bluuebunny/crossref_metadata_2025_split huggingface dataset:
The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. III. Functions having algebraic singularities
Authors: Lyness J. N. 
Published: June 2010

The purpose of this paper is to extend the MIPS theory described in Parts I and II to functions having algebraic singularities. As in the simpler cases, the theory is based on expressing the remainder term in the appropriate Fourier coefficient asymptotic expansion as an infinite series, each element of which is a remainder in the Euler-Maclaurin summation formula. In this way, an expression is found for

∫
a
b

f
(
x
)
cos
⁡
2
π
m
x

d
x

(
0
≦
a
>
b
≦
1
)

\smallint _a^bf(x)\cos 2\pi mx\;dx\;(0 \leqq a > b \leqq 1)

where

f
(
x
)
=

(
x
−
a

)
α

(
b
−
x

)
β

ϕ
(
x
)
,
ϕ
(
x
)

f(x) = {(x - a)^\alpha }{(b - x)^\beta }\phi (x),\phi (x)

being analytic and

α
,
β
>
−
1

\alpha ,\beta > - 1

. This expression, and variants of it form a convenient basis for the numerical calculation of a set of Fourier coefficients. The calculation requires approximate values of the first few derivatives of

ϕ
(
x
)

at

x
=
a

\phi (x)\;{\text {at}}\;x = a

and at

x
=
b

x = b

, together with trapezoidal rule sums over [0, 1] of

f
(
x
)

f(x)

. Some of the incidental constants are values of the generalized zeta function

ζ
(
s
,
a
)

\zeta (s,a)

.
Computation of a multivariate 𝐹 distribution
Authors: Amos D. E. , Bulgren W. G. 
Published: June 2010

Methods for evaluating the joint cumulative probability integral associated with random variables

F
k

=
(

X
k

/

r
k

)

/

(
Y

/

s
)
,
k
=
1
,
2
,
⋯
,
n

{F_k} = ({X_k}/{r_k})/(Y/s),k = 1,2, \cdots ,n

, are considered where the

X
k

{X_k}

and

Y
Y

are independently

χ
2

(

r
k

)

{\chi ^2}({r_k})

and

χ
2

(
s
)

{\chi ^2}(s)

, respectively. For

n
=
2

n = 2

, series representations in terms of incomplete beta distributions are given, while a quadrature with efficient procedures for the integrand is presented for

n
≧
2

n \geqq 2

. The results for

n
=
2

n = 2

are applied to the evaluation of the correlated bivariate

F
F

distribution.
The numerical computation of two transcendental functions related to the exponential integral
Authors: Chipman D. M. 
Published: June 2010

Algorithms for the computation of numerical values of the two transcendental functions
\[

∫
0
x

1
t

[
Ei
⁡
(
t
)
−
γ
−
ln
⁡

|
t
|

]
d
t

and

∫
0
x

1
t

[
Ei
⁡
(
t
)
−
γ
−
ln
⁡

|
t
|

]
exp
⁡
(
−
t
)
d
t
,

\int _0^x {\tfrac {1}{t}} [\operatorname {Ei} (t) - \gamma - \ln \left | t \right |]dt\quad {\text {and}}\quad \int _0^x {\tfrac {1}{t}} [\operatorname {Ei} (t) - \gamma - \ln \left | t \right |]\exp ( - t)dt,

\]
where

γ
\gamma

is Euler’s constant and

Ei
⁡
(
t
)

\operatorname {Ei} (t)

is the exponential integral, are presented for all ranges of the real variable

x
x

. A table of values of these functions is also given.
The number of distinct subsums of ∑₁^{𝑁}1/𝑖
Authors: Bleicher M. N. , Erdős P. 
Published: June 2010

In this paper we improve the lower bounds for the number,

S
(
N
)

S(N)

, of distinct values obtained as subsums of the first N terms of the harmonic series. We obtain a bound of the form
\[

S
(
N
)
⩾
e

(

N
log
⁡
2

log
⁡
N

∏
3

k
+
1

log

j

N

)

S(N) \geqslant e\left ( {\frac {{N\log 2}}{{\log N}}\prod \limits _3^{k + 1} {{{\log }_j}N} } \right )

