bluuebunny/crossref_metadata_2025 huggingface dataset:
Determinant expression of Selberg zeta functions. II
Authors: Koyama Shin-ya
Published: June 2012

This paper is the

PSL
⁡
(
2
,

C

)

\operatorname {PSL} (2,{\mathbf {C}})

-version of Part I. We show that for

PSL
⁡
(
2
,

C

)

\operatorname {PSL} (2,{\mathbf {C}})

and its subgroup

PSL
⁡
(
2
,
O
)

\operatorname {PSL} (2,O)

, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacians, where

O
O

is the integer ring of an imaginary quadratic field. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
An extension of Attouch’s theorem and its application to second-order epi-differentiation of convexly composite functions
Authors: Poliquin René A.
Published: June 2012

In 1977, Hedy Attouch established that a sequence of (closed proper) convex functions epi-converges to a convex function if and only if the graphs of the subdifferentials converge (in the Mosco sense) to the subdifferential of the limiting function and (roughly speaking) there is a condition that fixes the constant of integration. We show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping (in addition a constraint qualification is imposed). Using Attouch’s Theorem, Rockafellar showed that second-order epi-differentiation of a convex function and proto-differentiability of the subdifferential set-valued mapping are equivalent, moreover the subdifferential of one-half the second-order epi-derivative is the proto-derivative of the subdifferential mapping; we will extend this result to the convexly composite setting.
Cosmic approximate limits and fixed points
Authors: Segal J. , Watanabe T.
Published: June 2012

We define a condition for approximate inverse systems which implies that the limit of the system has the fixed point property. Moreover, this condition is categorical in the approximate shape category. We investigate the class of complex projective

n
n

-space like continua with respect to the fixed point property by means of this condition. As a further application we show that the hyperspace

C
(
X
)

C(X)

of nonempty subcontinua of an arc-like or circle-like Hausdorff continuum

X
X

has the fixed point property. We also prove that

2
X

{2^X}

and

C
(
X
)

C(X)

have the fixed point property for any locally connected Hausdorff continuum

X
X

.
Geometric aspects of reduction of order
Authors: Sherring James , Prince Geoff
Published: June 2012

Using the differential geometry of vectorfields and forms we reinterpret and extend the traditional idea of an integrating factor for a first order differential equation with symmetry. In particular, we provide a simple and manifestly geometric approach to reduction of order via symmetry for ordinary differential equations which largely obviates the necessity for canonical coordinates and the associated quotient manifolds. In so doing, some new results which generalise the class of Lie group actions which can be used to solve ordinary differential equations are developed.
Hausdorff dimension of wild fractals
Authors: Rushing T. B.
Published: June 2012

We show that for every

s
∈
[
n
−
2
,
n
]

s \in [n - 2,n]

there exists a homogeneously embedded wild Cantor set

C
s

{C^s}

in

R

n

,
n
≥
3

\mathbb {R}^n, n \geq 3

, of (local) Hausdorff dimension

s
s

. Also, it is shown that for every

s
∈
[
n
−
2
,
n
]

s \in [n - 2,n]

and for any integer

k
≠
n

k \ne n

such that

1
≤
k
≤
s

1 \leq k \leq s

, there exist everywhere wild

k
k

-spheres and

k
k

-cells, in

R

n

,
n
≥
3

\mathbb {R}^n, n \geq 3

, of (local) Hausdorff dimension

s
s

.
Subgroups of Bianchi groups and arithmetic quotients of hyperbolic 3-space
Authors: Grunewald Fritz , Schwermer Joachim
Published: June 2012

Let

O

\mathcal {O}

be the ring of integers in an imaginary quadratic number-field. The group

PSL

2

(

O

)

{\text {PSL}}_2(\mathcal {O})

acts discontinuously on hyperbolic

3
3

-space

H
H

. If

Γ
≤

PSL

2

(

O

)

\Gamma \leq {\text {PSL}}_2(\mathcal {O})

is a torsionfree subgroup of finite index then the manifold

Γ
∖
H

\Gamma \backslash H

can be compactified to a manifold

M
Γ

{M_\Gamma }

so that the inclusion

Γ
∖
H
≤

M
Γ

\Gamma \backslash H \leq {M_\Gamma }

is a homotopy equivalence.

M
Γ

{M_\Gamma }

is a compact with boundary. The boundary being a union of finitely many

2
2

-tori. This paper contains a computer-aided study of subgroups of low index in

PSL

2

(

O

)

{\text {PSL}}_2(\mathcal {O})

for various

O

\mathcal {O}

. The explicit description of these subgroups leads to a study of the homeomorphism types of the

M
Γ

{M_\Gamma }

.
The product of two normal initially 𝜅-compact spaces
Authors: van Douwen Eric K.
Published: June 2012

We prove that it is independent from

ZFC

{\text {ZFC}}

that for every cardinal

κ
\kappa

the following statements are equivalent: (a)

κ
\kappa

is singular; (b) initial

κ
\kappa

-compactness (defined above the introduction) is productive; (c) initial

κ
\kappa

-compactness is finitely productive; and (d) the product of two initially

κ
\kappa

-compact normal spaces is initially

κ
\kappa

-compact. In particular,

MA
\text {MA}

implies that there are two countably compact normal spaces whose product is not countably compact.
Imprimitive Gaussian sums and theta functions over a number field
Authors: Nemchenok Jacob
Published: June 2012

We obtain a reduction formula for an imprimitive Gaussian sum with a numerical character in an algebraic number field, i.e. a formula that expresses that sum as a product of several elementary factors times a primitive, proper, normed Gaussian sum (formulae (16) and (19)). We also introduce Gaussian sums with Hecke characters and derive a similar reduction formula for them. The derivation is based on an inversion formula for a multivariable theta function associated with the number field, twisted with the numerical character.
Toral actions on 4-manifolds and their classifications
Authors: Kim M. Ho
Published: June 2012

The existence of a cross-section is proved for some nonorientable

4
4

-manifolds with a

T
2

{T^2}

-action. Two

4
4

-manifolds with a

T
2

{T^2}

-action, which have the same previously known invariants, are constructed. By using a new homotopy invariant, they are proved to be homotopy inequivalent. Finally a stable diffeomorphism theorem is proved.
Obstructions and hypersurface sections (minimally elliptic singularities)
Authors: Behnke Kurt , Christophersen Jan Arthur
Published: June 2012

