bluuebunny/crossref_metadata_2025 huggingface dataset:
On the zero set of a holomorphic one-form on a compact complex manifold
Authors: Spurr Michael J.
Published: June 2010

On any compact complex surface

M
M

, divisors of nonnegative self-intersection which are contained in the zero set (or in the integral set) of a holomorphic

1
1

-form are shown to induce a fibration of

M
M

onto a Riemann surface. This result is extended to higher dimensions for

M
M

projective. Applications to zero sets of holomorphic

1
1

-forms on surfaces are given.
Some P.V.-equivalences and a classification of 2-simple prehomogeneous vector spaces of type 𝐼𝐼
Authors: Kimura Tatsuo , Kasai Shin-ichi , Taguchi Masanobu , Inuzuka Masaaki
Published: June 2010

A classification of

2
2

-simple prehomogeneous vector spaces is completed by using some P.V.-equivalences together with [3]. Some part is very different from the previous classification of the irreducible or simple cases [1, 2], and some new method is necessary. This result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces.
Representations of Hecke algebras
Authors: Gutkin Eugene
Published: June 2010

We find all operators of a certain type that satisfy the braid relations corresponding to any generalized Cartan matrix.
Kazhdan-Lusztig polynomials for Hermitian symmetric spaces
Authors: Boe Brian D.
Published: June 2010

A nonrecursive scheme is presented to compute the Kazhdan-Lusztig polynomials associated to a classical Hermitian symmetric space, extending a result of Lascoux-Schützenberger for grassmannians. The polynomials for the exceptional Hermitian domains are also tabulated. All the Kazhdan-Lusztig polynomials considered are shown to be monic.
Ideals associated to deformations of singular plane curves
Authors: Diaz Steven , Harris Joe
Published: June 2010

We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. These loci are the equisingular locus

E
S

ES

which parametrizes equisingular or topologically trivial deformations, the equigeneric locus

E
G

EG

which parametrizes deformations of constant geometric genus, and the equiclassical locus

E
C

EC

which parametrizes deformations of constant geometric genus and class. (The class of a reduced plane curve is the degree of its dual.) It was previously known that the tangent space to

E
S

ES

corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to

E
G

EG

corresponds to the conductor ideal. We show that the support of the tangent cone to

E
C

EC

corresponds to an ideal which we call the equiclassical ideal. By studying these ideals we are able to obtain information about the geometry and dimensions of

E
S

ES

,

E
C

EC

, and

E
G

EG

. This allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities.
On the nonlinear eigenvalue problem Δ𝑢+𝜆𝑒^{𝑢}=0
Authors: Suzuki Takashi , Nagasaki Ken’ichi
Published: June 2010

The structure of the set

C

\mathcal {C}

of solutions of the nonlinear eigenvalue problem

Δ
u
+
λ

e
u

=
0

\Delta u + \lambda {e^u} = 0

under Dirichlet condition in a simply connected bounded domain

Ω
\Omega

is studied. Through the idea of parametrizing the solutions

(
u
,

λ
)

(u,\,\lambda )

in terms of

s
=
λ

∫
Ω

e
u

d
x

s = \lambda \,\int _\Omega {{e^u}\,dx}

, some profile of

C

\mathcal {C}

is illustrated when

Ω
\Omega

is star-shaped. Finally, the connectivity of the branch of Weston-Moseley’s large solutions to that of minimal ones is discussed.
The existence of least area surfaces in 3-manifolds
Authors: Hass Joel , Scott Peter
Published: June 2010

This paper presents a new and unified approach to the existence theorems for least area surfaces in

3
3

-manifolds.
On the Wiener criterion and quasilinear obstacle problems
Authors: Heinonen Juha , Kilpeläinen Tero
Published: June 2010

We study the Wiener criterion and variational inequalities with irregular obstacles for quasilinear elliptic operators

A
A

,

A
(
x
,

∇
u
)
⋅
∇
u
≈

|

∇
u

|

p

A(x,\,\nabla u) \cdot \nabla u \approx |\nabla u{|^p}

, in

R

n

{{\mathbf {R}}^n}

. Local solutions are continuous at Wiener points of the obstacle function; if

p
>
n
−
1

p > n - 1

, the converse is also shown to be true. If

p
>
n
−
1

p > n - 1

, then a characterization of the thinness of a set at a point is given in terms of

A
A

-superharmonic functions.
On a class of functionals invariant under a 𝑍ⁿ action
Authors: Rabinowitz Paul H.
Published: June 2010

Consider a system of ordinary differential equations of the form

(

∗

)

({\ast })

\[

q
¨

+

V
q

(
t
,

q
)
=
f
(
t
)

\ddot q + {V_q}(t,\,q) = f(t)

\]
where

f
f

and

V
V

are periodic in

t
t

,

V
V

is periodic in the components of

q
=
(

q
1

,
…
,

q
n

)

q = ({q_1}, \ldots ,{q_n})

, and the mean value of

f
f

vanishes. By showing that a corresponding functional is invariant under a natural

Z

n

{{\mathbf {Z}}^n}

action, a simple variational argument yields at least

n
+
1

n + 1

distinct periodic solutions of (*). More general versions of (*) are also treated as is a class of Neumann problems for semilinear elliptic partial differential equations.
Homology of smooth splines: generic triangulations and a conjecture of Strang
Authors: Billera Louis J.
Published: June 2010

For

Δ
\Delta

a triangulated

d
d

-dimensional region in

R

d

{{\mathbf {R}}^d}

, let

S
m
r

(
Δ
)

S_m^r(\Delta )

denote the vector space of all

C
r

{C^r}

functions

F
F

on

Δ
\Delta

that, restricted to any simplex in

Δ
\Delta

, are given by polynomials of degree at most

m
m

. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds

Δ
\Delta

in the plane, getting lower bounds on the dimension of

S

m
r

(
Δ
)

