Real Integer Research Articles

Probability Generating Functions for Discrete Real-Valued Random Variables

The probability generating function is a powerful technique for studying the law of finite sums of independent discrete random variables taking integer positive values. For real-valued discrete random variables, the well-known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5, allow us to extend this technique to general discrete random variables. Being so, the purpose of this work is twofold. First, we show that discrete random variables taking real values, nonnecessarily integer or rational, may be studied with probability generating functions. Second, we intend to draw attention to some practical ways of performing the necessary calculations.

Random Unitary Beamforming With Partial Feedback for MISO Downlink Transmission Using Multiuser Diversity

In this paper, we study the problem of downlink transmission in multiple-input-single-output (MISO) wireless communication systems using multiuser diversity. There are antennas at the base station (BS) and single-antenna receivers. The performance of multiuser diversity depends on the number of users that have independent channel realizations. It is well known that increasing the number of users improves the performance at the expense of increasing feedback that is proportional to the number of users. For sufficiently large K, the capacity scales like M log log K. When K is large, increasing feedback limits practical applications of the multiuser diversity. Our approach is to reduce the feedback by selecting a threshold for the linfin-norm of the normalized cross-correlation (||zk||infin) between the users' channel and the beamforming matrix based on the random unitary beamforming. The average amount of feedback per time slot is real numbers and integers which does not change with K. To improve fairness, an equal ratio scheduling algorithm which could serve the users with different rate requirements is developed. Monte Carlo simulation results is provided to verify the performance of the proposed algorithm.

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

The dual simplex algorithm has become a strong contender in solving large scale LP problems. One key problem of any dual simplex algorithm is to obtain a dual feasible basis as a starting point. We give an overview of methods which have been proposed in the literature and present new stable and efficient ways to combine them within a state-of-the-art optimization system for solving real world linear and mixed integer programs. Furthermore, we address implementation aspects and the connection between dual feasibility and LP-preprocessing. Computational results are given for a large set of large scale LP problems, which show our dual simplex implementation to be superior to the best existing research and open-source codes and competitive to the leading commercial code on many of our most difficult problem instances.

Improved genetic algorithm for multi‐objective reactive power dispatch problem

AbstractThis paper presents an improved genetic algorithm (GA) approach for solving the multi‐objective reactive power dispatch problem. Loss minimization and maximization of voltage stability margin are taken as the objectives. Maximum L‐index of the system is used to specify the voltage stability level. Generator terminal voltages, reactive power generation of capacitor banks and tap changing transformer setting are taken as the optimization variables. In the proposed GA, voltage magnitudes are represented as floating point numbers and transformer tap‐setting and reactive power generation of capacitor bank are represented as integers. This alleviates the problems associated with conventional binary‐coded GAs to deal with real variables and integer variables with total number of permissible choices not equal to 25. Crossover and mutation operators which can deal with mixed variables are proposed. The proposed method has been tested on IEEE 30‐bus system and is compared with conventional methods and binary‐coded GA. The proposed method has produced the loss which is less than the value reported earlier and is well suitable for solving the mixed integer optimization problem. Copyright © 2007 John Wiley & Sons, Ltd.

On the existence of non‐supersymmetric black hole attractors for two‐parameter Calabi‐Yau's and attractor equations

