bluuebunny/crossref_metadata_2025 huggingface dataset:
Goodman’s conjecture and the coefficients of univalent functions
Authors: Lyzzaik Abdallah , Styer David
Published: June 2010

Goodman’s conjecture (for a bound on the modulus of the nth coefficient of a p-valent function as a linear combination of the moduli of the first p coefficients) is considered in the special case of functions which are polynomials of univalent functions. For such functions, it is shown that Goodman’s conjecture is equivalent to a set of coefficient conjectures for normalized univalent functions.
Extensions of 𝑍₂ bordism
Authors: Beem R. Paul
Published: June 2010

The purpose of this note is to characterize the image of any extension homomorphism from unoriented

Z
2

{Z_2}

bordism to G bordism, where G is a finite abelian group of even order.
Two embedding theorems for lattices
Authors: Grätzer G. , Platt C. R.
Published: June 2010

A lattice L satisfies

(
S

D
∧

)

(S{D_ \wedge })

if

a
∧
b
=
a
∧
c

a \wedge b = a \wedge c

implies that

a
∧
b
=
a
∧
(
b
∨
c
)
(
(

S
D

∨

)

a \wedge b = a \wedge (b \vee c) ((\mathrm {SD}_\vee )

is defined dually). Theorem. Every lattice can be embedded in the ideal lattice of a lattice satisfying

(

S
D

∧

)

(\mathrm {SD}_\wedge )

(respectively,

(

S
D

∨

)

(\mathrm {SD}_\vee )

). Call a lattice K transferable iff whenever K can be embedded in the ideal lattice of a lattice L, then K can be embedded in L. Corollary. Every transferable lattice satisfies

(

S
D

∧

)

(\mathrm {SD}_\wedge )

and

(

S
D

∨

)

(\mathrm {SD}_\vee )

.
Vector valued functions equivalent to measurable functions
Authors: Uhl J. J.
Published: June 2010

Let X be a Banach space with dual

X
∗

{X^\ast }

and let

(
Ω
,
Σ
,
μ
)

(\Omega ,\Sigma ,\mu )

be a finite measure space. Suppose

f
:
Ω
→
X

f:\Omega \to X

is weakly measurable. There exists a (norm) measurable

g
:
Ω
→
X

g:\Omega \to X

such that

⟨

x
∗

,
f
⟩
=
⟨

x
∗

,
g
⟩

\langle {x^\ast },f\rangle = \langle {x^\ast },g\rangle

a.e. for each

x
∗

∈

X
∗

{x^\ast } \in {X^\ast }

if and only if each set A of positive

μ
\mu

-measure has a subset B of positive

μ
\mu

-measure such that there is a weakly compact convex subset K of X with the property that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle x Superscript asterisk Baseline comma f mathematical right-angle less-than-or-slanted-equals sup Underscript x element-of upper K Endscripts mathematical left-angle x Superscript asterisk Baseline comma x mathematical right-angle">

⟨

x
∗

,
f
⟩
⩽

sup

x
∈
K

⟨

x
∗

,
x
⟩

\langle {x^\ast },f\rangle \leqslant \sup \limits _{x \in K} \;\langle {x^\ast },x\rangle

\]

μ
\mu

-almost everywhere on B for each

x
∗

∈

X
∗

{x^\ast } \in {X^\ast }

.
On the analytic continuation of Dirichlet series
Authors: Seeling Ronald
Published: June 2010

Given a continuable Dirichlet series, having as sequence of exponents

{

λ
n

}

n
=
1

∞

\{ {\lambda _n}\} _{n = 1}^\infty

, it is shown that any other Dirichlet series can be continued along the same paths away from its half plane of convergence provided that the two series have the same coefficients and that the difference of their exponents is eventually given by a function analytic at infinity evaluated at the

λ
n

{\lambda _n}

.
Complementary components of polynomial hulls
Authors: Basener Richard F.
Published: June 2010

Let X be a compact subset of the unit sphere S in

C

n

,
n
>
1

{{\mathbf {C}}^n},n > 1

. It is shown that if a point z is not in the polynomially convex hull of X, then there is a complementary component U of X in S such that z is not in the hull of S U.
An improvement theorem for Descartes systems
Authors: Smith Philip W.
Published: June 2010

An improvement (or comparison) theorem is proved for certain linear combinations of functions from a Descartes system. This theorem can then be applied to prove a conjecture of Lorentz, as well as more general results.
Some “almost-Dowker” spaces
Authors: Scott Brian M.
Published: June 2010

Call X an AD-space (for “almost-Dowker") if it is

T
3

{T_3}

but not countably metacompact. We construct, without set-theoretic assumptions, a class of zero-dimensional, orthocompact, nonnormal AD-spaces. Using the same techniques, we simplify an example due to Hayashi by showing that if

exp
⁡
(
exp
⁡
(
ω
)
)
=
exp
⁡
(

ω
1

)

\exp (\exp (\omega )) = \exp ({\omega _1})

, (e.g., if the continuum hypothesis holds), the “Cantor tree of height

ω
1

{\omega _1}

” is also such a space. Since

X
×
[
0
,
1
]

X \times [0,1]

is orthocompact iff X is orthocompact and countably metacompact, we now have “absolute” examples of orthocompact Tikhonov spaces whose products with [0, 1] are not orthocompact.
On the stable cohomotopy of 𝑅𝑃^{∞}
Authors: Snaith Victor
Published: June 2010

There is a conjecture of G. B. Segal concerning the relation between the Burnside ring of G and the stable cohomotopy of BG. When

G
=
Z

/

2

G = Z/2

this conjecture is shown to be equivalent to the triviality of the group of homotopy classes of

R

P
∞

R{P^\infty }

into the “cokernel of J".
Lattices of normally indecomposable modules
Authors: Brzezinski Juliusz
Published: June 2010

If M, N are finitely generated left R-modules, then M divides

N
1

{N^1}

if there is an epimorphism

M

(
r
)

→
N

{M^{(r)}} \to N

. M is normally indecomposable if

M
≅

M
1

⊕

M
2

M \cong {M_1} \oplus {M_2}

and

M
1

{M_1}

divides

M
2

{M_2}

imply

M
2

=
0

{M_2} = 0

. If R is an Artin algebra or an order over a complete discrete valuation ring in a semisimple algebra, the set of isomorphism classes of normally indecomposable R-modules (respectively R-lattices) is partially ordered by the divisibility relation. We show that for R of finite representation type this partially ordered set is a lattice satisfying the Jordan-Dedekind chain condition and the length of maximal chains is equal to the number of isomorphism classes of indecomposable R-modules (respectively R-lattices).
On functions subharmonic in a Lipschitz domain
Authors: Wu Jang-Mei Gloria
Published: June 2010

