bluuebunny/crossref_metadata_2025_split huggingface dataset:
Generalizations of a classical theorem in number theory
Authors: Hudson Richard H. 
Published: June 2010

A classical theorem conjectured by Jacobi asserts that for an odd prime p, the sum of the quadratic residues in the interval (0, p) is less than the sum of the quadratic nonresidues if and only if

p
≡
3

(
mod

4
)

p \equiv 3 \pmod 4

. We generalize Jacobi’s problem to kth powers

(
mod

p
)
,
k
>
2

\pmod p,k > 2

, and we consider in some detail a generalization of Jacobi’s conjecture to quadratic residues and nonresidues

(
mod

n
)

\pmod n

, n an arbitrary integer

>
2

> 2

. From the set of least positive residues

(
mod

n
)

\pmod n

, let

c
0

{c_0}

denote the subgroup of quadratic residues

(
mod

n
)

\pmod n

and let

c
1

,

c
2

,
…
,

c
t

{c_1},{c_2}, \ldots ,{c_t}

be the cosets which can be formed with respect to this subgroup. Computer data supports the following generalized Jacobi conjecture: The sum of the elements in

c
0

{c_0}

is less than or equal to the sum in any of the other cosets for every integer

n
>
2

n > 2

, a surprising conjecture, especially in view of the fact that counterexamples are easily obtained for

k
=
4
,
6
,
8
,
10

k = 4,6,8,10

, etc. (The coset sums are identical for odd k and prime modulus.) We resolve the generalized Jacobi conjecture in the affirmative when, for example, n is an integer admitting a primitive root, or

n
=

2
α

,
α
⩾
3

n = {2^\alpha },\alpha \geqslant 3

. (Here we give explicit formulae for the four coset sums.) For

n
=
2

p
α

n = 2{p^\alpha }

, our proof that the quadratic residues and the quadratic nonresidues

(
mod

n
)

\pmod n

have the same sum for odd prime p if and only if

p

≢

3

(
mod

8
)

p \nequiv 3 \pmod 8

is purely, elementary. On the other hand, we need Dirichlet’s class number formula for quadratic number fields with discriminant

−
p
≡
5

(
mod

8
)

- p \equiv 5 \pmod 8

to show that the sum of the quadratic nonresidues strictly exceeds the sum of the quadratic residues

(
mod

2

p
α

)

\pmod {2p^\alpha }

if

p
≡
3

(
mod

8
)

p \equiv 3 \pmod 8

. Computer data gives rise to a host of interesting problems we are unable to resolve. For example, if

n
=
2

p
1

α
1

p
2

α
2

⋯

p
r

α
r

,
1
⩽
i
⩽
r

n = 2p_1^{{\alpha _1}}p_2^{{\alpha _2}} \cdots p_r^{{\alpha _r}},1 \leqslant i \leqslant r

, we conjecture that a sufficient condition that the coset sums not be identical is that we have

p
i

≡
3

(
mod

8
)

{p_i} \equiv 3 \pmod 8

for every i. It is not hard to show that the coset sums are identical if every

p
i

≡
1

(
mod

4
)

{p_i} \equiv 1 \pmod 4

. However, the problem of finding a necessary condition is very difficult since, e.g., the coset sums are not identical for

n
⩽
1146

n \leqslant 1146

when

n
=
2
⋅
3
⋅
p

n = 2 \cdot 3 \cdot p

if

p
≡
23

(
mod

24
)

p \equiv 23 \pmod {24}

, but the sums are identical if

p
≡
7

(
mod

24
)

p \equiv 7 \pmod {24}

.
A bound on the 𝐿_{∞}-norm of 𝐿₂-approximation by splines in terms of a global mesh ratio
Authors: de Boor Carl 
Published: June 2010

