bluuebunny/crossref_metadata_2025 huggingface dataset:
The Atkinson-Wilcox theorem in thermoelasticity
Authors: Cakoni Fioralba , Dassios George
Published: December 2016

An incident disturbance propagates in a thermoelastic medium of the Biot type and it is scattered by a bounded discontinuity of the medium. On the surface of the scatterer any kind of boundary or transmission conditions, that secures well posedness, can hold. The scattered field consists of three kinds of displacement and two kinds of thermal waves. With the exception of one of the displacement waves, namely the transverse elastic wave, all other four scattered waves exhibit exponential attenuation as a result of the coupling between the longitudinal elastic and the thermal disturbances. We show that the displacement field can be expanded in three uniformly and absolutely convergent series in inverse powers of the distance between the observation point and the geometrical center of the scatterer. For the thermal wave a corresponding expansion with two series holds true. Each one of these three elastic and two thermal series describes the corresponding scattered wave and their validity is extended up to the sphere that circumscribes the scatterer. The leading coefficients in the two displacement series of the longitudinal type have only radial components which coincide with the corresponding radial scattering amplitudes. For the transverse displacement series the leading coefficient has only tangential components which coincide with the angular scattering amplitudes. An amazing result, which was not noticed before, is that the thermal scattering amplitudes, appearing as leading coefficients in the thermal expansions, are proportional to the corresponding radial longitudinal amplitudes of the elastic expansions. In other words, both scattering amplitudes of the two thermal waves carry no independent information about the scattering process. Finally, an analytic algorithm is provided which leads to the reconstruction of all five series from the knowledge of the three leading coefficients coming from the expansions for the displacement field alone. Consequently, if the radial and the tangential scattering amplitudes of the displacement field are given in the far field, then the exact displacement and thermal fields can be recovered all the way down to the smallest sphere containing the scatterer. In an equivalent component form we claim that the nine elastic and the two thermal expansions can be completely obtained once the two longitudinal and the two transverse elastic scattering amplitudes are given.
Breakdown of smooth solutions in one-dimensional magnetostriction
Authors: Hattori Harumi
Published: December 2016

We discuss a hyperbolic aspect of magnetostriction. The equations governing the longitudinal motion consist of the nonlinear wave equations and the rate equations for the motion of spin for the magnetic moment. We show that the breakdown of smooth solutions will take place in finite time even if the initial data are smooth.
Conservative motion of a discrete, tetrahedral top on a smooth horizontal plane
Authors: Greenspan Donald
Published: December 2016

Tetrahedral tops are simulated as discrete, rigid bodies in rotation by introducing a molecular mechanics formulation. The contact point of the top with the

(
X
,
Y
)

\left ( X, Y \right )

-plane is allowed to move in the plane. The conservative, dynamical differential equations are solved numerically in such a fashion that all the system invariants are preserved. Examples, which include cusp formation, and looping, are described and discussed.
Exact controllability for problems of transmission of the plate equation with lower-order terms
Authors: Liu Weijiu , Williams Graham H.
Published: December 2016

We consider the exact controllability for the problem of transmission of the plate equation with lower-order terms. Using Lions’ Hilbert Uniqueness Method (HUM for short), we show that the system is exactly controllable in

L
2

(
Ω
)

×

H

−
2

(
Ω
)

{L^2}\left ( \Omega \right ) \times {H^{ - 2}}\left ( \Omega \right )

. We also obtain some uniqueness theorems for the problem of transmission of the plate equation and for the operator

a

(
x
)

Δ
2

+
q

a\left ( x \right ){\Delta ^2} + q

.
Free energies in the frequency domain: the scalar case
Authors: Golden J. M.
Published: December 2016

A general closed expression is given for the isothermal minimum free energy of a linear viscoelastic material in terms of Fourier-transformed quantities. A one-parameter family of free energies is constructed, ranging continuously from the maximum to the minimum free energies.
The linear stability of traveling wave solutions for a reacting flow model with source term
Authors: Hsiao Ling , Pan Ronghua
Published: December 2016

In this paper, we consider a

3
×
3

3 \times 3

system for a reacting flow model with a source term in [7]. This model can be considered as a relaxation approximation to

2
×
2

2 \times 2

systems of conservation laws, which include the well-known

p
p

-system. From this viewpoint, by introducing the new waves through time-asymptotic expansion and using the

L

2

{L^{2}}

energy method, we establish the global existence and the linear stability of traveling wave solutions.
Two-dimensional Riemann problem for pressureless gas dynamics equations with functional solutions
Authors: Hu Jiaxin
Published: December 2016

The Riemann problem for a two-dimensional hyperbolic system of pressureless gas dynamics equations is considered here. Riemann solutions are constructed for any piecewise-constant initial data having two discontinuity rays with the origin as vertex. Non-classical waves (labelled as Dirac-shock waves) appear in some solutions. The entropy conditions for the Dirac-shock waves are given. Uniqueness of Riemann solutions containing Dirac-shock waves fails.
Periodic orbits in planar systems modelling neural activity
Authors: Kooij Robert E. , Giannakopoulos Fotios
Published: December 2016

