On the hilbert transform of bounded bandlimited signals

In this paper we analyze the Hilbert transform and existence of the analytical signal for the space B _?? ^?? of bandlimited signals that are bounded on the real axis. Originally, the theory was developed for signals in L ²(?) and then extended to larger signal spaces. While it is well known that the common integral representation of the Hilbert transform may diverge for some signals in B _?? ^?? and that the Hilbert transform is not a bounded operator on B _?? ^?? , it is nevertheless possible to define the Hilbert transform for the space B _?? ^?? . We use a definition that is based on the H ¹-BMO(?) duality. This abstract definition, which can be used for general bounded signals, gives no constructive procedure to compute the Hilbert transform. However, for the practically important special case of bounded bandlimited signals, we can provide such an explicit procedure by giving a closed-form expression for the Hilbert transform. Further, it is shown that the Hilbert transform of a signal in B _?? ^?? is still bandlimited but not necessarily bounded. With these results we continue the work of [1, 2].     

Continuation Along Bifurcation Branches for a Tumor Model with a Necrotic Core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R and 0<??<R, there exists a radially-symmetric stationary solution with tumor free boundary r=R and necrotic free boundary r=??. The system depends on a positive parameter ??, which describes tumor aggressiveness, and for a sequence of values ?? ₂<?? ₃<??, there exist branches of symmetry-breaking stationary solutions, which bifurcate from these values. Upon discretizing this model, we obtain a family of polynomial systems parameterized by tumor aggressiveness factor???. By continuously changing ?? using a homotopy, we are able to compute nonradial symmetric solutions. We additionally discuss linear and nonlinear stability of such solutions.     

A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ? _k ³ ?? _k?1 elements whereas the magnetic part of the equations is approximated by discontinuous ? _k ³ ?? _k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments.     

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f(x)=g(x)|x?t| ^?? , ?? being real. Depending on the value of ??, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g??C ^??[a,b], or g??C ^??(a,b) but can have arbitrary algebraic singularities at x=a and/or x=b, and letting h=(b?a)/n, n an integer, we derive asymptotic expansions for ${T}^{*}_{n}[f]=h\sum_{1\leq j\leq n-1,\ x_{j}\neq t}f(x_{j})$ , where x _j =a+jh and t??{x ₁,??,x _n?1}. These asymptotic expansions are based on some recent generalizations of the Euler?CMaclaurin expansion due to the author (A.?Sidi, Euler?CMaclaurin expansions for integrals with arbitrary algebraic endpoint singularities, in Math. Comput., 2012), and are used to construct our quadrature formulas, whose accuracies are then increased at will by applying to them the Richardson extrapolation process. We pay particular attention to the case in which ??=?2 and f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ , which arises in the context of periodic hypersingular integral equations. For this case, we propose the remarkably simple and compact quadrature formula $\widehat{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)-\pi^{2} g(t)h^{-1}$ , and show that $\widehat{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h??0 ???>0, and that it is exact for a class of singular integrals involving trigonometric polynomials of degree at most n. We show how $\widehat{Q}_{n}[f]$ can be used for solving hypersingular integral equations in an efficient manner. In the second part of this work, we derive the Euler?CMaclaurin expansion for integrals $I[f]=\int^{b}_{a} f(x)dx$ , where f(x)=g(x)(x?t) ^?? , with g(x) as before and ??=?1,?3,?5,??, from which suitable quadrature formulas can be obtained. We revisit the case of ??=?1, for which the known quadrature formula $\widetilde{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)$ satisfies $\widetilde{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h??0 ???>0, when f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ . We show that this formula too is exact for a class of singular integrals involving trigonometric polynomials of degree at most n?1. We provide numerical examples involving periodic integrands that confirm the theoretical results.     

Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions

We introduce an iterative method for computing the first eigenpair (?? _p ,e _p ) for the p-Laplacian operator with homogeneous Dirichlet data as the limit of (?? _q, u _q ) as q??p ^?, where u _q is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with the same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of u _q to e _p is in the C ¹-norm and the rate of convergence of ?? _q to ?? _p is at least O(p?q). Numerical evidence is presented.     

