bluuebunny/crossref_metadata_2025 huggingface dataset:
Holomorphic maps from 𝐶ⁿ to 𝐶ⁿ
Authors: Rosay Jean-Pierre , Rudin Walter
Published: June 2010

We study holomorphic mappings from

C

n

{{\mathbf {C}}^n}

to

C

n

{{\mathbf {C}}^n}

, and especially their action on countable sets. Several classes of countable sets are considered. Some new examples of Fatou-Bieberbach maps are given, and a nondegenerate map is constructed so that the volume of the image of

C

n

{{\mathbf {C}}^n}

is finite. An Appendix is devoted to the question of linearization of contractions.
Extremal analytic discs with prescribed boundary data
Authors: Chang Chin-Huei , Hu M. C. , Lee Hsuan-Pei
Published: June 2010

This paper concerns the existence and uniqueness of extremal analytic discs with prescribed boundary data in a bounded strictly linearly convex domain

D
D

in

C

n

{{\mathbf {C}}^n}

. We prove that for any two distinct points

p
p

,

q
q

in

∂
D

\partial D

(respectively,

p
∈
∂
D

p \in \partial D

and a vector

v
v

such that

−
1

v
∈

T
p

(
∂
D
)

\sqrt { - 1} v \in {T_p}(\partial D)

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle v comma ModifyingAbove nu With bar left-parenthesis p right-parenthesis mathematical right-angle equals sigma-summation Underscript 1 Overscript n Endscripts v Subscript j Baseline ModifyingAbove nu With bar Subscript j Baseline left-parenthesis p right-parenthesis greater-than 0">

⟨
v
,

ν
¯

(
p
)
⟩
=

∑
1
n

v
j

ν
¯

j

(
p
)
>
0

\langle v,\,\overline \nu (p)\rangle = \sum \nolimits _1^n {{v_j}{{\overline \nu }_j}(p) > 0}

where

ν
(
p
)

\nu (p)

is the outward normal to

∂
D

\partial D

at

p
p

) there exists an extremal analytic disc

f
f

passing through

p
p

,

q
q

if

∂
D
∈

C
k

\partial D \in {C^k}

,

k
⩾
3

k \geqslant 3

(respectively,

f
(
1
)
=
p

f(1) = p

,

f
′

(
1
)
=
v

f’ (1) = v

if

∂
D
∈

C
k

\partial D \in {C^k}

,

k
⩾
14

k \geqslant 14

). Consequently, we can foliate

D
¯

\overline D

with these extremal analytic discs.
Finite-codimensional invariant subspaces of Bergman spaces
Authors: Axler Sheldon , Bourdon Paul
Published: June 2010

For a large class of bounded domains in

C

\mathbb {C}

, we describe those finite codimensional subspaces of the Bergman space that are invariant under multiplication by

z
z

. Using different techniques for certain domains in

C

N

{\mathbb {C}^N}

, we describe those finite codimensional subspaces of the Bergman space that are invariant under multiplication by all the coordinate functions.
Curves of genus 2 with split Jacobian
Authors: Kuhn Robert M.
Published: June 2010

We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If

X
X

is a curve of genus

2
2

, and

f
:
X
→
E

f:X \to E

a map from

X
X

to an elliptic curve, then

X
X

has split jacobian. It is not true that a complement to

E
E

in the jacobian of

X
X

is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve

E
′

E’

and algebraic

f
:
X
→

E
′

f:X \to E’

, and we give an algorithm for finding that curve. The construction works in any characteristic other than two. Applications of the algorithm are given to give explicit examples in characteristics

0
0

and

3
3

.
First- and second-order epi-differentiability in nonlinear programming
Authors: Rockafellar R. T.
Published: June 2010

Problems are considered in which an objective function expressible as a max of finitely many

C
2

{C^2}

functions, or more generally as the composition of a piecewise linear-quadratic function with a

C
2

{C^2}

mapping, is minimized subject to finitely many

C
2

{C^2}

constraints. The essential objective function in such a problem, which is the sum of the given objective and the indicator of the constraints, is shown to be twice epi-differentiable at any point where the active constraints (if any) satisfy the Mangasarian-Fromovitz qualification. The epi-derivatives are defined by taking epigraphical limits of classical first-and second-order difference quotients instead of pointwise limits, and they reveal properties of local geometric approximation that have not previously been observed.
Coordinatization in superstable theories. II
Authors: Buechler Steven
Published: June 2010

In this paper we prove Theorem A. Suppose that

T
T

is superstable and

U
(
a

/

A
)
=
α
+
1

U(a/A) = \alpha + 1

, for some

α
\alpha

. Then in

T

eq

{T^{{\text {eq}}}}

there is a

c
∈
acl
⁡
(
A
a
)
∖
acl
⁡
(
A
)

c \in \operatorname {acl} (Aa)\backslash \operatorname {acl} (A)

such that one of the following holds. (i)

U
(
c

/

A
)
=
1

U(c/A) = 1

. (ii)

stp
⁡
(
c

/

A
)

\operatorname {stp} (c/A)

has finite Morley rank. In fact, this strong type is semiminimal with respect to a strongly minimal set.
Stable maps into free 𝐺-spaces
Authors: Greenlees J. P. C.
Published: June 2010

In this paper we introduce a systematic method for calculating the group of stable equivariant maps

[
X
,

Y

]
G

{[X,\,Y]^G}

[3, 18] into a

G
G

-free space or spectrum

Y
Y

. In fact the method applies without restriction on

X
X

whenever

G
G

is a

p
p

-group and

Y
Y

is

p
p

-complete and satisfies standard finiteness assumptions. The method is an Adams spectral sequence based on a new equivariant cohomology theory

c

∗

(
X
)

{c^{\ast }}(X)

which we introduce in

§
1

\S 1

. This spectral sequence is quite calculable and provides a natural generalisation of the classical Adams spectral sequence based on ordinary

mod
p

\bmod p

cohomology. It also geometrically realises certain inverse limits of nonequivariant Adams spectral sequences which have been useful in the study of the Segal conjecture [19, 5, 21, 9].
Markov-Duffin-Schaeffer inequality for polynomials with a circular majorant
Authors: Rahman Q. I. , Schmeisser G.
Published: June 2010

If

p
p

is a polynomial of degree at most

n
n

such that

|

p
(
x
)

|

⩽

1
−

x
2

|p(x)| \leqslant \sqrt {1 - {x^2}}

for

−
1
⩽
x
⩽
1

- 1 \leqslant x \leqslant 1

, then for each

k
k

,

max

|

p

(
k
)

