Showing papers on "Prime (order theory) published in 2003"
Posted Content•
TL;DR: The On-Line Encyclopedia of Integer Sequences (OEIS) as discussed by the authors provides a history of the OEIS and discusses recent sequences involving interesting unsolved problems and in many cases spectacular illustrations, such as Peaceable Queens, circles in the plane, the earliest cube-free binary sequence, the EKG and Yellowstone permutations, other lexicographically earliest sequences, iteration of number-theoretic functions, home primes and power trains, a memorable prime, a missing prime, Post's tag system, and coordination sequences.
Abstract: The recent history of The On-Line Encyclopedia of Integer Sequences (or OEIS), describing developments since 2009, and discussing recent sequences involving interesting unsolved problems and in many cases spectacular illustrations. These include: Peaceable Queens, circles in the plane, the earliest cube-free binary sequence, the EKG and Yellowstone permutations, other lexicographically earliest sequences, iteration of number-theoretic functions, home primes and power trains, a memorable prime, a missing prime, Post's tag system, and coordination sequences.
555 citations
Posted Content•
TL;DR: A Szemerédi-Trotter type theorem in finite fields is proved, and a new estimate for the Erdös distance problem in finite field, as well as the three-dimensional Kakeya problem in infinite fields is obtained.
Abstract: Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$. This is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.
343 citations
TL;DR: The prime minister's ability to access a series of personal and institutional power resources as mentioned in this paper enables the prime minister to lead, if not command, the core executive, and, in concert with others, to direct and control, its policy development.
Abstract: Prime ministerial predominance can enable the prime minister to lead, if not command, the core executive, and, in concert with others, to direct, if not control, its policy development. Leadership predominance facilitates prime ministerial predominance within the executive, and prime ministerial predominance reinforces leadership predominance within the party. Such predominance arises from the prime minister's ability to access a series of personal and institutional power resources. The more resources, the more powerful and predominant the prime minister is; the fewer resources, the less powerful and predominant they are. Such resources are necessarily transient, being accumulated and inevitably dispersed, acquired and lost, and are never permanent. When possessed, they can grant the prime minister considerable, if never overwhelming, intra-executive authority and influence, and the opportunity to be a stronger, but not the only element within the core executive.
140 citations
TL;DR: In this article, the authors present a translation of the Jordan theorem for finite groups of permutations in the context of number theory and topology, and present its translations in Number Theory and Topology.
Abstract: The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology. 1. Statements 1.1. Number theory. Let f = ∑n m=0 amx m be a polynomial of degree n, with coefficients in Z. If p is prime, let Np(f) be the number of zeros of f in Fp = Z/pZ. Theorem 1. Assume (i) n ≥ 2, (ii) f is irreducible in Q[x]. Then (a) There are infinitely many p’s with Np(f) = 0. (b) The set P0(f) of p’s with Np(f) = 0 has a density c0 = c0(f) which is > 0. [Recall that a subset P of the set of primes has density c if lim X→∞ number of p ∈ P with p ≤ X π(X) = c, where π(X) is as usual the number of primes ≤ X .] Moreover, Theorem 2. With the notation of Theorem 1, one has c0(f) ≥ 1 n , with strict inequality if n is not a power of a prime. Example. Let f = x + 1. One has p ∈ P0(f) if and only if p ≡ −1 (mod 4); this set is well-known to have density 1/2. We shall see more interesting examples in §5. 1.2. Topology. Let S1 be a circle. Let f : T → S be a finite covering of a topological space S. Assume: (i) f has degree n (i.e. every fiber of f has n elements), with n ≥ 2, (ii) T is arcwise connected and not empty. Theorem 3. There exists a continuous map φ : S1 → S which cannot be lifted to the covering T (i.e. there does not exist any continuous map ψ : S1 → T such that φ = f ◦ ψ). Received by the editors March 1, 2003. 2000 Mathematics Subject Classification. Primary 06-XX, 11-XX, 11F11. This text first appeared in Math Medley 29 (2002), 3–18. The writing was done with the help of Heng Huat Chan. c ©2002 Singapore Mathematical Society. Reprinted with permission.
137 citations
TL;DR: An algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory is introduced.
Abstract: We introduce an algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory. Tie algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms.
96 citations
TL;DR: Bourgain and Konyagin this article gave a lower bound on the exponential sums associated to subgroups of the multiplicative group F ∗ p for subsets A of the finite field F p, p prime.
94 citations
TL;DR: For each k⩾2, we exhibit infinite families of prime k-component links with Jones polynomial equal to that of the kcomponent unlink as mentioned in this paper, where Jones is the Jones constant.
91 citations
TL;DR: In this paper, the notion of multiplication modules over a commutative ring with identity was introduced, and the product of two submodules of such modules was characterized by the Nakayama lemma.
