bluuebunny/crossref_metadata_2025 huggingface dataset:
A random graph with a subcritical number of edges
Authors: Pittel B.
Published: June 2010

A random graphGn(prob⁡(edge)=p)(p=c/n,0>c>1){G_n}(\operatorname {prob} (\operatorname {edge} ) = p)\;(p = c/n,\,0 > c > 1)onnnlabelled vertices is studied. There are obtained limiting distributions of the following characteristics: the lengths of the longest cycle and the longest path, the total size of unicyclic components, the number of cyclic vertices, the number of distinct component sizes, and the middle terms of the component-size order sequence. For instance, it is proved that, with probability approaching(1−c)1/2exp⁡(∑j=1lcj/2j){(1 - c)^{1/2}}\exp (\sum \nolimits _{j = 1}^l {{c^j}/2j)}asn→∞n \to \infty, the random graph does not have a cycle of length>l> l. Another result is that, with probability approaching11, the size of theν\nuth largest component either equals an integer closest toalog⁡(bn/νlog5/2n)a\;\log (bn/\nu \,{\log ^{5/2}}n),a=a(c)a = a(c),b=b(c)b = b(c), or is one less than this integer, provided thatν→∞\nu \to \inftyandν=o(n/log5/2n)\nu = o(n/{\log ^{5/2}}n).
Zelevinski algebras related to projective representations
Authors: Bean M. , Hoffman P.
Published: June 2010

We define

L
L

-

PSH
\operatorname {PSH}

-algebras, and prove a classification theorem for such objects. The letters refer respectively to a ground ring

L
L

and to the positivity, selfadjointness and Hopf structures on an algebra, the basic example of which occurred in the study of projective representations of

S
n

{S_n}

. This is analogous to an idea over

Z

{\mathbf {Z}}

due to Zelevinski in connection with linear representations.
The fundamental module of a normal local domain of dimension 2
Authors: Yoshino Yuji , Kawamoto Takuji
Published: June 2010

The fundamental module

E
E

of a normal local domain

(
R
,

m

)

(R,\,\mathfrak {m})

of dimension

2
2

is defined by the nonsplit exact sequence

0
→
K
→
E
→

m

→
0

0 \to K \to E \to \mathfrak {m} \to 0

, where

K
K

is the canonical module of

R
R

. We prove that, if

R
R

is complete with

R

/

m

≃

C

R/\mathfrak {m} \simeq \mathbb {C}

, then

E
E

is decomposable if and only if

R
R

is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed.
Translates of exponential box splines and their related spaces
Authors: Ben-Artzi Asher , Ron Amos
Published: June 2010

Exponential box splines (

E
B

EB

-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space

H

\mathcal {H}

spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space

H

{\mathbf {H}}

of all entire functions spanned by the integer translates of an

E
B

EB

-spline. This investigation relies on a detailed description of

H

\mathcal {H}

and its discrete analog

S

\mathcal {S}

. The approach taken here is based on the observation that the structure of

H

\mathcal {H}

is relatively simple when

H

\mathcal {H}

is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex

E
B

EB

-splines. The first is when the Fourier transform of the

E
B

EB

-spline vanishes at some critical points, the second is when

H

\mathcal {H}

cannot be embedded in

S

\mathcal {S}

and the third is when

H

{\mathbf {H}}

is a proper subspace of

H

\mathcal {H}

. We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.
Algebras on the disk and doubly commuting multiplication operators
Authors: Axler Sheldon , Gorkin Pamela
Published: June 2010

We prove that a bounded analytic function

f
f

on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by

H
∞

{H^\infty }

and

f
¯

\overline f

does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if

f
f

and

g
g

are bounded analytic functions on the unit disk such that the commutator

T
f

T
g

∗

−

T
g

∗

T
f

{T_f}T_g^{\ast } - T_g^{\ast }{T_f}

(here

T
f

{T_f}

denotes the operator of multiplication by

f
f

on the Bergman space of the disk) is compact, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis min left-brace right-brace comma vertical-bar vertical-bar of ff prime left-parenthesis right-parenthesis z comma vertical-bar vertical-bar of gg prime left-parenthesis right-parenthesis z right-arrow 0">

(
1
−

|

z

|

2

)
min
{

|

f
′

(
z
)

|

,

|

g
′

(
z
)

|

}
→
0

(1 - |z{|^2})\min \{ |f’ (z)|,\;|g’ (z)|\} \to 0

as

|

z

|

↑
1

|z| \uparrow 1

.
Fast algorithms for multiple evaluations of the Riemann zeta function
Authors: Odlyzko A. M. , Schönhage A.
Published: June 2010

The best previously known algorithm for evaluating the Riemann zeta function,

ζ
(
σ
+
i
t
)

\zeta (\sigma + it)

, with

σ
\sigma

bounded and

t
t

large to moderate accuracy (within

±

t

−
c

\pm {t^{ - c}}

for some

c
>
0

c > 0

, say) was based on the Riemann-Siegel formula and required on the order of

t

1

/

2

{t^{1/2}}

operations for each value that was computed. New algorithms are presented in this paper which enable one to compute any single value of

ζ
(
σ
+
i
t
)

\zeta (\sigma + it)

with

σ
\sigma

fixed and

T
⩽
t
⩽
T
+

T

1

/

2

T \leqslant t \leqslant T + {T^{1/2}}

to within

±

t

−
c

\pm {t^{ - c}}

in

O
(

t
ε

)

O({t^\varepsilon })

operations on numbers of

O
(
log
⁡
t
)

O(\log t)

bits for any

ε
>
0

\varepsilon > 0

, for example, provided a precomputation involving

O
(

T

1

/

2
+
ε

)

O({T^{1/2 + \varepsilon }})

operations and

O
(

T

1

/

2
+
ε

)

