bluuebunny/crossref_metadata_2025 huggingface dataset:
𝑝-subgroups of compact Lie groups and torsion of infinite height in 𝐻*(𝐵𝐺)
Authors: Feshbach Mark
Published: June 2010

The relation between elementary abelian p-subgroups of a connected compact Lie group G and the existence of p-torsion in

H

∗

(
G
)

{H^ {\ast } }(G)

has been known for some time [B-S]. In this paper we prove that if G is any compact Lie group then

H

∗

(
B
G
)

{H^ {\ast } }(BG)

contains p-torsion of infinite height iff G contains an elementary abelian p-group not contained in a maximal torus. The hard direction is proven using the double coset theorem for the transfer. A third equivalent condition is also given.
On the boundary values of Riemann’s mapping function
Authors: Jackson R. J. V.
Published: June 2010

Classically, the calculus of variations is required to prove the existence of a biholomorphism from the unit disk to a given simply-connected, smooth domain in the complex plane. Here, the problem is reduced to the solution of an ordinary differential equation along the boundary of the domain. The sole coefficient in this equation is identified with the bounded term in the asymptotic expansion of the Bergman kernel function. It is shown that this coefficient can not depend upon any differential expression involving only the curvature function of the boundary.
The von Neumann kernel and minimally almost periodic groups
Authors: Rothman Sheldon
Published: June 2010

We calculate the von Neumann kernel

n
(
G
)

n(G)

of an arbitrary connected Lie group. As a consequence we see that the closed characteristic subgroup

n
(
G
)

n(G)

is also connected. It is shown that any Levi factor of a connected Lie group is closed. Then, various characterizations of minimal almost periodicity for a connected Lie group are given. Among them is the following. A connected Lie group G with radical R is minimally almost periodic (m.a.p.) if and only if

G

/

R

G/R

is semisimple without compact factors and

G

=

[
G
,

G

]
−

G\, = \,{[G,\,G]^ - }

. In the special case where R is also simply connected it is proven that

G

=

[
G
,

G
]

G\, = \,[G,\,G]

. This has the corollary that if the radical of a connected m.a.p. Lie group is simply connected then it is nilpotent. Next we prove that a connected m.a.p. Lie group has no nontrivial automorphisms of bounded displacement. As a consequence, if G is a m.a.p. connected Lie group, H is a closed subgroup of G such that

G

/

H

G/H

has finite volume, and

α
\alpha

is an automorphism of G with

disp

(
α
,

H
)

{\text {disp}}(\alpha ,\,H)

bounded, then

α
\alpha

is trivial. Using projective limits of Lie groups we extend most of our results on the characterization of m.a.p. connected Lie groups to arbitrary locally compact connected topological groups, and finally get a new and relatively simple proof of the Freudenthal-Weil theorem.
Some curvature properties of locally conformal Kähler manifolds
Authors: Vaisman Izu
Published: June 2010

Curvature identities and holomorphic sectional curvature of locally conformal Kähler manifolds are investigated. Particularly, sufficient conditions for such manifolds to be globally conformal Kähler are derived.
Branched extensions of curves in orientable surfaces
Authors: Ezell Cloyd L. , Marx Morris L.
Published: June 2010

Given a set of regular curves

f
1

,

…
,

f
ρ

{f_1}\,,\,\ldots ,\,{f_\rho }

in an orientable surface N, we are concerned with the existence and structure of all sense-preserving maps

F
:

M

→

N

F:\,M\, \to \,N

where (a) M is a bordered orientable surface with

ρ
\rho

boundary components

K
1

,
…
,

K
ρ

{K_1},\ldots ,\,{K_\rho }

, (b)

F

|

K
i

=

f
i

,

i

=

1
,

…
,

ρ

F|{K_i}\, = \,{f_i},\,i\, = \,1,\,\ldots ,\,\rho

, (c) at each interior point of M, there is an integer n such that F is locally topologically equivalent to the complex map

w

=

z
n

w\, = \,{z^n}

.
Effective 𝑝-adic bounds for solutions of homogeneous linear differential equations
Authors: Dwork B. , Robba P.
Published: June 2010