Let

L
k

f

{L_k}f

denote the least-squares approximation to

f
∈

L

1

f \in {{\mathbf {L}}_1}

by splines of order k with knot sequence

t

=
(

t
i

)
1

n
+
k

{\mathbf {t}} = ({t_i})_1^{n + k}

. In connection with their work on Galerkin’s method for solving differential equations, Douglas, Dupont and Wahlbin have shown that the norm

‖

L
k

‖

∞

{\left \| {{L_k}} \right \|_\infty }

, of

L
k

{L_k}

as a map on

L

∞

{{\mathbf {L}}_\infty }

can be bounded as follows,
\[

‖

L
k

‖

∞

⩽

const
k

M

t

,

{\left \| {{L_k}} \right \|_\infty } \leqslant {\operatorname {const}_k}{M_{\mathbf {t}}},

\]
with

M

t

{M_{\mathbf {t}}}

a global mesh ratio, given by
\[

M

t

:=

max
i

Δ

t
i

/

min

{
Δ

t
i

|

Δ

t
i

>
0
}
.

{M_{\mathbf {t}}}: = \max \limits _i \;\Delta {t_i}/\min \,\{ \Delta {t_i}|\Delta {t_i} > 0\}.

\]
Using their very nice idea together with some facts about B-splines, it is shown here that even
\[

‖

L
k

‖
∞

⩽

const
k

⁡
(

M

t

(
k
)

)

1

/

2

\| L_k \|_\infty \leqslant \operatorname {const}_k(M_{\mathbf {t}}^{(k)})^{1/2}

\]
with the smaller global mesh ratio

M

t

(
k
)

M_{\mathbf {t}}^{(k)}

given by
\[

M

t

(
k
)

:=

max

i
,
j

(

t

i
+
k

−

t
i

)

/

t

j
+
k

−

t
j

)
.

M_{\mathbf {t}}^{(k)}: = \max \limits _{i,j} ({t_{i + k}} - {t_i})/{t_{j + k}} - {t_j}).

\]
A mesh independent bound for

L

2

{{\mathbf {L}}_2}

-approximation by continuous piecewise polynomials is also given.
A necessary condition for the stability of a difference approximation to a hyperbolic system of partial differential equations
Authors: Burns Anne M. 
Published: June 2010

We are interested in the boundary conditions for a difference approximation to a hyperbolic system of partial differential equations

u
t

=
A

u
x

{u_t} = A{u_x}

,

u
(
x
,
0
)
=
F
(
x
)

u(x,0) = F(x)

,

K
u
(
0
,
t
)
=
0

Ku(0,t) = 0

in the quarter plane

x
⩾
0

x \geqslant 0

,

t
⩾
0

t \geqslant 0

. We consider approximations of the type:
\[

u
j

(
t
+
Δ
t
)
=

∑

k
=
−
r

p

C
k

u

j
+
k

(
t
)
,

j
=
1
,
2
,
…
,

{u_j}(t + \Delta t) = \sum \limits _{k = - r}^p {{C_k}{u_{j + k}}(t),\quad j = 1,2, \ldots ,}

\]

\[

u
j

+

∑

k
=
1

s

α

j
k

u
k

(
t
+
Δ
t
)
=
0
,

j
=
−
r
+
1
,
…
,
0.

{u_j} + \sum \limits _{k = 1}^s {{\alpha _{jk}}{u_k}(t + \Delta t) = 0,\quad j = - r + 1, \ldots ,0.}

\]
If N is the null space of K and E is the "negative" eigenspace of A, then the system of partial differential equations is well-posed if and only if

K
∩
E
=

{
0
}

K \cap E = \left \{ 0 \right \}

and Rank

K
=

K =

the number of negative eigenvalues of A. In direct analogy to this, we prove that for a difference scheme of the above type with

r
=
p
=
1

r = p = 1

,

K
′
=
I
+

Σ

k
=
1

s

α
k

K\prime = I + \Sigma _{k = 1}^s\;{\alpha _k}

and

N
′
=

N\prime =

null space of

K
′

K\prime

, a necessary condition for stability is

N
′
∩
E
=

{
0
}

N\prime \cap E = \left \{ 0 \right \}

. If, in addition, a condition proven by S. J. Osher to be sufficient for stability is not satisfied, then Rank

