bluuebunny/crossref_metadata_2025_split huggingface dataset:
Pure states on some group-invariant 𝐶*-algebras
Authors: Price Geoffrey L. 
Published: June 2010

Let

A

\mathfrak {A}

be a UHF algebra of Glimm type

n
∞

{n^\infty }

, i.e.,

A

=

⊗

k
⩾
1

∗

N
k

\mathfrak {A} = \otimes _{k \geqslant 1}^{\ast }{N_k}

, where

N
=

N
1

=

N
2

=
⋯

N = {N_1} = {N_2} = \cdots

are

n
×
n

n \times n

matrix algebras. We define an AF-subalgebra

A

G

{\mathfrak {A}^G}

of

A

\mathfrak {A}

, consisting of those elements of

A

\mathfrak {A}

invariant under a group of automorphisms

{

α
g

:
g
∈
G
=
SU
⁡
(
n
)
}

\{ {\alpha _g}:g \in G = \operatorname {SU} (n)\}

of product type.

A

G

{\mathfrak {A}^G}

is shown to be generated by an embedding of

S
(
∞
)

S(\infty )

, the discrete group of finite permutations on countably many symbols. Let

ω
\omega

be a pure product state on

A

\mathfrak {A}

,

ω
G

{\omega ^G}

its restriction to

A

G

{\mathfrak {A}^G}

. Let

e
∈
N

e \in N

be a one-dimensional projection with corresponding projections

e
k

∈

N
k

{e^k} \in {N_k}

. Then if both (i)

Σ

k
⩾
1

ω
(

e
k

)
=
∞

{\Sigma _{k \geqslant 1}}\omega ({e^k}) = \infty

, and (ii)

0
>

Σ

k
⩾
1

ω
(

e
k

)
[
1
−
ω
(

e
k

)
]
>
∞

0 > {\Sigma _{k \geqslant 1}}\omega ({e^k})[1 - \omega ({e^k})] > \infty

hold,

ω
G

{\omega ^G}

is not pure.

ω
G

{\omega ^G}

is shown to be pure if there exist orthogonal one-dimensional projections

{

p
i

:
1
⩽
i
⩽
n
}

\{ {p_i}:1 \leqslant i \leqslant n\}

of

N
N

with corresponding projections

p
i
k

∈

N
k

p_i^k \in {N_k}

such that

ω
(

p
i
k

)
=
0

\omega (p_i^k) = 0

or

1
1

,

1
⩽
i
⩽
n
,

k
⩾
1

1 \leqslant i \leqslant n,\,k \geqslant 1

, and

0
>

Σ

k
⩾
1

ω
(

p
i
k

)
>
∞

0 > {\Sigma _{k \geqslant 1}}\omega (p_i^k) > \infty

for at most one

i
i

.
Weak solutions of the Gellerstedt and the Gellerstedt-Neumann problems
Authors: Aziz A. K. , Schneider M. 
Published: June 2010

We consider the question of existence of weak and semistrong solutions of the Gellerstedt problem
\[

u

|

Γ
0

∪

Γ
1

∪

Γ
2

=
0

u{|_{{\Gamma _0} \cup {\Gamma _1} \cup {\Gamma _2}}} = 0

\]
and the Gellerstedt-Neumann problem
\[

(

d
n

u
=
k
(
y
)

u
x

d
y
−

u
y

d
x

|

Γ
0

=
0
,

u

|

Γ
1

∪

Γ
2

=
0
)

({d_n}u = k(y){u_x}dy - {u_y}dx{|_{{\Gamma _0}}} = 0,\qquad u{|_{{\Gamma _1} \cup {\Gamma _2}}} = 0)

\]
for the equation of mixed type
\[

L
[
u
]
≡
k
(
y
)

u

x
x

+

u

y
y

+
λ
u
=
f
(
x
,
y
)
,

λ
=
const
>
0

L[u] \equiv k(y){u_{xx}} + {u_{yy}} + \lambda u = f(x,y),\qquad \lambda = \operatorname {const} > 0

\]
in a region

G
G

bounded by a piecewise smooth curve

Γ
0

{\Gamma _0}

lying in the half-plane

y
>
0

y > 0

and intersecting the line

y
=
0

y = 0

at the points

A
(
−
1
,
0
)