O({T^{1/2 + \varepsilon }})

bits of storage is carried out beforehand. These algorithms lead to methods for numerically verifying the Riemann hypothesis for the first

n
n

zeros in what is expected to be

O
(

n

1
+
ε

)

O({n^{1 + \varepsilon }})

operations (as opposed to about

n

3

/

2

{n^{3/2}}

operations for the previous method), as well as improved algorithms for the computation of various arithmetic functions, such as

π
(
x
)

\pi (x)

. The new zeta function algorithms use the fast Fourier transform and a new method for the evaluation of certain rational functions. They can also be applied to the evaluation of

L
L

-functions, Epstein zeta functions, and other Dirichlet series.
The resolvent parametrix of the general elliptic linear differential operator: a closed form for the intrinsic symbol
Authors: Fulling S. A. , Kennedy G.
Published: June 2010

Nonrecursive, explicit expressions are obtained for the term of arbitrary order in the asymptotic expansion of the intrinsic symbol of a resolvent parametrix of an elliptic linear differential operator, of arbitrary order and algebraic structure, which acts on sections of a vector bundle over a manifold. Results for the conventional symbol are included as a special case.
Cobordism classes of manifolds with category four
Authors: Singh Harpreet
Published: June 2010

The Lusternik-Schnirelmann category of a manifold

M
M

is the smallest integer

k
k

such that

M
M

can be covered by

k
k

open sets each of which is contractible in

M
M

. The classification up to cobordism of manifolds with category

3
3

was completed by the author in 1985. The object of this paper is to attempt a similar classification of manifolds with category

4
4

.
On James’ type spaces
Authors: Sersouri Abderrazzak
Published: June 2010

We study the spaces

E
E

which are isometric to their biduals

E

∗

∗

{E^{{\ast }{\ast }}}

, and satisfy

dim
⁡
(

E

∗

∗

/

E
)
>
∞

\dim ({E^{{\ast }{\ast }}}/E) > \infty

. We show that these spaces have several common points with the usual James’ space. Our study leads to a kind of classification of these spaces and we show that there are essentially four different basic structures for such spaces in the complex case, and five in the real case.
Cyclic extensions of 𝐾(√-1)/𝐾
Authors: Arason Jón Kr. , Fein Burton , Schacher Murray , Sonn Jack
Published: June 2010

In this paper the height

ht

(
L

/

K
)

{\text {ht}}(L/K)

of a cyclic

2
2

-extension of a field

K
K

of characteristic

≠
2

\ne 2

is studied. Here

ht

(
L

/

K
)
≥
n

{\text {ht}}(L/K) \geq n

means that there is a cyclic extension

E
E

of

K
,
E
⊃
L

K,E \supset L

, with

[
E
:
L
]
=

2
n

[E:L] = {2^n}

. Necessary and sufficient conditions are given for

ht

(
L

/

K
)
≥
n

{\text {ht}}(L/K) \geq n

provided

K
(

−
1

)

K(\sqrt { - 1})

contains a primitive

2
n

{2^n}

th root of unity. Primary emphasis is placed on the case

L
=
K
(

−
1

)

L = K(\sqrt { - 1})

. Suppose

ht

(
K
(

−
1

)

/

K
)
≥
1

{\text {ht}}(K(\sqrt { - 1})/K) \geq 1

. It is shown that

ht

(
K
(

−
1

)

/

K
)
≥
2

{\text {ht}}(K(\sqrt { - 1})/K) \geq 2

and if

K
K

is a number field then

ht

(
K
(

−
1

)

/

K
)
≥
n

{\text {ht}}(K(\sqrt { - 1})/K) \geq n

for all

n
n

. For each

n
≥
2

n \geq 2

an example is given of a field

K
K

such that

ht

(
K
(

−
1

)

/

K
)
≥
n

{\text {ht}}(K(\sqrt { - 1})/K) \geq n

but

ht

(
K
(

−
1

)

/

K
)
≱
n
+
1

{\text {ht}}(K(\sqrt { - 1})/K) \ngeq n + 1

.
Bifurcation of critical periods for plane vector fields
Authors: Chicone Carmen , Jacobs Marc
Published: June 2010

A bifurcation problem in families of plane analytic vector fields which have a nondegenerate center at the origin for all values of a parameter

λ
∈

R

N

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

is studied. In particular, for such a family, the period function

(
ξ
,
λ
)
↦
P
(
ξ
,
λ
)

(\xi ,\lambda ) \mapsto P(\xi ,\lambda )

is defined; it assigns the minimum period to each member of the continuous band of periodic orbits (parametrized by

ξ
∈

R

\xi \in {\mathbf {R}}

) surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with

λ
\lambda

as bifurcation parameter. Generally, if the function

ρ
\rho

, given by

ξ
↦
P
(
ξ
,

λ
∗

)
−
P
(
0
,

λ
∗

)

\xi \mapsto P(\xi ,{\lambda _\ast }) - P(0,{\lambda _\ast })

, vanishes to order

2
k

2k

at the origin, then it is shown that the period function, after a perturbation of

λ
∗

{\lambda _\ast }

, has at most

k
k

critical points near the origin. If

ρ
\rho

vanishes to infinite order, i.e., the center is isochronous, it is shown that the number of critical points of

P
P

for perturbations of

λ
∗

{\lambda _\ast }

depends on the number of generators of the ideal of all Taylor coefficients of

ρ
(
ξ
,
λ
)

\rho (\xi ,\lambda )

, where the coefficients are considered elements of the ring of convergent power series in

