bluuebunny/crossref_metadata_2025 huggingface dataset:
Higher order Turán inequalities
Authors: Dimitrov Dimitar
Published: July 2002

The celebrated Turán inequalities
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis minus upper P Subscript n minus 1 Baseline left-parenthesis x right-parenthesis upper P Subscript n plus 1 Baseline left-parenthesis x right-parenthesis greater-than-or-equal-to 0">

P

n

2

(
x
)
−

P

n
−
1

(
x
)

P

n
+
1

(
x
)
≥
0

P_{n}^{2}(x) - P_{n-1}(x) P_{n+1}(x) \geq 0

,

x
∈
[
−
1
,
1
]

x \in [-1,1]

,

n
≥
1

n \geq 1

, where

P

n

(
x
)

P_{n}(x)

denotes the Legendre polynomial of degree

n
n

, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities

γ

n

2

−

γ

n
−
1

γ

n
+
1

≥
0

\gamma _{n}^{2} - \gamma _{n-1} \gamma _{n+1} \geq 0

,

n
≥
1

n \geq 1

, which hold for the Maclaurin coefficients of the real entire function

ψ
\psi

in the Laguerre-Pólya class,

ψ
(
x
)
=

∑

n
=
0

∞

γ

n

x

n

/

n
!

\psi (x) = \sum _{n=0}^{\infty } \gamma _{n} x^{n}/n!

.
Image multiple point spaces and rectified homotopical depth
Authors: Houston Kevin
Published: July 2002

We investigate the local topology of multiple point spaces in the image of a finite complex analytic map. In particular, we find examples of maps for which the constant sheaf on an image multiple point space of the map is perverse. These results are proved by the use of a spectral sequence which calculates the homology of the image of a continuous map.
Some structure of Borel locales
Authors: Isbell John
Published: July 2002

All Borel classes of sublocales of the real line after the first ambiguous class (in particular, the limit ambiguous classes) have proper (=irreducible) representatives.
Layered circlepackings and the type problem
Authors: Siders Ryan
Published: July 2002

We study the geometric type of a surface packed with circles. For circles packed in concentric layers of uniform degree, the circlepacking is specified by this sequence of degrees. We write an infinite sum whose convergence discerns the geometric type: if

h
i

h_i

layers of degree

6
6

follow the

i
i

th layer of degree

7
7

, and the

i
i

th layer of degree

7
7

has

c
i

c_i

circles, then

∑
log
⁡
(

h
i

)

/

c
i

\sum \log (h_i)/c_i

converges/diverges as the circlepacking is hyperbolic/Euclidean. We illustrate a hyperbolic circlepacking with surprisingly few layers of degree

>
6

>6

.
On conformal distortion and Sullivan’s sector theorem
Authors: de Faria Edson
Published: July 2002

Using general bounds on the conformal distortion of univalent maps, we prove a strong version of Sullivan’s sector theorem, which gives certain sufficient conditions for an arbitrarily long composition of univalent Herglotz functions to map the upper half-plane into a proper sub-sector.
Character degrees and local subgroups of 𝜋-separable groups
Authors: Navarro Gabriel , Wolf Thomas
Published: July 2002

Let

G
G

be a finite

{
p
,
q
}

\{p,q \}

-solvable group for different primes

p
p

and

q
q

. Let

P
∈

Syl

p

(
G
)

P \in \text {Syl}_{p}(G)

and

Q
∈

Syl

q

(
G
)

Q \in \text {Syl}_{q}(G)

be such that

P
Q
=
Q
P

PQ=QP

. We prove that every

χ
∈
Irr
(
G
)

\chi \in \text {Irr}(G)

of

p

′

p^{\prime }

-degree has

q

′

q^{\prime }

-degree if and only if

N

G

(
P
)
⊆

N

G

(
Q
)

\mathbf {N}_{G}(P) \subseteq \mathbf {N}_{G}(Q)

and

C

Q

′

(
P
)
=
1

\mathbf {C}_{Q^{\prime }}(P)=1

.
Some theta function identities related to the Rogers-Ramanujan continued fraction
Authors: Son Seung
Published: July 2002

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction

F
(
q
)

F(q)

. In order to prove some of these claims, G. N. Watson established two important theorems about

F
(
q
)

F(q)

that he found in Ramanujan’s notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.
Two finitely generated varieties having no infinite simple members
Authors: Willard Ross
Published: July 2002

Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote’s negative solution to Problem 11 from Hobby and McKenzie, The Structure of Finite Algebras.
Noncomplete linear systems on elliptic curves and Abelian varieties: Addendum to a paper by Ch. Birkenhake
Authors: Ballico E.
Published: July 2002

Here we give a result on the postulation (i.e. the 2-normality) of nonlinearly normal embeddings of Abelian varieties. This result improves some of the results proved in a recent paper by Ch. Birkenhake.
Continuity of Lie mappings of the skew elements of Banach algebras with involution
Authors: Berenguer M. , Villena A.
Published: July 2002

Let

A
A

and

B
B

be centrally closed prime complex Banach algebras with linear involution. If

A
A

is semisimple, then any Lie derivation of the skew elements of

A
A

is continuous and any Lie isomorphism from the skew elements of

B
B

onto the skew elements of

A
A

is continuous.
On the homotopy invariance of 𝐿² torsion for covering spaces
Authors: Mathai Varghese , Rothenberg Melvin
Published: July 2002

We prove the homotopy invariance of

L
2

L^2

torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the

L
2

L^2

cohomology of the covering space vanishes, the homotopy invariance was established by Lück. We also give some applications of our results.
Vector bundles on log terminal varieties
Authors: Mella Massimiliano
Published: July 2002

Let

X
X

be an

n
n

-dimensional variety and

E
E

an ample vector bundle on

X
X

of rank

e
e

. We give a complete classification of pairs

(
X
,
E
)

(X,E)

