bluuebunny/crossref_metadata_2025 huggingface dataset:
Global existence in 𝐿⁴(𝐑₊×𝐑) for a nonstrictly hyperbolic conservation law
Authors: Zhao Huijiang
Published: December 2016

We study the existence problem for the following nonstrictly hyperbolic system:
\[

u
t

+

1
2

(
3

u
2

+

v
2

)

x

=
0

{u_t} + \frac {1}{2}{\left ( 3{u^2} + {v^2} \right )_x} = 0

\]
,
\[

v
t

+

(
u
v
)

x

=
0

{v_t} + {\left ( uv \right )_x} = 0

\]
, with singular initial data, i.e.,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis u left-parenthesis t comma x right-parenthesis comma v left-parenthesis t comma x right-parenthesis right-parenthesis vertical-bar Subscript t equals 0 Baseline equals left-parenthesis u 0 left-parenthesis x right-parenthesis comma v 0 left-parenthesis x right-parenthesis right-parenthesis element-of upper L Superscript 4 Baseline left-parenthesis upper R comma upper R squared right-parenthesis">

(
u

(
t
,
x
)

,
v

(
t
,
x
)

)

|

t
=
0

=

(

u

0

(
x
)

,

v
0

(
x
)

)

∈

L

4

(
R
,

R

2

)

\left ( u\left ( t, x \right ), v\left ( t, x \right ) \right )\left | {_{t = 0}} \right . = \left ( {u_{0}}\left ( x \right ), {v_0}\left ( x \right ) \right ) \in {L^{4}}\left ( R, {R^{2}} \right )

\]
.
The bifurcation structure of a thin superconducting loop swith small variations in its thickness
Authors: Richardson G.
Published: December 2016

We study bifurcations between the normal and superconducting states, and between superconducting states with different winding numbers, in a thin loop of superconducting wire, of uniform thickness, to which a magnetic field is applied. We then consider the response of a loop with small thickness variations. We find that close to the transition between normal and superconducting states lies a region where the leading-order problem has repeated eigenvalues. This leads to a rich structure of possible behaviours. A weakly nonlinear stability analysis is conducted to determine which of these behaviours occur in practice.
On stability of shock waves in relativistic magnetohydrodynamics
Authors: Trakhinin Yu. L.
Published: December 2016

The structural stability of relativistic magnetohydrodynamic shock waves is studied. Stability results are obtained for the special case of fast parallel shock waves. It is proved that the instability and linear stability domains coincide with those of shock waves in relativistic gas dynamics. The domain of structural (nonlinear) stability, where the uniform Lopatinski condition is fulfilled for the stability problem, is found. It is shown that the structural stability domain is smaller than that of relativistic gas dynamic shock waves.
A note on exact particular solutions of the generalized shallow-water equations
Authors: Odulo Anatoly B.
Published: December 2016

This note presents a set of systems of two first-order quasi-linear partial differential equations, which can be reduced to the shallow-water equations. This set includes equations describing a two-layered fluid flow.
Uniform stabilization of the higher-dimensional system of thermoelasticity with a nonlinear boundary feedback
Authors: Liu Wei-Jiu , Zuazua Enrique
Published: December 2016

Using multiplier techniques and Lyapunov methods, we derive explicit decay rates for the energy in the higher-dimensional system of thermoelasticity with a nonlinear velocity feedback on part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary.
Maximal attractor for the system of one-dimensional polytropic viscous ideal gas
Authors: Zheng Songmu , Qin Yuming
Published: December 2016

In this paper, the dynamics for the system of polytropic viscous ideal gas is investigated. One of the important features of this problem is that the metric spaces

H

(
1
)

{H^{\left ( 1 \right )}}

and

H

(
2
)

{H^{\left ( 2 \right )}}

that we work with are two incomplete metric spaces, as can be seen from the constraints

θ
>
0

\theta > 0

and

u
>
0

u > 0

with

θ
\theta

and

u
u

begin absolute temperature and specific volume, respectively. For any constants

β
1

,

β
2

,

β
3

,

β
4

,

β
5

{\beta _1}, {\beta _2}, {\beta _3}, {\beta _4}, {\beta _5}

satisfying certain conditions, two sequences of closed subspaces

H
β

(
i
)

⊂

H

(
i
)

(
i
=
1
,
2
)

H_\beta ^{\left ( i \right )} \subset {H^{\left ( i \right )}} \left ( i = 1, 2 \right )

are found, and the existence of two maximal (universal) attractors in

H
β

(
1
)

H_\beta ^{\left ( 1 \right )}

and

H
β

(
2
)

H_\beta ^{\left ( 2 \right )}

is proved.
A nonconvex scalar conservation law with trilinear flux
Authors: Hayes Brian T. , Shearer Michael
Published: December 2016

The focus of this paper is on traveling wave solutions of the equation
\[

u
t

+
f

(
u
)

x

=
ϵ

u

x
x

+

ϵ
2

γ

u

x
x
x

{u_t} + f{\left ( u \right )_x} = \epsilon {u_{xx}} + {\epsilon ^2}\gamma {u_{xxx}}

\]
, in which the flux function

f
f

is trilinear and nonconvex. In particular, it is shown that for combinations of parameters in certain ranges, there are traveling waves that converge as

