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208
Efficient power control via pricing in wireless data networks
 IEEE Trans. on Commun
, 2002
"... Abstract—A major challenge in the operation of wireless communications systems is the efficient use of radio resources. One important component of radio resource management is power control, which has been studied extensively in the context of voice communications. With the increasing demand for wir ..."
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Cited by 331 (8 self)
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Abstract—A major challenge in the operation of wireless communications systems is the efficient use of radio resources. One important component of radio resource management is power control, which has been studied extensively in the context of voice communications. With the increasing demand for wireless data services, it is necessary to establish power control algorithms for information sources other than voice. We present a power control solution for wireless data in the analytical setting of a game theoretic framework. In this context, the quality of service (QoS) a wireless terminal receives is referred to as the utility and distributed power control is a noncooperative power control game where users maximize their utility. The outcome of the game results in a Nash equilibrium that is inefficient. We introduce pricing of transmit powers in order to obtain Pareto improvement of the noncooperative power control game, i.e., to obtain improvements in user utilities relative to the case with no pricing. Specifically, we consider a pricing function that is a linear function of the transmit power. The simplicity of the pricing function allows a distributed implementation where the price can be broadcast by the base station to all the terminals. We see that pricing is especially helpful in a heavily loaded system. Index Terms—Game theory, Pareto efficiency, power control, pricing, wireless data. I.
On Projection Algorithms for Solving Convex Feasibility Problems
, 1996
"... Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of the ..."
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Cited by 330 (44 self)
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Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated . Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given. 1991 M.R. Subject Classification. Primary 47H09, 49M45, 6502, 65J05, 90C25; Secondary 26B25, 41A65, 46C99, 46N10, 47N10, 52A05, 52A41, 65F10, 65K05, 90C90, 92C55. Key words and phrases. Angle between two subspaces, averaged mapping, Cimmino's method, computerized tomography, convex feasibility problem, convex function, convex inequalities, convex programming, convex set, Fej'er monotone sequence, firmly nonexpansive mapping, H...
Hermeneutics: Interpretation Theory
 in Schleiermacher, Dilthey, Heidegger and Gadamer, Northwestern University Studies in Phenomenology & Existential Philosophy
, 1969
"... Report on proposed doctoral thesis: ..."
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Entanglement of a pair of quantum bits
 Physical Review Letters
, 1997
"... The “entanglement of formation ” of a mixed state of a bipartite quantum system can be defined in terms of the number of pure singlets needed to create the state with no further transfer of quantum information. We find an exact formula for the entanglement of formation for all mixed states of two qu ..."
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Cited by 67 (0 self)
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The “entanglement of formation ” of a mixed state of a bipartite quantum system can be defined in terms of the number of pure singlets needed to create the state with no further transfer of quantum information. We find an exact formula for the entanglement of formation for all mixed states of two qubits having no more than two nonzero eigenvalues, and we report evidence suggesting that the formula is valid for all states of this system. PACS numbers: 03.65.Bz, 89.70.+c 1 Entanglement is the potential of quantum states to exhibit correlations that cannot be accounted for classically. For decades, entanglement has been the focus of much work in the foundations of quantum mechanics, being associated particularly with quantum nonseparability and the violation of Bell’s inequalities [1]. In recent years, however, it has begun to be viewed also as a potentially useful resource. The predicted capabilities of a quantum computer, for example, rely crucially on entanglement [2], and a proposed quantum cryptographic scheme achieves security by converting shared entanglement into a shared secret key [3]. For both theoretical and potentially practical reasons, it has become interesting to quantify entanglement, just as we quantify other resources such as energy and information. In this letter we adopt a recently proposed quantitative definition of entanglement and derive an explicit formula for the entanglement of a large class of states of a pair of binary quantum systems (qubits). The simplest kind of entangled system is a pair of qubits in a pure but nonfactorizable state. A pair of spin 1 particles in the singlet state 2 1 √ (  ↑↓ 〉 −  ↓↑〉) is perhaps the most familiar example, but one can also
A convergent incremental gradient method with constant step size
 SIAM J. OPTIM
, 2004
"... An incremental gradient method for minimizing a sum of continuously differentiable functions is presented. The method requires a single gradient evaluation per iteration and uses a constant step size. For the case that the gradient is bounded and Lipschitz continuous, we show that the method visits ..."