, with

X
X

log terminal and

e
≥
n

e\geq n

such that

K
X

+
d
e
t
E

K_X+det E

is not ample. The results we obtain were conjectured by Fujita, and recently by Zhang.
Unique decomposition of Riemannian manifolds
Authors: Eschenburg J.-H. , Heintze E.
Published: July 2002

We prove an extension of de Rham’s decomposition theorem to the non-simply connected case.
Optimal control of a functional equation associated with closed range selfadjoint operators
Authors: Gao S. , Pavel N.
Published: July 2002

Necessary and sufficient conditions for the optimality of a pair

(

y

∗

,

u

∗

)

(y^{*}, u^{*})

subject to

A

y

∗

=
B

u

∗

+
f

Ay^{*} = Bu^{*} + f

are given. Here

A
A

is a selfadjoint operator with closed range on a Hilbert space

H

\mathcal {H}

and

B
∈
L
(

H

)

B \in L(\mathcal {H})

. The case

B
B

– unbounded is also discussed, which leads to some open problems. This general functional scheme includes most of the previous results on the optimal control of the

T
T

–periodic wave equation for all

T
T

in a dense subset of

R

\mathbb {R}

. It also includes optimal control problems for some elliptic equations.
Nonsymmetric Osserman indefinite Kähler manifolds
Authors: Bonome A. , Castro R. , García-Río E. , Hervella L. , Vázquez-Lorenzo R.
Published: July 2002

The authors prove the existence of Osserman manifolds with indefinite Kähler metric of nonnegative or nonpositive holomorphic sectional curvature which are not locally symmetric.
A new characteristic of Möbius transformations by use of Apollonius quadrilaterals
Authors: Haruki Hiroshi , Rassias Themistocles
Published: July 2002

The purpose of this paper is to give a new invariant characteristic property of Möbius transformations from the standpoint of conformal mapping. To this end a new concept of “Apollonius quadrilaterals” is used.
The geography of irreducible 4-manifolds
Authors: Park Jongil
Published: July 2002

In this paper we investigate the existence and the uniqueness problems for simply connected irreducible

4
4

-manifolds. By taking fiber sums along an embedded surface of square

0
0

and by a rational blow-down procedure, we construct many new irreducible

4
4

-manifolds which have infinitely many distinct smooth structures. Furthermore, we prove that all but at most finitely many lattice points lying in the non-positive signature region with

2
e
+
3
s
i
g
n
≥
0

2e+3 sign \geq 0

are covered by these irreducible

4
4

-manifolds.
A metric space of A. H. Stone and an example concerning 𝜎-minimal bases
Authors: Bennett Harold , Lutzer David
Published: July 2002

In this paper we use a metric space

Y
Y

due to A. H. Stone and one of its completions

X
X

to construct a linearly ordered topological space

E
=
E
(
Y
,
X
)

E = E(Y,X)

that is Čech complete, has a

σ
\sigma

-closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a

σ
\sigma

-minimal base for its relative topology. However,

E
E

is not metrizable and is not quasi-developable. The construction of

E
(
Y
,
X
)

E(Y,X)

is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone’s metric space and our construction.
On an analogue of Selberg’s eigenvalue conjecture for 𝑆𝐿₃(𝐙)
Authors: Catto Sultan , Huntley Jonathan , Jorgenson Jay , Tepper David
Published: July 2002

Let

H

\mathcal {H}

be the homogeneous space associated to the group

PGL

3

(

R

)

\text {PGL}_{3}(\mathbf {R})

. Let

X
=
Γ

∖

H

X = \Gamma {\backslash \mathcal {H}}

where

Γ
=

SL

3

(

Z

)

\Gamma = \text {SL}_{3}(\mathbf {Z})

and consider the first nontrivial eigenvalue

λ

1

\lambda _{1}

of the Laplacian on

L

2

(
X
)

L^{2}(X)

. Using geometric considerations, we prove the inequality

λ

1

>
3

π

2

/

10

\lambda _{1} > 3\pi ^{2}/10

. Since the continuous spectrum is represented by the band

[
1
,
∞
)

[1,\infty )

, our bound on

λ

1

\lambda _{1}

can be viewed as an analogue of Selberg’s eigenvalue conjecture for quotients of the hyperbolic half space.
Unitarily-invariant linear spaces in C*-algebras
Authors: Marcoux L. , Murphy G.
Published: July 2002

Characterisations and containment results are given for linear subspaces of a unital C*-algebra that are invariant under conjugation by sets of unitary elements of the algebra. The (unitarily-invariant) linear span of the projections in a simple, unital C*-algebra having non-scalar projections is shown to contain all additive commutators of the algebra and, in certain cases, to be equal to the algebra.
Positive solutions of nonlinear elliptic equations in the Euclidean plane
Authors: Ufuktepe U. , Zhao Z.
Published: July 2002

In the present paper, we study the existence of solutions to the problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column normal upper Delta u plus f left-parenthesis x comma u right-parenthesis equals 0 2nd Column a m p semicolon in upper D 2nd Row 1st Column u greater-than 0 2nd Column a m p semicolon in upper D 3rd Row 1st Column u equals 0 2nd Column a m p semicolon on partial-differential upper D EndLayout">

{

Δ
u
+
f
(
x
,
u
)
=
0

a
m
p
;

in 

D

u
>
0

a
m
p
;

in 

D

u
=
0

a
m
p
;

on 

∂
D

\begin {cases} \Delta u+f(x,u)=0 & \text {in $D$}\\ u>0&\text {in $D$}\\ u=0 & \text {on $\partial D$} \end {cases}

\]
where

D
D

is an unbounded domain in

R

2

\mathbb {R}^2

with a compact nonempty boundary

∂
D

\partial D

consisting of finitely many Jordan curves. The goal is to prove an existence theorem for the above problem in a general setting by using Brownian path integration and potential theory.
Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type
Authors: Ha Chung-Wei
Published: July 2002

We consider the eigenvalue problem

u
+
λ
u
+
p
(
x
)
u
=
0

u+\lambda u +p(x)u=0

in

(
0
,
π
)