ϵ
→
0

\epsilon \to 0

to undercompressive shocks, in which the characteristics pass through the shock. The analysis is based on explicit solutions of the piecewise linear ordinary differential equation satisfied by traveling waves. The analytical results are illustrated by numerical solutions of the Riemann initial value problem, and are compared with corresponding explicit results for the case of a cubic flux function.
The effect of a small initial distortion of the basic flow on the subcritical transition in plane Poiseuille flow
Authors: Sun Pu
Published: December 2016

This paper presents an instability theory in which a mean flow and multiple wave interactions in the Poiseuille flow transition process are studied. It is shown that not only can this mean flow term come as the result of the Fourier decomposition of a general disturbance, it can also come as an exact solution to the unsteady Navier-Stokes equations. The presence of this term, though small, can produce totally different linear and nonlinear stability behavior for the flow at subcritical Reynolds numbers. In the linear stability case, with the presence of this mean flow perturbation term, the instabilities are obtained well below the critical value of 5772. When this mean flow term is introduced into the interactions with other harmonic perturbation waves, for the plane Poiseuille flow case with Reynolds numbers around 1200, the nonlinear interactions rapidly modify the total mean flow profile toward the mean flow profile observed in turbulence while the other two- and three-dimensional waves remain small. The initial energies needed to trigger the instabilities are much smaller than those reported by previous investigators. The intermittent character of the disturbance observed in transition experiments is also captured.
Effect of rugosity on a flow governed by stationary Navier-Stokes equations
Authors: Amirat Youcef , Bresch Didier , Lemoine Jérôme , Simon Jacques
Published: December 2016

We study the stationary flow of a fluid occupying a 3D infinite horizontal domain bounded by a rough wall that is at rest and by a plane that moves with a constant velocity. The rough wall is a plane covered with periodically distributed asperities of size

ε
\varepsilon

. We prove that, outside a neighbourhood of the rough region, the flow behaves asymptotically as a Couette flow, as

ε
→
0

\varepsilon \to 0

, up to an exponentially small error.
Stabilization of elastic plates with variable coefficients and dynamical boundary control
Authors: Guo Yuxia , Chai Shugen , Yao Pengfei
Published: December 2016

The aim of this paper is to investigate the stabilization of a hybrid system composed of a plate equation with variable coefficients and two ordinary differential equations under some suitable feedbacks. A rational energy decay rate is established by the multiplier method and the Riemannian geometry method, and the uniform energy decay rate for a simplified system is obtained.
Singular solutions to a nonlinear elliptic boundary value problem originating from corrosion modeling
Authors: Bryan Kurt , Vogelius Michael
Published: December 2016

We consider a nonlinear elliptic boundary value problem on a planar domain. The exponential type nonlinearity in the boundary condition is one that frequently appears in the modeling of electrochemical systems. For the case of a disk, we construct a family of exact solutions that exhibit limiting logarithmic singularities at certain points on the boundary. Based on these solutions, we develop two criteria that we believe predict the possible locations of the boundary singularities on quite general domains.
Integral representation in the hodograph plane of compressible flow
Authors: Hansen Erik B. , Hsiao George C.
Published: December 2016

Compressible flow is considered in the hodograph plane. The linearity of the equation determining the stream function is exploited to derive a representation formula involving boundary data only, and a fundamental solution to the adjoint equation. For subsonic flow, an efficient algorithm for computing the fundamental solution is developed.
Electron transport in semiconductor superlattices
Authors: Ben Abdallah N. , Degond P. , Mellet A. , Poupaud F.
Published: December 2016

In this paper, we rigorously derive a diffusion model for semiconductor superlattices, starting from a kinetic description of electron transport at the microscopic scale. Electron transport in the superlattice is modelled by a collisionless Boltzmann equation subject to a periodic array of localized scatters modeling the periodic heterogeneities of the material. The limit of a large number of periodicity cells combined with a large-time asymptotics leads to a homogenized diffusion equation which belongs to the class of so-called “SHE” models (for Spherical Harmonics Expansion). The rigorous convergence proof relies on fine estimates on the operator modeling the localized scatters.
Nonlinear surface waves on a tangential discontinuity in magnetohydrodynamics
Authors: Alì Giuseppe , Hunter John K.
Published: December 2016

We derive an asymptotic equation that describes the propagation of weakly nonlinear surface waves on a tangential discontinuity in incompressible magnetohydrodynamics. The equation is similar to, but simpler than, previously derived asymptotic equations for weakly nonlinear Rayleigh waves in elasticity, and is identical to a model equation for nonlinear Rayleigh waves proposed by Hamilton et al. The most interesting feature of the surface waves is that their nonlinear self-interaction is nonlocal. As a result of this nonlocal nonlinearity, smooth solutions break down in finite time, and appear to form cusps.
More-or-less-uniform sampling and lengths of curves
Authors: Noakes Lyle , Kozera Ryszard
Published: December 2016

More-or-less-uniform samples are introduced and used to estimate lengths of smooth regular strictly convex curves in

R

2

{\mathbb {R}^{2}}

. Quartic convergence is proved and illustrated by examples.
Stability of constant equilibrium state for dissipative balance laws system with a convex entropy
Authors: Ruggeri Tommaso , Serre Denis
Published: December 2016

For a one-dimensional system of dissipative balance laws endowed with a convex entropy, we prove, under natural assumptions, that a constant equilibrium state is asymptotically

L
2

{L^2}

-stable under a zero-mass initial disturbance. The technique is based on the construction of an appropriate Liapunov functional involving the entropy and a so-called compensation term.
The Schrödinger with variable mass model: mathematical analysis and semi-classical limit
Authors: Kefi Jihène
Published: December 2016

