bluuebunny/crossref_metadata_2025 huggingface dataset:
*-differential identities of prime rings with involution
Authors: Chuang Chen-Lian
Published: June 2010

Main Theorem. Let

R
R

be a prime ring with involution

∗

^{\ast }

. Suppose that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis x Subscript i Superscript normal upper Delta Super Subscript j Superscript Baseline comma left-parenthesis x Subscript i Superscript normal upper Delta Super Subscript j Superscript Baseline right-parenthesis Superscript asterisk Baseline right-parenthesis equals 0">

ϕ
(

x
i

Δ
j

,

(

x
i

Δ
j

)

∗

)
=
0

\phi (x_i^{{\Delta _j}},{(x_i^{{\Delta _j}})^{\ast }}) = 0

is a

∗

{\ast }

-differential identity for

R
R

, where

Δ
j

{\Delta _j}

are distinct regular words of derivations in a basis

M
M

with respect to a linear order

>
>

on

M
M

. Then

ϕ
(

z

i
j

,

z

i
j

∗

)
=
0

\phi ({z_{ij}},z_{ij}^{\ast }) = 0

is a

∗

{\ast }

-generalized identity for

R
R

, where

z

i
j

{z_{ij}}

are distinct indeterminates. Along with the Main Theorem above, we also prove the following: Proposition 1. Suppose that

∗

^{\ast }

is of the second kind and that

C
C

is infinite. Then

R
R

is special. Proposition 2. Suppose that

S
W

(
V
)
⊆
R
⊆

L
W

(
V
)

{S_W}(V) \subseteq R \subseteq {L_W}(V)

. Then

Q
Q

, the two-sided quotient ring of

R
R

, is equal to

L
W

(
V
)

{L_W}(V)

. Proposition 3 (Density theorem). Suppose that

D

V

{}_DV

and

W
D

{W_D}

are dual spaces with respect to the nondegenerate bilinear form

(
,
)

(,)

. Let

v
1

,
…
,

v
s

,

v
s
′

,
…
,

v
s
′

∈
V

{v_1}, \ldots ,{v_s},\;v_s^\prime , \ldots ,v_s^\prime \in V

and

u
1

,
…
,

u
t

,

u
1
′

,
…
,

u
t
′

∈
W

{u_1}, \ldots ,{u_t},\;u_1^\prime , \ldots ,u_t^\prime \in W

be such that

{

v
1

,
…
,

v
s

}

\{ {v_1}, \ldots ,{v_s}\}

is

D
D

-independent in

V
V

and

{

u
1

,
…
,

u
t

}

\{ {u_1}, \ldots ,{u_t}\}

is

D
D

-independent in

W
W

. Then there exists

a
∈

S
W

(
V
)

a \in {S_W}(V)

such that

v
i

a
=

v
i
′

(
i
=
1
,
…
,
s
)

{v_i}a = v_i^\prime \,(i = 1, \ldots ,s)

and

a

∗

u
j

=

u
j
′

(
j
=
1
,
…
,
t
)

{a^{\ast }}{u_j} = u_j^\prime \,(j = 1, \ldots ,t)

if and only if

(

v
i
′

,

u
j

)
=
(

v
i

,

u
j
′

)

(v_i’,{u_j}) = ({v_i},u_j’)

for

i
=
1
,
…
,
s

i = 1, \ldots ,s

and

j
=
1
,
…
,
t

j = 1, \ldots ,t

. Proposition 4. Suppose that

R
R

is a prime ring with involution

∗

^{\ast }

and that

f
f

is a

∗

{\ast }

-generalized polynomial. If

f
f

vanishes on a nonzero ideal of

R
R

, than

f
f

vanishes on

Q
Q

, the two-sided quotient ring of

R
R

.
Complemented subspaces of products of Banach spaces
Authors: Domański Paweł , Ortyński Augustyn
Published: June 2010

It is proved that: (i) every complemented subspace in an infinite product of

L
1

{L_1}

-predual Banach spaces

∏

i
∈
I

X
i

\prod \nolimits _{i \in I} {{X_i}}

is isomorphic to

Z
×

K

m

Z \times {{\mathbf {K}}^\mathfrak {m}}

, where

dim
⁡

K

=
1
,

m

⩽
card
⁡
I

\dim {\mathbf {K}} = 1,\;\mathfrak {m} \leqslant \operatorname {card} I

and

Z
Z

is isomorphic to a complemented subspace of

∏

i
∈
J

X
i

,

J
⊆
I
,

Z

\prod \nolimits _{i \in J} {{X_i},\;J \subseteq I,\;Z}

contains an isomorphic cop[ill] of

c
0

card
⁡
J

c_0^{\operatorname {card} J}

; (ii) every injective lcs (in particular, Fréchet) is of the form

Z
×

K

m

,

dim
⁡

K

=
1

Z \times {{\mathbf {K}}^\mathfrak {m}},\;\dim {\mathbf {K}} = 1

, where

Z
Z

has a fundamental family of seminorms of the cardinality

τ
\tau

and

Z
Z

contains an isomorphic copy of

l
∞

r

l_\infty ^\mathfrak {r}

(this is a generalization of Lindenstrauss’ theorem on injective Banach spaces); (iii) whenever

X
≃

l
p

,

1
⩽
p
⩽
∞

X \simeq {l_p},\;1 \leqslant p \leqslant \infty

, or

X
≃

c
0

X \simeq {c_0}

, then every complemented subspace in a power

X

m

{X^\mathfrak {m}}

(

m

\mathfrak {m}

is an arbitrary cardinal number) is isomorphic to

X

r

×

K

s

,

r

+
s
⩽

m

{X^\mathfrak {r}} \times {{\mathbf {K}}^\mathfrak {s}},\;\mathfrak {r} + s \leqslant \mathfrak {m}

(a generalization of the results due to Lindenstrauss and Pełczyński for

m

=
1

\mathfrak {m} = 1

).
Invariant measures and equilibrium states for piecewise 𝐶^{1+𝛼} endomorphisms of the unit interval
Authors: Bose Christopher J.
Published: June 2010