(
x
)

|

\max |{p^{(k)}}(x)|

on

[
−
1
,

1
]

[ - 1,\,1]

is maximized by the polynomial

(

x
2

−
1
)

U

n
−
2

(
x
)

({x^2} - 1){U_{n - 2}}(x)

where

U
m

{U_m}

is the

m
m

th Chebyshev polynomial of the second kind. The purpose of this paper is to investigate if it is enough to assume

|

p
(
x
)

|

⩽

1
−

x
2

|p(x)| \leqslant \sqrt {1 - {x^2}}

at some appropriately chosen set of

n
+
1

n + 1

points in

[
−
1
,

1
]

[ - 1,\,1]

. The problem is inspired by a well-known extension of Markov’s inequality due to Duffin and Schaeffer.
Subordination families and extreme points
Authors: Abu-Muhanna Yusuf , Hallenbeck D. J.
Published: June 2010

Let

s
(
F
)

s(F)

denote the set of functions subordinate to a univalent function

F
F

in

Δ
\Delta

the unit disk. Let

B
0

{B_0}

denote the set of functions

ϕ
(
z
)

\phi (z)

analytic in

Δ
\Delta

satisfying

|

ϕ
(
z
)

|

>
1

|\phi (z)| > 1

and

ϕ
(
0
)
=
0

\phi (0) = 0

. We prove that if

f
=
F
∘
ϕ

f = F \circ \phi

is an extreme point of

s
(
F
)

s(F)

, then

ϕ
\phi

is an extreme point of

B
0

{B_0}

. Let

D
=
F
(
s
)

D = F(s)

and

λ
(
w
,

∂
D
)

\lambda (w,\,\partial D)

denote the distance between

w
w

and

∂
D

\partial D

(boundary of

D
D

). We also prove that if

ϕ
\phi

is an extreme point of

B
0

{B_0}

and

|

ϕ
(

e

i
t

)

|

>
1

|\phi ({e^{it}})| > 1

for almost all

t
t

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 2 pi Baseline log lamda left-parenthesis upper F left-parenthesis phi left-parenthesis e Superscript i t Baseline right-parenthesis e Superscript i theta Baseline right-parenthesis comma partial-differential upper D right-parenthesis d t equals negative normal infinity">

∫
0

2
π

log
⁡
λ
(
F
(
ϕ
(

e

i
t

)

e

i
θ

)
,

∂
D
)

d
t
=
−
∞

\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}}){e^{i\theta }}),\,\partial D)\,dt = - \infty }

for almost all

θ
\theta

.
The blow-up surface for nonlinear wave equations with small spatial velocity
Authors: Friedman Avner , Oswald Luc
Published: June 2010

Consider the Cauchy problem for

u

t
t

−

ε
2

Δ
u
=
f
(
u
)

{u_{tt}} - {\varepsilon ^2}\Delta u = f(u)

in space dimension

⩽
3

\leqslant 3

where

f
(
u
)

f(u)

is superlinear and nonnegative. The solution blows up on a surface

t
=

ϕ
ε

(
x
)

t = {\phi _\varepsilon }(x)

. Denote by

t
=
ϕ
(
x
)

t = \phi (x)

the blow-up surface corresponding to

v
=
f
(
v
)

v = f(v)

. It is proved that

|

ϕ
ε

(
x
)
−
ϕ
(
x
)

|

⩽
C

ε
2

|{\phi _\varepsilon }(x) - \phi (x)| \leqslant C{\varepsilon ^2}

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue nabla left-parenthesis phi Subscript epsilon Baseline left-parenthesis x right-parenthesis minus phi left-parenthesis x right-parenthesis right-parenthesis EndAbsoluteValue less-than-or-slanted-equals upper C epsilon squared">

|

∇
(

ϕ
ε

(
x
)
−
ϕ
(
x
)
)

|

⩽
C

ε
2

|\nabla ({\phi _\varepsilon }(x) - \phi (x))| \leqslant C{\varepsilon ^2}

in a neighborhood of any point

x
0

{x_0}

where

ϕ
(

x
0

)
>
∞

\phi ({x_0}) > \infty

.
Differentiation theorem for Gaussian measures on Hilbert space
Authors: Tišer Jaroslav
Published: June 2010

It is shown that the differentiation theorem is valid in infinitely dimensional Hilbert space with certain Gaussian measures. The proof uses result from harmonic analysis concerning the behavior of Hardy-Littlewood maximal operator in highly dimensional space.
A sparse Graham-Rothschild theorem
Authors: Prömel Hans Jürgen , Voigt Bernd
Published: June 2010

The main result of this paper is a sparse version of the Graham-Rothschild partition theorem for

n
n

-parameter sets [R. L. Graham and B. L. Rothschild, Ramsey’s theorem for

n
n

-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257-292]. In particular, a sparse version of Hales-Jewett’s theorem is proved. We give several applications, e.g., for arithmetic progressions and finite sums of integers, confirming conjectures of J. Spencer and of J. Nešetřil and V. Rödl. We also consider graphs defined on parameter sets and prove a sparse and restricted induced partition theorem for such graphs, extending results from [H. J. Prömel, Induced partition properties of combinatorial cubes, J. Combin. Theory Ser. A 39 (1985), 177-208] and [P. Frankl, R. L. Graham, and V. Rödl, Induced restricted Ramsey theorems for spaces, J. Combin. Theory Ser. A 44 (1987), 120-128].
Valuations on meromorphic functions of bounded type
Authors: Nakai Mitsuru
Published: June 2010

The primary purpose of this paper is to show that every valuation on the field of meromorphic functions of bounded type on a finitely sheeted unlimited covering Riemann surface is a point valuation if and only if the same is true on its base Riemann surface. The result is then applied to concrete examples and some related results are obtained.
Symmetry diffeomorphism group of a manifold of nonpositive curvature
Authors: Eberlein Patrick
Published: June 2010

Let

M
~

\tilde M

denote a complete simply connected manifold of nonpositive sectional curvature. For each point

p
∈

M
~

p \in \tilde M

let

s
p

{s_p}

denote the diffeomorphism of

M
~

\tilde M

that fixes

p
p

and reverses all geodesics through

p
p

. The symmetry diffeomorphism group

G

∗

{G^{\ast }}

generated by all diffeomorphisms

{

s
p

:

p
∈

M
~

}

\{ {s_p}:\,p \in \tilde M\}

extends naturally to group of homeomorphisms of the boundary sphere

M
~

(
∞
)

