bluuebunny/crossref_metadata_2025_split huggingface dataset:
The lifting problem for affine structures in nilpotent Lie groups
Authors: Boyom Nguiffo B. 
Published: June 2010

Affine manifolds occur in several situations in pure and applied mathematics, (e.g. leaves of Lagrangian foliations, completely integrable Hamiltonian systems, linear representations of virtually polycyclic groups, geometric quantization and so on). This work is devoted to left invariant affinely flat structures in Lie groups. We are mainly concerned with the following situation. Let

G
G

and

G
0

{G_0}

be nilpotent Lie groups of dimension

n
+
1

n + 1

and

n
n

, respectively and let

h
:
G
→

G
0

h:G \to {G_0}

be a continuous homomorphism from

G
G

onto

G
0

{G_0}

. Given a left invariant affinely flat structure

(

G
0

,

∇
0

)

({G_0},{\nabla _0})

the lifting problem is to discover whether

G
G

has a left invariant affinely flat structure

(
G
,
∇
)

(G,\nabla )

such that

h
h

becomes an affine morphism. In the present work we answer positively when

(

G
0

,

∇
0

)

({G_0},{\nabla _0})

is "normal". Therefore the existence problem for a left invariant complete affinely flat structure in nilpotent Lie groups is solved by applying the following subsequent results. Let

A

f
(

G
0

)

\mathfrak {A}f({G_0})

be the set of left invariant affinely flat structures in the nilpotent Lie group

G
0

,
(

1
∘

)

A

f
(

G
0

)
≠
∅

{G_0},({1^ \circ })\;\mathfrak {A}f({G_0}) \ne \emptyset

implies the existence of normal structure

(

G
0

,

∇
0

)
∈

A

f
(

G
0

)
;
(

2
∘

)

h
:
G
→

G
0

({G_0},{\nabla _0}) \in \mathfrak {A}f({G_0});({2^ \circ })\;h:G \to {G_0}

being as above every normal structure

(

G
0

,

∇
0

)

({G_0},{\nabla _0})

has a normal lifted in

A

f
(
G
)

\mathfrak {A}f(G)

.
Relations between 𝐻^{𝑝}ᵤ and 𝐿^{𝑝}ᵤ in a product space
Authors: Strömberg Jan-Olov , Wheeden Richard L. 
Published: June 2010

Relations between

L
u
p

L_u^p

and

H
u
p

H_u^p

are studied for the product space

R

1

×

R

1

{{\mathbf {R}}^1} \times {{\mathbf {R}}^1}

in the case

1
>
p
>
∞

1 > p > \infty

and

u
(

x
1

,

x
2

)
=

|

Q
1

(

x
1

)

|

p

|

Q
2

(

x
2

)

|

p

w
(

x
1

,

x
2

)

u({x_1},{x_2}) = |{Q_1}({x_1}){|^p}|{Q_2}({x_2}){|^p}w({x_1},{x_2})

, where

Q
1

{Q_1}

and

Q
2

{Q_2}

are polynomials and

w
w

satisfies the

A
p

{A_p}

condition for rectangles. A description of the distributions in

H
u
p

H_u^p

is given. Questions about boundary values and about the existence of dense subsets of smooth functions satisfying appropriate moment conditions are also considered.
Higher-order Sugawara operators for affine Lie algebras
Authors: Goodman Roe , Wallach Nolan R. 
Published: June 2010

Let

g

^

\hat {\mathfrak {g}}

be the affine Lie algebra associated to a simple Lie algebra

g

\mathfrak {g}

. Representations of

g

^

\hat {\mathfrak {g}}

are described by current fields

X
(
ζ
)

X(\zeta )

on the circle

T

(
X
∈

g

{\mathbf {T}}\;(X \in \mathfrak {g}

and

ζ
∈

T

)

\zeta \in {\mathbf {T}})

. In this paper a linear map

σ
\sigma

from the symmetric algebra

S
(

g

)

S(\mathfrak {g})

to (formal) operator fields on a suitable category of

g

^

\hat {\mathfrak {g}}

modules is constructed. The operator fields corresponding to

g

\mathfrak {g}

-invariant elements of

S
(

g

)

S(\mathfrak {g})

are called Sugawara fields. It is proved that they satisfy commutation relations of the form

(
∗
)

