Extracting post-nonlinear signal with specific kurtosis range

Blind source extraction (BSE) is an important technique to extract a desired source from the mixed signals and the post-nonlinear (PNL) mixture is more realistic model in many situations. In this paper, we address the problem of extracting the source of interest from the PNL mixture. First, the prior knowledge about the desired source, such as its normalized kurtosis range, can be treated as a constraint and incorporated into the contrast function. Therefore, BSE from the PNL mixture can be formulated a constrained optimization problem. Second, the inverse of the unknown nonlinear function is approximated by the multi-layer perceptions (MLP) network because neural network can uniformly approximate any continuous function if there is sufficient number of neurons in the hidden layer. Finally, the source of interest can be extracted from the PNL mixture by minimizing the constrained optimization problem with standard gradient descent method. Extensive computer simulations and experiments demonstrate the validity of our algorithm.     

The Hausdorff separation axiom for fuzzy topological spaces

We develop a theory of α-Hausdorff fuzzy topological spaces which is compatible with α-compactness and fuzzy continuity, and for α a certain type of member of a given lattice we obtain characterizations of the α-Hausdorff subspaces of the fuzzy unit interval, the fuzzy open unit interval, and the fuzzy real line. In route we give an easy proof of the Fuzzy Tychonov Theorem for α-compactness and extend the theory of one-point α-compactifications.     

Frequency spanning homoclinic families

A family of maps or flows depending on a parameter ν which varies in an interval, spans a certain property if along the interval this property depends continuously on the parameter and achieves some asymptotic values along it. We consider families of periodically forced Hamiltonian systems for which the appropriately scaled frequency is spanned, namely it covers the semi-infinite line [0,∞). Under some natural assumptions on the family of flows and its adiabatic limit, we construct a convenient labelling scheme for the primary homoclinic orbits which may undergo a countable number of bifurcations along this interval. Using this scheme we prove that a properly defined flux function is C¹ in ν. Combining this proof with previous results of RK and Poje, immediately establishes that the flux function and the size of the chaotic zone depend on the frequency in a non-monotone fashion for a large class of families of Hamiltonian flows.     

On stability in terms of two measures of a hybrid system having partly invisible solutions

We consider three different dynamic systems. The first runs “smoothly” during a certain finite time interval, undergoes an abrupt change in the dynamics during the next (finite) time interval and is governed by the second system. The solution of this second system lies on a surface for a finite amount of time and becomes invisible. At the beginning of the third phase, the system is subjected to an impulse which causes the solution to leave the surface and we have the new hybrid impulsive system. In this paper, we employ two measures to find suitable conditions so that the new system again runs as smoothly as the first.     

Asymptotic analysis and asymptotic domain decomposition for an integral equation of the radiative transfer type

We consider an integral equation of the radiative transfer type stated in the interval [0,τ₀] with the length τ₀1. We construct an asymptotic solution of the problem and we give a method transforming this problem to some similar problems set in the interval with the length dτ₀. Error estimates are proved.     

Nearly monotone and nearly convex approximation by smooth splines in ,

Given a monotone or convex function on a finite interval we construct splines of arbitrarily high order having maximum smoothness which are “nearly monotone” or “nearly convex” and provide the rate of -approximation which can be estimated in terms of the third or fourth (classical or Ditzian–Totik) moduli of smoothness (for uniformly spaced or Chebyshev knots). It is known that these estimates are impossible in terms of higher moduli and are no longer true for “purely monotone” and “purely convex” spline approximation.     

On Inequalities of the Landau–Kolmogorov–Hörmander Type on a Segment and Real Straight Line

We prove inequalities of the Landau–Kolmogorov–Hörmander type for the uniform norms (on some subinterval) of positive and negative parts of intermediate derivatives of functions defined on a finite interval. By using the limit transition, we obtain a new proof or the well-known Hörmander result.     

Componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces

Summary In this paper we use interval arithmetic tools for the computation of componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces. We define a mapping for which under certain assumptions we can construct an interval vector which is mapped into itself. Using Brouwer's fixed point theorem we conclude the existence of a solution of the original equation in this interval vector. Under different assumptions we can construct an interval vector such that the range of the mapping has no point in common with this interval vector. This implies that there is no solution in this interval vector. Furthermore we consider an iteration method which improves componentwise errorbounds for a solution of a quadratic. The theoretical results of this paper are demonstrated by some numerical examples using the algebraic eigenvalue problem which is probably the best known example of a quadratic equation.This paper contains the main results of a talk given by the author on the occasion of the 25th anniversary of the founding of Numerische Mathematik, March 19–21, 1984 at the Technische Universität of Munich, Germany     

Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

In a canonical way, we establish an AZ-identity (see [2]) and its consequences, the LYM-inequality and the Sperner property, for the Boolean interval lattice. Furthermore, the Bollobas inequality for the Boolean interval lattice turns out to be just the LYM-inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice.Perhaps more surprising is that by our approach the conjecture of P. L. Erdöset al.[7] and Z. Füredi concerning an Erdös–Ko–Rado-type intersection property for the poset of Boolean chains could also be established. In fact, we give two seemingly elegant proofs.     

Creep of linear unoriented polymers in tension,compression, and torsion

The creep of polymethyl methacrylate, polystyrene, and polyethylene in tension, compression, and torsion has been investigated over a broad interval of temperature below the glass transition point. It is shown that the creep criterion changes depending on the temperature. This is associated with changes in molecular mobility.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 24–28, January–February, 1970.     

