bluuebunny/crossref_metadata_2025 huggingface dataset:
Traveling wave solutions of a gradient system: solutions with a prescribed winding number. II
Authors: Terman David
Published: June 2010

This paper completes the analysis begun in [2] concerning the existence of traveling wave solutions of a system of the form

u
t

=

u

x
x

+
∇
F
(
u
)

{u_t} = {u_{xx}} + \nabla F(u)

,

u
∈

R

2

u \in {{\mathbf {R}}^2}

. In [2] a notion of winding number for solutions was defined, and the proof that there exists a traveling wave solution with a prescribed winding number was reduced to a purely algebraic problem. In this paper the algebraic problem is solved.
Equivalence and strong equivalence of actions on handlebodies
Authors: Kalliongis John , Miller Andy
Published: June 2010

An algebraic characterization is given for the equivalence and strong equivalence classes of finite group actions on

3
3

-dimensional handlebodies. As one application it is shown that each handlebody whose genus is bigger than one admits only finitely many finite group actions up to equivalence. In another direction, the algebraic characterization is used as a basis for deriving an explicit combinatorial description of the equivalence and strong equivalence classes of the cyclic group actions of prime order on handlebodies with genus larger than one. This combinatorial description is used to give a complete closed-formula enumeration of the prime order cyclic group actions on such handlebodies.
Local 𝐻-maps of 𝐵𝑈 and applications to smoothing theory
Authors: Lance Timothy
Published: June 2010

When localized at an odd prime

p
p

, the classifying space

P
L

/

O

PL/O

for smoothing theory splits as an infinite loop space into the product

C
×
N

C \times N

where

C
=

Cokernel

(
J
)

C = {\text {Cokernel}}\,(J)

and

N
N

is the fiber of a

p
p

-local

H
H

-map

B
U
→
B
U

BU \to BU

. This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their

k
k

-invariants to properties of the defining self-maps of

B
U

BU

. If

Y
Y

is a smooth manifold, the set of homotopy classes

[
Y
,

N
]

[Y,\,N]

is a certain subgroup of resmoothings of

Y
Y

, and the

k
k

-invariants of

N
N

generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of

Y
Y

and frequently vanish.
Stochastic perturbations to conservative dynamical systems on the plane. II. Recurrency conditions
Authors: Wolansky G.
Published: June 2010

We consider a conservative system on the plane, subjected to a perturbation. The above perturbation is composed of a deterministic part and a random (white noise) part. We discuss the conditions under which there exists a unique, finite invariant measure to the perturbed system, and the weak compactness of the above measures for small enough perturbation’s parameter.
A canonical subspace of 𝐻*(𝐵𝑂) and its application to bordism
Authors: Pomerance Errol
Published: June 2010

A particularly nice canonical subspace of

H

∗

(
B
O
)

{H^{\ast }}(BO)

is defined. The bordism class of a map

f
:
X
→
Y

f:X \to Y

, where

X
X

and

Y
Y

are compact, closed manifolds, can be determined by the characteristic numbers corresponding to elements of this subspace, and these numbers can be easily calculated. As an application, we study the "fixed-point manifold" of a parameter family of self-maps

F
:
M
×
X
→
X

F:M \times X \to X

, thus refining to bordism the usual homological analysis of the diagonal which is the basis of the standard Lefschetz fixed point theorem.
Decompositions of continua over the hyperbolic plane
Authors: Rogers James T.
Published: June 2010

The following theorem is proved. Theorem. Let

X
X

be a homogeneous continuum such that

H
1

(
X
)
≠
0

{H^1}(X) \ne 0

. If

G

\mathcal {G}

is the collection of maximal terminal proper subcontinua of

X
X

, then (1) The collection

G

\mathcal {G}

is a monotone, continuous, terminal decomposition of

X
X

, (2) The nondegenerate elements of

G

\mathcal {G}

are mutually homeomorphic, indecomposable, cell-like, terminal, homogeneous continua of the same dimension as

X
X

, (3) The quotient space is a homogeneous continuum, and (4) The quotient space does not contain any proper, nondegenerate, terminal subcontinuum. This theorem is related to the Jones’ Aposyndetic Decomposition Theorem. The proof involves the hyperbolic plane and a subset of the circle at

∞
\infty

, called the set of ends of a component of the universal cover of

X
X

.
Nonconvex variational problems with general singular perturbations
Authors: Owen Nicholas C.
Published: June 2010

We study the effect of a general singular perturbation on a nonconvex variational problem with infinitely many solutions. Using a scaling argument and the theory of

Γ
\Gamma

-convergence of nonlinear functionals, we show that if the solutions of the perturbed problem converge in

L
1

{L^1}

as the perturbation parameter goes to zero, then the limit function satisfies a classical minimal surface problem.
Mappings of trees and the fixed point property
Authors: Marsh M. M.
Published: June 2010

We investigate weakly confluent, universal, and related mappings of trees and their relationships to the fixed point property for tree-like continua. This investigation leads to some new results, to generalizations of some known results, and to a partial solution of a question of H. Cook.
Some explicit cases of the Selberg trace formula for vector valued functions
Authors: Stopple Jeffrey
Published: June 2010

The trace formula for

S
L
(
2
,

Z

)

SL(2,{\mathbf {Z}})

can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation

π
\pi

. This paper makes this version explicit for the class of representations which can be realized as representations of the finite group

P
S
L
(
2
,

Z

/

q
)

PSL(2,{\mathbf {Z}}/q)

for some prime

q
q

. The body of the paper is devoted to computing, for the singular representations

π
\pi

, the determinant of the scattering matrix

Φ
(
s
,
π
)

\Phi (s,\pi )

on which the applications depend. The first application is a version of the Roelcke-Selberg conjecture. This follows from known results once the scattering matrix is given. The study of representations of

S
L
(
2
,

Z

)