(\ast )

\[

[
σ
(
u
)
(
ζ
)
,
X
(
η
)
]
=

c
∞

D
δ
(
ζ

/

η
)
σ
(

∇
X

u
)
(
ζ
)
+

higher-order

terms

[\sigma (u)(\zeta ),X(\eta )] = {c_\infty }D\delta (\zeta /\eta )\sigma ({\nabla _X}u)(\zeta ) + {\text {higher-order}}\;{\text {terms}}

\]
with the current fields, where

c
∞

{c_\infty }

is a renormalization of the central element in

g

^

\hat {\mathfrak {g}}

and

D
δ

D\delta

is the derivative of the Dirac delta function. The higher-order terms in

(
∗
)

(\ast )

are studied using results from invariant theory and finite-dimensional representation theory of

g

\mathfrak {g}

. For suitably normalized invariants

u
u

of degree

4
4

or less, these terms are shown to be zero. This vanishing is also proved for

g

=

sl

(
n
,

C

)

\mathfrak {g} = {\text {sl}}(n,{\mathbf {C}})

and

u
u

running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever

c
∞

{c_\infty }

is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for

g

^

\hat {\mathfrak {g}}

.
On the propagation of errors in the inversion of certain tridiagonal matrices
Authors: Lowan Arnold N. 
Published: June 2010

When the differential equation of heat conduction is replaced by the implicit difference analog, one is led to the solution of Ay = b where A is a tridiagonal matrix whose elements on the principal diagonal are = 2 + 2r and whose elements off the principal diagonal are = -r. The system of equations may be solved by the following algorithm:
\[

β
k

=

u

=

r
2

β

k
−
1

−
1

,

β
1

=

u
1

;

γ
k

=

−
r

β

−
1

;

z
k

=

(

b
k

+

r

z

k
−
1

)

β
k

−
1

,

z
1

=

b
1

u

−
1

;

y
k

=

z
k

−

γ
k

y

k
+
1

,

y
M

=

z
M

.

{\beta _k}\, = \,u\, = \,{r^2}\beta _{k - 1}^{ - 1},\,{\beta _1}\, = \,{u_1}\,;\qquad {\gamma _k}\, = \, - r{\beta ^{ - 1}};\qquad {z_k}\, = \,({b_k}\, + \,r{z_{k - 1}})\beta _k^{ - 1},\,{z_1}\, = \,{b_1}{u^{ - 1}};\qquad {y_k}\, = \,{z_k}\, - \,{\gamma _k}{y_{k + 1}},\qquad {y_M}\,=\,{z_M}.

\]
An upper bound of the round-off errors in the computed values of the

y
k

{y_k}

’s is obtained. An actual test case showed that the theoretical upper bound is about four times larger than the true round-off error. Moreover, the theoretical upper bound does not seem to vary appreciably with r.
Improved formulas for complete and partial summation of certain series.
Authors: Salzer Herbert E. , Kimbro Genevieve M. 
Published: June 2010

In two previous articles one of the authors gave formulas, with numerous examples, for summing a series either to infinity (complete) or up to a certain number n of terms (partial) by considering the sum of the first j terms

S
j

{S_j}

, or some suitable modification

S
j
′

{S’_j}

, closely related to

S
j

{S_j}

, as a polynomial in 1/j. Either

S
∞

{S_\infty }

or

S
n

{S_n}

was found by m-point Lagrangian extrapolation from

S

j

0

{{S}_{{{j}_{0}}}}

,

S

j

0

−
1

{{S}_{{{j}_{0}}-1}}

,

⋅
⋅
⋅

\cdot \cdot \cdot

,

S

j

0

−
m
+
1

{{S}_{{{j}_{0}}-m+1}}

to 1/j = 0 or 1/j = 1/n respectively. This present paper furnishes more accurate m-point formulas for sums (or sequences)

S
j

{S_j}

which behave as even functions of 1/j. Tables of Lagrangian extrapolation coefficients in the variable

1

/

j
2

1/{j^2}

are given for: complete summation, m = 2(1)7,

j
0

{j_0}

= 10, exactly and 20D, m = 11,

j
0

{j_0}

= 20, 30D; partial summation, m = 7,

j
0

{j_0}

= 10, n = 11(1)25(5)100, 200, 500, 1000, exactly. Applications are made to calculating

π
\pi

or the semi-perimeters of many-sided regular polygons, Euler’s constant,
\[

1
+

∑

r
=
1

j

{

−
1

(
4
r
−
1
)

2

+

1

(
4
r
+
1
)

