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Dimensions of the spaces of cusp forms and newforms on Γ0(N) and Γ1(N
 J. Number Theory
"... The study of modular forms on congruence groups was initiated by Hecke and Petersson in the 1930s and, at least when the weight k is an integer exceeding 1, is quite well understood. In particular, formulas for the dimensions of the spaces of modular forms and cusp forms on the congruence groups ..."
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The study of modular forms on congruence groups was initiated by Hecke and Petersson in the 1930s and, at least when the weight k is an integer exceeding 1, is quite well understood. In particular, formulas for the dimensions of the spaces of modular forms and cusp forms on the congruence groups
Publicly Verifiable Grouped Aggregation Queries on Outsourced Data Streams
"... Abstract—Outsourcing data streams and desired computations to a third party such as the cloud is a desirable option to many companies. However, data outsourcing and remote computations intrinsically raise issues of trust, making it crucial to verify results returned by third parties. In this context ..."
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Abstract—Outsourcing data streams and desired computations to a third party such as the cloud is a desirable option to many companies. However, data outsourcing and remote computations intrinsically raise issues of trust, making it crucial to verify results returned by third parties. In this context, we propose a novel solution to verify outsourced grouped aggregation queries (e.g., histogram or SQL Groupby queries) that are common in many business applications. We consider a setting where a data owner employs an untrusted remote server to run continuous grouped aggregation queries on a data stream it forwards to the server. Untrusted clients then query the server for results and efficiently verify correctness of the results by using a small and easytocompute signature provided by the data owner. Our work complements previous works on authenticating remote computation of selection and aggregation queries. The most important aspect of our solution is that it is publicly verifiable— unlike most prior works, we support untrusted clients (who can collude with other clients or with the server). Experimental results on real and synthetic data show that our solution is practical and efficient. I.
Zeroing the Baseball Indicator and the Chirality of Triples
, 2004
"... Starting with a common baseball umpire indicator, we consider the zeroing number for twowheel indicators with states (a; b) and threewheel indicators with states (a; b; c). Elementary number theory yields formulae for the zeroing number. The solution in the threewheel case involves a curiously ..."
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Starting with a common baseball umpire indicator, we consider the zeroing number for twowheel indicators with states (a; b) and threewheel indicators with states (a; b; c). Elementary number theory yields formulae for the zeroing number. The solution in the threewheel case involves a curiously nontrivial minimization problem whose solution determines the chirality of the ordered triple (a; b; c) of pairwise relatively prime numbers. We prove that chirality is in fact an invariant of the unordered triple fa; b; cg. We also show that the chirality of Fibonacci triples alternates between 1 and 2.
The Mathematica ® Journal Ordered and Unordered Factorizations of Integers
"... We study the number of ways of writing a positive integer n as a product of integer factors greater than one. We survey methods from the literature for enumerating and also generating lists of such factorizations for a given number n. In addition, we consider the same questions with respect to facto ..."
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We study the number of ways of writing a positive integer n as a product of integer factors greater than one. We survey methods from the literature for enumerating and also generating lists of such factorizations for a given number n. In addition, we consider the same questions with respect to factorizations that satisfy constraints, such as having all factors distinct. We implement all these methods in Mathematica and compare the speeds of various approaches to generating these factorizations in practice. To study the number of ways of writing a positive integer n as a product of integer factors greater than one, there are two basic cases to consider. First, we can regard rearrangements of factors as different. In the case of n = 12, this gives
NOTES ON SOME NEW KINDS OF PSEUDOPRIMES
"... Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow ppseudoprimes and elementary Abelian ppseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow ppseudoprime to two bases only, where ..."
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Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow ppseudoprimes and elementary Abelian ppseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow ppseudoprime to two bases only, where p = 2 or 3. In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavorable for Browkin’s observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow 2pseudoprimes to the same bases. In Section 3, we give examples of Sylow ppseudoprimes to the first several prime bases for the first several primes p. We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow ppseudoprime to the same bases for all p ∈{2, 3, 5, 7, 11, 13}. In Section 4, we define n to be a kfold Carmichael Sylow pseudoprime, ifitisaSylowppseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n − 1. We find and tabulate all three 3fold Carmichael Sylow pseudoprimes < 1016. In Section 5, we define a positive odd composite n to be a Sylow uniform pseudoprime to bases b1,...,bk, or a Sylupsp(b1,...,bk) for short, if it is a Sylppsp(b1,...,bk) for all the first ω(n − 1) − 1 small prime factors p of n − 1, where ω(n − 1) is the number of distinct prime factors of n − 1. We find and tabulate all the 17 Sylupsp(2, 3, 5)’s < 1016 and some Sylupsp(2, 3, 5, 7, 11)’s < 1024. Comparisons of effectiveness of Browkin’s observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6. 1.
Algorithmic Number Theory MSRI Publications
"... Quadratic nonresidues 36 Chinese remainder theorem 57 ..."
Algorithmic Number Theory MSRI Publications
"... Quadratic nonresidues 36 Chinese remainder theorem 57 ..."
FACTORING NEWPARTS OF JACOBIANS OF CERTAIN MODULAR CURVES
"... Abstract. We prove a conjecture of Yamauchi which states that the level N for which the new part of J0(N) is Qisogenous to a product of elliptic curves is bounded. We also state and partially prove a higherdimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we deriv ..."
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Abstract. We prove a conjecture of Yamauchi which states that the level N for which the new part of J0(N) is Qisogenous to a product of elliptic curves is bounded. We also state and partially prove a higherdimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we derive a formula for the trace of Hecke operators acting on spaces S new (N, k) of newforms of weight k and level N. We use this trace formula to study the equidistribution of eigenvalues of Hecke operators on these spaces. For any d ≥ 1, we estimate the number of normalized newforms of fixed weight and level, whose Fourier coefficients generate a number field of degree less than or equal to d. 1.
A Heuristic for the Prime Number Theorem This article appeared in The Mathematical Intelligencer 28:3 (2006) 6–9, and is copyright by Springer
"... Why does ‰ play such a central role in the distribution of prime numbers? Simply citing the Prime Number Theorem (PNT), which asserts that pHxL ~ x ê ln x, is not very illuminating. Here "~ " means "is asymptotic to " and pHxL is the number of primes less than or equal to x. So why do natural logs a ..."
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Why does ‰ play such a central role in the distribution of prime numbers? Simply citing the Prime Number Theorem (PNT), which asserts that pHxL ~ x ê ln x, is not very illuminating. Here "~ " means "is asymptotic to " and pHxL is the number of primes less than or equal to x. So why do natural logs appear, as opposed to another flavor of logarithm? The problem with an attempt at a heuristic explanation is that the sieve of Eratosthenes does not behave as one might guess from pure probabilistic considerations. One might think that sieving out the composites under x using primes up to è!!! x would lead to x P è!!! J1p< x 1 ÅÅÅÅ N as an asymptotic estimate of the count of p numbers remaining (the primes up to x; p always represents a prime). But this quantity turns out to be not asymptotic to x ê ln x. For F. Mertens proved in 1874 that the product is actually asymptotic to 2 ‰g ê ln x, or about 1.12 ê ln x. Thus the sieve is 11 % (from 1 ê 1.12) more efficient at eliminating composites than one might expect. Commenting on this phenomenon, which one might call the Mertens Paradox, Hardy and