S{}_m^r(\Delta )

for all

r
r

. For

r
=
1

r = 1

, we prove a conjecture of Strang concerning the generic dimension of the space of

C
1

{C^1}

splines over a triangulated manifold in

R

2

{{\mathbf {R}}^2}

. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.
The continuation theory for Morse decompositions and connection matrices
Authors: Franzosa Robert D.
Published: June 2010

The continuation theory for (

>
>

-ordered) Morse decompositions and the indices defined on them—the homology index braid and the connection matrices—is established. The equivalence between

>
>

-ordered Morse decompositions and

>
>

-consistent attractor filtrations is displayed. The spaces of (

>
>

-ordered) Morse decompositions for a product parametrization of a local flow are introduced, and the local continuation of (

>
>

-ordered) Morse decompositions is obtained via the above-described equivalence and the local continuation of attractors. The homology index braid and the connection matrices of an admissible ordering of a Morse decomposition are shown to be invariant on path components of the corresponding space of

>
>

-ordered Morse decompositions. This invariance is used to prove that the collection of connection matrices of a Morse decomposition is upper semicontinuous over the space of Morse decompositions (and over the parameter space) under local continuation.
The connectedness of symmetric and skew-symmetric degeneracy loci: even ranks
Authors: Tu Loring W.
Published: June 2010

A degeneracy locus is the set of points where a vector-bundle map has rank at most a given integer. Such a set is symmetric or skew-symmetric according as whether the vector-bundle map is symmetric or skew-symmetric. We prove a connectedness result, first conjectured by Fulton and Lazarsfeld, for skew-symmetric degeneracy loci and for symmetric degeneracy loci of even ranks.
Limitation topologies on function spaces
Authors: Bowers Philip L.
Published: June 2010

Four competing definitions for limitation topologies on the set of continuous functions

C
(
X
,
Y
)

C(X,Y)

are compared.
Isolated singularities of the Schrödinger equation with a good potential
Authors: Vázquez Juan Luis , Yarur Cecilia
Published: June 2010

We study the behaviour near an isolated singularity, say

0
0

, of nonnegative solutions of the Schrödinger equation

−
Δ
u
+
V
u
=
0

- \Delta u + Vu = 0

defined in a punctured ball

0
>

|

x

|

>
R

0 > |x| > R

. We prove that whenever the potential

V
V

belongs to the Kato class

K
n

{K_n}

the following alternative, well known in the case of harmonic functions, holds: either

|

x

|

n
−
2

u
(
x
)

|x{|^{n - 2}}u(x)

has a positive limit as

|

x

|

→
0

|x| \to 0

or

u
u

is continuous at

0
0

. In the first case

u
u

solves the equation

−
Δ
u
+
V
u
=
a
δ

- \Delta u + Vu = a\delta

in

{

|

x

|

>
R
}

\{ |x| > R\}

. We discuss the optimality of the class

K
n

{K_n}

and extend the result to solutions

u
≱
0

u \ngeq 0

of

−
Δ
u
+
V
u
=
f

- \Delta u + Vu = f

.
Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities
Authors: Castro Alfonso , Kurepa Alexandra
Published: June 2010

Let

p
,
φ
:
[
0
,
T
]
→
R

p,\varphi :[0,T] \to R

be bounded functions with

φ
>
0

\varphi > 0

. Let

g
:

R

→

R

g:{\mathbf {R}} \to {\mathbf {R}}

be a locally Lipschitzian function satisfying the superlinear jumping condition: (i)

lim

u
→
−
∞

(
g
(
u
)

/

u
)
∈

R

{\lim _{u \to - \infty }}(g(u)/u) \in {\mathbf {R}}

(ii)

lim

u
→
∞

(
g
(
u
)

/

u

1
+
ρ

)
=
∞

{\lim _{u \to \infty }}(g(u)/{u^{1 + \rho }}) = \infty

for some

ρ
>
0

\rho > 0

, and (iii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript u right-arrow normal infinity Endscripts left-parenthesis u slash g left-parenthesis u right-parenthesis right-parenthesis Superscript upper N slash 2 Baseline left-parenthesis upper N upper G left-parenthesis kappa u right-parenthesis minus left-parenthesis left-parenthesis upper N minus 2 right-parenthesis slash 2 right-parenthesis u dot g left-parenthesis u right-parenthesis right-parenthesis equals normal infinity">

lim

u
→
∞

(
u

/

g
(
u
)

)

N

/

2

(
N
G
(
κ
u
)
−
(
(
N
−
2
)

/

2
)
u
⋅
g
(
u
)
)
=
∞

{\lim _{u \to \infty }}{(u/g(u))^{N/2}}(NG(\kappa u) - ((N - 2)/2)u \cdot g(u)) = \infty

for some

κ
∈
(
0
,
1
]

\kappa \in (0,1]

where

G
G

is the primitive of

g
g

. Here we prove that the number of solutions of the boundary value problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta u plus g left-parenthesis u right-parenthesis equals p left-parenthesis double-vertical-bar x double-vertical-bar right-parenthesis plus c phi left-parenthesis double-vertical-bar x double-vertical-bar right-parenthesis">

Δ
u
+
g
(
u
)
=
p
(

‖
x
‖

)
+
c
φ
(

‖
x
‖

)

\Delta u + g(u) = p(\left \| x\right \|) + c\varphi (\left \| x\right \|)

for

x
∈

R

N

x \in {{\mathbf {R}}^N}

with

‖
x
‖

>
T
,
u
(
x
)
=
0

\left \| x\right \| > T,u(x) = 0

for

‖
x
‖

=
T

\left \| x\right \| = T

tends to

+
∞

+ \infty

when

c
c

tends to

+
∞

+ \infty

. The proofs are based on the "energy" and "phase plane" analysis.
Degree theory for equivariant maps. I
Authors: Ize J. , Massabò I. , Vignoli A.
Published: June 2010

A degree theory for equivariant maps is constructed in a simple geometrical way. This degree has all the basic properties of the usual degree theories and takes its values in the equivariant homotopy groups of spheres. For the case of a semifree

