bluuebunny/crossref_metadata_2025 huggingface dataset:
An integral representation for generalized temperatures in two space variables
Authors: Haimo Deborah Tepper
Published: June 2010

An integral representation is derived for a function which satisfies the generalized heat equation in one of the space variables and the adjoint generalized heat equation in the other space variable.
Extending uniformly continuous pseudo-ultrametrics and uniform retracts
Authors: Ellis Robert L.
Published: June 2010

It is first proved that any uniformly continuous pseudo-ultrametric on a subspace of a non-Archimedean uniform space X has a uniformly continuous extension to X (which preserves total boundedness or separability). Then it is proved that every complete subspace of an ultrametrizable space X is a uniform retract of X. This has consequences concerning the extension of uniformly continuous functions.
Stable homeomorphisms on infinite-dimensional normed linear spaces.
Authors: Curtis D. W. , McCoy R. A.
Published: June 2010

R. Y. T. Wong has recently shown that all homeomorphisms on a connected manifold modeled on infinite-dimensional separable Hilbert space are stable. In this paper we establish the stability of all homeomorphisms on a normed linear space

E
E

such that

E
E

is homeomorphic to the countable infinite product of copies of itself. The relationship between stability of homeomorphisms and a strong annulus conjecture is demonstrated and used to show that stability of all homeomorphisms on a normed linear space

E
E

implies stability of all homeomorphisms on a connected manifold modeled on

E
E

, and that in such a manifold collared

E
E

-cells are tame.
On the vanishing of 𝐸𝑥𝑡
Authors: Ramras Mark
Published: June 2010

In this paper we exhibit certain modules

A
A

over a commutative noetherian local ring

(
R
,

M

)

(R,\mathfrak {M})

which test projective dimension of finitely generated modules in the following sense: if

Ext
j

(
M
,
A
)
=
0

{\operatorname {Ext} ^j}(M,A) = 0

for all

j
≧
i

j \geqq i

, then pd

M
>
i

M > i

. We also show that the module

M

\mathfrak {M}

tests in a stronger way: if

Ext
i

(
M
,

M

)
=
0

{\operatorname {Ext} ^i}(M,\mathfrak {M}) = 0

, then pd

M
>
i

M > i

. In conclusion we show that if

(
R
,

M

)

(R,\mathfrak {M})

is artin, then

R
R

is self-injective if and only if

Ext
1

(
R

/

M

n

,
R
)
=
0

{\operatorname {Ext} ^1}(R/{\mathfrak {M}^n},R) = 0

, where the index of nilpotence of

M

\mathfrak {M}

is

n
+
1

n + 1

.
On the solutions of a squence of Lamé differential equations
Authors: Wong C. K.
Published: June 2010

In a paper by H. Triebel, the solutions of some Lamé differential equations are interpreted in terms of conformal mappings. In the present paper, they are interpreted and constructed as covering projections from the unit disk onto Riemann surfaces with signature. Furthermore, the continuous dependence of solutions on the coefficient is established.
A fixed point theorem for mappings with a nonexpansive iterate
Authors: Kirk W. A.
Published: June 2010

Let X be a reflexive Banach space which has strictly convex norm and suppose K is a nonempty, bounded, closed and convex subset of X. Suppose

T
:
K
→
K

T:K \to K

has the property that, for some positive integer

N
,

T
N

N,{T^N}

is nonexpansive (

‖

T
N

x
−

T
N

y

‖

≦

‖

x
−
y

‖

\left \| {{T^N}x - {T^N}y} \right \| \leqq \left \| {x - y} \right \|

for all

x
,
y
∈
K

x,y \in K

). A function

γ
(
N
)

\gamma (N)

is determined,

γ
(
N
)
>
1

\gamma (N) > 1

, such that if

‖

T
j

x
−

T
j

y

‖

≦
k

‖

x
−
y

‖

\left \| {{T^j}x - {T^j}y} \right \| \leqq k\left \| {x - y} \right \|

for all

x
,
y
∈
K
,
1
≦
j
≦
N
−
1

x,y \in K,1 \leqq j \leqq N - 1

, where

k
>
γ
(
N
)

k > \gamma (N)

, then T has a fixed point in K.
An extremal property of simplices
Authors: Butler G. J.
Published: June 2010

It is shown that the class of n-dimensional simplices is an extremal for the problem of maximizing the volume of the minimum circumscribing cylinder to an n-dimensional convex body of unit volume.
Associative 𝐻-spaces with small ranks
Authors: Hubbuck J. R.
Published: June 2010

Let X be a finite connected CW-complex which is a topological monoid. It is shown that if the rank of X is less than five, then the rational type of X is that of a Lie group.
Central separable algebras which are locally endomorphism rings of free modules
Authors: Auslander Bernice L.
Published: June 2010

The object of this paper is to study the kernel of the map of the Brauer group of an integrally closed noetherian domain A into the direct product of the Brauer groups of the localizations of A at prime ideals. It is shown that this kernel is isomorphically contained in the torsion subgroup of the first cohomology group of the sheaf of Cartier divisors over Spec A. As a consequence, the author describes several new sets of conditions on A which guarantee that the kernel is trivial.
On a theorem of Malcev
Authors: Lallement Gérard
Published: June 2010

For any pair of distinct elements a, b in a finitely generated abelian semigroup S, we indicate what are the homomorphisms

ϕ
\phi

of S onto a finite semigroup such that

ϕ
(
a
)
≠
ϕ
(
b
)

\phi (a) \ne \phi (b)

. This improves a previous result of Malcev which states that the considered semigroups are residually finite.
Property 𝐿 and direct integral decompositions of 𝑊-∗algebras
Authors: Willig Paul
Published: June 2010

If

A

\mathcal {A}

is a type

I
I

∞

W
-
∗

\mathrm {II}_\infty W\text {-}*

algebra on separable Hilbert space

H
,

A

H,\mathcal {A}

is spatially isomorphic to

B

⊗
B
(
K
)
,

B

\mathfrak {B} \otimes B(K),\mathfrak {B}

of type

I
I

1

\mathrm {II}_1

, K a separable Hilbert space. If

A

(
λ
)

\mathcal {A}(\lambda )

are the factors in the direct integral decomposition of

A

\mathcal {A}

, the set

L

=
{
λ

|

A

(
λ
)

\mathfrak {L} = \{ \lambda |\mathcal {A}(\lambda )

has property L} is

μ
\mu

-measurable, and

A

\mathcal {A}

has property L iff

μ
(
Λ
−

L

)
=
0

\mu (\Lambda - \mathfrak {L}) = 0

.
Minimal hypersurfaces in an 𝑚-sphere
Authors: Chen Bang-yen
Published: June 2010

