bluuebunny/crossref_metadata_2025_split huggingface dataset:
Left definite multiparameter eigenvalue problems
Authors: Binding Paul 
Published: June 2010

We study the problem
\[

(
∗
)

T
m

x
m

=

∑

n
=
1

k

λ
n

V

m
n

x
m

,

0
≠

x
m

∈

H
m

,

m
=
1
,
…
,
k
,

(\ast )\qquad {T_m}{x_m} = \sum \limits _{n = 1}^k {{\lambda _n}{V_{mn}}{x_m},\qquad 0 \ne } {x_m} \in {H_m},\,m = 1, \ldots ,k,

\]
where

T
m

{T_m}

and

V

m
n

{V_{mn}}

are selfadjoint linear operators on separable Hilbert spaces

H
m

{H_m}

, with

T
m

{T_m}

positive,

T
m

−
1

T_m^{ - 1}

compact and

V

m
n

{V_{mn}}

bounded. We assume “left definiteness” which involves positivity of certain linear combinations of cofactors in the determinant with

(
m
,

n
)

(m,\,n)

th entry

(

x
m

,

V

m
n

x
m

)

({x_m},\,{V_{mn}}{x_m})

. We establish a spectral theory for

(
∗
)

(\ast )

that is in some way simpler and more complete than those hitherto available for this case. In particular, we make use of operators

B
n

=

Δ
n

−
1

Δ
0

{B_n} = \Delta _n^{ - 1}{\Delta _0}

, where the

Δ
n

{\Delta _n}

are determinantal operators on

⊗

m
=
1

k

H
m

\otimes _{m = 1}^k{H_m}

. This complements an established approach to the alternative “right definite” problem (where

Δ
0

{\Delta _0}

is positive) via the operators

Γ
n

=

Δ
0

−
1

Δ
n

{\Gamma _n} = \Delta _0^{ - 1}{\Delta _n}

.
Nonlinear mappings that are globally equivalent to a projection
Authors: Plastock Roy 
Published: June 2010

The Rank theorem gives conditions for a nonlinear Fredholm map of positive index to be locally equivalent to a projection. In this paper we wish to find conditions which guarantee that such a map is globally equivalent to a projection. The problem is approached through the method of line lifting. This requires the existence of a locally Lipschitz right inverse,

F
↓

(
x
)

{F^ \downarrow }(x)

, to the derivative map

F
′

(
x
)

{F^\prime }(x)

and a global solution to the differential equation

P
′

(
t
)
=

F
↓

(
P
(
t
)
)
(
y
−

y
0

)

{P^\prime }(t) = {F^ \downarrow }(P(t))(y - {y_0})

. Both these problems are solved and the generalized Hadamard-Levy criterion
\[

∫
0
∞

inf

|

x

|

>
s

(

1

/

|

F
↓

(
x
)

|

)

d
s
=
∞

\int _0^\infty {\inf \limits _{|x| > s} \left ({1/|{F^ \downarrow }(x)|} \right )\,ds = \infty }

\]
is shown to be sufficient for

F
F

to be globally equivalent to a projection map (Theorem 3.2). The relation to fiber bundle mappings is explored in §4.
Subcontinua with degenerate tranches in hereditarily decomposable continua
Authors: Oversteegen Lex G. , Tymchatyn E. D. 
Published: June 2010

A hereditarily decomposable, irreducible, metric continuum

M
M

admits a mapping

f
f

onto

[
0
,
1
]

[0,1]

such that each

f

−
1

(
t
)

{f^{ - 1}}(t)

is a nowhere dense subcontinuum. The sets

f

−
1

(
t
)

{f^{ - 1}}(t)

are the tranches of

M
M

and

f

−
1

(
t
)

{f^{ - 1}}(t)

is a tranche of cohesion if

t
∈
{
0
,
1
}

t \in \{ 0,1\}

or

f

−
1

(
t
)
=

C1

(

f

−
1

(
[
0
,
t
)
)
)
∩

C1

(

f

−
1

(
(
t
,
1
]
)
)