Circular representation problem on hypergraphs

A hypergraph J=(X,E) is said to be circular representable, if its vertices can be placed on a circle, in such way that every edge of H induces an interval. This concept is a translation into the vocabulary of hypergraphs of the circular one's property for the (0, 1) matrices [6] studied by Tucker [9, 10]. We give here a characterization of the hypergraphs which are circular representable. We study when the associated representation is unique, and we characterize the possible transformations of a representation into another, a kind of problem which has already been treated from the algorithmic point of view by Booth and Lueker [1] or Duchet [2] in the case of the interval representable hypergraphs.Finally, we establish a connection between circular graphs and circular representable hypergraphs of the type of the Fulkerson-Gross connection between interval graphs and matrices having the consecutive one's property [5], in some special cases.     

Using Markov's interval arithmetic to evaluate Bessel-Ricatti functions

This paper deals with the application of interval arithmetic to Bessel-Ricatti functions. The extended interval arithmetic we have used is due to Markov and involves a check on monotonicity of functions in an attempt to try and get sharper bounds on computed intervals. The results we obtain are compared with those from Hansen's methods (based on bounding the Taylor series remainder) and those from Moore's technique of subdividing intervals. Two techniques are considered for evaluating derivatives of functions — one uses hand-coded derivatives and the other uses automatic differentiation. Numerical results are given, using Fortran 90 implementations of interval and automatic differentiation arithmetic.     

Optimal control of the motion of a quasilinear oscillatory system by small forces : PMM vol. 39, n 6, 1975, pp. 995–1005

We solve certain optimal control problems for the motion of a single-frequency oscillatory system which in the unperturbed state consists of an arbitrary number of oscillating elements. The solution is performed in the first approximation with respect to a small parameter . We assume that the frequency depends upon slow time, while the control goes only into the perturbing terms, so that the system is formally weakly controllable [1], But since the time interval over which the process evolves is a quantity ˜1/, all the controlled quantities are able to vary substantially [2, 3], i.e. we investigate the case, interesting in practice, of small but protracted control forces. As mechanical examples we calculate some optimal control problems for the oscillations of systems of the plane oscillator type, etc.     

Equivalence of the derived chains corresponding to a boundary-value problem on a finite interval,for polynomial operator pencils

Under investigation is the equivalence of derived chains constructed from root vectors of polynomial pencils of operators acting in a Hilbert space. These derived chains correspond to various boundary—value problems on a finite interval for an operator—differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 83–95, January, 1990.     

Ratio asymptotics for orthogonal rational functions on an interval

Let {α₁,α₂,…} be a sequence of real numbers outside the interval [−1,1] and μ a positive bounded Borel measure on this interval satisfying the Erd s–Turán condition μ′>0 a.e., where μ′ is the Radon–Nikodym derivative of the measure μ with respect to the Lebesgue measure. We introduce rational functions _n(x) with poles {α₁,…,α_n} orthogonal on [−1,1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of _n+1(x)/_n(x) as n tends to infinity under certain assumptions on the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation satisfied by the orthonormal functions.     

Interval availability analysis of a two-echelon,multi-item system

In this paper we analyze the interval availability of a two-echelon, multi-item spare part inventory system. We consider a scenario inspired by a situation that we encountered at Thales Netherlands, a manufacturer of naval sensors and naval command and control systems. Modeling the complete system as a Markov chain we analyze the interval availability and we compute in closed and exact form the expectation and the variance of the availability during a finite time interval [0, T]. We use these characteristics to approximate the survival function using a Beta distribution, together with the probability that the interval availability is equal to one. Comparison of our approximation with simulation shows excellent accuracy, especially for points of the distribution function below the mean value. The latter points are practically most relevant.     

Existence of Gompertz parameters with mortality deceleration rate and their asymptotic formulae for a large population

In this paper we provide an interval of existence of mortality rate parameters A and α, and their asymptotic expressions in a Gompertz survival model with mortality deceleration rate, in the absence of age specific mortality data.     

A one-parameter family of unimodal, concave, polynomial maps of the interval exhibiting multiplicities

This paper shows there exists a polynomial map, p, of the interval [0, 1] onto itself that is concave, symmetric about the point and such that, when parameterized {μp}, 0 ≤ μ ≤ 1, there exist three distinct values of the parameter μ₀ < μ₁ < μ₂ such that RLR³C = K(μ₀p) ≠ K(μ₁p) ≠ K(μ₂p) = RLR³C. There is also given an explicit construction of a C¹ family with the same properties.     

A spatial mixed Poisson framework for combination of excess-of-loss and proportional reinsurance contracts

In this paper a purely theoretical reinsurance model is presented, where the reinsurance contract is assumed to be simultaneously of an excess-of-loss and of a proportional type. The stochastic structure of the set of pairs (claim’s arrival time, claim’s size) is described by a Spatial Mixed Poisson Process. By using an invariance property of the Spatial Mixed Poisson Processes, we estimate the amount that the ceding company obtains in a fixed time interval in force of the reinsurance contract.     

Perturbed and non-perturbed Brownian taboo processes

In this paper we study the Brownian taboo process, which is a version of Brownian motion conditioned to stay within a finite interval, and the α-perturbed Brownian taboo process, which is an analogous version of an α-perturbed Brownian motion.We are particularly interested in the asymptotic behaviour of the supremum of the taboo process, and our main results give integral tests for upper and lower functions of the supremum as t→∞. In the Brownian case these include extensions of recent results in Lambert [4], but are proved in a quite different way.