In this paper, we propose and analyze a one-dimensional stationary quantum-transport model: the Schrödinger with variable mass. In the first part, we prove the existence of a solution for this model, with a self-consistent potential determined by the Poisson problem, whereas, in the second part, we rigorously study its semi-classical limit which gives us the kinetic model limit. The rigorous limit was based on the analysis of the support of the Wigner transform.
Asymptotic expansions of the Appell’s function 𝐹₁
Authors: Ferreira Chelo , López José L.
Published: December 2016

The first Appell’s hypergeometric function

F
1

(
a
,
b
,
c
,
d
;
x
,
y
)

{F_1}\left ( a, b, c, d; x, y \right )

is considered for large values of its variables

x
x

and/or

y
y

. An integral representation of

F
1

(
a
,
b
,
c
,
d
;
x
,
y
)

{F_1}\left ( a, b, c, d; x, y \right )

is obtained in the form of a generalized Stieltjes transform. Distributional approach is applied to this integral to derive four asymptotic expansions of this function in increasing powers of

1

/

(
1
−
x
)

1/\left ( 1 - x \right )

and/or

1

/

(
1
−
y
)

1/\left ( 1 - y \right )

. For certain values of the parameters

a
,
b
,
c

a, b, c

and

d
d

, two of these expansions also involve logarithmic terms in the asymptotic variables

1
−
x

1 - x

and/or

1
−
y

1 - y

. Coefficients of these expansions are given in terms of the Gauss hypergeometric function

2

F
1

(
α
,
β
,
γ
;
x
)

_2{F_1}\left ( \alpha , \beta , \gamma ; x \right )

and its derivative with respect to the parameter

α
\alpha

. All of the expansions are accompanied by error bounds for the remainder at any order of the approximation. These error bounds are obtained from the error test and, as numerical experiments show, they are considerably accurate.
Interpolations with elasticae in Euclidean spaces
Authors: Mio W. , Srivastava A. , Klassen E.
Published: December 2016

Motivated by interpolation problems arising in image analysis, computer vision, shape reconstruction, and signal processing, we develop an algorithm to simulate curve straightening flows under which curves in

R

n

{\mathbb {R}^{n}}

of fixed length and prescribed boundary conditions to first order evolve to elasticae, i.e., to (stable) critical points of the elastic energy

E
E

given by the integral of the square of the curvature function. We also consider variations in which the length

L
L

is allowed to vary and the flows seek to minimize the scale-invariant elastic energy

E

i
n
v

{E_{inv}}

, or the free elastic energy

E
λ

{E_\lambda }

.

E

i
n
v

{E_{inv}}

is given by the product of

L
L

and the elastic energy

E
E

, and

E
λ

{E_\lambda }

is the energy functional obtained by adding a term

λ
\lambda

-proportional to the length of the curve to

E
E

. Details of the implementations, experimental results, and applications to edge completion problems are also discussed.
Constrained quasiconvexification of the square of the gradient of the state in optimal design
Authors: Pedregal Pablo
Published: December 2016

We explicitly compute the constrained quasiconvexification of the integrand associated with the square of the gradient of the state in a typical optimal design problem in which a volume constraint is enforced.
Asymptotic behavior and strong convergence for hyperbolic systems of conservation laws with damping
Authors: Lattanzio Corrado , Rubino Bruno
Published: December 2016

A local type estimate will be proved here for general

2
×
2

2 \times 2

hyperbolic systems of conservation laws with strong dissipative term. Following the idea of [15], this result will be achieved by using as a preliminary step the convergence in the mean, which is an immediate consequence of the result of [11] obtained by using the compensated compactness theory.
Stability of the Riemann semigroup with respect to the kinetic condition
Authors: Colombo Rinaldo M. , Corli Andrea
Published: December 2016

This note deals with systems of hyperbolic conservation laws that are endowed with a generalized kinetic relation and develop phase transitions. The

L
1

{L^1}

-Lipschitzean continuous dependence of the solution from the kinetic relation is proved. Preliminarily, we rephrase several results known in the case of standard conservation laws to the case comprising phase boundaries.
Numerical study of quadratic area-preserving mappings
Authors: Hénon M.
Published: December 2016

Dynamical systems with two degrees of freedom can be reduced to the study of an area-preserving mapping. We consider here, as a model problem, the mapping given by the quadratic equations:

x
1

=
x
cos
⁡
α
−

(

y
−

x
2

)

sin
⁡
α

{x_1} = x\cos \alpha - \left ( {y - {x^2}} \right )\sin \alpha

,

y
1

=
x
sin
⁡
α
+

(

y
−

x
2

)

cos
⁡
α

{y_1} = x\sin \alpha + \left ( {y - {x^2}} \right ) \\ \cos \alpha

, which is shown to be in a sense the simplest nontrivial mapping. Some analytical properties are given, and numerical results are exhibited in Figs. 2 to 14.
Uniqueness in a melting slab with space- and time-dependent heating
Authors: Boley Bruno A.
Published: December 2016

Conditions for uniqueness of solution are derived for a slab which melts under an applied heat input dependent both on time and on the melted depth, the melted material being instantaneously removed. Some examples are constructed in which more than one solution is exhibited as well as one in which no melting solution exists. A theorem demonstrating monotonic dependence of solution on boundary data under certain conditions is proved.
Simplifications and reduction of the Sanders-Koiter linear shell equations for various midsurface geometries
Authors: Simmonds James G.
Published: December 2016