Abstract: By considering the notion of multiplication modules over a
commutative ring with identity, first we introduce the notion
product of two submodules of such modules. Then we use this
notion to characterize the prime submodules of a multiplication
module. Finally, we state and prove a version of Nakayama lemma
for multiplication modules and find some related basic results.
88 citations
TL;DR: In this paper, the authors initiated the study of generalized Jordan derivations and generalized Jordan triple derivations on prime rings and standard operator-algebras, and they initiated the work of generalized generalized Jordan triples.
Abstract: In this paper we initiate the study of generalized Jordan derivations
and generalized Jordan triple derivations on prime rings and standard operator
algebras.
Posted Content•
TL;DR: In this article, it was shown that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers.
Abstract: A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups.
In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers. Our approach in fact gives a largely self-contained proof of the entire classification theorem.
TL;DR: In this paper, a model of the Shimura variety at a prime p, with parahoric level structure at p, was investigated and it was shown that this model is flat, as conjectured by Rapoport and Zink.
07 Nov 2003
TL;DR: For rational elliptic curves in Weierstrass form, it was shown in this article that only nitely many primes will arise for a rational point under an isogeny.
Abstract: For a rational elliptic curve in Weierstrass form, Chud- novsky and Chudnovsky considered the likelihood that the denom- inators of the x-coordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a ratio- nal point under an isogeny, we use Siegel's Theorem to prove that only nitely many primes will arise. The same question is consid- ered for elliptic curves in homogeneous form prompting a visit to Ramanujan's famous taxi-cab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.
TL;DR: This paper presents a table of the orders of certain subgroups of the class groups of the real cyclotomic fields Q(ζl + ζl-1) for the primes l > 10,000 and argues that it is quite likely that these subgroups are in fact equal to theclass groups themselves, but there is at present no hope of proving this rigorously.
Abstract: The class numbers hl+ of the real cyclotomic fields Q(ζl + + ζl+-1) are notoriously hard to compute. Indeed, the number hl+ is not known for a single prime l ≥ 71. In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields Q(ζl + ζl-1) for the primes l > 10,000. It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue -on the basis of the Cohen-Lenstra heuristics-that the probability that our table is actually a table of class numbers hl+, is at least 98%.
TL;DR: In this paper, the density of the primes p < x for which E(F p ) is a cyclic group was shown to be an asymptotic formula for these primes.
Abstract: Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p < x for which E(F p ) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula.
TL;DR: Lexicality effect size was reduced in both nonword prime conditions, a result consistent with the lexical checking strategy described by S. A. Balota's pathway control hypothesis.
Abstract: The authors report 3 naming experiments using J. D. Zevin and D. A. Balota's (2000) multiple prime manipulation. They used 2 sets of nonword primes (fast and slow) and low-frequency exception word primes to separate the effects of prime speed from those of prime type. The size of the regularity effect was unaffected by prime type. Relative to the low-frequency exception word prime condition, the frequency effect was reduced in the fast, but not in the slow, nonword prime condition. Lexicality effect size was reduced in both nonword prime conditions, a result consistent with the lexical checking strategy described by S. J. Lupker, P. Brown, and L. Colombo (1997). The authors suggest that these results are better explained in terms of S. J. Lupker et al.'s time-criterion account than J. D. Zevin and D. A. Balota's pathway control hypothesis.
TL;DR: The authors used lexical decision in a dichotic listening situation and measured identity priming across channels to explore whether unattended stimuli can be processed lexically, and results are inconsistent with models of late filtering, which predict equal priming irrespective of prime saliency.
Abstract: The authors used lexical decision in a dichotic listening situation and measured identity priming across channels to explore whether unattended stimuli can be processed lexically. In 6 experiments, temporal synchronization of prime and target words was manipulated, and acoustic saliency of the unattended prime was varied by embedding it in a carrier sentence or in babble speech. When the prime was acoustically salient, a cross-channel priming effect emerged, and participants were aware of the prime. When the prime was less salient, no identity priming was found, and participants failed to notice the prime. Saliency was manipulated in ways that did not degrade the prime. Results are inconsistent with models of late filtering, which predict equal priming irrespective of prime saliency.
Posted Content•
TL;DR: In this article, the splitting dimension and the splitting ratio of a reduced F-finite ring R of characteristic p > 0 and q=p^e were studied in detail, and it was shown that R/P(R) is strongly F-regular.
Abstract: For a reduced F-finite ring R of characteristic p >0 and q=p^e one can write R^{1/q} = R^{a_q} \oplus M_q, where M_q has no free direct summands over R. We investigate the structure of F-finite, F-pure rings R by studying how the numbers a_q grow with respect to q. This growth is quantified by the splitting dimension and the splitting ratios of R which we study in detail. We also prove the existence of a special prime ideal P(R) of R, called the splitting prime, that has the property that R/P(R) is strongly F-regular. We show that this ideal captures significant information with regard to the F-purity of R.