A differentiable function is said to be

C

1
+
α

{C^{1 + \alpha }}

if its derivative is a Hölder continuous function with exponent

α
>
0

\alpha > 0

. We show that three well-known results about invariant measures for piecewise monotonic and

C
2

{C^2}

endomorphisms of the unit interval are in fact true for piecewise monotonic and

C

1
+
α

{C^{1 + \alpha }}

maps. We show the existence of unique, ergodic measures equivalent to Lebesgue measure for

C

1
+
α

{C^{1 + \alpha }}

Markov maps, extending a result of Bowen and Series for the

C
2

{C^2}

case. We present a generalization of Adler’s Folklore Theorem for maps which satisfy a restricted mixing condition, and we show that these

C

1
+
α

{C^{1 + \alpha }}

mixing endomorphisms possess unique equilibrium states, a result which was shown for the

C
2

{C^2}

case by P. Walters.
Regular transition functions and regular superprocesses
Authors: Dynkin E. B.
Published: June 2010

The class of regular Markov processes is very close to the class of right processes studied by Meyer, Getoor and others. We say that a transition function

p
p

is regular if it is the transition function of a well-defined regular Markov process. A characterization of regular transition functions is given which implies that, if

p
p

is regular, then the Dawson-Watanabe and the Fleming-Viot supertransition functions over

p
p

belong to the same class.
Global solvability of the derivative nonlinear Schrödinger equation
Authors: Lee Jyh-Hao
Published: June 2010

The derivative nonlinear Schrödinger equation

(
DNLS
)

(\text {DNLS})

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column i q Subscript t Baseline equals q Subscript x x Baseline plus-or-minus left-parenthesis q Superscript asterisk Baseline q squared right-parenthesis Subscript x Baseline comma 2nd Column a m p semicolon q equals q left-parenthesis x comma t right-parenthesis comma i equals StartRoot negative 1 EndRoot comma q Superscript asterisk Baseline left-parenthesis z right-parenthesis equals ModifyingAbove q left-parenthesis z right-parenthesis With bar comma EndLayout">

i

q
t

=

q

x
x

±

(

q
∗

q
2

)

x

,

a
m
p
;

q
=
q
(
x
,
t
)
,
i
=

−
1

,

q
∗

(
z
)
=

q
(
z
)

¯

,

\begin {array}{*{20}{c}} {i{q_t} = {q_{xx}} \pm {{({q^\ast }{q^2})}_x},} & {q = q(x,t), i = \sqrt { - 1} ,{q^\ast }(z) = \overline {q(z)} ,} \\ \end {array}

\]
was first derived by plasma physicists [9,10]. This equation was used to interpret the propagation of circular polarized nonlinear Alfvén waves in plasma. Kaup and Newell obtained the soliton solutions of

DNLS
\text {DNLS}

in 1978 [5]. The author obtained the local solvability of

DNLS
\text {DNLS}

in his dissertation [6]. In this paper we obtain global existence (in time

t
t

) of Schwartz class solutions of

DNLS
\text {DNLS}

if the

L
2

{L^2}

-norm of the generic initial data

q
(
x
,
0
)

q(x,0)

is bounded.
Mel′nikov transforms, Bernoulli bundles, and almost periodic perturbations
Authors: Meyer Kenneth R. , Sell George R.
Published: June 2010

In this paper we study nonlinear time-varying perturbations of an autonomous vector field in the plane

R
2

{R^2}

. We assume that the unperturbed equation, i.e. the given vector field has a homoclinic orbit and we present a generalization of the Melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory. The key to our generalization is the concept of the Melnikov transform, which is a linear transformation on the space of perturbation functions. The appropriate dynamical setting for studying these perturbation is the concept of a skew product flow. The concept of transversality we require is best understood in this context. Under conditions whereby the perturbed equation admits a transversal homoclinic trajectory, we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow. We show the dynamics near this trajectory can have the exotic behavior of the Bernoulli shift. The exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle, which we call, the Bernoulli bundle. We allow all perturbations which are bounded and uniformly continuous in time. Thus our theory includes the classical periodic perturbations studied by Melnikov, quasi periodic and almost periodic perturbations, as well as toroidal perturbations which are close to quasi periodic perturbations.
Sharp estimates for the nontangential maximal function and the Lusin area function in Lipschitz domains
Authors: Bañuelos Rodrigo , Moore Charles N.
Published: June 2010

Let

u
u

be a harmonic function on a domain of the form

D
=
{
(
x
,
y
)
:
x
∈

R

n

,
y
∈

R

,
y
>
ϕ
(
x
)
}

D = \{ (x,y):x \in {{\mathbf {R}}^n},y \in {\mathbf {R}},y > \phi (x)\}

where

ϕ
:

R

n

→

R

\phi :{{\mathbf {R}}^n} \to {\mathbf {R}}

is a Lipschitz function. The authors show a good-

λ
\lambda

inequality between

A
u

Au

, the Lusin area function of

u
u

, and

N
u

Nu

, the nontangential maximal function of

u
u

. This leads to an

L
p

{L^p}

inequality of the form

‖
A
u
‖

p

≤

C
p

‖
N
u
‖

p

\left \|Au\right \|_p \leq C_p\left \|Nu\right \|_p

which is sharp in the sense that

C
p

{C_p}

is of the smallest possible order in

p
p

as

p
→
∞

p \to \infty

. For

P
∈
∂
D

P \in \partial D

and

t
>
0

t > 0

we also consider the functions

A
u
(
P
+
(
0
,
t
)
)

Au(P + (0,t))

and

N
u
(
P
+
(
0
,
t
)
)

Nu(P + (0,t))

and show that a corollary of the good-

λ
\lambda

inequality is a law of the iterated logarithm involving these two functions as

t
→
0

t \to 0

. If

n
=
1

n = 1

and

ϕ
\phi

has a small Lipschitz constant the above results are shown valid with the roles of

N
u

Nu

and

A
u

Au

interchanged.
Index formulas for elliptic boundary value problems in plane domains with corners
Authors: Eskin Gregory
Published: June 2010

We derive the conditions for the operator corresponding to a general elliptic boundary value problem in a plane domain with corners to be Fredholm and give an explicit formula for the index of this operator.
Behavior of polynomials of best uniform approximation
Authors: Saff E. B. , Totik V.
Published: June 2010