S
1

{S^1}

-action, a complete computation of these groups is given, the range of the equivariant degree is determined, and the general

S
1

{S^1}

-action is reduced to that special case. Among the applications one recovers and unifies both the degree for autonomous differential equations defined by Fuller [F] and the

S
1

{S^1}

-degree for gradient maps introduced by Dancer [Da]. Also, a simple but very useful formula of Nirenberg [N] is generalized (see Theorem 4.4(ii)).
Harnack’s inequality for degenerate Schrödinger operators
Authors: Gutiérrez Cristian E.
Published: June 2010

We prove a Harnack inequality for nonnegative weak solutions of certain Schrödinger equations of the form

L
u
−
V
u
=
0

Lu - Vu = 0

where

L
L

is a second order degenerate elliptic operator in divergence form and

V
V

is a potential in certain class.
Quadrature and harmonic 𝐿¹-approximation in annuli
Authors: Armitage D. H. , Goldstein M.
Published: June 2010

Open sets

D
D

in

R
N

(
N
≥
3
)

{R^N}\;(N \geq 3)

with the property that

D
¯

\bar D

is a closed annulus

{
x
:

r
1

≤

‖
x
‖

≤

r
2

}

\{ x:{r_1} \leq \;\left \| x\right \| \; \leq {r_2}\}

are characterized by quadrature formulae involving mean values of certain harmonic functions. One such characterization is used to give a criterion for the existence of a best harmonic

L
1

{L^1}

approximant to a function which is subharmonic (and satisfies some other conditions) in an annulus.
Band-limited functions: 𝐿^{𝑝}-convergence
Authors: Barceló Juan A. , Córdoba Antonio
Published: June 2010

We consider the set

B
p

(
Ω
)

{B_p}(\Omega )

(functions of

L
p

(

R

)

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

whose Fourier spectrum lies in

[
−
Ω
,
+
Ω
]

[ - \Omega , + \Omega ]

). We prove that the prolate spheroidal wave functions constitute a basis of this space if and only if

4

/

3
>
p
>
4

4/3 > p > 4

. The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hilbert transform.
Summation, transformation, and expansion formulas for bibasic series
Authors: Gasper George
Published: June 2010

An indefinite bibasic sum containing three parameters is evaluated and used to derive bibasic extensions of Euler’s transformation formula and of a Fields and Wimp expansion formula. It is also used to derive a transformation formula involving four independent bases, a

q
q

-Lagrange inversion formula, and some quadratic, cubic and quartic summation formulas.
Periodic orbits of maps of 𝑌
Authors: Alsedà Lluís , Llibre Jaume , Misiurewicz Michał
Published: June 2010

We introduce some notions that are useful for studying the behavior of periodic orbits of maps of one-dimensional spaces. We use them to characterize the set of periods of periodic orbits for continuous maps of

Y
=
{
z
∈

C

:

z
3

∈
[
0
,
1
]
}

Y = \{ z \in {\mathbf {C}}:{z^3} \in [0,1]\}

into itself having zero as a fixed point. We also obtain new proofs of some known results for maps of an interval into itself.
The structure of some equivariant Thom spectra
Authors: Costenoble Steven R.
Published: June 2010

We show that the equivariant Thom spectra

M

O

Z

2

M{O_{{{\text {Z}}_2}}}

and

m

O

Z

2

m{O_{{{\text {Z}}_2}}}

do not split as wedges of equivariant Eilenberg-Mac Lane spectra, as they do nonequivariantly. This is done by finding two-stage Postnikov towers giving these spectra, and determining the nontrivial

k
k

-invariants. We also consider the question: In what sense is the spectrum

m

O

Z

2

m{O_{{{\text {Z}}_2}}}

representing unoriented bordism unique?
Piecewise linearization of real-valued subanalytic functions
Authors: Shiota Masahiro
Published: June 2010

We show that for a subanalytic function

f
f

on a locally compact subanalytic set

X
X

there exists a unique subanalytic triangulation (a simplicial complex

K
K

, a subanalytic homeomorphism

π
:

|

K

|

→
X

\pi :|K| \to X

) such that

f
∘
π

|

σ

,
σ
∈
K

f \circ \pi {|_\sigma }, \sigma \in K

, are linear.
On the Hopf index and the Conley index
Authors: McCord Christopher K.
Published: June 2010

The following generalization of the Poincaré-Hopf index theorem is proved: If

S
S

is an isolated invariant set of a flow on a manifold

M
M

, then the sum of the Hopf indices on

S
S

is equal (up to a sign) to the Euler characteristic of the homology Conley index of

S
S

.
Coupled points in the calculus of variations and applications to periodic problems
Authors: Zeidan Vera , Zezza Pier Luigi
Published: June 2010

The aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions, and to develop second order necessary conditions for optimality. When one of the end points is fixed, our necessary conditions reduce to the known ones involving conjugate points. We also apply our results to the periodic problems of the calculus of variations.
Prescribing zeros of functions in the Nevanlinna class on weakly pseudo-convex domains in 𝐶²
Authors: Shaw Mei-Chi
Published: June 2010

LetDDbe a bounded weakly pseudo-convex domain inC2{{\mathbf {C}}^2}of uniform strict type. For any positive divisorMMofDDwith finite area, there exists a holomorphic functionffin the Nevanlinna class such thatMMis the zero set offf. The proof is to study the solutions of∂¯\bar \partialwithL1(∂D){L^1}(\partial D)boundary values.
On the Cauchy problem and initial traces for a degenerate parabolic equation
Authors: DiBenedetto E. , Herrero M. A.
Published: June 2010