{f^{ - 1}}(t) = {\text {C1}}({f^{ - 1}}([0,t))) \cap {\text {C1}}\,({f^{ - 1}}((t,1]))

. The following answer a question of Mahavier and of E. S. Thomas, Jr. Theorem. Every hereditarily decomposable continuum contains a subcontinuum with a degenerate tranche. Corollary. If in an irreducible hereditarily decomposable continuum each tranche is nondegenerate then some tranche is not a tranche of cohesion. The theorem answers a question of Nadler concerning arcwise accessibility in hyperspaces.
Boundedness of fractional operators on 𝐿^{𝑝} spaces with different weights
Authors: Harboure Eleonor , Macías Roberto A. , Segovia Carlos 
Published: June 2010

Let

T
α

{T_\alpha }

be either the fractional integral operator

∫
f
(
y
)

|

x
−
y

|

α
−
n

d
y

\smallint f(y)|x - y{|^{\alpha - n}}\; dy

, or the fractional maximal operator

sup

{

r

α
−
n

∫

|

x
−
y

|

>
r

|

f
(
y
)

|

d
y
:

r
>
0

}

\sup \left \{ {{r^{\alpha - n}}{\smallint _{|x - y| > r}}|f(y)|dy:\,r > 0} \right \}

. Given a weight

w
w

(resp.

υ
\upsilon

), necessary and sufficient conditions are given for the existence of a nontrivial weight

υ
\upsilon

(resp.

w
w

) such that

(
∫

|

T
α

f

|

q

υ

d
x

)

1

/

q

⩽

(
∫

|

f

|

p

w

d
x

)

1

/

p

{(\smallint |{T_\alpha }f{|^q}\upsilon \;dx)^{1/q}} \leqslant {(\smallint |f{|^p}w\;dx)^{1/p}}

holds. Weak type substitutes in limiting cases are considered.
The distribution of solutions to equations over finite fields
Authors: Cochrane Todd 
Published: June 2010

Let

F

q

{\mathbb {F}_q}

be the finite field in

q
=

p
f

q = {p^f}

elements,

F
_

(

x
_

)

\underline F (\underline x )

be a

k
k

-tuple of polynomials in

F

q

[

x
1

,
…
,

x
n

]

{\mathbb {F}_q}[{x_1}, \ldots ,{x_n}]

,

V
V

be the set of points in

F

q
n

\mathbb {F}_q^n

satisfying

F
_

(

x
_

)
=

0
_

\underline F (\underline x ) = \underline 0

and

S
S

,

T
T

be any subsets of

F

q
n

\mathbb {F}_q^n

. Set

ϕ
(
V
,

0
_

)
=

|

V

|

−

q

n
−
k

\phi (V,\underline 0 ) = |V| - {q^{n - k}}

,
\[

ϕ
(
V
,

y
_

)
=

∑

x
_

∈
V

e

(

2
π
i

p

Tr
⁡
(

x
_

⋅

y
_

)

)

for

y
_

≠

0
_

,

\phi (V,\underline y ) = \sum \limits _{\underline x \in V} {e\left ( {\frac {{2\pi i}} {p}\operatorname {Tr} (\underline x \cdot \underline y )} \right )\quad {\text {for}}\;\underline y \ne \underline 0 ,}

\]
and

Φ
(
V
)
=

max

y
_

|

ϕ
(
V
,

y
_

)

|

\Phi (V) = {\max _{\underline y }}|\phi (V,\underline y )|

. We use finite Fourier series to show that

(
S
+
T
)
∩
V

(S + T) \cap V

is nonempty if

|

S

|

|

T

|

>

Φ
2

(
V
)

q

2
k

|S||T| > {\Phi ^2}(V){q^{2k}}

. In case

q
=
p

q = p

we deduce from this, for example, that if

C
C

is a convex subset of

R

n

{\mathbb {R}^n}

symmetric about a point in

Z

n

{\mathbb {Z}^n}

, of diameter

>
2
p

> 2p

(with respect to the sup norm), and

Vol
⁡
(
C
)
>

2

2
n

Φ
(
V
)

p
k

\operatorname {Vol} (C) > {2^{2n}}\Phi (V){p^k}

, then

C
C

contains a solution of

F
_

(

x
_

)
≡

0
_

(
mod
p
)

