A ring-like cascade connection and a class of NFSRs with the same cycle structures

Nonlinear feedback shift registers (NFSRs) are widely used in stream cipher designs. In this paper, we propose a variant of cascade connections of NFSRs, called ring-like cascade connections. It is shown that given an initial state of a ring-like cascade connection, each register outputs the sequence of the same period. Based on this configuration, a class of NFSRs with the same cycle structure can be derived. Moreover, inspired by this result, two more general types of NFSRs with the same cycle structures are also studied.     

Joint characteristic functions construction via copulas

When modelling dependent risks it is important to be able to generate joint distributions with given marginals. One of the ways which may be useful in connection with using the Fast Fourier Transform is to construct joint characteristic functions from marginal characteristic functions. In this paper a class of n-dimensional continuous copulas is presented which in turn lead to a simple construction of joint characteristic functions with given marginal characteristic functions. Bounds on various measures of correlation are also given.     

A necessary and sufficient condition for a class of nonsingular Galois NFSRs

Let n be a positive integer. An n-stage Galois NFSR is nonsingular if and only if the output sequences of each bit register are purely periodic. It is well known that a Galois NFSR used to build a stream cipher should be nonsingular. Recently, a useful concept that is the simplified feedback function is proposed for a Galois NFSR, which is a vectorial Boolean function. Generally, for a Galois NFSR, its simplified feedback function has less nonlinear terms than its traditional feedback function, and so is easier to analyze. Moreover, it has been shown that the nonsingularity of a Galois NFSR is decided by the invertibility of its simplified feedback function. Based on this observation, in this paper, we present a necessary and sufficient condition for the nonsingularity of a class of Galois NFSRs such that every component of its simplified feedback is linear or has a common nonlinear term.     

Reliable method for water reuse in multicontaminant batch plants: NIS policy

This paper presents a mathematical technique for minimisation of freshwater and wastewater in operations characterised by multiple contaminants. The minimisation of wastewater is achieved through the exploitation of recycle and reuse opportunities using a superstructure that entails all possible recycle and reuse possibilities. This eventually forms the basis for the mathematical formulation. The existence of an optimal production schedule, which provides starting and finishing times for water using operations, is not necessary in this formulation. The paper addresses the case where central reusable water storage is nonexistent, which implies that recycle and reuse opportunities are effected by direct stream routing. The contaminants in any given water stream need not be limiting at the same time which is characteristic of practical situations. This condition enforces the overall model to be cast as a mixed integer nonlinear programming (MINLP) problem which global optimality cannot be guaranteed for complex problems. A solution procedure to overcome structural problems and guarantee global optimality is also presented.     

Stability and Analysis of TCP Connections with RED Control and Exogenous Traffic

In this paper we study the stability and performance of a system involving several TCP connections passing through a tandem of RED controlled queues each of which has an incoming exogenous stream. The exogenous stream, representing the superposition of all incoming UDP connections into a queue, has been modeled as an MMPP stream. We consider both the TCP Tahoe and the TCP Reno versions. We start with the analysis of a single TCP connection sharing a RED controlled queue with an exogenous stream. The effect of the exogenous stream (which is almost always present in large networks) is to cause the system to converge to a stationary distribution from any initial conditions. This stability result holds good for any work conserving discipline. We also obtain the performance indices of the system; specifically the goodputs and the mean sojourn times of the various connections. The complexity involved in computation of performance indices for the system is reduced by approximating the evolution of the average queue length process of the RED queue by a deterministic ODE. Then, by using a decomposition approach of two time scales, we reduce the study of the system to that of a simplified one for which the performance measures can be obtained under stationarity. Finally, we extend the above results to the case when multiple TCP connections share a RED controlled queue with an exogenous stream and to the case when a TCP connection passes through several RED controlled tandem queues each of which has an incoming exogenous stream. We also consider an example of multiple TCPs passing through a tandem of queues. A number of simulation results have been provided which support the analysis.     

Characteristic time series and operation region of the system of two tank reactors (CSTR) with variable division of recirculation stream

The paper deals with a system of a cascade of two tank reactors, being characterized by the variable stream of recirculating fluid at each stage. The assumed mathematical model enables one to determine the system’s dynamics for the case when there is no time delay and for the opposite case. The time series of the conversion degree and of the dimensionless fluid temperature, characteristic for the system considered as well as the operation regions—the latter—basing on Feingenbaum diagrams with respect to the division ratio of the recirculating stream are presented.     