We study the obstruction space

T
2

{T^2}

for minimally elliptic surface singularities. We apply the main lemma of our previous paper [3] which relates

T
2

{T^2}

to deformations of hypersurface sections. To use this we classify general hypersurface sections of minimally elliptic singularities. As in the rational singularity case there is a simple formula for the minimal number of generators for

T
2

{T^2}

as a module over the local ring. This number is in many cases (e.g. for cusps of Hilbert modular surfaces) equal to the vector space dimension of

T
2

{T^2}

.
Functorial construction of Le Barz’s triangle space with applications
Authors: Keel Sean
Published: June 2012

We give a new functorial construction of the space of triangles introduced by Le Barz. This description is used to exhibit the space as a composition of smooth blowups, to obtain a space of unordered triangles, and to study how the space varies in a family.
On a theorem of Muckenhoupt and Wheeden and a weighted inequality related to Schrödinger operators
Authors: Pérez C.
Published: June 2012

We extend in several directions a theorem of B. Muckenhoupt and R. Wheeden relating the

L
p

{L^p}

-norms of Riesz potentials and fractional maximal operators. We apply these results to give a simple proof and sharpen a weighted inequality for Schrödinger operators of Chang, Wilson and Wolff.
Quadratic transformation formulas for basic hypergeometric series
Authors: Rahman Mizan , Verma Arun
Published: June 2012

Starting with some of the known transformation formulas for well-poised

2

ϕ
1

_2{\phi _1}

and very-well-poised

8

ϕ
7

_8{\phi _7}

basic hypergeometric series we obtain

q
q

-analogues of

36
36

quadratic transformation formulas given in

§
2.11

\S 2.11

of Higher transcendental functions, Vol. 1, edited by Erdélyi et al. We also derive some new quadratic transformation formulas that give rise to identities connecting very-well-poised but unbalanced

10

ϕ
9

_{10}{\phi _9}

series in base

q
q

with very-well-poised and balanced

12

ϕ

11

_{12}{\phi _{11}}

series in base

q
2

{q^2}

. A Rogers-Ramanujan type identity is also found as a limiting case.
Harmonic volume, symmetric products, and the Abel-Jacobi map
Authors: Faucette William M.
Published: June 2012

The author generalizes B. Harris’ definition of harmonic volume to the algebraic cycle

W
k

−

W
k
−

{W_k} - W_k^-

for

k
>
1

k > 1

in the Jacobian of a nonsingular algebraic curve

X
X

. We define harmonic volume, determine its domain, and show that it is related to the image

ν
\nu

of

W
k

−

W
k
−

{W_k} - W_k^-

in the Griffiths intermediate Jacobian. We derive a formula expressing harmonic volume as a sum of integrals over a nested sequence of submanifolds of the

k
k

-fold symmetric product of

X
X

. We show that

ν
\nu

, when applied to a certain class of forms, takes values in a discrete subgroup of

R

/

Z

{\mathbf {R}}/{\mathbf {Z}}

and hence, when suitably extended to complexvalued forms, is identically zero modulo periods on primitive forms if

k
≥
2

k \geq 2

. This implies that the image of

W
k

−

W
k
−

{W_k} - W_k^-

is identically zero in the Griffiths intermediate Jacobian if

k
≥
2

k \geq 2

. We introduce a new type of intermediate Jacobian which, like the Griffiths intermediate Jacobian, varies holomorphically with moduli, and we consider a holomorphic torus bundle on Torelli space with this fiber. We use the relationship mentioned above between

ν
\nu

and harmonic volume to compute the variation of

ν
\nu

when considered as a section of this bundle. This variational formula allows us to show that the image of

W
k

−

W
k
−

{W_k} - W_k^-

in this intermediate Jacobian is nondegenerate.
Irredundant sets in Boolean algebras
Authors: Todorčević Stevo
Published: June 2012

It is shown that every uncountable Boolean algebra

A
A

contains an uncountable subset

I
I

such that no

a
a

of

I
I

is in the subalgebra generated by

I
∖
{
a
}

I\backslash \{ a\}

using an additional axiom of set theory. It is also shown that a use of some such axiom is necessary.
Weak solutions of the porous medium equation
Authors: Dahlberg Björn E. J. , Kenig Carlos E.
Published: June 2012

We show that if

u
≥
0

u \geq 0

,

u
∈

L

loc

m

(
Ω
)

u \in L_{{\text {loc}}}^m(\Omega )

,

Ω
⊂

R

n
+
1

\Omega \subset {{\mathbf {R}}^{n + 1}}

solves

∂
u

/

∂
t
=
Δ

u
m

\partial u/\partial t = \Delta {u^m}

,

m
>
1

m > 1

, in the sense of distributions, then

u
u

is locally Hölder continuous in

Ω
\Omega

.
Weak solutions of the porous medium equation in a cylinder
Authors: Dahlberg Björn E. J. , Kenig Carlos E.
Published: June 2012

We show that if

D
⊂

R

n

D \subset {{\mathbf {R}}^n}

is a bounded domain with smooth boundary, and

u
∈

L
m

(
D
×
(
ε
,
T
)
)

u \in {L^m}(D \times (\varepsilon ,T))

,

u
≥
0

u \geq 0

, solves

∂
u

∂
t

=
Δ

u
m

\frac {{\partial u}} {{\partial t}} = \Delta {u^m}

,

m
>
1

m > 1

, in the sense of distributions on

D
×
(
0
,
T
)

D \times (0,T)

, and vanishes on

∂
D
×
(
0
,
T
)

\partial D \times (0,T)

in a suitable weak sense, then

u
u

is Hölder continuous in

D
¯

×
(
0
,
T
)

\overline D \times (0,T)

.
Conformal metrics with prescribed Gaussian curvature on 𝑆²
Authors: Cheng Kuo-Shung , Smoller Joel A.
Published: June 2012

