bluuebunny/crossref_metadata_2025 huggingface dataset:
Some new techniques in the dynamic programming solution of variational problems
Authors: Bellman Richard
Published: December 2016

In previous papers, it has been shown that the functional equation technique of dynamic programming may be applied to yield the numerical solution of a wide class of variational problems of the type occurring in mathematical physics, engineering, and economics.
Some integrated properties of solutions of the wave equation with non-planar boundaries
Authors: Ting Lu
Published: December 2016

Integrated properties of solutions of the wave equation with non-planar boundaries are found and applied to three dimensional supersonic flow problems and two dimensional diffraction problems.
On transfer functions and transients
Authors: Zemanian Armen H.
Published: December 2016

In the first part of this paper the concept of the positive real function is generalized so that it is applicable to transfer functions and the functions, satisfying this generalized concept, are arranged into classes. Some tests are then developed which may be used to determine whether a transfer function belongs to a particular class. It is also shown that if transfer functions have certain general forms then they will automatically be members of one of the classes. Finally, several properties of the phase functions for such system functions are developed.
On a routing problem
Authors: Bellman Richard
Published: December 2016

Given a set of

N
N

cities, with every two linked by a road, and the times required to traverse these roads, we wish to determine the path from one given city to another given city which minimizes the travel time. The times are not directly proportional to the distances due to varying quality of roads and varying quantities of traffic.
On finite expansion of a hole in a thin infinite plate
Authors: Hodge P. G. , Sankaranarayanan R.
Published: December 2016

A complete solution for all values of applied pressure is obtained for the expansion of a hole of zero radius in an initially uniform infinite sheet. The analysis is compared with those of previous investigators and found to be simpler and/or more complete. It is shown that the results are applicable to the expansion of a finite hole in a uniform sheet and to the expansion of a hole in a tapered sheet.
On the heat transfer to constant-property laminar boundary layer with power-function free-stream velocity and wall-temperature distributions
Authors: Imai Isao
Published: December 2016

The heat transfer to constant-property laminar boundary layer with power-function variations of free stream velocity

(

u
1

=
c

x
m

)

({u_1} = c{x^m})

and of temperature difference between wall and free stream

(

T
0

−

T
1

=
b

x
n

)

({T_0} - {T_1} = b{x^n})

is studied by means of an improved version of the WKB method developed by the author. It is found that the local heat-transfer coefficient

h
h

can be approximately given in the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction h x slash k Over left-parenthesis u 1 x slash v right-parenthesis Superscript 1 slash 2 Baseline EndFraction equals StartFraction 1 Over left-parenthesis 2 minus beta right-parenthesis Superscript 1 slash 2 Baseline EndFraction left-bracket StartFraction normal upper Gamma left-parenthesis 2 slash 3 right-parenthesis Over 3 Superscript 2 slash 3 Baseline normal upper Gamma left-parenthesis 4 slash 3 right-parenthesis EndFraction left-brace one half plus n left-parenthesis 2 minus beta right-parenthesis right-brace Superscript 1 slash 3 Baseline left-parenthesis sigma alpha right-parenthesis Superscript 1 slash 3 Baseline minus StartFraction beta Over 10 alpha EndFraction right-bracket comma">

h
x

/

k

(

u
1

x

/

v

)

1

/

2

=

1

(

2
−
β

)

1

/

2

[

Γ

(

2

/

3

)

3

2

/

3

Γ

(

4

/

3

)

{

1
2

+
n

(

2
−
β

)

}

1

/

3

(

σ
α

)

1

/

3

−

β

10
α

]

,

\frac {{hx/k}}{{{{\left ( {{u_1}x/v} \right )}^{1/2}}}} = \frac {1}{{{{\left ( {2 - \beta } \right )}^{1/2}}}}\left [ {\frac {{\Gamma \left ( {2/3} \right )}}{{{3^{2/3}}\Gamma \left ( {4/3} \right )}}{{\left \{ {\frac {1}{2} + n\left ( {2 - \beta } \right )} \right \}}^{1/3}}{{\left ( {\sigma \alpha } \right )}^{1/3}} - \frac {\beta }{{10\alpha }}} \right ],

\]
where

β
=
2
m

/

(
m
+
1
)

\beta = 2m/(m + 1)

,

α
\alpha

is the non-dimensional velocity gradient at the wall (usually expressed as

α
=
f
(
0
)

\alpha = f(0)

),

σ
\sigma

is the Prandtl number,

k
k

is the thermal conductivity, and

v
v

is the kinematic viscosity.
The thickness of cylindrical shocks and the PLK method
Authors: Levey H. C.
Published: December 2016

Cylindrical shocks occur when a viscous heat-conducting gas flows radially in a plane. This is a singular perturbation problem in which the perturbing parameter is the reciprocal of

R
e

{R_e}

, the Reynolds number of the flow. It is shown that for both inward facing shocks (source flow) and outward facing shocks (sink flow) the shock thickness is of order

R
e

−
1

S

−
1

log
⁡

(

R
e

S
3

)

