bluuebunny/crossref_metadata_2025 huggingface dataset:
Regularity of weak solutions of parabolic variational inequalities
Authors: Ziemer William P.
Published: June 2010

In this paper, parabolic operators of the form
\[

u
t

−
div
⁡
A
(
x
,

t
,

u
,

D
u
)
−
B
(
x
,

t
,

u
,

D
u
)

{u_t} - \operatorname {div} A(x,\,t,\,u,\,Du) - B(x,\,t,\,u,\,Du)

\]
are considered where

A
A

and

B
B

are Borel measurable and subject to linear growth conditions. Let

ψ
:

Ω
→

R
1

\psi :\,\Omega \to {R^1}

be a Borel function bounded above (an obstacle) where

Ω
⊂

R

n
+
1

\Omega \subset {R^{n + 1}}

. Let

u
∈

W

1
,
2

(
Ω
)

u \in {W^{1,2}}(\Omega )

be a weak solution of the variational inequality in the following sense: assume that

u
⩾
ψ

u \geqslant \psi

q.e. and
\[

∫
Ω

u
t

φ
+
A
⋅
D
φ
−
B
φ
⩾
0

\int _\Omega {{u_t}\varphi + A \cdot D\varphi - B\varphi \geqslant 0}

\]
whenever

φ
∈

W
0

1
,
2

(
Ω
)

\varphi \in W_0^{1,2}(\Omega )

and

φ
⩾
u
−
ψ

\varphi \geqslant u - \psi

q.e. Here q.e. means everywhere except for a set of classical parabolic capacity. It is shown that

u
u

is continuous even though the obstacle may be discontinuous. A mild condition on

ψ
\psi

which can be expressed in terms of the fine topology is sufficient to ensure the continuity of

u
u

. A modulus of continuity is obtained for

u
u

in terms of the data given for

ψ
\psi

.
Bifurcation phenomena associated to the 𝑝-Laplace operator
Authors: Guedda Mohammed , Véron Laurent
Published: June 2010

We determine the structure of the set of the solutions

u
u

of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus left-parenthesis StartAbsoluteValue u Subscript x Baseline EndAbsoluteValue Superscript p minus 2 Baseline u Subscript x Baseline right-parenthesis Subscript x Baseline plus f left-parenthesis u right-parenthesis equals lamda StartAbsoluteValue u EndAbsoluteValue Superscript p minus 2 Baseline u">

−

(

|

u
x

|

p
−
2

u
x

)
x

+
f
(
u
)
=
λ

|

u

|

p
−
2

u

- {(|{u_x}{|^{p - 2}}{u_x})_x} + f(u) = \lambda |u{|^{p - 2}}u

on

(
0
,

1
)

(0,\,1)

such that

u
(
0
)
=
u
(
1
)
=
0

u(0) = u(1) = 0

, where

p
>
1

p > 1

and

λ
∈

R

\lambda \in {\mathbf {R}}

. We prove that the solutions with

k
k

zeros are unique when

1
>
p
⩽
2

1 > p \leqslant 2

but may not be so when

p
>
2

p > 2

.
The classifying topos of a continuous groupoid. I
Authors: Moerdijk Ieke
Published: June 2010

We investigate some properties of the functor

B
B

which associates to any continuous groupoid

G
G

its classifying topos

B
G

BG

of equivariant

G
G

-sheaves. In particular, it will be shown that the category of toposes can be obtained as a localization of a category of continuous groupoids.
Local properties of secant varieties in symmetric products. II
Authors: Johnsen Trygve
Published: June 2010

Let

V
V

be a linear system on a curve

C
C

. In Part I we described a method for studying the secant varieties

V
d
r

V_d^r

locally. The varieties

V
d
r

V_d^r

are contained in the

d
d

-fold symmetric product

C

(
d
)

{C^{(d)}}

. In this paper (Part II) we apply the methods from Part I. We give a formula for local tangent space dimensions of the varieties

V
d
1

V_d^1

valid in all characteristics (Theorem 2.4). Assume

rk

V
=
n
+
1

\operatorname {rk}\;V = n + 1

and

char
⁡
K
=
0

\operatorname {char} K = 0

. In

§
§
3

\S \S 3

and

4
4

we describe in detail the projectivized tangent cones of the varieties

V
n
1

V_n^1

for a large class of points. The description is a generalization of earlier work on trisecants for a space curve. In

§
5

\S 5

we study the curve in

C

(
2
)

{C^{(2)}}

consisting of divisors

D
D

such that

2
D
∈

V
4
1

2D \in V_4^1

. We give multiplicity formulas for all points on this curve in

C

(
2
)

{C^{(2)}}

in terms of local geometrical invariants of

C
C

. We assume

char
⁡
K
=
0

\operatorname {char} K = 0

.
Determinacy of sufficiently differentiable maps
Authors: Selby Alan M.
Published: June 2010

Variants of the algebraic conditions of Mather are shown to be sufficient for the

k
k

-determinacy of

C
u

{C^u}

maps with respect to

j
j

-flat, contact (or right)

C
r

{C^r}

equivalence relations where

u
−
k
≤
r
≤
u
−
k
+
j
+
1

u - k \leq r \leq u - k + j + 1

and

0
≤
j
>
k
≤
u

0 \leq j > k \leq u

. The required changes of coordinates and matrix-valued functions are constructed from the variation of coefficients in polynomials. The main result follows from a finite-dimensional, polynomial pertubation argument which employs a parameter-dependent polynomial representation of functions based on Taylor’s formula. For

r
>
k

r > k

, the algebraic conditions are seen to be necessary.
On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains
Authors: Bandle Catherine , Levine Howard A.
Published: June 2010