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Cited by 63 (3 self)
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An incremental gradient method for minimizing a sum of continuously differentiable functions is presented. The method requires a single gradient evaluation per iteration and uses a constant step size. For the case that the gradient is bounded and Lipschitz continuous, we show that the method visits regions in which the gradient is small infinitely often. Under certain unimodality assumptions, global convergence is established. In the quadratic case, a global linear rate of convergence is shown. The method is applied to distributed optimization problems arising in wireless sensor networks, and numerical experiments compare the new method with the standard incremental gradient method.
CONVEXITY ACCORDING TO THE GEOMETRIC MEAN
"... (communicated by Zs. Páles) Abstract. We develop a parallel theory to the classical theory of convex functions, based on a change of variable formula, by replacing the arithmetic mean by the geometric one. It is shown that many interesting functions such as exp � sinh � cosh � sec � csc � arc sin � ..."
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Cited by 30 (5 self)
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(communicated by Zs. Páles) Abstract. We develop a parallel theory to the classical theory of convex functions, based on a change of variable formula, by replacing the arithmetic mean by the geometric one. It is shown that many interesting functions such as exp � sinh � cosh � sec � csc � arc sin � Γ etc illustrate the multiplicative version of convexity when restricted to appropriate subintervals of (0 � 1).Asa consequence, we are not only able to improve on a number of classical elementary inequalities but also to discover new ones. 1.
Hopfield models as generalized random mean field models. Mathematical aspects of spin glasses and neural networks
 3–89, Progr. Probab., 41 Birkhäuser
, 1998
"... Abstract: We give a comprehensive selfcontained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large devi ..."
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Cited by 28 (8 self)
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Abstract: We give a comprehensive selfcontained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete picture in the regime of “small α”; a particularly satisfactory result concerns a nontrivial regime of parameters in which we prove 1) the convergence of the local “mean fields ” to gaussian random variables with constant variance and random mean; the random means are from site to site independent gaussians themselves; 2) “propagation of chaos”, i.e. factorization of the extremal infinite volume Gibbs measures, and 3) the correctness of the “replica symmetric solution ” of Amit, Gutfreund and Sompolinsky [AGS]. This last result was first proven by M. Talagrand [T4], using different techniques.
A New Framework For Power Control In Wireless Data Networks: Games, Utility, And Pricing
, 1999
"... We develop a new framework for distributed power control for wireless data based on the economic principles of utility and pricing. Utility is defined as the measure of satisfaction that a user derives from accessing the wireless data network. Properties of utility functions are introduced and a spe ..."
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Cited by 27 (2 self)
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We develop a new framework for distributed power control for wireless data based on the economic principles of utility and pricing. Utility is defined as the measure of satisfaction that a user derives from accessing the wireless data network. Properties of utility functions are introduced and a specific function, based on throughput per terminal battery lifetime including forward error control, is presented and shown to conform to those properties. Users enter into a noncooperative game to maximize their individual utilities by adjusting their transmitter powers. A unique Nash equilibrium for the above game is shown to exist but is not Pareto efficient. A pricing function is then introduced which leads to Pareto improvements for the noncooperative game.
Generalized convexity and inequalities
 The University of Auckland, Report Series
, 2006
"... Abstract. Let R+ = (0, ∞) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 ∈ M, we say that a function f: R+ → R+ is (m1, m2)convex if f(m1(x, y)) ≤ m2(f(x), f(y)) for all x, y ∈ R+. The usual convexity i ..."
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Cited by 26 (11 self)
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Abstract. Let R+ = (0, ∞) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 ∈ M, we say that a function f: R+ → R+ is (m1, m2)convex if f(m1(x, y)) ≤ m2(f(x), f(y)) for all x, y ∈ R+. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1, m2)convexity on m1 and m2 and give sufficient conditions for (m1, m2)convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function. 1.