R_e^{ - 1}{S^{ - 1}}\log \left ( {{R_e}{S^3}} \right )

where

S
S

is the shock strength. This is contrary to results for sink flow which have been obtained by the use of Lighthill’s technique for rendering approximations uniformly valid—the

P
L
K

PLK

method. It is shown that this method fails when applied to singular perturbation problems of the type discussed here in which the small parameter multiplies the highest derivatives.
Eigenoscillations of an elastic cable
Authors: Laasonen Pentti
Published: December 2016

The eigenoscillations of a cable, supported at its end points, in a homogeneous gravitational field is investigated under the assumption that the bending stiffness is negligible but the behavior with respect to tension stresses is perfectly elastic. The problem involves two coupled second order equations and one independent second order equation; it is shown to be definitely self-adjoint and an iterative method for its solution is suggested.
Thermal shock in an elastic body with a spherical cavity
Authors: Sternberg Eli , Chakravorty J. G.
Published: December 2016

This investigation aims at the dynamic thermoelastic response of an infinite medium with a spherical cavity to a sudden uniform change in the temperature of its internal boundary. By means of the Laplace transform, a closed solution to this problem—exact within classical elastokinetics—is obtained in terms of error functions of real and complex arguments. The ensuing temperature stresses are compared with the corresponding quasi-static results.
Airfoil in a sonic shear flow jet: a mixed boundary value problem for the generalized Tricomi equation
Authors: Chang C. C. , Lundgren T. S.
Published: December 2016

In this paper, small perturbations of a non-uniform two-dimensional flow of a compressible inviscid fluid are considered. It is shown that for a particular class of main stream Mach number distributions, which are characterized by a sonic line along the

x
x

-axis, the linearized shear flow equation may be transformed into the generalized Tricomi equation. The mixed boundary value problem which results from considering perturbations generated by a two-dimensional camber surface is formulated and solved by utilizing the Wiener-Hopf technique.
Homogeneous solutions in elastic wave propagation
Authors: Miles John W.
Published: December 2016

Busemann’s method of conical flows is formulated for two-dimensional elastic wave propagation. The equations of motion are reduced to either Laplace’s equation in two dimensions or the wave equation in one dimension, and solutions then are obtained with the aid of complex variable or characteristics theory, respectively. Special attention is paid to that class of problems in which the hyperbolic domains (of the two-dimensional wave equation) are simple wave zones, in consequence of which the solutions may be continued into the elliptic domain (of Laplace’s equation) without explicitly posing the boundary conditions on the boundary separating the two domains. The method is applied to the diffraction of

P
P

- and

S
V

SV

-pulses by a perfectly weak half-plane.
Upper bounds and Saint-Venant’s principle in transient heat conduction
Authors: Boley Bruno A.
Published: December 2016

An investigation is carried out on transient heat-conduction problems with prescribed surface temperature, and the validity of Saint-Venant’s principle in parabolic boundary-value problems is discussed.
Duality in quadratic programming
Authors: Dorn W. S.
Published: December 2016

A proof, based on the duality theorem of linear programming, is given for a duality theorem for a class of quadratic programs. An illustrative application is made in the theory of elastic structures.
Vertex excited surface waves on one face of a right angled wedge
Authors: Karp S. N. , Karal F. C.
Published: December 2016

The problem of the propagation of electromagnetic waves by a magnetic line dipole source located at the corner of a right angled wedge is considered. It is assumed that an impedance or mixed boundary condition is prescribed on one of the wedge surfaces and that a homogeneous boundary condition is prescribed on the other. The impedance boundary condition is such that surface waves are generated. The amplitude of the surface wave generated is determined. A comparison is made between the magnitude of the surface wave for this problem and that of a magnetic-line dipole source located at the corner of a right angled wedge with the same impedance boundary condition prescribed on both* surfaces. The far field amplitude of the radiated electromagnetic field is also given as an elementary function of the angle of observation.
Dynamic programming approach to optimal inventory processes with delay in delivery
Authors: Bellman Richard
Published: December 2016

The usual dynamic programming approach to inventory processes with delays in delivery leads to functions of many variables. This multi-dimensionality prevents the straightforward utilization of digital computers.
Spin matrix exponentials and transmission matrices
Authors: Young L.
Published: December 2016

The three Pauli spin matrices

σ
i

(

i
=
1
,
2
,
3

)

{\sigma _i}\left ( {i = 1,2,3} \right )

occur in the mechanical, especially quantum mechanical, theory of rotation in three-dimensional space. The three spin matrix exponentials are here defined as exp

(

σ
i

x

)

\left ( {{\sigma _i}x} \right )

, where

x
x

is the independent variable. Transmission matrices can be expressed in terms of spin matrix exponentials, thereby permitting a more systematic treatment of transmission line circuits.
Condition-convection from a cylindrical source with increasing radius
Authors: Bailey H. R.
Published: December 2016

The problem of heat flow by conduction and convection from a cylindrical source with increasing radius is solved. A quasi stationary state solution is obtained for the case of a finite convection coefficient and with the radius increasing at a constant velocity. A transient solution is obtained for the case of an infinite convection coefficient and with the radius increasing at a rate proportional to the square root of time.
On the Sommerfeld half-plane problem
Authors: Lippmann Bernard A.
Published: December 2016