\]
whenever

log

k
+
1

N
⩾
k
+
1

{\log _{k + 1}}N \geqslant k + 1

, for

k
⩾
3

k \geqslant 3

. Slight modifications are needed for

k
=
1
,
2

k = 1,2

. We begin by discussing the number

Q
k

(
N
)

{Q_k}(N)

of integers

n
⩽
N
,
n
=

p
1

p
2

⋯

p
k

n \leqslant N,n = {p_1}{p_2} \cdots {p_k}

, where

p
i

>

e

α

p

i
−
1

,
i
=
2
,
⋯
,
k

{p_i} > {e^{\alpha {p_{i - 1}}}},i = 2, \cdots ,k

. We prove that
\[

N

log
⁡
N

∏

i
=
1

k
+
1

log

i

N
⩽

Q
k

(
N
)
⩽

(

1
+

k

log

k
+
1

N

)

N

log
⁡
N

∏

i
=
3

k
+
1

log

i

N

.

\frac {N}{{\log N}}\prod \limits _{i = 1}^{k + 1} {{{\log }_i}N \leqslant {Q_k}(N) \leqslant \left ( {1 + \frac {k}{{{{\log }_{k + 1}}N}}} \right )} \frac {N}{{\log N}}\prod \limits _{i = 3}^{k + 1} {{{\log }_i}N} .

\]
This bound is valid for

log

k
+
1

N
⩾
k
+
1

{\log _{k + 1}}N \geqslant k + 1

and for

1
⩽
α
⩽
2
(
1
−

e
2

(
4
)

/

e
3

(
4
)
)

1 \leqslant \alpha \leqslant 2(1 - {e_2}(4)/{e_3}(4))

. The symbols

log
i

x

{\log _i}x

and

e
i

(
x
)

{e_i}(x)

are defined by
\[

e
0

(
x
)
=
x
,

a
m
p
;

e

i
+
1

(
x
)
=

e

e
i

(
x
)

,

log

0

x
=
x
,

a
m
p
;

log

i
+
1

x
=
log
⁡
(

log

i

x
)
,

\begin {array}{*{20}{c}} {{e_0}(x) = x,} \hfill & {{e_{i + 1}}(x) = {e^{{e_i}(x)}},} \hfill \\ {{{\log }_0}x = x,} \hfill & {{{\log }_{i + 1}}x = \log ({{\log }_i}x),} \hfill \\ \end {array}

\]
where

log
⁡
x

\log x

denotes the logarithm to the base e.
A generating function for triangular partitions
Authors: Carlitz L. , Scoville Richard 
Published: June 2010

LetTk(n){T_k}(n)denote the number of solutions in nonnegative integersai{a_i}, of the equation\[n=∑i=1k∑j=1k−i+1aijn = \sum \limits _{i = 1}^k {\sum \limits _{j = 1}^{k - i + 1} {{a_{ij}}} }\]where theaij{a_{ij}}satisfy the inequalitiesaij⩾ai+1,j,aij⩾ai+1,j−1{a_{ij}} \geqslant {a_{i + 1,j}},{a_{ij}} \geqslant {a_{i + 1,j - 1}}. We show that\[∑n=1∞Tk(n)xn=(1−x)−k(1−x3)−k+1(1−x5)−k+2⋯(1−x2k−1)−1.\sum \limits _{n = 1}^\infty {{T_k}(n){x^n} = {{(1 - x)}^{ - k}}{{(1 - {x^3})}^{ - k + 1}}{{(1 - {x^5})}^{ - k + 2}} \cdots {{(1 - {x^{2k - 1}})}^{ - 1}}.}\]
A nearest point algorithm for convex polyhedral cones and applications to positive linear approximation
Authors: Wilhelmsen Don R. 
Published: June 2010