We consider the Cauchy problem (f)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column u Subscript t Baseline minus d i v left-parenthesis StartAbsoluteValue upper D u EndAbsoluteValue Superscript p minus 2 Baseline upper D u right-parenthesis equals 0 2nd Column a m p semicolon in bold upper R Superscript upper N Baseline times left-parenthesis 0 comma normal infinity right-parenthesis comma p greater-than 2 comma 2nd Row 1st Column u left-parenthesis x comma 0 right-parenthesis equals u 0 left-parenthesis x right-parenthesis comma 2nd Column a m p semicolon x element-of bold upper R Superscript upper N Baseline comma EndLayout">

{

u
t

−
div
⁡
(

|

D
u

|

p
−
2

D
u
)
=
0

a
m
p
;

in

R

N

×
(
0
,
∞
)
,
p
>
2
,

u
(
x
,
0
)
=

u
0

(
x
)
,

a
m
p
;

x
∈

R

N

,

\left \{ {\begin {array}{*{20}{c}} {{u_t} - \operatorname {div}(|Du{|^{p - 2}}Du) = 0} \hfill & {{\text {in}}\;{{\mathbf {R}}^N} \times (0,\infty ),p > 2,} \hfill \\ {u(x,0) = {u_0}(x),} \hfill & {x \in {{\mathbf {R}}^N},} \hfill \\ \end {array} } \right .

\]
and discuss existence of solutions in some strip

S
T

≡

R

N

×
(
0
,
T
)

{S_T} \equiv {{\mathbf {R}}^N} \times (0,T)

,

0
>
T
≤
∞

0 > T \leq \infty

, in terms of the behavior of

x
→

u
0

(
x
)

x \to {u_0}(x)

as

|

x

|

→
∞

|x| \to \infty

. The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of

L

loc

1

(

R

N

)

L_{{\text {loc}}}^1({{\mathbf {R}}^N})

.
Equivalent conditions to the spectral decomposition property for closed operators
Authors: Erdélyi I. , Wang Sheng Wang
Published: June 2010

The spectral decomposition property has been instrumental in developing a local spectral theory for closed operators acting on a complex Banach space. This paper gives some necessary and sufficient conditions for a closed operator to possess the spectral decomposition property.
Functional equations, tempered distributions and Fourier transforms
Authors: Baker John A.
Published: June 2010

This paper introduces a method for solving functional equations based on the Fourier transform of tempered distributions.
The generalized Borel conjecture and strongly proper orders
Authors: Corazza Paul
Published: June 2010

The Borel Conjecture is the statement that

C
=

[

R

]

>

ω
1

C = {[\mathbb {R}]^{ > {\omega _1}}}

, where

C
C

is the class of strong measure zero sets; it is known to be independent of ZFC. The Generalized Borel Conjecture is the statement that

C
=

[

R

]

>

c

C = {[\mathbb {R}]^{ > {\mathbf {c}}}}

. We show that this statement is also independent. The construction involves forcing with an

ω
2

{\omega _2}

-stage iteration of strongly proper orders; this latter class of orders is shown to include several well-known orders, such as Sacks and Silver forcing, and to be properly contained in the class of

ω
\omega

-proper,

ω
ω

{\omega ^\omega }

-bounding orders. The central lemma is the observation that A. W. Miller’s proof that the statement

(

∗

)

({\ast })

"Every set of reals of power c can be mapped (uniformly) continuously onto

[
0
,
1
]

[0,1]

" holds in the iterated Sacks model actually holds in several other models as well. As a result, we show for example that

(

∗

)

({\ast })

is not restricted by the presence of large universal measure zero

(

U

0

)

({{\text {U}}_0})

sets (as it is by the presence of large

C
C

sets). We also investigate the

σ
\sigma

-ideal

J

=
{
X
⊂

R

:
X

cannot be mapped uniformly continuously onto 

[
0
,
1
]
}

\mathcal {J} = \{ X \subset \mathbb {R}:X\;{\text {cannot be mapped uniformly continuously onto }}[0,1]\}

and prove various consistency results concerning the relationships between

J

,

U

0

\mathcal {J},\;{{\text {U}}_0}

, and AFC (where

AFC
=
{
X
⊂

R

:
X

is always first category} 

\operatorname {AFC} = \{ X \subset \mathbb {R}:X\;{\text {is always first category\} }}

). These latter results partially answer two questions of J. Brown.
Integral Dubrovin valuation rings
Authors: Morandi Patrick J. , Wadsworth Adrian R.
Published: June 2010

In the preceding paper, Dubrovin valuation rings integral over their centers in central simple algebras were characterized by value functions. Here, these value functions are used to give a method for extending integral Dubrovin valuation rings in generalized crossed product algebras. Several applications of this extension theorem are given, including new and more natural proofs of some theorems on valued division algebras over Henselian fields.
Value functions on central simple algebras
Authors: Morandi Patrick J.
Published: June 2010

In this paper we study noncommutative valuation rings as defined by Dubrovin. While there is in general no valuation associated to a Dubrovin valuation ring, we show that there is a value function associated to any Dubrovin valuation ring integral over its center. By using value functions we obtain information on Dubrovin valuation rings in a tensor product, both generalizing and giving a much simpler proof of a result about valued division algebras. By being able to work directly with central simple algebras we gain new information about division algebras over Henselian fields.
Hypergraphs with finitely many isomorphism subtypes
Authors: Kierstead Henry A. , Nyikos Peter J.
Published: June 2010

Let

H

=
(
H
,
E
)

\mathcal {H} = (H,E)

be an

n
n

-uniform infinite hypergraph such that the number of isomorphism types of induced subgraphs of

H

\mathcal {H}

of cardinality

λ
\lambda

is finite for some infinite

λ
\lambda

. We solve a problem due independently to Jamison and Pouzet, by showing that there is a finite subset

K
K

of

H
H

such that the induced subgraph on

H
−
K

H - K

is either empty or complete. We also characterize such hypergraphs in terms of finite (not necessarily uniform) hypergraphs.
An HNN-extension with cyclic associated subgroups and with unsolvable conjugacy problem
Authors: Lockhart Jody Meyer
Published: June 2010