We consider on

S
2

{S^2}

the problem of which functions

K
K

can be the scalar curvature of a metric conformal to the standard metric on

S
2

{S^2}

. We assume that

K
K

is a function of one variable, and we obtain a necessary and sufficient condition for the problem to be solvable. We also obtain several new sufficient conditions on

k
k

(which are easy to check), in order that the problem be solvable.
On Lagrange interpolation at disturbed roots of unity
Authors: Chui Charles K. , Shen Xie Chang , Zhong Le Fan
Published: June 2012

Letznk=eitnk{z_{nk}} = {e^{i{t_{nk}}}},0≤tn0>⋯>tnn>2π0 \leq {t_{n0}} > \cdots > {t_{nn}} > 2\pi,ffa function in the disc algebraAA, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript n Baseline equals max left-brace right-brace colon vertical-bar vertical-bar slash slash minus minus t times times nk times times times 2 k pi left-parenthesis right-parenthesis plus plus n 1 colon less-than-or-equal-to less-than-or-equal-to less-than-or-equal-to 0 kn">μn=max{|tnk−2kπ/(n+1)|:0≤k≤n}{\mu _n} = \max \{ |{t_{nk}} - 2k\pi /(n + 1)|:0 \leq k \leq n\}. Denote byLn(f;⋅){L_n}(f;\; \cdot )the polynomial of degreennthat agrees withffat{znk:k=0,…,n}\{ {z_{nk}}:k = 0, \ldots ,n\}. In this paper, we prove that for everypp,0>p>∞0 > p > \infty, there exists aδp>0{\delta _p} > 0, such that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper L Subscript n Baseline left-parenthesis f semicolon dot right-parenthesis minus f StartAbsoluteValue EndAbsoluteValue Subscript p Baseline equals upper O left-parenthesis omega left-parenthesis f semicolon StartFraction 1 Over n EndFraction right-parenthesis right-parenthesis">||Ln(f;⋅)−f||p=O(ω(f;1n))||{L_n}(f;\cdot ) - f|{|_p} = O(\omega (f;\frac {1} {n}))wheneverμn≤δp/(n+1){\mu _n} \leq {\delta _p}/(n + 1). It must be emphasized thatδp{\delta _p}necessarily depends onpp, in the sense that there exists a family{znk:k=0,…,n}\{ {z_{nk}}:k = 0, \ldots ,n\}withμn=δ2/(n+1){\mu _n} = {\delta _2}/(n + 1)and such that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper L Subscript n Baseline left-parenthesis f semicolon dot right-parenthesis minus f StartAbsoluteValue EndAbsoluteValue Subscript 2 Baseline equals upper O left-parenthesis omega left-parenthesis f semicolon StartFraction 1 Over n EndFraction right-parenthesis right-parenthesis">||Ln(f;⋅)−f||2=O(ω(f;1n))||{L_n}(f;\cdot ) - f|{|_2} = O(\omega (f;\frac {1} {n}))for allf∈Af \in A, but<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace StartAbsoluteValue EndAbsoluteValue upper L Subscript n Baseline left-parenthesis f semicolon dot right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Baseline colon f element-of upper A comma StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript normal infinity Baseline equals 1 right-brace">sup{||Ln(f;⋅)||p:f∈A,||f||∞=1}\sup \{ ||{L_n}(f;\cdot )|{|_p}:f \in A,||f|{|_\infty } = 1\}diverges for sufficiently large values ofpp. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for{znk}\{ {z_{nk}}\}.
The Noetherian property in rings of integer-valued polynomials
Authors: Gilmer Robert , Heinzer William , Lantz David
Published: June 2012

Let

D
D

be a Noetherian domain,

D
′

D\prime

its integral closure, and

Int
⁡
(
D
)

\operatorname {Int}(D)

its ring of integer-valued polynomials in a single variable. It is shown that, if

D
′

D\prime

has a maximal ideal

M
′

M\prime

of height one for which

D
′

/

M
′

D\prime /M\prime

is a finite field, then

Int
⁡
(
D
)

\operatorname {Int}(D)

is not Noetherian; indeed, if

M
′

M\prime

is the only maximal ideal of

D
′

D\prime

lying over

M
′
∩
D

M\prime \cap D

, then not even

Spec
⁡
(
Int
⁡
(
D
)
)

\operatorname {Spec}(\operatorname {Int}(D))

is Noetherian. On the other hand, if every height-one maximal ideal of

D
′

D\prime

has infinite residue field, then a sufficient condition for

Int
⁡
(
D
)

\operatorname {Int}(D)

to be Noetherian is that the global transform of

D
D

is a finitely generated

D
D

-module.
Functional equations satisfied by intertwining operators of reductive groups
Authors: Zhu Chen-bo
Published: June 2012

This paper generalizes a recent work of Vogan and Wallach [VW] in which they derived a difference equation satisfied by intertwining operators of reductive groups. We show that, associated with each irreducible finite-dimensional representation, there is a functional equation relating intertwining operators. In this way, we obtain natural relations between intertwining operators for different series of induced representations.
Rigidity of invariant complex structures
Authors: Dotti Miatello Isabel
Published: June 2012

A Kähler solvmanifold is a connected Kähler manifold

(
M
,
j
,

⟨
,
⟩

)

(M,j,\left \langle , \right \rangle )

admitting a transitive solvable group of automorphisms. In this paper we study the isomorphism classes of Kähler structures

(
j
,

⟨
,
⟩

)

(j,\left \langle , \right \rangle )

turning

M
M

into a Kähler solvmanifold. In the case when

(
M
,
j
,

⟨
,
⟩

)

(M,j,\left \langle , \right \rangle )

is irreducible and simply connected we show that any Kähler structure on

M
M

, having the same group of automorphisms, is isomorphic to

(
j
,

⟨
,
⟩

)

(j,\left \langle , \right \rangle )

.
Mixing properties of a class of Bernoulli-processes
Authors: Fiebig Doris
Published: June 2012

We prove that stationary very weak Bernoulli processes with rate

O
(
1

/

n
)

(

VWB

O
(
1

/

n
)
)

O(1/n)\;({\text {VWB}}\,O(1/n))

are strictly very weak Bernoulli with rate

O
(
1

/

n
)

O(1/n)

. Furthermore we discuss the relation between

VWB

O
(
1

/

n
)

{\text {VWB}}\;O(1/n)

and the classical mixing properties for countable state processes. In particular, we show that