In this paper we study the first initial-boundary value problem for

u
t

=
Δ
u
+

u
p

{u_t} = \Delta u + {u^p}

in conical domains

D
=
(
0
,
∞
)
×
Ω
⊂

R
N

D = (0,\infty ) \times \Omega \subset {R^N}

where

Ω
⊂

S

N
−
1

\Omega \subset {S^{N - 1}}

is an open connected manifold with boundary. We obtain some extensions of some old results of Fujita, who considered the case

D
=

R
N

D = {R^N}

. Let

λ
=
−

γ
−

\lambda = - {\gamma _ - }

where

γ
−

{\gamma _ - }

is the negative root of

γ
(
γ
+
N
−
2
)
=

ω
1

\gamma (\gamma + N - 2) = {\omega _1}

and where

ω
1

{\omega _1}

is the smallest Dirichlet eigenvalue of the Laplace-Beltrami operator on

Ω
\Omega

. We prove: If

1
>
p
>
1
+
2

/

(
2
+
λ
)

1 > p > 1 + 2/(2 + \lambda )

, there are no nontrivial global solutions. If

1
>
p
>
1
+
2

/

λ

1 > p > 1 + 2/\lambda

, there are no stationary solutions in

D
−
{
0
}

D - \{ 0\}

except

u
≡
0

u \equiv 0

. If

1
+
2

/

λ
>
p
>
(
N
+
1
)

/

(
N
−
3
)

1 + 2/\lambda > p > (N + 1)/(N - 3)

(if

N
>
3

N > 3

, arbitrary otherwise) there are singular stationary solutions

u
s

{u_s}

. If

u
(
x
,
0
)
⩽

u
s

(
x
)

u(x,0) \leqslant {u_s}(x)

, the solutions are global. If

1
+
2

/

λ
>
p
>
(
N
+
2
)

/

(
N
−
2
)

1 + 2/\lambda > p > (N + 2)/(N - 2)

and

u
(
x
,
0
)
⩽

u
s

u(x,0) \leqslant {u_s}

, with

u
(
x
,
0
)
∈
C
(

D
¯

)

u(x,0) \in C(\overline D )

, the solutions decay to zero. If

1
+
2

/

N
>
p

1 + 2/N > p

, there are global solutions. For

1
>
p
>
∞

1 > p > \infty

, there are

L
∞

{L^\infty }

data of arbitrarily small norm, decaying exponentially fast at

r
=
∞

r = \infty

, for which the solution is not global. We show that if

D
D

is the exterior of a bounded region, there are no global, nontrivial, positive solutions if

1
>
p
>
1
+
2

/

N

1 > p > 1 + 2/N

and that there are such if

p
>
1
+
2

/

N

p > 1 + 2/N

. We obtain some related results for

u
t

=
Δ
u
+

|

x

|

σ

u
p

{u_t} = \Delta u + |x{|^\sigma }{u^p}

in the cone.
Nonlinearly equivalent representations of quaternionic 2-groups
Authors: Mio Washington
Published: June 2010

We construct new examples of nonlinearly equivalent finite-dimensional real linear representations of quaternionic

2
2

-groups, which cannot be obtained from equivalent representations of cyclic groups by induction and composition techniques.
Totally categorical structures
Authors: Hrushovski Ehud
Published: June 2010

A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every

ℵ
0

{\aleph _0}

-stable,

ℵ
0

{\aleph _0}

-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.
A unified theory for real vs. complex rational Chebyshev approximation on an interval
Authors: Ruttan Arden , Varga Richard S.
Published: June 2010

A unified approach is presented for determining all the constants

γ

m
,
n

(
m
≥
0
,
n
≥
0
)

{\gamma _{m,n}}\;(m \geq 0,n \geq 0)

which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that

γ

m
,
m
+
2

=
1

/

3

(
m
≥
0
)

{\gamma _{m,m + 2}} = 1/3\;(m \geq 0)

, a problem which had remained open.
Digital representations using the greatest integer function
Authors: Reznick Bruce
Published: June 2010

LetSd(α){S_d}(\alpha )denote the set of all integers which can be expressed in the form∑εi[αi]\sum {{\varepsilon _i}[{\alpha ^i}]}, withεi∈{0,…,d−1}{\varepsilon _i} \in \{ 0, \ldots ,d - 1\}, whered≥2d \geq 2is an integer andα≥1\alpha \geq 1is real, and letId{I_d}denote the set ofα\alphaso thatSd(α)=Z+{S_d}(\alpha ) = {{\mathbf {Z}}^ + }. We show thatId=[1,rd)∪{d}{I_d} = [1,{r_d}) \cup \{ d\}, wherer2=131/4,r3=221/3{r_2} = {13^{1/4}},{r_3} = {22^{1/3}}andr2=(d2−d−2)1/2{r_2} = {({d^2} - d - 2)^{1/2}}ford≥4d \geq 4. Ifα∉Id\alpha \notin {I_d}we show thatTd(α){T_d}(\alpha ), the complement ofSd(α){S_d}(\alpha ), is infinite, and discuss the density ofTd(α){T_d}(\alpha )whenα>d\alpha > d. Ford≥4d \geq 4and a particular quadratic irrationalβ=β(d)>d\beta = \beta (d) > d, we describeTd(β){T_d}(\beta )explicitly and show that|Td(β)∩[0,n]||{T_d}(\beta ) \cap [0,n]|is of orderne(d){n^{e(d)}}, wheree(d)>1e(d) > 1.
Regularity of the metric entropy for expanding maps
Authors: Rychlik Marek
Published: June 2010