\tilde M(\infty )

. A subset

X
X

of

M
~

(
∞
)

\tilde M(\infty )

is called involutive if it is invariant under

G

∗

{G^{\ast }}

. Theorem. Let

X
⊆

M
~

(
∞
)

X \subseteq \tilde M(\infty )

be a proper, closed involutive subset. For each point

p
∈

M
~

p \in \tilde M

let

N
(
p
)

N(p)

denote the linear span in

T
p

M
~

{T_p}\tilde M

of those vectors at

p
p

that are tangent to a geodesic

γ
\gamma

whose asymptotic equivalence class

γ
(
∞
)

\gamma (\infty )

belongs to

X
X

. If

N
(
p
)

N(p)

is a proper subspace of

T
p

M
~

{T_p}\tilde M

for some point

p
∈

M
~

p \in \tilde M

, then

M
~

\tilde M

splits as a Riemannian product

M
~

1

×

M
~

2

{\tilde M_1} \times {\tilde M_2}

such that

N
N

is the distribution of

M
~

\tilde M

induced by

M
~

1

{\tilde M_1}

. This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if

M
~

\tilde M

is irreducible and

M
~

(
∞
)

\tilde M(\infty )

admits a proper, closed involutive subset

X
X

, then

M
~

\tilde M

is isometric to a symmetric space of noncompact type and rank

k
⩾
2

k \geqslant 2

.
Hausdorff dimension in graph directed constructions
Authors: Mauldin R. Daniel , Williams S. C.
Published: June 2010

We introduce the notion of geometric constructions in

R

m

{{\mathbf {R}}^m}

governed by a directed graph

G
G

and by similarity ratios which are labelled with the edges of this graph. For each such construction, we calculate a number

α
\alpha

which is the Hausdorff dimension of the object constructed from a realization of the construction. The measure of the object with respect to

H

α

{\mathcal {H}^\alpha }

is always positive and

σ
\sigma

-finite. Whether the

H

α

{\mathcal {H}^\alpha }

-measure of the object is finite depends on the order structure of the strongly connected components of

G
G

. Some applications are given.
Moduli spaces of Riemann surfaces of genus two with level structures. I
Authors: Lee Ronnie , Weintraub Steven H.
Published: June 2010

The cohomology of modular varieties defined by congruence subgroups of

Sp
4

(

Z

)

{\operatorname {Sp} _4}({\mathbf {Z}})

whose levels lie between

2
2

and

4
4

is studied. Using a counting argument and the techniques of zeta functions, the authors completely determine the cohomology of a particular variety of this type.
The 𝑞-Selberg polynomials for 𝑛=2
Authors: Kadell Kevin W. J.
Published: June 2010

We have conjectured that Selberg’s integral has a plethora of extensions involving the Selberg polynomials and proved that these are the Schur functions for

k
=
1

k = 1

. We prove this conjecture for

n
=
2

n = 2

and show that the polynomials are, in a formal sense, Jacobi polynomials. We conjecture an orthogonality relation for the Selberg polynomials which combines orthogonality relations for the Schur functions and Jacobi polynomials. We extend a basic Schur function identity. We give a

q
q

-analogue of the Selberg polynomials for

n
=
2

n = 2

using the little

q
q

-Jacobi polynomials.
On the regularity up to the boundary in the Dirichlet problem for degenerate elliptic equations
Authors: Bergamasco Adalberto P. , Gerszonowicz Jorge A. , Petronilho Gerson
Published: June 2010

We give a simple proof of the regularity up to the boundary of solutions of the Dirichlet problem for a class of second-order degenerate elliptic equations in the plane. We show that the method of transfer to the boundary via the associated heat equations, can be used to reduce the problem to proving the ellipticity or hypoellipticity of a pseudodifferential operator on the boundary.
The structure of quasimultipliers of 𝐶*-algebras
Authors: Lin Hua Xin
Published: June 2010

Let

A
A

be a

C
∗

{C^\ast }

-algebra and

A

∗
∗

{A^{\ast \ast }}

its enveloping

W
∗

{W^\ast }

-algebra. Let

LM

(
A
)

{\text {LM}}(A)

be the left multipliers of

A
A

,

RM

(
A
)

{\text {RM}}(A)

the right multipliers of

A
A

and

QM

(
A
)

{\text {QM}}(A)

the quasi-multipliers of

A
A

. A question was raised by Akemann and Pedersen [1] whether

QM

(
A
)
=

LM

(
A
)
+

RM

(
A
)

{\text {QM}}(A) = {\text {LM}}(A) + {\text {RM}}(A)

. McKennon [20] gave a nonseparable counterexample. L. Brown [6] shows the answer is negative for stable (separable)

C
∗

{C^\ast }

-algebras also. In this paper, we mainly consider

σ
\sigma

-unitial

C
∗

{C^\ast }

-algebras. We give a criterion for

QM

(
A
)
=

LM

(
A
)
+

RM

(
A
)

{\text {QM}}(A) = {\text {LM}}(A) + {\text {RM}}(A)

. In the case that

A
A

is stable, we give a necessary and sufficient condition for

QM

(
A
)
=

LM

(
A
)
+

RM

(
A
)

{\text {QM}}(A) = {\text {LM}}(A) + {\text {RM}}(A)

. We also give answers for other

C
∗

{C^\ast }

-algebras.
Deforming a PL submanifold of Euclidean space into a hyperplane
Authors: Vrabec Jože
Published: June 2010

Let

M
M

be a closed,

k
k

-connected,

m
m

-dimensional

PL

{\text {PL}}

submanifold of

R

2
m
−
k
−
1

(
1
≤
k
≤
m
−
4
)

{\mathbb {R}^{2m - k - 1}}\;(1 \leq k \leq m - 4)

. The main result of this paper states that if

m
−
k

m - k

is even, then every embedding of

M
M

into

R

2
m
−
k

{\mathbb {R}^{2m - k}}

can be isotopically deformed into

R

2
m
−
k
−
1

{\mathbb {R}^{2m - k - 1}}

, and specifies which embeddings of

M
M

into

R

2
m
−
k

{\mathbb {R}^{2m - k}}

can be deformed into

R

2
m
−
k
−
1

{\mathbb {R}^{2m - k - 1}}

in case

m
−
k

m - k

is odd.
The complex bordism of groups with periodic cohomology
Authors: Bahri Anthony , Bendersky Martin , Davis Donald M. , Gilkey Peter B.
Published: June 2010

