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175
Counting Modulo Quantifiers on Finite Structures
 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 builtin linear order are also considered. For instance, the class of linear orders of length divisible by n + 1 cannot be exp ..."
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Cited by 16 (2 self)
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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 builtin linear order are also considered. For instance, the class of linear orders of length divisible by n + 1 cannot
Counting subgraphs via homomorphisms
 In Automata, Languages and Programming: ThirtySixth 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, ..."
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Cited by 10 (3 self)
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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
"... 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 ..."
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Cited by 3 (3 self)
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) 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 wellknown 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
 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 ..."
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Cited by 36 (9 self)
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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
, 2004
"... We explain how the MeisselLehmerLagariasMillerOdlyzko 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 MeisselLehmerLagariasMillerOdlyzko 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
TimeSpace Tradeoffs for Counting NP Solutions Modulo Integers
 In Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km sat ..."
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Cited by 15 (5 self)
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the tradeoff. We prove that the same limitation holds for Sat and MOD6Sat, as well as MODmSat 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
, 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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Cited by 2 (0 self)
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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 Ktheory and Galois cohomology of the field F is presented by the norm residue homomorphism
THE FIBONACCI SEQUENCE MODULO N
"... 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 ..."
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Cited by 2 (0 self)
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problem of determining the period of repetition of the decimal representation of 1/p. If p is a prime other than2 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
 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 ^ p1. ..."
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Cited by 6 (0 self)
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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 ^ p1.
Subprincipal homomorphisms in positive characteristic
 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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Cited by 15 (8 self)
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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
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175