We investigate the asymptotic behavior of the polynomials

{

P
n

(
f
)

}
0
∞

\{ {P_n}(f)\} _0^\infty

of best uniform approximation to a function

f
f

that is continuous on a compact set

K
K

of the complex plane

C

{\mathbf {C}}

and analytic in the interior of

K
K

, where

K
K

has connected complement. For example, we show that for "most" functions

f
f

, the error

f
−

P
n

(
f
)

f - {P_n}(f)

does not decrease faster at interior points of

K
K

than on

K
K

itself. We also describe the possible limit functions for the normalized error

(
f
−

P
n

(
f
)
)

/

E
n

(f - {P_n}(f))/{E_n}

, where

E
n

:=

|

|

f
−

P
n

(
f
)

|

|

K

{E_n}: = ||f - {P_n}(f)|{|_K}

, and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to

f
f

on

K
K

exist that converge more rapidly at the interior points of

K
K

.
Lie sphere transformations and the focal sets of certain taut immersions
Authors: Buyske Steven G.
Published: June 2010

We study the images of certain taut or Dupin hypersurfaces, including their focal sets, under Lie sphere transformations (generalizations of conformal transformations of euclidean or spherical space). After the introduction, the method of studying hypersurfaces as Lie sphere objects is developed. In two recent papers, Cecil and Chern use submanifolds of the space of lines on the Lie quadric. Here we use submanifolds of the Lie quadric itself instead. The third section extends the concepts of tightness and tautness to semi-euclidean space. The final section shows that if a hypersurface is the Lie sphere image of certain standard constructions (tubes, cylinders, and rotations) over a taut immersion, the resulting family of curvature spheres is taut in the Lie quadric. The sheet of the focal set will be tight in euclidean space if it is compact. In particular, if a hypersurface in euclidean space is the Lie sphere image of an isoparametric hypersurface each compact sheet of the focal set will be tight.
Geometric quantization and the universal enveloping algebra of a nilpotent Lie group
Authors: Pedersen Niels Vigand
Published: June 2010

We study geometric quantization in connection with connected nilpotent Lie groups. First it is proved that the quantization map associated with a (real) polarized coadjoint orbit establishes an isomorphism between the space of polynomial quantizable functions and the space of polynomial quantized operators. Our methods allow noninductive proofs of certain basic facts from Kirillov theory. It is then shown how the quantization map connects with the universal enveloping algebra. This is the main result of the paper. Finally we show how one can explicitly compute global canonical coordinates on coadjoint orbits, and that this can be done simultaneously on all orbits contained in a given stratum of what we call "the fine

F

\mathcal {F}

-stratification of the dual of the Lie algebra". This is a generalization of a result of M. Vergne about simultaneous canonical coodinates for orbits in general position.
Isometric dilations for infinite sequences of noncommuting operators
Authors: Popescu Gelu
Published: June 2010

This paper develops a dilation theory for

{

T
n

}

n
=
1

∞

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

an infinite sequence of noncommuting operators on a Hilbert space, when the matrix

[

T
1

,

T
2

,
…
]

[{T_1},{T_2}, \ldots ]

is a contraction. A Wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for

{

T
n

}

n
=
1

∞

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

are obtained. Some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given. This results are used to extend the Sz.-Nagy-Foiaş lifting theorem to this noncommutative setting.
Holomorphic foliations in ruled surfaces
Authors: Gómez-Mont Xavier
Published: June 2010

We analyse the universal families of holomorphic foliations with singularities in a ruled surface. In terms of Chern classes we determine the general and the special families. We also classify all nonsingular foliations.
Negative scalar curvature metrics on noncompact manifolds
Authors: Bland John , Kalka Morris
Published: June 2010

In this paper we prove that every noncompact smooth manifold admits a complete metric of constant negative scalar curvature.
Integral representations of positive definite matrix-valued distributions on cylinders
Authors: Friedrich Jürgen
Published: June 2010

The notion of a

G
G

-continuous matrix-valued positive definite distribution on
\[

S
N

(
2
a
)
×

R

M

×
G

{S_N}(2a) \times {{\mathbf {R}}^M} \times G

\]
is introduced, where

G
G

is an abelian separable locally compact group and where

S
N

(
2
a
)

{S_N}(2a)

is an open ball around zero in

R

N

{\mathbf {R}^N}

with radius

2
a
>
0

2a > 0

. This notion generalizes that one of strongly continuous positive definite operator-valued functions. For these objects, a Bochner-type theorem gives a suitable integral representation if

N
=
1

N = 1

or if the matrix-valued distribution is invariant w.r.t. rotations in

R

N

{\mathbf {R}^N}

. As a consequence, appropriate extensions to the whole group are obtained. In particular, we show that a positive definite function on a certain cylinder in a separable real Hilbert space

H
H

may be extended to a characteristic function of a finite positive measure on

H
H

, if it is invariant w.r.t. rotations and continuous w.r.t. a suitable topology.
A Mandelbrot set whose boundary is piecewise smooth
Authors: Barnsley M. F. , Hardin D. P.
Published: June 2010

It is proved that the Mandelbrot set associated with the pair of mapsw1,2:C→C,w1(z)=sz+1,w2(z)=s∗z−1{w_{1,2}}:{\mathbf {C}} \to {\mathbf {C}}, {w_1}(z) = sz + 1, {w_2}(z) = {s^\ast }z - 1, with parameters∈Cs \in {\mathbf {C}}, is connected and has piecewise smooth boundary.
On the variety of plane curves of degree 𝑑 with 𝛿 nodes and 𝜅 cusps
Authors: Kang Pyung-Lyun
Published: June 2010

Let

P

N

{{\mathbf {P}}^N}

be the projective space which parametrizes all plane curves of degree

d
d

and

V
(
d
,
δ
,
κ
)

V(d,\delta ,\kappa )

the subvariety of

P

N

{{\mathbf {P}}^N}

consisting of all reduced and irreducible plane curves of degree

d
d

with

δ
\delta

nodes and

κ
\kappa

cusps as their only singularities. In this paper we prove that

V
(
d
,
δ
,
κ
)