Let D be a starlike Lipschitz domain in

R
n

,
n
⩾
2

{R^n},n \geqslant 2

. If w is a subharmonic function in D with positive harmonic majorant, then at almost every point on the boundary of D (surface measure), w has radial limit. Results on limits along certain

C
1

{C^1}

curves in general Lipschitz domains are also obtained.
A commutant of an unbounded operator algebra
Authors: Inoue Atsushi
Published: June 2010

A commutant

A

c

{\mathfrak {A}^c}

and bicommutant

A

c
c

{\mathfrak {A}^{cc}}

of an unbounded operator algebra

A

\mathfrak {A}

called a #-algebra are defined. The first purpose of this paper is to investigate whether the bicommutant

A

c
c

{\mathfrak {A}^{cc}}

of a #-algebra

A

\mathfrak {A}

is an

E

W
#

E{W^\# }

-algebra, as defined in [6], or not. The second purpose is to investigate the relation between

A

c
c

{\mathfrak {A}^{cc}}

and topologies on a #-algebra

A

\mathfrak {A}

.
A countably compact space and its products
Authors: Vaughan J. E.
Published: June 2010

It is known that if a topological space X is totally countably compact, then (a)

X

ω
1

{X^{{\omega _1}}}

(the product of

ω
1

{\omega _1}

copies of X) is countably compact, and (b) for every countably compact space Y, the product

X
×
Y

X \times Y

is countably compact. The main result of this paper is the construction of a space which satisfies (a) and (b), and is not totally countably compact. The example is

X
=
β
(
ω
)
∖
t
(
p
)

X = \beta (\omega )\backslash t(p)

, where

t
(
p
)

t(p)

is the type of a certain kind of ultrafilter on the natural numbers

ω
\omega

.
On the genus of graphs with Lick-White number 𝑘
Authors: Mitchem John
Published: June 2010

A graph is called n-degenerate if each of its subgraphs has a vertex of degree at most n. For each n the Lick-White number of graph G is the fewest number of sets into which

V
(
G
)

V(G)

can be partitioned such that each set induces an n-degenerate graph. An upper bound is obtained for the Lick-White number of graphs with given clique number. A number of estimates are derived for the number of vertices in triangle-free graphs with prescribed Lick-White number. These results are used to give lower bounds on the genus of such graphs.
The 𝑆𝐶_{𝑘+1}𝑃-integral and trigonometric series
Authors: Cross G. E.
Published: June 2010

Recently P. S. Bullen and C. M. Lee defined a scale of symmetric Cesàro-Perron integrals, but left open the question of whether their

S

C

k
+
1

P

S{C_{k + 1}}P

-integral solves the coefficient problem for (C, k) summable series. This paper gives an affirmative answer to that question under natural conditions.
A refinement of the arithmetic mean-geometric mean inequality
Authors: Cartwright D. I. , Field M. J.
Published: June 2010

Upper and lower bounds are given for the difference between the arithmetic and geometric means of n positive real numbers in terms of the variance of these numbers.
On the operator equation 𝐴𝑋+𝑋𝐵=𝑄
Authors: Goldstein Jerome A.
Published: June 2010

Consider the operator equation

(
∗
)
A
X
+
X
B
=
Q

(\ast )AX + XB = Q

; here A and B are (possibly unbounded) selfadjoint operators and Q is a bounded operator on a Hilbert space. The theory of one parameter semigroups of operators is applied to give a quick derivation of M. Rosenblum’s formula for approximate solutions of

(
∗
)

(\ast )

. Sufficient conditions are given in order that

(
∗
)

(\ast )

has a solution in the Schatten-von Neumann class

C

p

{\mathcal {C}_p}

if Q is in

C

p

{\mathcal {C}_p}

. Finally a sufficient condition for solvability of

(
∗
)

(\ast )

is given in terms of T. Kato’s notion of smoothness.
More on the connectivity of convex sets
Authors: Bridges D. S.
Published: June 2010

This note, written from a constructive viewpoint, shows how Mandelker’s results on connectivity of compact intervals [3] extend to open balls in Banach space, and complete, convex subsets of arbitrary normed spaces. This is achieved by proving the equivalence of Mandelker’s notion of connectivity with one of those discussed in [2].
Differential dimension polynomials of finitely generated extensions
Authors: Sit William
Published: June 2010

Let

G

=

F

⟨

η
1

,
…
,

η
n

⟩

\mathcal {G} = \mathcal {F}\langle {\eta _1}, \ldots ,{\eta _n}\rangle

be a finitely generated extension of a differential field

F

\mathcal {F}

with m derivative operators. Let d be the differential dimension of

G

\mathcal {G}

over

F

\mathcal {F}

. We show that the numerical polynomial
\[

ω

η

/

F

(
X
)
−
d

(

X
+
m

m

)

{\omega _{\eta /\mathcal {F}}}(X) - d\left ( {\begin {array}{*{20}{c}} {X + m} \\ m \\ \end {array} } \right )

\]
can be viewed as the differential dimension polynomial of certain extensions. We then give necessary and sufficient conditions for this numerical polynomial to be zero. An invariant (minimal) differential dimension polynomial for the extension

G

\mathcal {G}

over

F

\mathcal {F}

is defined and extensions for which this invariant polynomial is

d

(

X
+
M

m

)

d\left ( {\begin {array}{*{20}{c}} {X + M} \\ m \\ \end {array} } \right )

are characterised.
A nonstandard characterization of weak convergence
Authors: Anderson Robert M. , Rashid Salim
Published: June 2010

Let X be any topological space, and

C
(
X
)

C(X)

the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on

C
(
X
)

C(X)

to a tight Baire measure on X. This characterization applies whether or not the net or the individual functionals in the net are tight. Moreover, the characterization is expressed in terms of the values of an associated net of countably additive measures on all Baire sets of X; no distinguished family, such as the family of continuity sets of the limit, is involved. As a corollary, we obtain a new proof that a tight set of measures is relatively weakly compact.
A further refinement for coefficient estimates of univalent functions
Authors: Horowitz David
Published: June 2010

The coefficient inequalities of FitzGerald are used to show that if

f
(
z
)
=
z
+

a
2

z
2

+

a
3

z
3

+
…

f(z) = z + {a_2}{z^2} + {a_3}{z^3} + \ldots

is analytic and univalent in the unit disc, then

|

a
n

|

>
(
1.0657
)
n

|{a_n}| > (1.0657)n

. The technique used to obtain this bound cannot yield a result better than

|

a
n

|

>
(
1.0599
)
n

|{a_n}| > (1.0599)n

.
Stein manifolds on which the strong Poincaré problem can be solved
Authors: Ephraim Robert
Published: June 2010