A simple derivation of the Sommerfeld solution to the problem of the diffraction of a plane, scalar wave by a half-plane is given. The discusson is of interest mainly because of the simplicity of the argument; however, it is felt that the ansatz that forms the core of the argument is probably more generally applicable.
Discontinuities in integral-transform solutions
Authors: Boley Bruno A.
Published: December 2016

Criteria are derived for the determination of the magnitude and the location of discontinuities of solutions in the form of definite integrals obtained by means of integral-transform techniques. The types of integrals arising with the Fourier sine or cosine transforms and those arising with the Laplace transforms are considered in detail. Applications of the theory arise particularly with problems of wave propagation, where interest is centered on the location of wave fronts and the magnitude of jumps there; two illustrative examples of such problems relating to Timoshenko beams are included.
Translational addition theorems for spherical vector wave functions
Authors: Cruzan Orval R.
Published: December 2016

Translational addition theorems for spherical vector wave functions have been derived in a reduced form. The reduction has been accomplished by the use of formulas relating the coefficients that arise in the expansion of the product of two associated Legendre functions. These addition theorems should be useful, in general, in those cases involving electromagnetic sources and spherical bodies.
Stability of forced oscillations of a spherical pendulum
Authors: Miles John W.
Published: December 2016

The equations of motion for a lightly damped spherical pendulum that is subjected to harmonic excitation in a plane are approximated in the neighborhood of resonance by discarding terms of higher than the third order in the amplitude of motion. Steady-state solutions are sought in a four-dimensional phase space. It is found that: (a) planar harmonic motion is unstable over a major portion of the resonant peak, (b) non-planar harmonic motion is stable in a spectral neighborhood above resonance that overlaps neighborhoods of both stable and unstable planar motions, and (c) no stable, harmonic motions are possible in a finite neighborhood of the natural frequency. The spectral width of these neighborhoods is proportional to the two-thirds power of the amplitude of excitation. The steady-state motion in the last neighborhood is quasi-sinusoidal (at the forcing frequency) with slowly varying amplitude and phase. The waveform, as determined by an analog computer, is periodic but quite complex.
A theory for the square guarded hot plate—a solution of the heat conduction equation for a two layer system
Authors: Donaldson I. G.
Published: December 2016

A theoretical treatment is given for the action of the guarded hot plate apparatus used for measuring conductivities. The flow of heat through the metal hot plate itself, and the effects of temperature unbalance between hot plate and guard plate are both taken into account. The temperature at any point in the sample or in the heater plate may be determined. The theory is comprehensive and any of the dimensions or controlling parameters may be varied in the calculation.
On uniqueness in linear viscoelasticity
Authors: Breuer S. , Onat E. T.
Published: December 2016

It is shown that solutions of a class of boundary value problems in linear vicoelasticity are unique, if the relaxation moduli in shear and compression are steadily decreasing functions of time which are convex from below and tend to non-negative constant asymptotic values.
The small-field theory of the Joule and Wiedemann effects
Authors: Lewis J. A.
Published: December 2016

The simplest stress-strain-field relations valid for an isotropic, magnetostrictive medium are developed and applied to calculate the deformations of a wire in the case of the Joule effect (longitudinal magnetic field) and Wiedemann effect (helical magnetic field).
Remarks on the inertia instability of a rolling missile
Authors: Tideman Magnus
Published: December 2016

Lyapunov’s second method is applied to the question of stability when dynamic cross coupling is considered. The main result is a condition on the maximal non-symmetry that gives stable performance for any constant or non-constant angular velocity in roll. Methods are outlined for the treatment of some related questions.
An optimization of the phase-plane-delta method for the solution of non-linear differential equations
Authors: Ludeke Carl A. , Weber Richard R.
Published: December 2016

An improvement to the phase-plane-data method of solving non-linear differential equations of the type

d
2

x

/

d

t
2

+
H

(
x
)

=
0

{d^2}x/d{t^2} + H\left ( x \right ) = 0

is discussed. This improvement provides a means of determining an optimized value of the parameter

p
p

, frequency. In the conventional phase-plane-delta method the parameter

p
p

is chosen either arbitrarily or as the coefficient of the positive linear term. The phase-plane trajectory and period of oscillation can be more readily determined by this method than by the conventional phase-plane-delta method.
An approximate method in signal detection
Authors: Freiberger Walter F.
Published: December 2016

A theorem from the theory of Toeplitz forms (ref. 1) is applied to the problem of estimating the best test statistic for the detection of Gaussian signals in Gaussian noise.
Note on Green’s function in anisotropic elasticity
Authors: Gutzwiller Martin C.
Published: December 2016

Green’s function for an elastic, anisotropic medium is constructed with the help of the method used by Courant and Hilbert in the study of light propagation through an anisotropic dielectric. The amplitudes of the arrivals corresponding to each wave surface are expressed as functions of the curvature of the normal surfaces. Multi-valued solutions are expressed as integrals over the dislocation lines, whose motion is assumed to be known, but otherwise arbitrary. The formulae are worked out in detail for the isotropic medium.
Duality in nonlinear programming
Authors: Mangasarian O. L.
Published: December 2016