λ
\lambda

. Specifically, if the ideal is generated by the first

2
k

2k

Taylor coefficients, then a perturbation of

λ
∗

{\lambda _\ast }

produces at most

k
k

critical points of

P
P

near the origin. These theorems are applied to the quadratic systems with Bautin centers and to one degree of freedom "kinetic+potential" Hamiltonian systems with polynomial potentials. For the quadratic systems a complete solution of the bifurcation problem is obtained. For the Hamiltonian systems a number of results are proved independent of the degree of the potential and a complete solution is obtained for potentials of degree less than seven. Aside from their intrinsic interest, monotonicity properties of the period function are important in the question of existence and uniqueness of autonomous boundary value problems, in the study of subharmonic bifurcation of periodic oscillations, and in the analysis of the problem of linearization. In this regard it is shown that a Hamiltonian system with a polynomial potential of degree larger than two cannot be linearized. However, while these topics are the subject of a large literature, the spirit of this paper is more akin to that of N. Bautin’s treatment of the multiple Hopf bifurcation for quadratic systems and the work on various forms of the weakened Hilbert’s 16th problem first posed by V. Arnold.
Extending homeomorphisms and applications to metric linear spaces without completeness
Authors: Dobrowolski Tadeusz
Published: June 2010

A method of extending homeomorphisms between compacta metric spaces is presented. The main application is that homeomorphisms between compacta of an infinite-dimensional locally convex metric linear space extend to the whole space. A lemma used in the proof of this fact together with the method of absorbing sets is employed to show that every

σ
\sigma

-compact normed linear space is homeomorphic to a dense linear subspace of a Hilbert space. A discussion of the relative topological equivalence of absorbing sets in noncomplete spaces is included. The paper is concluded with some controlled versions of an isotopy extension theorem.
Superprocesses and their linear additive functionals
Authors: Dynkin E. B.
Published: June 2010

Let

X
=
(

X
t

,
P
)

X = ({X_t},P)

be a measure-valued stochastic process. Linear functionals of

X
X

are the elements of the minimal closed subspace

L
L

of

L
2

(
P
)

{L^2}(P)

which contains all

X
t

(
B
)

{X_t}(B)

with

∫

X
t

(
B
)

2

d
P

>
∞

\smallint {{X_t}{{(B)}^2}\;dP\; > \infty }

. Various classes of

L
L

-valued additive functionals are investigated for measure-valued Markov processes introduced by Watanabe and Dawson. We represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their Laplace transforms. For an important particular case—"weighted occupation times"—such equations have been established earlier by Iscoe. We consider Markov processes with nonstationary transition functions to reveal better the principal role of the backward equations. This is especially helpful when we derive the formula for the Laplace transforms.
A generalization of the Levine-Tristram link invariant
Authors: Smolinsky Lawrence
Published: June 2010

Invariants to

m
m

-component links are defined and are shown to be link cobordism invariants under certain conditions. Examples are given.
Realization of the level two standard 𝑠𝑙(2𝑘+1,𝐶)^{∼}-modules
Authors: Misra Kailash C.
Published: June 2010

In this paper we study the level two standard modules for the affine special linear Lie algebras. In particular, we give the vertex operator realizations of all level two standard modules for the affine special linear Lie algebras of odd rank.
Small zeros of quadratic forms over number fields. II
Authors: Vaaler Jeffrey D.
Published: June 2010

Let

F
F

be a nontrivial quadratic form in

N
N

variables with coefficients in a number field

k
k

and let

Z

\mathcal {Z}

be a subspace of

k
N

{k^N}

of dimension

M
,
1
≤
M
≤
N

M,1 \leq M \leq N

. If

F
F

restricted to

Z

\mathcal {Z}

vanishes on a subspace of dimension

L
,
1
≤
L
>
M

L,1 \leq L > M

, and if the rank of

F
F

restricted to

Z

\mathcal {Z}

is greater than

M
−
L

M - L

, then we show that

F
F

must vanish on

M
−
L
+
1

M - L + 1

distinct subspaces

X

0

,

X

1

,
…
,

X

M
−
L

{\mathcal {X}_0},{\mathcal {X}_1}, \ldots ,{\mathcal {X}_{M - L}}

in

Z

\mathcal {Z}

each of which has dimension

L
L

. Moreover, we show that for each pair

X

0

,

X

1

,
1
≤
l
≤
M
−
L

{\mathcal {X}_0},{\mathcal {X}_1},1 \leq l \leq M - L

, the product of their heights

H
(

X

0

)
H
(

X

1

)

H({\mathcal {X}_0})H({\mathcal {X}_1})

is relatively small. Our results generalize recent work of Schlickewei and Schmidt.
Interpolation between Sobolev and between Lipschitz spaces of analytic functions on starshaped domains
Authors: Straube Emil J.
Published: June 2010

We show that on a starshaped domain

Ω
\Omega

in

C
n

{\operatorname {C} ^n}

(actually on a somewhat larger, biholomorphically invariant class) the

L

p

{\mathcal {L}^p}

-Sobolev spaces of analytic functions form an interpolation scale for both the real and complex methods, for each

p
,

0
>
p
⩽
∞

p,\;0 > p \leqslant \infty

. The case

p
=
∞

p = \infty

gives the Lipschitz scale; here the functor

(
,

)

[
θ
]

{(,)^{[\theta ]}}

has to be considered (rather than

(
,

)

[
θ
]

{(,)_{[\theta ]}}

).
Processes disjoint from weak mixing
Authors: Glasner S. , Weiss B.
Published: June 2010

We show that the family

W

⊥

{\mathcal {W}^ \bot }

of ergodic measure preserving transformations which are disjoint from every weakly mixing m.p.t. properly contains the family

D

\mathcal {D}

of distal ergodic m.p.t. In the topological case we show that

P

I

\mathcal {P}\mathcal {I}

, the family of proximally isometric flows is properly contained in the family

M

(

W

⊥

)

