Integer Case Research Articles

Joint determination of the lot size and number of shipments for a family of integrated vendor–buyer systems considering defective products

In this paper we present two solution procedures to obtain the solution for a family of economic production quantity inventory models for an integrated vendor–buyer system considering that production system generates defective products, in which the number of shipments must be a discrete value and that the lot size can take continuous (case 1) or integer (case 2) values. These solution procedures provide a more realistic approach for solving a family of inventory models. Previous works only derive the optimal lot size considering the number of shipments as a given value or derive both the lot size and the number of shipments considering both decision variables as continuous. The number of shipments takes integer values in real life and thus it must be considered as a discrete variable. This paper revisits and solves previous inventory models considering the variables (lot size and number of shipments) according to their real-life nature.

Dynamical Properties of k-Free Lattice Points

We revisit the visible points of a lattice in Euclidean $n$-space together with their generalisations, the $k$th-power-free points of a lattice, and study the corresponding dynamical system that arises via the closure of the lattice translation orbit. Our analysis extends previous results obtained by Sarnak and by Cellarosi and Sinai for the special case of square-free integers and sheds new light on previous joint work with Peter Pleasants.

Comparison of Numerical Methods of the SEIR Epidemic Model of Fractional Order

In this paper, we consider the SEIR (Susceptible-Exposed-Infected-Recovered) epidemic model by taking into account both standard and bilinear incidence rates of fractional order. First, the nonnegative solution of the SEIR model of fractional order is presented. Then, the multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional order. Finally, the obtained results are compared with those obtained by the fourth-order Runge-Kutta method and non-standard finite difference (NSFD) method in the integer case.

Three-dimensional fractional-spin gravity

Abstract Using Wigner-deformed Heisenberg oscillators, we construct 3D Chern-Simons models consisting of fractional-spin fields coupled to higher-spin gravity and internal nonabelian gauge fields. The gauge algebras consist of Lorentz-tensorial Blencowe-Vasiliev higher-spin algebras and compact internal algebras intertwined by infinite-dimensional generators in lowest-weight representations of the Lorentz algebra with fractional spin. In integer or half-integer non-unitary cases, there exist truncations to gl(ℓ, ℓ ± 1) or gl(ℓ|ℓ ± 1) models. In all non-unitary cases, the internal gauge fields can be set to zero. At the semi-classical level, the fractional-spin fields are either Grassmann even or odd. The action requires the enveloping-algebra representation of the deformed oscillators, while their Fock-space representation suffices on-shell.The project was funded in part by F.R.S.-FNRS “Ulysse” Incentive Grant for Mobility in Scientific Research.

Epidemic Model of Leptospirosis Containing Fractional Order

We study an epidemic model of leptospirosis in fractional order numerically. The multistep generalized differential transform method is applied to find the accurate approximate solution of the epidemic model of leptospirosis disease in fractional order. A unique positive solution for the epidemic model in fractional order is presented. For the integer case derivative, the approximate solution of MGDTM is compared with the Runge-Kutta order four scheme. The numerical results are presented for the justification purpose.

Rigid Calabi-Yau threefolds, Picard Eisenstein series and instantons

Type IIA string theory compactified on a rigid Calabi-Yau threefold gives rise to a classical moduli space that carries an isometric action of U(2, 1). Various quantum corrections break this continuous isometry to a discrete subgroup. Focussing on the case where the intermediate Jacobian of the Calabi-Yau admits complex multiplication by the ring of quadratic imaginary integers d, we argue that the remaining quantum duality group is an arithmetic Picard modular group PU(2, 1; d). Based on this proposal we construct an Eisenstein series invariant under this duality group and study its non-Abelian Fourier expansion. This allows the prediction of non-perturbative effects, notably the contribution of D2- and NS5-brane instantons. The present work extends our previous analysis in 0909.4299 which was restricted to the special case of the Gaussian integers 1 = ℤ[i].