(1) A submanifold

M
n

{M^n}

of a euclidean space

E

n
+
2

{E^{n + 2}}

of codimension 2 is a pseudo-umbilical submanifold with constant mean curvature if and only if it is a minimal hypersurface of a hypersphere of

E

n
+
2

{E^{n + 2}}

. (2) A complete oriented minimal surface

M
2

{M^2}

of a 3-sphere

S
3

{S^3}

on which the Gauss curvature does not change its sign is either an equatorial sphere or a Clifford flat torus.
Bounds for the determinant of the sum of hermitian matrices
Authors: Fiedler Miroslav
Published: June 2010

Best possible lower and upper bounds for the determinant of the sum of two hermitian matrices in terms of the eigenvalues of both matrices are obtained.
The total symbol of a pseudo-differential operator
Authors: Petersen Bent E.
Published: June 2010

This note presents a direct completely coordinate-free proof that the symbols of a pseudo-differential operator are differential operators whose coefficients are functions on the higher order cotangent bundles.
Remark on Siegel domains of type 𝐼𝐼𝐼
Authors: Wolf Joseph A.
Published: June 2010

Bounded symmetric domains have standard realizations as “Siegel domains of type III.” Pjateckiĭ-Šapiro has introduced a more restrictive notion of “Siegel domain of type III.” Here we give a direct proof that those standard realizations satisfy the additional conditions of the new definition.
Koecher’s principle for quadratic Jordan algebras
Authors: McCrimmon Kevin
Published: June 2010

In this note we indicate two techniques for establishing identities in quadratic Jordan algebras. The first method, due to Professor M. Koecher, shows that to establish an identity in general it suffices to establish it when all the elements involved are invertible. The second technique involves interpreting a given identity in a Jordan algebra as a simpler identity in a homotope of that algebra. These two techniques are applied to derive some important identities.
An asymptotic property of the roots of polynomials
Authors: Flaschka Hermann
Published: June 2010

It is shown that if the imaginary parts of the roots

λ
j

(
s
)

{\lambda _j}(s)

of a polynomial

P
(
λ
,
s
)
,
s
∈

R
n

P(\lambda ,s),s \in {R^n}

, are unbounded for large

|

s

|

|s|

, then they are in fact unbounded along a one-parameter algebraic curve

s
=
s
(
R
)

s = s(R)

. The result may be used to reduce certain questions about polynomials in several variables to an essentially one-dimensional form; this is illustrated by an application to hyperbolic polynomials.
On 𝐻-spaces \modℭ_{𝔭}
Authors: Haslam H. B.
Published: June 2010

We show that if a space X is an H-space

mod

C

p

{\bmod \; \mathfrak {C}_p}

, then X is “dominated” by the reduced product space

X
∞

(
=
Ω
Σ
X
)

{X_\infty }( = \Omega \Sigma X)

of X and also by the component of the identity map in the space of all maps from X to X. We then deduce algebraic facts about the suspension homomorphism and the homomorphisms induced by the evaluation map. Making use of a new and pretty result of S. Weingram we give a short proof of the result of W. Browder that the

mod

p

\bmod \;p

Hurewicz homomorphism is zero in even dimensions for an H-space

mod

C

p

\bmod \; {\mathfrak {C}_p}

.
Quotient full linear rings
Authors: Hutchinson John J.
Published: June 2010

We define a ring

R
R

to be an FL (full linear) ring if

R
R

is isomorphic to the full ring of linear transformations of a left vector space over a division ring.

R
R

is QFL if its left maximal quotient ring is an FL ring. In this paper we give necessary and sufficient conditions for a ring to be a QFL ring. We also generalize some results of Chase and Faith concerning subdirect sum decompositions of rings whose left maximal quotient ring is the direct product of FL rings.
Principal ideals in the ideal lattice
Authors: Subramanian H.
Published: June 2010

We show that there cannot be a definition of “principal elements” in the theory of multiplicative lattices so that the notion of principal elements concurs with the notion of principal ideals when interpreted in the ideal lattices of rings.
On the smoothness of eigenfunctions of hyponormal singular integral operators
Authors: Clancey Kevin
Published: June 2010

Fix

φ
∈

L
∞

(
E
)

\varphi \in {L^\infty }(E)

and let

E
⊂
R

E \subset R

be bounded and measurable; for

1
>
p
>
∞

1 > p > \infty

consider the bounded linear operator
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T f left-parenthesis s right-parenthesis equals s f left-parenthesis s right-parenthesis plus StartFraction phi left-parenthesis s right-parenthesis Over pi EndFraction integral Subscript upper E Superscript asterisk Baseline StartFraction ModifyingAbove phi With bar left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis Over t minus s EndFraction d t a period e period s element-of upper E">

T
f
(
s
)
=
s
f
(
s
)
+

φ
(
s
)

π

∫
E
∗

φ
¯

(
t
)
f
(
t
)

t
−
s

d
t

a

.e

. 

s
∈
E

Tf(s) = sf(s) + \frac {{\varphi (s)}}{\pi }\int _E^\ast \frac {{\bar \varphi (t)f(t)}}{{t - s}}dt\;{\mkern 1mu} {\text {a}}{\text {.e}}{\text {. }}s \in E

\]
where

f
∈

L
p

(
E
)

f \in {L^p}(E)

. If

ν
=
λ
+
i
μ
∈
C

\nu = \lambda + i\mu \in C

then there are no nonzero

L
p

(
E
)

{L^p}(E)

solutions of

T
f
=
ν
f

Tf = \nu f

for

p
>
2

p > 2

in case

λ
\lambda

is a point of positive Lebesgue density in the complement of

E
E

.
Extending congruence relations
Authors: Krauss Peter
Published: June 2010

If

A

\mathfrak {A}

and

B

\mathfrak {B}

are algebras, where

A

⊆

B

\mathfrak {A} \subseteq \mathfrak {B}

, and

θ
\theta

is a congruence relation on

A

\mathfrak {A}

, let

θ

B

{\theta ^\mathfrak {B}}

be the smallest congruence relation on

B

\mathfrak {B}

containing

θ
\theta

.