\underline F (\underline x ) \equiv \underline 0 (\bmod p)

. We also show that if

B
B

is a box of points in

F

q
n

\mathbb {F}_q^n

not contained in any

(
n
−
1
)

(n - 1)

-dimensional subspace and

|

B

|

>
4
⋅

2

n
f

Φ
(
V
)

q
k

|B| > 4 \cdot {2^{nf}}\Phi (V){q^k}

, then

B
∩
V

B \cap V

contains

n
n

linearly independent points.
On the Hausdorff dimension of some graphs
Authors: Mauldin R. Daniel , Williams S. C. 
Published: June 2010

Consider the functions
\[

W
b

(
x
)
=

∑

n
=
−
∞

∞

b

−
α
n

[
Φ
(

b
n

x
+

θ
n

)
−
Φ
(

θ
n

)
]
,

{W_b}(x) = \sum \limits _{n = - \infty }^\infty {{b^{ - \alpha n}}[\Phi ({b^n}x + {\theta _n}) - \Phi ({\theta _n})],}

\]
where

b
>
1

b > 1

,

0
>
α
>
1

0 > \alpha > 1

, each

θ
n

{\theta _n}

is an arbitrary number, and

Φ
\Phi

has period one. We show that there is a constant

C
>
0

C > 0

such that if

b
b

is large enough, then the Hausdorff dimension of the graph of

W
b

{W_b}

is bounded below by

2
−
α
−
(
C

/

ln
⁡
b
)

2 - \alpha - (C/\ln b)

. We also show that if a function

f
f

is convex Lipschitz of order

α
\alpha

, then the graph of

f
f

has

σ
\sigma

-finite measure with respect to Hausdorff’s measure in dimension

2
−
α

2 - \alpha

. The convex Lipschitz functions of order

α
\alpha

include Zygmund’s class

Λ
α

{\Lambda _\alpha }

. Our analysis shows that the graph of the classical van der Waerden-Tagaki nowhere differentiable function has

σ
\sigma

-finite measure with respect to

h
(
t
)
=
t

/

ln
⁡
(
1

/

t
)

h(t) = t/\ln (1/t)

.
The Milne problem for the radiative transfer equations (with frequency dependence)
Authors: Golse François 
Published: June 2010

We study the following stationary frequency dependent transport equation:
\[

μ

∂
x

f
+
σ
(
ν
,

T
)
[
f
−

B
ν

(
T
)
]
=
0
,

x
>
0
,

ν
>
0
,

μ
∈
]
−
1
;

1
[
,

∬

R

+

×
]
−
1
;

1
[

σ
(
ν
,

T
)
[

B
ν

(
T
)
−
f
]

d
ν

d
μ

2

=
0
,

f
(
0
,

μ
,

ν
)
=
φ
(
μ
,

ν
)
,

ν
>
0
,

u
∈
]
0
;

1
[
,

\begin {array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu ,\,T)[f - {B_\nu }(T)] = 0,\qquad x > 0,\;\nu > 0,\;\mu \in ] - 1;\,1[,} \\ {\iint _{{{\mathbf {R}}^ + } \times ] - 1;\,1[} {\sigma (\nu ,\,T)[{B_\nu }(T) - f]\,d\nu \frac {{d\mu }} {2} = 0,}} \\ {f(0,\,\mu ,\,\nu ) = \varphi (\mu ,\,\nu ),\qquad \nu > 0,\;u \in ]0;\,1[,} \\ \end {array}

\]
where

B
ν

{B_\nu }

is the well-known Planck function appearing in astrophysics. We are able to describe the asymptotic behavior of

f
f

and

T
T

for

x
x

large, when

σ
(
ν
,

T
)

\sigma (\nu ,\,T)

is of the special form

σ
(
ν
,

T
)
=
σ
(
ν
)
k
(
T
)

\sigma (\nu ,\,T) = \sigma (\nu )k(T)

. Our method relies mainly on the monotonicity of the nonlinearity. The proof does not use any linearization of the equation; in particular, no smallness assumption on the data