K
=

K =

the number of negative eigenvalues of A is also necessary for stability. We then generalize this result to the case

r
>
1

r > 1

,

p
>
1

p > 1

. Together these conditions imply that "extrapolation" on "negative" eigenvectors leads to instability; "extrapolation" on "positive" eigenvectors is "almost necessary. "Extrapolation" on "positive" eigenvectors and not on "negative" eigenvectors is sufficient for stability.
Bernoulli related polynomials and numbers
Authors: Charalambides Ch. A. 
Published: June 2010

The polynomials

φ
n

(
x
;
a
,
b
)

{\varphi _n}(x;a,b)

of degree n defined by the equations
\[

Δ
a

φ
n

(
x
;
a
,
b
)
=

(
x
)

n
−
1
,
b

b

n
−
1

⋅
(
n
−
1
)
!

and

Δ
b

φ
n

(
x
;
a
,
b
)
=

φ

n
−
1

(
x
;
a
,
b
)

{\Delta _a}{\varphi _n}(x;a,b) = \frac {{{{(x)}_{n - 1,b}}}}{{{b^{n - 1}} \cdot (n - 1)!}}\quad {\text {and}}\quad {\Delta _b}{\varphi _n}(x;a,b) = {\varphi _{n - 1}}(x;a,b)

\]
where

(
x

)

n
,
b

=
x
(
x
−
b
)
(
x
−
2
b
)
⋯
(
x
−
n
b
+
b
)

{(x)_{n,b}} = x(x - b)(x - 2b) \cdots (x - nb + b)

is the generalized factorial and

Δ
a

f
(
x
)
=
f
(
x
+
a
)
−
f
(
x
)

{\Delta _a}f(x) = f(x + a) - f(x)

, are the subject of this paper. A representation of these polynomials as a sum of generalized factorials is given. The coefficients,

B
(
n
,
s
)

B(n,s)

,

s
=
a

/

b

s = a/b

, of this representation are given explicitly or by a recurrence relation. The generating functions of

φ
n

(
x
;
a
,
b
)

{\varphi _n}(x;a,b)

and

B
(
n
,
s
)

B(n,s)

are obtained. The limits of

φ
n

(
x
;
a
,
b
)

{\varphi _n}(x;a,b)

as

a
→
1

a \to 1

,

b
→
0

b \to 0

or

a
→
0

a \to 0

,

b
→
1

b \to 1

and the limits of

B
(
n
,
s
)

B(n,s)

as

s
→
±
∞

s \to \pm \infty

or

s
→
0

s \to 0

are shown to be the Bernoulli polynomials and numbers of the first and second kind, respectively. Finally, the generalized factorial moments of a discrete rectangular distribution are obtained in terms of

B
(
n
,
s
)

B(n,s)

in a form similar to that giving its usual moments in terms of the Bernoulli numbers.
The shooting method applied to a cyclic inequality
Authors: Troesch B. A. 
Published: June 2010

It is known that the cyclic sum
\[

S
n

(

x

)
=

∑

i
=
1

i
=
n

x
i

/

(

x

i
+
1

+

x

i
+
2

)

{S_n}({\mathbf {x}}) = \sum \limits _{i = 1}^{i = n} {{x_i}/({x_{i + 1}} + {x_{i + 2}})}

\]
where

x

n
+
1

=

x
1

{x_{n + 1}} = {x_1}

,

x

n
+
2

=

x
2

{x_{n + 2}} = {x_2}

,

x
i

⩾
0

{x_i} \geqslant 0

,

(

x

i
+
1

+

x

i
+
2

)
>
0

({x_{i + 1}} + {x_{i + 2}}) > 0

, can be made smaller than

n

/

2

n/2

for

n
⩾
24

n \geqslant 24

. The value of

λ
(
n
)
=
lim
inf

(
n
→
∞
)