A( - 1,0)

and

B
(
1
,
0
)

B(1,0)

. For

y
>
0

y > 0

,

G
G

is bounded by the characteristic curves

γ
1

(
x
>
0
)

{\gamma _1}(x > 0)

and

γ
2

(
x
>
0
)

{\gamma _2}(x > 0)

of (1) through the origin and the characteristics

Γ
1

{\Gamma _1}

and

Γ
2

{\Gamma _2}

through

A
A

and

B
B

which intersect

γ
1

{\gamma _1}

and

γ
2

{\gamma _2}

at the points

P
P

and

Q
Q

, respectively. Using a variation of the energy integral method, we give sufficient conditions for the existence of weak and semistrong solutions of the boundary value problems (Theorems 4.1, 4.2, 5.1).
Normal structure and weakly normal structure of Orlicz sequence spaces
Authors: Landes Thomas 
Published: June 2010

For a convex Orlicz functionφ:R+→R+∪{∞}\varphi :{{\bf {R}}_ + } \to {{\bf {R}}_ + } \cup \{ \infty \}and the associated Orlicz sequence spacelφ{l_\varphi }, we consider the following five properties: (1)lφ{l_\varphi }has a subspace isometric tol1{l_1}. (2)lφ{l_\varphi }is Schur. (3)lφ{l_\varphi }has normal structure. (4) Every weakly compact subset oflφ{l_\varphi }has normal structure. (5) Every bounded sequence inlφ{l_\varphi }has a subsequence(xn)({x_n})which is pointwise and almost convergent tox∈lφx \in {l_\varphi }, i.e.,limsupn→∞∥xn−x∥φ>liminfn→∞∥xn−y∥φ\lim \,{\sup _{n \to \infty }}\parallel {x_n} - x{\parallel _{\varphi }} > \lim \inf _{n \to \infty }\parallel {x_n} - y{\parallel _\varphi }for ally≠xy \ne x. Our results are: (1)⇔φ\Leftrightarrow \;\varphiis either linear at0(φ(s)/s=c>0,0>s⩽t)0\;(\varphi (s)/s = c > 0,0 > s \leqslant t)or does not satisfy theΔ2{\Delta _2}-condition at00. (2)⇔lφ\Leftrightarrow \;{l_\varphi }is isomorphic tol1⇔φ′(0)=limt→0φ(t)/t>0{l_1}\; \Leftrightarrow \;\varphi ’(0) = {\lim _{t \to 0}}\,\varphi \,(t)/t > 0. (3)⇔φ\Leftrightarrow \varphisatisfies theΔ2{\Delta _2}-condition at0,φ0, \varphiis not linear at00andC(φ)=sup{φ(t)>1}>12C(\varphi ) = \sup \,\{ \varphi \,(t) > 1\} > \frac {1}{2}. (4)⇔φ\Leftrightarrow \,\varphisatisfies theΔ2{\Delta _2}-condition at00andC(φ)>12orφ′(0)>0C\,(\varphi ) > \frac {1}{2}\;{\rm {or}}\;\varphi ’(0) > 0. (5)⇔φ\Leftrightarrow \;\varphisatisfies theΔ2{\Delta _2}-condition at00andC(φ)=1C(\varphi ) = 1. The last equivalence contains a result of Lami-Dozo [10].
Dirichlet series and automorphic forms on unitary groups
Authors: Orloff Tobias 
Published: June 2010

In a special case our unitary group takes the form
\[

G
=
{
g
∈

GL

(
p
+
2
,

C

)

|

t

g
¯

R
g
=
R
}
.

G = \{ g \in {\text {GL}}(p + 2,{\mathbf {C}}){|^t}\bar gRg = R\} .

\]
Here
\[

R
=

(

S

a
m
p
;
0

a
m
p
;
0

0

a
m
p
;
0

a
m
p
;
1

0

a
m
p
;

−
1

a
m
p
;
0

)

R = \left ( {\begin {array}{*{20}{c}} S & 0 & 0 \\ 0 & 0 & 1 \\ 0 & { - 1} & 0 \\ \end {array} } \right )

\]
is a skew-Hermitian matrix with entries in an imaginary quadratic number field

K
K

. We suppose that

−
i
R

- iR

has signature

(
p
+
1
,
1
)

(p + 1,1)

. This group acts naturally on the symmetric domain
\[

D
=

{

w
∈

C

p

,
z
∈

C

|

Im
⁡
(
z
)
>
−

1
2

t

w
¯

S
w

}

.