We consider a finite set of power series in one variable with coefficients in a field of characteristic zero having a chosen nonarchimedean valuation. We study the growth of these series near the boundary of their common “open” disk of convergence. Our results are definitive when the wronskian is bounded. The main application involves local solutions of ordinary linear differential equations with analytic coefficients. The effective determination of the common radius of convergence remains open (and is not treated here).
A relation between the coefficients in the recurrence formula and the spectral function for orthogonal polynomials
Authors: Geronimo Jeffrey S.
Published: June 2010

A relation is found between the rate of convergence of the coefficients in the recurrence formula for polynomials orthogonal on a segment of the real line and certain properties of the spectral function. The techniques of Banach algebras and scattering theory are used. The close connection between polynomials orthogonal on the unit circle and polynomials orthogonal on the real line is exploited.
A characterization of periodic automorphisms of a free group
Authors: McCool James
Published: June 2010

Let

θ
\theta

be an automorphism of finite order of a free group X. We characterise the action of

θ
\theta

on X by showing that X has a free basis which is the disjoint union of finite subsets

S
j

{S_j}

, where if

S
j

=

{

u
0

,

u
1

,

…
,

u
k

}

{S_j}\, = \,\{ {u_0},\,{u_1},\, \ldots ,\,{u_k}\}

then

u
i

θ

=

u

i

+

1

(
0

⩽

1

>

k
)

{u_i}\theta \, = \,{u_{i\, + \,1}}\,(0\, \leqslant \,1\, > \,k)

and

u
k

θ

=

A
j

u
0
e

B
j

{u_k}\theta \, = \,{A_j}u_0^e{B_j}

for some

A
j

{A_j}

,

B
j

{B_j}

in X and

ε

=

±

1

\varepsilon \, = \, \pm \,1

. As an application of this result, we obtain a list of the conjugacy classes of periodic automorphisms of the free group of rank three.
Inner product spaces associated with Poincaré complexes
Authors: Sasao Seiya , Takahashi Hideo
Published: June 2010

We consider the homotopy type classification of a certain kind of Poincaré complex. First we define an inner product space associated with such a Poincaré complex and we investigate the relation between the inner product space and the homotopy type of the Poincaré complex. As an application, some results for manifolds are proved.
Products in the Atiyah-Hirzebruch spectral sequence and the calculation of 𝑀𝑆𝑂_{∗}
Authors: Gray Brayton
Published: June 2010

It is possible to put a multiplicative structure in the Atiyah-Hirzebruch spectral sequence in certain cases even though the spectra involved are not both ring spectra. As a special case, an easy calculation of the homotopy of MSO is obtained.
On the computational complexity of determining the solvability or unsolvability of the equation 𝑋²-𝐷𝑌²=-1
Authors: Lagarias J. C.
Published: June 2010

The problem of characterizing thoseDfor which the Diophantine equationX2−DY2=−1{X^2}\, - \,D{Y^2}\, = \, - \,1is solvable has been studied for two hundred years. This paper considers this problem from the viewpoint of determining the computational complexity of recognizing suchD. For a givenD, one can decide the solvability or unsolvability ofX2−DY2=−1{X^2}\, - \,D{Y^2}\, = \, - \,1using the ordinary continued fraction expansion ofD\sqrt D, but for certainDthis requires more than13D(logD)−1\tfrac {1}{3}\,\sqrt D \,{(\log \,D)^{ - \,1}}computational operations. This paper presents a new algorithm for answering this question and proves that this algorithm always runs to completion inO(D1/4+ε)O({D^{1/4\, + \,\varepsilon }})bit operations. If the input to this algorithm includes a complete prime factorization ofDand a quadratic nonresidueni{n_i}for each primepi{p_i}dividingD, then this algorithm is guaranteed to run to completion inO((logD)5(loglogD)(logloglogD))O({(\log \,D)^5}\,(\log \,\log \,D)(\log \,\log \,\log \,D))bit operations. This algorithm is based on an algorithm that finds a basis of forms for the 2-Sylow subgroup of the class group of binary quadratic forms of determinantD.
A strong Stieltjes moment problem
Authors: Jones William B. , Thron W. J. , Waadeland Haakon
Published: June 2010