AbstractWe look for possible nonsupersymmetric black hole attractor solutions for type II compactification on (the mirror of) CY3(2,128) expressed as a degree‐12 hypersurface in WCP4[1,1,2,2,6]. In the process, (a) for points away from the conifold locus, we show that the existence of a non‐supersymmetric attractor along with a consistent choice of fluxes and extremum values of the complex structure moduli, could be connected to the existence of an elliptic curve fibered over C8 which may also be “arithmetic” (in some cases, it is possible to interpret the extremization conditions for the black‐hole superpotential as an endomorphism involving complex multiplication of an arithmetic elliptic curve), and (b) for points near the conifold locus, we show that existence of non‐supersymmetric black‐hole attractors corresponds to a version of A1‐singularity in the space Image(Z6→R2/Z2 (↪R3)) fibered over the complex structure moduli space. The (derivatives of the) effective black hole potential can be thought of as a real (integer) projection in a suitable coordinate patch of the Veronese map: CP5→CP20, fibered over the complex structure moduli space. We also discuss application of Kallosh's attractor equations (which are equivalent to the extremization of the effective black‐hole potential) for nonsupersymmetric attractors and show that (a) for points away from the conifold locus, the attractor equations demand that the attractor solutions be independent of one of the two complex structure moduli, and (b) for points near the conifold locus, the attractor equations imply switching off of one of the six components of the fluxes. Both these features are more obvious using the attractor equations than the extremization of the black hole potential.

New summation relations for the Stieltjes constants

The Stieltjes constants have been of interest for over a century, yet their detailed behaviour remains under investigation. These constants appear in the Laurent expansion of the Hurwitz zeta function about . We obtain novel single and double summatory relations for , including single summation relations for and , where a and b are real and p and q are positive integers. In addition, we obtain new integration formulae for the Hurwitz zeta function and a new expression for the Stieltjes constants . Portions of the presentation show an intertwining of the theory of the hypergeometric function with that of the Hurwitz zeta function.

OPTIMAL LOT SOLUTION TO CARDINALITY CONSTRAINED MEAN–VARIANCE FORMULATION FOR PORTFOLIO SELECTION

The pioneering work of the mean–variance formulation proposed by Markowitz in the 1950s has provided a scientific foundation for modern portfolio selection. Although the trade practice often confines portfolio selection with certain discrete features, the existing theory and solution methodologies of portfolio selection have been primarily developed for the continuous solution of the portfolio policy that could be far away from the real integer optimum. We consider in this paper an exact solution algorithm in obtaining an optimal lot solution to cardinality constrained mean–variance formulation for portfolio selection under concave transaction costs. Specifically, a convergent Lagrangian and contour‐domain cut method is proposed for solving this class of discrete‐feature constrained portfolio selection problems by exploiting some prominent features of the mean–variance formulation and the portfolio model under consideration. Computational results are reported using data from the Hong Kong stock market.

Some remarks on Nakagami-m parameter estimation using method of moments

This letter develops new estimators for the Nakagami-m parameter based on real (integer and fractional) sample moments. These moment-based estimators are asymptotically unbiased, efficient, and have lower computational budget compared to the best known integer and real moment estimators [Cheng, J et al., (2002), Eqs. (4) and (9)] but comparable to [Gaeddert, J et al., (2005), Eq. (5)]. Specifically, the asymptotic expansion of sample moments derived in this article provides a generalized closed-form expression for the Nakagami-m parameter without the need for coefficient optimization for different ratios of real moments as needed in [Gaeddert, J et al., (2005)].

Generalized moment estimators for the Nakagami fading parameter

Moment-based estimation of the Nakagami-m fading parameter is studied. A family of new, generalized, estimators, based on real (integer and possibly noninteger) sample moments, is derived. A new integer-moment estimator that outperforms known integer-moment estimators is presented. The new moment-based estimators are asymptotically unbiased and efficient.

Empirical comparison of search algorithms for discrete event simulation

Discrete-event simulation is a significant analysis tool for designing complex systems. In the research literature, several deterministic search algorithms have been linked with simulation for industrial applications; but there are few empirical comparisons of the various algorithms. This paper compares the Hooke–Jeeves pattern search, Nelder–Mead simplex, simulated annealing, and genetic algorithm optimization algorithms on variations of four industrial case study simulation problems. The simulation models include combinations of real variables, integer variables, non-numeric variables, deterministic constraints, and stochastic constraints. The genetic algorithm was the most robust, as it found near best solutions for all 25 test problems. However, it required the most replications of all the algorithms. The pattern search algorithm also found near best solutions to small- and medium-sized problems with no non-numeric variables, while requiring fewer replications than the genetic algorithm.