S
n

/

n

\lambda (n) = \lim \inf \;(n \to \infty )\;{S_n}/n

is investigated by the shooting method for two-point boundary value problems, and the analytical result

lim
(
n
→
∞
)

λ
(
n
)
⩽
0.49457

\lim (n \to \infty )\;\lambda (n) \leqslant 0.49457

is proved. The inherent difficulty in a straight-forward minimization of

S
n

(

x

)

{S_n}({\mathbf {x}})

is mentioned.
Exponential laws for fractional differences
Authors: Isaacs Godfrey L. 
Published: June 2010

In Math. Comp., v. 28, 1974, pp. 185-202, Diaz and Osler gave the following (formal) definition for

Δ
˙

α

f
(
z
)

{\dot \Delta ^\alpha }f(z)

, the

α
\alpha

th fractional difference of

f
(
z
)
:

Δ
˙

α

f
(
z
)
=

Σ

p
=
0

∞

A
p

−
α
−
1

f
(
z
+
α
−
p
)

f(z):{\dot \Delta ^\alpha }f(z) = \Sigma _{p = 0}^\infty A_p^{ - \alpha - 1}f(z + \alpha - p)

. They derived formulas and applications involving this difference. They asked whether their differences satisfied an exponent law and what the relation was between their differences and others, such as

Δ
α

f
(
z
)
=

Σ

p
=
0

∞

A
p

−
α
−
1

f
(
z
+
p
)

{\Delta ^\alpha }f(z) = \Sigma _{p = 0}^\infty A_p^{ - \alpha - 1}f(z + p)

. In this paper an exponent law for their differences is established and a relation found between the two differences mentioned above. Applications of these results are given.
A special class of Bell polynomials
Authors: Howard F. T. 
Published: June 2010

We examine the integersV(n,k)V(n,k)defined by means of\[k!∑n=0∞V(n,k)xn/n!=[x(ex+1)−2(ex−1)]k,k!\sum \limits _{n = 0}^\infty {V(n,k){x^n}/n! = {{[x({e^x} + 1) - 2({e^x} - 1)]}^k},}\]and, in particular, we show how these integers are related to the Bernoulli, Genocchi and van der Pol numbers, and the numbers generated by the reciprocal ofex−x−1{e^x} - x - 1. We prove that theV(n,k)V(n,k)are also related to the numbersW(n,k)W(n,k)defined by\[k!∑n=0∞W(n,k)xn/n!=[(x−2)(ex−1)]kk!\sum \limits _{n = 0}^\infty {W(n,k){x^n}/n! = {{[(x - 2)({e^x} - 1)]}^k}}\]in much the same way the associated Stirling numbers are related to the Stirling numbers. Finally, we examine, more generally, the Bell polynomialsBn,k(a1,a2,3−α,4−α,5−α,…){B_{n,k}}({a_1},{a_2},3 - \alpha ,4 - \alpha ,5 - \alpha , \ldots )and show how the methods of this paper can be used to prove several formulas involving the Bernoulli and Stirling numbers.
On some trigonometric integrals
Authors: Fettis Henry E. 
Published: June 2010

Expressions are obtained for the integrals
\[

I
λ

(
p
)

=

∫
0

π

/

2

(

sin
⁡
λ
θ

sin
⁡
θ

)

p

d
θ
,

J
λ

(
p
)

=

∫
0

π

/

2

(

1
−
cos
⁡
λ
θ

sin
⁡
θ

)

p

d
θ

I_\lambda ^{(p)} = \int _0^{\pi /2}{\left ( {\frac {{\sin \lambda \theta }}{{\sin \theta }}} \right )^p}d\theta ,\quad J_\lambda ^{(p)} = \int _0^{\pi /2}{\left ( {\frac {{1 - \cos \lambda \theta }}{{\sin \theta }}} \right )^p}d\theta