(0,\pi )

,

u
(
0
)
=
u
(
π
)
=
0

u(0)=u(\pi )=0

, where

p
∈

L

1

(
0
,
π
)

p\in L^{1}(0,\pi )

keeps a fixed sign and

‖
p

‖

L

1

>
0

\|p\|_{L^{1}}> 0

, and we obtain some lower and upper bounds for

‖
p

‖

L

1

\|p\|_{L^{1}}

in terms of its nonnegative eigenvalues

λ
\lambda

. Two typical results are: (1)

‖
p

‖

L

1

>

λ

|

sin
⁡

λ

π

|

\|p\|_{L^{1}}>\sqrt {\lambda }\,|\sin {\sqrt {\lambda }\,\pi }|

if

λ
>
1

\lambda > 1

and is not the square of a positive integer; (2)

‖
p

‖

L

1

≤
16

/

π

\|p\|_{L^{1}}\le 16/\pi

if

λ
=
0

\lambda =0

is the smallest eigenvalue.
The Dedekind-Mertens formula and determinantal rings
Authors: Bruns Winfried , Guerrieri Anna
Published: July 2002

We give a combinatorial proof of the Dedekind–Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank

1
1

Cohen–Macaulay modules over the determinantal rings

K
[
X
]

/

I
2

(
X
)

K[X]/I_2(X)

.
The covering numbers and “low 𝑀*-estimate" for quasi-convex bodies
Authors: Litvak A. , Milman V. , Pajor A.
Published: July 2002

This article gives estimates on the covering numbers and diameters of random proportional sections and projections of quasi-convex bodies in

R

\mathbb {R}

. These results were known for the convex case and played an essential role in the development of the theory. Because duality relations cannot be applied in the quasi-convex setting, new ingredients were introduced that give new understanding for the convex case as well.
Loewner’s theorem for kernels having a finite number of negative squares
Authors: Alpay D. , Rovnyak J.
Published: July 2002

By a theorem of Loewner, a continuously differentiable real-valued function on a real interval whose difference quotient is a nonnegative kernel is the restriction of a holomorphic function which has nonnegative imaginary part in the upper half-plane and is holomorphic across the interval. An analogous result is obtained when the difference-quotient kernel has a finite number of negative squares.
Local spectral theory and orbits of operators
Authors: Miller T. , Miller V.
Published: July 2002

ForT∈L(X)T\in \mathcal {L}(X), we give a condition that suffices forφ(T)\varphi (T)to be hypercyclic whereφ\varphiis a nonconstant function that is analytic on the spectrum ofTT. In the other direction, it is shown that a property introduced by E. Bishop restricts supercyclic phenomena: ifT∈L(X)T\in \mathcal {L}(X)is finitely supercyclic and has Bishop’s property(β)(\beta ), then the spectrum ofTTis contained in a circle.
On the excess of sets of complex exponentials
Authors: Fujii Nobuhiko , Nakamura Akihiro , Redheffer Ray
Published: July 2002

For

−
∞
>
n
>
∞

-\infty >n>\infty

let

μ
n

\mu _n

be complex numbers such that

μ
n

−
n

\mu _n-n

is bounded. For

n
>
0

n>0

define

λ
n

=

μ
n

+
a

\lambda _n=\mu _n+a

,

λ

−
n

=

μ

−
n

−
b

\lambda _{-n}=\mu _{-n}-b

where

a
,
b
≥
0

a,b\ge 0

. Then the excesses

E
E

in the sense of Paley and Wiener satisfy

E
(
{

λ
n

}
)
≤
E
(
{

μ
n

}
)

E(\{\lambda _n\})\le E(\{\mu _n\})

.
On symmetry of discrete polynomial hypergroups
Authors: Gebuhrer Marc-Olivier , Szwarc Ryszard
Published: July 2002

Let

H
H

be a discrete polynomial hypergoup on

N

\mathbb {N}

with Plancherel measure

μ
.

\mu .

If the hypergroup

H
H

is symmetric, the set of characters

H
^

\widehat {H}

can be identified with a compact subset of the real line which contains the support of

μ
.

\mu .

We show that the lower and upper bounds of

H
^

\widehat {H}

and

supp
⁡
μ

\operatorname {supp} \mu

coincide. In particular, the trivial character belongs to the support of the Plancherel measure.
Integrability of superharmonic functions, uniform domains, and Hölder domains
Authors: Gotoh Yasuhiro
Published: July 2002

Let

S
+

(
D
)

S^+(D)

denote the space of all positive superharmonic functions on a domain

D
⊂

R

n

D \subset \mathbf R^n

. Lindqvist showed that

log
⁡

S
+

(
D
)

\log S^+(D)

is a bounded subset of

B
M
O
(
D
)

BMO(D)

. Using this, we give a characterization of finitely connected

2
2

-dimensional uniform domains and remarks on Hölder domains.
An interpolation theorem related to the a.e. convergence of integral operators
Authors: Kiselev Alexander
Published: July 2002

We show that for integral operators of general form the norm bounds in Lorentz spaces imply certain norm bounds for the maximal function. As a consequence, the a.e. convergence for the integral operators on Lorentz spaces follows from the appropriate norm estimates.
Semi-invariants and weights of group algebras of finite groups
Authors: Passman D. , Wauters P.
Published: July 2002

We study the semi-invariants and weights of a group algebra

K
[
G
]

K[G]

over a field

K
K

of characteristic zero. Specifically, we show that certain basic results which hold when

G
G

is a polycyclic-by-finite group with

Δ

+

(
G
)
=
1

\Delta ^{+}(G) = 1

need not hold in the case of group algebras of finite groups. This turns out to be a purely group theoretic question about the existence of class preserving automorphisms.
On the finite dimensional unitary representations of Kazhdan groups
Authors: Rapinchuk A.
Published: July 2002

