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13
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 such a ..."
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Cited by 10 (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
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 6 (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
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 3 (2 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”
P.,Measures on finite concrete logics
 Proc. Amer.Math. Soc.,127
, 1999
"... Abstract. We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets. 1. ..."
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Abstract. We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets. 1.
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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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.
2003: What This Country Needs is an 18¢ Piece, in
 Mathematical Intelligencer
"... 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. 1 ..."
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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. 1
The Greedy Algorithms Class: Formalization, Synthesis and Generalization
, 1995
"... On the first hand, this report studies the class of Greedy Algorithms in order to find an as systematic as possible strategy that could be applied to the specification of some problems to lead to a correct program solving that problem. On the other hand, the standard formalisms underlying the G ..."
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On the first hand, this report studies the class of Greedy Algorithms in order to find an as systematic as possible strategy that could be applied to the specification of some problems to lead to a correct program solving that problem. On the other hand, the standard formalisms underlying the Greedy Algorithms (matroid, greedoid and matroid embedding) which are dependent on the particular type set are generalized to a formalism independent of any data type based on an algebraic specification setting.
LOCAL AUTOMORPHISMS OF SOME QUANTUM MECHANICAL STRUCTURES
, 2008
"... Abstract. Let H be a separable infinite dimensional complex Hilbert space. We prove that every continuous 2local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ..."
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Abstract. Let H be a separable infinite dimensional complex Hilbert space. We prove that every continuous 2local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented.
Running title: WIGNER’S UNITARYANTIUNITARY THEOREM FOR MODULES
, 1999
"... Let H be a Hilbert C∗module over a matrix algebra A. It is proved that any function T: H → H which preserves the absolute value of the (generalized) inner product is of the form Tf = ϕ(f)Uf (f ∈ H), where ϕ is a phasefunction and U is an Alinear isometry. The result gives a natural extension of W ..."
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Let H be a Hilbert C∗module over a matrix algebra A. It is proved that any function T: H → H which preserves the absolute value of the (generalized) inner product is of the form Tf = ϕ(f)Uf (f ∈ H), where ϕ is a phasefunction and U is an Alinear isometry. The result gives a natural extension of Wigner’s classical unitaryantiunitary theorem for Hilbert modules.
SEMIGROUP ENDOMORPHISMS OF B(H)
, 2000
"... Dedicated to the memory of Professor Béla SzőkefalviNagy Abstract. Let H be a complex separable infinite dimensional Hilbert space. We describe the form of all *semigroup endomorphisms φ of B(H) which are uniformly continuous on every commutative C ∗subalgebra. In particular, we obtain that if φ ..."
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Dedicated to the memory of Professor Béla SzőkefalviNagy Abstract. Let H be a complex separable infinite dimensional Hilbert space. We describe the form of all *semigroup endomorphisms φ of B(H) which are uniformly continuous on every commutative C ∗subalgebra. In particular, we obtain that if φ satisfies φ(0) = 0, then φ is additive. 1.