\mathcal {M}({\mathcal {W}^ \bot })

of multipliers for

W

⊥

{\mathcal {W}^ \bot }

.
Spectral decompositions of one-parameter groups of isometries on Hardy spaces
Authors: Karayannakis Dimitri
Published: June 2010

Spectral decompositions of strongly continuous one-parameter groups of surjective isometries on Hardy spaces of the disk

D

{\mathbf {D}}

and the torus

T

2

{{\mathbf {T}}^2}

are examined; a concrete description of the (pointwise) action of these decompositions is presented, mainly in the parabolic case, leading to a complete description of the action of the partial sum-operators of M. Riesz when carried from

L
p

(

R

)

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

to

H
p

(

D

)

{H^p}({\mathbf {D}})

,

1
>
p
≤
2

1 > p \leq 2

. The (pointwise) action of the spectral decompositions of these isometric groups on

H
p

(

T

2

)

{H^p}({{\mathbf {T}}^2})

,

1
>
p
>
∞

1 > p > \infty

is also examined and concrete descriptions are derived, mainly in the parabolic case.
On a maximal function on compact Lie groups
Authors: Cowling Michael , Meaney Christopher
Published: June 2010

Suppose that

G
G

is a compact Lie group with finite centre. For each positive number

s
s

we consider the

Ad
⁡
(
G
)

\operatorname {Ad}(G)

-invariant probability measure

μ
s

{\mu _s}

carried on the conjugacy class of

exp
⁡
(
s

H
ρ

)

\exp (s{H_\rho })

in

G
G

. This one-parameter family of measures is used to define a maximal function

M

f

\mathcal {M}\,f

, for each continuous function

f
f

on

G
G

. Our theorem states that there is an index

p
0

{p_0}

in

(
1
,
2
)

(1,2)

, depending on

G
G

, such that the maximal operator

M

\mathcal {M}

is bounded on

L
p

(
G
)

{L^p}(G)

when

p
p

is greater than

p
0

{p_0}

. When the rank of

G
G

is greater than one, this provides an example of a controllable maximal operator coming from averages over a family of submanifolds, each of codimension greater than one.
Realizing rotation vectors for torus homeomorphisms
Authors: Franks John
Published: June 2010

We consider the rotation set

ρ
(
F
)

\rho (F)

for a lift

F
F

of a homeomorphism

f
:

T
2

→

T
2

f:{T^2} \to {T^2}

, which is homotopic to the identity. Our main result is that if a vector

v
v

lies in the interior of

ρ
(
F
)

\rho (F)

and has both coordinates rational, then there is a periodic point

x
∈

T
2

x \in {T^2}

with the property that
\[

F
q

(

x
0

)
−

x
0

q

=
v

\frac {{{F^q}({x_0}) - {x_0}}}{q} = v

\]
where

x
0

∈

R
2

{x_0} \in {R^2}

is any lift of

x
x

and

q
q

is the least period of

x
x

.
Accessory parameters for punctured spheres
Authors: Kra Irwin
Published: June 2010

This paper contains some qualitative results about the accessory parameters for punctured spheres with signature. We show that the Fuchsian uniformizing connection, and hence also the accessory parameters, for the surface depends real analytically on moduli. We also show that the important invariants of a uniformization of a punctured sphere such as the accessory parameters, Fuchsian groups, Poincaré metrics, and covering maps vary continuously under degenerations such as coalescing of punctures.
Torsion points on abelian étale coverings of 𝑃¹-{0,1,∞}
Authors: Coleman Robert F.
Published: June 2010

Let

X
→

P

1

X \to {{\mathbf {P}}^1}

be an Abelian covering of degree

m
m

over

Q

(

μ
m

)

{\mathbf {Q}}({\mu _m})

unbranched outside

0
0

,

1
1

and

∞
\infty

. If the genus of

X
X

is greater than

1
1

embed

X
X

in its Jacobian

J
J

in such a way that one of the points above

0
0

,

1
1

or

∞
\infty

is mapped to the origin. We study the set of torsion points of

J
J

which lie on

X
X

. In particular, we prove that this set is defined over an extension of

Q

{\mathbf {Q}}

unramified outside

6
m

6m

. We also obtain information about the orders of these torsion points.
Conjugating homeomorphisms to uniform homeomorphisms
Authors: Sakai Katsuro , Wong Raymond Y.
Published: June 2010

Let

H
(
X
)

H(X)

denote the group of homeomorphisms of a metric space

X
X

onto itself. We say that

h
∈
H
(
X
)

h \in H(X)

is conjugate to

g
∈
H
(
X
)

g \in H(X)

if

g
=
f
h

f

−
1

{g = fhf^{-1}}

for some

f
∈
H
(
X
)

f \in H(X)

. In this paper, we study the questions: When is

h
∈
H
(
X
)

h \in H(X)

conjugate to

g
∈
H
(
X
)

g \in H(X)

which is a uniform homeomorphism or can be extended to a homeomorphism

g
~

\tilde {g}

on the metric completion of

X
X

Typically for a complete metric space

X
X

, we prove that

h
∈
H
(
X
)

h \in H(X)

is conjugate to a uniform homeomorphism if

H
H

is uniformly approximated by uniform homeomorphisms. In case

X
=

R

X = \mathbf {R}

, we obtain a stronger result showing that every homeomorphism on

R

\mathbf {R}

is, in fact, conjugate to a smooth Lipschitz homeomorphis. For a noncomplete metric space

X
X

, we provide answers to the existence of

g
~

\tilde {g}

under several different settings. Our results are concerned mainly with infinite-dimensional manifolds.
The universal von Staudt theorems
Authors: Clarke Francis
Published: June 2010