D = \left \{ {w \in {{\mathbf {C}}^p},z \in {\mathbf {C}}|\operatorname {Im} (z) > - {{\frac {1}{2}}^t}\bar wSw} \right \}.

\]
If

Γ
=
G
∩

SL

(
p
+
2
,

O

K

)

\Gamma = G \cap {\text {SL}}(p + 2,{\mathcal {O}_K})

with

O

K

{\mathcal {O}_K}

the ring of integers in

K
K

, then an automorphic form

f
(
w
,
z
)

f(w,z)

with respect to

Γ
\Gamma

has an expansion

Σ
r

g
r

(
w
)
⋅

e

2
π
i
r
z

{\Sigma _r}{g_r}(w) \cdot {e^{2\pi irz}}

. The functions

g
r

(
w
)

{g_r}(w)

are theta functions. Given another automorphic form

g
(
w
,
z
)

g(w,z)

with an expansion

Σ
s

h
s

(
w
)
⋅

e

2
π
i
s
z

{\Sigma _s}{h_s}(w) \cdot {e^{2\pi isz}}

we define a Dirichlet series

Σ
r

⟨

g
r

,

h
r

⟩

r

−
s

{\Sigma _r}\langle {g_r},{h_r}\rangle {r^{ - s}}

. Here

⟨

g
r

,

h
r

⟩

\langle {g_r},{h_r}\rangle

is a certain positive definite inner product on the space of theta functions. The series is obtained as an integral of Rankin type:
\[

∫

P
Γ

∖
D

f

g
¯

⋅

(

Im
⁡
(
z
)
+

1
2

i
t

w
¯

S
w

)

s

d
w

d

w
¯

d
z

d

z
¯

{\int _{{P_\Gamma }\backslash D}}f\bar g\cdot {\left ( {\operatorname {Im} (z) + \frac {1} {2}{i^t}\bar wSw} \right )^s}dw\;d\bar w\;dz\;d\bar z

\]
with

P
Γ

⊆
Γ

{P_\Gamma } \subseteq \Gamma

a subgroup of "translations". The series is analytically continued by studying the Eisenstein series arising when the above integral is transformed into an integral over

Γ
∖
D

\Gamma \backslash D

. In the case

p
=
1

p = 1

our results have an application to some recent work of Shintani, where the Euler product attached to an eigenfunction of the Hecke operators is obtained, up to some simple factors, as a series of the above type.
Equivariant bundles and cohomology
Authors: Kozlowski A. 
Published: June 2010

Let

G
G

be a topological group,

A
A

an abelian topological group on which

G
G

acts continuously and

X
X

a

G
G

-space. We define "equivariant cohomology groups" of

X
X

with coefficients in

A
A

,

H
G
i

(
X
;
A
)

H_G^i(X;A)

, for

i
≥
0

i \geq 0

which generalize Graeme Segal’s continuous cohomology of the topological group

G
G

with coefficients in

A
A

. In particular we have

H
G
1

(
X
;
A
)
≃

H_G^1(X;A) \simeq

equivalence classes of principal

(
G
,
A
)

(G,A)

-bundles over

X
X

. We show that when

G
G

is a compact Lie group and

A
A

an abelian Lie group we have for

i
>
1

H
G
i

(
X
;
A
)
≃

H
i

(
E
G

×
G

X
;
τ
A
)

i > 1\;H_G^i(X;A) \simeq {H^i}(EG{ \times _G}X;\tau A)

where

τ
A

\tau A

is the sheaf of germs of sections of the bundle

(
X
×
E
G
×
A
)

/

G
→
(
X
×
E
G
)

/

G

(X \times EG \times A)/G \to (X \times EG)/G

. For

i
=
1

i = 1

and the trivial action of

G
G

on

A
A

this is a theorem of Lashof, May and Segal.
Some results on locally finitely presentable categories
Authors: Makkai M. , Pitts A. M. 
Published: June 2010