A

\mathfrak {A}

is called a congruence subalgebra of

B

\mathfrak {B}

if

A

⊆

B

\mathfrak {A} \subseteq \mathfrak {B}

and, for every congruence relation

θ
\theta

on

A

,

θ

B

∩

|

A

|

2

=
θ

\mathfrak {A},{\theta ^\mathfrak {B}} \cap |\mathfrak {A}{|^2} = \theta

. Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous to those for elementary subalgebras. Consequently we obtain full analogues of the Jónsson-Morley-Vaught results concerning homogeneous-universal algebras, where the notion of “subalgebra” is everywhere replaced by “congruence subalgebra".
Monotonicity of positive semidefinite Hermitian matrices
Authors: Merris Russell , Pierce Stephen
Published: June 2010

Inequalities which compare elements of the convex cone of positive semidefinite hermitian matrices with products of roots of elements are proved. They yield inequalities for Schur functions (generalized matrix functions) which, when specialized to the determinant, give a result of R. Bellman and L. Mirsky.
Another fixed point theorem for plane continua
Authors: Hagopian Charles L.
Published: June 2010

A continuum

M
M

is said to be

λ
\lambda

connected if every two points of

M
M

can be joined by a hereditarily decomposable subcontinuum of

M
M

. Here we prove that a bounded plane continuum that does not have infinitely many complementary domains is

λ
\lambda

connected if and only if its boundary does not contain an indecomposable continuum. It follows that every

λ
\lambda

connected bounded nonseparating subcontinuum of the plane has the fixed point property.
Pseudo-isotopies of arcs and knots
Authors: Sikkema Carl D.
Published: June 2010

The purpose of this paper is to show that if an arc or simple closed curve contains a nontrivial Wilder arc, then it is not possible to transform a straight line segment onto the arc or a knot onto the simple closed curve. The proof uses the fact that every knot has a unique finite decomposition into prime knots.
A property of arithmetic sets
Authors: Tanaka Hisao
Published: June 2010

We shall show that every nonempty countable arithmetic subset of

N
N

{N^N}

contains at least one element

α
\alpha

such that the singleton

{
α
}

\{ \alpha \}

itself is arithmetic. The proof is carried out by using a method in classical descriptive set theory.
A generalization of Mori’s theorem
Authors: Lu Chin-pi
Published: June 2010

In this article, we consider a generalization of Mori’s theorem which is: Let

R
R

be a Zariski ring; if the completion of

R
R

is a unique factorization domain, then so is

R
R

.
Divinsky’s radical
Authors: Stewart Patrick N.
Published: June 2010

Let

F
F

and

R
R

be rings,

M
M

an

F
−
R

F - R

-bimodule, and

Δ
\Delta

the largest

F
F

-submodule

N
N

of

M
M

such that for each

x
∈
N
,
f
x
=
x

x \in N,fx = x

for some

f
∈
F

f \in F

. (1) If either

F
F

or

M
M

satisfies the minimum condition then

Δ
=

F
k

M

\Delta = {F^k}M

for some positive integer

k
k

; provided that whenever

x
∈

F
ω

M
=

∩

n
=
1

∞

(

F
n

M
)

x \in {F^\omega }M = \cap _{n = 1}^\infty ({F^n}M)

and

F
x
⊆
Δ

Fx \subseteq \Delta

, then

x
∈
Δ

x \in \Delta

. (2) If

M
M

satisfies the maximum condition and

F
=
(

f
1

,
⋯
,

f
n

)

F = ({f_1}, \cdots ,{f_n})

where

f
1

,
⋯
,

f
n

{f_1}, \cdots ,{f_n}

is a normalising set of generators (that is,
\[

f
i

F
=
F

f
i

modulo
⁡
(

f
1

,
⋯
,

f

i
−
1

)

{f_i}F = F{f_i}\operatorname {modulo} ({f_1}, \cdots ,{f_{i - 1}})

\]
for each

i
=
1
,
⋯
,
n
)

i = 1, \cdots ,n)

, then

Δ
=

F
ω

M

\Delta = {F^\omega }M

. (3) If

M
=
F
=
R
,
Δ
=
(
0
)

M = F = R,\Delta = (0)

,

R
R

satisfies the maximum condition, and

R
R

has a normalising set of generators, then

R
R

can be embedded in a Jacobson radical ring.
An extension of the Noether-Deuring theorem
Authors: Roggenkamp Klaus W.
Published: June 2010

Let

R
R

be a commutative semilocal noetherian ring,

Λ
\Lambda

a left noetherian

R
R

-algebra and

M
,
N

M,N

finitely generated left

Λ
\Lambda

-modules such that

End
Λ

(
M
)

{\operatorname {End} _\Lambda }(M)

is of finite type over

R
R

. By

R
^

\hat R

we denote the

(
rad
⁡
R
)

(\operatorname {rad} R)

-adic completion of

R
R

. Theorem.

M
M

is

Λ
\Lambda

-isomorphic to a direct summand of

N
N

iff

R
^

⊗
R

M

\hat R{ \otimes _R}M

is

R
^

⊗
R

Λ

\hat R{ \otimes _R}\Lambda

-isomorphic to a direct summand of

R
^

⊗
R

N

\hat R{ \otimes _R}N

. This result is used to prove a generalization of the Noether-Deuring theorem. Let

S
S

be a commutative

R
R

-algebra which is a faithful projective

R
R

-module of finite type; then

M
M

is

Λ
\Lambda

-isomorphic to direct summand of

N
N

iff

S

⊗
R

M

S{ \otimes _R}M

is

S

⊗
R

Λ

S{ \otimes _R}\Lambda

-isomorphic to a direct summand of

S

⊗
R

N

S{ \otimes _R}N

.
On asymptotic behavior of perturbed nonlinear systems
Authors: Fennell R. E. , Proctor T. G.
Published: June 2010

A version of the variation of constants formula for nonlinear systems is used to study the comparative asymptotic behavior of the systems

x
′

=
f
(
t
,
x
)

x’ = f(t,x)

and

y
′

=
f
(
t
,
y
)
+
g
(
t
,
y
)

y’ = f(t,y) + g(t,y)

.
An index for set-valued maps in infinite-dimensional spaces
Authors: Williams Stephen A.
Published: June 2010

Previous fixed point indexes defined for a set-valued map in an infinite-dimensional space have required the values of this map to be convex sets. The corresponding assumption of this paper is that the values be (co-)acyclic sets, i.e., that the reduced Alexander cohomology group of each of these sets be trivial in each dimension. Other assumptions are that the space is locally convex and that the map is compact and upper semicontinuous with no fixed points on the boundary of its domain. The index is defined, proved to be homotopy invariant, and proved to vanish in case there are no fixed points. The main methods used are finite-dimensional approximation and the Vietoris-Begle mapping theorem.
The Wall invariant of certain 𝑆¹ bundles
Authors: Anderson Douglas R.
Published: June 2010