Greatest Lower Bound of System Failure Probability on a Special Time Interval Under Incomplete Information About the Distribution Function of the Time to Failure of the System

The author solves the problem of finding greatest lower bounds for the probability F (??) – F (u),0 < u <, ?? < ∞, where \( u= m-{\upsigma}_{\mu}3\sqrt{3},\kern0.5em \upupsilon = m+{\upsigma}_{\mu}3\sqrt{3},\kern0.5em \mathrm{and}\kern0.5em {\upsigma}_{\mu} \) is a fixed dispersion in the set of distribution functions F (x) of non-negative random variables with unimodal differentiable density with mode m and two first fixed moments μ ₁ and μ ₂. The case is considered where the mode coincides with the first moment: m = μ ₁. The greatest lower bound of all possible greatest lower bounds for this problem is obtained and it is nearly one, namely, 0.98430.     

Sequential estimation of a threshold crossing time for a Gaussian random walk through correlated observations

Given a Gaussian random walk X with drift, we consider the problem of estimating its first-passage time ?? _A for a given level A from an observation process Y correlated to X. Estimators may be any stopping times ?? with respect to the observation process Y. Two cases of the process Y are considered: a noisy version of X and a process X with delay d. For a given loss function f(x), in both cases we find exact asymptotics of the minimal possible risk E f((?? ? ?? _A )/r) as A, d ?? ??, where r is a normalizing coefficient. The results are extended to the corresponding continuous-time setting where X and Y are Brownian motions with drift.     

Gravitational radiation in ultrarelativistic collisions

Classical ultrarelativistic gravitational bremsstrahlung in particle collisions is studied in the ADD model with d extra dimensions. The main goal in this consideration concerns trans-Planckian energies. The radiation efficiency ?? ?? E _rad /E _initial is computed in terms of the Schwarzschild radius r _S (E_collision), the impact parameter b and the center-of-mass Lorentz factor ?? _cm and is found to be ?? = C _d (r _S /b)^3d+3 times ?? _cm in some d-dependent power, larger than the previous estimates by powers of ?? _cm ? 1. The cubic graviton vertex is consistently taken into account, and the approximation is reliable for impact parameters in the range r _S < b < b _c , with b _c marking (for d ?? 0) the loss of the notion of classical trajectories. It follows that gravitational bremsstrahlung leads to extreme damping in trans-Planckian collisions, and the radiation reaction should be included in the analysis of black hole production.     

Algebraic approach to the interval linear static identification,tolerance, and control problems,or one more application of kaucher arithmetic

In this paper, theidentification problem, thetolerance problem, and thecontrol problem are treated for the interval linear equation Ax=b. These problems require computing an inner approximation of theunited solution set Σ_??(A, b)={x ∈ ? ⁿ | (?A ∈ A)(Ax ∈ b)}, of thetolerable solution set Σ_??(A, b)={x ∈ ? ⁿ | (?A ∈ A)(Ax ∈ b)}, and of thecontrollable solution set Σ_??(A, b)={x ∈ ? ⁿ | (?b ∈ b)(Ax ∈b)} respectively. Analgebraic approach to their solution is developed in which the initial problem is replaced by that of finding analgebraic solution of some auxiliary interval linear system in Kaucher extended interval arithmetic. The algebraic approach is proved almost always to give inclusion-maximal inner interval estimates of the solutionsets considered. We investigate basic properties of the algebraic solutions to the interval linear systems and propose a number of numerical methods to compute them. In particular, we present the simple and fastsubdifferential Newton method, prove its convergence and discuss numerical experiments.     

Mutually independent Hamiltonian cycles in alternating group graphs

The alternating group graph has been used as the underlying topology for many practical multicomputers, and has been extensively studied in the past. In this article, we will show that any alternating group graph AG _n , where n??3 is an integer, contains 2n?4 mutually independent Hamiltonian cycles. More specifically, let N=|V(AG _n )|, v _i ??V(AG _n ) for 1??i??N, and ??v ₁,v ₂,??,v _N ,v ₁?? be a Hamiltonian cycle of AG _n . We show that AG _n contains 2n?4 Hamiltonian cycles, denoted by $C_{l}=\langle v_{1},v_{2}^{l},\ldots,v_{N}^{l},v_{1}\rangle$ for 1??l??2n?4, such that $v_{i}^{l} \ne v_{i}^{l'}$ for all 2??i??N whenever l??l??. The result is optimal since each vertex of AG _n has exactly 2n?4 neighbors.     

The improved technique of electric and magnetic parameters measurements of powdered materials

This paper presents the measurement technique that allows to determine the relative permittivity and permeability of powdered materials. Measurements are realized in a coaxial transmission line which guarantees the broad band frequency characterization. Calculations utilize the scattering matrix parameters of the two-port formed by the sample of powdered material supported by two dielectric walls. The proposed measurement procedure is demonstrated in the example of ferrite powder - Yiitrium Garnet YIG class ferrite for which the relative permittivity (??^′,??^″) and permeability (??^′,??^″) are determined in the frequency range of 200-1200 MHz.     