It is shown that the twelve linear field equations of the Sanders-Koiter first approximation shell theory, when specialized to shells with midsurfaces of constant mean curvature, can be replaced by eight equations of particularly simple form. This is accomplished by adding certain negligibly small terms to the conventional uncoupled stress-strain relations. On the basis of order of magnitude estimates, it is argued that these equations are actually adequate for shells of arbitrary geometry. For shells of nonzero Gaussian curvature, the simplified field equations are reduced to four coupled scalar equations. For catenoidal and helicoidal shells, these equations are further reduced to two coupled fourth-order equations. Various special forms of these equations are shown to agree with results obtained by Lardner, Reissner, and Wan for shallow shells, shells of revolution undergoing axisymmetric and lateral deformation, and helicoidal shells undergoing axisymmetric deformation.
Higher approximations for transonic flows
Authors: Hayasi Nisiki
Published: December 2016

Using the results of the WKBI method, two hypothetical gases are introduced, whose graphs in the pressure-density plane and that of the polytropic gas have contact of order 4 and 5 at the sonic point. This is in contrast to the fact that such graphs for the Tricomi gas and the generalized Tricomi gas have contact of order 2 and 3, respectively, to that of the polytropic gas there. Various relations for these gases are derived and compared to those of the air, the Tricomi gas and the generalized Tricomi gas. Applicable range of the approximations to the airflow are

0.65
>
M
>
1.4

0.65 > M > 1.4

for the first approximation, and

0.5
>
M
>
1.5

0.5 > M > 1.5

for the second approximation,

M
M

being the local Mach number. This is compared to such ranges as

0.9
>
M
>
1.2

0.9 > M > 1.2

for the Tricomi gas, and

0.75
>
M
>
1.3

0.75 > M > 1.3

for the generalized Tricomi gas. Flow solutions for the hypothetical gases are expressed by the Airy functions.
The distance matrix of a graph and its tree realization
Authors: Patrinos A. N. , Hakimi S. L.
Published: December 2016

The results of Hakimi and Yau and others in the realization of a distance matrix are generalized to graphs (digraphs) whose branches (arcs) may have negative weights. Conditions under which such matrices have a tree, hypertree or directed tree realization are given, uniqueness of these realizations is discussed and algorithms for their construction are indicated.
Series equations involving Jacobi polynomials and mixed boundary-value problems of the Laplace equation
Authors: Shimasaki Masaaki , Kiyono Takeshi
Published: December 2016

A formal analysis of series equations involving Jacobi polynomials is given.

(

2
N
+
1

)

\left ( {2N + 1} \right )

series equations involving Jacobi polynomials are reduced to a set of

N
N

simultaneous Fredholm integral equations which can be solved numerically by the use of the Legendre-Gauss quadrature formula. In case of triple series equations the result is in agreement with that of Lowndes. Besides triple series equations, certain quadruple series equations can be also reduced to a single Fredholm integral equation of the second kind. Owing to the introduction of an arbitrary weight factor, the theory is feasible for the analysis of various many-part mixed boundary-value problems of the Laplace equation. As an example, special cases of certain trigonometric series equations are discussed in detail in connection with an electrostatic problem.
Boundary conditions and stability of inviscid plane-parallel flows
Authors: Rogers Joel C. W.
Published: December 2016

The problem of describing the manner in which disturbances in an incompressible flow propagate downstream, when the infinite speed of signal transmission would make it theoretically possible for the disturbances to travel upstream also, and indeed would make upstream values dependent on boundary conditions imposed at arbitrarily great distances downstream, arises in numerous cases of physical interest. One such case is the growth of perturbations on plane-parallel flows in long channels, where the nature of the perturbation at any particular location is “quasi-steady” in time, although the perturbation grows with distance downstream. Another such case arises with the inevitable numerical disturbances in numerical “experiments” with computers, where an infelicitous choice of downstream boundary conditions may lead to results wholly at variance with observations. The purpose of this paper is to indicate how the choice of boundary conditions can drastically influence the phenomena observed, and to select those boundary conditions most consistent with reality. Our ideas are applied in some detail to the description of the linear theory of growth of perturbations on a two-dimensional inviscid plane-parallel basic flow. In particular, we show that the stability or instability of the flow depends on the nature of the boundary conditions imposed, and we find boundary conditions whose application leads to results in conformity with actual experimental situations.
Finite Fourier self-transforms
Authors: Fedotowsky A. , Boivin G.
Published: December 2016

This paper considers the integral equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda gamma left-parenthesis t prime right-parenthesis equals left-parenthesis 2 pi right-parenthesis Superscript negative upper N Baseline integral Underscript normal upper Omega Endscripts exp left-parenthesis minus i omega dot t prime right-parenthesis integral Underscript upper T Endscripts gamma left-parenthesis t right-parenthesis exp left-parenthesis i omega dot t right-parenthesis d t d omega">

λ
γ

(

t
′

)

=

(

2
π

)

−
N

∫
Ω

exp
⁡

(

−
i
ω
⋅

t
′

)