Let

p
:
E
→
B

p:E \to B

be a principal

S
1

{S^1}

bundle with

B
B

dominated by a finite complex. Then it is easy to show that

E
E

is also dominated by a finite complex. In this paper we show, under suitable additional hypotheses, that in fact

E
E

has the homotopy type of a finite complex. The proof is carried out by computing Wall’s finiteness obstruction for

E
E

.
Elementary extensions of linear topological abelian groups
Authors: Phillips R. G. , Sperry P. L.
Published: June 2010

R. MacDowell and E. Specker obtain a structure theorem for elementary extensions of the integers by considering a certain residue mapping. In this paper we characterize those abelian groups in which an analogous situation exists and obtain the MacDowell-Specker result as a special case of our theory.
Starlike meromorphic functions
Authors: Miller James
Published: June 2010

In this paper we study meromorphic univalent functions which map the unit disk onto the exterior of a domain which is starlike with respect to some finite point different from the origin. We obtain bounds on the arc length, an integral representation, and bounds on the maximum modulus of starlike meromorphic functions.
A uniform asymptotic turning point theory for second order linear ordinary differential equations
Authors: Zauderer Erich
Published: June 2010

The method of Cherry for obtaining uniform asymptotic solutions for a second order linear ordinary differential equation with a single turning point of first order is formally extended to the case where the equation has an arbitrary number of turning points of various orders. This follows a recent extension by Lynn and Keller of Langer’s method to deal with the aforementioned more general problem.
On the countably generated 𝑧-ideals of 𝐶(𝑋)
Authors: De Marco G.
Published: June 2010

A necessary and sufficient condition for the countable generation of certain

z
z

-ideals of

C
(
X
)

C(X)

is given. In particular, for

X
X

compact, the countably generated

z
z

-ideals of

C
(
X
)

C(X)

are the sets of all functions which vanish on a neighborhood of some zero-set of

X
X

. Any finitely generated semiprime ideal of

C
(
X
)

C(X)

is generated by an idempotent.
Homotopical nilpotence of the seven sphere
Authors: Gilbert William J.
Published: June 2010

We prove that the homotopical nilpotence of

S
7

{S^7}

is 3, with respect to any of its 120 H-space multiplications.
Which abelian groups can support a directed, interpolation order?
Authors: Glass A. M. W.
Published: June 2010

We prove that an abelian group can support a directed, interpolation order if and only if it is torsion-free or its quotient by its torsion subgroup is noncyclic. The proof is of an elementary nature. As a consequence of the proof, it is also shown that an abelian group can support a directed, interpolation order if and only if it can support a directed, interpolation, weakly semi-isolated order. The paper is completely self-contained so as to be readable by nonspecialists.
Bi-unitary perfect numbers
Authors: Wall Charles R.
Published: June 2010

Let d be a divisor of a positive integer n. Then d is a unitary divisor if d and n/d are relatively prime, and d is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1. An integer is bi-unitaty perfect if it equals the sum of its proper bi-unitary divisors. The purpose of this paper is to show that there are only three bi-unitary perfect numbers, namely 6, 60 and 90.
On the exponent of the ideal class groups of complex quadratic fields.
Authors: Boyd David W. , Kisilevsky H.
Published: June 2010

Let

m
(
d
)

m(d)

denote the exponent of the ideal class group of the complex quadratic field

Q
(
√
d
)

Q(\surd d)

, where

d
>
0

d > 0

is a fundamental discriminant. It is shown that there are only finitely many

d
d

for which

m
(
d
)
=
3

m(d) = 3

. Assuming the extended Riemann Hypothesis, it is shown that

m
(
d
)
→
∞

 as 

|

d

|

→
∞

m(d) \to \infty {\text { as }}|d| \to \infty

.
On high order derivations of fields
Authors: Mordeson J. N. , Vinograde B.
Published: June 2010

Let

D

(
L

/

K
)

\mathcal {D}(L/K)

denote the derivation algebra of a field extension

L

/

K

L/K

of prime characteristic. If

L

/

K

L/K

is purely inseparable and has an exponent, then every intermediate field F of

L

/

K

L/K

equals the center of

D

(
L

/

F
)

\mathcal {D}(L/F)

. Here we prove the converse of this statement.
Positive transformations restricted to subspaces and inequalities among their proper values
Authors: Amir-Moéz A. R. , Perry C. R.
Published: June 2010

Let A be a positive Hermitian transformation on an n-dimensional unitary space

E
n

{E_n}

with proper values

a
1

≧
⋯
≧

a
n

{a_1} \geqq \cdots \geqq {a_n}

. Let

b
1

≧
⋯
≧

b
k

{b_1} \geqq \cdots \geqq {b_k}

be the proper values of

A

|

M

A|M

, where M is a proper subspace of

E
n

{E_n}

and

c
1

≧
⋯
≧

c
h

{c_1} \geqq \cdots \geqq {c_h}

be the proper values of

A

|

M
⊥

A|{M^ \bot }

. Let

i
1

>
⋯
>

i
r

{i_1} > \cdots > {i_r}

and

j
1

>
⋯
>

j
r

{j_1} > \cdots > {j_r}

be sequences of positive integers, with

i
r

≦
k

{i_r} \leqq k

and

j
r

≦
h

{j_r} \leqq h

. Then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis b Subscript i 1 Baseline midline-horizontal-ellipsis b Subscript i Sub Subscript r Subscript Baseline right-parenthesis dot left-parenthesis c Subscript j 1 Baseline midline-horizontal-ellipsis c Subscript j Sub Subscript r Subscript Baseline right-parenthesis greater-than-over-equals left-parenthesis a Subscript n minus r plus 1 Baseline midline-horizontal-ellipsis a Subscript n Baseline right-parenthesis left-parenthesis a Subscript i 1 plus j 1 minus 1 Baseline midline-horizontal-ellipsis a Subscript i Sub Subscript r Subscript plus j Sub Subscript r Subscript minus 1 Baseline right-parenthesis">

(

b

i
1

⋯

b

i
r

)
⋅
(

c

j
1

⋯

c

j
r

)
≧
(

a

n
−
r
+
1

⋯

a
n

)
(

a

i
1

+

j
1

−
1

⋯

a

i
r

+

j
r

−
1

)

({b_{{i_1}}} \cdots {b_{{i_r}}}) \cdot ({c_{{j_1}}} \cdots {c_{{j_r}}}) \geqq ({a_{n - r + 1}} \cdots {a_n})({a_{{i_1} + {j_1} - 1}} \cdots {a_{{i_r} + {j_r} - 1}})