In this paper we will prove certain properties of a planar dynamical system modelling the neural activity of a network consisting of two neurons. At first we show that for a certain region in parameter space (such that there exist three equilibria) the dynamical system has no periodic orbits. To this end we need a new criterion for the nonexistence of limit cycles in a system of Liénard type (Lemma 3.1). Next we derive conditions under which our model system has exactly one periodic orbit, which will be a stable limit cycle. Finally, we cover a part of the parameter space where we can prove that the dynamical system has three equilibria such that around two of the equilibria at most one limit cycle can exist.
Spatial heteroclinic bifurcation of time periodic solutions to the Ginzburg-Landau equation
Authors: Dang-Vu H. , Delcarte C.
Published: December 2016

In this paper we study the spatial structure of the time periodic solutions to the Ginzburg-Landau equation in various configurations (supercritical and subcritical). We show that spatially periodic bursting solutions or spatially heteroclinic solutions can occur depending upon the values of the coefficients. As a consequence of this study, we obtain an exact solution to the nonlinear system of Kapitula and Maier-Paape [16]. We then show that near a spatially heteroclinic solution there is an extremely intricate complex of spatially unstable solutions and KAM surfaces.
Relaxation of the isothermal Euler-Poisson system to the drift-diffusion equations
Authors: Junca S. , Rascle M.
Published: December 2016

We consider the one-dimensional Euler-Poisson system in the isothermal case, with a friction coefficient

ε

−
1

{\varepsilon ^{ - 1}}

. When

ε
→

0
+

\varepsilon \to {0_ + }

, we show that the sequence of entropy-admissible weak solutions constructed in [PRV] converges to the solution to the drift-diffusion equations. We use the scaling introduced in [MN2], who proved a quite similar result in the isentropic case, using the theory of compensated compactness. On the one hand, this theory cannot be used in our case; on the other hand, exploiting the linear pressure law, we can give here a much simpler proof by only using the entropy inequality and de la Vallée-Poussin criterion of weak compactness in

L

1

{L^{1}}

.
Exponential asymptotics with a small exponent
Authors: Kember G. C. , Fowler A. C. , Evans J. D. , O’Brien S. B. G.
Published: December 2016

Analytic and numerical solutions are considered to a simple model problem which contains a surprisingly complicated solution structure. Asymptotic solutions are sought when a parameter that appears as an exponent in the independent variable is small, the solution then exhibiting a sudden change in slope over a region that is exponentially thin. A straightforward approach using matched asymptotic expansions immediately reveals inadequacies of this method due to the requirement of an outer solution that needs to be evaluated beyond all orders in order to match to a suitable inner solution. This behaviour is elucidated by studying first the asymptotic structure of the solution using an exact integral, which explicitly reveals the need for the inclusion of exponentially small terms in the expansions. It is then shown how a direct asymptotic solution of the differential equation can be obtained by using Borel summation to evaluate the outer solution to exponential accuracy. Further, as a practical alternative, it is shown how these exponentially improved approximations can be made when an exact numerical solution is available and without recourse to the general term of the outer or inner expansions.
Uniqueness of limit cycles in a predator-prey system with Holling-type functional response
Authors: Sugie Jitsuro
Published: December 2016

This paper is concerned with the problem of uniqueness of limit cycles in a predator-prey system with Holling’s functional response

x
p

/

(
a
+

x
p

)

{x^p}/\left ( a + {x^p} \right )

, where

a
a

and

p
p

are positive parameters. The problem has not yet been settled only in the case

1
>
p
>
2

1 > p > 2

. This paper gives a sufficient condition under which the predator-prey system with

1
>

p
>
2

1 > \\ p > 2

has exactly one limit cycle by using a result of Zhang and Gao. Finally, the fact that our condition is also necessary is mentioned.
Slow decay in linear thermoelasticity
Authors: Koch Herbert
Published: December 2016

Energy estimates show that linearized thermoelasticity defines a contraction semigroup on a Hilbert space. We show that under a geometric condition this contraction is not strict, or, more precisely, the norm of the semigroup is 1 for all

t
≥
0

t \ge 0

. Convex domains always satisfy the geometric condition.
Shape recognition via Wasserstein distance
Authors: Gangbo Wilfrid , McCann Robert J.
Published: December 2016

The Kantorovich-Rubinstein-Wasserstein metric defines the distance between two probability measures

μ
\mu

and

ν
\nu

on

R

d
+
1

{R^{d + 1}}

by computing the cheapest way to transport the mass of

μ
\mu

onto

ν
\nu

, where the cost per unit mass transported is a given function

c

(
x
,
y
)

c\left ( x, y \right )

on

R

2
d
+
2

{R^{2d + 2}}

. Motivated by applications to shape recognition, we analyze this transportation problem with the cost

c

(
x
,
y
)

=

|

x
−
y

|

2

c\left ( x, y \right ) = {\left | {x - y} \right |^2}

and measures supported on two curves in the plane, or more generally on the boundaries of two domains

Ω
,
Λ
⊂

R

d
+
1

\Omega , \Lambda \subset {R^{d + 1}}

. Unlike the theory for measures that are absolutely continuous with respect to Lebesgue, it turns out not to be the case that