Self-affine scaling from non-integer phase-space partition in π+p and K+p collisions at 250 GeV/c

A factorial-moment analysis with real (integer and non-integer) phase space partition is applied to π+p and K+p collisions at 250 GeV/c. Clear evidence is shown for self-affine rather than self-similar power-law scaling in multiparticle production. The three-dimensional self-affine second-order scaling exponent is determined to be 0.061±0.010.

Applying interval arithmetic to real, integer, and boolean constraints

We present in this paper a unified processing for real, integer, and Boolean constraints based on a general narrowing algorithm which applies to any n-ary relation on R. The basic idea is to define, for every such relation ρ, a narrowing function ρ based on the approximation of ρ by a Cartesian product of intervals whose bounds are floating-point numbers. We then focus on nonconvex relations and establish several properties. The more important of these properties is applied to justify the computation of usual relations defined in terms of intersections of simpler relations. We extend the scope of the narrowing algorithm used in the language BNR-Prolog to integer and disequality constraints, to Boolean constraints, and to relations mixing numerical and Boolean values. As a result, we propose a new Constraint Logic Programming language called CLP(BNR), where BNR stands for Booleans, Naturals, and Reals. In this language, constraints are expressed in a unique structure, allowing the mixing of real numbers, integers, and Booleans. We end with the presentation of several examples showing the advantages of such an approach from the point of view of the expressiveness, and give some preliminary computational results from a prototype.

A CONSTANT TIME SORTING ALGORITHM FOR A THREE DIMENSIONAL RECONFIGURABLE MESH AND RECONFIGURABLE NETWORK

Sorting techniques have numerous applications in computer science. Current real number and integer sorting techniques for the reconfigurable mesh operate in constant time using a reconfigurable mesh of size n × n to sort n numbers. This paper presents a constant time algorithm to sort n items on a reconfigurable network with [Formula: see text] switches and [Formula: see text] processors. Also, new constant time selection and compression algorithms are given. All results may also be implemented on the 3-D reconfigurable mesh.

Experiments in Solving Mixed Integer Programming Problems on a Small Array of Transputers

The time-consuming process of solving large-scale Mixed Integer Programming problems using the branch-and-bound technique can be speeded up by introducing a degree of parallelism into the basic algorithm. This paper describes the development and implementation of a parallel branch-and-bound algorithm created by adapting a commercial MIP solver. Inherent in the design of this software are certain ad hoc methods, the use of which are necessary in the effective solution of real problems. The extent to which these ad hoc methods can successfully be transferred to a parallel environment, in this case an array of at most nine transputers, is discussed. Computational results on a variety of real integer programming problems are reported.

Experiments in Solving Mixed Integer Programming Problems on a Small Array of Transputers

The time-consuming process of solving large-scale Mixed Integer Programming problems using the branch-and-bound technique can be speeded up by introducing a degree of parallelism into the basic algorithm. This paper describes the development and implementation of a parallel branch-and-bound algorithm created by adapting a commercial MIP solver. Inherent in the design of this software are certain ad hoc methods, the use of which are necessary in the effective solution of real problems. The extent to which these ad hoc methods can successfully be transferred to a parallel environment, in this case an array of at most nine transputers, is discussed. Computational results on a variety of real integer programming problems are reported.

Chaotic rounding error in digital control systems

Chaotic behavior due to the round-off effect in digital control systems is called the chaotic rounding error. First, we model digital control systems with finite-wordlength digital compensators by mixed mappings. A mixed mapping system is described jointly by a point mapping and a cell mapping, that is, its state consists of real numbers and integers. After presenting some definitions and fundamental results of mixed mapping systems, we prove certain sufficient conditions for the existence of the chaotic rounding error. Finally, as an illustration, we discuss the chaotic rounding error of a system with a second-order plant and first-order digital compensator.