\]
for arbitrary real values of "

λ
\lambda

", and

p
=
1
,
2

p = 1,2

.
Numerical stability of the Halley-iteration for the solution of a system of nonlinear equations
Authors: Cuyt Annie A. M. 
Published: June 2010

Let

F
:

R

q

→

R

q

F:{{\mathbf {R}}^q} \to {{\mathbf {R}}^q}

and

x
∗

{x^ \ast }

a simple root in

R

q

{{\mathbf {R}}^q}

of the system of nonlinear equations

F
(
x
)
=
0

F(x) = 0

. Abstract Padé approximants (APA) and abstract Rational approximants (ARA) for the operator F have been introduced in [2] and [3]. The adjective “abstract” refers to the use of abstract polynomials [5] for the construction of the rational operators. The APA and ARA have been used for the solution of a system of nonlinear equations in [4]. Of particular interest was the following third order iterative procedure:
\[

x

i
+
1

=

x
i

+

a
i
2

a
i

+

1
2

F
i

′

−
1

F
i

a
i
2

,

{x_{i + 1}} = {x_i} + \frac {{a_i^2}}{{{a_i} + \frac {1}{2}F_i^{’- 1}F_i^{}a_i^2}},

\]
with

F
i
′

F_i’

the 1st Fréchet-derivative of F in

x
1

,

a
i

=
−

F
i

′

−
1

F
i

{x_1},{a_i} = - F_i^{’- 1}{F_i}

the Newton-correction where

F
i

=
F
(

x
i

)
,

F
i

{F_i} = F({x_i}),F_i^{}

the 2nd Fréchet-derivative of F in

x
i

{x_i}

where

F
i

a
i
2

F_i^{}a_i^2

is the bilinear operator

F
i

F_i^{}

evaluated in

(

a
i

,

a
i

)

({a_i},{a_i})

, and componentwise multiplication and division in

R

q

{{\mathbf {R}}^q}

. For

q
=
1

q = 1

this technique is known as the Halley-iteration [6, p. 91]. In this paper the numerical stability [7] of the Halley-iteration for the case

q
⩾
1

q \geqslant 1

is investigated and illustrated by a numerical example.
An algorithm for the solution of certain differential-difference equations of advanced type
Authors: Greaves G. 
Published: June 2010

The differential-difference problem
\[

d

d
s

{

s
J
(
s
)

}
+
J
(
s
+
1
)
=
0

(
Re
⁡
(
s
)
>
0
)
;

s
J
(
s
)
→
1

as

s
→
0
,

\frac {d}{{ds}}\;\{ {sJ(s)} \} + J(s + 1) = 0\;(\operatorname {Re} (s) > 0);\quad sJ(s) \to 1\;{\text {as}}\;s \to 0,

\]
which arises in the theory of numbers, is studied and an algorithm developed for the simultaneous numerical approximation of the derivatives

J

(
n
)

(
s
)

{J^{(n)}}(s)

. An error bound is established for the process described and its numerical performance and stability are discussed.
A Chebyshev polynomial rate-of-convergence theorem for Stieltjes functions
Authors: Boyd John P. 
Published: June 2010

The theorem proved here extends the author’s previous work on Chebyshev series [4] by showing that if

f
(
x
)

f(x)

is a member of the class of so-called "Stieltjes functions" whose asymptotic power series

Σ

a
n

x
n

\Sigma {a_n}{x^n}

about

x
=
0

x = 0

is such that
\[

lim

n
→
∞

¯

log
⁡

|

a
n

|

n
log
⁡
n

=
r
,

\overline {\lim \limits _{n \to \infty } } \frac {{\log |{a_n}|}}{{n\log n}} = r,

\]
then the coefficients of the series of shifted Chebyshev polynomials on

x
∈
[
0
,
a
]
,
Σ

b
n

T
n
∗

(
x

/

a
)

x \in [0,a],\Sigma {b_n}T_n^\ast (x/a)

, satisfy the inequality
\[

2

r
+
2

⩾

lim

n
→
∞

¯

log
⁡

|

(
log
⁡

|

b
n

|

)

|

log
⁡
n

⩾
1
−

r
2

.