We prove that any full subcategory of a locally finitely presentable (l.f.p.) category having small limits and filtered colimits preserved by the inclusion functor is itself l.f.p. Here "full" may be weakened to "full with respect to isomorphisms." Further, we characterize those left exact functors

I
:

C

→

D

I:{\mathbf {C}} \to {\mathbf {D}}

between small categories with finite limits for which the functor

I

∗

:

L
E
X

(

D

,

Set)

→

L
E
X

(

C

,Set)

{I^{\ast }}:{\mathbf {LEX}}({\mathbf {D}},{\text {Set)}} \to {\mathbf {LEX}}{\text {(}}{\mathbf {C}}{\text {,Set)}}

induced by composition is full and faithful. As an application, we prove a theorem on sheaf representations, a consequence of which is that, for any site

C

=
(

C

,
J
)

\mathcal {C} = ({\mathbf {C}},J)

on a category

C

{\mathbf {C}}

with finite limits, defined by a subcanonical Grothendieck topology

J
J

, the closure in

L
E
X

(

C

,

Set)

{\mathbf {LEX}}({\mathbf {C}},{\text {Set)}}

under small limits and filtered colimits of the models of

C

\mathcal {C}

is the whole of

L
E
X

(

C

,

Set)

{\mathbf {LEX}}({\mathbf {C}},{\text {Set)}}

.
On the theory of biorthogonal polynomials
Authors: Iserles A. , Nørsett S. P. 
Published: June 2010

Let

φ
(
x
,

μ
)

\varphi (x,\,\mu )

be a distribution in

x
∈

R

x \in {\mathbf {R}}

for every

μ
\mu

in a real parameter set

Ω
\Omega

. Subject to additional technical conditions, we study

m
m

th degree monic polynomials

p
m

{p_m}

that satisfy the biorthogonality conditions
\[

∫

−
∞

∞

p
m

(
x
)

d
φ
(
x
,

μ
l

)
=
0
,

l
=
1
,

2
,
…
,

m
,

m
⩾
1

\int _{ - \infty }^\infty {{p_m}(x)\,d\varphi (x,{\mu _l}) = 0,} \qquad l = 1,\,2, \ldots ,\,m,\;m \geqslant 1

\]
, for a distinct sequence

μ
1

,

μ
2

,

…

∈
Ω

{\mu _1},\,{\mu _2},\, \ldots \; \in \Omega \,

. Necessary and sufficient conditions for existence and uniqueness are established, as well as explicit determinantal and integral representations. We also consider loci of zeros, existence of Rodrigues-type formulae and reducibility to standard orthogonality. The paper is accompanied by several examples of biorthogonal systems.
Asymptotic periodicity of the iterates of positivity preserving operators
Authors: Miklavčič M. 
Published: June 2010

Assume that (A1)

X
X

is a real Banach space. (A2)

X
+

{X^ + }

is a closed subset of

X
X

with the following properties: (i) if

x
∈

X
+

x \in {X^ + }

,

y
∈

X
+

y \in {X^ + }

,

α
∈
[
0
,

∞
)

\alpha \in [0,\,\infty )

then

x
+
y
∈

X
+

x + y \in {X^ + }

and

α
x
∈

X
+

\alpha x \in {X^ + }

; (ii) there exists

M
0

∈
(
0
,

∞
)

{M_0} \in (0,\,\infty )

such that for each

x
∈
X

x \in X

there exist

x
+

∈

X
+

{x_ + } \in {X^ + }

and

x
−

∈

X
+

{x_ - } \in {X^ + }

which satisfy
\[

x
=

x
+

−

x
−

,

|

|

x
+

|

|

⩽

M
0

|

|

x

|

|

,

|

|

x
−

|

|

⩽

M
0

|

|

x

|

|

x = {x_ + } - {x_ - },\qquad ||{x_ + }|| \leqslant {M_0}||x||,\qquad ||{x_ - }|| \leqslant {M_0}||x||

\]
and if

x
=

y
+

−

y
−

x = {y_ + } - {y_ - }

for some

y
+

∈

X
+

{y_ + } \in {X^ + }

,

y
−

∈

X
+

{y_ - } \in {X^ + }

then

y
+

−

x
+

∈

X
+

{y_ + } - {x_ + } \in {X^ + }

; (iii) if

x
∈

X
+

x \in {X^ + }

,

y
∈

X
+

y \in {X^ + }

then

|

|

x

|

|

⩽

|

|

x
+
y

|

|

||x|| \leqslant ||x + y||

. (A3)