μ
−
a
.
e
.
x
∈
∂
Ω

\mu - a.e.x \in \partial \Omega

is transported to a single image

y
∈
∂
Λ

y \in \partial \Lambda

; however, we show that the images of

x
x

are almost surely collinear and parallel the normal to

∂
Ω

\partial \Omega

at

x
x

. If either domain is strictly convex, we deduce that the solution to the optimization problem is unique. When both domains are uniformly convex, we prove a regularity result showing that the images of

x
∈
∂
Ω

x \in \partial \Omega

are always collinear, and both images depend on

x
x

in a continuous and (continuously) invertible way. This produces some unusual extremal doubly stochastic measures.
Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping
Authors: Marcati Pierangelo , Mei Ming
Published: December 2016

In this paper we consider a model of hyperbolic balance laws with damping on the quarter plane

(
x
,
t
)
∈

R

+

×

R

+

(x, t) \in {\mathbb {R}_ + } \times {\mathbb {R}_ + }

. By means of a suitable shift function, which will play a key role to overcome the difficulty of large boundary perturbations, we show that the IBVP solutions converge time-asymptotically to the shifted nonlinear diffusion wave solutions of the Cauchy problem to the nonlinear parabolic equation given by the related Darcy’s law. We obtain also the time decay rates, which are the optimal ones in the

L

2

{L^{2}}

-sense. Our proof is based on the use of the classical energy method.
Elementary evaluation of certain infinite integrals involving Bessel functions
Authors: Fabrikant V. I. , Dôme G.
Published: December 2016

Although it is known theoretically that certain infinite integrals of Bessel functions can be expressed in terms of elementary functions, the practical evaluation of such integrals was quite difficult due to the algebraic complexity of the expressions involved. A simple and elegant algebra is introduced here which allows these integrals to be calculated in an elementary way in terms of elementary functions. Some relationships are shown between the integrals involving Bessel functions and two-dimensional integrals over a circle of elementary functions involving distances between points. A comparison is made with existing results, and some of them were found in error (or were misprints).
Global existence and asymptotic behaviour of the solution to the system in one-dimensional nonlinear thermoviscoelasticity
Authors: Qin Yuming
Published: December 2016

This paper is concerned with the global existence, uniqueness, and asymptotic behavior, as time tends to infinity, of the solution to the system in nonlinear one-dimensional thermoviscoelasticity. Our results show that the global solution approaches to the solution in the

H

1

{H^{1}}

norm to the corresponding stationary problem, as time tends to infinity.
Diffusion induced instability
Authors: Markenscoff Xanthippi
Published: December 2016

It is shown that a coupled mechanochemical system can be stable under no diffusion and unstable with the addition of diffusion.
Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics
Authors: Li Jiequan , Yang Hanchun
Published: December 2016

In this paper we study zero-pressure gas dynamics, which is a nonstrict hyperbolic system of nonlinear conservation laws with delta-shock waves in solutions. By using the generalized Rankine-Hugoniot relations to solve the Riemann problem with two pieces of constant initial data, multidimensional planar delta-shock waves dependent upon a family of one parameter are obtained. Furthermore, we choose a unique entropy solution through the process of a viscosity vanishing, and obtain a stability for delta-shocks in multidimensions.
On the dynamics of continuous distributions of dislocations
Authors: Binz Ernst , Schwarz Günter , Wenzelburger Jan
Published: December 2016

For materials with continuous distributions of dislocations a configuration space which unifies the continuum theory of defects with classical elasticity is given. Weak equations of motion are derived from the principle of virtual work. Using the Helmholtz decomposition theorem, this yields a coupled system of equations for the dynamics of dislocations and classical elasticity.
Multivalued solutions to the eikonal equation in stratified media
Authors: Izumiya S. , Kossioris G. T. , Makrakis G. N.
Published: December 2016

In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geometric solutions, using the notion of Legendrian immersions with an initial point source or an initial smooth front. Then, we study the singularities of the solutions in the case of a smooth or piecewise Hamiltonian in a boundaryless stratified medium. Finally, we study the singularities of the solutions in a domain with a boundary that describes the propagating field in a waveguide.
Energy decay for hyperbolic thermoelastic systems of memory type
Authors: Fatori Luci Harue , Muñoz Rivera Jaime E.
Published: December 2016

In this paper we study the hyperbolic thermoelastic system, which is obtained when, instead of Fourier’s law for the heat flux relation, we follow the linearized model proposed by Gurtin and Pipkin concerning the memory theory of heat conduction. In this case the thermoelastic model is fully hyperbolic. We show that the linear system is well posed and that the solution decays exponentially to zero as time goes to infinity.
Instabilities of a thermo-mechano-chemical system
Authors: Markenscoff Xanthippi
Published: December 2016

A Hadamard instability analysis of the partial differential equations governing a coupled thermo-mechano-chemical system predicts instability of the spinoidal decomposition type as well as instability of the “negative creep” type.
On a uniquely solvable integral equation in a mixed Dirichlet-Neumann problem of acoustic scattering
Authors: Krutitskii P. A.
Published: December 2016