. In this article generalizations of this inequality have been studied.
On groups of exponent four. II
Authors: Gupta C. K. , Gupta N. D.
Published: June 2010

C. R. B. Wright has shown that the nilpotency class of an

n
n

-generator group of exponent four is at most

3
n
−
1

3n - 1

. This bound is believed to be too large for higher values of

n
n

. In this note it is shown that if for some large enough integer

n
n

this bound can be improved to

[
2

1
2

n
]
−
1

[2\tfrac {1}{2}n] - 1

, then the free group of exponent four of infinite rank is solvable.
Injective hulls of certain 𝑆-systems over a semilattice
Authors: Johnson C. S. , McMorris F. R.
Published: June 2010

We construct, in the category of S-systems over a semilattice, the injective hulls of S-systems which are homomorphic images of S-subsystems of S.
An inequality of Turán type for Jacobi polynomials
Authors: Gasper George
Published: June 2010

For Jacobi polynomials

P
n

(
α
,
β
)

(
x
)
,
α
,
β
>
−
1

P_n^{(\alpha ,\beta )}(x),\alpha ,\beta > - 1

, let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript n Baseline left-parenthesis x right-parenthesis equals StartFraction upper P Subscript n Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis Over upper P Subscript n Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis 1 right-parenthesis EndFraction comma normal upper Delta Subscript n Baseline left-parenthesis x right-parenthesis equals upper R Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis minus upper R Subscript n minus 1 Baseline left-parenthesis x right-parenthesis upper R Subscript n plus 1 Baseline left-parenthesis x right-parenthesis period">

R
n

(
x
)
=

P
n

(
α
,
β
)

(
x
)

P
n

(
α
,
β
)

(
1
)

,

Δ
n

(
x
)
=

R
n
2

(
x
)
−

R

n
−
1

(
x
)

R

n
+
1

(
x
)
.

{R_n}(x) = \frac {{P_n^{(\alpha ,\beta )}(x)}}{{P_n^{(\alpha ,\beta )}(1)}},\quad {\Delta _n}(x) = R_n^2(x) - {R_{n - 1}}(x){R_{n + 1}}(x).

\]
We prove that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript n Baseline left-parenthesis x right-parenthesis greater-than-over-equals StartFraction left-parenthesis beta minus alpha right-parenthesis left-parenthesis 1 minus x right-parenthesis Over 2 left-parenthesis n plus alpha plus 1 right-parenthesis left-parenthesis n plus beta right-parenthesis EndFraction upper R Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis comma negative 1 less-than-over-equals x less-than-over-equals 1 comma n greater-than-over-equals 1 comma">

Δ
n

(
x
)
≧

(
β
−
α
)
(
1
−
x
)

2
(
n
+
α
+
1
)
(
n
+
β
)

R
n
2

(
x
)
,

−
1
≦
x
≦
1
,
n
≧
1
,

{\Delta _n}(x) \geqq \frac {{(\beta - \alpha )(1 - x)}}{{2(n + \alpha + 1)(n + \beta )}}R_n^2(x),\quad - 1 \leqq x \leqq 1,n \geqq 1,

\]
with equality only for

x
=
±
1

x = \pm 1

. This shows that the Turán inequality

Δ
n

(
α
)
≧
0
,
−
1
≦
x
≦
1

{\Delta _n}(\alpha ) \geqq 0, - 1 \leqq x \leqq 1

, holds if and only if

β
≧
α
>
−
1

\beta \geqq \alpha > - 1

.
A conjugacy criterion
Authors: Stevens Robert R.
Published: June 2010

A sufficient condition is given for the differential equation

x
+
p
(
t
)
x
=
0

x + p(t)x = 0

to have a nontrivial solution with more than one zero in a closed interval [a, b]. A partial converse is obtained and applications are given.
A representation theorem for 𝐿^{𝑝} spaces
Authors: Kanter Marek
Published: June 2010

Using the theory of symmetric stable process of index

p
∈
(
0
,
2
]

p \in (0,2]

, we prove that if a sepaéable Frechet space

L
L

has all its finite dimensional subspaces linearly isometric with a subspace of

L
p

[
0
,
1
]

{L^p}[0,1]

then

L
L

itself is linearly isometric with a subspace of

L
p

[
0
,
1
]

{L^p}[0,1]

.
Cyclic vectors of a cyclic operator span the space
Authors: Gehér L.
Published: June 2010

If a bounded linear operator on a Banach space has a cyclic vector, then the set of cyclic vectors spans the space.
Some fixed point results for 𝑈𝑉 decompositions of compact metric spaces
Authors: Cobb John , Voxman William
Published: June 2010

In this paper the preservation of the fixed point property under UV decompositions is studied. It is shown that if K is an n-dimensional complex with the fixed point property and G is

U

V

n
−
1

U{V^{n - 1}}

decomposition of K, then K/G also will have the fixed point property. Furthermore, if X is a compact metric space with the fixed point property, and G is a

U

V
n

U{V^n}

decomposition of X such that X/G may be embedded in a suitably small Euclidian space,

R
m

{R^m}

, then X/G retains the fixed point property.
A product theorem for 𝐻-group fibrations
Authors: Croom F. H.
Published: June 2010

Let (

(
E
,
p
,
B
)

(E,p,B)

) and (

(

E
′

,

p
′

,
B
)

(E’,p’,B)

) be

H
H

-group fibrations over

B
B

with basic fibers

F
F

and

F
′

F’

respectively. If there are base point preserving fiber maps

f
:
E
⇄

E
′

:
g

f:E \rightleftarrows E’:g

such that

f
f

is a fiber

H
H

-homomorphism, then

E
×

F
′

E \times F’

and

E
′

×
F

E’ \times F

have the same homotopy type.
A decomposition theorem for closed compact connected P. L. 𝑛-manifolds
Authors: Casler B. G.
Published: June 2010

Let

M
M

be a compact connected P.L.

n
n

-manifold without boundary. Then

M
M

is the union of 3 sets

E
1

,

E
2

{E_1},{E_2}

and

F
F

where

E
i

,
i
=
1
,
2

{E_i},i = 1,2

, is homomorphic to the interior of an

n
n

-ball and

F
F

is the P.L. image of an

(
n
−
1
)

(n - 1)

-sphere. Further each point of

F
F

is a limit point of

E
1

{E_1}

and

E
2

{E_2}

.
The independence of certain axioms of structures in sets
Authors: Hall Japheth
Published: June 2010