We use A. Weil’s criterion to prove that all finite dimensional unitary representations of a discrete Kazhdan group are locally rigid. It follows that any such representation is unitarily equivalent to a unitary representation over some algebraic number field.
Hamiltonian stationary normal bundles of surfaces in 𝐑³
Authors: Sakaki Makoto
Published: July 2002

A surface in

R

3

\mathbf R^3

has Hamiltonian stationary normal bundle if and only if it is either minimal, a part of a round sphere, or a part of a cone with vertex angle

π

/

2

\pi /2

.
Binomial free resolutions for normal toric surfaces
Authors: Gasharov Vesselin , Peeva Irena
Published: July 2002

We construct the minimal free resolution of the residue field over a normal toric surface.
Support functionals and smooth points in abstract 𝑀 spaces
Authors: Baoxiang Wang , Tingfu Wang
Published: July 2002

By presenting some properties of support functionals in abstract

M
M

spaces, we get some sufficient and necessary conditions for smooth points in abstract

M
M

(function) spaces. Moreover, the notion of the smallest support semi-norm is introduced and an explicit form for this functional in abstract

M
M

function spaces is also given.
Applications of Michael’s continuous selection theorem to operator extension problems
Authors: Zippin M.
Published: July 2002

A global approach and Michael’s continuous selection theorem are used to prove a slightly improved version of the Lindenstrauss - Pełczyński extension theorem for operators from subspaces of

c
0

c_0

into

C
(
K
)

C (K)

spaces.
Accelerations of Riemannian quadratics
Authors: Noakes Lyle
Published: July 2002

A Riemannian corner-cutting algorithm generalizing a classical construction for quadratics was previously shown by the author to produce a

C
1

C^1

curve

p
∞

p_\infty

whose derivative is Lipschitz. The present paper takes the analysis of

p
∞

p_\infty

a step further by proving that it possesses left and right accelerations everywhere. Two-sided accelerations are shown to exist on the complement of a countable dense subset

D
D

of the domain. The results are shown to be sharp in the following sense. For almost any scaled triple in Euclidean space there is a Riemannian perturbation of the Euclidean metric such that the two-sided accelerations of the resulting curve

p
∞

p_\infty

exist nowhere in

D
D

.
Singular measures with absolutely continuous convolution squares on locally compact groups
Authors: Bisbas Antonis
Published: July 2002

Saeki’s result states that on any locally compact nondiscrete group there exist continuous singular measures, with respect to the left Haar measure,

μ
\mu

with

μ
∗
μ

\mu * \mu

in

L

p

L^{p}

for all

p
,

1
≤
p
>
∞

p, \;1\leq p > \infty

. This paper gives a new and short proof of this using Rademacher-Riesz products.
Controlling and supporting sets in operator algebras
Authors: Burnap Charles , Lambert Alan
Published: July 2002

Two properties of subspaces of Banach algebras are introduced. These are closely related to Mathieu’s concept of ultraprime Banach algebras. Special attention is paid to nested

C
∗

C^*

and

W
∗

W^*

algebras.
Hardy’s inequality for 𝑊^{1,𝑝}₀-functions on Riemannian manifolds
Authors: Miklyukov Vladimir , Vuorinen Matti
Published: July 2002

We prove that for every Riemannian manifold

X

\mathcal {X}

with the isoperimetric profile of particular type there holds an inequality of Hardy type for functions of the class

W
0

1
,
p

(

X

)

W_0^{1,p}( \mathcal {X})

. We also study manifolds satisfying Hardy’s inequality and, in particular, we establish an estimate for the rate of growth of the weighted volume of the noncompact part of such a manifold.
Compactness of composition operators on BMOA
Authors: Smith Wayne
Published: July 2002

A function theoretic characterization is given of when a composition operator is compact on BMOA, the space of analytic functions on the unit disk having radial limits that are of bounded mean oscillation on the unit circle. When the symbol of the composition operator is univalent, compactness on BMOA is shown to be equivalent to compactness on the Bloch space, and a characterization in terms of the geometry of the image of the disk under the symbol of the operator results.
Solution of a functional equation arising from utility that is both separable and additive
Authors: Aczél János , Ger Roman , Járai Antal
Published: July 2002

The problem of determining all utility measures over binary gambles that are both separable and additive leads to the functional equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis v right-parenthesis equals f left-parenthesis v w right-parenthesis plus f left-bracket v upper Q left-parenthesis w right-parenthesis right-bracket comma v comma v upper Q left-parenthesis w right-parenthesis element-of left-bracket 0 comma k right-parenthesis comma w element-of left-bracket 0 comma 1 right-bracket period">

f
(
v
)
=
f
(
v
w
)
+
f
[
v
Q
(
w
)
]
,

v
,
v
Q
(
w
)
∈
[
0
,
k
)
,

w
∈
[
0
,
1
]

.

\begin{equation*}f(v) = f(vw) + f[vQ(w)], \qquad v, vQ(w) \in [0,k),\quad w \in [0,1]\,. \end{equation*}

The following conditions are more or less natural to the problem:

f
f

strictly increasing,

Q
Q

strictly decreasing; both map their domains onto intervals (

f
f

onto a

[
0
,
K
)

[0,K)

,

Q
Q

onto

[
0
,
1
]

[0,1]

); thus both are continuous,

k
>
1

k>1

,

f
(
0
)
=
0

f(0)=0

,

f
(
1
)
=
1

f(1)=1

,

Q
(
1
)
=
0

Q(1)=0

,

Q
(
0
)
=
1

Q(0)=1

. We determine, however, the general solution without any of these conditions (except

f
:
[
0
,
k
)
→

R

+

:=
[
0
,
∞
)

f: [0,k) \to \mathbb {R}_{+}:= [0, \infty )

,

Q
:
[
0
,
1
]
→

R

+

Q:[0,1]\to \mathbb {R}_{+}

, both into). If we exclude two trivial solutions, then we get as general solution

f
(
v
)
=
α

v

β

f(v) = \alpha v^{\beta }

(

β
>
0

\beta > 0

,

α
>
0

\alpha > 0

;