The main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the Perron-Frobenius operator for expanding mappings. As a consequence, we are able to show that the metric entropy of an expanding map has modulus of continuity

x
log
⁡
(
1

/

x
)

x\log (1/x)

on the space of

C
2

{C^2}

-expandings. We also give an explicit estimate of the rate of mixing for

C
1

{C^1}

-functions in terms of natural constants. It seems that the method we present can be generalized to other classes of dynamical systems, which have a distinguished invariant measure, like

Axiom
⁡
A

\operatorname {Axiom} \text {A}

diffeomorphisms. It also can be adopted to show that the entropy of the quadratic family

f
μ

(
x
)
=
1
−
μ

x
2

{f_\mu }(x) = 1 - \mu {x^2}

computed with respect to the absolutely continuous invariant measure found in Jakobson’s Theorem varies continuously (the last result is going to appear somewhere else).
Similarity, quasisimilarity, and operator factorizations
Authors: Curto Raúl E. , Fialkow Lawrence A.
Published: June 2010

We introduce and illustrate an operator factorization technique to study similarity and quasisimilarity of Hilbert space operators. The technique allows one to generate, in a systematic way, families of "test" operators, and to check for similarity and quasisimilarity with a given model. In the case of the unilateral shift

U
+

{U_ + }

, we obtain a one-parameter family of nonhyponormal, noncontractive, shift-like operators in the similarity orbit of

U
+

{U_ + }

. We also obtain new characterizations of quasisimilarity and similarity in terms of invariant operator ranges, and conditions for spectral and essential spectral inclusions.
A new 3-dimensional shrinking criterion
Authors: Daverman Robert J. , Repovš Dušan
Published: June 2010

We introduce a new shrinking criterion for cell-like upper semicontinuous decompositions

G
G

of topological

3
3

-manifolds, such that the embedding dimension (in the sense of Štan’ko) of the nondegeneracy set of

G
G

is at most one. As an immediate application, we prove a recognition theorem for

3
3

-manifolds based on a new disjoint disks property.
A separable space with no remote points
Authors: Dow Alan
Published: June 2010

In the model obtained by adding

ω
2

{\omega _2}

side-by-side Sacks reals to a model of

C
H

{\mathbf {CH}}

, there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size

ω
2

{\omega _2}

such that every uncountable subset has density

ω
1

{\omega _1}

. Furthermore assuming the existence of a measurable cardinal

κ
\kappa

with

2
κ

=

κ
+

{2^\kappa } = {\kappa ^ + }

, a space

X
X

is produced with no isolated points but with remote points in

υ
X
−
X

\upsilon X - X

. It is also shown that a pseudocompact space does not have remote points.
Cosmicity of cometrizable spaces
Authors: Gruenhage Gary
Published: June 2010

A space

X
X

is cometrizable if

X
X

has a coarser metric topology such that each point of

X
X

has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space

X
X

: (a)

X
X

is the continuous image of a separable metric space; (b)

X
ω

{X^\omega }

is hereditarily separable and hereditarily Lindelöf, (c)

X
2

{X^2}

has no uncountable discrete subspaces; (d)

X
X

is a Lindelöf semimetric space; (e)

X
X

has the pointed

ccc

{\text {ccc}}

. This result is a corollary to our main result which states that, under PFA, if

X
X

is a cometrizable space with no uncountable discrete subspaces, then either

X
X

is the continuous image of a separable metric space or

X
X

contains a copy of an uncountable subspace of the Sorgenfrey line.
Stable processes with drift on the line
Authors: Port Sidney C.
Published: June 2010

The stable processes on the line having a drift are investigated. Except for the symmetric Cauchy processes with drift these are all transient and points are nonpolar sets. Explicit information about the potential kernel is obtained and this is used to obtain specific results about hitting times and places for various sets.
On the Cauchy problem for the one-dimensional heat equation
Authors: Ginsberg F.
Published: June 2010

In this paper we show that the Cauchy problem for the one-dimensional heat equation, though non-well posed in the sense of Hadamard, can be solved numerically. It is shown that if we admit as solutions functions for which an a priori bound is assumed in some finite rectangle in x — t space then the solution depends Hölder continuously upon the given Cauchy data. The specific numerical scheme developed also exhibits the Hölder continuity so that we are sure of a meaningful numerical method.
A new algorithm for diagonalizing a real symmetric matrix
Authors: LaBudde C. Donald
Published: June 2010

The algorithm described in this paper is essentially a Jacobi-like procedure employing Householder and Jacobi orthogonal similarity transformations successively on a real symmetric matrix to obtain, in the limit, a diagonal matrix of eigenvalues. The columns of the product matrix of all the orthogonal transformations, taken in the proper order, form a complete orthonormal set of eigenvectors.
A method for the computation of the error function of a complex variable
Authors: Strand Otto Neall
Published: June 2010

This paper presents a method of computing

z
≡

(

2

/

π

)

∫
0
z

e

−

u
2

d
u

z \equiv \left ( {2/\sqrt \pi } \right )\int _0^z {{e^{ - {u^2}}}du}

, where

z
z

is complex. It is shown that

z
≡
1
−

erf 

z

z \equiv 1 - {\text {erf }}z

has no zeros in the right-hand half plane. An estimate of

|

erfc 

z

|

|{\text {erfc }}z|

is derived.
An iterative method for computing the generalized inverse of an arbitrary matrix
Authors: Ben-Israel Adi
Published: June 2010

The iterative process,

X

n
+
1

=

X
n

(
2
I
−
A

X
n

)

{X_{n + 1}} = {X_n}(2I - A{X_n})

, for computing

A

−
1

{A^{ - 1}}

, is generalized to obtain the generalized inverse.
Integration rules of hypercubic symmetry over a certain spherically symmetric space
Authors: Lyness J. N.
Published: June 2010

A theory of integration rules suitable for integration over a hypercube and having hypercubic symmetry has recently been published. In this paper it is found that, with minor modification, this theory may be directly applied to obtain integration rules of hypercubic symmetry suitable for integration over a complete

n
n

-dimensional space with the weight function

exp
⁡
(
−

x
1

2

−

x
2

2

⋅
⋅
⋅
−

x
n

2

)

\exp ( - {x_1}^2 - {x_2}^2 \cdot \cdot \cdot - {x_n}^2)

. As in the case of integration over hypercubes, an

n
n

-dimensional rule of degree

2
t
+
1

2t + 1

may be constructed requiring a number of function evaluations of order

2
t

n
t

/

t
!