∫
T

γ

(
t
)

exp
⁡

(

i
ω
⋅
t

)

d
t
d
ω

\lambda \gamma \left ( {t’} \right ) = {\left ( {2\pi } \right )^{ - N}}\int \limits _\Omega {\exp \left ( { - i\omega \cdot t’} \right )\int \limits _T {\gamma \left ( t \right )} \exp \left ( {i\omega \cdot t} \right )dt d\omega }

\]
as well as a more general one wherein the Fourier kernels are weighted. When

Ω
\Omega

and

T
T

are

N
N

-dimensional spherical domains, the eigenfunctions of the integral equation are generalized prolate spheroidal functions for which a new nomenclature is proposed. Many properties of the eigenfunctions are developed and summarized. Because of the importance of these functions in Fourier transform theory, old as well as new properties are included.
On the existence of solutions to optimization problems with eigenvalue constraints
Authors: Vepa Kosla
Published: December 2016

The optimum tapering of Bernoulli—Euler beams, i.e. the shape for which a given total mass yields the highest possible value of the first fundamental frequency of harmonic transverse small oscillations, is determined. The question of the existence of a solution to the optimization problem is considered. It is shown that, irrespective of the relationship between the flexural rigidity and linear mass density of the cantilever beam, the necessary conditions for optimality lead to a contradiction. This result is in partial disagreement with that obtained by earlier investigators. By imposing additional constraints on the optimization variable, a numerical solution for the case of the cantilever beam is obtained, using the formulation of the maximum principle of Pontryagin.
On uniqueness, stability and pointwise estimates in the Cauchy problem for coupled elliptic equations
Authors: Schaefer Philip W.
Published: December 2016

Results of the kind cited in the title are obtained for the improperly posed Cauchy problem for a system of two coupled elliptic partial differential equations. We assume one stabilizing condition is imposed on one of the dependent variables. The results follow from an a priori inequality which is derived as a consequence of the logarithmic convexity of a suitable functional.
On the connection between the elliptic equations of the Navier-Stokes type and the theory of harmonic functionals
Authors: Deshpande S. M.
Published: December 2016

It is established in this paper, under conditions more general than those used by Millikan, that the two-dimensional incompressible viscous flows past finite bodies cannot be equivalent to a variational problem of the Euler-Lagrange type. It has thereby been possible to obtain two harmonic functionals with a close relation to the Navier-Stokes equations.
On the non-uniqueness of elastic rotations for deformations of materials with elastic range
Authors: Owen David R.
Published: December 2016

In this note precise definitions of the concepts of elastic and permanent deformation are used to establish non-uniqueness of elastic rotations for deformations of materials with elastic range.
A Neumann series representation for solutions to boundary-value problems in dynamic elasticity
Authors: Ahner John F. , Hsiao George C.
Published: December 2016

A regularized integral equation formulation for two exterior fundamental boundary-value problems in elastodynamics is presented. In either case, the displacement vector is assumed to be harmonic in time with a small frequency. It is shown that the solution can be expressed as a Neumann series in terms of the prescribed function; moreover, a sufficient condition for the convergence of the series is established.
A variational formulation for constrained quasilinear vector systems
Authors: Geffen Nima
Published: December 2016

A variational formulation for multi-dimensional initial- and/or boundary-value problems for a system of quasilinear conservation equations with a rotationality condition in a vector form with the aid of a vector Lagrange multiplier is given. The duality between the physical and ’phase’ (or hodograph) spaces emerges, and the Lagrange multiplier turns out to be the vector potential for the conserved field, and hence of some interest in itself. Application is given to a family of transonic flows in the physical and hodograph planes, and to a problem in nonlinear sound propagation.
Dynamics of the Josephson junction
Authors: Levi M. , Hoppensteadt F. C. , Miranker W. L.
Published: December 2016

We study the sine-Gordon equation and systems of discrete approximations to it which are respectively a model of the Josephson junction and models of coupled-point Josephson junctions. We do this in the so-called current-driven case. The voltage response of these devices is the average of the time derivative of the solution of these equations and we demonstrate the existence of running periodic solutions for which the average exists. Static solutions are also studied. These together with the running solutions explain the multiple-valuedness in the response of a Josephson junction in several cases. The stability of the various solutions is described in some of the cases. Numerical results are displayed which give details of structure of solution types in the case of a single point junction and of the one-dimensional distributed junction.
On wave propagation in a deformed Mooney-Rivlin material
Authors: Hayes M.
Published: December 2016

For a homogeneously deformed incompressible Mooney—Rivlin material it is shown that there is a universal connection between the speeds of propagation of the plane transverse waves which may propagate in any direction in the material. Such connections are not valid, in general, for incompressible materials.
Bifurcations of the forced van der Pol oscillator
Authors: Holmes P. J. , Rand D. A.
Published: December 2016

We discuss the bifurcations of the variational equation of the forced van der Pol oscillator and prove the existence of bifurcations of saddle connection type as postulated by M. L. Cartwright [4] and A. W. Gillies [5].
On a functional equation arising in the stability theory of difference-differential equations
Authors: Castelan W. B. , Infante E. F.
Published: December 2016

The functional differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q prime left-parenthesis t right-parenthesis equals upper A upper Q left-parenthesis t right-parenthesis plus upper B upper Q Superscript upper T Baseline left-parenthesis tau minus t right-parenthesis comma negative normal infinity greater-than t greater-than normal infinity">

Q
′

(
t
)

=
A
Q

(
t
)

+
B

Q
T

(

τ
−
t

)