B
B

is a bounded linear operator on

X
X

. (A4)

B

X
+

⊂

X
+

B{X^ + } \subset {X^ + }

. (A5)

F
0

{F_0}

is a nonempty compact subset of

X
X

and

lim

n
→
∞

dist
⁡
(

B
n

x
,

F
0

)
=
0

{\lim _{n \to \infty }}\operatorname {dist} ({B^n}x,\,{F_0}) = 0

whenever

x
∈

X
+

x \in {X^ + }

and

|

|

x

|

|

=
1

||x|| = 1

. Then

B
n

x

{B^n}x

is asymptotically periodic for every

x
∈
X

x \in X

. This, and other properties of

B
B

, are proven in the paper.
Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities
Authors: Castro Alfonso , Kurepa Alexandra 
Published: June 2010

Let

p
,
φ
:
[
0
,
T
]
→
R

p,\varphi :[0,T] \to R

be bounded functions with

φ
>
0

\varphi > 0

. Let

g
:

R

→

R

g:{\mathbf {R}} \to {\mathbf {R}}

be a locally Lipschitzian function satisfying the superlinear jumping condition: (i)

lim

u
→
−
∞

(
g
(
u
)

/

u
)
∈

R

{\lim _{u \to - \infty }}(g(u)/u) \in {\mathbf {R}}

(ii)

lim

u
→
∞

(
g
(
u
)

/

u

1
+
ρ

)
=
∞

{\lim _{u \to \infty }}(g(u)/{u^{1 + \rho }}) = \infty

for some

ρ
>
0

\rho > 0

, and (iii)

lim

u
→
∞

(
u

/

g
(
u
)

)

N

/

2

(
N
G
(
κ
u
)
−
(
(
N
−
2
)

/

2
)
u
⋅
g
(
u
)
)
=
∞

{\lim _{u \to \infty }}{(u/g(u))^{N/2}}(NG(\kappa u) - ((N - 2)/2)u \cdot g(u)) = \infty

for some

κ
∈
(
0
,
1
]

\kappa \in (0,1]

where

G
G

is the primitive of

g
g

. Here we prove that the number of solutions of the boundary value problem

Δ
u
+
g
(
u
)
=
p
(

‖
x
‖

)
+
c
φ
(

‖
x
‖

)

\Delta u + g(u) = p(\left \| x\right \|) + c\varphi (\left \| x\right \|)

for

x
∈

R

N

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

with

‖
x
‖

>
T
,
u
(
x
)
=
0

\left \| x\right \| > T,u(x) = 0

for

‖
x
‖

=
T

\left \| x\right \| = T

tends to

+
∞

+ \infty

when

c
c

tends to

+
∞

+ \infty

. The proofs are based on the "energy" and "phase plane" analysis.
On the Cauchy problem and initial traces for a degenerate parabolic equation
Authors: DiBenedetto E. , Herrero M. A. 
Published: June 2010

We consider the Cauchy problem (f)
\[

{

u
t

−
div
⁡
(

|

D
u

|

p
−
2

D
u
)
=
0

a
m
p
;

in

R

N

×
(
0
,
∞
)
,
p
>
2
,

u
(
x
,
0
)
=

u
0

(
x
)
,

a
m
p
;

x
∈

R

N

,

\left \{ {\begin {array}{*{20}{c}} {{u_t} - \operatorname {div}(|Du{|^{p - 2}}Du) = 0} \hfill & {{\text {in}}\;{{\mathbf {R}}^N} \times (0,\infty ),p > 2,} \hfill \\ {u(x,0) = {u_0}(x),} \hfill & {x \in {{\mathbf {R}}^N},} \hfill \\ \end {array} } \right .