φ
\varphi

(in any sense) is required. Résumé. Nous étudions l’équation de transport stationnaire avec dépendance en fréquence:
\[

μ

∂
x

f
+
σ
(
ν
,

T
)
[
f
−

B
ν

(
T
)
]
=
0
,

x
>
0
,

ν
>
0
,

μ
∈
]
−
1
;

1
[
,

∬

R

+

×
]
−
1
;

1
[

σ
(
ν
,

T
)
[

B
ν

(
T
)
−
f
]

d
ν

d
μ

2

=
0
,

f
(
0
,

μ
,

ν
)
=
φ
(
μ
,

ν
)
;

ν
>
0
,

μ
∈
]
0
;

1
[
.

\begin {array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu ,\,T)[f - {B_\nu }(T)] = 0,\qquad x > 0,\;\nu > 0,\;\mu \in ] - 1;\,1[,} \\ {\iint _{{{\mathbf {R}}^ + } \times ] - 1;\,1[} {\sigma (\nu ,\,T)[{B_\nu }(T) - f]\,d\nu \frac {{d\mu }} {2} = 0,}} \\ {f(0,\,\mu ,\,\nu ) = \varphi (\mu ,\,\nu );\qquad \nu > 0,\;\mu \in ]0;\,1[.} \\ \end {array}

\]
Lorsque

σ
(
ν
,

T
)

\sigma (\nu ,\,T)

est de la forme particulière

σ
(
ν
,

T
)
=
σ
(
ν
)
k
(
T
)

\sigma (\nu ,\,T) = \sigma (\nu )k(T)

, nous savons décrire le comportement asymptotique de

f
f

et

T
T

pour

x
x

grand. Notre méthode repose principalement sur la monotonie de la non-linéarité. La preuve n’utilise aucune linéarisation de l’équation; en particulier, nous n’avons besoin d’aucune hypothèse de petitesse (d’aucune sorte) sur la donnée

φ
\varphi

.
A characterization of two weight norm inequalities for fractional and Poisson integrals
Authors: Sawyer Eric T. 
Published: June 2010

For

1
>
p
⩽
q
>
∞

1 > p \leqslant q > \infty

and

w
(
x
)

w(x)

,

v
(
x
)

v(x)

nonnegative functions on

R

n

{{\mathbf {R}}^n}

, we show that the weighted inequality
\[

(

∫

|

T
f

|

q

w

)

1

/

q

⩽
C

(

∫

f
p

v

)

1

/

p

{\left ( {\int {|Tf{|^q}w} } \right )^{1/q}} \leqslant C{\left ( {\int {{f^p}v} } \right )^{1/p}}

\]
holds for all

f
⩾
0

f \geqslant 0

if and only if both
\[

∫

[
T
(

χ
Q

v

1
−

p
′

)
]

q

w
⩽

C
1

(

∫
Q

v

1
−

p
′

)

q

/

p

>
∞

\int {{{[T({\chi _Q}{v^{1 - p’}})]}^q}w \leqslant {C_1}{{\left ( {\int _Q {{v^{1 - p’}}} } \right )}^{q/p}} > \infty }

\]
and
\[

∫

[
T
(

χ
Q

w
)
]

p
′

v

1
−

p
′

⩽

C
2

(

∫
Q

w

)

p
′

/

q
′

>
∞

{\int {{{[T({\chi _Q}w)]}^{p’}}{v^{1 - p’}} \leqslant {C_2}\left ( {\int _Q w } \right )} ^{p’/q’}} > \infty

\]
hold for all dyadic cubes

Q
Q

. Here

T
T

denotes a fractional integral or, more generally, a convolution operator whose kernel

K
K

is a positive lower semicontinuous radial function decreasing in

|

x

|

|x|

and satisfying

K
(
x
)
⩽
C
K
(
2
x
)

K(x) \leqslant CK(2x)