The mixed Dirichlet—Neumann problem for the Helmholtz equation in the exterior of several bodies (obstacles) is studied in 2 and 3 dimensions. The problem is investigated by a special modification of the boundary integral equation method. This modification can be called the “method of interior boundaries", because additional boundaries are introduced inside scattering bodies, where the Neumann boundary condition is given. The solution of the problem is obtained in the form of potentials on the whole boundary. The density in the potentials satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact, our method holds for any positive wave numbers. The Neumann and Dirichlet problems are particular cases of our problem.
Steady states for a one-dimensional model of the solar wind
Authors: Schaeffer Jack
Published: December 2016

A one (space) dimensional Vlasov equation is used to model the solar wind (a collisionless plasma) as it moves past an applied magnetic field (an obstacle). The goal is to understand physically reasonable steady states for this situation. When the applied magnetic field is sufficiently small, appropriate states are constructed.
Spatial decay of transient end effects in functionally graded heat conducting materials
Authors: Horgan C. O. , Quintanilla R.
Published: December 2016

The purpose of this research is to investigate the influence of material inhomogeneity on the spatial decay of end effects in transient heat conduction for isotropic inhomogeneous heat conducting solids. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The spatial decay of solutions to an initial-boundary value problem with variable coefficients on a semi-infinite strip is investigated. It is shown that the spatial decay of end effects in the transient problem is faster that that for the steady-state case. Qualitative methods involving second-order partial differential inequalities for quadratic functionals are first employed. Explicit decay estimates are then obtained by using comparison principle arguments involving solutions of the one-dimensional heat equation with constant coefficients. The results may be interpreted in terms of a Saint-Venant principle for transient heat conduction in inhomogeneous solids.
The singular perturbation limit of an elastic structure in a rapidly flowing nearly inviscid fluid
Authors: Artstein Zvi , Slemrod Marshall
Published: December 2016

The effective forces that govern the vertical oscillations of an elastic structure in a horizontally flowing fluid are displayed for the limit case where fluid velocity and viscosity tend to singular limits. The derived singular perturbation model is based on available representations that model the coupled dynamics as coupled oscillators. The fast dynamics of the system does not, in general, converge on the fast time scale to a stationary point; thus, the classical singular perturbation methods are not applicable. Rather, a method based on Young measures representation of the fast oscillations, and on averaging, is employed.
Dynamically accelerating cracks part 2: a finite length mode III crack in elastic material
Authors: Leise Tanya L. , Walton Jay R.
Published: December 2016

We consider a dynamically accelerating, finite length, mode III crack in an infinite elastic body. This initial boundary value problem has the nature of a free boundary problem since the crack tip motion is a priori unknown and must be found as part of the solution after imposition of a fracture criterion. Using an analog to a Dirichlet-to-Neumann map, we reduce the fracture problem to integrodifferential equations along the boundary that, for simplicity, we combine with a stress intensity factor fracture criterion. This approach has the advantage of being applicable to cases of multiple cracks as well as, in principle, to mode I cracks and to cracks in viscoelastic materials.
A self-similar viscosity approach for the Riemann problem in isentropic gas dynamics and the structure of the solutions
Authors: Kim Yong Jung
Published: December 2016

We study the Riemann problem for the system of conservation laws of one-dimensional isentropic gas dynamics in Eulerian coordinates. We construct solutions of the Riemann problem by the method of self-similar zero-viscosity limits, where the self-similar viscosity only appears in the equation for the conservation of momentum. No size restrictions on the data are imposed. The structure of the solutions obtained is also analyzed.
Conductive-radiative heat transfer in grey materials
Authors: Laitinen Mika , Tiihonen Timo
Published: December 2016

We study the well-posedness of a class of models describing heat transfer by conduction and radiation. For that purpose we propose an abstract mathematical framework that allows us to prove existence, uniqueness and the comparison principle for the weak solution with minimal or almost minimal a priori assumptions for the data. The theory covers different types of grey materials, that is, both semitransparent and opaque bodies as well as isotropic or nonisotropic scattering/reflection provided that the material properties do not depend on the wavelength of the radiation. To demonstrate the use of the abstract theory we consider in detail two examples, heat transfer between opaque bodies with diffuse-grey surfaces and a model with semitransparent material and specularly reflecting surfaces.
On the Helmholtz decomposition for polyadics
Authors: Dassios George , Lindell Ismo V.
Published: December 2016

A polyadic field of rank

n
n

is the tensor product of

n
n

vector fields. Helmholtz showed that a vector field, which is a polyadic field of rank 1, is nonuniquely decomposable into the gradient of a scalar function plus the rotation of a vector function. We show here that a polyadic field of rank

n
n

is, again nonuniquely, decomposable into a term consisting of

n
n

successive applications of the gradient to a scalar function, plus a term that consists of

(
n
−
1
)

\left ( n - 1 \right )

successive applications of the gradient to the rotation of a vector function, plus a term that consists of

(
n
−
2
)

\left ( n - 2 \right )

successive applications of the gradient to the rotation of the rotation of a dyadic function and so on, until the last

(
n
+
1
)