The main result of this work is a converse theorem to a duality theorem for nonlinear programming recently established by Wolfe. The conditions on the present theorem are slightly stronger than those imposed by Wolfe. Hanson has given a different proof of a similar theorem but without stating in the theorem some important assumptions made in the proof.†
On radial flow of a conducting gas in crossed fields
Authors: Dahlberg Erling
Published: December 2016

An earlier treatment of quasi-onedimensional flow of a conducting gas in crossed fields (Q.A.M., 19, 177 (1961)) is extended to radial flow with azimuthal magnetic field. The usual simplifying assumptions are made, including constant specific heats; and different flow situations are illustrated by means of phase diagrams (drawn for

γ
=
1.33

\gamma = 1.33

). One of the differences between radial flow (

E
E

= const.) and that in exponentially diverging channels (

E

/

B

E/B

= const.) is the possibility of a smooth transition from supersonic to subsonic motion that seems to exist for a certain range of strongly divergent radial channels.
On some questions arising in the approximate solution of nonlinear differential equations
Authors: Bellman R. , Richardson J. M.
Published: December 2016

A new approach to the approximate solution of nonlinear differential equations is explored. The basic idea is to rewrite the nonlinear equations in the form of an infinite sequence of coupled linear equations by application of the Carleman linearization process. The sequence is truncated at a finite stage by a linear closure approximation involving the minimization of the mean square error. Attention is given to the stability of the truncated sequence of linear equations with respect to propagation of error due to closure back to the earlier members of the sequence. The use of suitably defined orthogonal polynomials to simplify closure approximations is considered. The generalization of the general method to the multidimensional case is treated. Consideration is given to the concept of self-consistent closure methods in which the averaging of the squared closure error depends upon the approximate linear equations derived thereby. A specific example of the last is treated analytically in closed form and a numerical comparison is made with the exact solution.
Statistical properties of low-density traffic
Authors: Weiss George , Herman Robert
Published: December 2016

This paper considers an infinitely long line of traffic moving on a highway without traffic lights or other inhomogeneities. It is assumed that each car travels at a constant speed which is a random variable. A further assumption is that when one car overtakes another, passing is always possible and occurs without change of speed. it is shown that any initial headway distribution must relax to a negative exponential distribution in the limit of

t
t

becoming infinite. The statistics of passing events are examined, and it is shown that the probability of passing (or being passed by)

n
n

cars In time

t
t

is described by a Poisson distribution.
Nonlinear theories for thin shells
Authors: Sanders J. Lyell
Published: December 2016

Strain-displacement relations for thin shells valid for large displacements are derived. With these as a starting point approximate strain-displacement relations and equilibrium equations are derived by making certain simplifying assumptions. In particular the middle surface strains are assumed small and the rotations are assumed moderately small. The resulting equations are suitable as a basis for stability investigations or other problems in which the effects of deformation on equilibrium cannot be ignored, but in which the rotations are not too large.
Upper and lower bounds for the solution of a melting problem
Authors: Boley Bruno A.
Published: December 2016

A slab, melting under an arbitrary heat input on one face and insulated on the other is studied. Variable material properties are taken into account. After preliminary general considerations, it is shown that the solution to the stated problem is unique. It is then proved that higher rates of melting and higher temperatures will result from higher heating rates; from this intuitively plausible result upper and lower bounds for the solution are easily constructed. An example is worked out in detail.
An extension of Poincaré’s continuity theorem
Authors: Haseltine William R.
Published: December 2016

A theorem is derived which is useful in establishing the existence of periodic solutions of systems of nonlinear ordinary differential equations involving a small parameter in certain cases where first-order methods break down.
Camouflaging electrical 1-networks as graphs
Authors: Kron Gabriel
Published: December 2016

Lately it has become fashionable to develop the theory of electrical networks by starting first with a detailed, scholarly exposition of the theory of graphs (containing both nodes and branches), as given in textbooks on algebraic topology. However, when the graph analysis finally gets down to the study of actual electrically excited networks, it will invariably be found that all traces of the concepts of nodes and their associated “incidence-matrices” have disappeared from the final

e
=
z
i

e = zi

and other equations of state that are to be solved. In their place only the branches and the associated “connection-matrices” have been put to use, all still wrapped up however in an ill-fitting graph terminology. Of course such legerdemain had to be resorted to, since graph theorists happened to pick the wrong topological model for an electrical network. A graph possesses enough structure to propagate an electromagnetic wave and not an electrical current.
Singular cases in the optimum design of frames
Authors: Megarefs George J. , Hodge Philip G.
Published: December 2016

Many previous studies in optimum design have determined the parameters so as to set equal to zero the first variation of the property to be optimized. The present paper shows that in some simple cases the minimum value may not be a point of zero variation and that points of zero variation may be relative maxima as well as minima. A general theoretical explanation of such behavior is given and applications are made to simple frames.
On the solution of a non-linear parabolic equation with a floating boundary arising in a problem of plastic impact of a beam
Authors: Ting Thomas C. T.
Published: December 2016