{2^t}{n^t}/t!

, only.
Minimax polynomial approximation
Authors: Denman Harry H.
Published: June 2010

Some new methods for obtaining the minimax polynomial approximation of degree

n
n

to a continuous function are introduced, and applied to several simple functions. The amount of computation required is substantially reduced compared with that of previous methods.
Runge-Kutta methods with constrained minimum error bounds
Authors: King Richard
Published: June 2010

Optimum Runge-Kutta methods of orders

m
=
2
,
3

m = 2,3

, and

4
4

are developed for the differential equation

y
′

=
f
(
x
,
y
)

y’ = f(x,y)

under Lotkin’s conditions on the bounds for

f
f

and its partial derivatives, and with the constraint that the coefficient of

∂
m

f

/

∂

x
m

{\partial ^m}f/\partial {x^m}

in the leading error term be zero. The methods then attain higher order when it happens that

f
f

is independent of

y
y

.
Expansions for Coulomb wave functions
Authors: Boersma J.
Published: June 2010

In this paper we derive a number of expansions for Whittaker functions, regular and irregular Coulomb wave functions. The main result consists of a new expansion for the irregular Coulomb wave functions of orders zero and one in terms of regular Coulomb wave functions. The latter expansions are especially useful with a view to the numerical computation of irregular Coulomb wave functions.
On finite integrals involving trigonometric, Bessel, and Legendre functions
Authors: Lewis Richard L.
Published: June 2010

A finite integral involving the product of powers of trigonometric functions, up to two associated Legendre functions, and zero or one Bessel function is evaluated. When certain combinations of the otherwise complex function parameters are integers, the resulting expression becomes greatly simplified. So restricting the parameters, this still quite general case may be transformed into four canonical forms, each of which admits rapid convergence of the only nonterminating series in the expressions. Finally, closed form expressions are obtained for a number of special cases.
Further asymptotic expansions for the error functional
Authors: Ninham B. W. , Lyness J. N.
Published: June 2010

The generalised Euler-Maclaurin summation formula for the error functional

E
f
=
R
f
−
I
f

Ef = Rf - If

is not suitable for numerical computation if

f
(
x
)

f(x)

contains singularities in the complex plane which lie close to the interval of integration. In this paper, we use the methods and results of an earlier paper to construct alternative asymptotic expansions suitable in such cases.
Some computer-assisted topological models of Hilbert fundamental domains
Authors: Cohn Harvey
Published: June 2010

The Hubert modular group

H

{\text {H}}

for the integral domain

O

(

k

1

/

2

)

{\text {O}}({k^{1/2}})

has a fourdimensional fundamental domain

R

{\text {R}}

which should be represented geometrically (like the classic modular group). Computer assistance (by the Argonne CDC 3600) was used for outlining cross sections of the three-dimensional "floor" of

R

{\text {R}}

, which is a mosaic of an intractably large number of boundary pieces identified under

H

{\text {H}}

. The cross sections shown here might well contain enough information when

k
=
2
,
3
,
5
,
6

k = 2,3,5,6

to form some "incidence matrices" and see

R

{\text {R}}

(at least) combinatorially. For special symmetrized subgroups of

H

{\text {H}}

, it is plausible to see homologously independent

2
2

-spheres in (the corresponding)

R

{\text {R}}

. The program is a continuation of one outlined in two earlier issues of this journal v. 19, 1965, pp. 594–605, MR 33 #4016, and v. 21, 1967, pp. 76–86, MR 36 #5081.
Computation of minimal length full Steiner trees on the vertices of a convex polygon
Authors: Cockayne E. J.
Published: June 2010

A Steiner minimal tree is a tree of minimal length whose vertices are a given set of points

a
1

,
⋯
,

a
n

{a_1}, \cdots ,{a_n}

in

E
2

{E^2}

and any set of additional points

s
1

,
⋯
,

s
k

(
k
≧
0
)

{s_1}, \cdots ,{s_k}(k \geqq 0)

. In general, the introduction of extra points makes possible shorter trees than the minimal length tree whose vertices are precisely

a
1

,
⋯
,

a
n

{a_1}, \cdots ,{a_n}

and for which practical algorithms are known. A Steiner minimal tree is the union of special subtrees, known as full Steiner trees. This paper demonstrates the use of the computer in generating minimal length full Steiner trees on sets of points in

E
2

{E_2}

which are the vertices of convex polygons. The procedure given is a basis from which further research might proceed towards an ultimate practical algorithm for the construction of Steiner minimal trees.
A steepest ascent method for the Chebyshev problem
Authors: Meicler Marcel
Published: June 2010

In this paper we present an efficient ascent method for calculating the minimax solution of an overdetermined system of linear equations

A
x
=
b

Ax = b

. The algorithm makes best use of all the information available at each cycle in order to force a very steep path to the solution.
A generalization of a class of test matrices
Authors: Herbold Robert J.
Published: June 2010

We consider matrices of the following form:

G
n

(

a
1

,

a
2

,
⋯
,

a

n
−
1

,

b
1

,

b
2

⋯

b
n

)
=

{G_n}({a_1},{a_2}, \cdots ,{a_{n - 1}},{b_1},{b_2} \cdots {b_n}) =

(

β

i
,
j

)
,
1
≦
i

({\beta _{i,j}}),1 \leqq i

,

j
≦
n

j \leqq n

, where

a
1

,
⋯
,

a

n
−
1

,

b
1

,
⋯
,

b
n

{a_1}, \cdots ,{a_{n - 1}},{b_1}, \cdots ,{b_n}

are constants and
\[

β
i

,
j

=

b
j

,

 

j
≧
i
;

 

β

i
j

=

a
j

,

 

j
>
i
.