SL(2,{\mathbf {Z}})

in finite-dimensional vector spaces of (scalar-valued) holomorphic forms dates back to Hecke. Similar problems can be studied for vector spaces of Maass wave forms, with fixed level

q
q

and eigenvalue

λ
\lambda

. One would like to decompose the natural representation of

S
L
(
2
,

Z

)

SL(2,{\mathbf {Z}})

in this space, and count the multiplicities of its irreducible components. The eigenvalue estimate obtained for vector-valued forms is equivalent to an asymptotic count, as

λ
→
∞

\lambda \to \infty

, of these multiplicities.
Extendible sets in Peano arithmetic
Authors: Smith Stuart T.
Published: June 2010

Let

A

\mathcal {A}

be a structure and let

U
U

be a subset of

|

A

|

|\mathcal {A}|

. We say

U
U

is extendible if whenever

B

\mathcal {B}

is an elementary extension of

A

\mathcal {A}

, there is a

V
⊆

|

B

|

V \subseteq |\mathcal {B}|

such that

(

A

,
U
)
≺
(

B

,
V
)

(\mathcal {A},U) \prec (\mathcal {B},V)

. Our main results are: If

M

\mathcal {M}

is a countable model of Peano arithmetic and

U
U

is a subset of

|

M

|

|\mathcal {M}|

, then

U
U

is extendible iff

U
U

is parametrically definable in

M

\mathcal {M}

. Also, the cofinally extendible subsets of

|

M

|

|\mathcal {M}|

are exactly the inductive subsets of

|

M

|

|\mathcal {M}|

. The end extendible subsets of

|

M

|

|\mathcal {M}|

are not completely characterized, but we show that if

N

\mathcal {N}

is a model of Peano arithmetic of arbitrary cardinality and

U
U

is any bounded subset of

N

\mathcal {N}

, then

U
U

is end extendible.
On 𝐻_{*}(Ωⁿ⁺²𝑆ⁿ⁺¹;𝐹₂)
Authors: Hunter Thomas J.
Published: June 2010

In this paper, we study

H
∗

Ω

n
+
2

S

n
+
1

{H_\ast }{\Omega ^{n + 2}}{S^{n + 1}}

. Here

Ω
X

\Omega X

denotes the space of pointed maps

S
1

→
X

{S^1} \to X

, and

H
∗

{H_\ast }

represents homology modulo

2
2

. We show that the Eilenberg-Moore spectral sequence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T o r Subscript upper H Sub Superscript asterisk Subscript normal upper Omega 0 Sub Superscript n plus 1 Subscript upper S Sub Superscript n plus 1 Superscript asterisk asterisk Baseline left-parenthesis upper F Subscript 2 comma Baseline upper F 2 right-parenthesis right double arrow upper H Superscript asterisk Baseline normal upper Omega Superscript n plus 2 Baseline upper S Superscript n plus 1">

Tor

H
∗

Ω
0

n
+
1

S

n
+
1

∗
∗

⁡
(

F

2
,

F
2

)
⇒

H
∗

Ω

n
+
2

S

n
+
1

\operatorname {Tor}_{{H^\ast }\Omega _0^{n + 1}{S^{n + 1}}}^{\ast \ast }({F_{2,}}{F_2}) \Rightarrow {H^\ast }{\Omega ^{n + 2}}{S^{n + 1}}

collapses, and we identify the kernel of the Whitehead product map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Superscript n plus 1 Baseline p Subscript asterisk Baseline colon upper H Subscript asterisk Baseline normal upper Omega Superscript n plus 3 Baseline upper S Superscript 2 n plus 1 Baseline right-arrow upper H Subscript asterisk Baseline normal upper Omega Superscript n plus 1 Baseline upper S Superscript n">

Ω

n
+
1

p
∗

:

H
∗

Ω

n
+
3

S

2
n
+
1

→

H
∗

Ω

n
+
1

S
n

{\Omega ^{n + 1}}{p_\ast }:{H_\ast }{\Omega ^{n + 3}}{S^{2n + 1}} \to {H_\ast }{\Omega ^{n + 1}}{S^n}

. These observations yield two different descriptions of

H
∗

Ω

n
+
2

S

n
+
1

{H_\ast }{\Omega ^{n + 2}}{S^{n + 1}}

up to extension.
On some limit theorems for continued fractions
Authors: Samur Jorge D.
Published: June 2010

As a consequence of previous results on mixing random variables, some functional limit theorems for quantities related to the continued fraction expansion of a random number in

(
0
,
1
)

(0,1)

are given.
Proper holomorphic mappings from the two-ball to the three-ball
Authors: Cima J. A. , Suffridge T. J.
Published: June 2010

We prove that a proper mapping of the two ball in

C

n

{\mathbf {C}^{n}}

into the three ball, which is

C
2

{C^2}

on the closed two ball is equivalent to one of four normalized polynomial mappings. This improves the known result of Faran. The proof is basic using Taylor expansions.
The space of harmonic maps of 𝑆² into 𝑆⁴
Authors: Loo Bonaventure
Published: June 2010

Every branched superminimal surface of area

4
π
d

4\pi d

in

S
4

{S^4}

is shown to arise from a pair of meromorphic functions

(

f
1

,

f
2

)

({f_1},{f_2})

of bidegree

(
d
,
d
)

(d,d)

such that

f
1

{f_1}

and

f
2

{f_2}

have the same ramification divisor. Conditions under which branched superminimal surfaces can be generated from such pairs of functions are derived. For each

d
≥
1

d \geq 1

the space of harmonic maps (i.e branched superminimal immersions) of

S
2

{S^2}

into

S
4

{S^4}

of harmonic degree

d
d

is shown to be a connected space of complex dimension

2
d
+
4

2d + 4

.
Cell-like mappings and nonmetrizable compacta of finite cohomological dimension
Authors: Mardešić Sibe , Rubin Leonard R.
Published: June 2010