The independence of the axioms for spans and the independence of the axioms for closure structures are usually taken for granted. In this paper, the author establishes the independence of monotonicity, extensiveness, idempotence, the exchange property, the property of having

∅
\emptyset

as a fixed set and two covering properties (

α
\alpha

-character, with

α
\alpha

being some cardinal number, and a covering property with respect to generators). The independence of the axioms for closure structures and spans follow immediately. It is shown that any proof of the independence of a given axiom must involve an example with certain restrictions on the cardinal number

α
\alpha

.
On Liapunov’s inequality
Authors: Leighton Walter
Published: June 2010

An elementary proof is given of the Liapunov inequality

∫
a
b

p
+

(
x
)

d
x
>
4

/

(
b
−
a
)

\smallint _a^b{p_ + }(x)\;dx > 4/(b - a)

.
Properties of Stone-Čech compactifications of discrete spaces
Authors: Warren Nancy M.
Published: June 2010

Let

β
N

\beta N

be the Stone-Čech compactification of the integers N. It is shown that p is a P-point of

β
N
−
N

\beta N - N

, then

β
N
−
N
−
{
p
}

\beta N - N - \{ p\}

is not normal. Let D be an uncountable discrete set and

E
0

{E_0}

be the set of points in

β
D
−
D

\beta D - D

in the closures of countable subsets of D It is shown that there is a two-valued continuous function on

E
0

{E_0}

which cannot be extended continuously to

β
D

\beta D

.
A random Fredholm integral equation
Authors: Padgett W. J. , Tsokos Chris P.
Published: June 2010

The aim of this paper is the study of a random or stochastic integral equation of the Fredholm type given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t semicolon omega right-parenthesis equals h left-parenthesis t semicolon omega right-parenthesis plus integral Subscript 0 Superscript normal infinity Baseline k 0 left-parenthesis t comma tau semicolon omega right-parenthesis e left-parenthesis tau comma x left-parenthesis tau semicolon omega right-parenthesis right-parenthesis d tau comma t greater-than-over-equals 0">

x
(
t
;
ω
)
=
h
(
t
;
ω
)
+

∫
0
∞

k
0

(
t
,

τ
;
ω
)
e
(
τ
,
x
(
τ
;
ω
)
)

d
τ
,
t
≧
0

x(t;\omega ) = h(t;\omega ) + \smallint _0^\infty {{k_0}(t,} \tau ;\omega )e(\tau ,x(\tau ;\omega ))\;d\tau , t \geqq 0

, where

ω
∈
Ω

\omega \in \Omega

, the supporting set of the probability measure space

(
Ω
,
A
,
P
)

(\Omega ,A,P)

. The existence and uniqueness of a random solution to the above stochastic integral equation is considered. A random solution,

x
(
t
;
ω
)

x(t;\omega )

, of such a random equation is defined to be a random function which satisfies the equation almost surely. Several theorems and useful special cases are presented which give conditions such that a random solution exists.
On groups of exponent four. III
Authors: Gupta N. D. , Quintana R. B.
Published: June 2010

C. R. B. Wright has shown that the nilpotency class of an n-generator group of exponent four is at most

3
n
−
1

3n - 1

. In this paper, it is shown that if this bound can be improved to

3
n
−
3

3n - 3

then the free group of exponent four of infinite rank is solvable.
A note on zero divisors in group-rings
Authors: Lewin Jacques
Published: June 2010

Let

Z

G
1

Z{G_1}

and

Z

G
2

Z{G_2}

be the integral group rings of groups

G
1

{G_1}

and

G
2

{G_2}

with a common normal subgroup

H
H

and let

K
K

be a subgroup of

H
H

. Let

G
G

be the free product of

G
1

{G_1}

and

G
2

{G_2}

amalgamating

K
K

. If

Z

G
1

Z{G_1}

and

Z

G
2

Z{G_2}

are integral domains and if ZH has the Ore condition then ZG is again an integral domain.
Almost continuous real functions
Authors: Kellum K. R. , Garrett B. D.
Published: June 2010

A blocking set of a function f is a closed set which does not intersect f but which intersects each continuous function with domain the same as f. It is shown that for each function which is not almost continuous, there exists a minimal blocking set. Using this property it is shown that there exists an almost continuous function with domain [0, 1] which is a

G
δ

{G_\delta }

set but is not of Baire Class 1, and that there exists an almost continuous function dense in the unit square.
Mixtures of nonatomic measures
Authors: Rao K. P. S. Bhaskara , Rao M. Bhaskara
Published: June 2010

In this paper, we answer the following question: Is a mixture of nonatomic measures nonatomic? The answer turns out to be no, in general. Four sufficient conditions are given under which mixtures become nonatomic.
A fundamental matrix equation for finite sets
Authors: Ryser H. J.
Published: June 2010

Let

S
=
{

x
1

,

x
2

,
⋯
,

x
n

}

S = \{ {x_1},{x_2}, \cdots ,{x_n}\}

be an n-set and let

S
1

,

S
2

,
⋯
,

S
m

{S_1},{S_2}, \cdots ,{S_m}

be subsets of S. Let A of size m by n be the incidence matrix for these subsets of S. We now regard

x
1

,

x
2

,
⋯
,

x
n

{x_1},{x_2}, \cdots ,{x_n}

as independent indeterminates and define

X
=

diag

[

x
1

,

x
2

,
⋯
,

x
n

]

X = {\text {diag}}[{x_1},{x_2}, \cdots ,{x_n}]

. We then form the matrix product

A
X

A
T

=
Y

AX{A^T} = Y

, where

A
T

{A^T}

denotes the transpose of the matrix A. The symmetric matrix Y has in its (i,j) position the sum of the indeterminates in

S
i

∩

S
j

{S_i} \cap {S_j}

, and consequently Y gives us a complete description of the intersection patterns

S
i

∩

S
j

{S_i} \cap {S_j}

. The specialization

x
1

=

x
2

=
⋯
=

x
n

=
1

{x_1} = {x_2} = \cdots = {x_n} = 1

of this basic matrix equation has been used extensively in the study of block designs. We give some other interesting applications of the matrix equation that involve subsets with various restricted intersection patterns.
A lower bound for the permanent of a (0,1)-matrix
Authors: Gibson P. M.
Published: June 2010

Let

A
=
(

a

i
j

)

A = ({a_{ij}})

be an n-square fully indecomposable

(
0
,
1
)

(0,1)