Is is proved that if

B
G

BG

is the classifying space of a group

G
G

with periodic cohomology, then the complex bordism groups

M

U

∗

(
B
G
)

M{U_{\ast }}(BG)

are obtained from the connective

K
K

-theory groups

k

u

∗

(
B
G
)

k{u_{\ast }}(BG)

by just tensoring up with the generators of

M

U

∗

M{U_{\ast }}

as a polynomial algebra over

k

u

∗

k{u_{\ast }}

. The explicit abelian group structure is also given. The bulk of the work is the verification when

G
G

is a generalized quaternionic group.
Additive cohomology operations
Authors: Duflot Jeanne
Published: June 2010

The bigraded group

{

H
i

(

Σ
n

,

Z

/

p
)
}

\{ {H_i}({\Sigma _n},{\mathbf {Z}}/p)\}

becomes a Hopf algebra, if multiplication is induced by restriction, and comultiplication is induced by transfer. Using Steenrod’s method of considering elements of this bigraded group as

mod

 - 

p

\bmod {\text { - }}p

cohomology operations, the primitives of this Hopf algebra correspond to additive cohomology operations. In this paper we use the results known about the homology and cohomology of the symmetric groups and the operations they induce in

mod

 - 

p

\bmod {\text { - }}p

cohomology to write down two (additive) bases of the bigraded vector space of primitives of the above Hopf algebra.
The spectral measure and Hilbert transform of a measure-preserving transformation
Authors: Campbell James , Petersen Karl
Published: June 2010

V. F. Gaposhkin gave a condition on the spectral measure of a normal contraction on

L
2

{L^2}

sufficient to imply that the operator satisfies the pointwise ergodic theorem. We prove that unitary operators which come from measure-preserving transformations satisfy a stronger version of this condition. This follows from the fact that the rotated ergodic Hubert transform is a continuous function of its parameter. The maximal inequality on which the proof depends follows from an analytic inequality related to the Carleson-Hunt Theorem on the a.e. convergence of Fourier series.
Summability of Hermite expansions. II
Authors: Thangavelu S.
Published: June 2010

We study the summability of

n
n

-dimensional Hermite expansions where

n
>
1

n > 1

. We prove that the critical index for the Riesz summability is

(
n
−
1
)

/

2

(n - 1)/2

. We also prove analogues of the Fejér-Lebesgue theorem and Riemann’s localisation principle when the index

α
\alpha

of the Riesz means is

>
(
3
n
−
2
)

/

6

> (3n - 2)/6

.
Δ-closures of ideals and rings
Authors: Ratliff Louis J.
Published: June 2010

It is shown that if

R
R

is a commutative ring with identity and

Δ
\Delta

is a multiplicatively closed set of finitely generated nonzero ideals of

R
R

, then the operation

I
→

I
Δ

=

∪

K
∈
Δ

(
I
K
:
K
)

I \to {I_\Delta } = { \cup _{K \in \Delta }}(IK:K)

is a closure operation on the set of ideals

I
I

of

R
R

that satisfies a partial cancellation law, and it is a prime operation if and only if

R
R

is

Δ
\Delta

-closed. Also, if none of the ideals in

Δ
\Delta

is contained in a minimal prime ideal, then

I
Δ

⊆

I
a

{I_\Delta } \subseteq {I_a}

, the integral closure of

I
I

in

R
R

, and if

Δ
\Delta

is the set of all such finitely generated ideals and

I
I

contains an ideal in

Δ
\Delta

, then

I
Δ

=

I
a

{I_\Delta } = {I_a}

. Further,

R
R

has a natural

Δ
\Delta

-closure

R
Δ

,
A
→

A
Δ

{R^\Delta },A \to {A^\Delta }

is a closure operation on a large set of rings

A
A

that contain

R
R

as a subring,

A
→

A
Δ

A \to {A^\Delta }

behaves nicely under certain types of ring extension, and every integral extension overring of

R
R

is

R
Δ

{R^\Delta }

for an appropriate set

Δ
\Delta

. Finally, if

R
R

is Noetherian, then the associated primes of

I
Δ

{I_\Delta }

are also associated primes of

I
Δ

K

{I_\Delta }K

and

(
I
K

)
Δ

{(IK)_\Delta }

for all

K
∈
Δ

K \in \Delta

.
A note on local change of diffeomorphism
Authors: Masuda Mikiya
Published: June 2010

Let

D
(
M
)

D(M)

be the group of pseudo-isotopy classes of orientation preserving diffeomorphisms of a compact manifold

M
M

which restrict to the identity on

∂
M

\partial M

. If a compact manifold

N
N

of the same dimension as

M
M

is embedded in

M
M

, then extending maps in

D
(
N
)

D(N)

as the identity on the exterior of

N
N

defines a homomorphism

E
:
D
(
N
)
→
D
(
M
)

E:D(N) \to D(M)

. We ask if the kernel of

E
E

is finite and show that this is the case for special cases.
A calculus approach to hyperfunctions. II
Authors: Matsuzawa Tadato
Published: June 2010

We consider any hyperfunctions with the compact support as initial values of the solutions of the heat equation. The main aim of this paper is to unify the theory of distributions and hyperfunctions as well as simplify proofs of some important results via heat kernel.
On linear algebraic semigroups. II
Authors: Putcha Mohan S.
Published: June 2010

We continue from [11] the study of linear algebraic semigroups. Let S be a connected algebraic semigroup defined over an algebraically closed field K. Let

U

(
S
)

\mathcal {U}(S)

be the partially ordered set of regular

J

\mathcal {J}

-classes of S and let

E
(
S
)

E(S)

be the set of idempotents of S. The following theorems (among others) are proved. (1)

U

(
S
)

\mathcal {U}(S)

is a finite lattice. (2) If S is regular and the kernel of S is a group, then the maximal semilattice image of S is isomorphic to the center of

E
(
S
)

E(S)

. (3) If S is a Clifford semigroup and

f

∈

E
(
S
)

f\, \in \,E(S)

, then the set

{

e

|

e

∈

E
(
S
)
,

e

⩾

f
}

\{ \,e\,|\,e\, \in \,E(S),\,e\, \geqslant \,f\}

is finite. (4) If S is a Clifford semigroup, then there is a commutative connected closed Clifford subsemigroup T of S with zero such that T intersects each

J

\mathcal {J}

-class of S. (5) If S is a Clifford semigroup with zero, then S is commutative and is in fact embeddable in

(

K
n

,

⋅
)