Let M be a Stein manifold. Suppose every meromorphic function on M may be written as the quotient of two holomorphic functions which are pointwise relatively prime at every point of M. Then it will be shown that

H
2

(
M
,
Z
)
=
0

{H^2}(M,Z) = 0

. Thus, the solvability of the Strong Poincaré Problem is equivalent to the vanishing of the second integral cohomology, which in turn is equivalent to the solvability of Cousin II, all on a Stein manifold M. This closes a gap in the classically known theory.
A general result regarding the growth of solutions of first-order differential equations
Authors: Bank Steven B.
Published: June 2010

In this paper, we treat first-order algebraic differential equations whose coefficients are arbitrary complex-valued functions on an interval

[

x
0

,
+
∞
)

[{x_0}, + \infty )

, and we obtain an estimate on the growth of all real-valued solutions on

[

x
0

,
+
∞
)

[{x_0}, + \infty )

. Our result includes, as a very special case, the well-known result of Lindelöf for polynomial coefficients.
Another extremal property of perfect splines
Authors: Goodman T. N. T. , Lee S. L.
Published: June 2010

Let

t

=
{

t
i

}
,
i
=
1
,
2
,
…
,
n
+
k

{\mathbf {t}} = \{ {t_i}\} ,i = 1,2, \ldots ,n + k

, be a given sequence in [a, b],
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f 0 element-of upper W Subscript normal infinity Superscript k Baseline left-bracket a comma b right-bracket comma upper A greater-than double-vertical-bar f 0 Superscript left-parenthesis k right-parenthesis Baseline double-vertical-bar Subscript normal infinity Baseline">

f
0

∈

W
∞
k

[
a
,
b
]
,
A
>

‖

f
0

(
k
)

‖

∞

{f_0} \in W_\infty ^k[a,b],A > {\left \| {f_0^{(k)}} \right \|_\infty }

, and

F
=
{
f
∈

W
∞
k

[
a
,
b
]
:
f

|

t

=

f
0

|

t

F = \{ f \in W_\infty ^k[a,b]:f|{\mathbf {t}} = {f_0}|{\mathbf {t}}

and

‖

f

(
k
)

‖

∞

⩽
A
}

{\left \| {{f^{(k)}}} \right \|_\infty } \leqslant A\}

. We show that F contains precisely two perfect splines g, h of degree k with

|

g

(
k
)

|

=

|

h

(
k
)

|

=
A

|{g^{(k)}}| = |{h^{(k)}}| = A

and n interior nodes, and for all

f
∈
F

f \in F

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min left-parenthesis g left-parenthesis x right-parenthesis comma h left-parenthesis x right-parenthesis right-parenthesis less-than-or-slanted-equals f left-parenthesis x right-parenthesis less-than-or-slanted-equals max left-parenthesis g left-parenthesis x right-parenthesis comma h left-parenthesis x right-parenthesis right-parenthesis semicolon for-all x element-of left-bracket a comma b right-bracket">

min
(
g
(
x
)
,
h
(
x
)
)
⩽
f
(
x
)
⩽
max
(
g
(
x
)
,
h
(
x
)
)
;
∀
x
∈
[
a
,
b
]

\min (g(x),h(x)) \leqslant f(x) \leqslant \max (g(x),h(x));\forall x \in [a,b]

.
Ring extensions and essential monomorphisms
Authors: Würfel Tilmann
Published: June 2010

We study pairs of rings

R
⊂
S

R \subset S

such that

Hom
R

⁡
(
S
,
−
)
:
R
−
Mod
→
S
−
Mod

\operatorname {Hom}_R(S, - ):R - \operatorname {Mod} \to S - \operatorname {Mod}

preserves essential monomorphisms. We obtain a complete characterization of such a pair in case S is a torsion-free algebra over a Noetherian domain

R
≠

Q
u
o
t

(
R
)

R \ne \mathrm {Quot}(R)

; S is then a left ideally finite R-algebra. The rings R such that every ring extension

R
⊂
S

R \subset S

satisfies the above condition are subdirect sums of certain Artinian rings. Furthermore, we study a generalization of trivial ring extensions and show that the center of a semi-Artinian ring is again semi-Artinian.
A note on Helson’s existence theorem
Authors: Tanaka Jun-ichi
Published: June 2010

In the almost periodic context, Helson showed that there exists a unitary function in every simply invariant subspace. We provide a short proof of this result in more general setting and give some applications.
A theorem on 𝐶*-embedding
Authors: Dashiell F. K.
Published: June 2010

Theorem. In a totally nonmeager and regular space, every countable intersection of open, normal,

C
∗

{C^ \ast }

-embedded subsets is normal and

C
∗

{C^ \ast }

-embedded.
Local compactness and Hewitt realcompactifications of products
Authors: Ohta Haruto
Published: June 2010

In this note we prove McArthur’s conjecture [6]: If card X is nonmeasurable and if

v
(
X
×
Y
)
=
v
X
×
v
Y

v(X \times Y) = vX \times vY

holds for each space Y, then X is locally compact. Consequently, we can completely characterize the class of all spaces X such that for each space Y,

v
(
X
×
Y
)
=
v
X
×
v
Y

v(X \times Y) = vX \times vY

holds.
A counterexample to a conjecture of A. H. Stone
Authors: Bell Harold , Dickman R. F.
Published: June 2010

A. H. Stone has offered a sequence,

{
S
(
n
)
;
n
>
2
}

\{ S(n);n > 2\}

, of conjectures characterizing multicoherence for locally connected, connected, normal spaces. The conjecture

S
(
n
)

S(n)

is, “X is multicoherent if and only if X can be represented as the union of a circular chain of continua containing exactly n elements". It is known that

S
(
3
)

S(3)

always obtains and that

S
(
6
)

S(6)

obtains if the space is compact. In this paper, we construct a multicoherent plane Peano continuum C for which

S
(
7
)

S(7)

fails. Since

S
(
n
+
1
)

S(n + 1)

implies

S
(
n
)
,
n
>
2
,
S
(
n
)

S(n),n > 2,S(n)

fails for C for all

n
>
6

n > 6

. Furthermore we show that for any integer

n
⩾
3

n \geqslant 3

there exists a plane Peano continuum for which

S
(
2
n
)

S(2n)

obtains while

S
(
2
n
+
1
)

S(2n + 1)

fails.
A complete space of vector-valued measures
Authors: Kirk R. B. , Rehmer K.
Published: June 2010

Let X be a Hausdorff uniform space and E a Fréchet space (or more generally an LF-space) with dual F. Let

U
c

(
X
,
E
)

{U^c}(X,E)

denote the uniformly continuous functions from X into E which have a precompact range, and let

U
c

(
X
,
E
)

{U^c}(X,E)

have the topology of uniform convergence. Let

L
(
X
,
F
)