-matrix. It is shown that if each row sum of A is at least k then per
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A greater-than-over-equals sigma-summation Underscript i comma j equals 1 Overscript n Endscripts a Subscript i j Baseline minus 2 n plus 2 plus sigma-summation Underscript m equals 1 Overscript k minus 1 Endscripts left-parenthesis m factorial negative 1 right-parenthesis">

A
≧

∑

i
,
j
=
1

n

a

i
j

−
2
n
+
2
+

∑

m
=
1

k
−
1

(
m
!
−
1
)

A \geqq \sum \nolimits _{i,j = 1}^n{a_{ij}} - 2n + 2 + \sum \nolimits _{m = 1}^{k - 1}(m! - 1)

. This improves an inequality obtained by H. Minc.
Coalgebras, sheaves, and cohomology
Authors: Van Osdol D. H.
Published: June 2010

The category of sheaves on the topological space X with values in the algebraic category

A

\mathcal {A}

is shown to be cotripleable under the stalk category

A

|

X

|

{\mathcal {A}^{|X|}}

. If K is a field then the category of K-coalgebras is shown to be cotripleable under the category of K-vector spaces. This makes possible the interpretation of the first group of the associated triple cohomology complex. In particular, for coalgebras our

H
1

{H^1}

is isomorphic to Jonah’s

H
2

{H^2}

.
The conjugacy problem for the group of any tame alternating knot is solvable
Authors: Appel K. I. , Schupp P. E.
Published: June 2010

The theorem of the title is proved by using the techniques of Weinbaum along with the prime decomposition of knots and an extension of some small cancellation techniques of Lyndon and Schupp.
The graph extension theorem
Authors: Shult Ernest
Published: June 2010

A sufficient condition is given that a transitive permutation group G admits a transitive extension

G
∗

{G^\ast }

. The condition is graph-theoretic and does not involve any direct algebraic properties of the group being extended. The result accounts for a fairly wide class of doubly transitive groups, including the two doubly transitive representations of the groups

Sp

(
2
n
,
2
)

{\text {Sp}}(2n,2)

, and the doubly transitive representations of the Higman-Sims group, and the Conway group (.3).
Pure states with the restriction property
Authors: Barnes Bruce A.
Published: June 2010

Conditions are given which imply that a pure state of a

B
∗

{B^\ast }

-algebra A restricts to a pure state of some maximal commutative

∗

^\ast

-subalgebra of A.
Equivalence of projections
Authors: Berberian S. K.
Published: June 2010

Theorem: An

A

W
∗

A{W^\ast }

-algebra is the ring generated by its projections if and only if it has no abelian summand. Corollary: Every equivalence in an

A

W
∗

A{W^\ast }

-algebra may be implemented by a partial isometry in the ring generated by the projections of the algebra. The corollary is extended to certain finite Baer

∗
\ast

-rings.
New methods in coincidence theory
Authors: Mukherjea Kalyan K.
Published: June 2010

Cobordism theory is used to obtain a new type of result concerning the coincidence of maps between compact manifolds.
On a construction of Bredon
Authors: Roitberg Joseph
Published: June 2010

Using a homotopy-theoretical description of a geometric pairing due to Bredon, we show how to rederive Bredon’s results on the pairing. Furthermore, we are able to, in some sense, complete these results by combining the homotopy-theoretical approach with Sullivan’s determination of the 2-primary Postnikov decomposition of the space G/PL.
The Morse lemma on Banach spaces
Authors: Tromba A. J.
Published: June 2010

Let

f
:
U
→
R

f:U \to R

be a

C
3

{C^3}

map of an open subset U of a Banach space E. Let

p
∈
U

p \in U

be a critical point of

f
(
d

f
p

=
0
)

f(d{f_p} = 0)

. If E is a conjugate space

(
E
=

F
∗

)

(E = {F^ \ast })

we define what it means for p to be nondegenerate. In this case there is a diffeomorphism

γ
\gamma

of a neighborhood of p with a neighborhood of

0
∈
E
,
γ
(
p
)
=
0

0 \in E,\gamma (p) = 0

with
\[

f
∘

γ

−
1

(
x
)
=

1
2

d
2

f
p

(
x
,
x
)
+
f
(
p
)
.

f \circ {\gamma ^{ - 1}}(x) = \frac {1}{2}{d^2}{f_p}(x,x) + f(p).

\]
On the lower bound of the number of real roots of a random algebraic equation with infinite variance
Authors: Samal G. , Mishra M. N.
Published: June 2010

Let

N
n

{N_n}

be the number of real roots of a random algebraic equation

Σ
0
n

ξ
v

x
v

=
0

\Sigma _0^n{\xi _v}{x^v} = 0

, where the coefficients

ξ
v

{\xi _v}

’s are independent random variables with common characteristic function

exp
⁡
(
−
C

|

t

|

α

)

\exp ( - C|t{|^\alpha })

, C being a positive constant and

α
≧
1

\alpha \geqq 1

. It is proved that
\[

N
n

≧
(
μ
log
⁡
n
)

/

(
log
⁡
log
⁡
n
)
.

{N_n} \geqq (\mu \log n)/(\log \log n).

\]
The measure of the exceptional set tends to zero as n tends to infinity.
On a subclass of square integrable martingales
Authors: Isaacson Dean
Published: June 2010

Let

M

2
∗

\mathcal {M}_2^ \ast

denote the class of continuous, nowhere constant, square integrable martingales,

M
(
t
)
=
X
(

⟨
M

⟩
t

)

M(t) = X({\langle M\rangle _t})

, for which

⟨
M

⟩
t

{\langle M\rangle _t}

is a time change on the

σ
\sigma

-fields generated by the Brownian motion

X
(
t
)

X(t)

. It is shown that if

M
(
t
)
∈

M

2
∗

M(t) \in \mathcal {M}_2^ \ast

, then the family of

σ
\sigma

-fields generated by

M
(
t
)

M(t)

is a right continuous family. If

M
(
t
)
∈

M

2
∗

M(t) \in \mathcal {M}_2^ \ast

and if

σ
{
M
(
s
)
:
s
≦
t
}
=
σ
{
X
(
s
)
:
s
≦
t
}

\sigma \{ M(s):s \leqq t\} = \sigma \{ X(s):s \leqq t\}

for some Brownian motion

X
(
t
)

X(t)

, then

M
(
t
)
=

∫
0
t

Φ
(
s
)
d
X
(
s
)

M(t) = \smallint _0^t {\Phi (s)dX(s)}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript t Baseline left-parenthesis 1 slash normal upper Phi left-parenthesis s right-parenthesis right-parenthesis d upper M left-parenthesis s right-parenthesis">

X
(
t
)
=

∫
0
t

(
1

/

Φ
(
s
)
)
d
M
(
s
)