V(d,\delta ,\kappa )

is irreducible if

κ
⩽
3

\kappa \leqslant 3

, except possibly when

κ
=
3

\kappa = 3

and

d
=
5

d = 5

or

6
6

.
Expansion of the confluent hypergeometric function in series of Bessel functions
Authors: Luke Yudell L.
Published: June 2010

An expansion of the confluent hypergeometric function

Φ
(
a
,
c
,
z
)

\Phi (a,c,z)

in series of functions of the same kind has been given by Buchholz [1]. By specialization of some quantities, there is obtained an expansion in series of modified Bessel functions of the first kind,

I
υ

(
z
)

{I_\upsilon }(z)

, where

v
v

depends on the parameter

a
a

. Tricomi [2, 3] has developed two expansions of similar type where both the order and argument of the Bessel functions depend on the parameters

a
a

and

c
c

. In the present paper, we derive an expansion in series of Bessel functions of integral order whose argument is independent of

a
a

and

c
c

. Our expansion is advantageous for many purposes of computation since the parameters and variable of

Φ
(
a
,
c
,
z
)

\Phi (a,c,z)

appear in separated form. Also, for desk calculation, extensive tables of

I
n

(
z
)

{I_n}(z)

are available, while for automatic computation Bessel functions are easy to generate [4]. Special cases of the confluent function, such as the incomplete gamma function, are also studied. For the class of transcendents known as the error functions, including the Fresnel integrals, it is shown that our expansion coincides with that of Buchholz [1]. By specializing a parameter and passing to a limit, we derive expansions for the exponential integral and related functions. Other expansions for the error and exponential integrals are derived on altogether different bases. Finally, some numerical examples are presented to manifest the efficiency of our formulas.
Expansion of hypergeometric functions in series of other hypergeometric functions
Authors: Luke Yudell L. , Coleman Richard L.
Published: June 2010

In a previous paper [1] one of us developed an expansion for the confluent hypergeometric function in series of Bessel functions. A different expansion of the same kind given by Buchholz [2] was also studied. Since publication of [1], it was found that Rice [3] has also developed an expansion of this type, and yet a fourth expansion of this kind can be deduced from some recent work by Alavi and Wells [4]. In this note, we first deduce a multiplication formula for the Gaussian hypergeometric function which generalizes a statement of Chaundy, see (11), page 187 of [5], and includes a multiplication theorem for the confluent hypergeometric functions due to Erdélyi, see (7), page 283 of [5]. Our principal result is specialized to give an expansion of the confluent hypergeometric function in series of Bessel functions which includes the four above as special cases. With the aid of the Laplace transform, the latter result is used to derive an expansion of the Gaussian hypergeometric function in series of functions of the same kind with changed argument. This is advantageous since, throughout most of the unit disc, the change in argument leads to more rapidly converging series. For special values of the parameters, the expansion degenerates into known quadratic transformations.
The fairing of ship lines on a high-speed computer
Authors: Theilheimer Feodor , Starkweather William
Published: June 2010

Methods for using a digital high-speed computer to determine ship lines are presented. It is assumed that the offsets of a small number of points were taken from a preliminary design, and that it is desired to compute the offsets of an arbitrarily large number of points on the ship’s surface. Procedures for using a computer for the solution of this problem are described. Special emphasis is placed on the detection, by a computational criterion, of unwanted fluctuations and the correction of such fluctuations if they should occur. The method also includes a special procedure which takes care that those portions which are straight in the preliminary design remain straight in the final form. Illustrative examples of the methods are discussed.
Optimum-point formulas for osculatory and hyperosculatory interpolation
Authors: Salzer Herbert E.
Published: June 2010

Formulas are given for n-point osculatory and hyperosculatory (as well as ordinary) polynomial interpolation for

f
(
x
)

f(x)

, over (-1, 1), in terms of

f
(

x
i

)

f({x_i})

,

f
′

(

x
i

)

f’({x_i})

and

f
(

x
i

)

f({x_i})

at the irregularly-spaced Chebyshev points

x
i

=
−
cos
⁡
{
(
2
i
−
1
)
π

/

2
n
}

{x_i} = - \cos \{ (2i - 1)\pi /2n\}

,

i
=
1
,
…
,
n

i = 1,\ldots ,n

. The advantage over corresponding formulas for

x
i

{x_i}

equally spaced is in the squaring and cubing, in the respective osculatory and hyperosculatory formulas, of the approximate ratio of upper bounds for the remainder in ordinary interpolation using Chebyshev and equal spacing (e.g., for n = 10, the 15 per cent ratio for ordinary interpolation becoming 2.4 per cent and 0.37 per cent for osculatory and hyperosculatory interpolation). The upper bounds for the remainders in these optimum n-point r-ply confluent formulas (here r = 1 and 2) are around

2
r

2^r

times those of the optimum

{
(
r
+
1
)
n
}

\{ (r + 1)n\}

-point non-confluent formulas. But these present confluent formulas may require fewer computations for irregular arguments when

f
(
x
)

f(x)

satisfies a simple first or second-order differential equation. To facilitate computation, for n = 2(1)10, auxiliary quantities

a
i

{a_i}

,

b
i

{b_i}

and

c
i

{c_i}

,

i
=
1
,
…
,
n

i = 1,\ldots ,n

, independent of x, are tabulated exactly or to 15S, not precisely for the optimum points, but for those Chebyshev arguments rounded to 2D ("near-optimum” points). At the very worst (n = 9, hyperosculatory) this change about doubles the remainder, which is still less than

(

1
50

)

(\tfrac {1}{50})

th of the remainder in the corresponding equally-spaced formula.
Error analysis of Miller’s recurrence algorithm
Authors: Olver F. W. J.
Published: June 2010

Miller’s algorithm is a device for computing the most rapidly decreasing solution of a second-order linear difference equation. In this paper strict upper bounds are given for the errors in the values yielded by the algorithm, and general conclusions are drawn concerning the accuracy of the process.
Symmetric integration rules for hypercubes. III. Construction of integration rules using null rules
Authors: Lyness J. N.
Published: June 2010