\frac {2}{{r + 2}} \geqslant \overline {\lim \limits _{n \to \infty } } \frac {{\log |(\log |{b_n}|)|}}{{\log n}} \geqslant 1 - \frac {r}{2}.

\]
There is an intriguing relationship between this theorem and a similar rate-of-convergence theorem for Padé approximants of Stieltjes functions which is discussed below.
Three summation criteria for Fermat’s last theorem
Authors: Schwindt H. 
Published: June 2010

This paper extends the search for solutions of the congruences
\[

∑
1

[
p

/

6
]

1
i

≡
0
,

∑
1

[
p

/

6
]

1

i
2

≡
0

and

∑

[
p

/

6
]
+
1

[
p

/

5
]

1
i

≡
0

(
mod

p
)

\sum \limits _1^{[p/6]} {\frac {1}{i} \equiv 0,} \quad \sum \limits _1^{[p/6]} {\frac {1}{{{i^2}}} \equiv 0} \quad {\text {and}}\quad \sum \limits _{[p/6] + 1}^{[p/5]} {\frac {1}{i} \equiv 0\;\pmod p}

\]
to the limit

p
>
600000

p > 600000

. The only solutions found were

p
=
61

p = 61

in the first case, in the second

p
=
205129

p = 205129

, and in the third case

p
=
109

p = 109

and

p
=
491

p = 491

.
The maximum of a quasismooth function
Authors: Wilkins J. Ernest , Hatcher Theodore R. 
Published: June 2010

Let Z be the class of real-valued functions, defined and continuous on the closed interval

I
=
[
−
1
,
1
]

I = [ - 1,1]

, such that

f
(
±
1
)
=
0

f( \pm 1) = 0

and

|

f
(
ξ
)
−
2
f
{
(
ξ
+
η
)

/

2
}
+
f
(
η
)

|

⩽

|

ξ
−
η

|

|f(\xi ) - 2f\{ (\xi + \eta )/2\} + f(\eta )| \leqslant |\xi - \eta |

for all

ξ
\xi

and

η
\eta

in I. Let

K
=

sup

f
∈
Z

max

x
∈
I

|

f
(
x
)

|

K = {\sup _{f \in Z}}{\max _{x \in I}}|f(x)|

. We will prove that

13

/

10
⩽
K
⩽
1014

/

779
>
1.301669

13/10 \leqslant K \leqslant 1014/779 > 1.301669

.
More trigonometric integrals
Authors: Fettis Henry E. 
Published: June 2010

Integrals of the form
\[

∫
0

π

/

2

e

i
p
θ

cos

q

θ

d
θ
,

∫
0

π

/

2

e

i
p
θ

sin

q

θ

d
θ

\int _0^{\pi /2} {{e^{ip\theta }}{{\cos }^q}\theta \,d\theta ,\quad \int _0^{\pi /2} {{e^{ip\theta }}{{\sin }^q}\theta \,d\theta } }

\]
(p real,

Re
⁡
(
q
)
>
−
1

\operatorname {Re} (q) > - 1

) are expressed in terms of Gamma and hypergeometric functions for integer and noninteger values of q and p. The results include those of [2] as special cases.
Products and sums of powers of binomial coefficients mod 𝑝 and solutions of certain quaternary Diophantine systems
Authors: Hudson Richard H. 
Published: June 2010

In this paper we prove that certain products and sums of powers of binomial coefficients modulo

p
=
q
f
+
1

p = qf + 1

,

q
=

a
2

+

b
2

q = {a^2} + {b^2}

, are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition
\[

16

p
α

=

x
2

+
2
q

u
2

+
2
q

v
2

+
q

w
2

,

(
x
,
u
,
v
,
w
,
p
)
=
1
,

x
w
=
a

v
2

−
2
b
u
v
−
a

u
2

,

x
≡
4

(
mod

q
)
,
α
⩾
1.