\left ( n + 1 \right )

th term, which consists of

n
n

successive applications of the rotation operator to a polyadic function of rank

n
n

. Obviously, the

n
=
1

n = 1

case recovers the Helmholtz decomposition theorem. For dyadic fields a more symmetric representation is provided and formulae that provide the potential representation functions are given. The special cases of symmetric and antisymmetric dyadics are discussed in detail. Finally, the multidivergence type relations, which reduce the number of independent scalar representation functions to

n

2

{n^{2}}

, are presented.
Asymptotic and exact fundamental solutions in hereditary media with singular memory kernels
Authors: Hanyga Andrzej , Seredyńska M.
Published: December 2016

A method for constructing time-domain asymptotic solutions of hyperbolic partial differential equations with delay, with singular memory kernels, is presented. The asymptotic solutions are expressed in terms of basis functions that are regularizations of a sequence of distributions related by fractional integration.
Shock reflection for the damped 𝑃-system
Authors: Hsiao Ling , Li Hailiang
Published: December 2016

The global existence and the asymptotic behavior of the weak entropy solution, the piecewise smooth solution with one shock discontinuity, on a strip domain is investigated in the present paper. We show that, for small smooth initial data and boundary value with only one small jump at

(
x
,
t
)

=

(
0
,
0
)

\left ( x, t \right ) = \left ( 0, 0 \right )

, the piecewise smooth solution with one shock discontinuity exists globally in time. The shock discontinuity begins from

(
x
,
t
)

=

(
0
,
0
)

\left ( x, t \right ) = \left ( 0, 0 \right )

, moves forward and reflects in a finite time at the boundary

x
=
1

x = 1

to form a 1-shock, which goes backward and reflects at

x
=
0

x = 0

also in a finite time to create a new 2-shock. The shock strength decays exponentially and never disappears in finite time. As

t
→
∞

t \to \infty

, this solution converges to a constant state determined by the initial and the boundary conditions.
Viscosity solutions for dynamic problems with slip-rate dependent friction
Authors: Ionescu Ioan R.
Published: December 2016

The dynamic evolution of an elastic medium undergoing frictional slip is considered. The Coulomb law modeling the contact uses a friction coefficient that is a non-monotone function of the slip-rate. This problem is ill-posed, the solution is non-unique and shocks may be created on the contact interface. In the particular case of the one-dimensional shearing of an elastic slab, the (perfect) delay convention can be used to select a unique solution. Different solutions in acceleration and deceleration processes are obtained. To transform the ill-posed problem into a well-posed one and to justify the choice of the perfect delay criterion, a visco-elastic constitutive law with a small viscosity is used here. An existence and uniqueness result is obtained in three dimensions. The assumptions on the functions implied in the contact model are weak enough to include both the normal compliance and the Tresca model. The following conjecture, based on results of numerical simulations, is stated: in the elastic case, the solution chosen by the perfect delay convention is the one obtained from the solutions of the problem with viscosity, when the viscosity tends to zero.
Global attractivity in an RBC survival model of Wazewska and Lasota
Authors: Li Jing-Wen , Cheng Sui Sun
Published: December 2016

We obtain a condition for the positive equilibrium to be a global attractor of the survival model of red blood cells proposed by Wazewska and Lasota. Our technique is novel in the sense that a pair of nonlinear equations is utilized, and our result improves earlier results in [3] and [4].
Subsonic Lamb waves in anisotropic plates
Authors: Kuznetsov Sergey V.
Published: December 2016

A six-dimensional complex formalism for analysis of Lamb waves propagating with subsonic speed in anisotropic plates is formulated. Conditions for nonexistence of certain Lamb waves in anisotropic plates are obtained. An example of a transversely isotropic plate having “forbidden” speed at which no subsonic Lamb wave propagates is presented.
Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives
Authors: Van Buren Arnie L. , Boisvert Jeffrey E.
Published: December 2016

Alternative expressions for calculating the prolate spheroidal radial functions of the first kind

R

m
l

(
1
)

(
c
,
ξ
)

R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )

and their first derivatives with respect to

ξ
\xi

are shown to provide accurate values, even for low values of

l
−
m

l - m

where the traditional expressions provide increasingly inaccurate results as the size parameter

c
c

increases to large values. These expressions also converge in fewer terms than the traditional ones. They are obtained from the expansion of the product of

R

m
l

(
1
)

(
c
,
ξ
)

R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )

and the prolate spheroidal angular function of the first kind

S

m
l

(
1
)

(
c
,
η
)

S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right )

in a series of products of the corresponding spherical functions. King and Van Buren [12] had used this expansion previously in the derivation of a general addition theorem for spheroidal wave functions. The improvement in accuracy and convergence using the alternative expressions is quantified and discussed. Also, a method is described that avoids computer overflow and underflow problems in calculating

R

m
l

(
1
)

(
c
,
ξ
)

R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )

and its first derivative.
An analytical study of the Kelvin-Helmholtz instabilities of compressible, magnetized tangential velocity discontinuities with generalized polytrope laws
Authors: Brown Kevin G. , Choudhury S. Roy
Published: December 2016