A new operator, which we term a "Null Rule" is defined. Its properties which are analogous to those of an integration rule are investigated. It is found to be useful in the construction of high-dimensional integration rules of moderate degree. A set of integration rules

W

t
+
1

(
n
)

W_{t + 1}^{(n)}

are derived which are more economical in the number of required function evaluations than the previously published

G
¯

t
+
1

(
n
)

\bar G_{t + 1}^{(n)}

.
A starting method for solving nonlinear Volterra integral equations
Authors: Day J. T.
Published: June 2010

In this paper a fifth order starting method is given for Volterra equations of the form

y
(
t
)
=
f
(
t
)
+

∫

x
0

t

k
(
t
,
s
,
y
(
s
)
)

d
s

y(t) = f(t) + \int _{{x_0}}^t {k(t,s,y(s))} ds

. Computational examples are given for the method as a starting method for the Gregory-Newton method.
A continued fraction algorithm for the computation of higher transcendental functions in the complex plane
Authors: Gargantini I. , Henrici P.
Published: June 2010

This report deals with the numerical evaluation of a class of functions of a complex variable that can be represented as Stieltjes transforms of nonnegative real functions. The considered class of functions contains, among others, the confluent hypergeometric functions of Whittaker and the Bessel functions. The method makes it possible, in principle, to compute the values of the function with an arbitrarily small error, using one and the same algorithm in whole complex plane cut along the negative real axis. Detailed numerical data are given for the application of the algorithm to the modified Bessel function

K
0

(
z
)

{K_0}(z)

.
A method for the computation of the Fresnel integrals and related functions
Authors: Fleckner Oscar L.
Published: June 2010

A new method for computing the values of the Fresnel integrals and related functions is developed. Error estimates for the resulting power series are derived, and an application of the technique is discussed.
Numerical solution of the Dirichlet problem for systems of circular conductors between parallel ground lines
Authors: Kammler David W.
Published: June 2010

A Green’s function-integral equation technique is used to obtain a numerical solution to the two-dimensional Dirichlet problem for the multiply connected region between the ground fines

(
V
=
0
)

(V = 0)

at

y
=
0

y = 0

and

y
=
1

y = 1

and exterior to the

N
N

circular conductors with arbitrarily spaced centers

(

x
i

,

y
i

)

({x_i},{y_i})

and radii

R
i

,
i
=
1
,
2
,
⋯
,
N

{R_i},i = 1,2, \cdots ,N

. If the smallest distance between each pair of conducting surfaces exceeds .1, the capacitance matrix for this system of

N
N

conductors can be calculated to 6-place accuracy at a cost of about 15

N
2

{N^2}

seconds on the IBM 7044 computer.
The double points of Mathieu’s differential equation
Authors: Blanch G. , Clemm D. S.
Published: June 2010

Mathieu’s differential equation,

y
+
(
a
−
2
q
cos
⁡
2
x
)
y
=
0

y + (a - 2q\cos 2x)y = 0

, admits of solutions of period

π
\pi

or

2
π

2\pi

for four countable sets of characteristic values,

a
(
q
)

a(q)

, which can be ordered as

a
r

(
q
)

{a_r}(q)

,

r
=
0

r = 0

,

1
1

, …. The power series expansions for the

a
r

(
q
)

{a_r}(q)

converge up to the first double point for that order in the complex plane. [At a double point,

a
r

(
q
)
=

a
r

+
2
(
q
)

{a_r}(q) = {a_r} + 2(q)

.] The present work furnishes the double points for orders

r
r

up to and including 15. These double points are singular points, and the usual methods of determining the characteristic values break down at a singular point. However, it was possible to determine two smooth functions in which one could interpolate for both

q
q

and

a
r

(
q
)

{a_r}(q)

at the singular point. The method is quite general and can be used in other problems as well.
On designs of maximal (+1,-1)-matrices of order 𝑛≡2(𝑚𝑜𝑑4). II
Authors: Yang C. H.
Published: June 2010

Finding maximal

(
+
1
,
−
1
)

( + 1, - 1)

-matrices

M

2
m

{M_{2m}}

of order

2
m

2m

(with odd

m
m

) constructible in the standard form
\[

(

A

a
m
p
;
B

−

B
T

a
m
p
;

A
T

)

\left ( {\begin {array}{*{20}{c}} A & B \\ { - {B^T}} & {{A^T}} \\ \end {array} } \right )

\]
is reduced to the finding of two polynomials

C
(
w
)

C(w)

,

D
(
w
)

D(w)

(corresponding to the circulant submatrices

A
A

,

B
B

) satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel StartAbsoluteValue upper C left-parenthesis w right-parenthesis EndAbsoluteValue squared plus StartAbsoluteValue upper D left-parenthesis w right-parenthesis EndAbsoluteValue squared equals one half left-parenthesis m minus 1 right-parenthesis comma EndLayout">

(

∗

)

|

C
(
w
)

|

2

+

|

D
(
w
)

|

2

=

1
2

(
m
−
1
)
,

\begin{equation}\tag {$*$} |C(w)|^2 + |D(w){|^2} = \tfrac {1}{2}(m - 1),\end{equation}

, where

w
w

is any primitive

m
m

th root of unity. Thus, all

M

2
m

{M_{2m}}

constructible by the standard form (see [4]) can be classified by the formula

(
∗
)

\left ( * \right )

. Some new matrices

M

2
m

{M_{2m}}

for

m
=
25
,
27
,
31

m = 25,27,31

, were found by this method.
Remarks on the iterative solution of the Neumann problem on a rectangle by successive line over-relaxation
Authors: Dorr Fred W.
Published: June 2010

Successive line over-relaxation can be used to solve the equations for certain finite-difference analogs of the Neumann problem for Poisson’s equation on a rectangle. In this note, asymptotic estimates for the choice of relaxation parameter and rate of convergence of this method are collected. These results are then applied to some recent computational experiments carried out by John Gary.
A new type of Chebyshev quadrature
Authors: Barnhill R. E. , Dennis J. E. , Nielson G. M.
Published: June 2010