L(X,F)

be the space of all F-valued measures on X with finite support, and let

L
(
X
,
F
)

L(X,F)

be given the topology of uniform convergence over the uniformly equicontinuous subsets of

U
c

(
X
,
E
)

{U^c}(X,E)

having a common precompact range in E. The main result in the paper is a characterization of the completion of

L
(
X
,
F
)

L(X,F)

under this topology.
Moišezon spaces and positive coherent sheaves
Authors: Rabinowitz Joshua H.
Published: June 2010

In recent papers of Grauert and Riemenschneider, attempts have been made to generalize Kodaira’s embedding theorem to a characterization of Moišezon spaces. In this paper, we define a torsion-free coherent analytic sheaf of generic fiber dimension one as positive if its monoidal transform is positive. We prove: a normal irreducible compact complex space is Moišezon if and only if it carries a positive coherent sheaf of generic fiber dimension one.
The additive inverse eigenvalue problem and topological degree
Authors: Alexander J. C.
Published: June 2010

A short proof using topological degree is given of the additive inverse eigenvalue problem: The diagonal elements of any square complex matrix can be altered so as to cause the altered matrix to have any prescribed set of eigenvalues.
Imbeddings, immersions, and characteristic classes of differentiable manifolds
Authors: Papastavridis Stavros
Published: June 2010

Let

I
n
i

I_n^i

be the set of

mod

 - 

2

\bmod {\text { - }}2

characteristic classes which are of dimension i, and they are zero for all n-dimensional smooth manifolds. Let

I

n
,
k

i

I_{n,k}^i

be the set of i-dimensional

mod

 - 

2

\bmod {\text { - }}2

characteristic classes which are zero for all n-dimensional smooth manifolds which immerse in codimension k, (we are talking about normal characteristic classes). Let K be the (graded) ideal in

H
∗

(
B
O
,

Z
2

)

{H^ \ast }(BO,{Z_2})

generated by

w

k
+
1

,

w

k
+
2

,
…

{w_{k + 1}},{w_{k + 2}}, \ldots

. Then if

i
⩽
(
n
+
k
)

/

2

i \leqslant (n + k)/2

, we have

I

n
,
k

i

=

I
n
i

+

K
i

I_{n,k}^i = I_n^i + {K^i}

. We have some related results for imbedded manifolds, and also for manifolds which immerse or imbed with an SO, U, SU, Spin, etc. structure on the normal bundle.
Projective ideals in rings of dimension one
Authors: Carrig J. E. , Vasconcelos W. V.
Published: June 2010

In this paper we discuss a construction that shows the projective ideals of a one-dimensional ring as radically dense in the set of all faithful ideals. Its applications are to questions of existence of a classical ring of quotients, to counting equations, and to a trade-in between Noetherianess and low homological dimension.
Approximate units in ideals of group algebras
Authors: Rindler Harald
Published: June 2010

Several properties of locally compact groups can be characterized by the existence of certain approximate units in 3 ideals of the group algebra. We combine methods and results from [8], [7], [5].
A remark on the Grothendieck residue map
Authors: Carrell James B.
Published: June 2010

The purpose of this note is to give a direct proof that a global integral over a compact complex manifold X can be evaluated on the zero set of a meromorphic vector field on X with isolated zeros via a Grothendieck residue morphism. A special case of this evaluation is the meromorphic vector field theorem of Baum and Bott [1]. The present proof suggests some complements of the M.V.F. Theorem which are contained in Theorem 2.
On Markov stability
Authors: Risito Corrado
Published: June 2010

The concept of T-stability for vector-valued functions is introduced—a generalization of strong stability in the sense of Markov. Moreover, for solutions of T-periodic systems of differential equations, T-stability is compared with Liapunov stability and it is shown that boundedness and T-stability imply asymptotic almost periodicity.
Products of singular continuous measures
Authors: Maclean Alan
Published: June 2010

Let G be a compact Abelian group. It is shown that if the Fourier transform of

f
∈
A
(
G
)

f \in A(G)

satisfies certain lacunary conditions, then f may be factored as the convolution product of singular continuous measures.
Vector-valued continuous functions with strict topologies and angelic topological spaces
Authors: Khurana Surjit Singh
Published: June 2010

It is proved that if X is a metric space, E a Banach space containing a

σ
\sigma

-weakly-compact dense subset, then the space
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M Subscript tau Baseline left-parenthesis upper X comma upper E Superscript prime Baseline right-parenthesis comma sigma left-parenthesis upper M Subscript tau Baseline left-parenthesis upper X comma upper E Superscript prime Baseline right-parenthesis comma upper C Subscript b Baseline left-parenthesis upper X comma upper E right-parenthesis right-parenthesis right-parenthesis">

(

M
τ

(
X
,

E
′

)
,
σ
(

M
τ

(
X
,

E
′

)
,

C
b

(
X
,
E
)
)
)

({M_\tau }(X,E’),\sigma ({M_\tau }(X,E’),{C_b}(X,E)))

is angelic,

C
b

(
X
,
E
)

{C_b}(X,E)

being all bounded continuous functions from X into E and

M
τ

(
X
,

E
′

)

{M_\tau }(X,E’)

the dual of

C
b

(
X
,
E
)

{C_b}(X,E)

with the strict topology

β
\beta

.
Some applications of the Stone-Weierstrass theorem
Authors: de Paepe P. J.
Published: June 2010

Let A be a function algebra on a compact Hausdorff space X and let

f
∈
A

f \in A

. The Stone-Weierstrass theorem is used to obtain results on the function algebra on X generated by the elements of A and the function

f
¯

\bar f

.
Approximation of 𝐿¹-bounded martingales by martingales of bounded variation
Authors: Burkholder D. L. , Shintani T.
Published: June 2010

If

f
=
(

f
1

,

f
2

,
…
)

f = ({f_1},{f_2}, \ldots )

is a real

L
1

{L^1}

-bounded martingale and

ε
>
0

\varepsilon > 0

, then there is a martingale g of bounded variation satisfying

‖

f
−
g

‖

1

>
ε

{\left \| {f - g} \right \|_1} > \varepsilon

. The same result holds for X-valued martingales, where X is a Banach space, provided X has the Radon-Nikodým property. In fact, this characterizes Banach spaces having the Radon-Nikodým property. Theorem 1 identifies, for an arbitrary Banach space, the class of

L
1

{L^1}

-bounded martingales that converge almost everywhere.
On the distribution of maxima of martingales
Authors: Dubins Lester E. , Gilat David
Published: June 2010

Partial order the set of distributions on the real line by

ν
⩽

ν
′

\nu \leqslant \nu ’

if

ν
(
x
,
∞
)
⩽

ν
′

(
x
,
∞
)