VWB

O
(
1

/

n
)

{\text {VWB}}\,O(1/n)

implies

ϕ
\phi

-mixing.
Kloosterman sums for Chevalley groups
Authors: Dąbrowski Romuald
Published: June 2012

A generalization of Kloosterman sums to a simply connected Chevalley group

G
G

is discussed. These sums are parameterized by pairs

(
w
,
t
)

(w,t)

where

w
w

is an element of the Weyl group of

G
G

and

t
t

is an element of a

Q

{\mathbf {Q}}

-split torus in

G
G

. The

S
L
(
2
,

Q

)

SL(2,{\mathbf {Q}})

-Kloosterman sums coincide with the classical Kloosterman sums and

S
L
(
r
,

Q

)

SL(r,{\mathbf {Q}})

-Kloosterman sums,

r
≥
3

r \geq 3

, coincide with the sums introduced in [B-F-G,F,S]. Algebraic properties of the sums are proved by root system methods. In particular an explicit decomposition of a general Kloosterman sum over

Q

{\mathbf {Q}}

into the product of local

p
p

-adic factors is obtained. Using this factorization one can show that the Kloosterman sums corresponding to a toral element, which acts trivially on the highest weight space of a fundamental irreducible representation, splits into a product of Kloosterman sums for Chevalley groups of lower rank.
Twists of Hilbert modular forms
Authors: Shemanske Thomas R. , Walling Lynne H.
Published: June 2012

The theory of newforms for Hilbert modular forms is summarized including a statement of a strong multiplicity-one theorem and a characterization of newforms as eigenfunctions for a certain involution whose Dirichlet series has a prescribed Euler product. The general question of twisting Hilbert modular newforms by arbitrary Hecke characters is considered and the exact level of a character twist of a Hilbert modular form is determined. Conditions under which the twist of a newform is a newform are given. Applications include a strengthening in the elliptic modular case of a theorem of Atkin and Li’s regarding the characterization of imprimitive newforms as well as its generalization to the Hilbert modular case, and a decomposition theorem for certain spaces of newforms as the direct sum of twists of spaces of newforms of lower level.
Artinian subrings of a commutative ring
Authors: Gilmer Robert , Heinzer William
Published: June 2012

Given a commutative ring

R
R

, we investigate the structure of the set of Artinian subrings of

R
R

. We also consider the family of zero-dimensional subrings of

R
R

. Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite intersection, and the condition that the set of Artinian subrings of a ring forms a directed family.
On Dubrovin valuation rings in crossed product algebras
Authors: Haile Darrell , Morandi Patrick
Published: June 2012

Let

F
F

be a field and let

V
V

be a valuation ring in

F
F

. If

A
A

is a central simple

F
F

-algebra then

V
V

can be extended to a Dubrovin valuation ring in

A
A

. In this paper we consider the structure of Dubrovin valuation rings with center

V
V

in crossed product algebras

(
K

/

F
,
G
,
f
)

(K/F,G,f)

where

K

/

F

K/F

is a finite Galois extension with Galois group

G
G

unramified over

V
V

and

f
f

is a normalized two-cocycle. In the case where

V
V

is indecomposed in

K
K

we introduce a family of orders naturally associated to

f
f

, examine their basic properties, and determine which of these orders is Dubrovin. In the case where

V
V

is decomposed we determine the structure in the case of certain special discrete, finite rank valuations.
An extension theorem for closing maps of shifts of finite type
Authors: Ashley Jonathan
Published: June 2012

If there exists some right-closing factor map

π
:

Σ
A

→

Σ
B

\pi :{\Sigma _A} \to {\Sigma _B}

between aperiodic shifts of finite type, then any right-closing map

φ
:
X
→

Σ
B

\varphi :X \to {\Sigma _B}

from any shift of finite type

X
X

contained in

Σ
A

{\Sigma _A}

can be extended to a right-closing factor map from all of

Σ
A

{\Sigma _A}

onto

Σ
B

{\Sigma _B}

. We prove this and give some consequences.
Rational orbits on three-symmetric products of abelian varieties
Authors: Alzati A. , Pirola G. P.
Published: June 2012

Let

A
A

be an

n
n

-dimensional Abelian variety,

n
≥
2

n \geq 2

; let

CH
0

(
A
)

{\text {CH}_0}(A)

be the group of zero-cycles of

A
A

, modulo rational equivalence; by regarding an effective, degree

k
k

, zero-cycle, as a point on

S
k

(
A
)

{S^k}(A)

(the

k
k

-symmetric product of

A
A

), and by considering the associated rational equivalence class, we get a map

γ
:

S
k

(
A
)
→

CH
0

(
A
)

\gamma :{S^k}(A) \to {\text {CH}_0}(A)

, whose fibres are called

γ
\gamma

-orbits. For any

n
≥
2

n \geq 2

, in this paper we determine the maximal dimension of the

γ
\gamma

-orbits when

k
=
2

k = 2

or

3
3

(it is, respectively,

1
1

and

2
2

), and the maximal dimension of families of

γ
\gamma

-orbits; moreover, for generic

A
A

, we get some refinements and in particular we show that if

dim
⁡
(
A
)
≥
4

\dim (A) \geq 4

,

S
3

(
A
)

{S^3}(A)

does not contain any

γ
\gamma

-orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties:

A
t

=

E
t

×
B

{A_t} = {E_t} \times B

(

E
t

{E_t}

is an elliptic curve with varying moduli) and we have constructed suitable projections between

S
k

(

A
t

)

{S^k}({A_t})

and

S
k

(
B
)

{S^k}(B)

which preserve the dimensions of the families of

γ
\gamma

-orbits; then we have done induction on

n
n

. For

n
=
2

n = 2

the proof is based upon the papers of Mumford and Roitman on this topic.
Parametrizing smooth compactly supported wavelets
Authors: Wells Raymond O.
Published: June 2012

In this paper a concrete parameter space for the compactly supported wavelet systems of Daubechies is constructed. For wavelet systems with

N
N

(generic) nonvanishing coefficients the parameter space is a closed convex set in

R

(
N
−
2
)