α
=
1

\alpha = 1

for

f
(
1
)
=
1

f(1) = 1

), which satisfies all the above conditions. The paper concludes with a remark on the case where the equation is satisfied only almost everywhere.
Level sets of a typical 𝐶ⁿ function
Authors: Darji Udayan , Morayne Michał
Published: July 2002

We determine the level set structure of a typical

C
n

C^n

function.
A Moore space with a 𝜎-discrete 𝜋-base which cannot be densely embedded in any Moore space with the Baire property
Authors: Fearnley David
Published: July 2002

The author answers a question raised in the literature about twenty five years ago and raised again more recently in Open Problems in Topology, by G. M. Reed, concerning the conjecture that every Moore space with a

σ
\sigma

-discrete

π
\pi

-base can be densely embedded in a Moore space having the Baire property. Even though closely related results have made this conjecture seem likely to be true, the author shows that, surprisingly, the conjecture is false.
A Bessel function multiplier
Authors: Oberlin Daniel , Smith Hart
Published: July 2002

We obtain nearly sharp estimates for the

L

p

(

R

2

)

L^{p}({\mathbb {R}}^{2})

norms of certain convolution operators.
Decomposing a 4-manifold
Authors: Akbulut Selman
Published: July 2002

A splitting of a manifold in Kirby’s problem list (no. 4.97) is given.
An extension of the integral test
Authors: Durán Ana , Estrada Ricardo
Published: July 2002

We identify a class of functions for which an extension of the well known integral test holds, namely, the series

∑

k
=
1

∞

ϕ
(
k
)

\sum _{k=1}^{\infty }\phi (k)

is Cesàro summable if and only if the integral

∫

1

∞

ϕ
(
x
)

d
x

\int _{1}^{\infty }\phi (x)\,dx

is. We also give the corresponding multidimensional results.
On a theorem of Scott and Swarup
Authors: Mitra Mahan
Published: July 2002

Let

1
→
H
→
G
→

Z

→
1

1 \rightarrow H \rightarrow G \rightarrow \mathbb {Z} \rightarrow 1

be an exact sequence of hyperbolic groups induced by an automorphism

ϕ
\phi

of the free group

H
H

. Let

H
1

(
⊂
H
)

H_1 ( \subset H)

be a finitely generated distorted subgroup of

G
G

. Then there exist

N
>
0

N > 0

and a free factor

K
K

of

H
H

such that the conjugacy class of

K
K

is preserved by

ϕ
N

\phi ^N

and

H
1

H_1

contains a finite index subgroup of a conjugate of

K
K

. This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.
There are no denting points in the unit ball of 𝑊𝐶(𝐾,𝑋)
Authors: Rao T.
Published: July 2002

For an infinite compact set

K
K

and for any Banach space

X
X

we show that the unit ball of the space of

X
X

-valued functions that are continuous when

X
X

is equipped with the weak topology, has no denting points.
The multidimensional 𝑝-adic Green function
Authors: Chuong Nguyen , Van Co Nguyen
Published: July 2002

A proof much simpler than the one given by Bikulov (Investigation of the

p
p

-adic Green function, Teoret. Mat. Fiz. 87 (1991), 376–390) for properties of the 2-dimensional

p
p

-adic Green function is shown. By this method one can treat a multidimensional case, and some sharp estimates are obtained.
The Cremmer-Gervais solution of the Yang-Baxter equation
Authors: Hodges Timothy
Published: July 2002

A direct proof is given of the fact that the Cremmer-Gervais

R
R

-matrix satisfies the (Quantum) Yang-Baxter equation
Tensor products of subnormal operators
Authors: Feldman Nathan
Published: July 2002

We shall use a

C
∗

C^*

–algebra approach to study operators of the form

S
⊗
N

S \otimes N

where

S
S

is subnormal and

N
N

is normal. We shall determine the spectral properties for these operators, and find the minimal normal extension and the dual operator. We also give a necessary condition for

C
∗

(
S
⊗
N
)

C^*(S \otimes N)

to contain a compact operator and a sufficient condition for the algebraic equivalence of

S
⊗
N

S \otimes N

and

S
⊗
M

S \otimes M

. We also consider the existence of a

∗
−

*-

homomorphism

ϕ
:

C
∗

(
S
⊗
T
)
→

C
∗

(
S
)

\phi :C^*(S \otimes T) \to C^*(S)

satisfying

ϕ
(
S
⊗
T
)
=
S

\phi (S \otimes T) = S

. We shall characterize the operators

T
T

such that

ϕ
\phi

exists for every operator

S
S

. The problem of when

S
⊗
N

S \otimes N

is unitarily equivalent to

S
⊗
M

S \otimes M

is considered. Complete results are given when

N
N

and

M
M

are positive operators with finite multiplicity functions and

S
S

has compact self–commutator. Some examples are also given.
Lusin’s theorem for derivatives with respect to a continuous function
Authors: Aversa V. , Preiss D.
Published: July 2002

For a nowhere constant continuous function

g
g

on a real interval

I
I

and for a Borel measure

μ
\mu

on

I
I

, we give simple necessary and sufficient conditions guaranteeing, for any Borel function

f
f

on

I
I

, the existence of a continuous function

F
F

on

I
I

such that the derivative of

F
F

with respect to

g
g

is,

μ
\mu

almost everywhere, equal to

f
f

.
Porosity of ill-posed problems
Authors: Deville Robert , Revalski Julian
Published: July 2002

We prove that in several classes of optimization problems, including the setting of smooth variational principles, the complement of the set of well-posed problems is

σ
\sigma

-porous.
Asymptotic Eulerian expansions for binomial and negative binomial reciprocals
Authors: Marciniak Ewa , Wesołowski Jacek
Published: July 2002

Asymptotic expansions of any order for expectations of inverses of random variables with positive binomial and negative binomial distributions are obtained in terms of the Eulerian polynomials. The paper extends and improves upon an expansion due to David and Johnson (1956-7).
Kamienny’s criterion and the method of Coleman and Chabauty
Authors: Baker Matthew
Published: July 2002