2

}

=
1
−

1

3
2

+

1

5
2

−
⋯

for

j
=
∞

1 + \sum \limits _{r = 1}^j {\left \{ {\tfrac {{ - 1}}{{{{(4r - 1)}^2}}} + \tfrac {1}{{{{(4r + 1)}^2}}}} \right \} = 1 - \tfrac {1}{{{3^2}}} + \tfrac {1}{{{5^2}}} - \cdots } \,{\text {for}}\,j = \infty

\]
(Catalan’s constant), calculation of a definite integral as the limit of a suitably chosen sequence, determining later zeros of

J
v

(
x
)

{J_v}(x)

from earlier zeros for suitable v, etc. A useful device in many cases involving sums of odd functions, is to replace

S
j

{S_j}

by a trapezoidal-type

S
j
′

{S’_j}

which is seen, from the Euler-Maclaurin formula, to be formally a series in

1

/

j
2

1/{j^2}

. In almost every example, comparison with the earlier (1/j)-extrapolation showed these present formulas, for 7 points, to improve results by anywhere from around 4 to 9 places.
Multiple quadrature with central differences on one line
Authors: Salzer Herbert E. 
Published: June 2010

The coefficients

A

2
m

n

A_{2m}^n

in the n-fold quadrature formulas for the stepwise integration of (1)

y

(
n
)

=
f
(
x
,
y
,

y
′

,
⋯
,

y

(
n
−
1
)

)

{y^{(n)}} = f(x,y,y’, \cdots , {y^{(n - 1)}})

, at intervals of h, namely, for n even, (2)

δ
n

y
0

=

h
n

∑

m
=
1

10

(
1
+

A

2
m

n

δ

2
m

)

f
0

+
⋯

{\delta ^n}{y_0} = {h^n}\sum \nolimits _{m = 1}^{10} {(1 + A_{2m}^n{\delta ^{2m}}){f_0} + \cdots }

, for n odd, (3)

μ

δ
n

y
0

=

h
n

∑

m
=
1

10

(
1
+

A

2
m

n

δ

2
m

)

f
0

+
⋯

\mu {\delta ^n}{y_0} = {h^n}\sum \nolimits _{m = 1}^{10} {(1 + A_{2m}^n{\delta ^{2m}}){f_0} + \cdots }

, are tabulated exactly for n = 1(1)6, m = 1(1)10. They were calculated from the well-known symbolic formulas (4)

δ

n

y
=

(

δ

/

D

)

n

f

{{\delta }^{n}}y = {{({\delta }/{D})}^{n}}f

, (5)

(

δ

/

D

)

n

=

(

δ
h

/

2

sinh

−
1

(

δ

/

2

)
)

n

{{({\delta }/{D})}^{n}} = {{({{{\delta h}/{2\,\sinh }}^{-1}}({\delta }/{2}))}^{n}}

and (6)

μ
=

(
1
+

δ

2

/

4

)

1

/

2

=
1
+

δ

2

8

−

δ

4

128

+

δ

6

1024

−

5

δ

8

32768

+
⋯

\mu = {{(1 + {{{\delta }^{2}}}/{4})}^{{1}/{2}}} = 1 + \frac {{{\delta }^{2}}}{8}\,-\,\frac {{{\delta }^{4}}}{128} + \frac {{{\delta }^{6}}}{1024}\,-\,\frac {5{{\delta }^{8}}}{32768} + \cdots

. For calculating

y

(
r
)

{y^{(r)}}

, replace n by n - r in (2) and (3). Use of (2) and (3) avoids the solution of (1) by simultaneous lower-order systems for

n
>
1

n > 1

, as well as mid-interval tabular arguments, requires only even-order differences, on a single line, and provides great accuracy due to rapid decrease of

A

2
m

n

A_{2m}^n

as m increases. However, the integration may be slowed down by the need to estimate and refine iteratively the later values of

y
,

y
′

,
⋯
,

y

(
n
−
1
)

y,y’, \cdots , {y^{(n - 1)}}

required in

δ

2
m

f
0

{\delta ^{2m}}{f_0}

. Reference to earlier collected formulas of Legendre, Oppolzer, Thiele, Lindow, Salzer, Milne and Buckingham, reveals that Thiele and Buckingham come closest to (2), (3), as their works contain schemes that involve just tabular arguments throughout. For n odd, they give formulas that are based upon the series in

δ

2
m

{\delta ^{2m}}

for

(

1

/

μ

)