\]
and discuss existence of solutions in some strip

S
T

≡

R

N

×
(
0
,
T
)

{S_T} \equiv {{\mathbf {R}}^N} \times (0,T)

,

0
>
T
≤
∞

0 > T \leq \infty

, in terms of the behavior of

x
→

u
0

(
x
)

x \to {u_0}(x)

as

|

x

|

→
∞

|x| \to \infty

. The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of

L

loc

1

(

R

N

)

L_{{\text {loc}}}^1({{\mathbf {R}}^N})

.
The generalized Borel conjecture and strongly proper orders
Authors: Corazza Paul 
Published: June 2010

The Borel Conjecture is the statement that

C
=

[

R

]

>

ω
1

C = {[\mathbb {R}]^{ > {\omega _1}}}

, where

C
C

is the class of strong measure zero sets; it is known to be independent of ZFC. The Generalized Borel Conjecture is the statement that

C
=

[

R

]

>

c

C = {[\mathbb {R}]^{ > {\mathbf {c}}}}

. We show that this statement is also independent. The construction involves forcing with an

ω
2

{\omega _2}

-stage iteration of strongly proper orders; this latter class of orders is shown to include several well-known orders, such as Sacks and Silver forcing, and to be properly contained in the class of

ω
\omega

-proper,

ω
ω

{\omega ^\omega }

-bounding orders. The central lemma is the observation that A. W. Miller’s proof that the statement

(

∗

)

({\ast })

"Every set of reals of power c can be mapped (uniformly) continuously onto

[
0
,
1
]

[0,1]

" holds in the iterated Sacks model actually holds in several other models as well. As a result, we show for example that

(

∗

)

({\ast })

is not restricted by the presence of large universal measure zero

(

U

0

)

({{\text {U}}_0})

sets (as it is by the presence of large

C
C

sets). We also investigate the

σ
\sigma

-ideal

J

=
{
X
⊂

R

:
X

cannot be mapped uniformly continuously onto

[
0
,
1
]
}

\mathcal {J} = \{ X \subset \mathbb {R}:X\;{\text {cannot be mapped uniformly continuously onto }}[0,1]\}

and prove various consistency results concerning the relationships between

J

,

U

0

\mathcal {J},\;{{\text {U}}_0}

, and AFC (where

AFC
=
{
X
⊂

R

:
X

is always first category}

\operatorname {AFC} = \{ X \subset \mathbb {R}:X\;{\text {is always first category\} }}

). These latter results partially answer two questions of J. Brown.
Fast method for computing the number of primes less than a given limit
Authors: Mapes David C. 
Published: June 2010

"Fast Method for Computing the Number of Primes Less Than a Given Limit” describes three processes used during the course of calculation. In the first part of the paper the author proves:
\[

ϕ
(
x
,
a
)
=
ϕ
(
x
,
1
)
−
ϕ
(

x

p
2

,
1

)
−
ϕ
(

x

p
3

,
2

)
−
…
−
ϕ
(

x

p
a

,
a
−
1

)

\phi (x,a) = \phi (x,1) - \phi ({\frac {x}{{{p_2}}},1}) - \phi ({\frac {x}{{{p_3}}},2}) - \ldots - \phi ({\frac {x}{{{p_a}}},a - 1})

\]
where

ϕ
(
x
,
a
)

\phi (x,a)

represents the number of numbers less than or equal to x and not divisible by the first “a” primes. This identity is used to evaluate the formula

π
(
x
)
=
ϕ
(
x
,
a
)
+
a
−
1

\pi (x) = \phi (x,a) + a - 1

,

a
+
1
>
π
(

x

)

a + 1 > \pi (\sqrt x )

where resulting terms of the form

ϕ
(

x
′

,

a
′

)

\phi (x’,a’)

are broken down still further by the previously described method, or numerically evaluated using one or both of two other identities, the choice being dependent on

x
′

x’

and

a
′

a’

. Following the paper is a table of calculations made using this process which gives the values of

π
(
x
)

\pi (x)

for x at intervals of 10 million up to 1000 million, along with the Riemann and the Chebyshev approximations for

π
(
x
)

\pi (x)

and the amount they deviate from the true count.
Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators
Authors: Carasso Alfred 
Published: June 2010