The deformation of a cantilever beam with strain rate sensitivity subjected to impact loading at its base has been studied in [11] by an approximate method in which the inertia forces in the plastic region are neglected. If these forces are taken into account, the equation of motion in the plastic region is a fourth order non-linear parabolic differential equation with a floating boundary, i.e. one whose position varies with time and must be found as part of the solution. A numerical solution of this equation is presented here. The results show that the bending moment in the plastic region varies nearly linearly. This result implies that the shear force is nearly constant in the plastic region, and hence that the inertia forces in the plastic region are small in comparison with the shear force in the same region.
Lommel functions with imaginary argument
Authors: Rollinger Charles N.
Published: December 2016

To express the solution to certain physical problems in terms of a real function of a real variable a new function called a modified Lommel Function is defined. Some of its properties are presented along with some of the relations between this new function and modified Bessel functions.
The field of a dipole above an infinite corrugated plane
Authors: Chu T. S. , Karp S. N.
Published: December 2016

We investigate the excitation and propagation of the three dimensional electromagnetic field over an infinite corrugated plane which is approximated by an anisotropic impedance boundary condition. Emphasis is placed upon effects of surface anisotropy which are not evident in two dimensional treatments. In particular we consider the excitation by a magnetic point dipole in detail. It turns out that the fields are determined by a scalar wave function which satisfies a mixed boundary condition involving a linear combination of the wave function, its normal derivative and its second order tangential derivative. The exact formal solution is first derived, and then the radiated far field and the surface wave far field are evaluated separately. Both the phase and the amplitude of the excited surface wave are dependent upon the direction of observation. Numerical results are given. The physical significance of this solution is discussed. A comparison is made between this problem and the theory of ship waves.
On the stress distribution in anisotropic infinite wedges
Authors: Benthem J. P.
Published: December 2016

The differential equation for Airy’s stress function is given in oblique coordinates. The stresses are written as sums of four functions of complex variables. To these sums and to the boundary conditions the Mellin transform is applied. The inverse transform delivers the formal solution.
On the damping of a satellite motion
Authors: Morrison J. A.
Published: December 2016

The motion in a circular orbit of a gravitationally oriented satellite, whose angular motions about the local vertical are damped by a roll-vee, gyrostabilizer system, is considered, wherein the pitch axis of the satellite remains perpendicular to the orbital plane. It is shown that no matter how large the initial local angular velocity of the satellite is, this velocity reaches any given smaller value in a finite time. It is also shown how it is possible to obtain a bound on this damping time.
A wedge-flow approach to lubrication theory
Authors: Langlois W. E.
Published: December 2016

A theory of fluid-film lubrication is developed from an assumption of local wedge-flow, rather than local parallel-channel flow. This leads to a generalization of the Reynolds lubrication equation governing the pressure. A refinement of lubrication theory does not necessarily follow: unlike the parallel-channel case, the stress field may differ significantly from an isotropic pressure, so that imposing ambient conditions on the bearing periphery does not always yield boundary conditions for the pressure equation. If the bearing slope differs appreciably from zero only in the film interior, consistent boundary conditions are once more available.
The use of the perturbation method for determining the pumping frequencies of parametric amplifiers
Authors: Ludeke Carl A.
Published: December 2016

This paper demonstrates that an old but very powerful and very general mathematical technique, the perturbation method, may be used to investigate such a modern application as parametric amplification. All possible pumping frequencies are predicted without recourse to any specialized procedures.
Non-separable solutions of the Helmholtz wave equation
Authors: Moseley Donald S.
Published: December 2016

A set of solutions not obtainable by the method of separation of variables is presented for the vector Helmholtz wave equation in circular cylindrical coordinates limited to non-angular dependence. These are constructed of Bessel and trigonometric functions. For example, if A is the vector, the

r
r

-component of the simplest member of the set is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript r Baseline equals upper C 1 left-bracket m r upper J 0 left-parenthesis p r right-parenthesis cosine left-parenthesis m z right-parenthesis plus p z upper J 1 left-parenthesis p r right-parenthesis sine left-parenthesis m z right-parenthesis right-bracket e Superscript minus i w t Baseline comma">

A
r

=

C
1

[

m
r

J
0

(

p
r

)

cos
⁡

(

m
z

)

+
p
z

J
1

(

p
r

)

sin
⁡

(

m
z

)

]

e

−
i
w
t

,

{A_r} = {C_1}\left [ {mr{J_0}\left ( {pr} \right )\cos \left ( {mz} \right ) + pz{J_1}\left ( {pr} \right )\sin \left ( {mz} \right )} \right ]{e^{ - iwt}},

\]
where

C
1

{C_1}

is an arbitrary constant,

m
m

and

p
p

are propagation constants, and

ω
\omega

is angular frequency. Brief reference is made to three-dimensional solutions in rectangular coordinates.
Radiative transfer effects in natural convection above fires—opaque approximation
Authors: Murgai M. P. , Varma R. K.
Published: December 2016