Canonical D. C. programming techniques for solving a convex program with an additional constraint of multiplicative type

We consider a nonconvex programming problem of minimizing a linear functioncx over a convex setX?? ⁿ with an additional constraint ofmultiplicative type \(\prod _{i = 1}^p \psi _i (x) \leqslant 1\) , where the functionsψ _i are convex and positive onX. The main idea of our approach is to transform this problem, by usingp additional variables, into acanonical d.c. programming problem with the special structure that thereverse convex constraint involved does only depend on the newly introduced variables. This special structure suggests modifying certain techniques in d.c. programming in a way that the operations handling the nonconvexity are actually performed in the space of the additional variables. The resulting algorithm works very well whenp is small (in comparison withn).     

Spin-spin interaction in general relativity and induced geometries with nontrivial topology

We consider the dynamics of a self-gravitating spinor field and a self-gravitating rotating perfect fluid. It is shown that both these matter distributions can induce a vortex field described by the curl 4-vector of a tetrad: θ ⁱ = ½ε ^iklm e _(a)k e _l;m ^(a) , where e _k ^(a) are components of the tetrad. The energy-momentum tensor T _ik (ω) of this field has been found and shown to violate the strong and weak energy conditions which leads to possible formation of geometries with nontrivial topology like wormholes. The corresponding exact solutions to the equations of general relativity have been found. It is also shown that other vortex fields, e.g., the magnetic field, can also possess such properties.     

Shorthand Universal Cycles for Permutations

The set of permutations of ??n??={1,??,n} in one-line notation is ??(n). The shorthand encoding of a ₁?a _n ????(n) is a ₁?a _n?1. A shorthand universal cycle for permutations (SP-cycle) is a circular string of length n! whose substrings of length n?1 are the shorthand encodings of ??(n). When an SP-cycle is decoded, the order of ??(n) is a Gray code in which successive permutations differ by the prefix-rotation ?? _i =(1 2 ? i) for i??{n?1,n}. Thus, SP-cycles can be represented by n! bits. We investigate SP-cycles with maximum and minimum ??weight?? (number of ?? _n?1s in the Gray code). An SP-cycle n a n b?n z is ??periodic?? if its ??sub-permutations?? a,b,??,z equal ??(n?1). We prove that periodic min-weight SP-cycles correspond to spanning trees of the (n?1)-permutohedron. We provide two constructions: B(n) and C(n). In B(n) the spanning trees use ??half-hunts?? from bell-ringing, and in C(n) the sub-permutations use cool-lex order by Williams (SODA, 987?C996, 2009). Algorithmic results are: (1)?memoryless decoding of B(n) and C(n), (2)?O((n?1)!)-time generation of B(n) and C(n) using sub-permutations, (3)?loopless generation of B(n)??s binary representation n bits at a time, and (4)?O(n+??(n))-time ranking of B(n)??s permutations where ??(n) is the cost of computing a permutation??s inversion vector. Results (1)?C(4) improve on those for the previous SP-cycle construction D(n) by Ruskey and Williams (ACM Trans. Algorithms 6(3):Art.?45, 2010), which we characterize here using ??recycling??.     

On horizons and wormholes in k-essence theories

We study the properties of possible static, spherically symmetric configurations in k-essence theories with the Lagrangian functions of the form F(X), X ≡ ?,_α ?,^α. A no-go theorem has been proved, claiming that a possible black-hole-like Killing horizon of finite radius cannot exist if the function F(X) is required to have a finite derivative dF/dX. Two exact solutions are obtained for special cases of kessence: one for F(X) = F ₀ X ^1/3, another for F(X) = F ₀|X|^1/2 ? 2Λ, where F ₀ and Λ are constants. Both solutions contain horizons, are not asymptotically flat, and provide illustrations for the obtained nogo theorem. The first solution may be interpreted as describing a black hole in an asymptotically singular space-time, while in the second solution two horizons of infinite area are connected by a wormhole.     

Non-fragile robust finite-time H ∞ control for nonlinear stochastic it? systems using neural network

This paper deals with the problem of non-fragile robust finite-time H _?? control for a class of uncertain nonlinear stochastic It? systems via neural network. First, applying multi-layer feedback neural networks, the nonlinearity is approximated by linear differential inclusion (LDI) under statespace representation. Then, a sufficient condition is proposed for the existence of non-fragile state feedback finite-time H _?? controller in terms of matrix inequalities. Furthermore, the problem of nonfragile robust finite-time H _?? control is reduced to the optimization problem involving linear matrix inequalities (LMIs), and the detailed solving algorithm is given for the restricted LMIs. Finally, an example is given to illustrate the effectiveness of the proposed method.     