This paper is concerned with double sequences of complex numbers

C

=

{

c
n

}

−
∞

∞

C\, = \,\{ {c_n}\} _{ - \infty }^\infty

and with formal Laurent series

L
0

(
C
)

=

Σ
1
∞

−

c

−
m

z
m

{L_0}(C)\, = \,\Sigma _1^\infty \, - \,{c_{ - m}}{z^m}

and

L
∞

(
C
)

=

Σ
0
∞

c
m

z

−
m

{L_\infty }(C)\, = \,\Sigma _0^\infty \,{c_m}{z^{ - m}}

generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having

L
0

(
C
)

{L_0}(C)

and

L
∞

(
C
)

{L_\infty }(C)

as asymptotic expansions at

z

=

0

z\, = \,0

and

z

=

∞

z\, = \,\infty

, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function

ψ
(
t
)

\psi (t)

with infinitely many points of increase on

[
0
,

∞
)

[0,\,\infty )

such that, for every integer n,

c
n

=

∫
0
∞

(
−
t
)

n

d
ψ
(
t
)

{c_n}\, = \,\int _0^\infty {{{( - t)}^n}\,d\psi (t)}

? The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function

ψ
(
t
)

\psi (t)

has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive T-fractions) which correspond to

L
0

(
C
)

{L_0}(C)

at

z

=

0

z\, = \,0

and

L
∞

(
C
)

{L_\infty }(C)

at

z

=

∞

z\, = \,\infty

. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have

L
0

(
C
)

{L_0}(C)

and

L
∞

(
C
)

{L_\infty }(C)

as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions

ψ

(
0
)

(
t
)

{\psi ^{(0)}}(t)

and

ψ

(
1
)

(
t
)

{\psi ^{(1)}}(t)

solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both

L
0

(
C
)

{L_0}(C)

and

L
∞

(
C
)

{L_\infty }(C)

. References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.
On a 4-manifold homology equivalent to a bouquet of surfaces
Authors: Kawauchi Akio
Published: June 2010

This paper gives some algebraic invariants for a piecewise linear imbedding of a surface into some 4-manifold inducing a Z or Q-homology isomorphism. Several examples are obtained by using these invariants.
Invariance of solutions to invariant parametric variational problems
Authors: Brothers John E.
Published: June 2010

Let G be a compact Lie group of diffeomorphisms of a connected orientable manifold M of dimension

n
+
1

n + 1

. Assume the orbits of highest dimension to be connected. Let

Ψ
\Psi

be a convex positive even parametric integrand of degree n on M which is invariant under the action of G. Let T be a homologically

Ψ
\Psi

-minimizing rectifiable current of dimension n on M, and assume there exists a G-invariant rectifiable current

T
′

T’

which is homologous to T. It is shown that T is G-invariant provided

Ψ
\Psi

satisfies a symmetry condition which makes it no less efficient for the tangent planes of T to lie along the orbits. This condition is satisfied by the area integrand in case G is a group of isometries of a Riemannian metric on M. Consequently, one obtains the corollary that if a rectifiable current T is a solution to the n-dimensional Plateau problem in M with G-invariant boundary

∂
T

\partial T

, and if

∂
T

\partial T

bounds a G-invariant rectifiable current

T
′

T’

such that

T
−

T
′

T - T’

is a boundary, then T is G-invariant. An application to the Plateau problem in

S

3

{{\textbf {S}}^3}

is given.
Noninvariance of an approximation property for closed subsets of Riemann surfaces
Authors: Scheinberg Stephen
Published: June 2010

A closed subset E of an open Riemann surface M is said to have the approximation property

a

\mathcal {a}

if each continuous function on E which is analytic at all interior points of E can be approximaed uniformly on E by functions which are everywhere analytic on M. It is known that

a

\mathcal {a}

is a topological invariant (i.e., preserved by homeomorphisms of the pair

(
M
,
E
)