(

δ

/

D

)

n

({1}/{\mu }){{({\delta }/{D})}^{n}}

instead of

μ

(

δ

/

D

)

n

\mu {{({\delta }/{D})}^{n}}

as in the present arrangement.
Factorization of polynomials over finite fields
Authors: McEliece Robert J. 
Published: June 2010

If

f
(
x
)

f(x)

is a polynomial over

G
F
(
q
)

GF(q)

, we observe (as has Berlekamp) that if

h

(
x

)
q

≡
h
(
x
)
(
mod
f
(
x
)
)

h{(x)^q} \equiv h(x)(\bmod f(x))

, then

f
(
x
)
=
∏

a

∈
G
F
(
q
)

gcd
(
f
(
x
)
,
h
(
x
)
−
a
)

f(x) = \prod {{{_a}_{ \in GF(q)}}\gcd (f(x),h(x) - a)}

. The object of this paper is to give an explicit construction of enough such

h
h

’s so that the repeated application of this result will succeed in separating all irreducible factors of

f
f

. The

h
h

’s chosen are loosely defined by

h
i

(
x
)
≡

x
i

+

x

i
q

+

x

i

q
2

+
⋯
(
mod
f
(
x
)
)

{h_i}(x) \equiv {x^i} + {x^{iq}} + {x^{i{q^2}}} + \cdots (\bmod f(x))

. A detailed example over

G
F
(
2
)

GF(2)

is given, and a table of the factors of the cyclotomic polynomials

Φ
n

(
x
)
(
mod
p
)

for

p
=
2
,
n
≦
250
;
p
=
3
,
n
≦
100
;
p
=
5
,
7
,
n
≦
50

{\Phi _n}(x)(\bmod p){\text { for }}p = 2,n \leqq 250;p = 3,n \leqq 100;p = 5,7,n \leqq 50

, is included.
Movable singularities and quadrature
Authors: Goodrich R. F. , Stenger F. 
Published: June 2010

A general procedure is described for treating a movable singularity in an integral. This enabĺes us to change the original integral

I
0

{I_0}

into

G

I
1

G{I_1}

, where

G
G

depends only on the parameters of the singularity and

I
1

{I_1}

, is a new integral which exists for all values of the parameters. The results are then applied to the particular problem of evaluating
\[

∫

−
1

1

f
(
x
)
d
x

{
(
1
−

x
2

)
(
1
−

k
2

x
2

)
}

1

/

2

,

\int _{ - 1}^1 {\frac {{f(x)dx}} {{{{\{ (1 - {x^2})(1 - {k^2}{x^2})\} }^{1/2}}}}} ,

\]
where

f
f

is entire and

k
k

varies between

0
0

and

1
1

. Some new quadrature schemes and new effective methods of evaluating incomplete elliptic integrals are derived.
New error coefficients for estimating quadrature errors for analytic functions
Authors: Rabinowitz Philip , Richter Nira 
Published: June 2010

Since properly normalized Chebyshev polynomials of the first kind

T
~

n

(
Z
)

{\tilde T_n}(Z)

satisfy
\[

(

T
~

m

,

T
~

n

)
=

∫

∈
ρ

T
~

m

(

z

)

T
n

(

z

)

¯

|

1
−

z

2

|

−
1

/

2

|

d

z

|

=

δ

m
n

({\tilde T_m},{\tilde T_n}) = \int _{ \in \rho } {{{\tilde T}_m}({\text {z}})} \overline {{T_n}({\text {z}})} |1 - {{\text {z}}^2}{|^{ - 1/2}}|d{\text {z}}| = {\delta _{mn}}

\]
for ellipses

∈
ρ

\in \rho

with foci at

±
1

\pm 1

, any function analytic in

∈
ρ

\in \rho

has an expansion,

f
(

z

)
=
∑

a
n

T
~

n

(

z

)

f({\text {z}}) = \sum {{a_n}{{\tilde T}_n}({\text {z}})}

with

a
n

=
(
f
,

T
~

n

)

{a_n} = (f,{\tilde T_n})

. Applying the integration error operator

E
E

yields

E
(
f
)
=
∑

a
n

E
(

T
~

n

)

E(f) = \sum {{a_n}E({{\tilde T}_n})}

. Applying the Cauchy-Schwarz inequality to

E
(
f
)

E(f)

leads to the inequality
\[

|

E
(
f
)

|

2

≦
∑

|

a
n

|

2

∑

|

E
(

T
~

n

)

|

2

=

|

|

f

|

|

2

|

|

E

|

|

2

.

|E(f){|^2} \leqq \sum {|{a_n}{|^2}} \sum {|E({{\tilde T}_n})} {|^2} = ||f|{|^2}||E|{|^2}.