We prove general forms of von Staudt’s theorems on the Bernoulli numbers. As a consequence we are able to deduce strong versions of a number of congruences involving various generalisations of the Bernoulli numbers. For example we obtain an improved form of a congruence due to Hurwitz involving the Laurent series coefficients of the Weierstrass elliptic function associated with a square lattice.
A higher order invariant of differential manifolds
Authors: Fredricks Gregory A. , Gilkey Peter B. , Parker Phillip E.
Published: June 2010

We discuss conditions under which a lens space is

s
s

th order flat.
Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces
Authors: Nakamura Gen , Maeda Yoshiaki
Published: June 2010

Local smooth isometric embedding problems of low dimensional Riemannian manifolds into Euclidean spaces are studied. Namely, we prove the existence of a local smooth isometric embedding of a smooth Riemannian

3
3

-manifold with nonvanishing curvature into Euclidean

6
6

-space. For proving this, we give a local solvability theorem for a system of a nonlinear PDE of real principal type. To obtain the local solvability theorem, we need a tame estimate for the linearized equation corresponding to the given PDE, which is presented by two methods. The first is based on the result of Duistermaat-Hörmander which constructed the exact right inverse for linear PDEs of real principal type by using Fourier integral operators. The second method uses more various properties of Fourier integral operators given by Kumano-go, which seems to be a simpler proof than the above.
Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds
Authors: Chi Quo-Shin , Zheng Yunbo
Published: June 2010

Rigidity of minimal immersions of constant curvature in harmonic sequences generated by holomorphic curves in Grassmann manifolds is studied in this paper by lifting them to holomorphic curves in certain projective spaces. We prove that for such curves the curvature must be positive, and that all such simply connected curves in

C

P
n

C{P^n}

are generated by Veronese curves, thus generalizing Calabi’s counterpart for holomorphic curves in

C

P
n

C{P^n}

. We also classify all holomorphic curves from the Riemann sphere into

G
(
2
,
4
)

G(2,4)

whose curvature is equal to

2
2

into two families, which illustrates pseudo-holomorphic curves of positive constant curvature in

G
(
m
,
N
)

G(m,N)

are in general not unitarily equivalent, constracting to the fact that generic isometric complex submanifolds in a Kaehler manifold are congruent.
Characterizations of normal quintic 𝐾-3 surfaces
Authors: Yang Jin Gen
Published: June 2010

If a normal quintic surface is birational to a

K
K

-

3
3

surface then it must contain from one to three triple points as its only essential singularities. A

K
K

-

3
3

surface is the minimal model of a normal quintic surface with only one triple point if and only if it contains a nonsingular curve of genus two and a nonsingular rational curve crossing each other transversally. The minimal models of normal quintic

K
K

-

3
3

surfaces with several triple points can also be characterized by the existence of some special divisors.
Variational problems on contact Riemannian manifolds
Authors: Tanno Shukichi
Published: June 2010

We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable

CR

{\text {CR}}

manifolds. Then the torsion and the generalized Tanaka-Webster scalar curvature are defined properly. Furthermore, we define gauge transformations of contact Riemannian structure, and obtain an invariant under such transformations. Concerning the integral related to the invariant, we define a functional and study its first and second variational formulas. As an example, we study this functional on the unit sphere as a standard contact manifold.
Tables for the evaluation of ∫₀^{∞}𝑥^{𝛽}𝑒^{-𝑥}𝑓(𝑥)𝑑𝑥 by Gauss-Laguerre quadrature
Authors: Concus P. , Cassatt D. , Jaehnig G. , Melby E.
Published: June 2010

Tables of abscissae and weight coefficients to fifteen places are presented for the Gauss-Laguerre quadrature formula
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline x Superscript beta Baseline e Superscript negative x Baseline f left-parenthesis x right-parenthesis d x tilde sigma-summation Underscript k equals 1 Overscript n Endscripts upper H Subscript k Baseline f left-parenthesis a Subscript k Baseline right-parenthesis">

∫
0
∞

x
β

e

−
x

f
(
x
)
d
x
∼

∑

k
=
1

n

H
k

f
(

a
k

)

\int _0^\infty {{x^\beta }{e^{ - x}}f(x)dx \sim \sum \limits _{k = 1}^n {{H_k}f({a_k})} }

for

β
=
−

1
4

,
−

1
2

,

and

−

3
4

\beta = - \tfrac {1}{4}, - \tfrac {1}{2},{\text {and}} - \tfrac {3}{4}

and n = 1(1)15.
A nonlinear alternating direction method
Authors: Kellogg R. B.
Published: June 2010

An alternating direction iteration method is formulated, and convergence is proved, for the solution of certain systems of nonlinear equations. The method is applied to a heat conduction problem with a nonlinear boundary condition.
An extension of the Hille-Hardy formula
Authors: Srivastava H. M.
Published: June 2010

While attempting to give extensions of the well-known Hille-Hardy formula for the generalized Laguerre polynomials

{

L
n

(
α
)

(
x
)
}

\{ {L_n}^{(\alpha )}(x)\}

defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 minus t right-parenthesis Superscript negative 1 minus alpha Baseline exp left-bracket minus StartFraction x t Over 1 minus t EndFraction right-bracket equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper L Subscript n Baseline Superscript left-parenthesis alpha right-parenthesis Baseline left-parenthesis x right-parenthesis t Superscript n">

(
1
−
t

)

−
1
−
α

exp
⁡

[

−

x
t

1
−
t

]

=

∑

n
=
0

∞

L
n

(
α
)

(
x
)

t
n

{(1 - t)^{ - 1 - \alpha }}\exp \left [ { - \frac {{xt}} {{1 - t}}} \right ] = \sum \limits _{n = 0}^\infty {{L_n}^{(\alpha )}} (x){t^n}