In this paper, we consider the conjugacy problem for

HNN

{\text {HNN}}

-extensions of groups with solvable conjugacy problem for which the associated subgroups are cyclic. An example of such a group with unsolvable conjugacy problem is constructed. A similar construction is given for free products with amalgamation.
Conjugation and the prime decomposition of knots in closed, oriented 3-manifolds
Authors: Miyazaki Katura
Published: June 2010

In this paper we consider the prime decomposition of knots in closed, oriented

3
3

-manifolds. (For classical knots one can easily prove the uniqueness of prime decomposition by using a standard innermost disk argument.) We define a new relation, conjugation, between oriented knots in closed, oriented

3
3

-manifolds and prove the following results. (1) The prime decomposition is, roughly speaking, uniquely determined up to conjugation, (2) there is a prime knot

R

\mathcal {R}

in

S
1

×

S
2

{S^1} \times {S^2}

such that

R

#

K

1

=

R

#

K

2

\mathcal {R}\# {\mathcal {K}_1} = \mathcal {R}\# {\mathcal {K}_2}

if

K

1

{\mathcal {K}_1}

is a conjugation of

K

2

{\mathcal {K}_2}

, and (3) if a knot

K

\mathcal {K}

has a prime decomposition which does not contain

R

\mathcal {R}

, then it is the unique prime decomposition of

K

\mathcal {K}

.
Conjugacy classes whose square is an infinite symmetric group
Authors: Moran Gadi
Published: June 2010

Let

X
ν

{X_\nu }

be the set of all permutations

ξ
\xi

of an infinite set

A
A

of cardinality

ℵ
ν

{\aleph _\nu }

with the property: every permutation of

A
A

is a product of two conjugates of

ξ
\xi

. The set

X
0

{X_0}

is shown to be the set of permutations

ξ
\xi

satisfying one of the following three conditions: (1)

ξ
\xi

has at least two infinite orbits. (2)

ξ
\xi

has at least one infinite orbit and infinitely many orbits of a fixed finite size

n
n

. (3)

ξ
\xi

has: no infinite orbit; infinitely many finite orbits of size

k
,
l

k,l

and

k
+
l

k + l

for some positive integers

k
,
l

k,l

; and infinitely many orbits of size

>
2

> 2

. It follows that

ξ
∈

X
0

\xi \in {X_0}

iff some transposition is a product of two conjugates of

ξ
\xi

, and

ξ
\xi

is not a product

σ
i

\sigma i

, where

σ
\sigma

has a finite support and

i
i

is an involution. For

ν
>
0
,

ξ
∈

X
ν

\nu > 0,\;\xi \in {X_\nu }

iff

ξ
\xi

moves

ℵ
ν

{\aleph _\nu }

elements, and satisfies (1), (2) or

(

3
′

)

(3’)

, where

(

3
′

)

(3’)

is obtained from (3) by omitting the requirement that

ξ
\xi

has infinitely many orbits of size

>
2

> 2

. It follows that for

ν
>
0
,

ξ
∈

X
ν

\nu > 0,\;\xi \in {X_\nu }\;

iff

ξ
\xi

moves

ℵ
ν

{\aleph _\nu }

elements and some transposition is the product of two conjugates of

ξ
\xi

. The covering number of a subset

X
X

of a group

G
G

is the smallest power of

X
X

(if any) that equals

G
G

[AH]. These results complete the classification of conjugacy classes in infinite symmetric groups with respect to their covering number.
On the numerical treatment of heat conduction problems with mixed boundary conditions
Authors: Lowan Arnold N.
Published: June 2010

The two-dimensional problem of heat conduction in a rectangle where the temperature is prescribed over a portion of the boundary while the temperature gradient is prescribed over the remainder of the boundary, may be treated numerically by replacing the differential equation of heat conduction and the equations expressing the given initial and boundary conditions by their difference analogs and solving the resulting system. It is shown that if the scheme is to be stable the intervals

Δ
x

\Delta x

and

Δ
y

\Delta y

must be chosen so that

k
Δ
t

/

(
Δ
x

)
2

+
k
Δ
t

/

(
Δ
y

)
2

≦

1
2

k\Delta t/{(\Delta x)^2} + k\Delta t/{(\Delta y)^2} \leqq \tfrac {1}{2}

.
On stability criteria of explicit difference schemes for certain heat conduction problems with uncommon boundary conditions
Authors: Lowan Arnold N.
Published: June 2010

Stability criteria are derived for the explicit difference schemes appropriate to the following problems: 1) heat conduction in a slab in contact with a well stirred liquid; 2) heat conduction in a slab radiating to one face of a thin slab with infinite thermal conductivity, the other face of which radiates into a medium at prescribed temperature; 3) heat conduction in a cylinder radiating to the inner surface of a thin coaxial cylindrical shell with infinite thermal conductivity, the outer surface of which radiates into a medium at prescribed temperature.
Fast method for computing the number of primes less than a given limit
Authors: Mapes David C.
Published: June 2010

"Fast Method for Computing the Number of Primes Less Than a Given Limit” describes three processes used during the course of calculation. In the first part of the paper the author proves:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis x comma a right-parenthesis equals phi left-parenthesis x comma 1 right-parenthesis minus phi left-parenthesis StartFraction x Over p 2 EndFraction comma 1 right-parenthesis minus phi left-parenthesis StartFraction x Over p 3 EndFraction comma 2 right-parenthesis minus ellipsis minus phi left-parenthesis StartFraction x Over p Subscript a Baseline EndFraction comma a minus 1 right-parenthesis">

ϕ
(
x
,
a
)
=
ϕ
(
x
,
1
)
−
ϕ
(

x

p
2

,
1

)
−
ϕ
(

x

p
3

,
2

)
−
…
−
ϕ
(

x

p
a

,
a
−
1

)

\phi (x,a) = \phi (x,1) - \phi ({\frac {x}{{{p_2}}},1}) - \phi ({\frac {x}{{{p_3}}},2}) - \ldots - \phi ({\frac {x}{{{p_a}}},a - 1})