The linear Kelvin-Helmholtz instability of tangential velocity discontinuities in high velocity magnetized plasmas with isotropic or anisotropic pressure is investigated. A new analytical technique applied to the magnetohydrodynamic equations with generalized polytrope laws (for the pressure parallel and perpendicular to the magnetic field) yields the complete structure of the unstable, standing waves in the (inverse plasma beta, Mach number) plane for modes at arbitrary angles to the flow and the magnetic field. The stable regions in the (inverse plasma beta, propagation angle) plane are mapped out via a level curve analysis, thus clarifying the stabilizing effects of both the magnetic field and the compressibility. For polytrope indices corresponding to the double adiabatic and magnetohydrodynamic equations, the results reduce to those obtained earlier using these models. Detailed numerical results are presented for other cases not considered earlier, including the cases of isothermal and mixed waves. Also, for modes propagating along or opposite to the magnetic field direction and at general angles to the flow, a criterion is derived for the absence of standing wave instability—in the isotropic MHD case, this condition corresponds to (plasma beta)

≤
1

\le 1

.
Model-based tracking of moving objects in cluttered environments
Authors: Gidas Basilis , Gomes Fernando Carvalho , Robertson Christopher
Published: December 2016

We explore a coherent statistical/Bayesian framework for tracking rigid or non-rigid objects in highly cluttered environments. The procedure involves three basic models: (i) an object representation, (ii) a dynamic model, and (iii) a data or observation model. We employ two object representations — deformable templates, and a hierarchical/syntactic model; the deformations of the template are parametrized by a finite number of global parameters viewed as generalized coordinates of the model; they determine the state space of the tracking problem. The dynamic model describes the evolution of the generalized coordinates (or the corresponding state vector); our dynamic model is a non-Gaussian Markov chain whose transition probabilities are derived from Lagrangian mechanics. We introduce two data models: one nonlinear and one linear (Gaussian); the design of the two models uses different image processing techniques. The nonlinear filtering problem induced by three models is solved both by the Extended Kalman Filter (EKF), and an iterative algorithm—to be referred to as the Monte Carlo Filter (MCF)— introduced in the statistic literature and first employed in Computer Vision by Blake and Isard. We present several tracking experiments with real video sequences. In each experiment we implement both MCF and EKF, and their performances are compared. Overall, we find that MCF is more robust than EKF in situations with considerable clutter and/or occlusions, but EKF is comparable and sometimes more accurate than MCF in situations with less occlusion and degradation. Our studies also demonstrate that when the motion has sharp discontinuities in direction, nonlinearities in the dynamic model are key for successful tracking; put differently, the deficiencies of linear dynamics models cannot be easily compensated with more sophisticated data models.
Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations
Authors: Li Hailiang , Markowich Peter , Mei Ming
Published: December 2016

The hydrodynamic model for semiconductors in one dimension is considered. For perturbated Riemann data, global subsonic (weak) entropy solutions, piecewise continuous and piecewise smooth solutions with shock discontinuities are constructed and their asymptotic behavior is analyzed. In subsonic domains, the solution is smooth and, exponentially as

t
→
∞

t \to \infty

, tends to the corresponding stationary solution due to the influence of Poisson coupling. Along the shock discontinuity, the shock strength and the difference of derivatives of solutions decay exponentially affected by the relaxation mechanism.
A mathematical representation of biological variability in medical images
Authors: Matejic Larisa
Published: December 2016

Medical image ensembles exhibit variability and it is the aim of computational anatomy to represent such variabilities mathematically and to exploit these knowledge representations by inference algorithms implemented through code. The variability is caused by several factors and our pattern theoretic approach rests on the assumption that they can be understood in terms of groups of transformations and probability measures on such groups. We shall arrange the similarity groups in a cascade, typically starting with the more rigid transformations and continuing with more flexible ones. Most importantly, however, we attach great significance to the physical and biological interpretation of the similarity groups.
Estimating stresses in a partially inflated high altitude balloon using a relaxed energy
Authors: Collier William G.
Published: December 2016

During ascent and at other times during flight, the lifting gas of a high altitude balloon is compressed and only able to partially inflate the balloon. In this condition the surface of the balloon will sag to form folds and wrinkles which are difficult to analyze. Previous numerical work to analyze these types of balloons was based on minimizing extrema of potential energy of balloon shapes that included an explicit representation of excess material as folds. These models used the conventional strain energy for linear isotropic membranes and permitted compressive states to enter the solutions. This paper explores the application of the energy relaxation method to the earlier models to produce solutions free of compressive states. Numerical results computed using the relaxed energy are presented and compared with results computed using the standard strain energy for a membrane.
A simple model for phase transitions: from the discrete to the continuum problem
Authors: Pagano S. , Paroni R.
Published: December 2016

In this paper we study a one-dimensional model simulating the shear in a two-dimensional body. We analyse the discrete system and we deduce the continuum limit of the lattice model as the lattice parameter goes to zero. Different energies are introduced and linked together.
Optimal control of a chemotaxis system
Authors: Fister K. Renee , McCarthy C. Maeve
Published: December 2016