A Chebyshev quadrature is of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline w left-parenthesis x right-parenthesis f left-parenthesis x right-parenthesis d x asymptotically-equals c sigma-summation Underscript k equals 1 Overscript n Endscripts f left-parenthesis x Subscript k Baseline right-parenthesis">

∫

−
1

1

w
(
x
)
f
(
x
)
d
x
≃

c

∑

k
=
1

n

f
(

x
k

)

\int _{ - 1}^1 {w(x)f(x)dx \simeq } c\sum \limits _{k = 1}^n {f({x_k})}

\]
It is usually desirable that the nodes

x
k

{x_k}

be in the interval of integration and that the quadrature be exact for as many monomials as possible (i.e., the first

n
+
1

n + 1

monomials). For

n
=
1
,
⋅
⋅
⋅
,
7

n = 1, \cdot \cdot \cdot ,7

and

9
9

, such a choice of nodes is possible, but for

n
=
8

n = 8

and

n
>
9

n > 9

, the nodes are complex. In this note, the idea used is that the

l
2

{l^2}

-norm of the deviations of the first

n
+
1

n + 1

monomials from their moments be a minimum. Numerical calculations are carried out for

n
=
8
,
10

n = 8,10

, and

11
11

and one interesting feature of the numerical results is that a “multiple” node at the origin is required. The above idea is then generalized to a minimization of the

l
2

{l^2}

-norm of the deviations of the first

k
k

monomials,

k
≧
n
+
1

k \geqq n + 1

, including

k
=
∞

k = \infty

, and corresponding numerical results are presented.
Sterbenz, P. H
Authors: Ganžela I. F. , Fike C. T.
Published: June 2010

Various writers have dealt with the subject of optimal starting approximations for square-root calculation by Newton’s method. Three optimality criteria that have been used can be shown to lead to closely related approximations. This fact makes it surprisingly easy to choose a starting approximation of some prescribed form so that the maximum relative error after any number of Newton iterations is as small as possible.
Stability of difference approximations of dissipative type for mixed initial-boundary value problems. I
Authors: Osher Stanley
Published: June 2010

H. O. Kreiss, [2], has recently extended the stability theory of difference approximations to include the hyperbolic system
\[

u
t

=
A

u
x

,
0
≦
x
,
t

u_t = A{u_x},0 \leqq x,t

\]
, with

A
A

a diagonal matrix. Appropriate boundary and initial conditions are given. The amplification matrix

Q
^

(
ξ
)

\hat Q(\xi )

need not be diagonal. However, he required that

|

Q
^

(
ξ
)

|

≦
1

|\hat Q(\xi )| \leqq 1

. We use certain results in matrix theory and Wiener-Hopf factorization to replace this restrictive assumption by certain reasonable assumptions on accuracy of

Q
^

(
ξ
)

\hat Q(\xi )

and smoothness of an associated positive-definite symmetric matrix. This technique will be important in half-space problems in many space variables since for such problems the amplification matrix will certainly not be diagonal.
On the approximate minimization of functionals
Authors: Daniel James W.
Published: June 2010

This paper considers in general the problem of finding the minimum of a given functional

f
(
u
)

f(u)

over a set

B
B

by approximately minimizing a sequence of functionals

f
n

(

u
n

)

{f_n}({u_n})

over a "discretized" set

B
n

{B_n}

; theorems are given proving the convergence of the approximating points

u
n

{u_n}

in

B
n

{B_n}

to the desired point

u
u

in

B
B

. Applications are given to the Rayleigh-Ritz method, regularization, Chebyshev solution of differential equations, and the calculus of variations.
Some limiting cases of the 𝐺-transformation
Authors: Gray H. L. , Schucany W. R.
Published: June 2010

In this paper some new nonlinear transformations are introduced. They arise from considering the limit of the

G
G

-transformation as a particular parameter approaches its limiting value. The primary purpose of these transformations is to increase the rate of convergence of an improper integral. However, by introduction of an iteration method it is shown that they may also be used to produce approximating functions for the tail of an improper integral. Several examples are included.
Asymptotic behavior of solutions to the finite-difference wave equation
Authors: Pearson Carl E.
Published: June 2010

A stable finite-difference scheme for the wave equation may possess features not shared by the underlying partial differential equation. These discrepancies are explored; in particular, an asymptotic estimate for the magnitude of precursor effects is obtained.
Amicable numbers and their distribution
Authors: Bratley Paul , Lunnon Fred , McKay John
Published: June 2010

All the amicable pairs whose smaller member is less than108{10^8}are now known. Using this data a conjecture is made about the distribution of amicable pairs. A result on "crowds" (sociable numbers of order 3) is also given.
Computer investigation of Coulomb wave functions
Authors: Pexton Robert L.
Published: June 2010

Numerical examples and heuristic reasoning are employed to illustrate the difficulties encountered in computing irregular Coulomb wave functions by Boersma’s formulae.
Computation of best one-sided 𝐿₁ approximation
Authors: Lewis James T.
Published: June 2010

A computational procedure based on linear programming is presented for finding the best one-sided

L
1

{L_1}

approximation to a given function. A theorem which ensures that the computational procedure yields approximations which converge to the best approximation is proved. Some numerical examples are discussed.
Chebyshev approximations for the Coulomb phase shift
Authors: Cody W. J. , Hillstrom K. E.
Published: June 2010

This note presents nearly-best rational approximations for the Coulomb phase shift

σ
0

(
η
)
=
arg
⁡
Γ
(
1
+
i
η
)

{\sigma _0}(\eta ) = \arg \Gamma (1 + i\eta )

. Maximal relative errors range down to between

4.24
×

10

−
19

4.24 \times {10^{ - 19}}

and

1.09
×

10

−
20

1.09 \times {10^{ - 20}}

. The nontrivial zero of

σ
0

(
η
)

{\sigma _0}(\eta )

is also given.
An error analysis for numerical multiple integration. III
Authors: Barnhill Robert E. , Nielson Gregory M.
Published: June 2010

Asymptotic results of two types are given for minimum norm cubatures, defined on a certain Hubert space of analytic functions. Cubatures with a high degree of precision are realized as the limits of minimum norm cubatures, and an algebraic approach for the determination of efficient rules is derived. Numerical methods and examples are also included.
Cross-product cubature error bounds
Authors: Lether Frank G.
Published: June 2010