/

2

{{\mathbf {R}}^{(N - 2)/2}}

, which can be explicitly described in the Fourier transform domain. The moment-free wavelet systems are subsets obtained by the intersection of the parameter space and an affine subspace of

R

(
N
−
2
)

/

2

{{\mathbf {R}}^{(N - 2)/2}}

.
2-weights for unitary groups
Authors: An Jian Bei
Published: June 2012

This paper gives a description of the local structures of

2
2

-radical subgroups in a finite unitary group and proves Alperin’s weight conjecture for finite unitary groups when the characteristic of modular representation is even.
Chebyshev type estimates for Beurling generalized prime numbers. II
Authors: Zhang Wen-Bin
Published: June 2012

Let

N
(
x
)

N(x)

be the distribution function of the integers in a Beurling generalized prime system. The Chebyshev type estimates for Beurling generalized prime numbers in the general case
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis x right-parenthesis equals x sigma-summation Underscript nu equals 1 Overscript n Endscripts upper A Subscript nu Baseline log Superscript rho Super Subscript nu Superscript minus 1 Baseline x plus upper O left-parenthesis x log Superscript negative gamma Baseline x right-parenthesis">

N
(
x
)
=
x

∑

ν
=
1

n

A
ν

log

ρ
ν

−
1

x
+
O
(
x

log

−
γ

x
)

N(x) = x\sum \limits _{\nu = 1}^n {{A_\nu }} {\log ^{{\rho _\nu } - 1}}x + O(x{\log ^{ - \gamma }}x)

\]
is a long standing question. In this paper we shall give an affirmative answer to the question by proving that the Chebyshev type estimates
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than limit inf Underscript x right-arrow normal infinity Endscripts StartFraction psi left-parenthesis x right-parenthesis Over x EndFraction comma limit sup Underscript x right-arrow normal infinity Endscripts StartFraction psi left-parenthesis x right-parenthesis Over x EndFraction greater-than normal infinity">

0
>
lim

inf

x
→
∞

ψ
(
x
)

x

,

lim

sup

x
→
∞

ψ
(
x
)

x

>
∞

0 > \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}}{x},\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}}{x} > \infty

\]
hold even under weaker condition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 1 Superscript normal infinity Baseline x Superscript negative 1 Baseline left-brace sup Underscript x greater-than normal infinity Endscripts y Superscript negative 1 Baseline StartAbsoluteValue upper N left-parenthesis y right-parenthesis minus y sigma-summation Underscript nu equals 1 Overscript n Endscripts upper A Subscript nu Baseline log Superscript rho Super Subscript nu Superscript minus 1 Baseline y EndAbsoluteValue right-brace d x greater-than normal infinity">

∫
1
∞

x

−
1

{

sup

x
>
∞

y

−
1

|

N
(
y
)
−
y

∑

ν
=
1

n

A
ν

log

ρ
ν

−
1

y

|

}

d
x
>
∞

\int _1^\infty {{x^{ - 1}}} \left \{ {\sup \limits _{x > \infty } {y^{ - 1}}\left | {N(y) - y\sum \limits _{\nu = 1}^n {{A_\nu }} {{\log }^{{\rho _\nu } - 1}}y} \right |} \right \}\,dx > \infty

\]
with

ρ
n

=
τ
≥
1

\rho _n=\tau \geq 1

,

0
>

ρ
1

>

ρ
2

>
⋯
>

ρ
n

0>\rho _1>\rho _2 >\cdots > \rho _n

, and

A
n

>
0

A_n > 0

. This generalizes a result of Diamond and a result of the present author.
Varieties of topological geometries
Authors: Groh Hansjoachim
Published: June 2012

A variety of topological geometries is either A. a projective variety

L

(
F
)

\mathcal {L}(F)

over some topological field

F
F

, or B. a matchstick variety

M

(
X
)

\mathcal {M}(X)

over some topological space

X
X

. As a main tool for showing this, we prove a structure theorem for arbitrary topological geometries.
The limiting behavior of the Kobayashi-Royden pseudometric
Authors: Kaliman Shulim
Published: June 2012

We study the limit of the sequence of Kobayashi metrics of Riemann surfaces (when these Riemann surfaces form an analytic fibration in such a way that the total space of fibration becomes a complex surface), as the fibers approach the center fiber which is not in general smooth. We prove that if the total space is a Stein surface and the smooth part of the center fiber contains a component biholomorphic to a quotient of the disk by a Fuchsian group of first kind, then the Kobayashi metrics of the near-by fibers converge to the Kobayashi metric of this component as fibers tend to the center fiber.
Rees algebras of ideals having small analytic deviation
Authors: Huckaba Sam , Huneke Craig
Published: June 2012

In this article we identify two large families of ideals of a Cohen-Macaulay (sometimes Gorenstein) local ring whose Rees algebras are Cohen-Macaulay. Our main results imply, for example, that if

(
R
,
M
)

(R,M)

is a regular local ring and

P
P

is a prime ideal of

R
R

such that

P
n

{P^n}

is unmixed for all

n
≥
1

n \geq 1

, then the Rees algebra

R
[
P
t
]

R[Pt]

is Cohen-Macaulay if either

dim
⁡
(
R

/

P
)
=
2

\dim (R/P) = 2

, or

dim
⁡
(
R

/

P
)
=
3
,
R

/

P

\dim (R/P) = 3,R/P

is Cohen-Macaulay, and

R

/

P

R/P

is integrally closed.
Holomorphic extension and decomposition from a totally real manifold
Authors: Ye Zai Fei
Published: June 2012

This paper is to develop an elementary cohomological approach for decomposing a function into boundary values of holomorphic functions and for discussing the corresponding microlocal analysis and hyperfunction theory.
Estimates for operator norms on weighted spaces and reverse Jensen inequalities
Authors: Buckley Stephen M.
Published: June 2012

We examine the dependence on theAp{A_p}norm ofwwof the operator norms of singular integrals, maximal functions, and other operators inLp(w){L^p}(w). We also examine connections between some fairly general reverse Jensen inequalities and theAp{A_p}andRHpR{H_p}weight conditions.
Classification of the Tor-algebras of codimension four almost complete intersections
Authors: Kustin Andrew R.
Published: June 2012

Let

(
R
,
m
,
k
)