,

x
∈

R

n

x \in {{\mathbf {R}}^n}

. Applications to degenerate elliptic differential operators are indicated. In addition, a corresponding characterization of those weights

v
v

on

R

n

{{\mathbf {R}}^n}

and

w
w

on

R

+

n
+
1

{\mathbf {R}}_ + ^{n + 1}

for which the Poisson operator is bounded from

L
p

(
v
)

{L^p}(v)

to

L
q

(
w
)

{L^q}(w)

is given.
Local uncertainty inequalities for locally compact groups
Authors: Price John F. , Sitaram Alladi 
Published: June 2010

Let

G
G

be a locally compact unimodular group equipped with Haar measure

m
m

,

G
^

\hat G

its unitary dual and

μ
\mu

the Plancherel measure (or something closely akin to it) on

G
^

\hat G

. When

G
G

is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given

θ
∈

[

0
,

1
2

)

\theta \in \left [ {0,\tfrac {1} {2}} \right .)

there exists a constant

K
θ

{K_\theta }

such that for all

f
f

in a certain class of functions on

G
G

and all measurable

E
⊆

G
^

E \subseteq \hat G

,
\[

(

∫
E

Tr
⁡
(
π

(
f
)

∗

π
(
f
)
)

d
μ
(
π
)

)

1

/

2

⩽

K
θ

μ

(
E

)
θ

|

|

ϕ
θ

f

|

|

2

{\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f))\,d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2}

\]
where

ϕ
θ

{\phi _\theta }

is a certain weight function on

G
G

(for which an explicit formula is given). When

G
=

R

k

G = {{\mathbf {R}}^k}

the inequality has been established with

ϕ
θ

(
x
)
=

|

x

|

k
θ

{\phi _\theta }(x) = |x{|^{k\theta }}

.
Stable manifolds in the method of averaging
Authors: Schecter Stephen 
Published: June 2010

Consider the differential equation

z
˙

=
ε
f
(
z
,

t
,

ε
)

\dot z = \varepsilon f(z,\,t,\,\varepsilon )

, where

f
f

is

T
T

periodic in

t
t

and

ε
>
0

\varepsilon > 0

is a small parameter, and the averaged equation

z
˙

=

f
¯

(
z
)
:=
(
1

/

T
)

∫
0
T

f
(
z
,

t
,

0
)

d
t

\dot z = \overline f (z): = (1/T)\,\int _0^T {\,f(z,\,t,\,0)\,dt}

. Suppose the averaged equation has a hyperbolic equilibrium at

z
=
0

z = 0

with stable manifold

W
¯

\overline W

. Let

β
ε

(
t
)

{\beta _\varepsilon }(t)

denote the hyperbolic

T
T

-periodic solution of

z
˙

=
ε
f
(
z
,

t
,

ε
)

\dot z = \varepsilon f(z,\,t,\,\varepsilon )

near

z
≡
0

z \equiv 0

. We prove a result about smooth convergence of the stable manifold of

β
ε

(
t
)

{\beta _\varepsilon }(t)

to

W
¯

×

R

\overline W \times {\mathbf {R}}

as

ε
→
0

\varepsilon \to 0

. The proof uses ideas of Vanderbauwhede and van Gils about contractions on a scale of Banach spaces.
On 𝐻_{*}(Ωⁿ⁺²𝑆ⁿ⁺¹;𝐹₂)
Authors: Hunter Thomas J. 
Published: June 2010

In this paper, we study

H
∗

Ω

n
+
2

S

n
+
1

{H_\ast }{\Omega ^{n + 2}}{S^{n + 1}}

. Here

Ω
X

\Omega X

denotes the space of pointed maps

S
1

→
X

{S^1} \to X

, and

H
∗

{H_\ast }

represents homology modulo

2
2

. We show that the Eilenberg-Moore spectral sequence

Tor

H
∗

Ω
0

n
+
1

S

n
+
1

∗
∗

⁡
(

F

2
,

F
2

)
⇒

H
∗

Ω

n
+
2

S

n
+
1

\operatorname {Tor}_{{H^\ast }\Omega _0^{n + 1}{S^{n + 1}}}^{\ast \ast }({F_{2,}}{F_2}) \Rightarrow {H^\ast }{\Omega ^{n + 2}}{S^{n + 1}}

collapses, and we identify the kernel of the Whitehead product map

Ω

n
+
1

p
∗

:

H
∗

Ω

n
+
3

S

2
n
+
1

→

H
∗

Ω

n
+
1

S
n

{\Omega ^{n + 1}}{p_\ast }:{H_\ast }{\Omega ^{n + 3}}{S^{2n + 1}} \to {H_\ast }{\Omega ^{n + 1}}{S^n}

. These observations yield two different descriptions of

H
∗

Ω

n
+
2

S

n
+
1

{H_\ast }{\Omega ^{n + 2}}{S^{n + 1}}

up to extension.
Translation semigroups and their linearizations on spaces of integrable functions
Authors: Grabosch Annette 
Published: June 2010

Of concern is the unbounded operatorAΦf=f′{A_\Phi }f = f’with nonlinear domainD(AΦ)={f∈W1,1:f(0)=Φ(f)}D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\}which is considered on the Banach spaceEEof Bochner integrable functions on an interval with values in a Banach spaceFF. Under the assumption thatΦ\Phiis a Lipschitz continuous operator fromEEtoFF, it is shown thatAΦ{A_{\Phi }}generates a strongly continuous translation semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. For linear operatorsΦ\Phiseveral properties such as essential-compactness, positivity, and irreducibility of the semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}depending on the operatorΦ\Phiare studied. It is shown that ifFFis a Banach lattice with order continuous norm, then(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}is the modulus semigroup of(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. Finally spectral properties ofAΦ{A_\Phi }are studied and the spectral bounds(AΦ)s({A_\Phi })is determined. This leads to a result on the global asymptotic behavior in the case whereΦ\Phiis linear and to a local stability result in the case whereΦ\Phiis Fréchet differentiable.
Product integration for the generalized Abel equation
Authors: Weiss Richard 
Published: June 2010

The solution of the generalized Abel integral equation
\[

g
(
t
)
=

∫
0
t

{
k
(
t
,
s
)

/

(
t
−
s
)

α

}
f
(
s
)

d
s
,

0
>
α
>
1
,

g(t) = \int _0^t {\{ k(t,s)/{{(t - s)}^\alpha }\} f(s)} ds,\;0 > \alpha > 1,

\]
where

k
(
t
,
s
)

k(t,s)

is continuous, by the product integration analogue of the trapezoidal method is examined. It is shown that this method has order two convergence for

α
∈
[

α
1

,
1
)

\alpha \in [{\alpha _1},1)

with

α
1

≑
0.2117

{\alpha _1} \doteqdot 0.2117

. This interval contains the important case

α
=

1
2

\alpha = \tfrac {1}{2}

. Convergence of order two for

α
∈
(
0
,

α
1

)

\alpha \in (0,{\alpha _1})

is discussed and illustrated numerically. The possibility of constructing higher order methods is illustrated with an example.
How slowly can quadrature formulas converge?
Authors: Lipow Peter R. , Stenger Frank 
Published: June 2010

Let

{

Q
n

}

n
=
1

∞

\{ {Q_n}\} _{n = 1}^\infty

denote a sequence of quadrature formulas,

Q
n

(
f
)
≡

∑

i
=
1

k
n

w
i

(
n
)

f
(

x
i

(
n
)

)

{Q_n}(f) \equiv \sum _{i = 1}^{{k_n}}w_i^{(n)}f(x_i^{(n)})

, such that

Q
n

(
f
)
→

∫
0
1

f
(
x
)
d
x

{Q_n}(f) \to \int _0^1 f (x)dx

for all

f
∈
C
[
0
,
1
]

f \in C[0,1]

. Let

0
>
ε
>

1
4

0 > \varepsilon > \frac {1}{4}

and a sequence

{

a
n

}

n
=
1

∞

\{ {a_n}\} _{n = 1}^\infty

be given, where

a
1

≧

a
2

≧

a
3

≧
.
.
.

{a_1} \geqq {a_2} \geqq {a_3} \geqq ...