Equational theory of positive numbers with exponentiation

A. Tarski asked if all true identities involving 1, addition, multiplication, and exponentiation can be derived from certain so-called high-school identities (and a number of related questions). I prove that equational theory of (N, 1, +, , T) is decidable (a T b means ac' for positive a, b) and that entailment relation in this theory is decidable (and present a similar result for inequalities). A. J. Wilkie found an identity not derivable from axioms with a difficult proof-theoretic argument of nonderivability. I present a model of axioms consisting of 59 elements in which Wilkie's identity fails. 1. This note is related to Tarski's school problem and a number of other model-theoretic questions concerning exponentiation of positive real numbers and positive integers (see e.g. [1]). Let a T b = a b for positive a, b, and L = the set of terms in signature (1, +, *, T). As always, R+ is the set of positive reals. We give proofs of decidability for two problems about identities, and we also present a 59-element model in which high school algebra identities are true, while Wilkie's identity is false. Our first result gives a new proof of a theorem of A. Macintyre [3] (proved for terms in one variable by Richardson [4]). THEOREM 1. Let X be any subset of R+ containing 1 and closed under addition, multiplication, and exponentiation. Then the set of valid equalities T = { t1 = t2l tl, t2 E L, X W= t1 = t2 } is decidable and does not depend on X. The proof is based on the following lemma, which is proved in ??2 and 3. LEMMA 1. There is a recursive function M: L X L -N such that, for any t1(r, s), t2(r, s) E L, for any positive real (values of ) -Fif card{s E R+ It1(r, s) = t2Qr, s)} > M(t1, t2) then Vs E R+: t1(r, s) = t2(r, s). PROOF OF THEOREM 1. Proceed by induction on the number of variables: Vs E X: t1(r, s) = t2(r, s) is equivalent to &m 1t1(r, k) = t2(r-, k), where M = M(t1, t2). Received by the editors February 4, 1983 and, in revised form, April 2, 1984. 1980 Mathematics Subject Classification. Primary 03B25, 03C05, 03C13. Kev words and phrases. Exponentiation of positive reals, exponentiation of positive integers, school problem, decidability of equational theory, decidability of entailment relation, differential ring, finite model of axioms. ?01985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

Time-dependent Aharonov-Bohm Hamiltonian and admissibility criteria of quantum wave functions

Self-adjointness of the time-independent Aharonov-Bohm Hamiltonian is shown to allow a continuous family of different dynamics including those following from Pauli's criterion of rotational invariance, Aharonov-Bohm criterion of single valuedness and a version of Pauli's criterion appropriate to cylindrical symmetry suggested by Henneberger. A time-dependent fluxF(t) linking the Aharonov-Bohm solenoid leads to the time-dependent AB Hamiltonian. Explicit solutions in cases with and without inaccessible regions for the charged particle rule out applicability of both versions of the Pauli criterion. The solutions contain one time-independent parameter α, integer values of which correspond to single-valued wave functions. Any real (integer or noninteger) value of α is allowed. Charge and current densities depend on α andF(t) only through the combination α−eF(t)/2π and lead to the conclusion that physical effects of changing the flux during an experiment can be understood as local effects of the electric field inevitably associated with changing magnetic flux.

On the class number of a unit lattice over a ring of real quadratic integers

Let K be a totally real algebraic number field. In a positive definite quadratic space over K a lattice En is called a unit lattice of rank n if En has an orthonormal basis {e1 …, en}. The class number one problem is to find n and K for which the class number of En is one. Dzewas ([1]), Nebelung ([3]), Pfeuffer ([6], [7]) and Peters ([5]) have settled this problem.

Recurrence relations for the indefinite integrals of the associated Legendre functions

Two recurrence relations are derived for the computation of the integral of the associated Legendre functions of real argument and integer order and degree.