({K^n},\, \cdot )

for some

n

∈

Z

+

n\, \in \,{\textbf {Z}^ + }

. (6) If

ch

⋅

K

=

0

{\text {ch}}\, \cdot \,K\, = \,0

and S is a commutative Clifford semigroup, then S is isomorphic to a direct product of an abelian connected unipotent group and a closed connected subsemigroup of

(

K
n

,

⋅
)

({K^n},\, \cdot )

for some

n

∈

Z

+

n\, \in \,{\textbf {Z}^ + }

. (7) If S is a regular semigroup and

dim

⋅

S

⩽

2

{\text {dim}}\, \cdot \,S\, \leqslant \,2

, then

|

U

(
S
)

|

⩽

4

\left | {\mathcal {U}(S)} \right |\, \leqslant \,4

. (8) If S is a Clifford semigroup with zero and

dim

⋅

S

=

3

{\text {dim}}\, \cdot \,S\, = \,3

, then

|

E
(
S
)

|

=

|

U

(
S
)

|

\left | {E(S)} \right |\, = \,\left | {\mathcal {U}(S)} \right |

can be any even number

⩾

8

\geqslant \,8

. (9) If S is a Clifford semigroup then

U

(
S
)

\mathcal {U}(S)

is a relatively complemented lattice and all maximal chains in

U

(
S
)

\mathcal {U}(S)

have the same number of elements.
The reduction of an arbitrary real square matrix to tridiagonal form using similarity transformations
Authors: LaBudde C. Donald
Published: June 2010

In this paper a new algorithm for reducing an arbitrary real square matrix to tri-diagonal form using real similarity transformations is described. The method is essentially a generalization of a method due to A. S. Householder for accomplishing the same reduction in the case where the matrix is real and symmetric.
Numerical integration over the 𝑛-dimensional spherical shell
Authors: Mustard D.
Published: June 2010

The n-dimensional generalisation of a theorem by W. H. Peirce [1] is given, providing a method for constructing product type integration rules of arbitrarily high polynomial precision over a hyperspherical shell region and using a weight function

r
s

{r^s}

. Table I lists orthogonal polynomials, coordinates and coefficients for integration points in the angular rules for 3rd and 7th degree precision and for

n
=
3
(
1
)
8

n = 3(1)8

. Table II gives the radial rules for a shell of internal radius R and outer radius 1: (i) a formula for the coordinate and coefficient in the 3rd degree rule for arbitrary n, R; (ii) a formula for the coordinates and coefficients for the 7th degree rule for arbitrary n and R = 0 and (iii) a table of polynomials, coordinates and coefficients to 9D for n = 4, 5 and

R
=
0
,

1
4

,

1
2

,

3
4

R = 0,\tfrac {1}{4},\tfrac {1}{2},\tfrac {3}{4}

.
Numerical solution for flux components in potential flow
Authors: von Rosenberg Dale U.
Published: June 2010

Values of the flux components are often desired in potential flow problems. Second-order correct finite-difference analogs are developed for the differential equations defining these flux components. Two iterative methods of solving the resulting finite-difference equations are presented. Experimental results indicate the most efficient value of the iteration parameter and demonstrate that the number of iterations required is approximately proportional to the square root of the number of points in the grid.
The numerical integration of ordinary differential equations
Authors: Gear C. W.
Published: June 2010

Multistep methods for initial value problems are expressed in a matrix form. The application of such methods to higher-order equations is studied with the result that new techniques for both first- and higher-order equations are found. The direct approach to higher-order equations is believed to offer speed and accuracy advantages; some numerical evidence is presented. The new technique applied to first-order equations is a slight extension of the conventional multistep method and avoids the Dahlquist [2] stability theorem, that is, these new

k
k

-step methods are of order

2
k

2k

and yet convergent. The matrix formalism introduced provides an easy mechanism for examining the equivalence of methods as introduced by Descloux [3]. It is pointed out that the new first-order method on

k
k

- steps, Adams’ method on

(
2
k
−
1
)

(2k - 1)

-steps and Nordsieck’s [7] method with

2
k

2k

components are equivalent to each other. In fact, all methods discussed can be placed in equivalence classes so that theorems need only be proved for one member of each class. The choice between the members of a class can be made on the basis of round-off errors and amount of computation only. Arguments are given in favor of the extension of Nordsieck’s method for general use because of its speed and applicability to higher order problems directly. The theorems ensuring convergence and giving the asymptotic form of the error are stated. The proofs can be found in a cited report.
Numerical analysis of boundary-layer problems in ordinary differential equations
Authors: Murphy W. D.
Published: June 2010

We categorize some of the finite-difference methods that can be used to treat the initial-value problem for the boundary-layer differential equation
\[

(
1
)

 

μ

y
′

=
f
(
y
,
x
)
;
y
(
0
)
=

y
0

.

(1){\text { }}\mu y’ = f(y,x);y(0) = {y^0}.

\]
These methods take the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 right-parenthesis sigma-summation Underscript i equals 0 Overscript k Endscripts alpha Subscript i Baseline upper Y Subscript n plus i Baseline equals h Superscript 1 minus gamma Baseline sigma-summation Underscript i equals 0 Overscript k Endscripts beta Subscript i Baseline f left-parenthesis upper Y Subscript n plus i Baseline comma x Subscript n plus i Baseline right-parenthesis plus upper R Subscript n Baseline comma">

(
2
)

 

∑

i
=
0

k

α
i

Y

n
+
i

=

h

1
−
γ

∑

i
=
0

k

β
i

f

(

Y

n
+
i

,

x

n
+
i

)
+

R
n

,

(2){\text { }}\sum \limits _{i = 0}^k {{\alpha _i}{Y_{n + i}} = {h^{1 - \gamma }}} \sum \limits _{i = 0}^k {{\beta _i}f} ({Y_{n + i}},{x_{n + i}}) + {R_n},

\]
where

α
ν

{\alpha _\nu }

and

β
ν

(
ν
=
0
,
1
,
⋯
,
k
)

{\beta _\nu }(\nu = 0,1, \cdots ,k)

denote real constants which do not depend upon

n
,

R
n

n,{R_n}

is the round-off error,

μ
=

h
r

,
0
>
γ
>
1

\mu = {h^r},0 > \gamma > 1

, and

h
h

is the mesh size. We define a new kind of stability called

μ
\mu

stability and prove that under certain conditions

μ
\mu

stability implies convergence of the difference method. We investigate

μ
\mu

stability and the optimal methods which it allows, i.e., methods of maximum accuracy. The idea of relating