, and where

a
n

→
0

{a_n} \to 0

as

n
→
∞

n \to \infty

. Then there exists a function

f
∈
C
[
0
,
1
]

f \in C[0,1]

and a sequence

{

n
k

}

k
=
1

∞

\{ {n_k}\} _{k = 1}^\infty

such that

|

f
(
x
)

|

≦
2

a
1

/

|

(
1
−
4
)

|

|f(x)| \leqq 2{a_1}/|(1 - 4)|

, and such that

∫
0
1

f
(
x
)
d
x
−

Q

n
k

(
f
)
=

a

k
,

k
=
1
,
2
,
3
,
.
.
.

\int _0^1 f (x)dx - {Q_{{n_k}}}(f) = {a_{k,}}k = 1,2,3,...

.
Numerical methods for computing angles between linear subspaces
Authors: Björck Ȧke , Golub Gene H. 
Published: June 2010

Assume that two subspaces F and G of a unitary space are defined as the ranges (or null spaces) of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles

θ
k

(
F
,
G
)

{\theta _k}(F,G)

and orthogonal sets of principal vectors

u
k

∈
F

{u_k} \in F

and

v
k

∈
G
,
k
=
1
,
2
,
⋯
,
q
=
dim
⁡
(
G
)
≦
dim
⁡
(
F
)

{v_k} \in G,k = 1,2, \cdots ,q = \dim (G) \leqq \dim (F)

. An important application in statistics is computing the canonical correlations

σ
k

=
cos
⁡

θ
k

{\sigma _k} = \cos {\theta _k}

between two sets of variates. A perturbation analysis shows that the condition number for

θ
k

{\theta _k}

essentially is

max
(
κ
(
A
)
,
κ
(
B
)
)

\max (\kappa (A),\kappa (B))

, where

κ
\kappa

denotes the condition number of a matrix. The algorithms are based on a preliminary QR-factorization of A and B (or

A
H

{A^H}

and

B
H

{B^H}

), for which either the method of Householder transformations (HT) or the modified Gram-Schmidt method (MGS) is used. Then

cos

θ
k

\cos \;{\theta _k}

and

sin

θ
k

\sin \;{\theta _k}

are computed as the singular values of certain related matrices. Experimental results are given, which indicates that MGS gives

θ
k

{\theta _k}

with equal precision and fewer arithmetic operations than HT. However, HT gives principal vectors, which are orthogonal to working accuracy, which is not generally true for MGS. Finally, the case when A and/or B are rank deficient is discussed.
A class of quadrature formulas
Authors: Kumar Ravindra 
Published: June 2010

It is proved that there exists a set of polynomials orthogonal on

[
−
1
,
1
]

[ - 1,1]

with respect to the weight function

(

1

)

w
(
t
)

/

(
t
−
x
)

\begin{equation}\tag {$1$} w(t)/(t - x)\end{equation}

corresponding to the polynomials orthogonal on

[
−
1
,
1
]

[ - 1,1]

with respect to the weight function w. Simplified forms of such polynomials are obtained for the special cases

(

2

)

w
(
t
)
=

(
1
−

t
2

)

−
1

/

2

,

=

(
1
−

t
2

)

1

/

2

,

=

(
(
1
−
t
)

/

(
1
+
t
)
)

1

/

2

,

\begin{equation}\tag {$2$} \begin {array}{*{20}{c}} {w(t) = {{(1 - {t^2})}^{ - 1/2}},} \\ { = {{(1 - {t^2})}^{1/2}},} \\ { = {{((1 - t)/(1 + t))}^{1/2}},} \\ \end{array} \end{equation}

and the generating functions and the recurrence relation are also given. Subsequently, a set of quadrature formulas given by

(

3

)

∫

−
1

1

(
1
+
t
)

p
−
1

/

2

(
1
−
t
)

q
−
1

/

2

(
1
+

a
2

+
2
a
t
)

−
1

f
(
t
)
d
t
=

∑

k
=
1

n

H
k

f
(

t
k

)
+

E
n

(
f
)

\begin{equation}\tag {$3$} \int _{ - 1}^1 {{{(1 + t)}^{p - 1/2}}{{(1 - t)}^{q - 1/2}}{{(1 + {a^2} + 2at)}^{ - 1}}f(t)dt = \sum \limits _{k = 1}^n {{H_k}f({t_k}) + {E_n}(f)} } \end{equation}

for

(
p
,
q
)
=
(
0
,
0
)
,
(
0
,
1
)