\]
, the author applies here certain operational techniques and the method of finite mathematical induction to derive several bilinear generating functions associated with various classes of generalized hypergeometric polynomials. It is observed that the earlier works of Brafman [2], [3], [4], Chaundy [5], Meixner [12], Weisner [16], and others quoted in the literature, are only specialized or limiting forms of the results presented here.
Calculation of the first factor of the class number of the cyclotomic field
Authors: Metsänkylä Tauno
Published: June 2010

The values of the first factorH1(m){H_1}(m)of the class number of themmth cyclotomic field are tabulated for 120 compositemm’s larger than 100. A conjecture concerning a divisibility property ofH1(m){H_1}(m)is stated.
Block implicit one-step methods
Authors: Shampine L. F. , Watts H. A.
Published: June 2010

A class of one-step methods which obtain a block of

r
r

new values at each step are studied. The asymptotic behavior of both implicit and predictor-corrector procedures is examined.
On amicable and sociable numbers
Authors: Cohen Henri
Published: June 2010

An exhaustive search has yielded 236 amicable pairs of which the lesser number is smaller than108{10^8}, 57 pairs being new. It has also yielded 9 new sociable groups of order 10 or less, of which the lesser number is smaller than6.107{6.10^7}; the 9 sociable groups are all of order 4. The sequence of iterates of the functions(n)=σ(n)−ns(n) = \sigma (n) - nstarting with 276 has also been extended to 119 terms.
The explicit inverse of a tridiagonal matrix
Authors: Schlegel P.
Published: June 2010

The closed form inverse of a tridiagonal matrix, which is a slight generalization of a matrix considered by D. Kershaw (Math. Comp., v. 23, 1969, pp. 189–191), is given in this note. If the matrix has integer elements, an integer multiple of the inverse can be computed by integer arithmetic, that is, without machine roundoff error.
Conditioning of quasi-Newton methods for function minimization
Authors: Shanno D. F.
Published: June 2010

Quasi-Newton methods accelerate the steepest-descent technique for function minimization by using computational history to generate a sequence of approximations to the inverse of the Hessian matrix. This paper presents a class of approximating matrices as a function of a scalar parameter. The problem of optimal conditioning of these matrices under an appropriate norm as a function of the scalar parameter is investigated. A set of computational results verifies the superiority of the new methods arising from conditioning considerations to known methods.
Error bounds for polynomial spline interpolation
Authors: Schultz Martin H.
Published: June 2010

New upper and lower bounds for the

L
2

{L^2}

and

L
∞

{L^\infty }

norms of derivatives of the error in polynomial spline interpolation are derived. These results improve corresponding results of Ahlberg, Nilson, and Walsh, cf. [1], and Schultz and Varga, cf. [5].
Control and estimation of computational errors in the evaluation of interpolation formulae and quadrature rules
Authors: Gustafson Sven-Ȧke
Published: June 2010

Approximate rules for evaluating linear functionals are often obtained by requiring that the rule shall give exact value for a certain linear class of functions. The parameters of the rule appear hence as the solution of a system of equations. This can generally not be solved exactly but only "numerically." Sometimes large errors occur in the parameters defining the rule, but the resultant error in the computed value of the functional is small. In the present paper we shall develop efficient methods of computing a strict bound for this error in the case when the parameters of the rule are determined from a linear system of equations.
Trigonometric interpolation and curve-fitting
Authors: Newbery A. C. R.
Published: June 2010

Some algorithms are introduced, whereby a function defined on an arbitrarily spaced set of abscissas may be interpolated or approximated by trigonometric or hyperbolic polynomials. The interpolation may be ordinary or osculatory. Least squares approximation is included; the approximant may be a pure sine series or a cosine series or a balanced trigonometric or hyperbolic polynomial. An application to a periodicity-search is described.
Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature
Authors: Lyness J. N.
Published: June 2010

The conventional Fourier coefficient asymptotic expansion is derived by means of a specific contour integration. An adjusted expansion is obtained by deforming this contour. A corresponding adjustment to the Euler-Maclaurin expansion exists. The effect of this adjustment in the error functional for a general quadrature rule is investigated. It is the same as the effect of subtracting out a pair of complex poles from the integrand, using an unconventional subtraction function. In certain applications, the use of this subtraction function is of practical value. An incidental result is a direct proof of Erdélyi’s formula for the Fourier coefficient asymptotic expansion, valid when

f
(
x
)

f(x)

has algebraic or logarithmic singularities, but is otherwise analytic.
Diophantine approximation of ternary linear forms
Authors: Cusick T. W.
Published: June 2010

The paper gives an efficient method for finding arbitrarily many solutions in integers x, y, z of the Diophantine inequality

|

x
+
α
y
+
β
z

|

max
(

y
2

,

z
2

)
>
c

|x + \alpha y + \beta z|\max ({y^2},{z^2}) > c

, where

α
\alpha

defines a totally real cubic field F over the rationals, the numbers 1,

α
,
β

\alpha ,\beta

form an integral basis for F, and c is a constant which can be calculated in terms of parameters of the method. For certain values of c, the method generates all solutions of the inequality.
On descent from local minima
Authors: Goldstein A. A. , Price J. F.
Published: June 2010

When a local minimum of a function of several variables has been found by use of an algorithm for finding such minima numerically, one often runs the same algorithm many times with different starting values in the hopes of finding a lower minimum. Here, under the assumption that a local minimum is known, a process with analytical criteria is described which sometimes finds smaller local minima in an algorithmic manner.
Rate of convergence of Lawson’s algorithm
Authors: Cline A. K.
Published: June 2010

The algorithm of Charles L. Lawson determines uniform approximations of functions as limits of weighted

L
2

{L_2}

approximations. Lawson noticed from experimental evidence that the algorithm seemed to converge linearly and convergence was related to a factor which was the ratio of the largest nonmaximum error of the best uniform approximation to the maximum error. This paper proves the linear convergence and explores the relation of the rate of convergence to this ratio.
A predictor-corrector method for a certain class of stiff differential equations
Authors: Guderley Karl G. , Hsu Chen-chi
Published: June 2010