Compact Hausdorff spaces

X
X

of cohomological dimension

dim
Z

X
≤
n

{\dim _Z}X \leq n

are characterized as cell-like images of compact Hausdorff spaces

Z
Z

with covering dimension

Z
≤
n

Z \leq n

. The proof essentially uses the newly developed techniques of approximate inverse systems.
Complex approximation of real functions by reciprocals of polynomials
Authors: Wulbert Daniel
Published: June 2010

Characterizations are given for local and global best rational approximations to a real function. The characterizations are specialized to reciprocals of polynomials, where they are used to settle some conjectures and questions.
On Cournot-Nash equilibrium distributions for games with a nonmetrizable action space and upper semi-continuous payoffs
Authors: Khan M. Ali
Published: June 2010

We report results on the existence of a Cournot-Nash equilibrium distribution for games in which the action space is not necessarily metrizable and separable and the payoff functions are not necessarily continuous. Our work relies on the theory of Radon measures as developed by Schwartz-Topsoe and on the epitopology as developed by Dolecki-Salinetti-Wets
Topological entropy of homoclinic closures
Authors: Mendoza Leonardo
Published: June 2010

In this paper we study the topological entropy of certain invariant sets of diffeomorphisms, namely the closure of the set of transverse homoclinic points associated with a hyperbolic periodic point, in terms of the growth rate of homoclinic orbits. First we study homoclinic closures which are hyperbolic in

n
n

-dimensional compact manifolds. Using the pseudo-orbit shadowing property of basic sets we prove a formula similar to Bowen’s one on the growth of periodic points. For the nonuniformly hyperbolic case we restrict our attention to compact surfaces.
Translation semigroups and their linearizations on spaces of integrable functions
Authors: Grabosch Annette
Published: June 2010

Of concern is the unbounded operatorAΦf=f′{A_\Phi }f = f’with nonlinear domain<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D left-parenthesis upper A Subscript normal upper Phi Baseline right-parenthesis equals StartSet f element-of upper W Superscript 1 comma 1 Baseline colon f left-parenthesis 0 right-parenthesis equals normal upper Phi left-parenthesis f right-parenthesis EndSet">D(AΦ)={f∈W1,1:f(0)=Φ(f)}D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\}which is considered on the Banach spaceEEof Bochner integrable functions on an interval with values in a Banach spaceFF. Under the assumption thatΦ\Phiis a Lipschitz continuous operator fromEEtoFF, it is shown thatAΦ{A_{\Phi }}generates a strongly continuous translation semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. For linear operatorsΦ\Phiseveral properties such as essential-compactness, positivity, and irreducibility of the semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}depending on the operatorΦ\Phiare studied. It is shown that ifFFis a Banach lattice with order continuous norm, then(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}is the modulus semigroup of(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. Finally spectral properties ofAΦ{A_\Phi }are studied and the spectral bounds(AΦ)s({A_\Phi })is determined. This leads to a result on the global asymptotic behavior in the case whereΦ\Phiis linear and to a local stability result in the case whereΦ\Phiis Fréchet differentiable.
Variations of Hodge structure, Legendre submanifolds, and accessibility
Authors: Carlson James A. , Toledo Domingo
Published: June 2010

Variations of Hodge structure of weight two are integral manifolds for a distribution in the tangent bundle of a period domain. This distribution has dimension

h

2
,
0

h

1
,
1

{h^{2,0}}{h^{1,1}}

and is nonintegrable for

h

2
,
0

>
1

{h^{2,0}} > 1

. In this case it is known that the dimension of an integral manifold does not exceed

1
2

h

2
,
0

h

1
,
1

\frac {1} {2}{h^{2,0}}{h^{1,1}}

. Here we give a new proof, based on an analogy between Griffiths’ horizontal differential system of algebraic geometry and the contact system of classical mechanics. We show also that any two points in such a domain can be joined by a horizontal curve which is piecewise holomorphic.
Ergodic attractors
Authors: Pugh Charles , Shub Michael
Published: June 2010

Using the graph transform method, we give a geometric treatment of Pesin’s invariant manifold theory. Beyond deriving the existence, uniqueness, and smoothness results by Fathi, Herman, and Yoccoz our method allows us to do four things: optimally conserve smoothness, deal with endomorphisms, prove absolute continuity of the Pesin laminations, and produce ergodic attractors.
A Poisson-Plancherel formula for the universal covering group with Lie algebra of type 𝐵_{𝑛}
Authors: Dourmashkin Peter
Published: June 2010

A proof is given for the Poisson-Plancherel formula for Lie groups of type

B
n

{B_n}

using the recurrence relations for the Plancherel function on adjacent Cartan subalgebras given in [12] and the recurrence relations for the discrete series constants which determine a

G
G

-invariant generalized function on

g

∗

{{\mathbf {g}}^\ast }

appearing in the formula.
Multiresolution approximations and wavelet orthonormal bases of 𝐿²(𝑅)
Authors: Mallat Stephane G.
Published: June 2010

A multiresolution approximation is a sequence of embedded vector spaces

(

V

j

)

j
∈

z

{({{\mathbf {V}}_j})_{j \in {\text {z}}}}

for approximating

L

2

(

R

)

{{\mathbf {L}}^2}({\mathbf {R}})

functions. We study the properties of a multiresolution approximation and prove that it is characterized by a

2
π

2\pi

-periodic function which is further described. From any multiresolution approximation, we can derive a function

ψ
(
x
)

\psi (x)

called a wavelet such that

(

2
j

ψ
(

2
j

x
−
k
)

)

(
k
,
j
)
∈

z

2

{(\sqrt {{2^j}} \psi ({2^j}x - k))_{(k,j) \in {{\text {z}}^2}}}

is an orthonormal basis of

L

2

(

R

)

{{\mathbf {L}}^2}({\mathbf {R}})

. This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space

H

s

{{\mathbf {H}}^s}

.
Calculation of transient motion of submerged cables
Authors: Walton Thomas S. , Polachek Harry
Published: June 2010