\]
where

ϕ
(
x
,
a
)

\phi (x,a)

represents the number of numbers less than or equal to x and not divisible by the first “a” primes. This identity is used to evaluate the formula

π
(
x
)
=
ϕ
(
x
,
a
)
+
a
−
1

\pi (x) = \phi (x,a) + a - 1

,

a
+
1
>
π
(

x

)

a + 1 > \pi (\sqrt x )

where resulting terms of the form

ϕ
(

x
′

,

a
′

)

\phi (x’,a’)

are broken down still further by the previously described method, or numerically evaluated using one or both of two other identities, the choice being dependent on

x
′

x’

and

a
′

a’

. Following the paper is a table of calculations made using this process which gives the values of

π
(
x
)

\pi (x)

for x at intervals of 10 million up to 1000 million, along with the Riemann and the Chebyshev approximations for

π
(
x
)

\pi (x)

and the amount they deviate from the true count.
Exponential differences
Authors: Delves L. M.
Published: June 2010

The concept of functional differences is described, and the calculus of functional differences developed for the particular case of the exponential function.
Error analysis of recurrence equations
Authors: Tait R.
Published: June 2010

An error analysis of the Miller algorithm for computation from threeterm recurrence equations is given. Bounds are supplied in terms of the known coefficients and a method of finding suitable starting values for prescribed relative error is investigated.
Generating pseudo-random numbers by shuffling a Fibonacci sequence
Authors: Gebhardt Friedrich
Published: June 2010

M. D. MacLaren and G. Marsaglia [2] have proposed to mix two pseudo-random number generators in the following way: The first generator is used at the beginning to fill an array with pseudo-random numbers; whenever a random number is needed, the second generator determines which element of the array is to be used and replaced by a new number from the first generator. In this study, only one generator is utilized for both purposes; moreover, the generator chosen (a Fibonacci sequence) is by itself a rather poor one. Nevertheless, the final sequence of pseudo-random numbers passed all statistical tests applied to it, including

χ
2

−

{\chi ^2} -

tests of the maximum and minimum of two to ten succeeding numbers and tests applied to sequences immediately following a small number or two almost equal ones.
Parallel methods for the numerical integration of ordinary differential equations
Authors: Miranker Willard L. , Liniger Werner
Published: June 2010

In this paper we derive a class of numerical integration formulas of a parallel type for ordinary differential equations. These formulas may be used simultaneously on a set of arithmetic processors to increase the integration speed. Conditions for the convergence of such formulas are formulated. Explicit examples for two and four processor cases are derived. Results of numerical experiments are given which show an effective improvement in computation speed.
Minimising truncation error in finite difference approximations to ordinary differential equations
Authors: Osborne M. R.
Published: June 2010

It is shown that the error in setting up a class of finite difference approximations is of two kinds: a quadrature error and an interpolation error. In many applications the quadrature error is dominant, and it is possible to take steps to reduce it. In the concluding section an attempt is made to answer the question of how to find a finite difference formula which is best in the sense of minimising the work which has to be done to obtain an answer to within a specified tolerance.
Some limiting values and two error estimation procedures for power series approximations
Authors: Mangad Moshe
Published: June 2010

Several limiting values concerned with the various remainder forms in the Taylor series expansion for one and two variables are developed. In addition, two error estimation procedures which are applicable to power series approximations of differential equations are also obtained.
Comparison of the method of averages with the method of least squares.
Authors: Dahlquist G. , Sjöberg B. , Svensson P.
Published: June 2010

It is shown that the computationally simple method of averages can yield a surprisingly good solution of an overdetermined system of linear equations, provided that the grouping of the equations is done in an appropriate way. The notion of angle between linear subspaces is applied in a general comparison of this method and the method of least squares. The optimal application of the method is treated for the test problem of fitting a polynomial of degree less than six.
A new approximation for the chi-square integral
Authors: Gray H. L. , Thompson R. W. , McWilliams G. V.
Published: June 2010

In this paper a new method for estimating the value of an improper integral by a finite sum is introduced. In particular, the method is applied to the Chi-Square integral and proves to be of some value in estimating the value of this integral for values equal or greater than .9.
Extremal properties of balanced tri-diagonal matrices
Authors: Businger Peter A.
Published: June 2010

If

A
A

is a square matrix with distinct eigenvalues and

D
D

a nonsingular matrix, then the angles between row- and column-eigenvectors of

D

−
1

A
D

{D^{ - 1}}AD

differ from the corresponding quantities of

A
A

. Perturbation analysis of the eigenvalue problem motivates the minimization of functions of these angles over the set of diagonal similarity transforms; two such functions which are of particular interest are the spectral and the Euclidean condition numbers of the eigenvector matrix

X
X

of

D

−
1

A
D

{D^{ - 1}}AD

. It is shown that for a tri-diagonal real matrix

A
A

both these condition numbers are minimized when

D
D

is chosen such that the magnitudes of corresponding sub- and super-diagonal elements are equal.
Some extensions of Legendre quadrature
Authors: Newbery A. C. R.
Published: June 2010

The

m
m

-point Gauss-Legendre formula gives an exact expression for the integral of an algebraic polynomial of maximum degree

2
m
−
1

2m - 1

in terms of

m
m

ordinates. It is shown that analogous formulas can be derived for exponential and trigonometric polynomials.
Approximate solutions for mixed boundary value problems by finite-difference methods
Authors: Thuraisamy V.
Published: June 2010

For mixed boundary value problems of Poisson and/or Laplace’s equations in regions of the Euclidean space

E
n

{E_n}

,

n
≧
2

n \geqq 2

, finite-difference analogues are formulated such that the matrix of the resulting system is of positive type. Discretization errors are established in a manner to reveal the continuous dependence of the rate of convergence on the smoothness of the solution. Isolated data singularities and their application to exterior problems are also discussed.
Iterates of a number-theoretic function
Authors: Lal Mohan
Published: June 2010