This paper is concerned with cross-product cubature rules. We use Sard’s Kernel Theorem

[
10
]

[10]

to express the cross-product cubature error in terms of one variable kernels. This simplified representation of the error is then used to derive cubature error bounds analogous to those obtained by Secrest and Stroud

[
13
]

[13]

, for quadrature rules.
A procedure for conformal maps of simply connected domains by using the Bergman function
Authors: Burbea J.
Published: June 2010

Conformal maps of simply connected domains onto the unit circle are computed by means of the Bergman function of the domain. Ellipses and squares are mapped by this method. Further various parameters of the Schwarz-Christoffel formula are computed in terms of the Bergman function.
An analysis of “boundary-value techniques” for parabolic problems.
Authors: Carasso Alfred , Parter Seymour V.
Published: June 2010

Finite-difference methods for parabolic initial boundary problems are usually treated as marching procedures. However, if the solution reaches a known steady state value as

t
→
∞

t \to \infty

, one may provide approximate values on a line

t
=
T

t = T

for a preselected

T
T

suitably large. With this extra data, it is feasible to consider the use of elliptic boundary-value techniques for the numerical computation of such problems. In this report we give a complete analysis of this method for the linear second-order case with time-independent coefficients. We also discuss iterative methods for solving the difference equations. Finally, we give an example where the method fails.
Chains of quadratic residues
Authors: Gupta Hansraj
Published: June 2010

The problem considered in this paper is that of finding longest chains of the type:

r
1

,

r
2

,

r
3

,
⋯
,

r
m

{r_1},{r_2},{r_3}, \cdots ,{r_m}

, for which the

m
(
m
+
1
)

/

2

m(m + 1)/2

sums

r
i

+

r

i
+
1

+

r

i
+
2

+
⋯
+

r
j

,
1
≦
i
≦
j
≦
m

{r_i} + {r_{i + 1}} + {r_{i + 2}} + \cdots + {r_j},1 \leqq i \leqq j \leqq m

, will be distinct quadratic residues of a given prime p.
𝑃𝐸𝐶𝐸 algorithms for the solution of stiff systems of ordinary differential equations
Authors: Klopfenstein R. W. , Davis C. B.
Published: June 2010

This paper presents a study of a class of PECE algorithms consisting of an application of a predictor followed by application of one iteration of a pseudo Newton-Raphson method to a corrector. Such algorithms require precisely two evaluations of the derivative function for each forward step. Theorems 1 and 4 show that the stability properties of such algorithms compare favorably with those obtained with application of the Newton-Raphson method to the corrector iterated to convergence. A subclass of these algorithms have local truncation error of second order and some have local truncation error of third order. Theorems 2 and 3 exhibit members of this subclass wherein an estimate of the local truncation error is explicit in the algorithm at each step. Initially (in Theorem 1) these algorithms are characterized in terms of their stability properties in the limit as the interval of integration becomes indefinitely large. In Section 5, their properties for other intervals of integration are discussed through the study of some enclosure properties.
Large intervals between consecutive primes
Authors: Cadwell J. H.
Published: June 2010

Some results in number theory, including the Prime Number Theorem, can be obtained by assuming a random distribution of prime numbers. In addition, conjectural formulae, such as Cherwell’s for the density of prime pairs

(
p
,
p
+
2
)

(p,p + 2)

obtained in this way, have been found to agree well with the available evidence. Recently, primes have been determined over ranges of 150,000 numbers with starting points up to

10

15

{10^{15}}

. Statistical arguments are used to obtain a formula for the largest interval between consecutive primes in such a range, and it is found to agree well with recorded values. The same method is applied to predict the maximum interval between consecutive primes occurring below a given integer.
Minimax approximations subject to a constraint
Authors: Fike C. T. , Sterbenz P. H.
Published: June 2010

A class of approximation problems is considered in which a continuous, positive function

φ
(
x
)

\varphi (x)

is approximated by a rational function satisfying some identity. It is proved under certain hypotheses that there is a unique rational approximation satisfying the constraint and yielding minimax relative error and that the corresponding relative-error function does not have an equal-ripple graph. This approximation is, moreover, just the rational approximation to

φ
(
x
)

\varphi (x)

yielding minimax logarithmic error. This approximation, in turn, is just a constant multiple of the rational approximation to

φ
(
x
)

\varphi (x)

yielding minimax relative error but not necessarily satisfying the constraint.
The linear two-point boundary-value problem on an infinite interval
Authors: Robertson T. N.
Published: June 2010

A numerical method, using finite-difference approximations to the second-order differential equation, is given which tests the suitability of the finite point chosen to represent infinity before computing the numerical solution. The theory is illustrated with examples and suggestions for further applications of the method are presented.
On the computation of some Grunsky coefficients relevant to the Bieberbach conjecture
Authors: Ross George G.
Published: June 2010

The Bieberbach coefficients

a
8

,

a

10

,

a

12

{a_8},{a_{10}},{a_{12}}

are parametrized in terms of the Grunsky coefficients. A digital computer is used to carry out the necessary integer calculations.
Stability of parabolic difference approximations to certain mixed initial boundary value problems
Authors: Osher Stanley
Published: June 2010

We consider the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline minus a left-parenthesis x comma t right-parenthesis u Subscript x x Baseline minus b left-parenthesis x comma t right-parenthesis u Subscript x Baseline minus c left-parenthesis x comma t right-parenthesis u equals f left-parenthesis x comma t right-parenthesis">

u
t

−
a
(
x
,
t
)

u

x
x

−
b
(
x
,
t
)

u
x

−
c
(
x
,
t
)
u
=
f
(
x
,
t
)

{u_t} - a(x,t){u_{xx}} - b(x,t){u_x} - c(x,t)u = f(x,t)

\]
in a region

0
≦
x
≦
1
,
t
≧
0

0 \leqq x \leqq 1,t \geqq 0

, with inhomogeneous initial and boundary data. We are concerned with stability and estimates on divided differences in the maximum norm for solutions of consistent implicit, multistep, parabolic difference approximations to this problem. Using a parametrix approach, we give sufficient conditions for certain estimates to be valid.
An improved method for numerical conformal mapping
Authors: Hayes John K. , Kahaner David K. , Kellner Richard G.
Published: June 2010

