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464
PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 1278 (4 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
Some properties of partitions
 Proc. London Math. Soc
, 1954
"... 1. WE denote by p{n) the number of unrestricted partitions of a positive integer n. Ramanujan discovered, and later proved, three striking arithmetical properties of p{n), namely: #(571+4) = 0 (mod 5), (1.1) p{ln+5) = = 0(mod7), (1.2) p{Un+6) = 0 (mod 11). (1.3) All existing proofs of these result ..."
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Cited by 115 (1 self)
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1. WE denote by p{n) the number of unrestricted partitions of a positive integer n. Ramanujan discovered, and later proved, three striking arithmetical properties of p{n), namely: #(571+4) = 0 (mod 5), (1.1) p{ln+5) = = 0(mod7), (1.2) p{Un+6) = 0 (mod 11). (1.3) All existing proofs of these results appeal to the theory of generating functions, and provide no method of actually separating the partitions concerned into q equal classes {q = 5, 7, or 11). Dyson (1) discovered empirically a remarkable combinatorial method of dividing the partitions of 5w+4 and ln\5 into 5 and 7 equal classes respectively. Defining the rank of a partition as the largest part minus the number of parts, he divided the partitions of any number into 5 classes according to their ranks modulo 5. For numbers of the form 5w+4, these 5 classes are all equal, while for numbers of other forms some but not all of the classes are equal; similar
Quantum Dynamics and Decompositions of Singular Continuous Spectra
 J. Funct. Anal
, 1995
"... . We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (wi ..."
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Cited by 105 (10 self)
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. We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (with k/k = 1). The spectral measure ¯/ of / (and H ) is uniquely defined by [24]: h/ ; f(H)/i = Z oe(H) f(x) d¯/ (x) ; (1:1) for any measurable (Borel) function f . The time evolution of the state / , in the Schrodinger picture of quantum mechanics, is given by /(t) = e \GammaiHt / : (1:2) The relations between various properties of the spectral measure ¯/ (with an emphasis on "fractal" properties) and the nature of the time evolution have been the subject of several recent papers [7,13,1518,20,22,33,36,39]. Our purpose in this paper is twofold: First, we use a theory, due to Rogers and Taylor [28,29], of decomposing singular continuous measures with respect to Hausdorff measures to i...
Modified Prüfer and EFGP Transforms and the Spectral Analysis of OneDimensional Schrödinger Operators
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1997
"... Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete halfline Schrödinger operators with slowly decaying potentials. Among our results we show if V (x) = ∑∞ n=1 anW (x − xn), where W has compact support and xn/xn+1 → 0, then H has purely ..."
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Cited by 86 (24 self)
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Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete halfline Schrödinger operators with slowly decaying potentials. Among our results we show if V (x) = ∑∞ n=1 anW (x − xn), where W has compact support and xn/xn+1 → 0, then H has purely a.c. (resp. purely s.c.) spectrum on (0, ∞) if ∑ a2 n <∞(resp. ∑ a2 n = ∞). For λn−1/2an potentials, where an are independent, identically distributed random variables with E(an) =0,E(a2 n)=1,and λ < 2, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.
Some integer factorization algorithms using elliptic curves
 Australian Computer Science Communications
, 1986
"... Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order ..."
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Cited by 55 (13 self)
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Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard’s “p − 1” factorization algorithm. 1
Apollonian circle packings: Number theory
, 2003
"... Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper st ..."
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Cited by 49 (3 self)
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Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: x2 þ y2 þ z2 þ w2 1 2ðx þ y þ z þ wÞ2: Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. We present evidence suggesting that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by Corresponding author.
Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary
 Sém. Lotharingien de Combinatoire 42 (1999), 63 pp.; in The Andrews Festschrift
, 2001
"... When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s taufunction τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising mat ..."
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Cited by 45 (18 self)
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When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s taufunction τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 44 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Strict selfassembly of discrete Sierpinski triangles
 Proceedings of The Third Conference on Computability in Europe
"... Winfree (1998) showed that discrete Sierpinski triangles can selfassemble in the Tile Assembly Model. A striking molecular realization of this selfassembly, using DNA tiles a few nanometers long and verifying the results by atomicforce microscopy, was achieved by Rothemund, Papadakis, and Winfree ..."
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Cited by 42 (15 self)
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Winfree (1998) showed that discrete Sierpinski triangles can selfassemble in the Tile Assembly Model. A striking molecular realization of this selfassembly, using DNA tiles a few nanometers long and verifying the results by atomicforce microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above selfassemblies tile completely filledin, twodimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict selfassembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else. We first prove that the standard discrete Sierpinski triangle cannot strictly selfassemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly selfassembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, nonstrict selfassemblies, our strict selfassembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and cornerturning operations. We verify our strict selfassembly using the local determinism method of Soloveichik and Winfree (2007). 1