The system of nonlinear partial differential equations governing the transient motion of a cable immersed in a fluid is solved by finite difference methods. This problem may be considered a generalization of the classical vibrating string problem in the following respects: a) the motion is two dimensional, b) large displacements are permitted, c) forces due to the weight of the cable, buoyancy, drag and virtual inertia of the medium are included, and d) the properties of the cable need not be uniform. The numerical solution of this system of equations presents a number of interesting mathematical problems related to: a) the nonlinear nature of the equations, b) the determination of a stable numerical procedure, and c) the determination of an effective computational method. The solution of this problem is of practical significance in the calculation of the transient forces acting on mooring and towing lines which are subjected to arbitrarily prescribed motions.
A method for the numerical integration of coupled first-order differential equations with greatly different time constants
Authors: Treanor Charles E.
Published: June 2010

Coupled differential equations which describe the simultaneous relaxation of different components at greatly different rates present a difficulty in numerical integration, since the integration interval is determined by the fastest rate, and the region of integration is determined by the slowest rate. In the present paper an integration formula is derived from the approximation that within an interval the first derivative can be expressed as

d
y

/

d
x
=
−
P
y
+
Q
(
x
)

dy/dx = - Py + Q(x)

. The method is exact if the differential equation is of the form shown, where

P
P

is constant and

Q
(
x
)

Q(x)

is a quadratic in

x
x

. The algorithm utilizes only the first derivative and thus has a parallel to the Runge-Kutta method. For

P
h

Ph

small (where

h
h

is the integration interval) the method is identical to fourth-order Runge-Kutta and thus is correct to order

h
4

{h^4}

. Results for the coupled chemistry of high-temperature air are compared with results obtained from the usual Runge-Kutta procedure.
Computing multiplicative inverses in 𝐺𝐹(𝑝)
Authors: Collins George E.
Published: June 2010

Two familiar algorithms, the extended Euclidean algorithm and the Fermat algorithm (based on Fermat’s theoremap≡a(modp){a^p}\equiv a \pmod p), are analyzed and compared as methods for computing multiplicative inverses inGF(p){\text {GF}}(p). Using Knuth’s results on the average number of divisions in the Euclidean algorithm, it is shown that the average number of arithmetic operations required by the Fermat algorithm is nearly twice as large as the average number for the extended Euclidean algorithm. For each of the two algorithms, forward and backward versions are distinguished. It is shown that all numbers computed in the forward extended Euclidean algorithm are bounded by the larger of the two inputs, a property which was previously established by Kelisky for the backward version.
On the convergence of discrete approximations to the Navier-Stokes equations
Authors: Chorin Alexandre Joel
Published: June 2010

A class of useful difference approximations to the full nonlinear Navier-Stokes equations is analyzed; the convergence of these approximations to the solutions of the corresponding differential equations is established and the rate of convergence is estimated.
Rational Chebyshev approximations for the error function
Authors: Cody W. J.
Published: June 2010

This note presents nearly-best rational approximations for the functions erf

(
x
)

(x)

and erfc

(
x
)

(x)

, with maximal relative errors ranging down to between

6
×

10

−
19

6 \times {10^{ - 19}}

and

3
×

10

−
20

3 \times {10^{ - 20}}

.
Optimal conditioning of quasi-Newton methods
Authors: Shanno D. F. , Kettler P. C.
Published: June 2010

Quasi-Newton methods accelerate gradient methods for minimizing a function by approximating the inverse Hessian matrix of the function. Several papers in recent literature have dealt with the generation of classes of approximating matrices as a function of a scalar parameter. This paper derives necessary and sufficient conditions on the range of one such parameter to guarantee stability of the method. It further shows that the parameter effects only the length, not the direction, of the search vector at each step, and uses this result to derive several computational algorithms. The algorithms are evaluated on a series of test problems.
Some results on sparse matrices
Authors: Brayton Robert K. , Gustavson Fred G. , Willoughby Ralph A.
Published: June 2010

A comparison in the context of sparse matrices is made between the Product Form of the Inverse PFI (a form of Gauss-Jordan elimination) and the Elimination Form of the Inverse EFI (a form of Gaussian elimination). The precise relation of the elements of these two forms of the inverse is given in terms of the nontrivial elements of the three matricesLL,UU,U−1{U^{ - 1}}associated with the triangular factorization of the coefficient matrixAA; i.e.,A=L⋅UA = L \cdot U, whereLLis lower triangular andUUis unit upper triangular. It is shown that the zerononzero structure of the PFI always has more nonzeros than the EFI. It is proved that Gaussian elimination is a minimal algorithm with respect to preserving sparseness if the diagonal elements of the matrixAAare nonzero. However, Gaussian elimination is not necessarily minimal ifAAhas some zero diagonal elements. The same statements hold for the PFI as well. A probabilistic study of fill-in and computing times for the PFI and EFI sparse matrix algorithms is presented. This study suggests quantitatively how rapidly sparse matrices fill up for increasing densities, and emphasizes the necessity for reordering to minimize fill-in.
Error of the Newton-Cotes and Gauss-Legendre quadrature formulas
Authors: Kambo N. S.
Published: June 2010

It was shown by P. J. Davis that the Newton-Cotes quadrature formula is convergent if the integrand is an analytic function that is regular in a sufficiently large region of the complex plane containing the interval of integration. In the present paper, a bound on the error of the Newton-Cotes quadrature formula for analytic functions is derived. Also the bounds on the Legendre polynomial and the Legendre function of the second kind are obtained. These bounds are employed to derive a bound on the error of the Gauss-Legendre quadrature formula for analytic functions.
Factoring polynomials over large finite fields
Authors: Berlekamp E. R.
Published: June 2010

This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field

GF

(

p
m

)

{\text {GF}}({p^m})

to the problem of finding the roots in

GF

(
p
)

{\text {GF}}(p)

of certain other polynomials over

GF

(
p
)

{\text {GF}}(p)