,
−
∞
>
t
>
∞

Q’\left ( t \right ) = AQ\left ( t \right ) + B{Q^T}\left ( {\tau - t} \right ), - \infty > t > \infty

\]
, where

A
A

,

B
B

are

n
×
n

n \times n

constant matrices,

τ
≥
0

\tau \ge 0

,

Q

(
t
)

Q\left ( t \right )

is a differentiable

n
×
n

n \times n

matrix and

Q
T

(
t
)

{Q^T}\left ( t \right )

is its transpose, is studied. Existence, uniqueness and an algebraic representation of its solutions are given.
The Stefan problem with arbitrary initial and boundary conditions
Authors: Tao L. N.
Published: December 2016

The paper is concerned with the free boundary problem of a semi-infinite body with an arbitrarily prescribed initial condition and an arbitrarily prescribed boundary condition at its face. An analytically exact solution of the problem is established, which is expressed in terms of some functions and polynomials of the similarity variable

x

/

t

1

/

2

x/{t^{1/2}}

and time

t
t

. Convergence of the series solution is considered and proved. Hence the solution also serves as an existence proof. Some special initial and boundary conditions are discussed, which include the Neumann problem and the one-phase problem as special cases.
Nonlinear analysis with an arbitrary stimulus ensemble
Authors: Victor Jonathan D. , Knight Bruce W.
Published: December 2016

A family of Wiener-type methods is discussed in a general context. These methods share the concept of expansion of an unknown transducer as an orthogonal series. The terms of the series are drawn from a hierarchy of subspaces of transducers that are orthogonal with respect to a particular stimulus ensemble. Choices of specific stochastic ensembles lead to previously described analytical methods, including the classical one of Wiener.
Accurate approximations to the polygamma functions
Authors: Kelley C. Stuart
Published: December 2016

Empirically obtained approximations to the polygamma functions

ψ

(
n
)

(
x
)

{\psi ^{\left ( n \right )}}\left ( x \right )

and their associated accuracies are presented. These simple functions are quite accurate, this accuracy increasing with

x
x

and being best for small

n
n

. These approximate expressions are shown to be circuitously related to the Stirling approximation to the parent gamma function.
Flows with condensation
Authors: Yih Chia Shun
Published: December 2016

Some simple flows with condensation are considered and their solutions given. For the vapor phase, the nonlinearity of the equations of motion and of heat diffusion, and of the equation of state, and the dissipation due to shear and volume viscosities are taken into account. For the liquid phase the density is assumed constant. At the interface, where condensation takes place, the velocity, the stress, and the temperature gradient are all discontinuous. The same approach can be used for flows with evaporation.
Theoretical analysis of steady nonadiabatic premixed laminar flames
Authors: Margolis Stephen B.
Published: December 2016

A one-dimensional, steady, non-adiabatic, premixed laminar flame is assumed semi-infinite with the burner at the origin, and the investigation centers on the asymptotic behavior of the temperature and species mass fractions in the burned region near infinity. After a consideration of the general

N
N

-species problem, specific results are obtained for the global two-step reaction

v
r

R
→

μ
i

I

{v_r}R \to {\mu _i}I

,

v
r

I
→

μ
p

P

{v_r}I \to {\mu _p}P

, where

R
R

,

I
I

,

P
P

denote reactant, intermediate, and product species, respectively, and

v
r

{v_r}

,

v
i

{v_i}

,

μ
i

{\mu _i}

,

μ
p

{\mu _p}

are stoichiometric coefficients. Assuming Arrhenius kinetics, it is shown that the classical linearized asymptotic theory is not applicable unless

v
r

=

v
i

=
1

{v_r} = {v_i} = 1

, in which case the approach to burned equilibrium is an exponential decay. Consequently, a nonlinear theory applicable to arbitrary

v
r

{v_r}

and

v
i

{v_i}

is presented which shows that in general the asymptotic decay is algebraic. It is further shown that boundedness of the solution at infinity permits the arbitrary specification of only three boundary conditions on the original sixth-order differential system. This result is illustrated by a comprehensive analytical example and the computational implications for the general

N
N

-species problem are discussed.
On stability and periodicity in phosphorus nutrient dynamics
Authors: Arnold E. M.
Published: December 2016

The stability of an equilibrium and the existence of limit cycles in a three-dimensional dynamical system arising in predator-prey-nutrient dynamics are demonstrated, using center manifold theory. Some implications of this result for limnological applications are discussed.
Nonlinear surface waves in self-gravitating fluids
Authors: Malik S. K. , Singh M.
Published: December 2016

The weakly nonlinear disturbances in a self-gravitating, incompressible, inviscid fluid slab are studied. When the wave number is equal to the critical wave number, the amplitude modulation results in nonlinear Schrodinger equation. The finite-amplitude standing wave is stable against modulation. The nonlinear cutoff wave number is also obtained.
Penalty/finite-element approximations of a class of unilateral problems in linear elasticity
Authors: Kikuchi Noboru , Song Young Joon
Published: December 2016

The present paper is concerned with a development of a penalty/finite-element approximation of a class of unilateral problems in linear elasticity. A penalty method is applied to resolve the inequality constraint due to contact, and convergence with respect to the penalty parameter is discussed. Then finite-element approximations are introduced to the penalized formulation with a priori error estimates in terms of the penalty and mesh parameters. Several numerical examples are also given in the end of the paper.
A free boundary problem arising in a kinematic wave model of channel flow with infiltration
Authors: Sherman B.
Published: December 2016