{\beta _i}_{,j} = {b_j},{\text { }}j \geqq i;{\text { }}{\beta _{ij}} = {a_j},{\text { }}j > i.

\]
We deduce in analytic form the determinant, inverse matrix, characteristic equation, and eigenvectors of

G
n

{G_n}

. Knowing these properties enables us to generate valuable test matrices by appropriately selecting the order and elements of

G
n

{G_n}

.
A new method for computing toroidal harmonics
Authors: Fettis Henry E.
Published: June 2010

A method for computing Legendre functions of integer order and half-odd degree is presented. The method is based on the theory of quadratic transformation of the argument, and is a generalization of Gauss’ or Landen’s transformation for computing elliptic integrals.
On the stability of Friedrichs’ scheme and the modified Lax-Wendroff scheme
Authors: Vaillancourt Rémi
Published: June 2010

Necessary and sufficient stability criteria for Friedrichs’ scheme and the modified Lax-Wendroff scheme with smooth coefficients are derived by means of Kreiss’ Matrix Theorem and the first Stability Theorem of Lax and Nirenberg.
A note on solid partitions
Authors: Knuth Donald E.
Published: June 2010

The problem of enumerating partitions which satisfy a given partial order relation is reduced to the problem of enumerating permutations satisfying that relation. This theorem is applied to the enumeration of solid partitions; existing tables of solid partitions are extended.
Consecutive power residues or nonresidues
Authors: Rabung J. R. , Jordan J. H.
Published: June 2010

For any positive integers

k
k

and

l
l

, A. Brauer [1] has shown that there exists a number

z
(
k
,
l
)

z(k,l)

such that, for any prime number

p
>
z
(
k
,
l
)

p > z(k,l)

, a sequence of

l
l

consecutive numbers occurs in at least one

k
k

th-power class modulo

p
p

. For particular

k
k

and

l
l

, one is sometimes able to find a least bound,

Λ
∗
(
k
,
l
)

\Lambda *(k,l)

, before, or at which, the first member of such a sequence must appear. In this paper, we describe a method used to compute

Λ
∗
(
8
,
2
)

\Lambda *(8,2)

and

Λ
∗
(
3
,
3
)

\Lambda *(3,3)

.
Chebyshev iteration methods for integral equations of the second kind.
Authors: Sag T. W.
Published: June 2010

In this paper the numerical solution of Fredholm integral equations of the second kind using an iterative method in which the solution is represented by a Chebyshev series is discussed. A description of a technique of Chebyshev reduction of the norm of the kernel for use in cases when the iterations converge slowly or not at all is also given. Finally, the application of the methods to other types of second-kind equations is considered.
On iteration procedures for equations of the first kind, 𝐴𝑥=𝑦, and Picard’s criterion for the existence of a solution
Authors: Diaz J. B. , Metcalf F. T.
Published: June 2010

Suppose that the (not identically zero) linear operator

A
A

, on a real Hilbert space

H
H

to itself, is compact, selfadjoint, and positive semidefinite; that

y
y

is a vector of

H
H

which is perpendicular to the null space of

A
A

; and that

μ
\mu

is a real number such that

0
>
μ
>
2

/

|

|

A

|

|

0 > \mu > 2/||A||

. Then, the "iteration scheme"

x

n
=
+
1

=

x
n

+
μ
(
y
−
A

x
n

)
,
n
=
0
,
1
,
2
,
⋅
⋅
⋅

{x_{n = + 1}} = {x_n} + \mu (y - A{x_n}),n = 0,1,2, \cdot \cdot \cdot

, yields a strongly convergent sequence of vectors

{

x
n

}

n
=
0

∞

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

if and only if "Picard’s criterion" for the existence of a solution of

A
x
=
y

Ax = y

holds (i.e., if and only if

y
y

is perpendicular to the null space of

A
A

, and

∑

k
=
1

∞

(
y
,

u
k

)

2

/

λ
k
2

>
∞

\sum \nolimits _{k = 1}^\infty {{{(y,{u_k})}^2}} /\lambda _k^2 > \infty

, where the

u
k

{u_k}

and the

λ
k

{\lambda _k}

are the orthonormalized eigenvectors, and the corresponding eigenvalues, of

A
A

, respectively). An analogous result holds when

A
A

is only required to be compact.
Generalized logarithmic error and Newton’s method for the 𝑚th root.
Authors: Phillips David L.
Published: June 2010

The problem of obtaining optimal starting values for the calculation of integer roots using Newton’s method is considered. It has been shown elsewhere that if relative error is used as the measure of goodness of fit. then optimal results are not obtained when the initial approximation is a best fit. Furthermore, if the so-called logarithmic error instead of the relative error is used in the square root case, then a best initial fit is optimal for both errors It is shown here that for each positive integer

m
m

,

m
≧
3

m \geqq 3

, and each negative integer

m
m

, there is a certain generalized logarithmic error for which a best initial fit to the mth root is optimal. It is then shown that an optimal fit can be found by just multiplying a best relative error fit by a certain constant. Also, explicit formulas are found for the optimal initial linear fit.
On the construction of Gaussian quadrature rules from modified moments.
Authors: Gautschi Walter
Published: June 2010

