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33
Vinogradov’s mean value theorem via efficient congruencing
"... Abstract. We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural ..."
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Cited by 38 (11 self)
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Abstract. We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural numbers whenever s � 2k 2 + 2k − 3. 1.
A Fast and Simple Algorithm for the Money Changing Problem
 ALGORITHMICA
, 2007
"... The Money Changing Problem (MCP) can be stated as follows: Given k positive integers a1 < ···< ak and a query integer M, is there a linear combination ∑ i ci ai = M with nonnegative integers ci,a decomposition of M? If so, produce one or all such decompositions. The largest integer without ..."
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Cited by 15 (5 self)
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The Money Changing Problem (MCP) can be stated as follows: Given k positive integers a1 < ···< ak and a query integer M, is there a linear combination ∑ i ci ai = M with nonnegative integers ci,a decomposition of M? If so, produce one or all such decompositions. The largest integer without such a decomposition is called the Frobenius number g(a1,...,ak). A data structure called the residue table of a1 words can be used to compute the Frobenius number in time O(a1). We present an intriguingly simple algorithm for computing the residue table which runs in time O(ka1), with no additional memory requirements, outperforming the best previously known algorithm. Simulations show that it performs well even on “hard ” instances from the literature. In addition, we can employ the residue table to answer MCP decision instances in constant time, and a slight modification of size O(a1) to compute one decomposition for a query M. Note that since both computing the Frobenius number and MCP (decision) are NPhard, one cannot expect to find an algorithm that is polynomial in the size of the input, i.e., in k, log ak, and log M. We then give an algorithm which, using a modification of the residue table, also constructible in O(ka1) time, computes all decompositions of a query integer M. Its worstcase running time is O(ka1) for each
Approximating the main conjecture in Vinogradov’s mean value theorem, submitted
"... Abstract. We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree k for moments of order 2s, establishing strongly diagonal behaviour for 1 6 s 6 12k(k + 1) − 13k + o(k). In particular, as k → ∞, we confirm the main conjecture in Vinogradov’s mean value theorem for 1 ..."
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Abstract. We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree k for moments of order 2s, establishing strongly diagonal behaviour for 1 6 s 6 12k(k + 1) − 13k + o(k). In particular, as k → ∞, we confirm the main conjecture in Vinogradov’s mean value theorem for 100 % of the critical interval 1 6 s 6 12k(k + 1). 1.
Influencefree states on compound quantum systems
, 2005
"... Consider two spatially separated agents, Alice and Bob, each of whom is able to make local quantum measurements, and who can communicate with each other over a purely classical channel. As has been pointed out by a number of authors, the set of mathematically consistent joint probability assignments ..."
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Cited by 10 (7 self)
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Consider two spatially separated agents, Alice and Bob, each of whom is able to make local quantum measurements, and who can communicate with each other over a purely classical channel. As has been pointed out by a number of authors, the set of mathematically consistent joint probability assignments (“states”) for such a system is properly larger than the set of quantummechanical mixed states for the joint AliceBob system. Indeed, it is canonically isomorphic to the set of positive (but not necessarily completely positive) linear maps L(HA) → L(HB) from the bounded linear operators on Alice’s Hilbert space to those on Bob’s, satisfying Tr (φ(1)) = 1. The present paper explores the properties of these states. We review what is known, including the fact that allowing classical communication between parties is equivalent to enforcing “noinstantaneoussignalling” (“no–influence”) in the direction opposite the communication. We establish that in the subclass of “decomposable”
A Relational Approach To Optimization Problems
, 1996
"... The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming s ..."
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Cited by 7 (0 self)
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The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other nonrelational calculi. The graph
ACTIVE LATTICES DETERMINE AW*ALGEBRAS
"... Abstract. We prove that AW*algebras are determined by their projections, their symmetries, and the action of the latter on the former. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*algebras is equivalent to a full subc ..."
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Abstract. We prove that AW*algebras are determined by their projections, their symmetries, and the action of the latter on the former. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*algebra that enables recovering its total structure. 1.
Measures on finite concrete logics
 PROC. AMER.MATH. SOC.,127
, 1999
"... We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets. ..."
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Cited by 3 (0 self)
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We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets.
Translation invariance, exponential sums, and Waring’s problem
"... Abstract. We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the translation invariance of associated systems of Diophan ..."
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Abstract. We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the translation invariance of associated systems of Diophantine equations to derive powerful congruence constraints on the underlying variables. There are applications to Weyl sums, the distribution of polynomials modulo 1, and other Diophantine problems such as Waring’s problem.
A KowalskiS lodkowski theorem for 2local ∗homomorphisms on von Neumann algebras, preprint 2014
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What This Country Needs is an 18¢ Piece
, 2002
"... We consider sets of coin denominations which permit change to be made using as few coins as possible, on average, and explain why the United States should adopt an 18¢ piece. ..."
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Cited by 2 (0 self)
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We consider sets of coin denominations which permit change to be made using as few coins as possible, on average, and explain why the United States should adopt an 18¢ piece.