime p, so that np is the number of powers of 2 which are distinct mod p. We have the elementary bounds logp < £ np ^ p-1.

Sub-principal homomorphisms in positive characteristic

by George J. Mcninch - Math. Zeitschrift
"... ABSTRACT. Let G be a reductive group over an algebraically closed field of characteristic p, and let u ∈ G be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism φ: SL2/k → G whose image contains u. This result was first obtained b ..."
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ABSTRACT. Let G be a reductive group over an algebraically closed field of characteristic p, and let u ∈ G be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism φ: SL2/k → G whose image contains u. This result was first obtained
Results 1 - 10 of 175