Suppose K is a convex polyhedral cone in

E
n

{E_n}

and is defined in terms of some generating set

{

e
1

,

e
2

,
…
,

e
N

}

\{ {e_1},{e_2}, \ldots ,{e_N}\}

. A procedure is devised so that, given any point

q
∈

E
n

q \in {E_n}

, the nearest point p in K to q can be found as a positive linear sum of

N
∗

⩽
n

{N^\ast } \leqslant n

points from the generating set. The procedure requires at most finitely many linear steps. The algorithm is then applied to find a positive representation
\[

L
f
=

∑

i
=
1

N
∗

λ
i

f
(

x
i

)
,

λ
i

>
0
,
f
∈
Φ
,

Lf = \sum \limits _{i = 1}^{{N^\ast }} {{\lambda _i}f({x_i}),} \quad {\lambda _i} > 0,f \in \Phi ,

\]
for a positive linear functional L acting on a suitable finite-dimensional function space

Φ
\Phi

.
Numbers generated by the reciprocal of 𝑒^{𝑥}-𝑥-1
Authors: Howard F. T. 
Published: June 2010

In this paper we examine the polynomialsAn(z){A_n}(z)and the rational numbersAn=An(0){A_n} = {A_n}(0)defined by means of\[exzx2(ex−x−1)−1=2∑n=0∞An(z)xn/n!.{e^{xz}}{x^2}{({e^x} - x - 1)^{ - 1}} = 2\sum \limits _{n = 0}^\infty {{A_n}(z){x^n}/n!} .\]We prove that the numbersAn{A_n}are related to the Stirling numbers and associated Stirling numbers of the second kind, and we show that this relationship appears to be a logical extension of a similar relationship involving Bernoulli and Stirling numbers. Other similarities betweenAn{A_n}and the Bernoulli numbers are pointed out. We also reexamine and extend previous results concerningAn{A_n}andAn(z){A_n}(z). In particular, it has been conjectured thatAn{A_n}has the same sign as−cos⁡nθ- \cos n\theta, wherereiθr{e^{i\theta }}is the zero ofex−x−1{e^x} - x - 1with smallest absolute value. We verify this for1⩽n⩽143291 \leqslant n \leqslant 14329and show that if the conjecture is not true forAn{A_n}, then|cos⁡nθ|>10−(n−1)/5|\cos n\theta | > {10^{ - (n - 1)/5}}. We also show thatAn(z){A_n}(z)has no integer roots, and in the interval[0,1][0,1],An(z){A_n}(z)has either two or three real roots.
Modifying singular values: existence of solutions to sytems of nonlinear equations having a possibly singular Jacobian matrix
Authors: Gay David 
Published: June 2010

We show that if a certain nondegeneracy assumption holds, it is possible to guarantee the existence of a solution to a system of nonlinear equations

f
(
x
)
=
0

f(x) = 0

whose Jacobian matrix

J
(
x
)

J(x)

exists but may be singular. The main idea is to modify small singular values of

J
(
x
)

J(x)

in such a way that the modified Jacobian matrix

J
^

(
x
)

\hat J(x)

has a continuous pseudoinverse

J
^

+

(
x
)

{\hat J^ + }(x)

and that a solution

x
∗

{x^\ast }

of

f
(
x
)
=
0

f(x) = 0

may be found by determining an asymptote of the solution to the initial value problem

x
(
0
)
=

x
0

,
x
′
(
t
)
=
−

J
^

+

(
x
)
f
(
x
)

x(0) = {x_0},x\prime (t) = - {\hat J^ + }(x)f(x)

. We briefly discuss practical (algorithmic) implications of this result. Although the nondegeneracy assumption may fail for many systems of interest (indeed, if the assumption holds and

J
(

x
∗

)

J({x^\ast })

is nonsingular, then

x
∗

{x^\ast }

is unique), algorithms using

J
^

+

(
x
)

{\hat J^ + }(x)

may enjoy a larger region of convergence than those that require (an approximation to)

J

−
1

(
x
)

{J^{ - 1}}(x)

.
A method of virtual displacements for the degenerate discrete 𝑙₁ approximation problem
Authors: Fraser W. , Bennett J. M. 
Published: June 2010