X(t) = \smallint _0^t {(1/\Phi (s))dM(s)}

for some process

Φ
(
s
)

\Phi (s)

with

Φ
(
s
)
≠
0

\Phi (s) \ne 0

a.e.

d
t
×
d
P

dt \times dP

.
Isomorphism types of infinite symmetric graphs
Authors: Bang C. M.
Published: June 2010

Professor Bjarni Jónsson asked about the cardinality of isomorphism types of infinite symmetric graphs of order m, for each infinite cardinal m. We show that there are

2
m

{2^m}

pairwise nonisomorphic infinite symmetric graphs of order m, for each infinite cardinal m.
Nonnegative matrices with nonnegative inverses
Authors: DeMarr Ralph
Published: June 2010

We generalize a result stating that a nonnegative finite square matrix has a nonnegative inverse if and only if it is the product of a permutation matrix by a diagonal matrix. We consider column-finite infinite matrices and give a simple proof using elementary ideas from the theory of partially ordered linear algebras.
𝐿^{𝑝}-convolution operators supported by subgroups
Authors: Dunkl Charles F. , Ramirez Donald E.
Published: June 2010

Let G be a compact nonabelian group and H be a closed subgroup of G. Then H is a set of spectral synthesis for the Fourier algebra

A
(
G
)

A(G)

(and indeed for

A
p

(
G
)
,
1
≦
p
>
∞

{A^p}(G),1 \leqq p > \infty

). For

1
≦
p
>
∞

1 \leqq p > \infty

, each

L
p

(
G
)

{L^p}(G)

-multiplier T corresponds to a

L
p

(
H
)

{L^p}(H)

-multiplier S by the rule

(
T
f
)

|

H
=
S
(
f

|

H
)
,
f
∈
A
(
G
)

(Tf)|H = S(f|H),f \in A(G)

, if and only if the support of T is contained in H.
Perturbed asymptotically stable sets
Authors: McCann Roger C.
Published: June 2010

Perturbations of a dynamical system are defined and the behavior of compact asymptotically stable sets under these perturbations is determined. The occurrence of critical points in a perturbed planar dynamical system is also investigated.
Isometries of 𝐻^{𝑝} spaces of the torus
Authors: Lal Nand , Merrill Samuel
Published: June 2010

Denote by

H
p

(
1
≦
p
≦
∞
)

{H^p}(1 \leqq p \leqq \infty )

the Banach spaces of complex-valued functions in

L
p

{L^p}

of the torus whose Fourier coefficients vanish off a half plane determined by a lexicographic ordering. The surjective isometries of the spaces

H
p

(
p
≠
2
)

{H^p}(p \ne 2)

are characterized in terms of unimodular functions on the circle and conformal maps of the disc. For

1
>
p
>
∞
(
p
≠
2
)

1 > p > \infty (p \ne 2)

the proof depends upon a characterization of certain invariant subspaces previously given by the authors.
On closed curves in Minkowski spaces
Authors: Witsenhausen H. S.
Published: June 2010

The minimum pseudo-diameter d and the length L of a simple closed rectifiable curve in Minkowski space satisfy

L
≧
g
d

L \geqq gd

where g is the half-girth of the unit ball. The bound is sharp.
Irreducible 3-manifolds whose orientable covers are not prime
Authors: Row W. H.
Published: June 2010

J. L. Tollefson has asked if every closed covering space of a prime 3-manifold is prime. In the present paper, the author provides a negative answer by constructing infinitely many topologically distinct, irreducible, closed 3-manifolds with the property that none of their orientable covering spaces are prime. These 3-manifolds are distinguished by the maximum number of disjoint, nonparallel, 2-sided projective planes that they contain. The author does not know if every closed covering space of a prime, orientable 3-manifold is prime.
Error bounds for Galerkin’s method for monotone operator equations
Authors: Schultz Martin H.
Published: June 2010

An abstract theorem, generalizing a result of Nitsche, is proved. This gives sharp error bounds for the Galerkin method for approximating the solutions of a large class of non-linear operator equations in Hilbert spaces.
On an induction theorem for relative Grothendieck groups
Authors: Gustafson William H.
Published: June 2010

We present an improvement in the proof of Dress’ induction theorem for relative Grothendieck rings.
Functions with real part greater than 𝛼
Authors: McCarty Carl P.
Published: June 2010

Let

P

b

(
α
)

{\mathcal {P}_b}(\alpha )

denote the class of functions

P
(
z
)
=
1
+
b
(
1
−
α
)
z
+
⋯

P(z) = 1 + b(1 - \alpha )z + \cdots

which are analytic and satisfy

Re
⁡
{
P
(
z
)
}
>
α

\operatorname {Re} \{ P(z)\} > \alpha

for

|

z

|

>
1

|z| > 1

where

α
∈
[
0
,
1
)

\alpha \in [0,1)

and

b
∈
[
0
,
2
]

b \in [0,2]

. We demonstrate some inequalities involving

|

P
′

(
z
)

|

|P’(z)|

and

|

P
′

(
z
)

/

P
(
z
)

|

|P’(z)/P(z)|

dependent on b and

α
\alpha

which are subsequently applied to the class of functions whose derivative lies in

P

b

(
α
)

{\mathcal {P}_b}(\alpha )

to obtain distortion, covering, and radius of convexity properties.
On invariant sets and on a theorem of Ważewski
Authors: Hartman Philip
Published: June 2010

The first part of the paper treats the question of the existence of a solution

x
=
x
(
t
)

x = x(t)

of an ordinary differential equation which exists for

t
≧

t
0

t \geqq {t_0}

and remains in a given closed set F for every assigned initial point

(

t
0

,
x
(

t
0

)
)
∈
F

({t_0},x({t_0})) \in F

or, in the autonomous case,

x
(

t
0

)
∈
F

x({t_0}) \in F

. The results involve conditions which, for the autonomous case, reduce to

dist

(

x
0

+
h
f
(

x
0

)
,
F
)

/

h
→
0

{\text {dist}}({x^0} + hf({x^0}),F)/h \to 0

as

h
→
+
0

h \to + 0

for all

x
0

∈
F

{x^0} \in F

. The second part of the paper deals with theorems of the Ważewski type which, in some situations, permit the relaxation of the hypothesis that egress points are strict egress points.
Existence of a seminormal basis in 𝐶[0,1]
Authors: Samn Sherwood
Published: June 2010

The existence of a basis in

C
[
0
,
1
]

C[0,1]

consisting of pairwise orthogonal elements is established.