This paper gives a new proof of Kamienny’s Criterion using the method of Coleman and Chabauty.
Strict regular completions of Cauchy spaces
Authors: Brock Paul , Richardson Gary
Published: July 2002

A diagonal condition is defined and used in characterizing the Cauchy spaces which have a strict, regular completion.
Successional stability of vector fields in dimension three
Authors: Schreiber Sebastian
Published: July 2002

A topological invariant, the community transition graph, is introduced for dissipative vector fields that preserve the skeleton of the positive orthant. A vector field is defined to be successionally stable if it lies in an open set of vector fields with the same community transition graph. In dimension three, it is shown that vector fields for which the origin is a connected component of the chain recurrent set can be approximated in the

C
1

C^1

Whitney topology by a successionally stable vector field.
A counterexample to the Fredholm alternative for the 𝑝-Laplacian
Authors: Drábek Pavel , Takáč Peter
Published: July 2002

The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional

p
p

-Laplacian with

1
>
p
>
∞

1 > p > \infty

is considered:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel minus left-parenthesis StartAbsoluteValue u Superscript prime Baseline EndAbsoluteValue Superscript p minus 2 Baseline u Superscript prime Baseline right-parenthesis Superscript prime Baseline minus lamda StartAbsoluteValue u EndAbsoluteValue Superscript p minus 2 Baseline u equals f left-parenthesis x right-parenthesis for 0 greater-than x greater-than upper T semicolon u left-parenthesis 0 right-parenthesis equals u left-parenthesis upper T right-parenthesis equals 0 comma EndLayout">

(*)

−
(

|

u
′

|

p
−
2

u
′

)
′

−
λ

|

u

|

p
−
2

u
=
f
(
x
)

 for 

0
>
x
>
T
;

u
(
0
)
=
u
(
T
)
=
0
,

\begin{equation*} - (|u’|^{p-2} u’)’ - \lambda |u|^{p-2} u = f(x) \;\mbox { for } 0 > x > T ; \quad u(0) = u(T) = 0 , \tag {*}\end{equation*}

where

f
≡
1
+
h

f\equiv 1 + h

with

h
∈

L
∞

(
0
,
T
)

h\in L^\infty (0,T)

small enough. Solvability properties of Problem (*) with respect to the spectral parameter

λ
∈

R

\lambda \in \mathbb {R}

are investigated. We focus our attention on some fundamental differences between the cases

p
≠
2

p\neq 2

and

p
=
2

p=2

. For

p
≠
2

p\neq 2

we give a counterexample to the classical Fredholm alternative (which is valid for the linear case

p
=
2

p=2

).
Completeness in the set of wavelets
Authors: Garrigós Gustavo , Speegle Darrin
Published: July 2002

We study the completeness properties of the set of wavelets in

L

2

(

R

)

L^{2}(\mathbb {R})

. It is well-known that this set is not closed in the unit ball of

L

2

(

R

)

L^{2}(\mathbb {R})

. However, if one considers the metric inherited as a subspace (in the Fourier transform side) of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis double-struck upper R comma d xi right-parenthesis intersection upper L squared left-parenthesis double-struck upper R Subscript asterisk Baseline comma StartFraction d xi Over StartAbsoluteValue xi EndAbsoluteValue EndFraction right-parenthesis">

L

2

(

R

,
d
ξ
)
∩

L

2

(

R

∗

,

d
ξ

|

ξ

|

)

L^{2}(\mathbb {R},d\xi ) \cap L^{2}(\mathbb {R}_{*},{\frac {{d\xi }}{{|\xi |}}})

, we do obtain a complete metric space.
Mistake in Hirsch’s proof of the Brouwer Fixed Point Theorem
Authors: Joshi Kapil
Published: July 2002

A mistake in Hirsch’s proof of the Brouwer Fixed Point Theorem, based on the Simplicial Approximation Theorem, is pointed out along with some elaboration of its nature and possible ways to salvage Hirsch’s proof.
Strongly exposed points in uniform algebras
Authors: Beneker Paul , Wiegerinck Jan
Published: July 2002

In this paper we show that the unit ball of an infinite dimensional function algebra has no strongly exposed points.
Frobenius properties and Maschke-type theorems for entwined modules
Authors: Brzeziński Tomasz
Published: July 2002

Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems known for Doi-Hopf modules are extended to the case of entwined modules.
Continued fractions with bounded partial quotients
Authors: Stambul Pierre
Published: July 2002

This paper gives the exact bound of the continued fraction expansion of

a
θ
+
b

c
θ
+
d

\frac {a\theta +b}{c\theta +d}

when

θ
\theta

has bounded partial quotients and

h
:
x
↦

a
x
+
b

c
x
+
d

h\colon x\mapsto \frac {ax+b}{cx+d}

is a Möbius transformation where all entries are integers.
A wild minimal plane in ℝ³
Authors: Andrade Plácido
Published: July 2002

The main object of this article is to construct a complete minimal immersed plane in

R

3

\mathbb {R}^3

whose closure has nonempty interior but it is not dense in the whole space. Furthermore, its Gaussian curvature is bounded.
The spectral properties of certain linear operators and their extensions
Authors: Barnes Bruce
Published: July 2002

Let

H
H

be a Hilbert space with inner-product

(
x
,
y
)

(x,y)

, and let

R
R

be a bounded positive operator on

H
H

which determines an inner-product,

⟨
x
,
y
⟩
=
(
R
x
,
y
)
,
x
,
y
∈
H

\langle x,y\rangle =(Rx,y), x, y\in H

. Denote by

H
−

H^-

the completion of

H
H

with respect to the norm

‖
x
‖
=
⟨
x
,
x

⟩

1

/

2

\|x\|=\langle x,x\rangle ^{1/2}

. In this paper, operators having certain relationships with

R
R

are studied. In particular, if

T
=
S

R

1

/

2

T=SR^{1/2}

where

S
∈
B
(
H
)