Given the system of equations
\[

∑

j
=
1

n

a

i
j

x
j

=

b
i

,

i
=
1
,
…
,
m
,

\sum \limits _{j = 1}^n {{a_{ij}}{x_j} = {b_i},\quad i = 1, \ldots ,m,}

\]
let

A
i

=
(

a

i
1

,
…
,

a

i
n

)

{A_i} = ({a_{i1}}, \ldots ,{a_{in}})

. It is known that if the matrix

A
=
(

a

i
j

)

A = ({a_{ij}})

has rank

k
⩽
n

k \leqslant n

, then there is a point X which provides a minimum of
\[

R
(
X
)
=

∑

i
=
1

m

|

r
i

(
X
)

|

=

∑

i
=
1

m

|

(

A
i

,
X
)
−

b
i

|

R(X) = \sum \limits _{i = 1}^m {|{r_i}(X)| = } \sum \limits _{i = 1}^m {|({A_i},X) - {b_i}|}

\]
such that

r
i

(
X
)
=
0

{r_i}(X) = 0

for at least k values of the index i. If

r
i

(
X
)
=
0

{r_i}(X) = 0

for exactly k values of the index i, the point or vertex is called ordinary, while if

r
i

(
X
)
=
0

{r_i}(X) = 0

for more than k values of i, the vertex is termed degenerate. A necessary and sufficient condition to determine if X minimizes R is valid if X is an ordinary vertex but not if X is degenerate. A degeneracy at X can be removed by applying perturbations to an appropriate number of the

b
i

{b_i}

so that X becomes an ordinary vertex of a modified problem. By noting that the test uses only values of the

A
i

{A_i}

, it is possible to avoid actual introduction of the perturbations to the

b
i

{b_i}

with a resulting substantial improvement of the efficiency of the computation.
On an integral summable to 2𝜉(𝑠)/(𝑠(𝑠-1))
Authors: Walker P. L. 
Published: June 2010

Let

ψ
(
x
)
=

Σ

n
=
1

∞

e

−

n
2

π
x

\psi (x) = \Sigma _{n = 1}^\infty {e^{ - {n^2}\pi x}}

, and

χ
(
u
)
=

e

u

/

2

(
1
+
2
ψ
(

e

2
u

)
)

\chi (u) = {e^{u/2}}(1 + 2\psi ({e^{2u}}))

. The divergent integral

2

∫
0
∞

χ
(
u
)
cos

2\smallint _0^\infty \chi (u)\cos

tu du is shown to be summable for certain complex values of t to the function

2
ξ
(
s
)

/

s
(
s
−
1
)
=

π

−
s

/

2

Γ
(
1

/

2
s
)
ζ
(
s
)

2\xi (s)/s(s - 1) = {\pi ^{ - s/2}}\Gamma (1/2s)\zeta (s)

, where

s
=
1

/

2
+
i
t

s = 1/2 + it

, and

ζ
(
s
)

\zeta (s)

is the zeta-function of Riemann. The values of a resulting approximation to

2
ξ
(
s
)

/

s
(
s
−
1
)

2\xi (s)/s(s - 1)

are computed and its zeros located.
Minimax approximate solutions of linear boundary value problems
Authors: Schmidt Darrell , Wiggins Kenneth L. 
Published: June 2010

Define the operator

D
:
C
[
0
,
τ
]
→
C
[
0
,
τ
]

D:C[0,\tau ] \to C[0,\tau ]

by

D
[
u
]
=
u
−

a
0

u
′
−

a
1

u

D[u] = u - {a_0}u\prime - {a_1}u

where

a
0

,

a
1

∈
C
[
0
,
τ
]

{a_0},{a_1} \in C[0,\tau ]

and consider the two point boundary value problem

(

BVP

)

D
[
y
]
(
x
)
=

a
2

(
x
)

({\text {BVP}})\;D[y](x) = {a_2}(x)

,

x
∈
[
0
,
τ
]

x \in [0,\tau ]

,

N
0

[
y
]
=

α
0

y
(
0
)
+

α
1

y
′
(
0
)
=

α
2

{N_0}[y] = {\alpha _0}y(0) + {\alpha _1}y\prime (0) = {\alpha _2}

,

N
τ

[
y
]
=

β
0

y
(
τ
)
+

β
1

y
′
(
τ
)
=

β
2

{N_\tau }[y] = {\beta _0}y(\tau ) + {\beta _1}y\prime (\tau ) = {\beta _2}

where

a
2

∈
C
[
0
,
τ
]