\nu (x,\infty ) \leqslant \nu ’(x,\infty )

for all x. Then, for each

μ
\mu

with a finite first moment, the family

M
(
μ
)

M(\mu )

of all

ν
\nu

which are distributions of (essential) suprema of martingales closed on the right by a

μ
\mu

-distributed random number, has a least upper bound

μ
∗

{\mu ^ \ast }

, and is, therefore, a tight family. In fact,

μ
∗

{\mu ^ \ast }

is

μ
¯

\bar \mu

, the distribution of the Hardy-Littlewood extremal maximal function associated with

μ
\mu

. Moreover,

μ
∗

{\mu ^ \ast }

is itself an element of

M
(
μ
)

M(\mu )

. For each

p
>
1

p > 1

, the classical moment inequality that the

L
p

{L_p}

norm of

μ
¯

\bar \mu

(and of

μ
∗

{\mu ^ \ast }

) is at most

p

/

(
p
−
1
)

p/(p - 1)

times the

L
p

{L_p}

norm of

μ
\mu

is shown to be sharp.
The binary digits of a power
Authors: Stolarsky Kenneth B.
Published: June 2010

Let

B
(
m
)

B(m)

denote the number of ones in the binary expansion of the integer

m
⩾
1

m \geqslant 1

and let

r
h

(
m
)
=
B
(

m
h

)

/

B
(
m
)

{r_h}(m) = B({m^h})/B(m)

where h is a positive integer. The maximal order of magnitude of

r
h

(
m
)

{r_h}(m)

is

c
(
h
)

(
log
⁡
m

)

(
h
−
1
)

/

h

c(h){(\log m)^{(h - 1)/h}}

where

c
(
h
)
>
0

c(h) > 0

depends only on h. That this is best possible follows from the Bose-Chowla theorem. The minimal order of magnitude of

r
2

(
m
)

{r_2}(m)

is at most

c

(
log
⁡
log
⁡
m

)
2

/

log
⁡
m

c{(\log \log m)^2}/\log m

where

c
>
0

c > 0

is an absolute constant.
Certain idempotents lying in the centralizer of the group of units
Authors: Hunter R. P.
Published: June 2010

Let S be a compact connected monoid of dimension n having G as a connected group of units. Let B be a closed subgroup outside of the minimal ideal. The maximum dimension possible for the product BG is

n
−
1

n - 1

. If this maximum is attained by BG and GB and both are Lie groups then B meets the centralizer of G.
Two theorems on boundary values of analytic functions
Authors: Barker S. R.
Published: June 2010

We give a new admissible maximal inequality for analytic functions of several complex variables, and also show that an analytic function which is admissibly bounded at almost all boundary points of a domain D must have an admissible limit a.e. These results are then applied to the boundary behaviour of the Nevanlinna class of D.
A metric inequality characterizing barycenters and other Pettis integrals
Authors: Bilyeu Russell G.
Published: June 2010

Certain Pettis integrals, including barycenters of probability measures on weakly compact subsets of Banach spaces, are characterized by an integral inequality which refers only to distances between points, avoiding any reference to the linear structure of the Banach space. This is an elaboration of the Mazur-Ulam discovery that the metric determines the linear structure.
On spectra of Hausdorff operators on 𝑙²₊
Authors: Deddens James A.
Published: June 2010

In this note we compute the spectra of all Hausdorff operators on

l
+
2

l_ + ^2

. We also compute the norm and point spectra of the adjoint. Our technique uses standard operator theory and analytic function theory.
Harmonic nonholomorphic maps from a surface to a sphere
Authors: Lemaire Luc
Published: June 2010

For every p and

D

\mathcal {D}

such that

|

D

|

⩽
p
−
1

|\mathcal {D}| \leqslant p - 1

, there exist a Riemann surface of genus p and a harmonic nonholomorphic map of degree

D

\mathcal {D}

from that surface to the sphere. For some of the degrees, a holomorphic map exists for the same Riemann surface.
The large condition for rings with Krull dimension
Authors: Boyle Ann K.
Published: June 2010

A module M with Krull dimension is said to satisfy the large condition if for any essential submodule L of M, the Krull dimension of

M

/

L

M/L

is strictly less than the Krull dimension of M. For a right noetherian ring R with Krull dimension

α
\alpha

this is equivalent to the condition that every f.g. uniform submodule of

E
(

R
R

)

E({R_R})

with Krull dimension

α
\alpha

is critical. It is also shown that if R is right noetherian with Krull dimension

α
\alpha

and if

I
0

{I_0}

is a right ideal maximal with respect to K

dim
⁡

I
0

>
α

\dim {I_0} > \alpha

, then R satisfies the large condition if and only if

I
0

{I_0}

is a finite intersection of cocritical right ideals and

I
0

{I_0}

is closed.
Finite-dimensional perturbations
Authors: Behncke Horst
Published: June 2010

Let A be a normal operator on the Hilbert space

H

\mathcal {H}

and let B be an operator of finite rank, rank

B
=
m

B = m

, such that

A
+
B

A + B

is normal. Moreover let E (resp. F) denote the spectral projections of A (resp.

A
+
B

A + B

) for the set

{
ζ
∈

C

|

|

ζ
−
λ

|

⩽
α
}

\{ \zeta \in {\mathbf {C}}||\zeta - \lambda | \leqslant \alpha \}

. Then

dim

E
−
m
⩽
dim
⁡
F
⩽
dim
⁡
E
+
m

\dim \;E - m \leqslant \dim F \leqslant \dim E + m

.
Bundle shifts and Ahlfors functions
Authors: Abrahamse M. B. , Bastian J. J.
Published: June 2010

If S is a bundle shift over R with multiplicity k, if the boundary of R has n components, and if

ϕ
\phi

is the Ahlfors function for a point in R, then

ϕ
(
S
)

\phi (S)

is a unilateral shift of multiplicity kn. It follows that a reductive algebra containing the rational algebra of a bundle shift of finite multiplicity is selfadjoint.
The Cauchy problem for hyperbolic operators with variable multiple characteristics
Authors: Yamamoto Kazuhiro
Published: June 2010

Let

P
(
t
,
x
,

D
t

,

D
x

)

P(t,x,{D_t},{D_x})

be a hyperbolic differential operator with the principal symbol

p
m

(
t
,
x
,
τ
,
ξ
)

{p_m}(t,x,\tau ,\xi )

. We assume that

P
m

{P_m}

is denoted by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi Subscript j equals 1 Superscript s Baseline left-parenthesis tau minus lamda Subscript j Baseline right-parenthesis Superscript m Super Subscript j Baseline normal upper Pi Subscript j equals s plus 1 Superscript m minus upper N plus s Baseline left-parenthesis tau minus lamda Subscript j Baseline right-parenthesis">