\begin {array}{*{20}{c}} {16{p^\alpha } = {x^2} + 2q{u^2} + 2q{v^2} + q{w^2},\quad (x,u,v,w,p) = 1,} \\ {xw = a{v^2} - 2buv - a{u^2},\quad x \equiv 4\pmod q,\alpha \geqslant 1.} \\ \end {array}

\]
The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field
\[

K
=
Q

(

i

2
q
+
2
a

q

)

,

K = Q\left ( {i\sqrt {2q + 2a\sqrt q } } \right ),

\]
as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field

Q
(

e

2
π
i
p

/

q

)

Q({e^{2\pi ip/q}})

. Let the four cosets of the subgroup A of quartic residues be given by

c
j

=

2
j

A
,
j
=
0
,
1
,
2
,
3

{c_j} = {2^j}A,j = 0,1,2,3

, and let
\[

s
j

=

1
q

∑

t
∈

c
j

t
,

j
=
0
,
1
,
2
,
3.

{s_j} = \frac {1}{q}\sum \limits _{t \in {c_j}} {t,\quad j = 0,1,2,3.}

\]
Let

s
m

{s_m}

and

s
n

{s_n}

denote the smallest and next smallest of the

s
j

{s_j}

respectively. We give new, and unexpectedly simple determinations of

Π

k
∈

c
n

k
f
!

{\Pi _{k \in {c_n}}}kf!

and

Π

k
∈

c

n
+
2

k
f
!

{\Pi _{k \in {c_{n + 2}}}}kf!

, in terms of the parameters x in the above partition of

16

p
α

16{p^\alpha }

, in the complicated case that arises when the class number of K is

>
1

> 1

and

s
m

≠

s
n

{s_m} \ne {s_n}

.
Asymptotic expansions of multiple integrals of rapidly oscillating functions
Authors: Iwaniec T. , Lutoborski A. 
Published: June 2010

Expansions of multiple integrals
\[

∫

a
1

b
1

⋯

∫

a
n

b
n

w
(

σ
1

x
1

,
…
,

σ
n

x
n

)
g
(

x
1

,
…
,

x
n

)

d

x
1

⋯
d

x
n

,

\int _{{a_1}}^{{b_1}} \cdots \int _{{a_n}}^{{b_n}} {w({\sigma _1}{x_1}, \ldots ,{\sigma _n}{x_n})g({x_1}, \ldots ,{x_n})\;d{x_1} \cdots d{x_n},}

\]
where w is a function on

R

n

{{\mathbf {R}}^n}

which is

(

b
k

−

a
k

)

({b_k} - {a_k})

-periodic in the kth variable,

k
=
1
,
…
,
n

k = 1, \ldots ,n

, and g is smooth, are given in terms of negative powers of the integers

σ
1

,
…
,

σ
n

{\sigma _1}, \ldots ,{\sigma _n}

. Estimates of the remainder term in the expansion are also given.
Elliptic pseudoprimes
Authors: Miyamoto I. , Murty M. Ram 
Published: June 2010