S\in B(H)

, then

T
T

has an extension

T
−

∈
B
(

H
−

)

T^-\in B(H^-)

, and

T
T

and

T
−

T^-

have essentially the same spectral and Fredholm properties.
Chern-Osserman inequality for minimal surfaces in 𝐇ⁿ
Authors: Qing Chen , Yi Cheng
Published: July 2002

We obtain Chern-Osserman’s inequality of a complete properly immersed minimal surface in hyperbolic

n
n

-space, provided the

L

2

L^{2}

-norm of the second fundamental form of the surface is finite.
A {𝐾}-energy characterization of extremal Kähler metrics
Authors: Simanca Santiago
Published: July 2002

We show that for any polarized compact Kähler manifold, the extremal Kähler metrics that represent the given cohomology class can be characterized as critical points of a suitably defined K-energy functional.
Splittings of Banach spaces induced by Clifford algebras
Authors: Carothers N. , Dilworth S. , Sobecki David
Published: July 2002

Let

H
H

be an infinite-dimensional Hilbert space of density character

m

\mathfrak {m}

. By representing

H
H

as a module over an appropriate Clifford algebra, it is proved that

H
H

possesses a family

{

A

α

}

α
∈

m

\{A_{\alpha }\}_{\alpha \in \mathfrak {m}}

of proper closed nonzero subspaces such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis upper S Subscript upper A Sub Subscript alpha Subscript Baseline comma upper S Subscript upper A Sub Subscript beta Subscript Baseline right-parenthesis equals d left-parenthesis upper S Subscript upper A Sub Subscript alpha Sub Superscript up-tack Subscript Baseline comma upper S Subscript upper A Sub Subscript beta Subscript Baseline right-parenthesis equals d left-parenthesis upper S Subscript upper A Sub Subscript alpha Sub Superscript up-tack Subscript Baseline comma upper S Subscript upper A Sub Subscript beta Sub Superscript up-tack Subscript Baseline right-parenthesis equals NestedStartRoot 2 minus StartRoot 2 EndRoot NestedEndRoot left-parenthesis alpha not-equals beta right-parenthesis period">

d
(

S

A

α

,

S

A

β

)
=
d
(

S

A

α

⊥

,

S

A

β

)
=
d
(

S

A

α

⊥

,

S

A

β

⊥

)
=

2
−

2

(
α
≠
β
)
.

\begin{equation*}d(S_{A_{\alpha }},S_{A_{\beta }})=d(S_{A^{\perp }_{\alpha }},S_{A_{\beta }}) =d(S_{A^{\perp }_{\alpha }},S_{A^{\perp }_{\beta }})=\sqrt {2-\sqrt 2}\qquad (\alpha \ne \beta ).\end{equation*}

Analogous results are proved for

L

p

L_{p}

spaces and for

c

0

(
X
)

c_{0}(X)

and

ℓ

p

(
X
)

\ell _{p}(X)

(

1
≤
p
≤
∞

1 \le p \le \infty

) when

X
X

is an arbitrary nonzero Banach space.
On the semisimplicity of pure sheaves
Authors: Fu Lei
Published: July 2002

We obtain a criteria for a pure sheaf to be semisimple. As a corollary, we prove the following: Let

X
0

X_0

and

S
0

S_0

be two schemes over a finite field

F

q

\mathbf {F}_q

, and let

f
0

:

X
0

→

S
0

f_0: X_0\rightarrow S_0

be a proper smooth morphism. Assume

S
0

S_0

is normal and geometrically connected, and assume there exists a closed point

s
s

in

S
0

S_0

such that the Frobenius automorphism

F
s

F_s

acts semisimply on

H
i

(

X

s
¯

,

Q

l

¯

)

H^i(X_{\bar s}, {\overline {\mathbf {Q}_l}})

, where

X

s
¯

X_{\bar s}

is the geometric fiber of

f
0

f_0

at

s
s

(this last assumption is unnecessary if the semisimplicity conjecture is true). Then

R
i

f

0
∗

Q

l

¯

R^if_{0\ast } {\overline {\mathbf {Q}_l}}

is a semisimple sheaf on

S
0

S_0

. This verifies a conjecture of Grothendieck and Serre provided the semisimplicity conjecture holds. As an application, we prove that the galois representations of function fields associated to the

l
l

-adic cohomologies of

K
3

K3

surfaces are semisimple. We also get a theorem of Zarhin about the semisimplicity of the Galois representations of function fields arising from abelian varieties. The proof relies heavily on Deligne’s work on Weil conjectures.
Euler characteristic of the Milnor fibre of plane singularities
Authors: Melle-Hernández A.
Published: July 2002

We give a formula for the Euler characteristic of the Milnor fibre of any analytic function

f
f

of two variables. This formula depends on the intersection multiplicities, the Milnor numbers and the powers of the branches of the germ of the curve defined by

f
.

f.

The goal of the formula is that it use neither the resolution nor the deformations of

f
.

f.

Moreover, it can be use for giving an algorithm to compute it.
Non-semistable Arakelov bound and hyperelliptic Szpiro ratio for function fields
Authors: Nguyen Khac
Published: July 2002

We prove a complex function field analogue of Szpiro’s conjecture for hyperelliptic curves and some applications. The cases of function fields of positive characteristic and number fields are discussed briefly.
A generalization of the Lefschetz fixed point theorem and detection of chaos
Authors: Srzednicki Roman
Published: July 2002

We consider the problem of existence of fixed points of a continuous map

f
:
X
→
X

f:X\to X

in (possibly) noninvariant subsets. A pair

(
C
,
E
)

(C,E)

of subsets of

X
X

induces a map

f
†

:
C

/

E
→
C

/

E

f^\dagger :C/E\to C/E

given by

f
†

(
[
x
]
)
=
[
f
(
x
)
]

f^\dagger ([x])=[f(x)]

if

x
,
f
(
x
)
∈
C
∖
E

x,f(x)\in C\setminus E

and

f
†

(
[
x
]
)
=
[
E
]

f^\dagger ([x])=[E]

elsewhere. The following generalization of the Lefschetz fixed point theorem is proved: If