Gas dynamics applications of a characteristic Cauchy problem

Some initial-boundary-value problems for a system of quasilinear partial differential equations of gas dynamics with the initial data prescribed on the characteristic surface (characteristic Cauchy problem) are considered. The following three-dimensional flow problems are investigated: the flow produced by a motion of an impermeable piston; the flow produced by a permeable piston with a given pressure; and the flow produced by the moving free boundary. In the first two problems, the piston motion is prescribed; in the last problem, the free boundary motion cannot be prescribed in advance and must be determined as a part of the problem. It is shown that those problems can be reduced to a characteristic Cauchy problem of a certain standard type that satisfies the analog of Cauchy-Kowalewski's existence theorem in the class of analytical functions (Differential Equations 12 (1977) 1438-1444). Thus, it is proved that, in the case of the analyticity of the input data, the considered problems have unique piecewise analytic solutions which may be expressed by infinite power series (the procedure of constructing the power series solution is described in Differential Equations 12 (1977) 1438-1444 as a part of the proof of the theorem).     

On affine sub-families of Grain-like structures

Grain is one of eSTREAM hardware-oriented finalists. It contains a cascade connection of an 80-bit primitive linear feedback shift registers (\({{\mathrm{LFSR}}}\)) into an 80-bit nonlinear feedback shift register (\({{\mathrm{NFSR}}}\)). The variant Grain-128 has a cascade connection with both \({{\mathrm{LFSR}}}\) and \({{\mathrm{NFSR}}}\) of order 128. We consider Grain-like structures, i.e., the cascade connection of a primitive \({{\mathrm{LFSR}}}\) into an \({{\mathrm{NFSR}}}\) of the same order. It is easy to know that in such a structure, all the affine sub-families of the \({{\mathrm{NFSR}}}\) are also the affine sub-families of the cascade connection. We prove that if the degree of the characteristic function of the \({{\mathrm{NFSR}}}\) is bigger than 2, then affine sub-families of the cascade connection must also be affine sub-families of the \({{\mathrm{NFSR}}}\). The same result holds if the order of the primitive \({{\mathrm{LFSR}}}\) is bigger than the order of the \({{\mathrm{NFSR}}}\).     

On the Euler characteristic of finite unions of convex sets

The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces, its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper.     

On an iterative method for a class of integral equations of the first kind

In this paper, we investigate an iterative method which has been proposed [1] for the numerical solution of a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Integral equations of this special type occur in experimental physics, astronomy, medical tomography and other fields where density functions cannot be measured directly, but are related to observable functions via integral equations. In order to take into account the non-negativity of density functions, the proposed iterative scheme was defined in such a way that only non-negative solutions can be approximated. The first part of the paper presents a motivation for the iterative method and discusses its convergence. In particular, it is shown that there is a connection between the iterative scheme and a certain concave functional associated with integral equations of this type. This functional can be interpreted as a generalization of the log-likelihood function of a model from emission tomography. The second part of the paper investigates the convergence properties of the discrete analogue of the iterative method associated with the discretized equation. Sufficient conditions for local convergence are given; and it is shown that, in general, convergence is very slow. Two numerical examples are presented.     

The inverse problem for the characteristic functions of several commuting operators

In this paper we derive conditions for an operator valued function to be the characteristic function of several commuting operators in a Hilbert space. We use the connection of this problem to some problems in partial differential equation to get a solution for a class of operator valued functions.     

Some Local Properties of Spectrum of Linear Dynamical Systems in Hilbert Space

We prove some local properties of the spectrum of a linear dynamical system in Hilbert space. The semigroup generator, the control operator and the observation operator may be unbounded. We consider (i) the PBH test, (ii) the correspondence between the poles of the resolvent of the semigroup generator and the poles of the transfer function, and (iii) pole-zero cancellation between two transfer functions of the cascade connection of two dynamical systems. For our investigation we take well-posed linear systems and a subclass of them called weakly regular systems as the most general setting.     

Numerical integration of Hamiltonian problems by G-symplectic methods

It is the purpose of this paper to consider the employ of General Linear Methods (GLMs) as geometric numerical solvers for the treatment of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, we exploit here a concept of near conservation for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve an accurate conservation of the Hamiltonian. In this paper we focus our attention on the connection between order of convergence and Hamiltonian deviation by multivalue methods. Moreover, we derive a semi-implicit GLM which results competitive to symplectic Runge-Kutta methods, which are notoriously implicit.     

Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method

Spectral discretization methods are well established methods for the computation of characteristic roots of time-delay systems. In this paper a method is presented for computing all characteristic roots in a given right half plane. In particular, a procedure for the automatic selection of the number of discretization points is described. This procedure is grounded in the connection between a spectral discretization and a rational approximation of exponential functions. First, a region that contains all desired characteristic roots is estimated. Second, the number of discretization points is selected in such a way that in this region the rational approximation of the exponential functions is accurate. Finally, the characteristic roots approximations, obtained from solving the discretized eigenvalue problem, are corrected up to the desired precision by a local method. The effectiveness and robustness of the procedure are illustrated with several examples and compared with DDE-BIFTOOL.     