Water flowing down a dry channel and infiltrating into the channel bed constitutes a free boundary problem. The free boundary is the time history of the water edge or front. In this paper we discuss a kinematic wave model of the problem. The problem is formulated in Sec. 1 and the results summarized in Sec. 2. In Secs. 3 and 4 the mathematical details are carried out, and in Sec. 5 a model using the continuity and momentum equations of hydraulics is discussed.
On the local form of the second law of thermodynamics in continuum mechanics
Authors: Woods L. C.
Published: December 2016

The Clausius-Duhem inequality

σ

(

r
,
t

)

≥
0

\sigma \left ( {r,t} \right ) \ge 0

, a widely adopted axiom in continuum mechanics, leads to the conclusion that for many materials the entropy

s
s

cannot depend on gradients like the temperature gradient g and the velocity gradient e. But this is at variance with the received view (since Gibbs) that entropy is a function of thermodynamic state, however detailed that state description may be. Gradients, and even higher derivatives of macroscopic variables, may be included as state variables (although only on macroscopic time scales shorter than or comparable with their natural relaxation times), and the fundamental property of entropy is its convexity—the more detailed the specification of state, the smaller is the corresponding value of entropy.
Weight functions for a class of Liapunov functions in the plane
Authors: Anderson Larry R. , Ducich Sarah E.
Published: December 2016

In this paper we give a class of weight functions which generate Liapunov functions for a general second-order differential system. In the special case of a Lienard equation we give conditions under which these weight functions may be chosen so as to improve certain known estimates of regions of asymptotic stability. The procedure is applied to a well-known equation and new estimates are obtained.
On the boundary value problem associated with a general twisted tube with a uniform nonrotating section
Authors: Murray J. C.
Published: December 2016

An orthogonal curvilinear coordinate system is used to formulate the Dirichlet problem of potential theory associated with the interior of a general twisted tube with a uniform non-rotating section. Two solution schemes are presented for a class of finite and infinite tube geometries. The boundary-value problems associated with a tube of uniform circular section are discussed as an example.
On the convergence of solutions of Volterra equations to almost-periodic functions
Authors: Gripenberg Gustaf
Published: December 2016

The rate of convergence of solutions of a certain Volterra integral equation and a system of two Volterra equations to almost-periodic limit functions is studied. The equations considered arise from some diffusion problems with nonlinear and almost-periodic boundary conditions.
Elastic-plastic field at the tip of a propagating shear crack
Authors: Lo K. K.
Published: December 2016

Asymptotic stress and strain fields are presented for cracks propagating dynamically in incompressible elastic-perfectly plastic materials. Perfect plasticity solutions for stationary cracks are recovered upon setting the velocity of propagation of the crack tip equal to zero. The stress field near the crack tip is found to consist of a combination of uniform and non-uniform stress regions.
The Stefan problem with a convective boundary condition
Authors: Solomon A. D. , Alexiades V. , Wilson D. G.
Published: December 2016

We study the one-phase Stefan problem on a semi-infinite strip

x
≥
0

x \ge 0

, with the convective boundary condition

−
K

T
x

(

0
,
t

)

=
h

[

T
L

−
T
(
0
,
t
)

]

- K{T_x}\left ( {0, t} \right ) = h\left [ {{T_L} - T(0, t)} \right ]

. Points of interest include: a) behavior of the surface temperature

T

(

0
,
t

)

T\left ( {0, t} \right )

; b) asymptotic behavior as

h
→
∞

h \to \infty

; c) uniqueness, and d) bounds on the phase change front and total system energy.
On a certain class of elastic materials with nonelliptic energy densities
Authors: Kikuchi N. , Triantafyllidis N.
Published: December 2016

In this paper we are concerned with the behavior of a certain class of hyperelastic materials with non-elliptic strain energy density functions that satisfy very weak growth conditions at infinity. It is shown, by means of an example, that solutions of boundary-value problems involving these materials do not exist within the usual admissible spaces in view of the severe discontinuities involved (displacement discontinuities).
On a geophysical inviscid vortex
Authors: Granger Robert A.
Published: December 2016

We investigate the special case of an inviscid axisymmetric vortex rotating steadily in an atmosphere having an adiabatic lapse rate. We show that the analytic solution of the governing equations depends on the source strength of the vortex. The technique used is based largely on an unpublished report of Lilly [1].
A Hopf bifurcation in single-loop positive-feedback systems
Authors: Selgrade James F.
Published: December 2016

This paper gives sufficient conditions for a Hopf bifurcation in a five-dimensional system of ordinary differential equations which provides a model for positive feedback in biochemical control circuits. These conditions only depend on the feedback function and its first and second derivative. The conditions are used to exhibit Hopf bifurcations for the Griffith equations and the Tyson-Othmer equations.
Passivity and linear system stability
Authors: Venkatesh Y. V.
Published: December 2016

Using the network concept of passivity (or positive realness), new criteria for stability and instability of linear systems (with time-varying coefficients) are derived.
Spatially periodic, force-free magnetic fields with resistive decay
Authors: Meyer Rudolf X.
Published: December 2016

Force-free magnetic fields, i.e. fields that are either parallel or antiparallel to the electric current, occur in the solar chromosphere and also have been used in certain magnetic confinement schemes in controlled thermonuclear fusion research. In this paper, we derive a general expression for force-free fields that decay resistively and are spatially periodic.
A Liapunov functional for linear Volterra integro-differential equations
Authors: Abrahamson D. L. , Infante E. F.
Published: December 2016