X
X

is metrizable,

C
C

and

E
E

are compact ANRs, and

f
†

f^\dagger

is continuous, then

f
f

has a fixed point in

C
∖
E

¯

\overline {C\setminus E}

provided the Lefschetz number of

H
~

∗

(

f
†

)

\widetilde H^\ast (f^\dagger )

is nonzero. Actually, we prove an extension of that theorem to the case of a composition of maps. We apply it to a result on the existence of an invariant set of a homeomorphism such that the dynamics restricted to that set is chaotic.
A relation between certain interpolated Cuntz algebras and interpolated free group factors
Authors: Ueda Yoshimichi , Watatani Yasuo
Published: July 2002

We investigate von Neumann algebras generated by the real parts of generators of Toeplitz extensions of interpolated Cuntz algebras

O

β

\mathcal O_{\beta }

on sub-Fock spaces. We show that some of them are isomorphic to interpolated free group factors

L
(

F
r

)

L(F_r)

. For example, in case of the golden number

β
=

1
+

5

2

\beta = \frac {1+\sqrt {5}}{2}

the corresponding number

r
r

is

3
2

\frac {3}{2}

.
Galois embeddings for linear groups
Authors: Abhyankar Shreeram
Published: July 2002

A criterion is given for the solvability of a central Galois embedding problem to go from a projective linear group covering to a vectorial linear group covering.
Center manifolds for smooth invariant manifolds
Authors: Chow Shui-Nee , Liu Weishi , Yi Yingfei
Published: July 2002

We study dynamics of flows generated by smooth vector fields in

R

n

{\mathbb {R}}^n

in the vicinity of an invariant and closed smooth manifold

Y
Y

. By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of

Y
Y

) based on the information of the linearization along

Y
Y

, which contains every locally bounded solution and is persistent under small perturbations.
The Markov spectra for Fuchsian groups
Authors: Vulakh L.
Published: July 2002

Applying the Klein model

D
2

D^2

of the hyperbolic plane and identifying the geodesics in

D
2

D^2

with their poles in the projective plane, the author develops a method of determining infinite binary trees in the Markov spectrum for a Fuchsian group. The method is applied to a maximal group commensurable with the modular group and other Fuchsian groups.
Some properties of minimal surfaces in singular spaces
Authors: Mese Chikako
Published: July 2002

This paper involves the generalization of minimal surface theory to spaces with singularities. Let

X
X

be an NPC space, i.e. a metric space of non-positive curvature. We define a (parametric) minimal surface in

X
X

as a conformal energy minimizing map. Using this definition, many properties of classical minimal surfaces can also be observed for minimal surfaces in this general setting. In particular, we will prove the boundary monotonicity property and the isoperimetric inequality for minimal surfaces in

X
X

.
Unipotent groups associated to reduced curves
Authors: Penniston David
Published: July 2002

Let

X
X

be a curve defined over an algebraically closed field

k
k

with

char
⁡
(
k
)
=
p
>
0

\operatorname {char}(k)=p>0

. Assume that

X

/

k

X/k

is reduced. In this paper we study the unipotent part

U
U

of the Jacobian

J

X

/

k

J_{X/k}

. In particular, we prove that if

p
p

is large in terms of the dimension of

U
U

, then

U
U

is isomorphic to a product of additive groups

G

a

\mathbb {G}_a

.
The density of rational lines on cubic hypersurfaces
Authors: Parsell Scott
Published: July 2002

We provide a lower bound for the density of rational lines on the hypersurface defined by an additive cubic equation in at least

57
57

variables. In the process, we obtain a result on the paucity of non-trivial solutions to an associated system of Diophantine equations.
Resonance problems with respect to the Fucík spectrum
Authors: Schechter Martin
Published: July 2002

We study semilinear boundary value problems which have asymptotic resonance with respect to the linear part. The difficulties for Fučík resonance problems are compounded by the fact that there is no eigenspace with which to work. The present paper uses new linking theorems which can deal with the sets required to obtain critical points.
Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations
Authors: Chen Gui-Qiang , Frid Hermano
Published: July 2002

We prove the uniqueness of Riemann solutions in the class of entropy solutions in

L
∞

∩
B

V

l
o
c

L^\infty \cap BV_{loc}

for the

3
×
3

3\times 3

system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global

L
2

L^2

-stability of the Riemann solutions even in the class of entropy solutions in

L
∞

L^\infty

with arbitrarily large oscillation for the

3
×
3

3\times 3

system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under

L
1

L^1

perturbation of the Riemann initial data, as long as the corresponding solutions are in

L
∞

L^\infty

and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions

U
(
x
,
t
)

U(x,t)

, piecewise Lipschitz in

x
x

, for any

t
>
0

t>0

.
Algebraic isomorphisms of limit algebras
Authors: Donsig A. , Hudson T. , Katsoulis E.
Published: July 2002

We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras.
Two-weight norm inequalities for Cesàro means of Laguerre expansions
Authors: Muckenhoupt Benjamin , Webb David
Published: July 2002

Two-weight

L

p

L^{p}

norm inequalities are proved for Cesàro means of Laguerre polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and “unweighted” cases, by including all values of

p
≥
1

p\geq 1

for all positive orders of the Cesàro summation and all values of the Laguerre parameter

α
>
−
1

\alpha >-1

. Almost everywhere convergence results are obtained as a corollary. For the Cesàro means the hypothesized conditions are shown to be necessary for the norm inequalities. Necessity results are also obtained for the norm inequalities with the supremum of the Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.
Minimal projective resolutions
Authors: Green E. , Solberg Ø. , Zacharia D.
Published: July 2002

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the

Ext
\operatorname {Ext}

-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the “no loop” conjecture.