In continuation of a previous paper on a similar subject, we examine here, in detail, the opaque approximation of the radiative transfer equation by retaining the vertical diffusivity term within the boundary layer theory. The results seem to indicate the presence of a layer very near the horizontal boundary of the plume where the radiation effects are dominant. The thickness of this layer tends to decrease with increasing absorption coefficient.
Conditions for the causality of nonlinear operators defined on a function space
Authors: Sandberg I. W.
Published: December 2016

This note considers nonlinear operators

T
T

with both range and domain subsets of the space of complex

n
n

-vector-valued functions of the real variable

t
t

for

−
∞
>
t
>
∞

- \infty > t > \infty

. Conditions, in which energy-type quantities play a key role, are presented under which

T
T

is causal in the following sense:
Difference methods for a nonlinear elliptic system of partial differential equations
Authors: McAllister G. T.
Published: December 2016

This paper proves the existence and uniqueness of non-negative solutions to the Dirichlet problem associated with the nonlinear elliptic system
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta u Subscript k Baseline equals b Subscript k Baseline product Underscript l equals 1 Overscript m Endscripts u Subscript l Superscript n left-parenthesis l right-parenthesis Baseline comma k equals 1 comma period period period comma m left-parenthesis asterisk right-parenthesis">

Δ

u
k

=

b
k

∏

l
=
1

m

u
l

n

(
l
)

,
k
=
1
,
.
.
.
,
m

(
∗
)

\Delta {u_k} = {b_k}\prod \limits _{l = 1}^m {u_l^{n\left ( l \right )}} ,k = 1,...,m\left ( * \right )

\]
where the

b
k

{b_k}

and the Dirichlet data

u
k

=

φ
k

{u_k} = {\varphi _k}

are non-negative.
Variational principles in anisotropic and nonhomogeneous elastokinetics
Authors: Ben-Amoz M.
Published: December 2016

Variational principles are formulated in terms of polarization stresses and strains in anisotropic and nonhomogeneous elastokinetics. The principles correspond to the well-known principles of Reissner and Washizu and are shown to produce, under suitable subsidiary conditions, functional which are a generalization to elastokinetics of the Hashin-Shtrikman functional in elastostatics.
A deformation bounding theorem for flow-law plasticity
Authors: Hodge Philip G.
Published: December 2016

It is shown that the elastic principle of minimum complementary energy can be generalized to provide a bounding theorem for an elastic/perfectly-plastic material. For certain special classes of problems the bounding principle becomes a true minimum principle.
On asymptotic stability of nonlinear hereditary phenomena
Authors: Distéfano J. N. , Sackman J. L.
Published: December 2016

Conditions for long-term stability of nonlinear hereditary phenomena are derived from two Tauberian theorems. When the hereditary phenomena are time invariable (i.e., of the closed cycle type), then the asymptotic limits of the phenomena are evaluated. An application is made to the investigation of the stability of a rigid column restrained by a nonlinear viscoelastic element.
Uniform asymptotic solutions of discontinuous initial value problems for dispersive hyperbolic equations
Authors: Boersma J.
Published: December 2016

This paper is concerned with a discontinuous initial value problem for a dispersive hyperbolic equation where the dispersive term of the equation contains a large parameter.
New analytic solutions of the one-dimensional heat equation for temperature and heat flow rate both prescribed at the same fixed boundary (with applications to the phase change problem)
Authors: Langford David
Published: December 2016

New solutions of the heat equation are exhibited for the case in which both the temperature and heat flow rate are prescribed at a single fixed boundary. The prescribed temperature and heat flow rate may be any arbitrary infinitely differentiable functions of time. The new solutions are applicable for one-dimensional (radial) heat flow in spheres, cylinders, and slabs.
On certain boundary-value problems with strong vanishing on the boundary
Authors: McCrea J. , Synge J. L.
Published: December 2016

In Newtonian statics of a continuum we have a symmetric stress tensor with six components and a body-force with three components, and the divergence of the stress tensor equals the body-force vector reversed. If the body-force vector as assigned in some finite domain

I
I

with boundary

B
B

, we have three equations to be satisfied by six stress components. The equations of equilibrium, coupled with conditions on

B
B

, cannot determine the stress, but they do define a class of stress distributions, provided the body-force and the conditions on

B
B

are consistent. The purpose of this paper is to show that, if the body-force satisfies the usual conditions of equilibrium and vanishes strongly on

B
B

in the sense that this body-force, and all its derivatives up to order

N
N

, vanish on

B
B

, then there exists a stress distribution which also vanishes strongly on

B
B

, the order of vanishing being greater by one than that of the body-force.
A difference method for plane problems in dymanic elasticity
Authors: Clifton R. J.
Published: December 2016