LetEbe an elliptic curve overQwith complex multiplication by an order in an imaginary quadratic field. Letψn{\psi _n}denote thenth division polynomial, and letPbe a rational point ofEof infinite order. A natural numbernis called anelliptic pseudoprimeifn|ψn+1(P)n|{\psi _{n + 1}}(P)andnis composite. LetN(x)N(x)denote the number of elliptic pseudoprimes up tox. We show thatN(x)≪x(log⁡log⁡x)7/2/(log⁡x)3/2N(x) \ll x{(\log \log x)^{7/2}}/{(\log x)^{3/2}}. More generally, ifP1,…,Pr{P_1}, \ldots ,{P_r}arerindependent rational points ofEwhich have infinite order, andΓ\Gammais the subgroup generated by them, denote byNΓ(x){N_\Gamma }(x)the number of compositen≤xn \leq xsatisfyingn|ψn+1(Pi)n|{\psi _{n + 1}}({P_i}),1≤i≤r1 \leq i \leq r. Forr≥2r \geq 2, we proveNΓ(x)≪xexp⁡(−c(log⁡x)(log⁡log⁡x)){N_\Gamma }(x) \ll x\exp ( - c\sqrt {(\log x)(\log \log x))}for some positive constantc.
Collocation methods for nonlinear Volterra integro-differential equations with infinite delay
Authors: Brunner Hermann 
Published: June 2010

In this paper we study the numerical solution of nonlinear Volterra integro-differential equations with infinite delay by spline collocation and related Runge-Kutta type methods. The kernel function in these equations is of the form

k
(
t
,
s
,
y
(
t
)
,
y
(
s
)
)

k(t,s,y(t),y(s))

, with a representative example given by Volterra’s population equation, where we have

k
(
t
,
s
,
y
(
t
)
,
y
(
s
)
)
=
a
(
t
−
s
)
⋅
G
(
y
(
t
)
,
y
(
s
)
)

k(t,s,y(t),y(s)) = a(t - s) \cdot G(y(t),y(s))

.
On a Diophantine equation related to perfect codes
Authors: Alter Ronald 
Published: June 2010

A necessary condition for the existence of perfect double Hamming-error-correcting codes on q symbols, for q a prime power, is that the Diophantine equation
\[

(

n

0

)

+

(

n

1

)

(
q
−
1
)
+

(

n

2

)

(
q
−
1

)
2

=

q
k

\left ( {\begin {array}{*{20}{c}} n \\ 0 \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} n \\ 1 \\ \end {array} } \right )(q - 1) + \left ( {\begin {array}{*{20}{c}} n \\ 2 \\ \end {array} } \right ){(q - 1)^2} = {q^k}

\]
have a nontrivial solution in positive integers. In this paper this equation is considered for all q, and by applying Newton’s method for approximating the roots of a polynomial, it is established that it has no nontrivial solutions for all n, odd k, and q of the form

q
=
2

s
2

q = 2{s^2}

.
Mining metabolic pathways through gene expression
Authors: Hancock Timothy , Takigawa Ichigaku , Mamitsuka Hiroshi 
Published: June 2010

AbstractMotivation: An observed metabolic response is the result of the coordinated activation and interaction between multiple genetic pathways. However, the complex structure of metabolism has meant that a compete understanding of which pathways are required to produce an observed metabolic response is not fully understood. In this article, we propose an approach that can identify the genetic pathways which dictate the response of metabolic network to specific experimental conditions.Results: Our approach is a combination of probabilistic models for pathway ranking, clustering and classification. First, we use a non-parametric pathway extraction method to identify the most highly correlated paths through the metabolic network. We then extract the defining structure within these top-ranked pathways using both Markov clustering and classification algorithms. Furthermore, we define detailed node and edge annotations, which enable us to track each pathway, not only with respect to its genetic dependencies, but also allow for an analysis of the interacting reactions, compounds and KEGG sub-networks. We show that our approach identifies biologically meaningful pathways within two microarray expression datasets using entire KEGG metabolic networks.Availability and implementation: An R package containing a full implementation of our proposed method is currently available from http://www.bic.kyoto-u.ac.jp/pathway/timhancockContact: timhancock@kuicr.kyoto-u.ac.jpSupplementary information: Supplementary data are available at Bioinformatics online.