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About 10,000 papers for “Change-making problem”

The change-making problem

L. J. Bunce, J. D. Maitl · 1975

Abstract. Let A be a von Neumann algebra with no direct summand of Type I2, and let P(A) be its lattice of projections. Let X be a Banach space. Let m: P(A) → X be a bounded function such that m(p + q) = m(p) + m(q) whenever p and q are orthogonal projections. The main theorem states that m has a unique extension to a bounded linear operator from A to X. In particular, each bounded complex-valued finitely additive quantum measure on P(A) has a unique extension to a bounded linear functional on A. Physical background In von Neumann’s approach to the mathematical foundations of quantum mechanics, the bounded observables of a physical system are identified with a real linear space, L, of bounded selfadjoint operators on a Hilbert space H. It is reasonable to assume that L is closed in the weak operator topology and that whenever x ∈ L then x 2 ∈ L. (Thus L is a Jordan algebra and contains spectral projections.) Then the projections in L form a complete orthomodular lattice, P, otherwise known as the lattice of “questions ” or the quantum logic of the physical system. A quantum measure is a map µ: P → R such that whenever p and q are orthogonal projections µ(p + q) = µ(p) + µ(q). In Mackey’s formulation of quantum mechanics [11] his Axiom VII makes the assumption that L = L(H)sa. Mackey states, that in contrast to his other axioms, Axiom VII has no physical justification; it is adopted for mathematical convenience. One of the technical advantages of this axiom was that, by Gleason’s Theorem, a completely additive positive quantum measure on the projections of L(H) is the restriction of a bounded linear functional (provided H is not two-dimensional). In order to weaken Axiom VII it was desirable to strengthen Gleason’s Theorem.

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Combinatorics of the change-making problem

Adamaszek, Anna, Adamaszek, Michał · Warwick Research Archive Portal (University of Warwick) · 2010

We investigate the structure of the currencies (systems of coins) for which the greedy change-making algorithm always finds an optimal solution (that is, a one with minimum number of coins). We present a series of necessary conditions that must be satisfied by the Values of coins in such systems. We also uncover some relations between such currencies and their sub-currencies.

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Making Change Happen: A Problem-based Computer Simulation Making Change Happen: A Problem-based Computer Simulation

Prof Dr, Philip Hallinger · 2008

Globalization and related trends have created an era of rapid change that reaches into the lives of all organizations. The capacity to change has become a necessary core competency for all organizations. Organizational leaders need to be able to produce more successful change by making better use of the human resources at the organization’s disposal. Leaders must learn how to make organizations work smarter. This involves becoming more adept at leading change. This paper presents the design and content of a problem-based learning module designed around a computer-based simulation intended to help organizational leaders learn to lead change. In the simulation, learners are confronted with the problem of implementing new information technology (e.g., a new ERM system) in a company. Students learn through an interactive process of team learning through playing the computer simulation, periodic instructor debriefings, readings, and written reflection exercises. They use the problem-based simulation as a means of constructing and being able to apply relevant theories of change management.

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Optimal Bounds for the Change-Making Problem

Dexter Kozen, Shmuel Zaks · DAIMI Report Series · 1991

<p>The change-making problem is the problem of representing a given value with the fewest coins possible. We investigate the problem of determining whether the greedy algorithm produces an optimal representation of all amounts for a given set of coin denominations 1 = c_1 < c_2 < ... < c_m. Chang and Gill show that if the greedy algorithm is not always optimal, then there exists a counterexample x in the range</p><p>c_3 <= x < (c_m(c_m c_m-1 + c_m - 3c_m-1)) \ (c_m - c_m-1).</p><p>To test for the existence of such a counterexample, Chang and Gill propose computing and comparing the greedy and optimal representations of all x in this range. In this paper we show that if a counterexample exists, then the smallest one lies in the range c_3 + 1 < x < c_m + c_m-1, and these bounds are tight. Moreover, we give a simple test for the existence of a counterexample that does not require the calculation of optimal representations.</p><p>In addition, we give a complete characterization of three-coin systems and an efficient algorithm for all systems with a fixed number of coins. Finally, we show that a related problem is <em>co</em>NP-complete.</p>

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On a variant of the change-making problem

Adam N. Letchford, Licong Cheng · Operations Research Letters · 2024

The change-making problem (CMP), introduced in 1970, is a classic problem in combinatorial optimisation. It was proven to be NP -hard in 1975, but it can be solved in pseudo-polynomial time by dynamic programming. In 1999, Heipcke presented a variant of the CMP which, at first glance, looks harder than the standard version. We show that, in fact, her variant can be solved in polynomial time.

Cited by 0Open access