\]
.

|

|

E

|

|

||E||

can be computed for any integration rule and approximated quite accurately for Gaussian integration rules. The bound for

|

E
(
f
)

|

|E(f)|

using this norm is compared to that using a previously studied norm based on Chebyshev polynomials of the second kind and is shown to be superior in practical situations. Other results involving the latter norm are carried over to the new norm.
Miniaturized tables of Bessel functions. II
Authors: Luke Yudell L. 
Published: June 2010

In a previous study, we discussed the expansion of two-parameter functions in a double series of Chebyshev polynomials, and, in particular, we presented coefficients for the evaluation of the modified Bessel function

(
2
z

/

π

)

1

/

2

e
z

K
v

(
z
)

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

to 20 decimals for all

z
≧
5

z \geqq 5

and all

v
,
0
≦
v
≦
1

v,0 \leqq v \leqq 1

. In the present study, we give similar coefficients for the evaluation of

g

e

−
z

z

−
μ

I
v

(
z
)

g{e^{ - z}}{z^{ - \mu }}{I_v}(z)

to at least 20 decimals where

I
v

(
z
)

{I_v}(z)

is the modified Bessel function of the first kind and g and

μ
\mu

are certain constants which depend on the range of the parameter and variable for four different situations. The ranges are

(
1
)

0
>
z
≦
8
,
0
≦
v
≦
4
;
(
2
)

0
>
z
≦
8
,
4
≦
v
≦
8
;
(
3
)

z
≧
8
,
−
1
≦
v
≦
0
;
(
4
)

z
≧
8
,
0
≦
v
≦
1

(1)\;0 > z \leqq 8,0 \leqq v \leqq 4;(2)\;0 > z \leqq 8,4 \leqq v \leqq 8;(3)\;z \geqq 8, - 1 \leqq v \leqq 0;(4)\;z \geqq 8,0 \leqq v \leqq 1

.
Some properties of a class of band matrices
Authors: Hoskins W. D. , Ponzo P. J. 
Published: June 2010

Let

A
(
2
r
+
1
,
n
)

A(2r + 1,n)

denote the

n
×
n

n \times n

band matrix, of bandwidth

2
r
+
1

2r + 1

, with the binomial coefficients in the expansion of

(
x
−
1

)

2
r

{(x - 1)^{2r}}

as the elements in each row and column. Using the fact that the rows of

A
(
2
r
+
1
,
n
)

A(2r + 1,n)

provide the coefficients for the

2
r

2r

th central difference, a number of properties of

A
(
2
r
+
1
,
n
)

A(2r + 1,n)

are obtained for all positive integers

r
r

and

n
n

. These include obtaining explicit formulas for

det
A
(
2
r
+
1
,
n
)
,

A

−
1

(
2
r
+
1
,
n
)
,

|

|

A

−
1

(
2
r
+
1
,
n
)

|

|

∞

\det A(2r + 1,n),{A^{ - 1}}(2r + 1,n),||{A^{ - 1}}(2r + 1,n)|{|_\infty }

and an upper triangular matrix

U
U

such that

A
(
2
r
+
1
,
n
)
U

A(2r + 1,n)U

is lower triangular.
A unified presentation of certain classical polynomials
Authors: Srivastava H. M. , Singhal J. P. 
Published: June 2010

This paper attempts to present a unified treatment of the classical orthogonal polynomials, viz. Jacobi, Laguerre and Hermite polynomials, and their generalizations introduced from time to time. The results obtained here include a number of linear, bilinear and bilateral generating functions and operational formulas for the polynomials

{

T
n

(
α
,
β
)

(
x
,
a
,
b
,
c
,
d
,
p
,
r
)

|

n
=
0
,
1
,
2
,
⋯
}

\{ T_n^{(\alpha ,\beta )}(x,a,b,c,d,p,r)|n = 0,1,2, \cdots \}

, defined by Eq. (3) below.*
The Euler-Maclaurin expansion for the simplex
Authors: Lyness J. N. , Puri K. K. 
Published: June 2010