A new technique for the numerical conformal mapping of a planar region onto the unit disk has been presented and tested by Symm. By elaborating on his methods, we have improved the accuracy of the numerical results by up to four orders of magnitude. For illustration, our methods have been applied to several of the same regions considered in the literature by Symm and Rabinowitz. A flexible FORTRAN code and User’s Guide are reproduced on the microfiche card in this issue.
Uniform approximation through partitioning
Authors: Weinstein S. E.
Published: June 2010

In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set

X
X

is approached through the partitioning of

X
X

and the definition of norms corresponding to the partition and each of the standard

L
p

{L_p}

norms

1
≦
p
>
∞

1 \leqq p > \infty

. For computational convenience, a pseudo norm is defined corresponding to each partition. When the partition is chosen appropriately, the corresponding best approximations (using both the norms and the pseudo norm) are arbitrarily close to a best uniform approximation. A chracterization theorem for best pseudo norm approximation is presented, along with an alternation theorem for best pseudo norm approximation to a univariate function.
Table of irreducible polynomials over 𝐺𝐹[2] of degrees 10 through 20
Authors: Mossige Svein
Published: June 2010

The construction of the tables was based on linear recurring sequences over

GF[2]

{\text {GF[2]}}

. For each degree

n
n

, the polynomials are sorted with respect to their periods. Each polynomial is listed in octal representation with period and decimation.
Monotonicity and iterative approximations involving rectangular matrices
Authors: Plemmons Robert J.
Published: June 2010

A new characterization of row-monotone matrices is given and is related to the Moore-Penrose generalized inverse. The

M
M

-matrix concept is extended to rectangular matrices with full column rank. A structure theorem is provided for all matrices

A
A

with full column rank for which the generalized inverse

A
+

≧
0

{A^ + } \geqq 0

. These results are then used to investigate convergent splittings of rectangular matrices in relation to iterative techniques for computing best least squares solutions to rectangular systems of linear equations.
Diophantine approximation of ternary linear forms. II
Authors: Cusick T. W.
Published: June 2010

Let

θ
\theta

denote the positive root of the equation

x
3

+

x
2

−
2
x
−
1
=
0

{x^3} + {x^2} - 2x - 1 = 0

; that is,

θ
=
2
cos
⁡
(
2
π

/

7
)

\theta = 2\cos (2\pi /7)

. The main result of the paper is the evaluation of the constant

lim

sup

M
→
∞

min

M
2

|

x
+
θ
y
+

θ
2

z

|

\lim {\sup _{M \to \infty }}\min {M^2}|x + \theta y + {\theta ^2}z|

, where the min is taken over all integers

x
,
y
,
z

x,y,z

satisfying

1
≦
max
(

|

y

|

,

|

z

|

)
≦
M

1 \leqq \max (|y|,|z|) \leqq M

. Its value is

(
2
θ
+
3
)

/

7
≈
.78485

(2\theta + 3)/7 \approx .78485

. The same method can be applied to other constants of the same type.
On 𝑘-free sequences of integers
Authors: Wagstaff Samuel S.
Published: June 2010

Let

A

(
k
)

(
n
)

{A^{(k)}}(n)

denote the cardinality of the largest subsequence of

0
,
1
,
2
,
⋯
,
n
−
1

0,1,2, \cdots ,n - 1

, which contains no

k
k

numbers in arithmetical progression. (Such a sequence is called

k
k

-free.)

A

(
k
)

(
n
)

{A^{(k)}}(n)

is computed (on an IBM 360/65) for

3
≦
k
≦
8

3 \leqq k \leqq 8

, and various values of

n
n

to about 50. The results support the old conjecture that for all

k
k

, the limit
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Superscript left-parenthesis k right-parenthesis Baseline equals limit Underscript n right-arrow normal infinity Endscripts left-parenthesis upper A Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis n right-parenthesis right-parenthesis slash n equals 0">

τ

(
k
)

=

lim

n
→
∞

(

A

(
k
)

(
n
)
)

/

n
=
0

{\tau ^{(k)}} = {\lim _{n \to \infty }}({A^{(k)}}(n))/n = 0

. The results
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Superscript left-parenthesis 5 right-parenthesis Baseline greater-than .649 comma tau Superscript left-parenthesis 6 right-parenthesis Baseline greater-than .721 comma tau Superscript left-parenthesis 7 right-parenthesis Baseline greater-than .776">

τ

(
5
)

>
.649
,

τ

(
6
)

>
.721
,

τ

(
7
)

>
.776

{\tau ^{(5)}} > .649,{\tau ^{(6)}} > .721,{\tau ^{(7)}} > .776

, and

τ

(
8
)

>
.8071

{\tau ^{(8)}} > .8071

are obtained. Several cases of a (disproved) conjecture of G. Szekeres are verified, including

A

(
5
)

(
94
)
=
64

{A^{(5)}}(94) = 64

.
On the nonexistence of simplex integration rules for infinite integrals
Authors: Davis P. J. , Rabinowitz P.
Published: June 2010

It is shown that there do not exist integration rules of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline f left-parenthesis x right-parenthesis d x equals sigma-summation Underscript i equals 1 Overscript n Endscripts w Subscript Sub Subscript i Subscript Baseline f left-parenthesis x Subscript i Baseline right-parenthesis plus upper C Subscript n Baseline f Superscript left-parenthesis m right-parenthesis Baseline left-parenthesis xi right-parenthesis comma 0 greater-than xi greater-than normal infinity period">

∫
0
∞

f
(
x
)

d
x
=

∑

i
=
1

n

w

i

f
(

x
i

)
+

C
n

f

(
m
)

(
ξ
)
,

0
>
ξ
>
∞
.

\int _0^\infty {f(x)} dx = \sum \limits _{i = 1}^n {{w_{_i}}f({x_i}) + {C_n}{f^{(m)}}(\xi ),\quad 0 > \xi > \infty .}

\]