μ
\mu

to

h
h

allows us to study the nature of the difference equation for very small

μ
\mu

. We can, however, look at this in another way. Given a differential equation in the form of Eq. (1) we ask how can we choose

h
h

so that the associated difference equation will give an accurate approximation. If

μ
\mu

is sufficiently small, choose

h
h

by the formula

h
=

μ

1

/

γ

h = {\mu ^{1/\gamma }}

where

0
>
γ
>
1

0 > \gamma > 1

.
Sylvester’s identity and multistep integer-preserving Gaussian elimination
Authors: Bareiss Erwin H.
Published: June 2010

A method is developed which permits integer-preserving elimination in systems of linear equations,

A
X
=
B

AX = B

, such that

(
a
)

(a)

the magnitudes of the coefficients in the transformed matrices are minimized, and

(
b
)

(b)

the computational efficiency is considerably increased in comparison with the corresponding ordinary (single-step) Gaussian elimination. The algorithms presented can also be used for the efficient evaluation of determinants and their leading minors. Explicit algorithms and flow charts are given for the two-step method. The method should also prove superior to the widely used fraction-producing Gaussian elimination when

A
A

is nearly singular.
Explicit inverses and condition numbers of certain circulants
Authors: Charmonman S. , Julius R. S.
Published: June 2010

Explicit inverses and condition numbers of two test-circulants with first rows

{
a
,
a
+
h
,
⋯
,
a
+
(
n
−
1
)
h
}

\{ a,a + h, \cdots ,a + (n - 1)h\}

and

{
a
,
a
h
,
⋯
,
a

h

n
−
1

}

\{ a,ah, \cdots ,a{h^{n - 1}}\}

respectively are given in terms of the parameters defining the circulants.
On the resolvent of a linear operator associated with a well-posed Cauchy problem
Authors: Miller John
Published: June 2010

We show how local estimates may be obtained for holomorphic functions of a class of linear operators on a finite-dimensional linear vector space. This is accomplished by classifying the spectrum of each operator and then estimating its resolvent on certain contours in the left half-plane. We apply these methods to prove some known theorems, and in addition we obtain new estimates for the inverse of these operators. Analogous results for power-bounded operators are given in [3].
The Riemann hypothesis and pseudorandom features of the Möbius sequence
Authors: Good I. J. , Churchhouse R. F.
Published: June 2010

A study of the cumulative sums of the Möbius function on the Atlas Computer of the Science Research Council has revealed certain statistical properties which lead the authors to make a number of conjectures. One of these is that any conjecture of the Mertens type, viz.
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper M left-parenthesis upper N right-parenthesis EndAbsoluteValue equals StartAbsoluteValue sigma-summation Underscript n equals 1 Overscript upper N Endscripts mu left-parenthesis n right-parenthesis EndAbsoluteValue greater-than k left-parenthesis square-root upper N right-parenthesis">

|

M
(
N
)

|

=

|

∑

n
=
1

N

μ
(
n
)

|

>
k

(

√
N

)

|M(N)| = \left | {\sum \limits _{n = 1}^N {\mu (n)} } \right | > k\left ( {\surd N} \right )

\]
where

k
k

is any positive constant, is false, and indeed the authors conjecture that
\[

Lim

sup

{

M
(
x
)

(
x
log
⁡
log
⁡
x
)

−
1

/

2

}

=

√

(

12

)

/

√

(

12

)

π

π

{\text {Lim}}\sup \left \{ {M(x){{(x\log \log x)}^{ - 1/2}}} \right \} = {{\surd \left ( {12} \right )} \left / {\vphantom {{\surd \left ( {12} \right )} \pi }} \right . \pi }

\]
.
Summation of series of positive terms by condensation transformations.
Authors: Daniel James W.
Published: June 2010

The condensation transformation, which maps series of positive terms into more conveniently summed alternating series, each term

v
j

{v_j}

of which is itself an infinite series, is discussed with examples. It is shown that for a large class of extremely slowly convergent series (essentially those dominated by the “logarithmic scale") the series defining the terms

v
j

{v_j}

are more easily summed than the original and may in fact be transformed further if desired. Numerical examples reveal the power of the method.
On the condition of a matrix arising in the numerical inversion of the Laplace transform
Authors: Gautschi Walter
Published: June 2010

Bellman, Kalaba, and Lockett recently proposed a numerical method for inverting the Laplace transform. The method consists in first reducing the infinite interval of integration to a finite one by a preliminary substitution of variables, and then employing an

n
n

-point Gauss-Legendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of

n
n

linear algebraic equations. Luke suggests the possibility of using Gauss-Jacobi quadrature (with parameters

α
\alpha

and

β
\beta

) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters

α
\alpha

,

β
\beta

may have a beneficial influence on the condition of the linear system of equations. The object of this note is to investigate the condition number cond

(
n
,
α
,
β
)

(n,\alpha ,\beta )

of this system as a function of

n
n

,

α
\alpha

, and

β
\beta

. It is found that cond

(
n
,
α
,
β
)

(n,\alpha ,\beta )

is usually larger than cond

(
n
,
β
,
α
)

(n,\beta ,\alpha )

if

β
>
α

\beta > \alpha

, at least asymptotically as

n
→
∞

n \to \infty

. Lower bounds for cond

(
n
,
α
,
β
)

(n,\alpha ,\beta )

are obtained together with their asymptotic behavior as

n
→
∞

n \to \infty

. Sharper bounds are derived in the special cases

n
n

,

n
n

odd, and

α
=
β
=
±

1
2

\alpha = \beta = \pm \frac {1} {2}

,

n
n

arbitrary. There is also a short table of cond

(
n
,
α
,
β
)

(n,\alpha ,\beta )

for

α
\alpha

,

β
=
−
.8
(
.2
)
0
,
.5
,
1
,
2
,
4
,
8
,
16
,
β
≦
α

\beta = - .8(.2)0,.5,1,2,4,8,16,\beta \leqq \alpha

, and

n
=
5
,
10
,
20
,
40

n = 5,10,20,40

. The general conclusion is that cond

(
n
,
α
,
β
)

(n,\alpha ,\beta )

grows at a rate which is something like a constant times

(
3
+
√
8

)
n

{(3 + \surd 8)^n}

, where the constant depends on

α
\alpha

and

β
\beta

, varies relatively slowly as a function of

α
\alpha

,

β
\beta

, and appears to be smallest near

α
=
β
=
−
1

\alpha = \beta = - 1

. For quadrature rules with equidistant points the condition grows like

(
2
√
2

/

3
π
)