A natural extension of the one-dimensional trapezoidal rule to the simplex

0
≦

x
i

≦
1
,
∑

x
i

≦
1

0 \leqq {x_i} \leqq 1, \sum {x_i} \leqq 1

, is a rule Rf which uses as abscissas all those points on a hyper-rectangular lattice of spacing

h
=
1

/

m

h = 1/m

which lie within the simplex, assigning an equal weight to each interior point. In this paper, rules of this type are defined and some of their properties are derived. In particular, it is shown that the error functional satisfies an Euler-Maclaurin expansion of the type
\[

R
f
−
I
f
∼

A
1

h
+

A
2

h
2

+
⋯
+

A
p

h
p

+
O
(

h

p
+
1

)

Rf - If \sim {A_1}h + {A_2}{h^2} + \cdots + {A_p}{h^p} + O({h^{p + 1}})

\]
so long as

f
(

x

)

f({\text {x}})

and its partial derivatives of order up to p are continuous. Conditions under which this asymptotic series terminates are given, together with the condition for odd terms to drop out leaving an even expansion. The application to Romberg integration is discussed.
Factorization tables for 𝑥ⁿ-1 over 𝐺𝐹(𝑞)
Authors: Beard Jacob T. B. , West Karen I. 
Published: June 2010

These tables give the complete factorization of

x
n

−
1

{x^n} - 1

over

GF

(
q
)
,
q
=

p
a

,
2
⩽
n
⩽
d

{\text {GF}}(q),q = {p^a},2 \leqslant n \leqslant d

as below, together with the Euler

Φ
\Phi

-function of

x
n

−
1

{x^n} - 1

whenever

Φ
(

x
n

−
1
)
>

10
8

\Phi ({x^n} - 1) > {10^8}

.
\[

q
=
2
;
d
=
32

a
m
p
;

q
=
3
;
d
=
27

a
m
p
;

q
=
11
;
d
=
15

q
=

2
2

;
d
=
16

a
m
p
;

q
=

3
2

;
d
=
15

a
m
p
;

q
=
13
;
d
=
15

q
=

2
3

;
d
=
16

a
m
p
;

q
=
5
;
d
=
25
,
n
≠

23

†

a
m
p
;

q
=
17
;
d
=
15

q
=

2
4

;
d
=
16

a
m
p
;

q
=

5
2

;
d
=
10

a
m
p
;

q
=
19
;
d
=
12

q
=

2
5

;
d
=
12

a
m
p
;

q
=
7
;
d
=
15

a
m
p
;

q
=
23
;
d
=
10

\begin {array}{*{20}{c}} {q = 2;d = 32} \hfill & {q = 3;d = 27} \hfill & {q = 11;d = 15} \hfill \\ {q = {2^2};d = 16} \hfill & {q = {3^2};d = 15} \hfill & {q = 13;d = 15} \hfill \\ {q = {2^3};d = 16} \hfill & {q = 5;d = 25,n \ne {{23}^\dagger }} \hfill & {q = 17;d = 15} \hfill \\ {q = {2^4};d = 16} \hfill & {q = {5^2};d = 10} \hfill & {q = 19;d = 12} \hfill \\ {q = {2^5};d = 12} \hfill & {q = 7;d = 15} \hfill & {q = 23;d = 10} \hfill \\ \end {array}

\]
On Laplace transforms near the origin
Authors: Wong R. 
Published: June 2010

Let

f
(
t
)

f(t)

be locally integrable on

[
0
,
∞
)

[0,\infty )

and let

L
{
f
}
(
s
)

L\{ f\} (s)

denote the Laplace transform of

f
(
t
)

f(t)

. In this note, we prove that if

f
(
t
)
∼

t

−
β

Σ

n
=
0

∞

a
n

(
log
⁡
t

)

−
n

f(t) \sim {t^{ - \beta }}\Sigma _{n = 0}^\infty {a_n}{(\log t)^{ - n}}

as

t
→
∞

t \to \infty

, where

0
⩽
Re
⁡
β
>
1

0 \leqslant \operatorname {Re} \beta > 1

, then

L
{
f
}
(
s
)
∼

s

β
−
1

Σ

n
=
0

∞

c
n

(
log
⁡
1

/

s

)

−
n

L\{ f\} (s) \sim {s^{\beta - 1}}\Sigma _{n = 0}^\infty {c_n}{(\log 1/s)^{ - n}}

as

s
→
0

s \to 0

in

|

arg
⁡
s

|

⩽
π

/

2
−
Δ

|\arg s| \leqslant \pi /2 - \Delta

, the

c
n

{c_n}

being constants.
Uniform convergence of Galerkin’s method for splines on highly nonuniform meshes
Authors: Natterer Frank 
Published: June 2010