Π

j
=
1

s

(
τ
−

λ
j

)

m
j

Π

j
=
s
+
1

m
−
N
+
s

(
τ
−

λ
j

)

\Pi _{j = 1}^s{(\tau - {\lambda _j})^{{m_j}}}\Pi _{j = s + 1}^{m - N + s}(\tau - {\lambda _j})

and

(

λ
i

−

λ
j

)
(
t
,
x
,
ξ
)
≠
0

({\lambda _i} - {\lambda _j})(t,x,\xi ) \ne 0

if

(
i
,
j
)
≠
(
k
,
m
−
N
+
k
)

(
k
=
1
,
…
,
s
)

(i,j) \ne (k,m - N + k)\;(k = 1, \ldots ,s)

, where

N
=

Σ

j
=
1

s

m
j

N = \Sigma _{j = 1}^s{m_j}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript j Baseline left-parenthesis t comma x comma xi right-parenthesis element-of upper C Superscript normal infinity Baseline left-parenthesis left-bracket 0 comma upper T right-bracket times upper R Superscript n Baseline times left-parenthesis upper R Superscript n Baseline minus 0 right-parenthesis right-parenthesis">

λ
j

(
t
,
x
,
ξ
)
∈

C
∞

(
[
0
,
T
]
×

R
n

×
(

R
n

∖
0
)
)

{\lambda _j}(t,x,\xi ) \in {C^\infty }([0,T] \times {R^n} \times ({R^n}\backslash 0))

. Under a generalized condition of E. E. Levi, we shall show that the Cauchy problem

P
u
=
f

Pu = f

in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma upper T right-bracket times upper R Superscript n Baseline comma upper D Subscript t Superscript j Baseline u Subscript vertical-bar t equals 0 Baseline equals g Subscript j Baseline left-parenthesis j equals 1 comma ellipsis comma m minus 1 right-parenthesis">

[
0
,
T
]
×

R
n

,

D
t
j

u

|

t
=
0

=

g
j

(
j
=
1
,
…
,
m
−
1
)

[0,T] \times {R^n},D_t^j{u_{|t = 0}} = {g_j}(j = 1, \ldots ,m - 1)

is well posed. When

m
j

=
1
(
j
=
1
,
…
,
s
)

{m_j} = 1(j = 1, \ldots ,s)

, our result coincides those of Ohya and Petkov.
Convergence of certain cosine sums in a metric space — 𝐿
Authors: Singh Niranjan , Sharma K. M.
Published: June 2010

In this paper a generalization of a theorem of Garrett and Stanojevic [6] has been obtained.
Closed geodesics on Riemann surfaces
Authors: Jørgensen Troels
Published: June 2010

This note deals with an elementary property of closed geodesics on surfaces with a complete metric of constant negative curvature.
Simplicial Schreier systems and the commutator subgroup of the free group on the circle
Authors: Ghullam Sabah A.
Published: June 2010

It is shown that the commutator subgroup of the free simplicial group on the circle has a simplicial Schreier system and is a free simplicial group on a pointed simplicial set.
Characterizing series for faithful d.g. near-rings
Authors: Meldrum J. D. P. , Lyons C.
Published: June 2010

Let (R, S) be a distributively generated near ring satisfying

(
R
,
S
)
⊆
(
E
(
G
)
,

End

(
G
)
)

(R,S) \subseteq (E(G),{\text {End}}(G))

and

S
⊆

End

(
G
)

S \subseteq {\text {End}}(G)

for some group G, endomorphism near ring

E
(
G
)

E(G)

, and subsemigroup S of the endomorphisms of G,

End

(
G
)

{\text {End}}(G)

. The radicals

J
(
R
)

J(R)

of (R, S) are characterized in terms of series of subgroups of G. We assume S contains the inner automorphisms of G and obtain two main results on characterizing series. (1) If G satisfies both chain conditions on S subgroups then a unique minimal characterizing series exists. (2) If G is finite, then both maximal and minimal characterizing series exist, are unique, and are themselves characterized in G.
Wendroff type inequalities
Authors: Shastri R. P. , Kasture D. Y.
Published: June 2010

Results (1) and (3) of Wendroff on two-variable scalar integral inequalities, quoted without proof by Beckenback and Bellman [1, p. 154] have been generalized to cover a system of integral inequalities by following an approach different from that of Jagdish Chandra and Davis [3].
Periodic solutions of perturbed conservative systems
Authors: Ward James R.
Published: June 2010

The existence of

2
π

2\pi

-periodic solutions to the system

x
+
grad

G
(
x
)
=
p
(
t
,
x
)

x + \text {grad}\; G(x) = p(t,x)

, p being

2
π

2\pi

-periodic in t, is established under conditions at infinity on the Hessian matrix of G. The condition used is weaker than earlier known conditions of a similar nature.
On the stability of the linear mapping in Banach spaces
Authors: Rassias Themistocles M.
Published: June 2010

Let

E
1

,

E
2

{E_1},{E_2}

be two Banach spaces, and let

f
:

E
1

→

E
2

f:{E_1} \to {E_2}

be a mapping, that is “approximately linear". S. M. Ulam posed the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam’s problem.
Capacities and spans on Riemann surfaces
Authors: Burbea Jacob
Published: June 2010

Let

K
(
z
,
z
)
,
R
(
z
,
z
)

K(z,z),R(z,z)

, and

C
(
z
)

C(z)

be the values of the Bergman kernel, the reduced Bergman kernel and the analytic capacity on an open Riemann surface

Ω
\Omega

(with respect to a local parameter z). Let

M
(
z
)
=
π
K
(
z
,
z
)

M(z) = \pi K(z,z)

and

S
(
z
)
=

π
R
(
z
,
z
)

S(z) = \sqrt {\pi R(z,z)}

. For

Ω
∉

O
G

\Omega \notin {O_G}

and for each integer

n
⩾
0

n \geqslant 0

, it is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript left-parenthesis n plus 1 right-parenthesis left-parenthesis n plus 2 right-parenthesis Baseline less-than-or-slanted-equals left-parenthesis n plus 1 right-parenthesis factorial left-parenthesis product Underscript k equals 0 Overscript n plus 1 Endscripts k factorial right-parenthesis Superscript negative 2 Baseline det double-vertical-bar upper M Subscript j k overbar Baseline double-vertical-bar Subscript j comma k equals 0 Superscript n Baseline comma">

C

(
n
+
1
)
(
n
+
2
)

⩽
(
n
+
1
)
!

(

∏

k
=
0

n
+
1

k
!