{a_2} \in C[0,\tau ]

,

α
0
2

+

α
1
2

≠
0

\alpha _0^2 + \alpha _1^2 \ne 0

and

β
0
2

+

β
1
2

≠
0

\beta _0^2 + \beta _1^2 \ne 0

. Let

Π
k

{\Pi _k}

denote the set of polynomials of degree at most k and define the approximating set

P

k

=
{
p
∈

Π
k

:

N
0

[
p
]
=

α
2

,

N
τ

[
p
]
=

β
2

}

{\mathcal {P}_k} = \{ p \in {\Pi _k}:{N_0}[p] = {\alpha _2},{N_\tau }[p] = {\beta _2}\}

. Then for each

k
⩾
3

k \geqslant 3

there exists

p
k

∈

P

k

{p_k} \in {\mathcal {P}_k}

satisfying

‖

D
[

p
k

]
−

a
2

‖

=

inf

p
∈

P

k

‖

D
[
p
]
−

a
2

‖

=

δ
k

\left \| {D[{p_k}] - {a_2}} \right \| = {\inf _{p \in {\mathcal {P}_k}}}\left \| {D[p] - {a_2}} \right \| = {\delta _k}

, where

‖
⋅
‖

\left \| \cdot \right \|

denotes the uniform norm on

C
[
0
,
τ
]

C[0,\tau ]

. If the homogeneous BVP

D
[
y
]
=
0

D[y] = 0

,

N
0

[
y
]
=

N
τ

[
y
]
=
0

{N_0}[y] = {N_\tau }[y] = 0

has no nontrivial solutions, then the nonhomogeneous BVP has a unique solution y and

lim

k
→
∞

‖

p
k

(
i
)

−

y

(
i
)

‖

=
0

{\lim _{k \to \infty }}\left \| {p_k^{(i)} - {y^{(i)}}} \right \| = 0

for

i
=
0
,
1
,
2

i = 0,1,2

. If X denotes a closed subset of

[
0
,
τ
]

[0,\tau ]

and
\[

δ

k
,
X

=

inf

p
∈

P

k

max

x
∈
X

|

D
[
p
]
(
x
)
−

a
2

(
x
)

|

,

{\delta _{k,X}} = \inf \limits _{p \in {\mathcal {P}_k}} \max \limits _{x \in X} |D[p](x) - {a_2}(x)|,

\]
then for each

ε
>
0

\varepsilon > 0

there exists

δ
>
0

\delta > 0

such that

d
(
x
)
⩽
δ

d(x) \leqslant \delta

implies that

0
⩽

δ
k

−

δ

k
,
X

⩽
ε

0 \leqslant {\delta _k} - {\delta _{k,X}} \leqslant \varepsilon

, where

d
(
X
)

d(X)

denotes the density of X in

[
0
,
τ
]

[0,\tau ]

. Several numerical examples are given.
An asymptotic expansion for the upper percentage points of the 𝜒²-distribution
Authors: Fettis Henry E. 
Published: June 2010

An asymptotic development is given for estimating the value of the variable

χ
\chi

for which the

χ
2

{\chi ^2}

-distribution
\[

Q
(

χ
2

,
v
)
=

1

Γ
(
v

/

2
)

∫

χ
2

/

2

∞

t

v

/

2
−
1

e

−
t

d
t

Q({\chi ^2},v) = \frac {1}{{\Gamma (v/2)}}\int _{{\chi ^2}/2}^\infty {t^{v/2 - 1}}{e^{ - t}}dt

\]
assumes a preassigned value

α
\alpha

, in the region where the quantity

η
=
−
ln
⁡
[
Γ
(
v

/

2
)
α
]

\eta = - \ln [\Gamma (v/2)\alpha ]

satisfies
\[

η
>>
ln
⁡
η
.

\eta > > \ln \eta .