Connections on Lagrangian submanifolds and some quasiclassical approximation problems. I

In this paper we construct a flat connection (possibly with torsion) on Lagrangian submanifolds which is connected with formulas for asymptotic wave functions generated by this submanifold. The new formalism naturally includes the Bohr-Sommerfeld condition and Maslov characteristic class and lets us establish new global formulas for asymptotic wave functions in the subtlest cases (motion in the neighborhood of a separatrix).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 41–54, 1989.     

Probing Option Prices for Information

We present a methodology for extracting information from option prices when the market is viewed as knowledgeable. By expanding the information filtration judiciously and determining conditional characteristic functions for the log of the stock price, we obtain option pricing formulae which when fit to market data may reveal this information. In particular, we consider probing option prices for knowledge of the future stock price, instantaneous volatility, and the asymptotic dividend stream. Additionally the bridge laws developed are also useful for simulation based on stratified sampling that conditions on the terminal values of paths.      

Random functions of Poisson type

Let $\text{(X,}\textcolor{red}{}\text{, μ)}$ be a σ-finite nonatomic measure space. We think of the customary analysis based upon $\text{(X,}\textcolor{red}{}\text{, μ)}$ as continuum analysis. By contrast discrete analysis is based upon an arbitrary countable subset of X, rather than upon X itself, and all countable subsets are treated alike with a Poisson process used to distinguish among them probabilisticly. The sort of functions appropriate for discrete analysis are the Campbell functions, or, as they are called in the present paper, the random functions of Poisson type. The paper presents an account of the ideas underlying discrete analysis and treats briefly the specifies of representation, stochastic integrals, and duality theory for random functions of Poisson type. It is chiefly concerned, however, with those random functions which occur in connection with the discrete analysis of Brownian motion, (for example, with Gaussian noise). In particular it shows that there is a completely positive map which carries such discrete processes onto an algebraic version of Wiener's Brownian motion process, and that under this map, random functions of Poisson type go over to the appropriate random functions of Wiener type. It also shows that the map carries random variables into noncommuting operators characteristic of quantum theory.     

Generic cryptographic weakness of k-normal Boolean functions in certain stream ciphers and cryptanalysis of grain-128

This paper considers security implications of k-normal Boolean functions when they are employed in certain stream ciphers. A generic algorithm is proposed for cryptanalysis of the considered class of stream ciphers based on a security weakness of k-normal Boolean functions. The proposed algorithm yields a framework for mounting cryptanalysis against particular stream ciphers within the considered class. Also, the proposed algorithm for cryptanalysis implies certain design guidelines for avoiding certain weak stream cipher constructions. A particular objective of this paper is security evaluation of stream cipher Grain-128 employing the developed generic algorithm. Contrary to the best known attacks against Grain-128 which provide complexity of a secret key recovery lower than exhaustive search only over a subset of secret keys which is just a fraction (up to 5%) of all possible secret keys, the cryptanalysis proposed in this paper provides significantly lower complexity than exhaustive search for any secret key. The proposed approach for cryptanalysis primarily depends on the order of normality of the employed Boolean function in Grain-128. Accordingly, in addition to the security evaluation insights of Grain-128, the results of this paper are also an evidence of the cryptographic significance of the normality criteria of Boolean functions.     

Inversion of Analytic Characteristic Functions and?Infinite Convolutions of Exponential and Laplace Densities

We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized Dirichlet series, which in turn is an infinite linear combination of exponential or Laplace densities. These results are applied to several examples.     

The asymptotic blow-up of a surface in Euclidean 3-space

Given a smoothly immersed surface in Euclidean (or affine) 3-space, the asymptotic directions define a subset in the Grassmann bundle of unoriented one-dimensional subspaces over the surface. This links the Euler characteristic of the region where the Gauss curvature is nonpositive with the index of singularities in a natural line field defined on this subset. To apply this we need only identify mechanisms which restrict the index of the singularities. In Section 2.1 we show that specific configurations of nonpositive Gauss curvature cannot be realized by an immersed surface and that specific configurations in 2-sphere cannot be realized as Gauss images of surfaces. In Section 2.2 we prove an existence theorem for surfaces which satisfy regularity conditions and a Symplectic Monge Ampere PDE. In general, a PDE of this type will not restrict the indices of the singularities over a solution. However, we show that over a surface of nonzero constant mean curvature the indices are restricted and, hence, that specific configurations of nonpositive Gauss curvature cannot be realized by a constant mean curvature surface.