. The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the logarithm of the order of the finite field. Certain observations on the application of these methods to the factorization of polynomials over the rational integers are also included.
Triangular elements in the finite element method
Authors: Bramble James H. , Zlámal Miloš
Published: June 2010

For a plane polygonal domain

Ω
\Omega

and a corresponding (general) triangulation we define classes of functions

p
m

(
x
,
y
)

{p_m}(x,y)

which are polynomials on each triangle and which are in

C

(
m
)

(
Ω
)

{C^{(m)}}(\Omega )

and also belong to the Sobolev space

W
2

(
m
+
1
)

(
Ω
)

W_2^{(m + 1)}(\Omega )

. Approximation theoretic properties are proved concerning these functions. These results are then applied to the approximate solution of arbitrary-order elliptic boundary value problems by the Galerkin method. Estimates for the error are given. The case of second-order problems is discussed in conjunction with special choices of approximating polynomials.
Solution of Vandermonde systems of equations
Authors: Björck Ȧke , Pereyra Victor
Published: June 2010

We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. This is achieved by observing that a part of the earlier algorithm is equivalent to Newton’s interpolation method. This allows also to produce a progressive algorithm which is significantly more efficient than previous available methods. Algol-60 programs and numerical results are included. Confluent Vandermonde systems are also briefly discussed.
Uniform asymptotic expansions of the Jacobi polynomials and an associated function
Authors: Elliott David
Published: June 2010

Asymptotic expansions have been obtained using two theorems due to Olver for the Jacobi polynomials and an associated function. These expansions are uniformly valid for complex arguments over certain regions, for large values of the order.
Miniaturized tables of Bessel functions
Authors: Luke Yudell L.
Published: June 2010

In this report, we discuss the representation of bivariate functions in double series of Chebyshev polynomials. For an application, we tabulate coefficients which are accurate to 20 decimals for the evaluation of

(
2
z

/

π

)

1

/

2

e
z

K
v

(
z
)

{(2z/\pi )^{1/2}}{e^z}{K_v}(z)

for all

z
≧
5

z \geqq 5

and all

v
,
0
≦
v
≦
1

v,0 \leqq v \leqq 1

. Since

K
v

(
z
)

{K_v}(z)

is an even function in v and satisfies a three-term recurrence formula in v which is stable when used in the forward direction, we can readily evaluate

K
v

(
z
)

{K_v}(z)

for all

z
≧
5

z \geqq 5

and all

v
≧
0

v \geqq 0

. Only 205 coefficients are required to achieve an accuracy of about 20 decimals for the z and v ranges described. Extension of these ideas for the evaluation of all Bessel functions and other important bivariate functions is under way.
Uniform computation of the error function and other related functions
Authors: Matta F. , Reichel A.
Published: June 2010

Uniform methods of computation, to any required degree of accuracy, for the error and other closely related functions are given.
Product integration for the generalized Abel equation
Authors: Weiss Richard
Published: June 2010

The solution of the generalized Abel integral equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript t Baseline left-brace k left-parenthesis t comma s right-parenthesis slash left-parenthesis t minus s right-parenthesis Superscript alpha Baseline right-brace f left-parenthesis s right-parenthesis d s comma 0 greater-than alpha greater-than 1 comma">

g
(
t
)
=

∫
0
t

{
k
(
t
,
s
)

/

(
t
−
s
)

α

}
f
(
s
)

d
s
,

0
>
α
>
1
,

g(t) = \int _0^t {\{ k(t,s)/{{(t - s)}^\alpha }\} f(s)} ds,\;0 > \alpha > 1,

\]
where

k
(
t
,
s
)

k(t,s)

is continuous, by the product integration analogue of the trapezoidal method is examined. It is shown that this method has order two convergence for

α
∈
[

α
1

,
1
)

\alpha \in [{\alpha _1},1)

with

α
1

≑
0.2117

{\alpha _1} \doteqdot 0.2117

. This interval contains the important case

α
=

1
2

\alpha = \tfrac {1}{2}

. Convergence of order two for

α
∈
(
0
,

α
1

)

\alpha \in (0,{\alpha _1})

is discussed and illustrated numerically. The possibility of constructing higher order methods is illustrated with an example.
A note on the evaluation of the complementary error function
Authors: Hunter D. B. , Regan T.
Published: June 2010

A modification is proposed to a method of Matta and Reichel for evaluating the complementary error function of a complex variable, so as to improve the numerical stability of the method in certain critical regions.
How slowly can quadrature formulas converge?
Authors: Lipow Peter R. , Stenger Frank
Published: June 2010

Let

{

Q
n

}

n
=
1

∞

\{ {Q_n}\} _{n = 1}^\infty

denote a sequence of quadrature formulas,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Subscript n Baseline left-parenthesis f right-parenthesis identical-to sigma-summation Underscript i equals 1 Overscript k Subscript n Baseline Endscripts w Subscript i Superscript left-parenthesis n right-parenthesis Baseline f left-parenthesis x Subscript i Superscript left-parenthesis n right-parenthesis Baseline right-parenthesis">

Q
n

(
f
)
≡

∑

i
=
1

k
n

w
i

(
n
)

f
(

x
i

(
n
)

)

{Q_n}(f) \equiv \sum _{i = 1}^{{k_n}}w_i^{(n)}f(x_i^{(n)})

, such that

Q
n

(
f
)
→

∫
0
1

f
(
x
)
d
x

{Q_n}(f) \to \int _0^1 f (x)dx

for all

f
∈
C
[
0
,
1
]

f \in C[0,1]

. Let

0
>
ε
>

1
4

0 > \varepsilon > \frac {1}{4}

and a sequence

{

a
n

}

n
=
1

∞

\{ {a_n}\} _{n = 1}^\infty

be given, where

a
1

≧

a
2

≧

a
3

≧
.
.
.

{a_1} \geqq {a_2} \geqq {a_3} \geqq ...