)

−
2

det

‖

M

j

k
¯

‖

j
,
k
=
0

n

,

{C^{(n + 1)(n + 2)}} \leqslant (n + 1)!{\left ( {\prod \limits _{k = 0}^{n + 1} {k!} } \right )^{ - 2}}\det \left \| {{M_{j\bar k}}} \right \|_{j,k = 0}^n,

\]
where

C
=
C
(
z
)

C = C(z)

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript j k overbar Baseline equals left-parenthesis partial-differential Superscript j plus k Baseline slash partial-differential z Superscript j Baseline partial-differential z overbar Superscript k Baseline right-parenthesis upper M left-parenthesis z right-parenthesis">

M

j

k
¯

=
(

∂

j
+
k

/

∂

z
j

∂

z
¯

k

)
M
(
z
)

{M_{j\bar k}} = ({\partial ^{j + k}}/\partial {z^j}\partial {\bar z^k})M(z)

. Equality occurs if and only if

Ω
\Omega

is conformally equivalent to the unit disk less (possibly) a closed set of inner capacity zero. The special case of this result, namely when

n
=
0

n = 0

, is due to Hejhal and Suita. Let

κ
(
z
)

\kappa (z)

be the curvature of the “span metric”

S
(
z
)

|

d
z

|

S(z)|dz|

. As an attempt to resolve a conjecture of Suita, we also show that for

Ω
∉

O

A
D

,
κ
(
z
)
⩽
−
2

\Omega \notin {O_{AD}},\kappa (z) \leqslant - 2

for each

z
∈
Ω

z \in \Omega

. Both results are proved by studying suitable extremal problems.
An elementary solution of the monotone mapping problem
Authors: Martin Joseph , Rosenholtz Ira
Published: June 2010

A simple example of a monotone, noncompact mapping from

R

3

{{\mathbf {R}}^3}

to

R

3

{{\mathbf {R}}^3}

is constructed.
A functional characterization of primitive base
Authors: Wicke Howard H.
Published: June 2010

All published formulations of the concept of primitive base involve the concept of well ordering in a prominent way. This paper presents various conditions, not involving well ordering, on a function

h
:
N
×
X
→
τ

h:N \times X \to \tau

such that if a space

(
X
,
τ
)

(X,\tau )

has a function satisfying one of these conditions, then it may be proved (using the axiom of choice) that it has a primitive base. These conditions are used in some characterizations of base of countable order. Examples show nonequivalence of the conditions with primitive base if the axiom of choice does not hold.
Differentiable pseudo-free circle actions on homotopy spheres
Authors: Liang Chao Chu
Published: June 2010

Let G denote the circle group,

φ
\varphi

a differentiable pseudo-free G-action of type

(

p
1

,
…
,

p
k

)

({p_1}, \ldots ,{p_k})

on a homotopy sphere

Σ

2
n
+
1

{\Sigma ^{2n + 1}}

, and X the vector field induced by

φ
\varphi

. If w is a G-invariant 1-form satisfying

w
(
X
)
=
1

w(X) = 1

, then we will prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript normal upper Sigma Endscripts w logical-and left-parenthesis d w right-parenthesis Superscript n Baseline equals plus-or-minus left-parenthesis p 1 p 2 midline-horizontal-ellipsis p Subscript k Baseline right-parenthesis Superscript negative 1">

∫
Σ

w
∧

(
d
w

)
n

=
±

(

p
1

p
2

⋯

p
k

)

−
1

{\smallint _\Sigma }w \wedge {(dw)^n} = \pm {({p_1}{p_2} \cdots {p_k})^{ - 1}}

.
Reductions of 𝑛-fold covers
Authors: Stahl Saul
Published: June 2010

Motivated by L. Lovász’s recent proof of the Kneser conjecture [3], this paper offers another result which relates topological and graph theoretical concepts. A method for converting n-fold covers to

(
n
−
1
)

(n - 1)

-fold covers is presented. This yields a strengthening of the classical Borsuk, Lusternik and Schnirelmann theorem on covers of spheres. The same conversion also has applications to multicolorings of graphs.
A characterization of nonatomic Hilbert algebras
Authors: Figà-Talamanca Alessandro , Mauceri Giancarlo
Published: June 2010

We say that a Hilbert algebra is atomic if its fulfillment is generated by its minimal projections. We prove that Hilbert algebra

A

\mathcal {A}

is not atomic if and only if there is an infinite group

G

\mathcal {G}

of unitary elements of the von Neumann algebra generated by

A

\mathcal {A}

, and an element

ξ
0

{\xi _0}

of the fulfillment of

A

\mathcal {A}

, which commutes with every element of

G

\mathcal {G}

, and such that set

{
U

ξ
0

:
U
∈

G

}

\{ U{\xi _0}:U \in \mathcal {G}\}

is orthonormal. This result is then applied to gain information on the Plancherel measure of certain unimodular groups.
On 𝐿¹ convergence of Fourier series with quasi-monotone coefficients
Authors: Garrett J. W. , Rees C. S. , Stanojević Č. V.
Published: June 2010

For the class of Fourier series with quasi-monotone coefficients, it is proved that

‖

s
n

−

σ
n

‖

=
o
(
1
)
,
n
→
∞

\left \| {{s_n} - {\sigma _n}} \right \| = o(1),n \to \infty

, if and only if

a
n

lg
⁡
n
=
o
(
1
)
,
n
→
∞

{a_n}\lg n = o(1),n \to \infty

. This generalizes a theorem for monotone coefficients and provides a new proof for a result due to Telyakovskii and Fomin.
On the fundamental group of a compact negatively curved manifold
Authors: Chen Su Shing
Published: June 2010

A question of Hirsch and Thurston on the fundamental group of a compact negatively curved manifold is investigated.
Tensor product of commutative unions of groups
Authors: Anderson James A. , Kimura Naoki
Published: June 2010

In this paper a description is given of the tensor product of two arbitrary commutative unions of groups.
Products of reflections in the unitary group
Authors: Djoković Dragomir Ž. , Malzan Jerry
Published: June 2010

Let

A
∈
U
(
n
)
,
det
(
A
)
=
±
1

A \in U(n),\det (A) = \pm 1

and let

exp
⁡
(
i

α
k

)
,
1
⩽
k
⩽
n

\exp (i{\alpha _k}),1 \leqslant k \leqslant n

be the eigenvalues of A where

0
⩽

α
1

⩽

α
2

⩽
⋯
⩽

α
n

>
2
π

0 \leqslant {\alpha _1} \leqslant {\alpha _2} \leqslant \cdots \leqslant {\alpha _n} > 2\pi

. Then

k
(
A
)
=
(

α
1

+
⋯
+

α
n

)