Iterates of a function defined by the sum of the prime divisors of a number, where the multiple factors are counted multiply, are considered. The process of iteration is terminated at a prime. The density distribution of these primes is investigated empirically, for

N
≦
60000

N \leqq 60000

and it is found to be quite constant
On a problem of Hasse
Authors: Zassenhaus H. , Liang J.
Published: June 2010

A

p
p

-adic method to construct explicitly a generating automorphism of the Hubert classfield over

Q
(
√
−
47
)

Q(\surd - 47)

and to perform Tshirnhausen transformations for generating equations of the real subfield is developed.
On systems of difference equations with wrong boundary conditions
Authors: Osher Stanley
Published: June 2010

Kreiss and Lundquist

[
1
]

[1]

, have recently obtained certain results about the decay of influence of wrong boundary conditions in overdetermined difference approximations to a hyperbolic partial differential equation. We have extended this result to include systems of hyperbolic partial differential equations in several space variables and their corresponding overdetermined difference approximations. The technique of proof involves examining sections of the resolvent of the difference operator for analyticity within the unit circle as in Kreiss

[
6
]

[6]

and Osher

[
2
]

[2]

.
Some new results on equal sums of like powers
Authors: Brudno Simcha
Published: June 2010

The Diophantine equation

∑

i
=
1

M

x
i
n

=

∑

i
=
1

M

y
i
n

\sum \nolimits _{i = 1}^M {x_i^n = \sum \nolimits _{i = 1}^M {y_i^n} }

is examined for

n
=
3
,
4

n = 3,4

and

6
6

and

M
=
[
(
n
+
1
)

/

2
]

M = [(n + 1)/2]

. A method for generating parametric solutions for

n
=
4

n = 4

is derived and several new numerical examples for

n
=
4
,
6

n = 4,6

are given. The method also applies for all other values of

M
M

and possibly for values of

n
n

greater than 6, too.
Integral relations among Bessel functions
Authors: Schulz-DuBois E. O.
Published: June 2010

Inverse Laplace transformation of seemingly trivial identities is used to derive novel integral relations among ordinary and modified Bessel functions.
Reducing a matrix to Hessenberg form
Authors: Businger P. A.
Published: June 2010

It has been an open problem whether the reduction of a matrix to Hessenberg (almost triangular) form by Gaussian similarity transformations is numerically stable [

2
2

, p. 364]. We settle this question by exhibiting a class of matrices for which this process is unstable.
Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators
Authors: Carasso Alfred
Published: June 2010

In this paper we analyze the convergence of a centered finite-difference approximation to the nonselfadjoint Sturm-Liouville eigenvalue problem where[unk][{\text {unk}}]has smooth coefficients anda(x)≧a0>0a(x) \geqq {a_0} > 0on [0, 1]. We show that the rate of convergence isO(Δx2)O(\Delta {x^2})as in the selfadjoint case for a scheme of the same accuracy. We also establish discrete analogs of theSturmoscillation and comparison theorems. As a corollary we obtain the result\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup Underscript upper M right-arrow normal infinity semicolon normal upper Delta Subscript x Baseline right-arrow 0 semicolon left-parenthesis upper M plus 1 right-parenthesis normal upper Delta Subscript x Baseline equals 1 Endscripts left-brace sigma-summation Underscript p equals 1 Overscript upper M Endscripts StartFraction StartAbsoluteValue EndAbsoluteValue upper V Superscript p Baseline StartAbsoluteValue EndAbsoluteValue normal infinity Over normal upper Lamda Subscript p Baseline EndFraction right-brace greater-than normal infinity">limsupM→∞;Δx→0;(M+1)Δx=1{∑p=1M||Vp||∞Λp}>∞\lim \sup \limits _{M \to \infty ;{\Delta _x} \to 0;(M + 1){\Delta _x} = 1} \left \{ {\sum \limits _{p = 1}^M {\frac {{||{V^p}||\infty }} {{{\Lambda _p}}}} } \right \} > \infty\]) whereΔx=1/(M+1)\Delta x = 1/(M + 1)is the mesh size andΛp,Vp{\Lambda _p},{V^p}are the characteristic pairs ofLL, theM×MM \times Mmatrix which approximates[unk][{\text {unk}}], andVp{V^p}is normalized so that||Vp||2=1||{V^p}|{|_2} = 1.
The solution of integral equations in Chebyshev series
Authors: Scraton R. E.
Published: June 2010

If the solution of an integral equation can be expanded in the form of a Chebyshev series, the equation can be transformed into an infinite set of algebraic equations in which the unknowns are the coefficients of the Chebyshev series. The algebraic equations are solved by standard iterative procedures, in which it is not necessary to determine beforehand how many coefficients are significant. The method is applicable to equations of either Fredholm or Volterra types.
Lucasian criteria for the primality of 𝑁=ℎ⋅2ⁿ-1
Authors: Riesel Hans
Published: June 2010

Let

v
i
=

v

i
−
1

2

−
2

vi = v_{i - 1}^2 - 2

with

v
0

{v_0}

given. If

v

n
−
2

≡
0
(
mod
N
)

{v_{n - 2}} \equiv 0(\bmod N)

is a necessary and sufficient criterion that

N
=
h
⋅

2
n

−
1

N = h \cdot {2^n} - 1

be prime, this is called a Lucasian criterion for the primality of

N
N

. Many such criteria are known, but the case

h
=
3
A

h = 3A

has not been treated in full generality earlier. A theorem is proved that (by aid of computer) enables the effective determination of suitable numbers

v
0

{v_0}

for any given

N
N

, if

h
>

2
n

h > {2^n}

. The method is used on all

N
N

in the domain

h
=
3
(
6
)
105
,
n
≦
1000

h = 3(6)105,n \leqq 1000

. The Lucasian criteria thus constructed are applied, and all primes

N
=
h
⋅

2
n

−
1

N = h \cdot {2^n} - 1

in the domain are tabulated.
On Lehmer’s method for finding the zeros of a polynomial
Authors: Stewart G. W.
Published: June 2010