TL;DR: In this article, Mestre and Serre presented an algorithm for computing the space of weight one forms mod 2 on X_0(N/2) for p arbitrary and N>4 prime to p, and a way to compute the Hecke algebra of mod p modular forms of weight two on Gamma_1(N) is presented.
Abstract: Two integral structures on the Q-vector space of modular forms of weight two on X_0(N) are compared at primes p exactly dividing N. When p=2 and N is divisible by a prime that is 3 mod 4, this comparison leads to an algorithm for computing the space of weight one forms mod 2 on X_0(N/2). For p arbitrary and N>4 prime to p, a way to compute the Hecke algebra of mod p modular forms of weight one on Gamma_1(N) is presented, using forms of weight p, and, for p=2, parabolic group cohomology with mod 2 coefficients.
Appendix A is a letter from Mestre to Serre, of October 1987, where he reports on computations of weight one forms mod 2 of prime level.
Appendix B reports on an implementation for p=2 in Magma, using Stein's modular symbols package, with which Mestre's computations are redone and slightly extended.
Posted Content•
TL;DR: In this paper, an exact sequence relating the relative equivariant cohomologies of the skeletons of an equivariantly formal T-space is presented, which goes back to Atiyah and Bredon, generalizing the so-called Chang-Skjelbred lemma.
Abstract: Let T be a torus. We present an exact sequence relating the relative equivariant cohomologies of the skeletons of an equivariantly formal T-space. This sequence, which goes back to Atiyah and Bredon, generalizes the so-called Chang-Skjelbred lemma. As coefficients, we allow prime fields and subrings of the rationals, including the integers. We extend to the same coefficients a generalization of this "Atiyah-Bredon sequence" for actions without fixed points which has recently been obtained by Goertsches and Toeben.
TL;DR: In this paper, a nonseparable C∗-algebra that is prime but not primitive is constructed, which solves an old problem of Dixmier, and is shown to be NP-hard.
TL;DR: In this paper, it was shown that right noetherian algebras of finite Gelfand-Kirillov dimension defined by homogeneous semigroup relations satisfy a polynomial identity.
15 Sep 2003
TL;DR: For an odd prime p and integers n, m, and k such that n = 2m+1)k, a new family of p-ary Helleseth-Gong sequences of period p/sup n/-1 with optimal correlation property is constructed in this paper.
Abstract: For an odd prime p and integers n, m, and k such that n=(2m+1)k, a new family of p-ary sequences of period p/sup n/-1 with optimal correlation property is constructed using the p-ary Helleseth-Gong sequences with ideal autocorrelation, where the size of the sequence family is p/sup n/. That is, the maximum nontrivial correlation value R/sub max/ of all pairs of distinct sequences in the family does not exceed p/sup n/2/+1, which means the family has optimal correlation in terms of Welch's lower bound. The symbol distribution of the sequences in the family is enumerated. It is also shown that the linear span of the sequences in the family is (m+2)n except for the m-sequence in the family.
Journal Article•
TL;DR: In this article, the ΔI = 1/2 rule in K→πππ amplitudes was shown to have an analytic matching between short and long-distance scale dependences within dimensional renormalization schemes.
Abstract: We present new results for the matrix elements of the Q 6 and Q 4 penguin operators, evaluated in a large- N c approach which incorporates important (N c 2 n f /N c ) unfactorized contributions. Our approach shows analytic matching between short- and long-distance scale dependences within dimensional renormalization schemes, such as . Numerically, we find that there is a large positive contribution to the ΔI = 1/2 matrix element of Q 6 and hence to the direct CP-violation parameter e'/e. We also present results for the ΔI = 1/2 rule in K→ππ amplitudes, which incorporate the related and important ``eye-diagram'' contributions of (N c 21/N c ) from the Q 2 operator (i.e. the penguin-like contraction). The results lead to an enhancement of the ΔI = 1/2 effective coupling. The origin of the large unfactorized contributions which we find is discussed in terms of the relevant scales of the problem.
TL;DR: In this paper, the authors studied the prime and maximal spectra of a BL-algebra and proved that the prime spectrum is a compact T − 0 topological space and the maximal spectrum is compact Hausdorff topology.
Abstract: In this paper we study the prime and maximal spectra of a BL-algebra, proving that the prime spectrum is a compact T
0 topological space and that the maximal spectrum is a compact Hausdorff topological space. We also define and study the reticulation of a BL-algebra.
Posted Content•
TL;DR: The result of as discussed by the authors extends the result of J. Bourgain, N. Katz, and T. Tao, where q is a prime, A is a subset of a finite field, and c>0.
Abstract: Let q be a prime, A be a subset of a finite field $F=\Bbb Z/q\Bbb Z$, $|A| 0$ and c>0. This extends the result of J. Bourgain, N. Katz, and T. Tao.
TL;DR: In this paper, it is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov.
Abstract: It is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov [Phys. Lett. A284, 1 -5 (2001)].