(R,m,k)

be a local ring in which

2
2

is a unit. Assume that every element of

k
k

has a square root in

k
k

. We classify the algebras

Tor
∙
R

⁡
(
R

/

J
,
k
)

\operatorname {Tor}_ \bullet ^R(R/J,k)

as

J
J

varies over all grade four almost complete intersection ideals in

R
R

. The analogous classification has already been found when

J
J

varies over all grade four Gorenstein ideals [21], and when

J
J

varies over all ideals of grade at most three [5, 30]. The present paper makes use of the classification, in [21], of the Tor-algebras of codimension four Gorenstein rings, as well as the (usually nonminimal)

DG

{\text {DG}}

-algebra resolution of a codimension four almost complete intersection which is produced in [25 and 26].
Quantitative rectifiability and Lipschitz mappings
Authors: David Guy , Semmes Stephen
Published: June 2012

The classical notion of rectifiability of sets in

R

n

{{\mathbf {R}}^n}

is qualitative in nature, and in this paper we are concerned with quantitative versions of it. This issue arises in connection with

L
p

{L^p}

estimates for singular integral operators on sets in

R

n

{{\mathbf {R}}^n}

. We give a criterion for one reasonably natural quantitative rectifiability condition to hold, and we use it to give a new proof of a theorem in [D3]. We also give some results on the geometric properties of a certain class of sets in

R

n

{{\mathbf {R}}^n}

which can be viewed as generalized hypersurfaces. Along the way we shall encounter some questions concerning the behavior of Lipschitz functions, with regard to approximation by affine functions in particular. We shall also discuss an amusing variation of the classical Lipschitz and bilipschitz conditions, which allow some singularities forbidden by the classical conditions while still forcing good behavior on substantial sets.
Loop space homology of spaces of small category
Authors: Félix Yves , Thomas Jean-Claude
Published: June 2012

Only little is known concerning

H
∗

(
Ω
X
;

k

)

{H_\ast }(\Omega X;{\mathbf {k}})

, the loop space homology of a finite

CW

{\text {CW}}

complex

X
X

with coefficients in a field

k

{\mathbf {k}}

. A space

X
X

is called an

r
r

-cone if there exists a filtration

∗
=

X
0

⊂

X
1

⊂
⋯
⊂

X
r

=
X

\ast = {X_0} \subset {X_1} \subset \cdots \subset {X_r} = X

, such that

X
i

{X_i}

has the homotopy type of the cofibre of a map from a wedge of sphere into

X

i
−
1

{X_{i - 1}}

. Denote by

A
X

{A_X}

the sub-Hopf algebra image of

H
∗

(
Ω

X
1

)

{H_\ast }(\Omega {X_1})

. We prove then that for a graded

r
r

-cone,

r
≤
3

r \leq 3

, there exists an isomorphism
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript upper X Baseline circled-times upper T left-parenthesis upper U right-parenthesis ModifyingAbove right-arrow With approximately-equals upper H Subscript asterisk Baseline left-parenthesis normal upper Omega upper X right-parenthesis">

A
X

⊗
T
(
U
)

→

≅

H
∗

(
Ω
X
)

{A_X} \otimes T(U)\xrightarrow { \cong }{H_\ast }(\Omega X)

.
Combinatorics of triangulations of 3-manifolds
Authors: Luo Feng , Stong Richard
Published: June 2012

In this paper, we study the average edge order of triangulations of closed

3
3

-manifolds and show in particular that the average edge order being less than

4.5
4.5

implies that triangulation is on the

3
3

-sphere.
On Klein’s combination theorem. IV
Authors: Maskit Bernard
Published: June 2012

This paper contains an expansion of the combination theorems to cover the following problems. New rank

1
1

parabolic subgroups are produced, while, as in previous versions, all elliptic and parabolic elements are tracked. A proof is given that the combined group is analytically finite if and only if the original groups are; in the analytically finite case, we also give a formula for the hyperbolic area of the combined group (i.e., the hyperbolic area of the set of discontinuity on the

2
2

-sphere modulo

G
G

) in terms of the hyperbolic areas of the original groups. There is also a new variation on the first combination theorem in which the common subgroup has finite index in one of the two groups.
Removing index 0 fixed points for area preserving maps of two-manifolds
Authors: Slaminka Edward E.
Published: June 2012

Using the method of free modifications developed by M. Brown and extended to area preserving homeomorphisms, we prove the following fixed point removal theorem. Theorem. Let

h
:
M
→
M

h:M \to M

be an orientation preserving, area preserving homeomorphism of an orientable two-manifold

M
M

having an isolated fixed point

p
p

of index

0
0

. Given any open neighborhood

N
N

of

p
p

such that

N
∩
Fix
⁡
(
h
)
=
p

N \cap \operatorname {Fix}(h) = p

, there exists an area preserving homeomorphism

h
^

\hat h

such that (i)
\[

h
^

=
h

o
n

M
−
N

¯

\hat h = h\;on\;\overline {M - N}

\]
and (ii)

h
^

\hat h

is fixed point free on

N
N

. Two applications of this theorem are the second fixed point for the topological version of the Conley-Zehnder theorem on the two-torus, and a new proof of the second fixed point for the Poincaré-Birkhoff Fixed Point Theorem.
Higher-dimensional analogues of Fuchsian subgroups of 𝑃𝑆𝐿(2,𝔬)
Authors: Vulakh L. Ya.
Published: June 2012

The problem of classification of

2
×
2

2 \times 2

indefinite Hermitian matrices over orders in Clifford algebras is considered. The unit groups of these matrices are analogous to maximal arithmetic Fuchsian subgroups of

PSL

(
2
,

o

)

{\text {PSL}}(2,\mathfrak {o})

where

o

\mathfrak {o}

is an order in a quadratic number field.
Supercuspidal representations and Poincaré series over function fields
Authors: Bump Daniel , Takahashi Shuzo
Published: June 2012

In this paper, we will give a new construction of certain cusp forms on

G
L
(
2
)

GL(2)

over a rational function field. The forms which we construct are analogs of holomorphic modular forms, in that the local representations at the infinite place are in the discrete series. The novelty of our approach is that we are able to give a very explicit construction of these forms as certain ’Poincaré series.’ We will also study the exponential sums which arise in the Fourier expansions of these Poincaré series.
Pseudobases in direct powers of an algebra
Authors: Bankston Paul
Published: June 2012