\]
This development generalizes a similar one given by Blair and coauthors [2] for the case

v
=
1

v = 1

. It is also shown how the estimates thus obtained may be used in conjunction with various iterative schemes to give more accurate values.
On the roles of “stability” and “convergence” in semidiscrete projection methods for initial-value problems
Authors: Parter Seymour V. 
Published: June 2010

Consider the initial value problem (1.1)
\[

d

d
t

u
(
t
)
=
A
u
(
t
)
+
f
(
t
)
,

t
>
0
,

\frac {d}{{dt}}u(t) = Au(t) + f(t),\quad t > 0,

\]
(1.2)
\[

u
(
0
)
=

u
0

,

u(0) = {u_0},

\]
where A is a linear operator taking

D
(
A
)
⊂
X

D(A) \subset X

into X, where X is a Banach space. Consider also semidiscrete numerical methods of the form: find

U
N

(
t
)
:
[
0
,
T
]
→

X
N

{U_N}(t):[0,T] \to {X_N}

such that

(
1.1
′
)

(1.1\prime )

\[

d

U
N

d
t

=

A
N

U
N

+

P
N

f
,

\frac {{d{U_N}}}{{dt}} = {A_N}{U_N} + {P_N}f,

\]

(
1.2
′
)

(1.2\prime )

\[

U
N

(
0
)
=

U
N
0

∈

X
N

,

{U_N}(0) = U_N^0 \in {X_N},

\]
where

X
N

{X_N}

is a finite dimensional subspace and

P
N

{P_N}

is a projector onto

X
N

{X_N}

. The study of such numerical methods may be related to the approximation of semigroups and Laplace transform methods making use of the resolvent operators

(
A
−
λ
I

)

−
1

,

(

A
N

−
λ

I
N

)

−
1

{(A - \lambda I)^{ - 1}},{({A_N} - \lambda {I_N})^{ - 1}}

. The basic results require stability or weak stability and give convergence rates of the same order as in the steady state problems.
Majorizing sequences and error bounds for iterative methods
Authors: Miel George J. 
Published: June 2010

Given a sequence

{

x
n

}

n
=
0

∞

\{ {x_n}\} _{n = 0}^\infty

in a Banach space, it is well known that if there is a sequence

{

t
n

}

n
=
0

∞

\{ {t_n}\} _{n = 0}^\infty

such that

‖

x

n
+
1

−

x
n

‖

⩽

t

n
+
1

−

t
n

\left \| {{x_{n + 1}} - {x_n}} \right \| \leqslant {t_{n + 1}} - {t_n}

and

lim

t
n

=

t
∗

>
∞

\lim {t_n} = {t^\ast } > \infty

, then

{

x
n

}

n
=
0

∞

\{ {x_n}\} _{n = 0}^\infty

converges to some

x
∗

{x^\ast }

and the error bounds

‖

x
∗

−

x
n

‖

⩽

t
∗

−

t
n

\left \| {{x^\ast } - {x_n}} \right \| \leqslant {t^\ast } - {t_n}

hold. It is shown that certain stronger hypotheses imply sharper error bounds,
\[

‖

x
∗

−

x
n

‖

⩽

t
∗

−

t
n

(

t
n

−

t

n
−
1

)

μ

‖

x
n

−

x

n
−
1

‖

μ

⩽

t
∗

−

t
n

(

t
1

−

t
0

)

μ

‖

x
1

−

x
0

‖

μ

,

μ
⩾
0.

\left \| {{x^\ast } - {x_n}} \right \| \leqslant \frac {{{t^\ast } - {t_n}}}{{{{({t_n} - {t_{n - 1}})}^\mu }}}{\left \| {{x_n} - {x_{n - 1}}} \right \|^\mu } \leqslant \frac {{{t^\ast } - {t_n}}}{{{{({t_1} - {t_0})}^\mu }}}{\left \| {{x_1} - {x_0}} \right \|^\mu },\quad \mu \geqslant 0.