(p,q) = (0,0),(0,1)

and (1, 1) is established; these formulas are valid for analytic functions. Convergence of the quadrature rules is discussed, using a technique based on the generating functions. This method appears to be simpler than the one suggested by Davis [2, pp. 311-312] and used by Chawla and Jain [3]. Finally, bounds on the error are obtained.
Roots of two transcendental equations as functions of a continuous real parameter
Authors: Pexton Robert L. , Steiger Arno D. 
Published: June 2010

The roots,

λ
\lambda

and

η
\eta

, of the transcendental equations

j
l

(
α
λ
)

y
l

(
λ
)
=

j
l

(
λ
)

y
l

(
α
λ
)

{j_l}(\alpha \lambda ){y_l}(\lambda ) = {j_l}(\lambda ){y_l}(\alpha \lambda )

and
\[

[
x

j
l

(
x
)

]

x
=
α
η

′

[
x

y
l

(
x
)

]

x
=
η

′

=
[
x

j
l

(
x
)

]

x
=
η

′

[
x

y
l

(
x
)

]

x
=
α
η

′

[x{j_l}(x)]_{x = \alpha \eta }^\prime [x{y_l}(x)]_{x = \eta }^\prime = [x{j_l}(x)]_{x = \eta }^\prime [x{y_l}(x)]_{x = \alpha \eta }^\prime

\]
where

l
=
1
,
2
,
…

l = 1,2, \ldots

are considered as functions of the continuous real parameter

α
\alpha

. The symbols

j
l

{j_l}

and

y
l

{y_l}

denote the spherical Bessel functions of the first and second kind. The two transcendental equations are invariant under the transformations

λ
→
−
λ

\lambda \to - \lambda

and

η
→
−
η

\eta \to - \eta

, respectively. Therefore, only positive roots are discussed. All the

λ
\lambda

-roots increase monotonically as

α
\alpha

increases in the open interval (0, 1). For each order l, the smallest

η
\eta

-root decreases monotonically as

α
\alpha

increases in (0, 1), tending towards

l
(
l
+
1
)

\sqrt {l(l + 1)}

as

α
\alpha

approaches unity. For

α
∈
(
0
,
1
)

\alpha \in (0,1)

all the other

η
\eta

-roots have a minimum value equal to

l
(
l
+
1
)

/

α

\sqrt {l(l + 1)} /\alpha \,

.
The rate of convergence of Hermite function series
Authors: Boyd John P. 
Published: June 2010

Let

α
>
0

\alpha > 0

be the least upper bound of

γ
\gamma

for which
\[

f
(
z
)
∼
O
(

e

−
q

|

z

|

γ

)

f(z) \sim O({e^{ - q|z|\gamma }})

\]
for some positive constant q as

|

z

|

→
∞

|z| \to \infty

on the real axis. It is then proved that at least an infinite subsequence of the coefficients

{

a
n

}

\{ {a_n}\}

in
\[

f
(
z
)
=

e

−

z
2

/

2

∑

n
=
0

∞

a
n

H
n

(
z
)
,

f(z) = {e^{ - {z^2}/2}}\sum \limits _{n = 0}^\infty {{a_n}{H_n}(z),}

\]
where the

H
n

{H_n}

are the normalized Hermite polynomials, must satisfy certain lower bounds. The theorems show two striking facts. First, the convergence rate of a Hermite series depends not only upon the order

ρ
\rho

for an entire function or the location of the nearest singularity for a singular function as for a power series but also upon

α
\alpha

, thus making the convergence theory of Hermitian series more complicated (and interesting) than that for any ordinary Taylor expansion. Second, the poorer the match between the asymptotic behavior of

f
(
z
)

f(z)

and

exp
⁡
(
−
1

/

2

z
2

)

\exp (-1/2 z^2)

the poorer the convergence of the Hermite series will be.