A subsetPPof an abstract algebraAAis apseudobasisif every function fromPPintoAAextends uniquely to an endomorphism onAA.AAis calledκ\kappa-freehas a pseudobasis of cardinalityκ\kappa;AAisminimally freeifAAhas a pseudobasis. (The00-free algebras are "rigid" in the strong sense; the11-free groups are always abelian, and are precisely the additive groups ofEE-rings.) Our interest here is in the existence of pseudobases in direct powersAI{A^I}of an algebraAA. On the positive side, ifAAis a rigid division ring,κ\kappais a cardinal, and there is no measurable cardinalμ\muwith|A|>μ≤κ|A| > \mu \leq \kappa, thenAI{A^I}isκ\kappa-free whenever|I|=|Aκ||I| = |{A^\kappa }|. On the negative side, ifAAis a rigid division ring and there is a measurable cardinalμ\muwith|A|>μ≤|I||A| > \mu \leq |I|, thenAI{A^I}is not minimally free.
On a conjecture of Lin-Ni for a semilinear Neumann problem
Authors: Adimurthi  , Yadava S. L.
Published: June 2012

Let

Ω
\Omega

be a bounded domain in

R

n

{\mathbb {R}^n}

(
n
≥
3
)

(n \geq 3)

and

λ
>
0

\lambda > 0

. We consider
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column minus normal upper Delta u plus lamda u equals u Superscript left-parenthesis n plus 2 right-parenthesis slash left-parenthesis n minus 2 right-parenthesis Baseline 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u greater-than 0 2nd Column a m p semicolon in normal upper Omega comma 3rd Row 1st Column StartFraction partial-differential u Over partial-differential nu EndFraction equals 0 2nd Column a m p semicolon on partial-differential normal upper Omega comma EndLayout">

−
Δ
u
+
λ
u
=

u

(
n
+
2
)

/

(
n
−
2
)

a
m
p
;

in

Ω
,

u
>
0

a
m
p
;

in

Ω
,

∂
u

∂
ν

=
0

a
m
p
;

on

∂
Ω
,

\begin {array}{*{20}{c}} { - \Delta u + \lambda u = {u^{(n + 2)/(n - 2)}}} & {{\text {in}}} \; {\Omega ,} \\ {u > 0} & {{\text {in}}} \; {\Omega ,} \\ {\frac {{\partial u}} {{\partial \nu }} = 0} & {{\text {on}}} \; {\partial \Omega ,} \\ \end {array}

\]
and show that for

λ
\lambda

sufficiently small, the minimal energy solutions are only constants.
A structural criterion for the existence of infinite central Λ(𝑝) sets
Authors: Hare Kathryn E. , Wilson David C.
Published: June 2012

We classify the compact, connected groups which have infinite central

Λ
(
p
)

\Lambda (p)

sets, arithmetically characterize central

Λ
(
p
)

\Lambda (p)

sets on certain product groups, and give examples of

Λ
(
p
)

\Lambda (p)

sets which are non-Sidon and have unbounded degree. These sets are intimately connected with Figà-Talamanca and Rider’s examples of Sidon sets, and stem from the existence of families of tensor product representations of almost simple Lie groups whose decompositions into irreducibles are rank-independent.
Multivariate discrete splines and linear Diophantine equations
Authors: Jia Rong Qing
Published: June 2012

In this paper we investigate the algebraic properties of multivariate discrete splines. It turns out that multivariate discrete splines are closely related to linear diophantine equations. In particular, we use a solvability condition for a system of linear diophantine equations to obtain a necessary and sufficient condition for the integer translates of a discrete box spline to be linearly independent. In order to understand the local structure of discrete splines we develop a general theory for certain systems of linear partial difference equations. Using this theory we prove that the integer translates of a discrete box spline are locally linearly independent if and only if they are linearly independent.
A tom Dieck theorem for strong shape theory
Authors: Günther Bernd
Published: June 2012

We consider an appropriate class of locally finite closed coverings of spaces, for which the strong shape of the elements of the covering and of their intersections determine the strong shape of the whole space. Conclusions concerning shape dimension and spaces having the strong shape of

CW

{\text {CW}}

-complexes are drawn, and a Leray spectral sequence for strong homology is given.
Hyponormal Toeplitz operators and extremal problems of Hardy spaces
Authors: Nakazi Takahiko , Takahashi Katsutoshi
Published: June 2012

The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos’s question about a subnormal Toeplitz operator when the self-commutator is finite rank.
Solutions to the nonautonomous bistable equation with specified Morse index. I. Existence
Authors: Alikakos Nicholas D. , Bates Peter W. , Fusco Giorgio
Published: June 2012

We investigate the existence of unstable solutions of specified Morse index for the equation

ε
2

u

x
x

−
f
(
x
,
u
)
=
0

{\varepsilon ^2}{u_{xx}} - f(x,u) = 0

on a finite interval and Neumann boundary conditions.
Set convergences. An attempt of classification
Authors: Sonntag Yves , Zălinescu Constantin
Published: June 2012

We endow families of nonempty closed subsets of a metric space with uniformities defined by semimetrics. Such structure is completely determined by a class (which is a family of closed sets) and a type (which is a semimetric). Two types are sufficient to define (and classify) almost all convergences known till now. These two types offer the possibility of defining other set convergences.
Approximation of approximate fibrations by bundle maps
Authors: Im Y. H.
Published: June 2012

In this paper, we give some conditions under which approximate fibrations can be approximated by locally trivial bundle maps.
An inverse boundary value problem for Schrödinger operators with vector potentials
Authors: Sun Zi Qi
Published: June 2012

We consider the Schrödinger operator for a magnetic potential

A
→

\vec A

and an electric potential

q
q

, which are supported in a bounded domain in

R

n

{\mathbb {R}^n}

with

n
≥
3

n \geq 3

. We prove that knowledge of the Dirichlet to Neumann map associated to the Schrödinger operator determines the magnetic field

rot
⁡
(

A
→

)

\operatorname {rot}(\vec A)

and the electric potential

q
q

simultaneously, provided

rot
⁡
(

A
→

)