\]
Representative applications to infinite series and to iterates of types

x
n

=
G

x

n
−
1

{x_n} = G{x_{n - 1}}

and

x
n

=
H
(

x
n

,

x

n
−
1

)

{x_n} = H({x_n},{x_{n - 1}})

are given for

μ
=
1

\mu = 1

. Error estimates with

0
⩽
μ
⩽
2

0 \leqslant \mu \leqslant 2

are shown to be valid and optimal for Newton iterates under the hypotheses of the Kantorovich theorem. The unified convergence theory of Rheinboldt is used to derive error bounds with

0
⩽
μ
⩽
1

0 \leqslant \mu \leqslant 1

for a class of Newton-type methods, and these bounds are shown to be optimal for a subclass of methods. Practical limitations of the error bounds are described.
The convergence and partial convergence of alternating series
Authors: Philip J. R. 
Published: June 2010

The alternating series is

Σ

n
=
1

∞

j
n

f
(
n
)
=
[
j
]
f

\Sigma _{n = 1}^\infty {j_n}f(n) = [j]f

, with f a single-signed monotonic function of the real variable x. The

j
n

{j_n}

are

±
1

\pm 1

, their sign fixed by repetition of the ’template’ [j] of finite length 2p. [j] constitutes a difference scheme of ’differential order’ D, which can be determined. The principal theorem is that

[
j
]
f

[j]f

is ’partially convergent’ if and only if

lim

x
→
∞

f

(
D
−
1
)

(
x
)

{\lim _{x \to \infty }}{f^{(D - 1)}}(x)

is bounded. A series is partially convergent when the limit as

M
→
∞

M \to \infty

of the sum of 2pM terms exists. For [j] ’pure’, the improved Euler-Maclaurin expansion (IEM) gives the compact representation

(A)

S

(
p
)

∼
−

μ
(
D
)

2
p

∑

r
=
0

∞

(
2
p

)

2
r

B

2
r

(
1

/

2
)

(
2
r
!
)

f

(
2
r
+
D
−
1
)

(

θ
r

)
,

1
−
p
⩽

θ
r

⩽
p
.

\begin{equation}\tag {A} S^{(p)} \sim - \frac {\mu (D)}{2p} \sum _{r = 0}^\infty (2p)^{2r} \frac {B_{2r}(1/2)}{(2r!)} f^{(2r+D-1)} (\theta _r),\quad 1 - p \leqslant \theta _r \leqslant p. \end{equation}

S

(
p
)

S^{(p)}

is the sum,

μ
(
D
)

\mu (D)

is the Dth ’template moment’, and the

B

2
r

{B_{2r}}

are Bernoulli numbers. Efficient means for practical summation of these series follow also from IEM. In illustration, 10 alternating series with D ranging from 1 to 3 are summed using IEM. It is found that the leading term of (A) with

θ
0

=
1

/

2

\theta _0 = 1/2

gives a simple but effective estimate of sums. The paper also gives a comparison with Euler’s transformation in the case

p
=
1

p = 1

and discusses sums to N terms with

N

/

2
p

N/2p

nonintegral and finite but large.
On determination of best-possible constants in integral inequalities involving derivatives
Authors: Neta Beny 
Published: June 2010

This paper is concerned with the numerical approximation of the best possible constants

γ

n
,
k

{\gamma _{n,k}}

in the inequality
\[

‖

F

(
k
)

‖

2

⩽

γ

n
,
k

−
1

{

‖
F
‖

2

+

‖

F

(
n
)

‖

2

}

,

{\left \| {{F^{(k)}}} \right \|^2} \leqslant \gamma _{n,k}^{ - 1}\;\left \{ {{{\left \| F \right \|}^2} + {{\left \| {{F^{(n)}}} \right \|}^2}} \right \},

\]
where
\[

‖
F
‖

2

=

∫
0
∞

|

F
(
x
)

|

2

d
x
.

{\left \| F \right \|^2} = \int _0^\infty |F(x){|^2}\;dx.