Hanbury Brown and Twiss correlations in quantum Hall systems

We study a Hanbury Brown and Twiss (HBT) interferometer formed with chiral edge channels of a quantum Hall system. HBT cross correlations are calculated for a device operating both in the integer and fractional quantum Hall regimes, the latter at Laughlin filling fractions. We find that in both cases, when the current is dominated by electron tunneling, current-current correlations show antibunching, characteristic of fermionic correlations. When the current-current correlations are dominated by quasiparticle tunneling, the correlations reveal bunching, characteristic of bosons. For electron tunneling, we use the Keldysh technique, and show that the result for fractional filling factors can be obtained in a simple way from the results of the integer case. It is shown that quasiparticle-dominated cross-current correlations can be analyzed by means of a quantum master-equation approach. We present here a detailed derivation of the results [Campagnano et al., Phys. Rev. Lett. 109, 106802 (2012)] and generalize them to all Laughlin fractions.

Formula omitted] control of fractional linear systems

In this paper, the standard H∞ control problem for continuous-time fractional linear time-invariant single-input–single-output systems is solved. The adopted approach consists of extending to the fractional case the procedure followed within the classical solution for the integer case. According to the classical route, we first consider the generalization to fractional systems of the standard Youla parameterization of all the stabilizing controllers. The H∞ optimal controller is then found among the class of stabilizing controllers, recasting the control problem into a model-matching one. The results obtained naturally extend well-established results to a fractional setting, including both the commensurate and noncommensurate case, thus providing a framework where H∞ design can be recast along the well-known and fruitful lines of the integer case. A worked-through example is discussed in detail to illustrate the design procedure.

Existence of deviating fractional differential equation

In this paper we shall establish sufficient conditions for the existence of solutions of a class of fractional differential equation (Cauchy type ) and its solvability in a subset of the Banach space. The main tool used in our study is the non-expansive operator technique. The non integer case is taken in sense of Riemann-Liouville fractional operators. Applications are illustrated.

The number of finite index subgroups of Baumslag–Solitar groups

Gelman obtained a simple formula for the number of finite index subgroups of Baumslag–Solitar groups BS(p, q) = 〈a, t | t −1 a p t = a q 〉, where p and q are coprime integers. We generalize this formula to the case of arbitrary nonzero integers. The proof is obtained by calculating the number of permutations y ∈ S n such that the subgroup of S n generated by x and y, where x is given, is transitive. Bibliography: 4 titles.

Performance Analysis of Digital Communication Systems Over Composite $\eta{-}\mu$/Gamma Fading Channels

In this paper, we analyze and evaluate the performance of digital communication systems that operate over a composite fading channel following the η-μ/gamma distribution. More specifically, new analytical expressions for the channel capacity under different transmission schemes are derived in the form of infinite series. For the special case of integer μ values, simplified closed-form expressions are also obtained. Moreover, we deduce exact tractable formulas for the average bit error rate (BER) of various modulation schemes. In the high-power regime, simplified capacity and BER expressions are provided, revealing the implications of the model parameters on system performance. The relationship of the presented results with previously reported results on generalized- K and K fading channels is also established. We note that the theoretical results presented herein are given in analytical form, which can efficiently be evaluated.

On Integral Well-rounded Lattices in the Plane

We investigate distribution of integral well-rounded lattices in the plane, parameterizing the set of their similarity classes by solutions of the family of Pell-type Diophantine equations of the form $x^2+Dy^2=z^2$ where $D>0$ is squarefree. We apply this parameterization to the study of the greatest minimal norm and the highest signal-to-noise ratio on the set of such lattices with fixed determinant, also estimating cardinality of these sets (up to rotation and reflection) for each determinant value. This investigation extends previous work of the first author in the specific cases of integer and hexagonal lattices and is motivated by the importance of integral well-rounded lattices for discrete optimization problems. We briefly discuss an application of our results to planar lattice transmitter networks.

On some generalizations of the implicit Euler method for discontinuous fractional differential equations

We discuss the numerical solution of differential equations of fractional order with discontinuous right-hand side. Problems of this kind arise, for instance, in sliding mode control. After applying a set-valued regularization, the behavior of some generalizations of the implicit Euler method is investigated. We show that the scheme in the family of fractional Adams methods possesses the same chattering-free property of the implicit Euler method in the integer case. A test problem is considered to discuss in details some implementation issues and numerical experiments are presented.