/

π

k(A) = ({\alpha _1} + \cdots + {\alpha _n})/\pi

is an integer and

0
⩽
k
(
A
)
⩽
2
n
−
1

0 \leqslant k(A) \leqslant 2n - 1

. Denote by

l
(
A
)

l(A)

the length of A with respect to the set of all reflections, i.e.,

l
(
A
)

l(A)

is the smallest integer m such that A is a product of m reflections. A reflection is a matrix conjugate to

diag

(
−
1
,
1
,
…
,
1
)

{\text {diag}}( - 1,1, \ldots ,1)

. Our main result is the formula

l
(
A
)
=
max
(
k
(
A
)
,
k
(

A
∗

)
)

l(A) = \max (k(A),k({A^\ast }))

.
Two applications of change of rings theorems for Poincaré series
Authors: Herzog Jürgen , Steurich Manfred
Published: June 2010

Let (R, m, k) be an artinian Gorenstein ring with

dim
k

m

/

m
2

⩾
2

{\dim _k}m/{m^2} \geqslant 2

. If

P
R

{P^R}

denotes the Poincaré-series and

ϕ
R

{\phi _R}

denotes the Bass-series of R, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript upper R Baseline equals left-parenthesis upper P Superscript upper R Baseline minus 1 right-parenthesis x Superscript negative 1 Baseline left-parenthesis 1 minus upper P Superscript upper R Baseline x squared right-parenthesis Superscript negative 1">

ϕ
R

=
(

P
R

−
1
)

x

−
1

(
1
−

P
R

x
2

)

−
1

{\phi _R} = ({P^R} - 1){x^{ - 1}}{(1 - {P^R}{x^2})^{ - 1}}

with

R
¯

=
R

/

0
:
m

\bar R = R/0:m

, see Proposition 1. With similar methods as in the proof of Proposition 1 we calculate the Poincaré-series of a certain class of local rings: Let (R, m, k) be a local ring,

(

α

i
j

)

({\alpha _{ij}})

a

n
−
1
×
n

n - 1 \times n

-matrix with entries in m,

n
⩾
3

n \geqslant 3

. Let

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

be elements in m and

A

\mathfrak {A}

the ideal generated by the maximal minors of

(

α

i
j

)

({\alpha _{ij}})

and by

Σ
j

α

i
j

x
j

(
i
=
1
,
…
,
n
−
1
)

{\Sigma _j}{\alpha _{ij}}{x_j}(i = 1, \ldots ,n - 1)

. If depth

A

=
n

\mathfrak {A} = n

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Superscript upper R slash German upper A Baseline dot left-parenthesis left-parenthesis 1 minus upper X right-parenthesis Superscript n minus 1 Baseline minus upper X right-parenthesis left-parenthesis 1 plus upper X right-parenthesis Superscript n Baseline equals upper P Superscript upper R">

P

R

/

A

⋅
(

(
1
−
X

)

n
−
1

−
X
)

(
1
+
X

)
n

=

P
R

{P^{R/\mathfrak {A}}} \cdot ({(1 - X)^{n - 1}} - X){(1 + X)^n} = {P^R}

.
A Helly-type theorem for the dimension of the kernel of a starshaped set
Authors: Breen Marilyn
Published: June 2010

This study will investigate the dimension of the kernel of a starshaped set, and the following result will be obtained: Let S be a compact set in some linear topological space L. For

1
⩽
k
⩽
n

1 \leqslant k \leqslant n

, the dimension of

ker
⁡
S

\ker S

is at least k if and only if for some

ε
>
0

\varepsilon > 0

and some n-dimensional flat

F
n

{F^n}

in L, every

f
(
n
,
k
)

f(n,k)

points of S see via S a common k-dimensional neighborhood in

F
n

{F^n}

having radius

ε
\varepsilon

. The number

f
(
n
,
k
)

f(n,k)

is defined inductively as follows:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row f left-parenthesis 2 comma 1 right-parenthesis equals 4 comma 2nd Row f left-parenthesis n comma k right-parenthesis equals f left-parenthesis n minus 1 comma k right-parenthesis plus n plus 2 for 3 less-than-or-slanted-equals n and 1 less-than-or-slanted-equals k less-than-or-slanted-equals n minus 1 comma 3rd Row f left-parenthesis n comma n right-parenthesis equals n plus 1 period EndLayout">

f
(
2
,
1
)
=
4
,

f
(
n
,
k
)
=
f
(
n
−
1
,
k
)
+
n
+
2

for

3
⩽
n

and

1
⩽
k
⩽
n
−
1
,

f
(
n
,
n
)
=
n
+
1.

\begin {array}{*{20}{c}} {f(2,1) = 4,} \hfill \\ {f(n,k) = f(n - 1,k) + n + 2\quad {\text {for}}\;3 \leqslant n\;{\text {and}}\;1 \leqslant k \leqslant n - 1,} \hfill \\ {f(n,n) = n + 1.} \hfill \\ \end {array}

\]
On subvarieties of a Hermitian manifold
Authors: Ferraris Carlos J.
Published: June 2010

We make use of the variational properties of the geodesic distance function of a Riemannian manifold and the technique of the “blowing-up” (

σ
\sigma

-process) on a complex manifold to derive the nonexistence of compact complex analytic subvarieties in a simply connected, complete Hermitian manifold with nonpositive sectional curvature.
Compactification of a convergence space
Authors: Kumar Vinod
Published: June 2010

A characterization for the class of convergence spaces having the largest Hausdorff compactification is given, and regularity and

λ
\lambda

-Hausdorffness of modified Richardson compactification are discussed.
Crystallisations of 2-fold branched coverings of 𝑆³
Authors: Ferri Massimo
Published: June 2010

We describe the construction of a crystallisation of a 2-fold cyclic covering space of

S
3

{S^3}

branched over a link, from a bridge-presentation of the branch set.
Reflexive modules over certain differential polynomial rings
Authors: Haghany A.
Published: June 2010

Let K be a commutative Noetherian integral domain with a derivation d and let

R
=
K
[
x
,
d
]

R = K[x,d]

. When K is quasi-normal and d is suitably restricted we shall give several equivalent conditions for an R-module to be reflexive. The relations between K and R in the context of reflexivity are also investigated.
An 𝐿¹ remainder theorem for an integrodifferential equation with asymptotically periodic solution
Authors: Hannsgen Kenneth B.
Published: June 2010

For a certain integrodifferential equation of Volterra type on

(
0
,
∞
)

(0,\infty )

, with piecewise linear convolution kernel, it is shown that the solution is

u
(
t
)
=
α

cos
⁡
β
t
+
ρ
(
t
)

u(t) = \alpha \;\cos \beta t + \rho (t)

, with

ρ
∈

L
1

(
0
,
∞
)

\rho \in {L^1}(0,\infty )

and

α
\alpha

and

β
\beta

constant;

u
′

u’

is represented similarly.