Given a weight function

ω
(
x
)

\omega (x)

on

(
α
,
β
)

(\alpha ,\beta )

, and a system of polynomials

{

p
k

(
x
)

}

k
=
0

∞

\left \{ {{p_k}(x)} \right \}_{k = 0}^\infty

, with degree

p
k

(
x
)
=
k

{p_k}(x) = k

, we consider the problem of constructing Gaussian quadrature rules
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript alpha Superscript beta Baseline f left-parenthesis x right-parenthesis omega left-parenthesis x right-parenthesis d x equals sigma-summation Underscript r equals 1 Overscript n Endscripts lamda Subscript r Baseline Superscript left-parenthesis n right-parenthesis Baseline f left-parenthesis xi Subscript r Baseline Superscript left-parenthesis n right-parenthesis Baseline right-parenthesis">

∫
α
β

f
(
x
)
ω
(
x
)
d
x
=

∑

r
=
1

n

λ
r

(
n
)

f
(

ξ
r

(
n
)

)

\int _\alpha ^\beta {f(x)\omega (x)dx = \sum \nolimits _{r = 1}^n {{\lambda _r}^{(n)}f({\xi _r}^{(n)})} }

from "modified moments"

v
k

=

∫
α
β

p
k

(
x
)
ω
(
x
)
d
x

{v_k} = \int _\alpha ^\beta {{p_k}(x)\omega (x)dx}

. Classical procedures take

p
k

(
x
)
=

x
k

{p_k}(x) = {x^k}

, but suffer from progressive ill-conditioning as

n
n

increases. A more recent procedure, due to Sack and Donovan, takes for

{

p
k

(
x
)
}

\{ {p_k}(x)\}

a system of (classical) orthogonal polynomials. The problem is then remarkably well-conditioned, at least for finite intervals

[
α
,
β
]

[\alpha ,\beta ]

. In support of this observation, we obtain upper bounds for the respective asymptotic condition number. In special cases, these bounds grow like a fixed power of

n
n

. We also derive an algorithm for solving the problem considered, which generalizes one due to Golub and Welsch. Finally, some numerical examples are presented.
Two simple algorithms for discrete rational approximation
Authors: Barrodale I. , Mason J. C.
Published: June 2010

This paper reports on computational experience with algorithms due to Loeb and Appel for rational approximation on discrete point sets. Following a brief review of the linear discrete approximation problem, the two rational algorithms are stated in a general setting. Finally, several numerical examples of applications to

l
1

{l_1}

,

l
2

{l_2}

, and

l
∞

{l_\infty }

approximation are supplied and discussed.
A note on a conjecture of L. J. Mordell
Authors: Malcolm Michael A.
Published: June 2010

A computer proof is described for a previously unsolved problem concerning the inequality

∑

i
=
1

n

x
i

/

(

x

i
+
1

+

x

i
+
2

)
≧
n

/

2

\sum \nolimits _{i = 1}^n {{x_i}/({x_{i + 1}} + {x_{i + 2}}) \geqq n/2}

.
Practical throw-back interpolation
Authors: Burgoyne F. D.
Published: June 2010

Precise conditions are determined for the validity of some frequently-used throwback interpolation formulae.
A numerical determination of the modulus of doubly connected domains by using the Bergman curvature
Authors: Burbea J.
Published: June 2010

The moduli of doubly connected domains are computed by means of the Bergman curvature associated with the Bergman function of the domain. The moduli of nonconcentric annuli, elliptic rings, confocal elliptic rings, squares inside circles and squares inside squares are determined by this method.
An interior a priori estimate for parabolic difference operators and an application
Authors: Bondesson Magnus
Published: June 2010

A general class of finite-difference approximations to a parabolic system of differential equations in a bounded domain

Ω
\Omega

is considered. It is shown that if a solution

U
h

{U_h}

of the discrete problem converges in a discrete

L
2

{L^2}

norm to a solution U of the continuous problem as the mesh size h tends to zero, then the difference quotients of

U
h

{U_h}

converge to the corresponding derivatives of U, the convergence being uniform on any compact subset of

Ω
\Omega

. In particular,

U
h

{U_h}

converges uniformly on compact subsets to U as h tends to zero, provided there is convergence in the discrete

L
2

{L^2}

norm. The main part of the paper is devoted to the establishment of an a priori estimate for the solutions of the discrete problem. This estimate is then used to derive the stated result.
The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. II. Piecewise continuous functions and functions with poles near the interval [0,1]
Authors: Lyness J. N.
Published: June 2010

In Part I, the MIPS method for calculating Fourier coefficients was introduced, and applied to functions

f
∈

C

(
p
)

[
0
,
1
]

f \in {C^{(p)}}[0,1]

. In this part two extensions of the theory are described. One modification extends the theory to piecewise continuous functions,

f
∈
P

C

(
p
)

[
0
,
1
]

f \in P{C^{(p)}}[0,1]

. Using these results the method may be used to calculate approximations to trigonometrical integrals (in which the length of the interval need not coincide with a period of the trigonometrical weighting function). The other modification treats functions which are analytic, but whose low-order derivatives vary rapidly due to poles in the complex plane near the interval of integration. Essentially these poles are ’subtracted out’ but this is done implicitly by the inclusion of additional terms in the standard series. The practical application of these modified methods requires that the nature and location of the discontinuities—or poles—be known at least approximately.
A stable, rational QR algorithm for the computation of the eigenvalues of an Hermitian, tridiagonal matrix
Authors: Reinsch Christian H.
Published: June 2010