8
n

(2\surd 2/3\pi ){8^n}

.
On a theorem of Piatetsky-Shapiro and approximation of multiple integrals
Authors: Haber Seymour , Osgood Charles F.
Published: June 2010

Let

f
f

be a function of

s
s

real variables which is of period

1
1

in each variable, and let the integral

I
I

of

f
f

over the unit cube in

s
s

-space be approximated by
\[

Q
(
f
)
=

1
N

∑

r
=
1

N

f
(
r

x)

Q(f) = \frac {1} {N}\sum \limits _{r = 1}^N {f(r} {\text {x)}}

\]
(where

x = x(N)

{\text {x = x(N)}}

is a point in

s
s

-space). For certain classes of

f
′

s

f’{\text {s}}

’s, defined by conditions on their Fourier coefficients, it is shown using methods of N. M. Korobov, that

x’s

{\text {x’s}}

’s can be found for which error bounds of the form

|

I
(
f
)
−
Q
(
f
)

|

>
K
(
f
)

N

−
p

\left | {I(f) - Q(f)} \right | > K(f){N^{ - p}}

will be true. However, for the class of all

f
′

s

f’{\text {s}}

’s with absolutely convergent Fourier series, it is shown that there are no

x

′

s

{\text {x}}’{\text {s}}

’s for which a bound of the form

|

I
(
f
)
−
Q
(
f
)

|

=
O
(
F
(
N
)
)

\left | {I(f) - Q(f)} \right | = O(F(N))

will hold, for any

F
(
N
)

F(N)

which approaches zero as

N
N

goes to infinity.
On the computation of the eigenvalues of a tridiagonal matrix
Authors: Gargantini I.
Published: June 2010

A recent algorithm for the simultaneous approximation of all zeros of a polynomial is applied to the computation of the eigenvalues of a tridiagonal matrix. The method works in the presence of multiplicity and degeneracy and has been tested in a multitude of cases ; its practical limitation on a computer is the large number of locations required for matrices of high order.
Some factors of the numbers 𝐺_{𝑛}=6²ⁿ+1 and 𝐻_{𝑛}=10²ⁿ+1
Authors: Riesel Hans
Published: June 2010

All numbers

G
n

=

6

2
n

+
1

G_n = 6^{2n} + 1

and

H
n

=

10

2
n

+
1

H_n = 10^{2n} + 1

are searched for factors of the form

p
=
u
⋅

2
8

+
1
>
3.88
⋅

10

11

p = u \cdot 2^8 + 1 > 3.88 \cdot 10^{11}

for

s
≧
n
+
1

s \geqq n + 1

, and odd

u
u

. The search limit for

u
u

was 60000 for

G
n

{G_n}

, and 156250 for

H
n

{H_n}

. A number of factors are found in this range. The numbers

G
6

{G_6}

and

H
6

{H_6}

, lacking small factors, are proved composite by calculating

5

(

G
6

−
1
)

/

2

(
mod

G
6

)

5^{(G_6 - 1)/2} (\bmod G_6)

and

3

(

H
6

−
1
)

/

2

(
mod

H
6

)

3^{(H_6 - 1)/2} (\bmod H_6)

, the residues found being different from

±
1

\pm 1

. The smallest numbers

G
n

G_n

and

H
n

H_n

with unknown characters are

G

11

G_{11}

and

H

10

H_{10}

.
An algorithm for the determination of space groups
Authors: Brown Harold
Published: June 2010

This semiexpository paper presents in detail a numerical formulation of an algorithm for the determination of all

n
n

-dimensional, crystallographic space groups. The implementation of this algorithm on a digital computer is also described.
Uniform asymptotic solution of second order linear differential equations without turning varieties
Authors: Stengle Gilbert
Published: June 2010

The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We obtain uniform asymptotic solutions for certain equations of the form
\[

ϵ

2
n

y
=
a
(
t
,
ϵ
)
y
,

(

)
′

=
d

/

d
t
,

\epsilon ^{2n} y = a(t, \epsilon )y, \quad (\quad )’ = d/dt,

\]
where

n
n

is a positive integer,

t
t

and

ϵ
\epsilon

are real variables ranging over

|
t
|

≦

t
0

,
0
>
ϵ
≦

ϵ
0

\left | t \right | \leqq {t_0},0 > \epsilon \leqq { \epsilon _0}

, and

a
a

is a function infinitely differentiable on the closure of this domain. We require that

a
(
t
,
ϵ
)

a(t, \epsilon )

satisfy conditions which can be regarded as generalized nonturning-point conditions. These conditions imply the absence of secondary turning points, and reduce in the simplest case to the condition

a
(
t
,
0
)
≠
0

a(t,0) \ne 0

, but also include cases (the interesting ones) in which

a
(
0
,
0
)
=
0

a(0,0) = 0
Osculatory interpolation
Authors: Kahng S. W.
Published: June 2010

An explicit method of osculatory interpolation with a function of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column upper R left-parenthesis x right-parenthesis equals 2nd Column a m p semicolon f 00 left-parenthesis a 0 Subscript 0 Baseline plus g 0 left-parenthesis x right-parenthesis f 0 Subscript 1 Baseline left-parenthesis a 0 Subscript 1 Baseline plus g 0 left-parenthesis x right-parenthesis f 0 Subscript 2 Baseline left-parenthesis a 0 Subscript 2 Baseline plus midline-horizontal-ellipsis plus g 0 left-parenthesis x right-parenthesis 2nd Row 1st Column Blank 2nd Column a m p semicolon dot f Subscript 0 comma m 0 Baseline left-parenthesis a Subscript 0 comma m 0 Baseline plus g 0 left-parenthesis x right-parenthesis f 10 left-parenthesis a 10 plus g 1 left-parenthesis x right-parenthesis f 11 left-parenthesis a 11 3rd Row 1st Column Blank 2nd Column a m p semicolon plus midline-horizontal-ellipsis plus g 1 left-parenthesis x right-parenthesis f Subscript 1 comma m 1 Baseline left-parenthesis a Subscript 1 comma m 1 Baseline plus g 1 left-parenthesis x right-parenthesis f 20 left-parenthesis a 20 plus midline-horizontal-ellipsis plus g Subscript n minus 1 Baseline left-parenthesis x right-parenthesis 4th Row 1st Column Blank 2nd Column a m p semicolon dot f Subscript n Baseline 0 Baseline left-parenthesis a Subscript n Baseline 0 Baseline plus g Subscript n Baseline left-parenthesis x right-parenthesis f Subscript n Baseline 1 Baseline left-parenthesis a Subscript n Baseline 1 Baseline plus midline-horizontal-ellipsis plus g Subscript n Baseline left-parenthesis x right-parenthesis f Subscript n comma m Sub Subscript n Subscript Baseline left-parenthesis a Subscript n comma m Sub Subscript n Subscript Baseline right-parenthesis right-parenthesis midline-horizontal-ellipsis right-parenthesis EndLayout">