In this paper we analyze the convergence of a centered finite-difference approximation to the nonselfadjoint Sturm-Liouville eigenvalue problem where[unk][{\text {unk}}]has smooth coefficients anda(x)≧a0>0a(x) \geqq {a_0} > 0on [0, 1]. We show that the rate of convergence isO(Δx2)O(\Delta {x^2})as in the selfadjoint case for a scheme of the same accuracy. We also establish discrete analogs of theSturmoscillation and comparison theorems. As a corollary we obtain the result\[limsupM→∞;Δx→0;(M+1)Δx=1{∑p=1M||Vp||∞Λp}>∞\lim \sup \limits _{M \to \infty ;{\Delta _x} \to 0;(M + 1){\Delta _x} = 1} \left \{ {\sum \limits _{p = 1}^M {\frac {{||{V^p}||\infty }} {{{\Lambda _p}}}} } \right \} > \infty\]) whereΔx=1/(M+1)\Delta x = 1/(M + 1)is the mesh size andΛp,Vp{\Lambda _p},{V^p}are the characteristic pairs ofLL, theM×MM \times Mmatrix which approximates[unk][{\text {unk}}], andVp{V^p}is normalized so that||Vp||2=1||{V^p}|{|_2} = 1.
Some relations between the values of a function and its first derivative at 𝑛 abscissa points
Authors: Huddleston Robert E. 
Published: June 2010

For a polynomial, P, of degree

2
n
−
2

2n - 2

, there exists a relation between the values of P and the values of its first derivative,

P
′

P’

, at the n abscissa points

x
1

,
⋯
,

x
n

{x_1}, \cdots ,{x_n}

,
\[

∑

i
=
1

n

[

a
i

P
(

x
i

)
+

b
i

P
′

(

x
i

)
]
=
0.

\sum \limits _{i = 1}^n {[{a_i}P({x_i}) + {b_i}P’({x_i})] = 0.}

\]
Replacing P by a differentiable function y yields
\[

∑

i
=
1

n

[

a
i

y
(

x
i

)
+

b
i

y
′

(

x
i

)
]
=
E
(
y
,
x
)
.

\sum \limits _{i = 1}^n {[{a_i}y({x_i}) + {b_i}y’({x_i})] = E(y,x).}

\]
These relations are obtained and the error function

E
(
y
,
x
)

E(y,x)

is given explicitly.
A new error analysis for a cubic spline approximate solution of a class of Volterra integro-differential equations
Authors: Guzek Joseph A. , Kemper Gene A. 
Published: June 2010

In this paper a third-order numerical method is considered which utilizes a twice continuously differentiable third degree spline to approximate the solution of
\[

x
˙

(
t
)
=
F

(

t
,
x
(
t
)
,

∫
a
t

K
(
t
,
u
,
x
(
u
)
)

d
u

)

,

x
(
a
)
=

x
0

,

\begin {array}{*{20}{c}} {\dot x(t) = F\left ( {t,x(t),\int _a^t {K(t,u,x(u))\;du} } \right ),} \hfill \\ {x(a) = {x_0},} \hfill \\ \end {array}

\]
at discrete points in the interval [a, b]. The error analysis uses a technique usually associated with linear multistep methods.
Irregularities in the distribution of primes and twin primes
Authors: Brent Richard P. 
Published: June 2010

The maxima and minima of

⟨
L
(
x
)
⟩
−
π
(
x
)
,
⟨
R
(
x
)
⟩
−
π
(
x
)

\langle L(x)\rangle - \pi (x),\langle R(x)\rangle - \pi (x)

, and

⟨

L
2

(
x
)
⟩
−

π
2

(
x
)

\langle {L_2}(x)\rangle - {\pi _2}(x)

in various intervals up to

x
=
8
×

10

10

x = 8 \times {10^{10}}

are tabulated. Here

π
(
x
)

\pi (x)

and

π
2

(
x
)

{\pi _2}(x)

are respectively the number of primes and twin primes not exceeding

x
,
L
(
x
)

x,L(x)

is the logarithmic integral,

R
(
x
)

R(x)

is Riemann’s approximation to

π
(
x
)

\pi (x)

, and

L
2

(
x
)

{L_2}(x)

is the Hardy-Littlewood approximation to

π
2

(
x
)

{\pi _2}(x)

. The computation of the sum of inverses of twin primes less than

8
×

10

10

8 \times {10^{10}}

gives a probable value

1.9021604
±
5
×

10

−
7

1.9021604 \pm 5 \times {10^{ - 7}}

for Brun’s constant.