Liapunov functionals of quadratic form have been used extensively for the study of the stability properties of linear ordinary, functional and partial differential equations. In this paper, a quadratic functional

V
V

is constructed for the linear Volterra integrodifferential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove x With dot left-parenthesis t right-parenthesis equals upper A x left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript upper T Baseline upper B left-parenthesis t minus tau right-parenthesis x left-parenthesis tau right-parenthesis d t comma t greater-than-or-equal-to t 0 comma x left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis comma 0 less-than-or-equal-to t less-than-or-equal-to t 0">

x
˙

(
t
)

=
A
x

(
t
)

+

∫
0
T

B

(

t
−
τ

)

x

(
τ
)

d
t
,

t
≥

t
0

,

x

(
t
)

=
f

(
t
)

,

0
≤
t
≤

t
0

\dot x\left ( t \right ) = Ax\left ( t \right ) + \int _0^T {B\left ( {t - \tau } \right )x\left ( \tau \right ) dt, \qquad t \ge {t_0}, \\ x\left ( t \right ) = f\left ( t \right ), \qquad 0 \le t \le {t_0}}

\]
. This functional, and its derivative

V
˙

\dot V

, is more general than previously constructed ones and still retains desirable computational qualities; moreover, it represents a natural generalization of the Liapunov function for ordinary differential equations. The method of construction used suggests functionals which are useful for more general equations.
A boundary integral equation for the solution of a class of problems in anisotropic inhomogeneous thermostatics and elastostatics
Authors: Clements David L. , Rogers C.
Published: December 2016

The solution of an important class of boundary-value problems in anisotropic inhomogeneous thermostatics and elastostatics is obtained in terms of a boundary integral equation. The equation may be used as a basis for the numerical solution of particular boundary-value problems.
Numerical conformal mapping and analytic continuation
Authors: Bisshopp Frederic
Published: December 2016

A numerical method for determination of least-square approximations of an arbitrary complex mapping function is derived here and implemented with fast Fourier transforms (FFTs). An essential feature of the method is the factoring of a discrete Hilbert transform in a pair of Fourier transforms in order to reduce the operation count of the longest computation to

O

(

N
log
⁡
N

)

O\left ( {N\log N} \right )

. A similar factoring of the discrete Poisson integral formula allows an explicit inversion of it in

O

(

N
log
⁡
N

)

O\left ( {N\log N} \right )

operations instead of

O

(

N
3

)

O\left ( {{N^3}} \right )

(N

3

^{3}

). The resulting scheme for analytic continuation appears to be considerably more reliable than the evaluation of polynomials. Examples are treated, and APL implementations of algorithms are provided.
Near critical free surface flow past an obstacle
Authors: Cole Susan L.
Published: December 2016

This paper describes the nonlinear effects produced by arbitrarily small bumps in two-dimensional free surface flows with Fraude numbers close to 1

+

^{+}

. These effects are determined by asymptotically matching (approximate) solutions to the ideal flow equations.
On the boundary-value problem associated with a general twisted tube with a slowly varying circular section
Authors: Murray J. C.
Published: December 2016

A non-orthogonal curvilinear coordinate system is used to formulate the Dirichlet problem of potential theory associated with the interior of a general twisted tube with a slowly varying circular section. A solution scheme is presented for two cases of a tube of finite length.
Some novel infinite series of spherical Bessel functions
Authors: Vavreck Andrew N. , Thompson William
Published: December 2016

By a double application of the translational additional theorem for spherical wave functions, whereby one shifts an

n
n

th order axisymmetric wave function from some origin to another and then in turn back to the first, one obtains a mathematical identity in the form of the

n
n

th order spherical wave function equated to an infinite series containing every order spherical wave function. The coefficients of the terms in this infinite series are themselves infinite series of spherical Bessel functions of arbitrary argument. These latter series must either sum to zero or unity to satisfy the mathematical identity. Following this reasoning, a collection of infinite series involving spherical Bessel functions has been generated. Some of the low mode order results are presented.
Nonuniform motion of an edge dislocation in an anisotropic solid. II
Authors: Markenscoff X. , Ni L.
Published: December 2016

The nonuniform motion of an edge dislocation in a singularly hyperbolic cubic (or hexagonal) crystal is analyzed, both for general nonuniform motion and for a motion starting from rest with constant velocity. In the latter case the solution is obtained in closed form from which the stress behaviour near the wavefront is also derived.
On stability diagrams for damped Hill equations
Authors: Pedersen Pauli
Published: December 2016

In previous works, the Galerkin approach is shown to be most efficient for quantitative stability analysis of the solutions to Hill equations. This approach, furthermore, should be recognized to be a theoretical tool as well, which enables us in a most simple way to prove a number of theorems on decoupling, symmetry and other similarities related to the stability diagrams.
Velocity and vorticity correlations
Authors: Berger B. S.
Published: December 2016

It is shown that one point velocity correlations may be expressed as a volume integral of the product of the singular part of Green’s function and a function,

h

j
M

(

x
,
X

)

{h_{jM}}\left ( {x,X} \right )

, which satisfies Poisson’s equation and vanishes for points on the boundary. An explicit expression is found for

h

j
M

(

x
,
X

)

{h_{jM}}\left ( {x,X} \right )

. These results provide a computational method for determining the velocity correlations.