\operatorname {rot}(\vec A)

is small in the

L
∞

{L^\infty }

topology.
The Alexander and Markov theorems via diagrams for links in 3-manifolds
Authors: Sundheim Paul A.
Published: June 2012

Let

M
M

be a

3
3

-manifold with an open book decomposition. We obtain a new proof that a link in

M
M

has a braided form and that two braided forms are related by a sequence of two Markov moves for

M
M

by generalizing Morton’s approach for links in

S
3

{S^3}

.
On the hyperbolic Kac-Moody Lie algebra 𝐻𝐴₁⁽¹⁾
Authors: Kang Seok-Jin
Published: June 2012

In this paper, using a homological theory of graded Lie algebras and the representation theory of

A
1

(
1
)

A_1^{(1)}

, we compute the root multiplicities of the hyperbolic Kac-Moody Lie algebra

H

A
1

(
1
)

HA_1^{(1)}

up to level

4
4

and deduce some interesting combinatorial identities.
Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups
Authors: Fraenkel Aviezri S.
Published: June 2012

For a real number

α
\alpha

, the floor function

⌊
α
⌋

\left \lfloor \alpha \right \rfloor

is the integer part of

α
\alpha

. The sequence

{

⌊

m
α

⌋

:
m
=
1
,
2
,
3
,
…
}

\{ \left \lfloor {m\alpha } \right \rfloor :m = 1,2,3, \ldots \}

is the Beatty sequence of

α
\alpha

. Identities are proved which express the sum of the iterated floor functional

A
i

{A^i}

for

1
≤
i
≤
n

1 \leq i \leq n

, operating on a nonzero algebraic number

α
\alpha

of degree

≤
n

\leq n

, in terms of only

A
1

=

⌊

m
α

⌋

,
m

{A^1} = \left \lfloor {m\alpha } \right \rfloor ,m

and a bounded term. Applications include discrete chaos (discrete dynamical systems), explicit construction of infinite nonchaotic subsequences of chaotic sequences, discrete order (identities), explicit construction of nontrivial Beatty subsequences, and certain arithmetical semigroups. (Beatty sequences have a large literature in combinatorics. They have also been used in nonperiodic tilings (quasicrystallography), periodic scheduling, computer vision (digital lines), and formal language theory.)
The structure of a random graph at the point of the phase transition
Authors: Łuczak Tomasz , Pittel Boris , Wierman John C.
Published: June 2012

Consider the random graph models

G
(
n
,
#

edges

=
M
)

G(n,\# \;{\text {edges}} = M)

and

G
(
n
,
Prob
⁡
(

edge

)
=
p
)

G(n,\operatorname {Prob}({\text {edge}}) = p)

with

M
=
M
(
n
)
=
(
1
+
λ

n

−
1

/

3

)
n

/

2

M = M(n) = (1 + \lambda {n^{ - 1/3}})n/2

and

p
=
p
(
n
)
=
(
1
+
λ

n

−
1

/

3

)

/

n

p = p(n) = (1 + \lambda {n^{ - 1/3}})/n

. For

l
≥
−
1

l \geq - 1

define an

l
l

-component of a random graph as a component which has exactly

l
l

more edges than vertices. Call an

l
l

-component with

l
≥
1

l \geq 1

a complex component. For both models, we show that when

λ
\lambda

is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order

n

2

/

3

{n^{2/3}}

, and the maximum value of

l
l

is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order

n

2

/

3

{n^{2/3}}

, and deletion of all suspended trees results in a "smoothed" graph of size of order

n

1

/

3

{n^{1/3}}

, with the maximum vertex degree

3
3

. The total number of branching vertices, i.e., of degree

3
3

, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a

3
3

-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order

n

1

/

3

{n^{1/3}}

. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdös and Rényi back in 1960. It is proven that there exists

p
(
λ
)

p(\lambda )

, the limiting planarity probability, with

0
>
p
(
λ
)
>
1

0 > p(\lambda ) > 1

,

p
(
−
∞
)
=
1

p( - \infty ) = 1

,

p
(
∞
)
=
0

p(\infty ) = 0

. In particular,

G
(
n
,
M
)

(
G
(
n
,
p
)
,

resp

.
)

G(n,M)\quad (G(n,p),{\text {resp}}.)

is almost surely nonplanar iff
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M minus n slash 2 right-parenthesis n Superscript negative 2 slash 3 Baseline right-arrow normal infinity left-parenthesis left-parenthesis n p minus 1 right-parenthesis n Superscript negative 1 slash 3 Baseline right-parenthesis right-arrow normal infinity comma resp period right-parenthesis">

(
M
−
n

/

2
)

n

−
2

/

3

→
∞

(
(
n
p
−
1
)

n

−
1

/

3

)
→
∞
,

resp

.
)

(M - n/2){n^{ - 2/3}} \to \infty \;((np - 1){n^{ - 1/3}}) \to \infty ,{\text {resp}}.)

.
Uniqueness theorems for parametrized algebraic curves
Authors: Hall Peter
Published: June 2012

Let

L
1

,
…
,

L
n

{L_1}, \ldots ,{L_n}

be lines in

P

2

{\mathbb {P}^2}

and let

f
,
g
:

P

1

→

P

2

f,g:{\mathbb {P}^1} \to {\mathbb {P}^2}

be nonconstant algebraic maps. For certain configurations of lines

L
1

,
…
,

L
n

{L_1}, \ldots ,{L_n}

, the hypothesis that, for

i
=
1
,
…
,
n

i = 1, \ldots ,n

, the inverse images

f

−
1

(

L
i

)

{f^{ - 1}}({L_i})

and

g

−
1

(

L
i

)

{g^{ - 1}}({L_i})

are equal, not necessarily with the same multiplicities, implies that

f
f

is identically equal to

g
g

.
Noether’s theorem for Hopf orders in group algebras
Authors: Weinraub David M.
Published: June 2012

LetKbe a local field with valuation ringRof residue characteristicpcontaining a primitivepth root of unityζp{\zeta _p}. We state an analog to Noether’s Theorem for modules overR-Hopf algebras and use induction techniques to deduce a criterion for this analog to hold. We then construct a family of noncommutative Hopf algebras which satisfy the criterion.