In stiff systems of linear ordinary differential equations, certain elements of the matrix describing the system are very large. Sometimes, e.g., in treating partial differential equations, the problem can be formulated in such a manner that large elements appear only in the main diagonal. Then the elements causing stiffness can be taken into account analytically. This is the basis of the predictor-corrector method presented here. The truncation error can be estimated in terms of the difference between predicted and corrected values in nearly the same manner as for the customary predictor-corrector method. The question of stability, which is crucial for stiff equations, is first studied for a single equation; as expected, the method is much more stable than the usual predictor- corrector method. For systems of equations, sufficient conditions for stability are derived which require less work than a detailed stability analysis. The main tool is a matrix norm which is consistent with a weighted infinity vector norm. The choice of the weights is critical. Their determination leads to the question whether a certain matrix has a positive inverse.
On the solution of systems of equations by the epsilon algorithm of Wynn
Authors: Gekeler E.
Published: June 2010

The

ϵ
\epsilon

-algorithm has been proposed by Wynn on a number of occasions as a convergence acceleration device for vector sequences; however, little is known concerning its effect upon systems of equations. In this paper, we prove that the algorithm applied to the Picard sequence

x

i
+
1

=
F
(

x

i

)

{{\text {x}}_{i + 1}} = F({{\text {x}}_i})

of an analytic function

F
:

R

n

⊃
D
→

R

n

F:{{\text {R}}^n} \supset D \to {{\text {R}}^n}

provides a quadratically convergent iterative method; furthermore, no differentiation of

F
F

is needed. Some examples illustrate the numerical performance of this method and show that convergence can be obtained even when

F
F

is not contractive near the fixed point. A modification of the method is discussed and illustrated.
The integral analogue of the Leibniz rule
Authors: Osler Thomas J.
Published: June 2010

This paper demonstrates that the classical Leibniz rule for the derivative of the product of two functions
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript upper N Baseline u upsilon equals sigma-summation Underscript k equals 0 Overscript upper N Endscripts left-parenthesis Subscript k Baseline Superscript upper N Baseline right-parenthesis upper D Superscript upper N minus k Baseline u upper D Superscript k Baseline upsilon">

D
N

u
υ
=

∑

k
=
0

N

(
k
N

)

D

N
−
k

u

D
k

υ

{D^N}u\upsilon = \sum \limits _{k = 0}^N {(_k^N)} {D^{N - k}}u{D^k}\upsilon

\]
has the integral analog
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript alpha Baseline u upsilon equals integral Subscript negative normal infinity Superscript normal infinity Baseline left-parenthesis Subscript omega Baseline Superscript alpha Baseline right-parenthesis upper D Superscript alpha minus omega Baseline u upper D Superscript omega Baseline upsilon d omega period">

D
α

u
υ
=

∫

−
∞

∞

(
ω
α

)

D

α
−
ω

u

D
ω

υ
d
ω
.

 

{D^\alpha }u\upsilon = \int _{ - \infty }^\infty {(_\omega ^\alpha )} {D^{\alpha - \omega }}u{D^\omega }\upsilon d\omega .{\text { }}

\]
The derivatives occurring are “fractional derivatives.” Various generalizations of the integral are given, and their relationship to Parseval’s formula from the theory of Fourier integrals is revealed. Finally, several definite integrals are evaluated using our results.
Optimal multiplication chains for computing a power of a symbolic polynomial
Authors: Gentleman W. Morven
Published: June 2010

This paper shows that in a certain model of symbolic manipulation of algebraic formulae, the simple method of computing a power of a symbolic polynomial by repeated multiplication by the original polynomial is, in essence, the optimal method.
Complex zeros of the error function and of the complementary error function
Authors: Fettis Henry E. , Caslin James C. , Cramer Kenneth R.
Published: June 2010

The first one hundred zeros of the error function and of the complementary error function are given. An asymptotic formula for the higher zeros is also derived.
Accurate evaluation of Wiener integrals
Authors: Chorin Alexandre Joel
Published: June 2010

A new quadrature formula for an important class of Wiener integrals is presented, in which the Wiener integrals are approximated by n-fold integrals with an error

O
(

n

−
2

)

O({n^{ - 2}})

. The resulting n-fold integrals can then be approximated by ordinary finite sums of remarkably simple structure. An example is given.
Generalized local maximum principles for finite-difference operators
Authors: Brandt Achi
Published: June 2010

The generalized local maximum principle for a difference operator

L
h

{L_h}

asserts that if

L
h

u
(
x
)
>
0

{L_h}u(x) > 0

then

Γ
u

\Gamma u

cannot attain its positive maximum at the net-point x. Here

Γ
\Gamma

is a local net-operator such that

Γ
u
=
u
+
O
(
h
)

\Gamma u = u + O(h)

for any smooth function u. This principle, with simple forms of

Γ
\Gamma

, is proved for some quite general classes of second-order elliptic operators

L
h

{L_h}

, whose associated global matrices are not necessarily monotone. It is shown that these generalized principles can be used for easy derivation of global a priori estimates to the solutions of elliptic difference equations and to their difference-quotients. Some examples of parabolic difference equations are also treated.
Local extrapolation in the solution of ordinary differential equations
Authors: Shampine L. F.
Published: June 2010

The local errors being estimated in the solution of an initial value problem can be added in to make the solution more accurate but this is not always advisable. A rule for deciding when to extrapolate is studied for one-step methods. Some observations about the correctness of local error estimators and extrapolation of multistep methods are also made.
Mesh refinement
Authors: Browning Gerald , Kreiss Heinz-Otto , Oliger Joseph
Published: June 2010