, and where

a
n

→
0

{a_n} \to 0

as

n
→
∞

n \to \infty

. Then there exists a function

f
∈
C
[
0
,
1
]

f \in C[0,1]

and a sequence

{

n
k

}

k
=
1

∞

\{ {n_k}\} _{k = 1}^\infty

such that

|

f
(
x
)

|

≦
2

a
1

/

|

(
1
−
4
)

|

|f(x)| \leqq 2{a_1}/|(1 - 4)|

, and such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline f left-parenthesis x right-parenthesis d x minus upper Q Subscript n Sub Subscript k Subscript Baseline left-parenthesis f right-parenthesis equals a Subscript k comma Baseline k equals 1 comma 2 comma 3 comma period period period">

∫
0
1

f
(
x
)
d
x
−

Q

n
k

(
f
)
=

a

k
,

k
=
1
,
2
,
3
,
.
.
.

\int _0^1 f (x)dx - {Q_{{n_k}}}(f) = {a_{k,}}k = 1,2,3,...

.
One-step piecewise polynomial Galerkin methods for initial value problems
Authors: Hulme Bernie L.
Published: June 2010

A new approach to the numerical solution of systems of first-order ordinary differential equations is given by finding local Galerkin approximations on each subinterval of a given mesh of size

h
h

. One step at a time, a piecewise polynomial, of degree

n
n

and class

C
0

{C^0}

, is constructed, which yields an approximation of order

O
(

h

2
n

)

O({h^{2n}})

at the mesh points and

O
(

h

n
+
1

)

O({h^{n + 1}})

between mesh points. In addition, the

j
j

th derivatives of the approximation on each subinterval have errors of order

O
(

h

n
−
j
+
1

)
,
1
≦
j
≦
n

O({h^{n - j + 1}}),1 \leqq j \leqq n

. The methods are related to collocation schemes and to implicit Runge-Kutta schemes based on Gauss-Legendre quadrature, from which it follows that the Galerkin methods are

A
A

-stable.
The modified successive overrelaxation method with fixed parameters
Authors: Kincaid David R. , Young David M.
Published: June 2010

Expressions for the spectral radius and for certain norms of the modified successive overrelaxation method with fixed parameters are derived. Also established are expressions for the virtual spectral radius and for certain virtual norms of this method. Parameter restrictions are determined so that the spectral radius and the norms coincide with the virtual spectral radius and the virtual norms, respectively. Optimum parameters which minimize these expressions are obtained. These results extend those of Young [11], [12].
Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections
Authors: Bramble James H. , Dupont Todd , Thomée Vidar
Published: June 2010

Consider Dirichlet’s problem in a plane domain

Ω
\Omega

with smooth boundary

∂
Ω

\partial \Omega

. For the purpose of its approximate solution, an approximating domain

Ω
h

,
0
>
h
≦
1

{\Omega _h},0 > h \leqq 1

, with polygonal boundary

∂

Ω
h

\partial {\Omega _h}

is introduced where the segments of

∂

Ω
h

\partial {\Omega _h}

have length at most

h
h

. A projection method introduced by Nitsche [6] is then applied on

Ω
h

{\Omega _h}

to give an approximate solution in a finite-dimensional subspace of functions

S
h

{S_h}

, for instance a space of splines defined on a triangulation of

Ω
h

{\Omega _h}

. The boundary terms in the bilinear form associated with Nitsche’s method are modified to correct for the perturbation of the boundary.
Difference approximations for boundary and eigenvalue problems for ordinary differential equations
Authors: Kreiss Heinz-Otto
Published: June 2010

The boundary value problem for ordinary differential equations is considered and a general theory for difference approximation is developed. In particular, the influence of extra boundary conditions is investigated and the eigenvalue problem is considered in detail.
Primitive binary polynomials
Authors: Stahnke Wayne
Published: June 2010

One primitive polynomial modulo two is listed for each degree n through

n
=
168

n = 168

. Each polynomial has the minimum number of terms possible for its degree. The method used to generate the list is described.
Quadratic fields with four invariants divisible by 3
Authors: Shanks Daniel , Serafin Richard
Published: June 2010

Imaginary quadratic fields are developed that have four invariants divisible by 3. Their associated real fields are found to differ in one significant respect: one case has two elementary generators and the other has only one.
Implementation and analysis of the Todd-Coxeter algorithm
Authors: Cannon John J. , Dimino Lucien A. , Havas George , Watson Jane M.
Published: June 2010

A recent form of the Todd-Coxeter algorithm, known as the lookahead algorithm, is described. The time and space requirements for this algorithm are shown experimentally to be usually either equivalent or superior to the Felsch and Haselgrove-Leech-Trotter algorithms. Some findings from an experimental study of the behaviour of Todd-Coxeter programs in a variety of situations are given.
Numerical methods for computing angles between linear subspaces
Authors: Björck Ȧke , Golub Gene H.
Published: June 2010

Assume that two subspaces F and G of a unitary space are defined as the ranges (or null spaces) of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles

θ
k

(
F
,
G
)

{\theta _k}(F,G)

and orthogonal sets of principal vectors

u
k

∈
F

{u_k} \in F

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v Subscript k Baseline element-of upper G comma k equals 1 comma 2 comma midline-horizontal-ellipsis comma q equals dimension left-parenthesis upper G right-parenthesis less-than-over-equals dimension left-parenthesis upper F right-parenthesis">

v
k

∈
G
,
k
=
1
,
2
,
⋯
,
q
=
dim
⁡
(
G
)
≦
dim
⁡
(
F
)

{v_k} \in G,k = 1,2, \cdots ,q = \dim (G) \leqq \dim (F)

. An important application in statistics is computing the canonical correlations

σ
k

=
cos
⁡

θ
k

{\sigma _k} = \cos {\theta _k}

between two sets of variates. A perturbation analysis shows that the condition number for

θ
k

{\theta _k}

essentially is

max
(
κ
(
A
)
,
κ
(
B
)
)

\max (\kappa (A),\kappa (B))