Chemotaxis is the process by which cells aggregate under the force of a chemical attractant. The cell and chemoattractant concentrations are governed by a coupled system of parabolic partial differential equations. We investigate the optimal control of the proportion of cells being generated in two settings. One involves harvesting the actual cells and the other depicts removing a proportion of the chemoattractant. The optimality system for each problem contains forward and backward reaction-diffusion and convection-diffusion equations. Numerical results are presented.
An inverse problem for the three-dimensional multi-connected vibrating membrane with Robin boundary conditions
Authors: Zayed E. M. E.
Published: December 2016

This paper deals with the very interesting problem concerning the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in

R
3

{R^3}

. The trace of the heat semigroup

θ

(
t
)

=

∑

v
=
1

∞

exp
⁡

(
−
t

μ
v

)

\theta \left ( t \right ) = \sum \nolimits _{v = 1}^\infty {\exp \left ( - t{\mu _v} \right )}

, where

{

μ
v

}

v
=
1

∞

\left \{ {{\mu _v}} \right \}_{v = 1}^\infty

are the eigenvalues of the negative Laplacian
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus nabla squared equals minus sigma-summation Underscript beta equals 1 Overscript 3 Endscripts left-parenthesis StartFraction partial-differential Over partial-differential x Superscript beta Baseline EndFraction right-parenthesis squared">

−

∇
2

=
−

∑

β
=
1

3

(

∂

∂

x
β

)

2

- {\nabla ^2} = - {\sum \nolimits _{\beta = 1}^3 {\left ( {\frac {\partial }{{\partial {x^\beta }}}} \right )} ^2}

in the

(

x
1

,

x
2

,

x
3

)

\left ( {x^1}, {x^2}, {x^3} \right )

-space, is studied for a general multiply-connected bounded domain

Ω
\Omega

in

R
3

{R^3}

surrounding by simply connected bounded domains

Ω
j

{\Omega _j}

with smooth bounding surfaces

S
j

(
j
=
1
,
.
.
.
,
n
)

{S_j}\left ( j = 1,...,n \right )

, where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components

S
i
∗

(
i
=
1
+

k

j
−
1

,
.
.
.
,

k
j

)

S_i^* \left ( i = 1 + {k_{j - 1}},...,{k_j} \right )

of the bounding surfaces

S
j

{S_j}

is considered, such that

S
j

=

∪

i
=
1
+

k

j
−
1

k
j

S
i
∗

{S_j} = \cup _{i = 1 + {k_{j - 1}}}^{{k_j}} S_i^*

, where

k
0

=
0

{k_0} = 0

. Some applications of

θ

(
t
)

\theta \left ( t \right )

for an ideal gas enclosed in the multiply-connected bounded container

Ω
\Omega

with Robin boundary conditions are given. We show that the asymptotic expansion of

θ

(
t
)

\theta \left ( t \right )

for short-time

t
t

plays an important role in investigating the influence of the finite container

Ω
\Omega

on the thermodynamic quantities of an ideal gas.
Stability of Jin-Xin relaxation shocks
Authors: Humpherys Jeffrey
Published: December 2016

We examine the spectrum of shock profiles for the Jin-Xin relaxation scheme for systems of hyperbolic conservation laws in one spatial dimension. By using a weighted norm estimate, we prove that these shock profiles exhibit strong spectral stability in the weak shock limit.
The principle of exchange of stabilities for Couette flow
Authors: Herron Isom H. , Ali Halima N.
Published: December 2016

The eigenvalue problem for the linear stability of Couette flow between two rotating concentric cylinders to axisymmetric disturbances is considered. It is proved that the principle of exchange of stabilities holds when the cylinders rotate in the same direction and the circulation decreases outwards. The proof is based on the notion of a positive operator which is analogous to a positive matrix. Such operators have a spectral property which implies the principle of exchange of stabilities.
Limiting current singularities in electro diffusion
Authors: Dubi C.
Published: December 2016

This paper concerns the phenomenon of limiting current in the local electro neutrality approximation of electrodiffusion. The limiting current is the value of the control parameter, the electric current, for which the straightforward split of an electrodiffusional system into a quasi-electro-neutral bulk and charged boundary (electric double) layer valid near the equilibrium is no longer expected to stand. Singularities in the solution appearing at the limiting current are studied.
Rearrangements and fourth order equations
Authors: Ferone Vincenzo , Kawohl Bernd
Published: December 2016

The paper contains a priori estimates for the deformation of plates and beams. In particular we investigate the “worst cases” for the maximum deformation depending on where a load is placed on a beam or plate. The methods of proof use rearrangement argument.
Local and global existence for the coupled Navier-Stokes-Poisson problem
Authors: Donatelli Donatella
Published: December 2016

In this paper we investigate the Cauchy Problem for coupled Navier-Stokes-Poisson equation. The global existence of weak solutions in Sobolev framework is proved by using some compactification properties deduced from the Poisson equation.
Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source
Authors: Chan C. Y. , Tian H. Y.
Published: December 2016

Let

q
q

be a nonnegative real number, and

T
T

be a positive real number. This article studies the following degenerate semilinear parabolic first initial-boundary value problem:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript q Baseline u Subscript t Baseline left-parenthesis x comma t right-parenthesis minus u Subscript x x Baseline left-parenthesis x comma t right-parenthesis equals a squared delta left-parenthesis x minus b right-parenthesis f left-parenthesis u left-parenthesis x comma t right-parenthesis right-parenthesis f o r Baseline 0 greater-than x greater-than 1 comma 0 greater-than t less-than-or-equal-to upper T">

x
q

u
t

(
x
,
t
)