An efficient compact quadratic convex reformulation for general integer quadratic programs

We address the exact solution of general integer quadratic programs with linear constraints. These programs constitute a particular case of mixed-integer quadratic programs for which we introduce in Billionnet et al. (Math. Program., 2010) a general solution method based on quadratic convex reformulation, that we called MIQCR. This reformulation consists in designing an equivalent quadratic program with a convex objective function. The problem reformulated by MIQCR has a relatively important size that penalizes its solution time. In this paper, we propose a convex reformulation less general than MIQCR because it is limited to the general integer case, but that has a significantly smaller size. We call this approach Compact Quadratic Convex Reformulation (CQCR). We evaluate CQCR from the computational point of view. We perform our experiments on instances of general integer quadratic programs with one equality constraint. We show that CQCR is much faster than MIQCR and than the general non-linear solver BARON (Sahinidis and Tawarmalani, User's manual, 2010) to solve these instances. Then, we consider the particular class of binary quadratic programs. We compare MIQCR and CQCR on instances of the Constrained Task Assignment Problem. These experiments show that CQCR can solve instances that MIQCR and other existing methods fail to solve.

The Karatsuba integer middle product

We study the problem of computing middle products of multiple-precision integers. In particular we adapt the Karatsuba polynomial middle product algorithm to the integer case, showing how to efficiently mitigate the failure of bilinearity of the integer middle product noted by Hanrot, Quercia and Zimmermann.

The Homotopy Analysis Method for Approximating of Giving Up Smoking Model in Fractional Order

In this paper, we consider the giving up smoking model. First, we present the giving up smoking model in fractional order. Then the homotopy analysis method (HAM) is employed to compute an approximate and analytical solution of the model in fractional order. The obtained results are compaired with those obtained by forth order Runge-Kutta method and nonstandard numerical method in the integer case. Finally, we present some numerical results.

Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions

In this paper, we study a boundary value problem of nonlinear fractional differential equations of order q (1<q ≤ 2) with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.

Matrix completion problems over integral domains: The case with a diagonal of prescribed blocks

Let R be an arbitrary integral domain, let ∧ = { λ 1 , … , λ n } be a multiset of elements of R , let σ be a permutation of { 1 , … , k } let n 1 , … , n k be positive integers such that n 1 + ⋯ + n k = n , and for r = 1 , … , k let A r ∈ R n r × n σ ( r ) . We are interested in the problem of finding a block matrix Q = Q rs r , s = 1 k ∈ R n × n with spectrum Λ and such that Q r σ ( r ) = A r for r = 1 , … , k . Cravo and Silva completely characterized the existence of such a matrix when R is a field. In this work we construct a solution matrix Q that solves the problem when R is an integral domain with two exceptions: (i) k = 2 ; (ii) k ≥ 3 , σ ( r ) = r and n r > n / 2 for some r . What makes this work quite unique in this area is that we consider the problem over the more general algebraic structure of integral domains, which includes the important case of integers. Furthermore, we provide an explicit and easy to implement finite step algorithm that constructs an specific solution matrix (we point out that Cravo and Silva’s proof is not constructive).

On convergence in mixed integer programming

Let $${P \subseteq {\mathbb R}^{m+n}}$$ be a rational polyhedron, and let P I be the convex hull of $${P \cap ({\mathbb Z}^m \times {\mathbb R}^n)}$$ . We define the integral lattice-free closure of P as the set obtained from P by adding all inequalities obtained from disjunctions associated with integral lattice-free polyhedra in $${{\mathbb R}^m}$$ . We show that the integral lattice-free closure of P is again a polyhedron, and that repeatedly taking the integral lattice-free closure of P gives P I after a finite number of iterations. Such results can be seen as a mixed integer analogue of theorems by Chvatal and Schrijver for the pure integer case. One ingredient of our proof is an extension of a result by Owen and Mehrotra. In fact, we prove that for each rational polyhedron P, the split closures of P yield in the limit the set P I .

Cyclotomic matrices and graphs over the ring of integers of some imaginary quadratic fields

We determine all Hermitian O Q ( d ) -matrices for which every eigenvalue is in the interval [ − 2 , 2 ] , for each d ∈ { − 2 , − 7 , − 11 , − 15 } . To do so, we generalise charged signed graphs to L -graphs for appropriate finite sets L , and classify all L -graphs satisfying the same eigenvalue constraints. We find that, as in the integer case, any such matrix/graph is contained in a maximal example with all eigenvalues ±2.