The most efficient program for finding all the eigenvalues of a symmetric matrix is a combination of the Householder tridiagonalization and the QR algorithm. The latter, if carried out in a natural way, requires 4n additions, 10n multiplications, 2n divisions, and n square roots per iteration (n the order of the matrix). In 1963, Ortega and Kaiser showed that the process can be carried out using no square roots (and saving 7n multiplications). However, their algorithm is unstable and several modifications were suggested to increase its accuracy. We, too, want to give such a modification together with some examples demonstrating the achieved accuracy.
Optimal quadrature formulas using generalized inverses. I. General theory and minimum variance formulas
Authors: Duris C. S.
Published: June 2010

This paper is the first of two papers concerning the derivation of optimal quadrature formulas. In Part I, we develop results concerning generalized inverses and use these results to derive some minimum variance quadrature formulas. The formulas are obtained by inverting appropriate systems of numerical differentiation formulas. The second paper, Part II, will use the same results concerning generalized inverses to derive Sard “best” quadrature formulas.
On the divisibility of an odd perfect number by the sixth power of a prime
Authors: McDaniel Wayne L.
Published: June 2010

It is shown that any odd perfect number less than

10

9118

{10^{9118}}

is divisible by the sixth power of some prime.
Error bounds for compound quadrature of weakly singular integrals
Authors: Feldstein Alan , Miller Richard K.
Published: June 2010

This paper studies the convergence of numerical quadratures of singular integrands. The singularities are ignored in the sense that whenever a singularity occurs the integrand is redefined to be zero. Several convergence theorems are proved under the assumption that the integrand can be dominated near each singularity by a monotone, integrable function.
On Jacobi and Jacobi-like algorithms for a parallel computer
Authors: Sameh Ahmed H.
Published: June 2010

Many existing algorithms for obtaining the eigenvalues and eigenvectors of matrices would make poor use of such a powerful parallel computer as the ILLIAC IV. In this paper, Jacobi’s algorithm for real symmetric or complex Hermitian matrices, and a Jacobi-like algorithm for real nonsymmetric matrices developed by P. J. Eberlein, are modified so as to achieve maximum efficiency for the parallel computations.
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals left-parenthesis x minus a right-parenthesis Superscript alpha Baseline left-parenthesis b minus x right-parenthesis Superscript beta Baseline phi left-parenthesis x right-parenthesis comma phi left-parenthesis x right-parenthesis">

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)

.
On the remainder in quadrature rules
Authors: Tuan P. D.
Published: June 2010

An expression is obtained for the remainder in quadrature rules applied to functions whose Hubert transforms exist. The estimation of the remainder is illustrated by means of a particular example.
The method of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences
Authors: Cryer Colin W.
Published: June 2010

A method for solving free boundary problems for journal bearings by means of finite differences has been proposed by Christopherson. We analyse Christopherson’s method in detail for the case of an infinite journal bearing where the free boundary problem is as follows: Given

T
>
0

T > 0

and

h
(
t
)

h(t)

find

τ
∈
(
0
,
T
]

\tau \in (0,T]

and

p
(
t
)

p(t)

such that (i)

[

h
3

p
′

]
′

=

h
′

[{h^3}p’]’ = h’

for

t
∈
(
0
,
τ
)

t \in (0,\tau )

, (ii)

p
(
0
)
=
0

p(0) = 0

, (iii)

p
(
t
)
=
0

p(t) = 0

for

t
∈
[
τ
,
T
]

t \in [\tau ,T]

, and (iv)

p
′

(
τ
−
0
)
=
0

p’(\tau - 0) = 0

. First, it is shown that the discrete approximation is accurate to

O
(

[
Δ
t

]
2

)

O({[\Delta t]^2})

where

Δ
t

\Delta t

is the step size. Next, it is shown that the discrete problem is equivalent to a quadratic programming problem. Then, the iterative method for computing the discrete approximation is analysed. Finally, some numerical results are given.
Stable difference schemes with uneven mesh spacings
Authors: Ciment Melvyn
Published: June 2010

We consider a finite-difference approximation to the Cauchy problem for a firstorder hyperbolic partial differential equation using different mesh spacings in different portions of the domain. By reformulating our problem as a difference approximation to an initial-boundary value problem, we are able to use the theory of H. O. Kreiss and S. Osher to analyze the

L
2

{L_2}

stability of our scheme.
Numerical solution of a parabolic free boundary problem arising in statistical decision theory
Authors: Sackett Gary G.
Published: June 2010

A parabolic free boundary problem which arises in a statistical decision setting is reduced to a free boundary problem for the heat equation which is amenable to numerical solution by the method of lines. An algorithm is given and the apparently globally converging results are compared with the asymptotic expansions of H. Chernoff and J. Breakwell.
Series expansions of 𝑊_{𝑘,𝑚}(𝑍) involving parabolic cylinder functions
Authors: Wong R. , Rosenbloom E.
Published: June 2010

In this paper, an explicit error bound is obtained for an expansion of the Whittaker function,

W

k
,
m

(
z
)

{W_{k,m}}(z)

, in series of parabolic cylinder functions. It is also shown that the Whittaker function may be asymptotically represented as the sum of two products where one product involves a parabolic cylinder function and the other product involves the first-order derivative of this function.
A generalized interpolation algorithm
Authors: Newbery A. C. R.
Published: June 2010

An interpolation algorithm is derived, which will construct an

(
n
+
1
)

(n + 1)

-point interpolant based on any sequence of interpolatory functions that can be defined by a three-term linear recursion. By suitable parameter choice, a single algorithm can be made to interpolate in terms of the classical polynomial sequences or in terms of trigonometric or hyperbolic series etc. An analysis of truncation error is included.
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript k Baseline equals left-parenthesis upper X Subscript k Baseline slash r Subscript k Baseline right-parenthesis slash left-parenthesis upper Y slash s right-parenthesis comma k equals 1 comma 2 comma midline-horizontal-ellipsis comma n">