Different sets of conditions for an estimate of the form
\[

‖

y
−

y
π

‖

L
∞

(
a
,
b
)

⩽
C

max
i

h
i

r
+
1

‖

y

(
r
+
1
)

‖

L
∞

(

I
i

)

{\left \| {y - {y^\pi }} \right \|_{{L_\infty }(a,b)}} \leqslant C\max \limits _i h_i^{r + 1}{\left \| {{y^{(r + 1)}}} \right \|_{{L_\infty }({I_i})}}

\]
to hold are given. Here,

y
π

{y^\pi }

is the Galerkin approximation to the solution y of a boundary value problem for an ordinary differential equation, the trial functions being polynomials of degree

⩽
r

\leqslant r

on the subintervals

I
i

=
[

x
i

,

x

i
+
1

]

{I_i} = [{x_i},{x_{i + 1}}]

of length

h
i

{h_i}

. The sequence of subdivisions

π
:

x
0

>

x
1

>
⋯
>

x
n

\pi :{x_0} > {x_1} > \cdots > {x_n}

need not be quasi-uniform.
To compute the optimal interpolation formula
Authors: Gaffney P. W. 
Published: June 2010

The purpose of this paper is to explain how to compute the function

Ω
\Omega

which interpolates values of a function of one variable

f
(
x
)

f(x)

at n distinct points

x
1

>

x
2

>
⋯
>

x

n
−
1

>

x
n

{x_1} > {x_2} > \cdots > {x_{n - 1}} > {x_n}

and which, whenever

‖

f

(
k
)

‖

∞

\left \|{f^{(k)}}\right \|_\infty

is bounded and the value of the bound is unknown, provides the smallest possible value of

B
(
x
)

B(x)

in the error bound
\[

|

f
(
x
)
−
Ω
(
x
)

|

⩽
B
(
x
)

‖

f

(
k
)

‖

∞
.

|f(x) - \Omega (x)| \leqslant B(x)\left \|{f^{(k)}}\right \|_{\infty .}

\]
A dispersion analysis for difference schemes: tables of generalized Airy functions
Authors: Chin R. C. Y. , Hedstrom G. W. 
Published: June 2010

This paper contains graphs and tables of the function
\[

A

i

p
,
q

(
α
,
x
)
=

∫

−
∞

∞

(
2
π
)

−
1

exp
⁡
{
i

y
p

/

p
−
α

y
q

/

q
+
i
x
y
}

d
y

A{i_{p,q}}(\alpha ,x) = \int _{ - \infty }^\infty {{{(2\pi )}^{ - 1}}\exp \{ i{y^p}/p - \alpha {y^q}/q + ixy\} \;dy}

\]
and its indefinite integral for

p
=
3
,
5
,
7

p = 3,5,7

, for

q
=
2
,
4
,
6

q = 2,4,6

, and for several values of

α
\alpha

with

α
⩾
0

\alpha \geqslant 0

. It is shown how these tables should influence the choice of an artificial viscosity for a difference scheme for a linear hyperbolic equation.
An asymptotic approximation for a type of Fourier integral
Authors: Schmidt Paul W. 
Published: June 2010

A uniform asymptotic approximation which can be used for all

q
h
⩾
0

qh \geqslant 0

is developed for the Fourier integral
\[

I
(
h
)
=

∫
q
z

f
(

y
2

−

q
2

)

(

y
2

−

q
2

)

1

/

2

sin
⁡
y
h

d
y

I(h) = \int _q^z {\frac {{f(\sqrt {{y^2} - {q^2})} }}{{{{({y^2} - {q^2})}^{1/2}}}}\sin yh\,dy}

\]
under the assumptions that

h
z
>>
1

hz > > 1

, that the first

L
+
2

L + 2

derivatives of

f
(
y
)

f(y)

are continuous for

0
⩽
y
⩽

(

z
2

−

q
2

)

1

/

2

0 \leqslant y \leqslant {({z^2} - {q^2})^{1/2}}

, and that the first

2
L
+
2

2L + 2

derivatives of

f
(
y
)

f(y)

are continuous at

y
=
0

y = 0

. The approximation is especially convenient when

z
>>
q

z > > q

.