R
(
x
)
=

a
m
p
;

f

00

(

a
0

0

+

g
0

(
x
)

f
0

1

(

a
0

1

+

g
0

(
x
)

f
0

2

(

a
0

2

+
⋯
+

g
0

(
x
)

a
m
p
;
⋅

f

0
,

m
0

(

a

0
,

m
0

+

g
0

(
x
)

f

10

(

a

10

+

g
1

(
x
)

f

11

(

a

11

a
m
p
;
+
⋯
+

g
1

(
x
)

f

1
,

m
1

(

a

1
,

m
1

+

g
1

(
x
)

f

20

(

a

20

+
⋯
+

g

n
−
1

(
x
)

a
m
p
;
⋅

f

n
0

(

a

n
0

+

g
n

(
x
)

f

n
1

(

a

n
1

+
⋯
+

g
n

(
x
)

f

n
,

m
n

(

a

n
,

m
n

)
)
⋯
)

\begin{align*} R(x) = & {f_{00}}({a_0}_0 + {g_0}(x){f_0}_1({a_0}_1 + {g_0}(x){f_0}_2({a_0}_2 + \cdots + {g_0}(x) \\ & \cdot f_{0, m_0}(a_{0, m_0} + g_0(x)f_{10}(a_{10} + g_1(x)f_{11}(a_{11} \\ & + \cdots + g_1(x)f_{1,m_1}(a_{1,m_1} + g_1(x)f_{20}(a_{20} + \cdots + {g_{n - 1}}(x) \\ & \cdot f_{n0}(a_{n0} + g_n(x)f_{n1}(a_{n1} + \cdots + g_n(x)f_{n, m_n}(a_{n,m_n})) \cdots ) \end{align*}

is described. Error terms for the interpolation are determined.
An iterative finite-difference method for hyperbolic systems
Authors: Abarbanel S. , Zwas G.
Published: June 2010

An iterative finite-difference scheme for initial value problems is presented. It is applied to the quasi-linear hyperbolic system representing the one-dimensional time dependent flow of a compressible polytropic gas. The emphasis in this research was on the handling of discontinuities, such as shock waves, and overcoming the post-shock oscillations resulting from nonlinear instabilities. The linear stability is investigated as well. The success of the method is indicated by the monotonic profiles which were obtained for almost all the cases tested.
Eberlein measure and mechanical quadrature formulae II. Numerical results
Authors: Sarma V. L. N. , Stroud A. H.
Published: June 2010

In a previous paper it was shown how a probability measure (Eberlein measure) on the closed unit ball of the sequence space,

l
1

{l_1}

, can be used to find the variance

σ
2

{\sigma ^2}

of the error functional for a quadrature formula for the

k
k

-dimensional cube, regarded as a random variable. Here we give values of

σ
\sigma

for some specific formulae.
On Gauss’s class number problems
Authors: Shanks Daniel
Published: June 2010

Let

h
h

be the class number of binary quadratic forms (in Gauss’s formulation). All negative determinants having some

h
=
6
n
±
1

h = 6n \pm 1

can be determined constructively: for

h
=
5

h = 5

there are four such determinants; for

h
=
7

h = 7

, six; for

h
=
11

h = 11

, four; and for

h
=
13

h = 13

, six. The distinction between class numbers for determinants and for discriminants is discussed and some data are given. The question of one class/genus for negative determinants is imbedded in the larger question of the existence of a determinant having a specific Abelian group as its composition group. All Abelian groups of order

>
25

> 25

so exist, but the noncyclic groups of order 25, 49, and 121 do not occur. Positive determinants are treated by the same composition method. Although most positive primes of the form

n
2

−
8

{n^2} - 8

have

h
=
1

h = 1

, an interesting subset does not. A positive determinant of an odd exponent of irregularity also appears in the investigation. Gauss indicated that he could not find one.
A note on the stability of predictor-corrector techniques
Authors: Case James
Published: June 2010

The theory of schlicht mappings is used to estimate stepsizes which guarantee the stability of a linear multistep method for the integration of ordinary differential equations.
Statistical study of digits of some square roots of integers in various bases
Authors: Beyer W. A. , Metropolis N. , Neergaard J. R.
Published: June 2010

Some statistical tests of randomness are made of the first 88062 binary digits (or equivalent in other bases) of

√
n

\surd n

in various bases

b
b

,

2
≦
n
≦
15

2 \leqq n \leqq 15

(

n
n

square-free) with

b
=
2
,
4
,
8
,
16

b = 2,4,8,16

and

n
=
2
,
3
,
5

n = 2,3,5

with

b
=
3
,
5
,
6
,
7

b = 3,5,6,7

, and

10
10

. The statistical tests are the

χ
2

{\chi ^2}

test for cumulative frequency distribution of the digits, the lead test, and the gap test. The lead test is an examination of the distances over which the cumulative frequency of a digit exceeded its expected value. It is related to the arc sine law. The gap test (applied to the binary digits) consists of an examination of the distribution of runs of ones. The conclusion of the study is that no evidence of the lack of randomness or normality appears for the digits of the above mentioned

√
n

\surd n

in the assigned bases

b
b

. It seems to be the first statistical study of the digits of any naturally occurring number in bases other than decimal or binary (octal).
The initial-value problem for the equation (𝑡𝑢_{𝑡})_{𝑡}=𝑢ₓₓ
Authors: Solomon Alan , Solomon Faiza
Published: June 2010

It is shown that the initial-value problem of the equation

(
t

u
t

)
t

=

u

x
x

{(t{u_t})_t} = {u_{xx}}

with the value of

u
u

prescribed for

t
=
0

t = 0

has a unique solution satisfying a maximum principle. In addition, several numerical schemes for its solution are proposed.
Error estimates for a Chebyshev quadrature method
Authors: Basu N. K.
Published: June 2010

Filippi [1] has proposed a quadrature scheme for any function

f
(
x
)

f(x)

in

[
−
1
,
1
]

[ - 1,1]

, based on expanding the integrand in a series of Chebyshev polynomials of the second kind. In this paper the error associated with this quadrature method when applied to analytic functions has been investigated in detail.