, where

κ
\kappa

denotes the condition number of a matrix. The algorithms are based on a preliminary QR-factorization of A and B (or

A
H

{A^H}

and

B
H

{B^H}

), for which either the method of Householder transformations (HT) or the modified Gram-Schmidt method (MGS) is used. Then

cos

θ
k

\cos \;{\theta _k}

and

sin

θ
k

\sin \;{\theta _k}

are computed as the singular values of certain related matrices. Experimental results are given, which indicates that MGS gives

θ
k

{\theta _k}

with equal precision and fewer arithmetic operations than HT. However, HT gives principal vectors, which are orthogonal to working accuracy, which is not generally true for MGS. Finally, the case when A and/or B are rank deficient is discussed.
Stability and convergence of difference approximations to pseudo-parabolic partial differential equations
Authors: Ford William H. , Ting T. W.
Published: June 2010

Two difference approximations to the solution of a pseudo-parabolic problem are constructed and shown by means of stability analysis to converge in the "discrete"

L
2

{L_2}

norm. A relation between parabolic and pseudo-parabolic difference schemes is discussed, and the stability of difference approximations to backward time parabolic and pseudo-parabolic problems is also considered.
On weird and pseudoperfect numbers
Authors: Benkoski S. J. , Erdős P.
Published: June 2010

If n is a positive integer and

σ
(
n
)

\sigma (n)

denotes the sum of the divisors of n, then n is perfect if

σ
(
n
)
=
2
n

\sigma (n) = 2n

, abundant if

σ
(
n
)
≧
2
n

\sigma (n) \geqq 2n

and deficient if

σ
(
n
)
>
2
n

\sigma (n) > 2n

. n is called pseudoperfect if n is the sum of distinct proper divisors of n. If n is abundant but not pseudoperfect, then n is called weird. The smallest weird number is 70. We prove that the density of weird numbers is positive and discuss several related problems and results. A list of all weird numbers not exceeding

10
6

{10^6}

is given.
A class of quadrature formulas
Authors: Kumar Ravindra
Published: June 2010

It is proved that there exists a set of polynomials orthogonal on

[
−
1
,
1
]

[ - 1,1]

with respect to the weight function

(

1

)

w
(
t
)

/

(
t
−
x
)

\begin{equation}\tag {$1$} w(t)/(t - x)\end{equation}

corresponding to the polynomials orthogonal on

[
−
1
,
1
]

[ - 1,1]

with respect to the weight function w. Simplified forms of such polynomials are obtained for the special cases
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel StartLayout 1st Row w left-parenthesis t right-parenthesis equals left-parenthesis 1 minus t squared right-parenthesis Superscript negative 1 slash 2 Baseline comma 2nd Row equals left-parenthesis 1 minus t squared right-parenthesis Superscript 1 slash 2 Baseline comma 3rd Row equals left-parenthesis left-parenthesis 1 minus t right-parenthesis slash left-parenthesis 1 plus t right-parenthesis right-parenthesis Superscript 1 slash 2 Baseline comma EndLayout EndLayout">

(

2

)

w
(
t
)
=

(
1
−

t
2

)

−
1

/

2

,

=

(
1
−

t
2

)

1

/

2

,

=

(
(
1
−
t
)

/

(
1
+
t
)
)

1

/

2

,

\begin{equation}\tag {$2$} \begin {array}{*{20}{c}} {w(t) = {{(1 - {t^2})}^{ - 1/2}},} \\ { = {{(1 - {t^2})}^{1/2}},} \\ { = {{((1 - t)/(1 + t))}^{1/2}},} \\ \end{array} \end{equation}

and the generating functions and the recurrence relation are also given. Subsequently, a set of quadrature formulas given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 3 right-parenthesis EndLabel integral Subscript negative 1 Superscript 1 Baseline left-parenthesis 1 plus t right-parenthesis Superscript p minus 1 slash 2 Baseline left-parenthesis 1 minus t right-parenthesis Superscript q minus 1 slash 2 Baseline left-parenthesis 1 plus a squared plus 2 a t right-parenthesis Superscript negative 1 Baseline f left-parenthesis t right-parenthesis d t equals sigma-summation Underscript k equals 1 Overscript n Endscripts upper H Subscript k Baseline f left-parenthesis t Subscript k Baseline right-parenthesis plus upper E Subscript n Baseline left-parenthesis f right-parenthesis EndLayout">

(

3

)

∫

−
1

1

(
1
+
t
)

p
−
1

/

2

(
1
−
t
)

q
−
1

/

2

(
1
+

a
2

+
2
a
t
)

−
1

f
(
t
)
d
t
=

∑

k
=
1

n

H
k

f
(

t
k

)
+

E
n

(
f
)

\begin{equation}\tag {$3$} \int _{ - 1}^1 {{{(1 + t)}^{p - 1/2}}{{(1 - t)}^{q - 1/2}}{{(1 + {a^2} + 2at)}^{ - 1}}f(t)dt = \sum \limits _{k = 1}^n {{H_k}f({t_k}) + {E_n}(f)} } \end{equation}

for

(
p
,
q
)
=
(
0
,
0
)
,
(
0
,
1
)

(p,q) = (0,0),(0,1)

and (1, 1) is established; these formulas are valid for analytic functions. Convergence of the quadrature rules is discussed, using a technique based on the generating functions. This method appears to be simpler than the one suggested by Davis [2, pp. 311-312] and used by Chawla and Jain [3]. Finally, bounds on the error are obtained.
A variable order finite difference method for nonlinear multipoint boundary value problems
Authors: Lentini M. , Pereyra V.
Published: June 2010

An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied earlier by H. B. Keller. Variable order is provided through deferred corrections, while a built-in natural asymptotic estimator is used to automatically refine the mesh in order to achieve a required tolerance. Extensive numerical experimentation and a FORTRAN program are included.
The resultant of the cyclotomic polynomials 𝐹_{𝑚}(𝑎𝑥) and 𝐹_{𝑛}(𝑏𝑥)
Authors: Apostol Tom M.
Published: June 2010