\]
A list of all constants

γ

n
,
k

{\gamma _{n,k}}

for

n
⩽
10

n \leqslant 10

is given.
Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations
Authors: Layton William J. 
Published: June 2010

We consider the approximate solution of the initial value problem
\[

∂
u

∂
t

=

∂
u

∂
x

,

u
(
x
,
0
)
=
v
(
x
)
,

\frac {{\partial u}}{{\partial t}} = \frac {{\partial u}}{{\partial x}},\quad u(x,0) = v(x),

\]
by a dissipative Galerkin method. When v is taken to have a jump discontinuity at zero, that discontinuity will propagate along

x
+
t
=
0

x + t = 0

, in the true solution u. Estimates in

L
2

{L_2}

and

L
∞

{L_\infty }

of the pollution effects of the discontinuity are found. These estimates show those effects to decay exponentially in

h

−
1

{h^{ - 1}}

in regions a fixed distance d from the discontinuity and exponentially in d for fixed h.
A general method of approximation. I
Authors: Wrigge Staffan , Fransén Arne 
Published: June 2010

In this paper we study two families of functions, viz. F and H, and show how to approximate the functions considered in the interval [0,1 ]. The functions are assumed to be real when the argument is real. We define
\[

F
=
{
f
;
(

i

)

f

(

1
2

+
x

)

=
f

(

1
2

−
x

)

,
(

ii

)

f
(
0
)
=
f
(
1
)
=
0
,
(

iii

)

f
(
x
)

is analytic in a sufficiently large neighborhood of

x
=
0
}
,

F = \{ f;({\text {i}})\,f\left ( {\frac {1}{2} + x} \right ) = f\left ( {\frac {1}{2} - x} \right ),({\text {ii}})\,f(0) = f(1) = 0,({\text {iii}})\;f(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\},

\]

\[

H
=
{
h
;
(

j

)

h

(

1
2

+
x

)

=
−
h

(

1
2

−
x

)

,
(

jj

)

h
(
0
)
=
h
(
1
)
=
0
,
(

jjj

)

h
(
x
)

is analytic in a sufficiently large neighborhood of

x
=
0
}
.

H = \{ h;({\text {j}})\;h\left ( {\frac {1}{2} + x} \right ) = - h\left ( {\frac {1}{2} - x} \right ),({\text {jj}})\;h(0) = h(1) = 0,({\text {jjj}})\;h(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\}.

\]
The approximations are defined in the interval [0,1 ] by
\[

min

∫
0
1

(

f
(
x
)
−

∑

n
=
1

k

c

n
,
k

[
x
(
1
−
x
)
]

n

)

2

x
q

(
1
−
x
)

q

d
x

\min \int _0^1 {{{\left ( {f(x) - \sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx}

\]
and
\[

min

∫
0
1

(

h
(
x
)
−
(
1
−
2
x
)

∑

n
=
1

k

c

n
,
k

[
x
(
1
−
x
)
]

n

)

2

x
q

(
1
−
x
)

q

d
x

,

\min \int _0^1 {{{\left ( {h(x) - (1 - 2x)\sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx} ,

\]
where

q
∈
{
0
,
1
,
2
,
…
}

q \in \{ 0,1,2, \ldots \}

. The associated matrices are analyzed using the theory of orthogonal polynomials, especially the Jacobi polynomials

G
n

(
p
,
q
,
x
)

{G_n}(p,q,x)

. We apply the general theory to the basic trigonometric functions

sin
⁡
(
x
)

\sin (x)

and

cos
⁡
(
x
)

\cos (x)

.
An algorithm for the second immanant
Authors: Grone Robert , Merris Russell 
Published: June 2010

Let

χ
\chi

be an irreducible character of the symmetric group

S
n

{S_n}

. For

A
=
(

a

i
j

)

A = ({a_{ij}})

an n-by-n matrix, define the immanant of A corresponding to

χ
\chi

by
\[

d
(
A
)
=

∑

σ
∈

S
n

χ
(
σ
)

∏

t
=
1

n

a

t
σ
(
t
)

.

d(A) = \sum \limits _{\sigma \in {S_n}} {\chi (\sigma )\prod \limits _{t = 1}^n {{a_{t\sigma (t)}}.} }

\]
The article contains an algorithm for computing

d
(
A
)

d(A)

when

χ
\chi

corresponds to the partition (2,

1

n
−
2

{1^{n - 2}}

).