Lehmer’s method for finding a zero of a polynomial is a procedure for searching the complex plane in such a way that a zero is isolated in a sequence of disks of decreasing radii. In this paper modifications of the method that improve its stability are given. The convergence of the method and the use of the resulting approximate zero to deflate the polynomial are discussed.
Numerical techniques for finding 𝜈-zeros of Hankel functions
Authors: Cochran James Alan , Hoffspiegel Judith N.
Published: June 2010

This paper is concerned with numerical procedures for the evaluation of the zeros, with respect to order, of Hankel functions and their derivatives in cases when the arguments of these functions are held fixed. Using Olver’s asymptotic expansions, two auxiliary tables have been computed, one appropriate for real and the other for purely imaginary argument. These tables, included herein, permit the calculation of rather accurate approximations to the desired

ν
\nu

-zeros for wide ranges of argument and index. Moreover, from the given tabular entries, the errors attendant with any approximate

ν
\nu

-zero so determined can be easily estimated.
Trigonometric and Gaussian quadrature
Authors: Knight C. J. , Newbery A. C. R.
Published: June 2010

Some relationships are established between trigonometric quadrature and various classical quadrature formulas. In particular Gauss-Legendre quadrature is shown to be a limiting case of trigonometric quadrature.
Discrete least squares and quadrature formulas
Authors: Wilson M. Wayne
Published: June 2010

The purpose of this paper is two-fold. Firstly, we explore some of the intimate connections between discrete least squares processes and quadratures. Secondly, we present an algorithm to construct Gauss-type integration formulas, and consider briefly the method proposed by Gautschi [2].
A note on trapezoidal methods for the solution of initial value problems
Authors: Gourlay A. R.
Published: June 2010

The trapezoidal rule for the numerical integration of first-order ordinary differential equations is shown to possess, for a certain type of problem, an undesirable property. The removal of this difficulty is shown to be straightforward, resulting in a modified trapezoidal rule. Whilst this latent difficulty is slight (and probably rare in practice), the fact that the proposed modification involves negligible additional programming effort would suggest that it is worthwhile. A corresponding modification for the trapezoidal rule for the Goursat problem is also included.
Accelerating convergence of trigonometric approximations
Authors: Jones William B. , Hardy G.
Published: June 2010

Lanczos has recently developed a method for accelerating the convergence of trigonometric approximations for smooth, nonperiodic functions by modifying their boundary behavior. The method is reformulated here in terms of interpolation theory and is shown to be related to the theory of Lidstone interpolation. Extensions given include a new type of modifying function and the establishment of criteria for the convergence of associated interpolation series. Applications are given for the error function and its derivative.
The convergence of single-rank quasi-Newton methods
Authors: Broyden C. G.
Published: June 2010

Analyses of the convergence properties of general quasi-Newton methods are presented, particular attention being paid to how the approximate solutions and the iteration matrices approach their final values. It is further shown that when Broyden’s algorithm is applied to linear systems, the error norms are majorised by a superlinearly convergent sequence of an unusual kind.
A posteriori bounds in the numerical solution of mildly nonlinear parabolic equations
Authors: Carasso Alfred
Published: June 2010

We derive a posteriori bounds for

(
V
−

V
^

)

(V - \hat V)

and its difference quotient

(
V
−

V
^

)
x

{(V - \hat V)_x}

, where

V
V

and

V
^

\hat V

are, respectively, the exact and computed solution of a difference approximation to a mildly nonlinear parabolic initial boundary problem, with a known steadystate solution. It is assumed that the computation is over a long interval of time. The estimates are valid for a class of difference approximations, which includes the CrankNicolson method, and are of the same magnitude for both

(
V
−

V
^

)

(V - \hat V)

and

(
V
−

V
^

)
x

(V - \hat V)x

.
Highly accurate numerical solution of casilinear elliptic boundary-value problems in 𝑛 dimensions
Authors: Pereyra Victor
Published: June 2010

The method of Iterated Deferred Corrections, whose theory was developed by the author, is applied to the problems of the title. The necessary asymptotic expansions are obtained and the way in which the corrections are produced by means of numerical differentiation is described in detail. Numerical results and comparisons with the variationalsplines methods are given.
Improved Newton iteration for integral roots
Authors: King Richard F.
Published: June 2010

An improved Newton iteration procedure for computing pth roots from best Chebyshev or Moursund initial approximations is developed. It differs from the usual Newton method by a multiplicative factor at each step. This multiplier halves the relative error by translating the usual one-sided error curve into a two-sided one, and then adjusting to make a Moursund-like fit. The generalized logarithmic error is used in determining this set of factors.
Strongly asymmetric sequences generated by four elements
Authors: Fiala František
Published: June 2010

Some general properties of strongly asymmetric sequences generated by

m
≧
1

m \geqq 1

elements (m-SAS) are given. Computational experience with two algorithms—for listing of all 4-SASs of a given length n and for generating the smallest 4-SAS of length

n
=
1
,
2
,
⋯

n = 1,2, \cdots

—supports the conjecture that there exists an infinite 4-SAS. The smallest 4-SAS of length 592 is presented.
Chebyshev approximations for the Riemann zeta function
Authors: Cody W. J. , Hillstrom K. E. , Thacher Henry C.
Published: June 2010

This paper presents well-conditioned rational Chebyshev approximations, involving at most one exponentiation, for computation of eitherζ(s)\zeta (s)orζ(s)−1,.5≦s≦55\zeta (s) - 1,.5 \leqq s \leqq 55, for up to 20 significant figures. The logarithmic error is required in one case. An algorithm for the Hurwitz zeta function, and an example of nearly double degeneracy are also given.
Calculation of the gamma function by Stirling’s formula
Authors: Spira Robert
Published: June 2010

In this paper, we derive a simple error estimate for the Stirling formula and also give numerical coefficients.