−

u

x
x

(
x
,
t
)

=

a
2

δ

(
x
−
b
)

f

(
u

(
x
,
t
)

)

f
o
r
0
>
x
>
1
,
0
>
t
≤
T

{x^q}{u_t}\left ( x, t \right ) - {u_{xx}}\left ( x, t \right ) = {a^2}\delta \left ( x - b \right )f\left ( u\left ( x, t \right ) \right ) \qquad for 0 > x > 1, 0 > t \le T

\]
,
\[

u

(
x
,
0
)

=
ψ

(
x
)

f
o
r
0
≤
x
≤
1

u\left ( x, 0 \right ) = \psi \left ( x \right ) \qquad for 0 \le x \le 1

\]
,
\[

u

(
0
,
t
)

=
u

(
1
,
t
)

=
0

f
o
r
0
>
t
≤
T

u\left ( 0, t \right ) = u\left ( 1, t \right ) = 0 \qquad for 0 > t \le T

\]
, where

δ

(
x
)

\delta \left ( x \right )

is the Dirac delta function, and

f
f

and

ψ
\psi

are given functions. It is shown that the problem has a unique solution before a blow-up occurs,

u
u

blows up in a finite time, and the blow-up set consists of the single point

b
b

. A lower bound and an upper bound of the blow-up time are also given. To illustrate our main results, an example is given. A computational method is also given to determine the finite blow-up time.
On the Geselowitz formula in biomagnetics
Authors: Dassios George , Kariotou Fotini
Published: December 2016

A detailed analysis of the Geselowitz formula for the magnetic induction and for the electric potential fields, due to a localized dipole current density, is provided. It is shown that the volume integral, which describes the contribution of the conductive tissue to the magnetic field, exhibits a hyper-singular behaviour at the point where the dipole source is located. This singularity is handled both via local regularization of the volume integral as well as through calculation of the total flux it generates. The analysis reveals that the contribution of the primary dipole to the volume integral is equal to the one third of the magnetic field generated by the primary dipole while the rest is due to the distributed conductive tissue surrounding the singularity. Furthermore, multipole expansion is introduced, which expresses the magnetic field in terms of polyadic moments of the electric potential over the surface of the conductor.
Remark on indirect matching of singularly perturbed boundary value problems
Authors: Karwowski Andrzej Joachim
Published: December 2016

We examine four singularly perturbed boundary value problems. We show that it is possible to simplify the standard matching procedure by studying the boundary layer equations with the gauge function

η
\eta

as a new independent variable.
A new Weber-type transform
Authors: Thambynayagam R. K. Michael , Habashy Tarek M.
Published: December 2016

In this paper we introduce a new Weber-type transform pair for the representation of a function

f

(
r
)

f\left ( r \right )

defined over the domain

a
≤
r
>
∞

a \le r > \infty

and which satisfies the Robin mixed boundary condition

f

(
a
)

+
λ

f
′

(
a
)

=
0

f\left ( a \right ) + \lambda f’\left ( a \right ) = 0

. The orthogonality relationships of the transform kernels are derived in both the spatial and the spectral domains as well as Parseval’s theorem. We apply this new Weber-type transform pair to solve a mixed boundary value problem in a system of planar layers.
Asymptotic behaviour for a partially diffusive relaxation system
Authors: Escobedo Miguel , Laurençot Philippe
Published: December 2016

The time asymptotics of nonnegative and integrable solutions to a partially diffusive relaxation system is investigated. Under suitable assumptions on the relaxation term, the convergence to a self-similar source type solution, either of the heat equation or of the viscous Burgers equation, is proved. The proof relies on optimal decay rates and classical scaling arguments.
Analysis of a class of nonlinear integro-differential equations arising in a forestry application
Authors: Kraemer Michael A. , Kalachev Leonid V.
Published: December 2016

Certain models describing the age dynamics of a natural forest give rise to nonlinear integro-differential equations for the seedlings density as a function of time. The special feature of the problem is that corresponding solutions have non-smooth second derivatives. Since the biological model contains a small parameter, a perturbation method can be used to find an asymptotic solution. Banach’s fixed point theorem is used to prove existence and uniqueness of the solution, the convergence of a numerical scheme, and the validity of the asymptotic approximation. In an example numerical and asymptotic approximations are compared for various choices of time steps.
On similarity solutions and blow-up spectra for a semilinear wave equation
Authors: Galaktionov V. A. , Pohozaev S. I.
Published: December 2016

We construct countable spectra of different asymptotic patterns of self-similar and approximate self-similar types for global and blow-up solutions for the semilinear wave equation
\[

u

t
t

=
Δ
u
+

|
u
|

p
−
1

u
,

x
∈

R
N

,
t
>
0
,

{u_{tt}} = \Delta u + {\left | u \right |^{p - 1}}u, \qquad x \in {R^N}, t > 0,

\]
in different ranges of exponent

p
p

and dimension

N
N

.