Efficient heat transfer enhancement by elastic turbulence with polymer solution in a curved microchannel

Heat and mass transfer in microscale flows are limited due to extremely low Reynolds number (Re). In a curved microchannel, however, complex flow behaviors, such as elastic instability and elastic turbulence, can be induced via viscoelastic fluid at vanishingly low-Re conditions, which is of great potential to enhance the heat transfer performance. The influence of elastic instabilities and turbulence on heat dissipation of exothermic components is experimentally investigated in this study. The heat transfer performance of both viscoelastic (polymer solutions) and Newtonian (sucrose solutions) fluid flows in a curved microchannel with a square cross section is experimentally characterized. Titanium–platinum (Ti–Pt) thin films embedded at the bottom wall of the polydimethylsiloxane (PDMS) microchannel serve as both microheater and temperature sensor. For viscoelastic fluids, the spectrum of outlet temperature fluctuation in broad frequency (f) region fits the power law of f ^?1.1. Heat transfer enhancement due to the elastic turbulence in a curved microchannel is thereby identified by the drastic growth of the Nusselt number (Nu, the ratio of convective to conductive heat transfer normal to the boundary) with the increase in the Weissenberg number (Wi, the ratio of elastic stress to viscous stress). The mechanism of heat transfer enhanced by the convection effect of elastic turbulence is also elucidated.     

Can bigravity gravitational baryogenesis explain the CMB 143 GHz excess line?

It has been recently claimed [arXiv: 1510.00126], that the 143 GHz excess line in the Cosmic MicrowaveBackground (CMB) spectrumcould be explained by a collision of our Universe with an alternate Universe in which the baryon to entropy ratio is 65 times larger than the corresponding value measured for ourUniverse. This 143 GHz excess line is due to baryons, as was claimed, since the excess line corresponds to the recombination epoch, while the rest of the CMB signal is free of such excess lines. Thus the excess line is ascribed to an effect of collision of our Universe with a parallel universe. In this paper, we propose an alternative mechanism to explain the 143 GHz excess CMB line by using a simple bimetric gravity model which makes use of two metrics, the foreground metric g _μν and the background metric f _μν. The foreground Universe describes our Universe, and the background Universe is assumed to be underlying. The metrics are chosen to satisfy \({f_{\mu \nu }} = D_{{g_{\mu \nu }}}^2\), and the bimetric gravity model is constrained in such a way that the resulting Einstein equations for the background and foreground Universe are identical. In effect, the foreground and background Universe are indistinguishable at the cosmological solutions level. However, for the choices of the metrics we made, the scalar curvatures of the foreground and background Universes, namely R _g and R _f, are related by R _f = 1/D ², which can effectively result in different baryon to entropy ratios for the two Universes, via the gravitational baryogenesis mechanism. According to the gravitational baryogenesis mechanism, the baryon to entropy ratio is \(\eta B = \dot R/M_*^2T\), which means that the baryon to entropy ratio for the foreground and the background Universes we chose, namely ηB(g) and ηB(f), satisfy the relation \(\eta B\left( f \right) = \frac{1}{{{D^2}}}\eta B\left( g \right)\), and if D is chosen as \(D = 1/\sqrt {65} \), this could explain the 143 GHz excess line. The resulting phenomenological picture is quite appealing, since in the context of our bimetric gravity model, the foreground and background Universes coexist and are indistinguishable at the cosmological solutions level, but they are distinguishable only via the gravitational baryogenesis mechanism, which results in a baryon to entropy ratio for the background Universe, which is 65 time larger from the one corresponding to our Universe.     

The role of free work identity in viscous compressible flows

Some consequences of energy identity are discussed, on assumption that there exists a neighborhood of S_b of radius η where the total energy is a minimum. For fluid phase transition the neighborhood where the rest state S_b results in isolated minimum for internal energy has finite radius r that will restrict to zero as basic density ϱ_b approaches a critical value ϱ_*. Nonlinear asymptotic stability for barotropic viscous fluids is proved by use of free work identity which enables us to provide a stronger generalized energy inequality. The stability theorem is proved in a class of regular unsteady flows which are supposed to exist. Nonlinear instability for fluid phase change with zero external forces is proved. The goal is reached assuming by absurdum that ϱ is stable in L^∞ norm.