The equations governing dynamic elastic deformation under conditions of plane strain are written as a system of symmetric, hyperbolic, first order, partial differential equations with constant coefficients. A system of explicit difference equations with second order accuracy are presented for the solution of mixed initial and boundary value problems for regions composed of rectangles. At interior points the difference scheme is the same as the scheme proposed by Lax and Wendroff. They established the schemes conditional stability for initial value problems. At boundary points Butler’s procedure based on integration along bicharacteristics is used to derive the appropriate difference equations. The method is applied to a problem for which the exact solution is known. Numerical evidence indicates that the method is stable for a mesh ratio

k

/

h

k/h

approximately twice as large as required by the Lax—Wendroff condition. The error in total energy is

O

(

k
3

)

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

and increases linearly with increasing

t
t

.
On the construction of Hermitian from Lagrangian difference approximations
Authors: Tanrikulu Mahmut , Prager William
Published: December 2016

It is shown how simple finite difference approximations to the Laplace operator in two- or three-dimensions can be combined to construct Hermitian finite difference approximations to the two- or three-dimensional Poisson equation

Δ
u
=
f

\Delta u = f

.
On some mathematical aspects of kinetic model equations
Authors: Sirovich Lawrence , Thurber James K.
Published: December 2016

A class of integro-differential equations used in the kinetic theory of gases is investigated. The existence and uniqueness of solutions to the initial value problem associated with these equations is proved under relatively weak conditions. Under somewhat stronger assumptions the validity of the application of Fourier—Laplace transforms to this problem is demonstrated. Finally the asymptotic approach to hydrodynamics is shown.
Approximations with rational polynomials
Authors: Kovach Ladis D. , Larson Ronald V.
Published: December 2016

One of the cornerstones of mathematical analysis is the use of Maclaurin’s series for the representation of transcendental functions. The advent of the electronic computer, however, has caused attention to be focussed on truncated, series. A number of techniques have been developed with the object of finding the “best” approximation to a transcendental function. The present paper describes a new technique by means of which such functions can be represented with a minimum number of terms.
On nonlinear steady-state solutions to moving load problems
Authors: Hegemier G. A.
Published: December 2016

In certain moving load problems involving elastic solids or structures with an idealized infinite or semi-infinite spatial domain, steady-state solutions of the governing equations of motion are frequently sought as an alternative to a more complex transient analysis. In seeking a solution of this type the implication is, of course, that it represents the limit of a transient problem. While this appears to be generally true of well-posed linear elastic problems, the same cannot be said of nonlinear elastic formulations. To substantiate this point the case of an infinite-length string supported by a nonlinear elastic foundation and subjected to a concentrated load moving with constant velocity is studied in this paper. It is shown that although the steady-state solution of the problem is unique for load velocities both above and below the signal velocity of the string, that above is locally unstable and therefore cannot represent the limit of transient motions.
Variational principles for linear coupled thermoelasticity
Authors: Nickell R. E. , Sackman J. L.
Published: December 2016

Several variational principles are derived for the initial-boundary-value problem of fully coupled linear thermoelasticity for an inhomogeneous, anisotropic continuum. A consistent set of field variables is employed and a method based on the Laplace transform is used to incorporate the initial conditions explicitly into the formulation. These principles lend themselves readily to numerical solutions based on an extended Ritz method.
On some inverse problems in potential theory
Authors: Klamkin M. S. , Newman D. J.
Published: December 2016

It is a well known result that the inverse square law has the property that the attraction, at an external point, due to a sphere of uniform density is the same as if the sphere was concentrated at its center. The most general law of attraction having this property for all spheres among the class of central force laws is known to be the inverse square plus linear. For the corresponding internal problem, it is well known that there is no attractive force acting on a particle inside a uniform spherical shell and, inversely, this property characterizes the inverse square law. We establish the first inverse result under the weaker condition that the desired property holds for just two thin spherical shells of any radii. For the corresponding internal problem, we need three incommensurable radii.
The completeness of Biot’s solution of the coupled thermoelastic problem
Authors: Verruijt A.
Published: December 2016

In one of his papers on the theory of thermoelasticity, M. A. Biot [2] has presented a solution to the differential equations very similar to the well-known Boussinesq—Papcovitch solution in the theory of elasticity. In this note it is proved that this solution is complete, the proof being based upon Mindlin’s theorem of completeness of the Boussinesq—Papcovitch solution in elasticity.
A note on a solution of Possio’s integral equation for an oscillating airfoil in subsonic flow
Authors: Miles John W.
Published: December 2016

Following a procedure analogous to that used by Schwarz for solving the integral equation for an oscillating airfoil in incompressible flow, a method of attack on the same problem for subsonic, compressible flow is outlined. The circulation and kernel to the circulation integral equations are expanded in series with the coefficients

M

2
n

{M^{2n}}

and

M

2
n

{M^{2n}}

ln

M
M

, and a corresponding expansion for the Green’s function is thereby implicitly determined. The solution up to (but not including) terms of

O

(

M
4

,

M
4

ln
⁡
M

)

O\left ( {{M^4},{M^4}\ln M} \right )

is stated. A method of iteration for improving a given solution is also indicated.
Currents on the surface of an infinite cylinder excited by an axial slot
Authors: Papas C. H. , King Ronold
Published: December 2016