We study mesh refinement techniques for first-order hyperbolic equations. A refinement method for use with the leap-frog scheme is defined and its stability established. The remainder of the paper is devoted to a discussion of the effects of nonuniform grids and the circumstances under which they may be used.
Rate of convergence estimates for nonselfadjoint eigenvalue approximations
Authors: Bramble J. H. , Osborn J. E.
Published: June 2010

In this paper, a general approximation theory for the eigenvalues and corresponding subspaces of generalized eigenfunctions of a certain class of compact operators is developed. This theory is then used to obtain rate of convergence estimates for the errors which arise when the eigenvalues of nonselfadjoint elliptic partial differential operators are approximated by Rayleigh-Ritz-Galerkin type methods using finite-dimensional spaces of trial functions, e.g. spline functions. The approximation methods include several in which the functions in the space of trial functions are not required to satisfy any boundary conditions.
Error analysis for Fourier series evaluation
Authors: Newbery A. C. R.
Published: June 2010

A floating-point error analysis is given for the standard recursive method of evaluating trigonometric polynomials. It is shown that, by introducing a phase-shift, one can hold the error growth down to an essentially linear function of the degree. Explicit computable error bounds are derived and numerically verified.
Eigenfrequencies of an elliptic membrane
Authors: Troesch B. A. , Troesch H. R.
Published: June 2010

The first few eigenfrequencies of a homogeneous elliptic membrane, which is fixed along its boundary, are given in a graph. It is explained in detail, how more accurate results can readily be obtained for special purposes. The known expansion of the eigenfrequencies for small and large eccentricities are summarized. As an application some nodal patterns for a membrane with a double eigenvalue are presented.
The distribution of small gaps between successive primes
Authors: Brent Richard P.
Published: June 2010

For

r
≧
1

r \geqq 1

and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes

p
≦
N

p \leqq N

such that

p
+
2
r

p + 2r

is the least prime greater than p is asymptotic to
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 2 Superscript upper N Baseline left-parenthesis sigma-summation Underscript k equals 1 Overscript r Endscripts StartFraction upper A Subscript r comma k Baseline Over left-parenthesis log x right-parenthesis Superscript k plus 1 Baseline EndFraction right-parenthesis d x comma">

∫
2
N

(

∑

k
=
1

r

A

r
,
k

(
log
⁡
x
)

k
+
1

)

d
x
,

\int _2^N {\left ( {\sum \limits _{k = 1}^r {\frac {{{A_{r,k}}}}{{{{(\log x)}^{k + 1}}}}} } \right )} \;dx,

\]
where the

A

r
,
k

{A_{r,k}}

are certain constants. We describe a method for computing these constants. Related constants are given to 10D for

r
=
1
(
1
)
40

r = 1(1)40

, and some empirical evidence supporting the conjecture is mentioned.
On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations
Authors: Prothero A. , Robinson A.
Published: June 2010

The stiffness in some systems of nonlinear differential equations is shown to be characterized by single stiff equations of the form
\[

y
′

=

g
′

(
x
)
+
λ
{
y
−
g
(
x
)
}
.

y’ = g’(x) + \lambda \{y - g(x)\}.

\]
The stability and accuracy of numerical approximations to the solution

y
=
g
(
x
)

y = g(x)

, obtained using implicit one-step integration methods, are studied. An S-stability property is introduced for this problem, generalizing the concept of A-stability. A set of stiffly accurate one-step methods is identified and the concept of stiff order is defined in the limit

Re
⁡
(
−
λ
)
→
∞

\operatorname {Re}(-\lambda ) \to \infty

. These additional properties are enumerated for several classes of A-stable one-step methods, and are used to predict the behaviour of numerical solutions to stiff nonlinear initial-value problems obtained using such methods. A family of methods based on a compromise between accuracy and stability considerations is recommended for use on practical problems.
Iterates of the unitary totient function
Authors: Lal Mohan
Published: June 2010

The iterates of the unitary analogue of Euler’s totient function

φ
∗

(
n
)

{\varphi ^\ast }(n)

are investigated empirically for

n
≦

10
5

n \leqq {10^5}

.
Rational Chebyshev approximations for the modified Bessel functions 𝐼₀(𝑥) and 𝐼₁(𝑥)
Authors: Blair J. M.
Published: June 2010

This note presents nearly-best rational approximations for the functions

I
0

(
x
)

{I_0}(x)

and

I
1

(
x
)

{I_1}(x)

, with relative errors ranging down to

10

−
23

{10^{ - 23}}

.
Methods for modifying matrix factorizations
Authors: Gill P. E. , Golub G. H. , Murray W. , Saunders M. A.
Published: June 2010

In recent years, several algorithms have appeared for modifying the factors of a matrix following a rank-one change. These methods have always been given in the context of specific applications and this has probably inhibited their use over a wider field. In this report, several methods are described for modifying Cholesky factors. Some of these have been published previously while others appear for the first time. In addition, a new algorithm is presented for modifying the complete orthogonal factorization of a general matrix, from which the conventional QR factors are obtained as a special case. A uniform notation has been used and emphasis has been placed on illustrating the similarity between different methods.
Differences of fractional order
Authors: Díaz J. B. , Osler T. J.
Published: June 2010

Derivatives of fractional order,

D
α

f

{D^\alpha }f

, have been considered extensively in the literature. However, little attention seems to have been given to finite differences of fractional order,

Δ
α

f

{\Delta ^\alpha }f

. In this paper, a definition of differences of arbitrary order is presented, and

Δ
α

f

{\Delta ^\alpha }f

is computed for several specific functions f (Table 2.1). We find that the operator

Δ
α

{\Delta ^\alpha }

is closely related to the contour integral which defines Meijer’s G-function. A Leibniz rule for the fractional difference of the product of two functions is discovered and used to generate series expansions involving the special functions.