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript x Baseline StartFraction 1 Over t EndFraction left-bracket upper E i left-parenthesis t right-parenthesis minus gamma minus ln StartAbsoluteValue t EndAbsoluteValue right-bracket d t and integral Subscript 0 Superscript x Baseline StartFraction 1 Over t EndFraction left-bracket upper E i left-parenthesis t right-parenthesis minus gamma minus ln StartAbsoluteValue t EndAbsoluteValue right-bracket exp left-parenthesis negative t right-parenthesis d t comma">

∫
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 orders of approximation of the first derivative of cubic splines at the knots
Authors: Kershaw D.
Published: June 2010

The order of approximation of the first derivative of four types of interpolating cubic splines are found. The splines are defined by a variety of endpoint conditions and include the natural cubic spline and the periodic cubic spline. It is found that for two types there is an increase in the order of approximation when equal intervals are used, and that for a special distribution of knots the same order can be realized for the natural spline.
Convergence of singular difference approximations for the discrete ordinate equations in 𝑥-𝑦 geometry
Authors: Madsen N. K.
Published: June 2010

The solutions to two well-known finite difference approximations are shown to converge to the solution of the discrete ordinate equations which are an approximation to the linear Boltzmann equation. These difference schemes are the diamond approximation of Carlson, and the central difference approximation. These schemes are known to give singular systems of algebraic equations in certain cases. Despite this singularity, convergence is shown for all cases when solutions exist.
On a condition for the nonsingularity of a matrix
Authors: Householder Alston S.
Published: June 2010

A new derivation of a nonsingularity condition is given and related to conditions for the convergence of a matrix interation.
Some 𝐴-stable methods for stiff ordinary differential equations
Authors: Jain R. K.
Published: June 2010

This paper gives some

A
A

-stable methods of order

2
n

2n

, with variable coefficients, based on Hermite interpolation polynomials, for the stiff system of ordinary differential equations, making use of

n
n

starting values. The method is exact if the problem is of the form

y
′

(
t
)
=
P
y
(
t
)
+
Q
(
t
)

y’(t) = Py(t) + Q(t)

, where

P
P

is a constant and

Q
(
t
)

Q(t)

is a polynomial of degree

2
n

2n

.
An explicit addition formula for the inverse Jacobian elliptic functions
Authors: Fettis Henry E.
Published: June 2010

An explicit addition formula is derived for the inverse sine-amplitude function. This is used to obtain an explicit formula for the inverse sine-amplitude function with complex argument.
On the distribution of pseudo-random numbers generated by the linear congruential method
Authors: Niederreiter Harald
Published: June 2010

The discrepancy of sequences of pseudo-random numbers generated by the linear congruential method is estimated, thereby improving a result of Jagerman. Applications to numerical integration are mentioned.
Density bounds for Euler’s function
Authors: Wall Charles R.
Published: June 2010

Let

φ
\varphi

be Euler’s function. Upper and lower bounds are presented for

D
(
x
)

D(x)

, the density of the integers

n
n

for which

φ
(
n
)

/

n
≦
x

\varphi (n)/n \leqq x

. The bounds, for

x
=
0
(
.01
)
1

x = 0(.01)1

, have an average spread of less than 0.0203.
The asymptotic expansion of the Meijer 𝐺-function
Authors: Fields Jerry L.
Published: June 2010

Gamma function identities are integrated to expand the Meijer

G
G

-function in a basic set of functions, each of which is simply characterized asymptotically.
Numerical Chebyshev approximation by interpolating rationals
Authors: Williams Jack
Published: June 2010

The paper is concerned with the Chebyshev approximation of decay-type functions

f
(
x
)

f(x)

by interpolating rationals. The interpolating points are chosen to be the zeros of

f
(
x
)

f(x)

. Existence, uniqueness and characterization of best approximations are first shown. An exchange algorithm is then described for computing the best approximation.
A search procedure and lower bound for odd perfect numbers
Authors: Tuckerman Bryant
Published: June 2010

An infinite tree-generating "q-algorithm" is defined, which if executed would enumerate all odd perfect numbers (opn’s). A truncated execution shows that any opn has either some component

p
a

>

10

18

{p^a} > {10^{18}}

, with a even, or no divisor

>
7

> 7

; hence any opn must be

>

10

36

> {10^{36}}

.
Numerical results for Waring’s problem in 𝐺𝐹[𝑞,𝑥]
Authors: Webb William A.
Published: June 2010

Let p be a prime and let K and

A
i

{A_i}

denote polynomials whose coefficients are elements of the finite field with p elements. Problems concerning the expression of an arbitrary polynomial K as sums of a small number of squares or cubes of polynomials

A
i

{A_i}

are discussed. In the problems treated the degrees of the

A
i

{A_i}

are restricted to be as small as possible. In particular, it is shown that at least five cubes are necessary and that three squares seem to suffice in all but one special case.
A winding number algorithm for closed polygonal paths
Authors: Petty J. V.
Published: June 2010

A winding number algorithm for closed polygonal paths (not necessarily simple) is derived using classical complex analysis results and techniques. The algorithm is designed specifically to handle large cases efficiently. The performance of a computer program based on the algorithm is discussed and compared with the performance of a computer program which obtains the winding number directly by antidifferentiation.
Chebyshev approximation by exponentials on finite subsets
Authors: Braess Dietrich
Published: June 2010

This paper is concerned with Chebyshev approximation by exponentials on finite subsets. We take into account that varisolvency does not hold for exponentials in general. A bound for the derivatives of exponentials is established and convergence of the solutions for the discrete problems is proved in the topology of compact convergence on the open interval.