The distribution of surface-current density is determined on an axially infinite, perfectly conducting cylinder. A constant distribution of electric field across a narrow axial slot is assumed given.
A note on Riemann’s method applied to the diffusion equation
Authors: Miles John W.
Published: December 2016

The method of Riemann (or Green) is applied to the integration of the parabolic equation in

m

{m}

dimensions and two variables. The result is applied to the two phase diffusion problem. The reduction to tractable equations is not simple, however, and the results of the investigation appear to be essentially negative.
On torsion of prisms with longitudinal holes
Authors: Ling Chih-Bing
Published: December 2016

This paper presents a method of solution, called the method of images, for the torsion of prisms having one or more longitudinal holes. The method is applicable to prisms of the following four, and only four, sections: a rectangle, an equilateral triangle, an isosceles triangle and a

30
∘

−

60
∘

−

90
∘

{30^ \circ } - {60^ \circ } - {90^ \circ }

triangle. These four sections form a group by themselves. The solution is obtained by adding to the known solution of a corresponding solid prism without holes a system of harmonic functions which vanish on the entire external boundary of the given section, and besides possess a singularity at the centre of each hole. Such a system of functions may be constructed from Weierstrass’ Sigma function and its allied functions. The solution is illustrated by applying it in detail to a rectangular prism having a central longitudinal hole. Numerical results are shown for the special case of a square prism.
A method of variation for flow problems. II
Authors: Manwell A. R.
Published: December 2016

The method of variation of reference [1] is developed afresh in a slightly different manner which enables the main principle used in [1] to be derived directly and also makes the actual calculations much simpler. It is shown how a variety of problems concerning aerofoils possessing minimal properties may be reduced to the solution of integro-differential equations which determine the mapping of the aerofoil onto a circular region. It is briefly indicated how the method may be extended to three dimensional flows.
On double-pulse stability criteria with damping
Authors: Morduchow M. , Galowin L.
Published: December 2016

A simple necessary condition for the stability of a linear dynamic system with elastic and damping characteristics which vary periodically with the same period is derived. General explicit necessary and sufficient conditions for stability are then developed for a double-pulse system. Such a system can be characterized by a pair of eigenvalues, or complex frequencies, corresponding to each half-period, and the stability of this system depends only on these complex frequencies. It is shown that a necessary, though not sufficient, condition for the stability of any such system is that the arithmetic mean of the real parts of all four of the complex frequencies over an entire period be negative or zero. This is shown to be true, more generally, for an

n

\mathcal {n}

-pulse system. The physical significance of the results is discussed, and numerical examples are given.
Torsion of a circular cylinder having a spherical cavity
Authors: Ling Chih-Bing
Published: December 2016

This paper presents a torsion solution, based on the Michell-Föppl theory, for an infinite circular cylinder having a symmetrically-located spherical cavity. Numerical values are also given to show, in particular, the effect of the cavity on the maximum shear stress in the cylinder.
Mixed boundary value problems in soil mechanics
Authors: Shield R. T.
Published: December 2016

The stress-strain law for an ideal soil formulated in a recent paper [1]

3

^{3}

is applied here to obtain the velocity equations referred to the stress characteristic lines in plane strain problems. Simple velocity fields associated with families of straight characteristic lines are then examined, together with discontinuities in the velocity field. The results are applied to obtain the incipient velocity field for the indentation of a semi-infinite mass of material by a flat punch or footing, and to solve the problem of indentation by a lubricated wedge.
A method of solution of the equations of classical gas-dynamics using Einstein’s equations
Authors: McVittie G. C.
Published: December 2016

It is known that Einstein’s equations in general relativity provide explicit expressions for the density, pressure and velocity of a perfect gas in terms of the coefficients of the metric (the potentials) and hence in terms of the coordinates. Using orthogonal space-times, the expressions involve four potentials only between which consistency relations hold. It is shown how degeneration of the Einstein equations to Newtonian hydrodynamics provides general solutions of the equations of classical gas-dynamics for motions which may be either of constant or of variable entropy. The consistency relations are obtained in the general case. As an illustration, one-dimensional gas-dynamics are discussed and it is shown how the consistency relations are manipulated. The solution in which one or other of the Riemann variables is constant is obtained as a special case and motions of variable entropy are also attained.
On the expansion of functions in terms of their moments
Authors: Green H. S.
Published: December 2016

A general method is devised for the reconstruction of functions of a continuous variable from their moments. An analogue is given for functions of a discrete variable. An application is given to the solution of partial differential equations with a given initial condition.
Plastic flow in a deeply notched bar with semi-circular root
Authors: Wang Alexander J.
Published: December 2016

The unsteady motion problem of a circular-notched bar pulled in tension in plane strain is considered. The theory of perfectly plastic solids is used. Large strains are analyzed so that the material can also be considered as plastic-rigid. The basic equations governing stress and velocity are integrated independently in the characteristic plane. The results are used to construct the boundary change in a step-by-step manner. The problem is greatly simplified because at each step the new free boundary of the plastic region can be approximated by a circle. The final shape of the boundary of an initially semi-circular notch is presented when plastic flow has reduced the initial connection at the root to a line contact between the shanks.