The resultantρ(Fm(ax),Fn(bx))\rho ({F_m}(ax),{F_n}(bx))is calculated for arbitrary positive integersmandn, and arbitrary nonzero complex numbersaandb. An addendum gives an extended bibliography of work on cyclotomic polynomials published since 1919.
Continued fractions and linear recurrences
Authors: Mills W. H.
Published: June 2010

Let

t
0

,

t
1

,

t
2

,
⋯

{t_0},{t_1},{t_2}, \cdots

be a sequence of elements of a field F. We give a continued fraction algorithm for

t
0

x
+

t
1

x
2

+

t
2

x
3

+
⋯

{t_0}x + {t_1}{x^2} + {t_2}{x^3} + \cdots

. If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence. More generally the algorithm produces a nontrivial solution of the system
\[

∑

j
=
0

s

t

i
+
j

λ
j

,

0
⩽
i
⩽
s
−
1
,

\sum \limits _{j = 0}^s {{t_{i + j}}{\lambda _j},\quad 0 \leqslant i \leqslant s - 1,}

\]
for every positive integer s.
Estimating the efficiency of backtrack programs
Authors: Knuth Donald E.
Published: June 2010

One of the chief difficulties associated with the so-called backtracking technique for combinatorial problems has been our inability to predict the efficiency of a given algorithm, or to compare the efficiencies of different approaches, without actually writing and running the programs. This paper presents a simple method which produces reasonable estimates for most applications, requiring only a modest amount of hand calculation. The method should prove to be of considerable utility in connection with D. H. Lehmer’s branch-and-bound approach to combinatorial optimization.
A collocation method for two-point boundary value problems
Authors: Ahlberg J. H. , Ito T.
Published: June 2010

This article is concerned with the use of collocation by splines to numerically solve two-point boundary value problems. The problem is analyzed in terms of cubic splines first and then extended to the use of quintic and septic splines. Consideration is given both to convergences as the mesh is refined and to the bandwidth of the matrices involved. Comparisons are made to a similar approach using the Galerkin method rather than collocation.
Diagonalization and simultaneous symmetrization of the gas-dynamic matrices
Authors: Warming R. F. , Beam Richard M. , Hyett B. J.
Published: June 2010

The hyperbolicity of the unsteady, inviscid, gas-dynamic equations implies the existence of a similarity transformation which diagonalizes an arbitrary linear combination

Σ

k
j

A
j

\Sigma {k_j}{A_j}

of coefficient matrices

A
j

{A_j}

. The matrix T that accomplishes this transformation is given explicitly, and the spectral norms of T and

T

−
1

{T^{ - 1}}

are computed. It is also shown that the individual matrices

A
j

{A_j}

are simultaneously symmetrized by the same similarity transformation. Applications of the transformations T and

T

−
1

{T^{ - 1}}

and their norms include the well-posedness of the Cauchy problem, linear stability theory for finite-difference approximations, construction of difference schemes based on characteristic relations, and simplification of the solution of block-tridiagonal systems that arise in implicit time-split algorithms.
Computation of 𝜋 using arithmetic-geometric mean
Authors: Salamin Eugene
Published: June 2010

A new formula for

π
\pi

is derived. It is a direct consequence of Gauss’ arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre’s relation for elliptic integrals. The error analysis shows that its rapid convergence doubles the number of significant digits after each step. The new formula is proposed for use in a numerical computation of

π
\pi

, but no actual computational results are reported here.
The largest degrees of irreducible characters of the symmetric group
Authors: McKay John
Published: June 2010

The largest irreducible degrees and the partitions associated with them are tabulated for the symmetric group

Σ
n

{\Sigma _n}

for n up to 75. Analytic upper and lower bounds are derived for the largest degree.
Numerical applications of reflection to partial differential equations
Authors: Snider Arthur David
Published: June 2010

Recent papers have reported results on the numerical solution of nonlinear free boundary problems wherein a conformal transformation (which must be determined) maps the unknown flow region onto a known domain; the boundary conditions are handled by the method of steepest descent. The present paper discusses the use of the reflection property of solutions of elliptic equations to determine these boundary conditions. The procedure is applied to the vena contracta models, and it is seen that it converges about ten times faster than the steepest-descent method.
On the efficiency of algorithms for polynomial factoring
Authors: Moenck Robert T.
Published: June 2010

Algorithms for factoring polynomials over finite fields are discussed. A construction is shown which reduces the final step of Berlekamp’s algorithm to the problem of finding the roots of a polynomial in a finite field

Z
p

{Z_p}

. It is shown that if the characteristic of the field is of the form

p
=
L
⋅

2
l

+
1

p = L \cdot {2^l} + 1

, where

l
≃
L

l \simeq L

, then the roots of a polynomial of degree n may be found in

O
(

n
2

log
⁡
p
+
n

log
2

p
)

O({n^2}\log p + n{\log ^2}p)

steps. As a result, a modification of Berlekamp’s method can be performed in

O
(

n
3

+

n
2

log
⁡
p
+
n

log
2

p
)

O({n^3} + {n^2}\log p + n{\log ^2}p)

steps. If n is very large then an alternative method finds the factors of the polynomial in

O
(

n
2

log
2

n
+

n
2

log
⁡
n
log
⁡
p
)

O({n^2}{\log ^2}n + {n^2}\log n\log p)

. Some consequences and empirical evidence are discussed.
Finite element methods for elliptic equations using nonconforming elements
Authors: Baker Garth A.
Published: June 2010

A finite element method is developed for approximating the solution of the Dirichlet problem for the biharmonic operator, as a canonical example of a higher order elliptic boundary value problem. The solution is approximated by special choices of classes of discontinuous functions, piecewise polynomial functions, by virtue of a special variational formulation of the boundary value problem. The approximating functions are not required to satisfy the prescribed boundary conditions. Optimal error estimates are derived in Sobolev spaces.