(M,E)

) when M is of finite genus but not in general, not even for

C
∞

{C^\infty }

quasi-conformal automorphisms of M. The principal result of this paper is that

a

\mathcal {a}

is not invariant even under a real-analytic isotopy of quasi-conformal automorphisms (of a certain M). M is constructed as the two-sheeted unbranched cover of the plane minus a certain discrete subset of the real axis, and the isotopy is induced by

(
x
+

i
y
,

t
)
↦
x
+

i
t
y

(x + \,iy,\,t) \mapsto x + \,ity

, for

t
>
0

t > 0

; E can be taken to be that portion of M which lies over a horizontal strip.
Applications of the Fourier-Wiener transform to differential equations on infinite-dimensional spaces. I
Authors: Lee Yuh Jia
Published: June 2010

Let

(
H
,
i
,
B
)

(H,i,B)

be an abstract Wiener space and

p
t

{p_t}

be the Wiener measure on B with variance t. Let [B] be the complexification of B and

E

a

{\mathcal {E}_a}

be the class of exponential type analytic functions defined on [B]. We define the Fourier-Wiener c-transform for any f in

E

a

{\mathcal {E}_a}

by
\[

F
c

f
(
y
)

=

∫

B

f
(
x

+

i
y
)

p
c

(
d
x
)

{F_c}f(y)\, = \,\int _\textbf {B} {f(x\, + \,iy){p_c}(dx)}

\]
and the inverse transform by

F

c

−
1

f
(
y
)

=

F

c

f
(
−
y
)

\mathcal {F}_c^{ - 1}f(y)\, = \,{\mathcal {F}_c}f( - y)

. Then the inversion formula holds and

F

2

{\mathcal {F}_2}

extends to

L
2

(
B
,

p
1

)

{L^2}(B,{p_1})

as a unitary operator. Next, we apply the above transform to investigate the existence, uniqueness and regularity of solutions for Cauchy problems associated with the following two equations: (1)

u
t

=

−

N

k

u

{u_t}\, = \, - {\mathcal {N}^k}u

, (2)

u

t
t

=

−

N

k

u

{u_{tt}}\, = \, - {\mathcal {N}^k}u

; and the elliptic type equation (3)

−

N
k

u

=

f
(
k

⩾

1
)

- {N^k}u\, = \,f(k\, \geqslant \,1)

, where

Δ
\Delta

is the Laplacian and

N

u
(
x
)

=

−
Δ
u
(
x
)

+

(
x
,
D
u
(
x
)
)

\mathcal {N}u(x)\, = \, - \Delta u(x)\, + \,(x,Du(x))

.
Families of real and symmetric analytic functions
Authors: Abu-Muhanna Yusuf , MacGregor Thomas H.
Published: June 2010

We introduce families of functions analytic in the unit disk and having rotational symmetries. The families include the

k
k

-fold symmetric univalent functions which have real coefficients. We relate the families to special classes of functions with a positive real part and then determine their extreme points. The case

k
=
2

k = 2

corresponds to the odd functions which "preserve quadrants" and the extreme points of this set are characterized by having a radial limit which is real or imaginary almost everywhere. We also find estimates on the initial coefficients of functions in the families.
Subnormal operators, Toeplitz operators and spectral inclusion
Authors: Keough Gerard E.
Published: June 2010

Let

S
S

be a subnormal operator on the Hilbert space

H
H

, and let

N
=
∫
z

d
E
(
z
)

N = \int z \;dE(z)

be its minimal normal extension on

K
K

. Let

μ
\mu

be a scalar spectral measure for

N
N

. If

f
∈

L
∞

(
μ
)

f \in {L^\infty }(\mu )

, define

T
f

=
P
f
(
N
)

|

H

:

H
→
H

{T_f} = Pf(N){|_H}:\;H \to H

, where

P
:
K
→
H

P:K \to H

denotes orthogonal projection.

S
S

has the

C
∗

{C^ \ast }

-Spectral Inclusion Property (

C
∗

{C^ \ast }

-SIP) if

σ
(
f
(
N
)
)
⊆
σ
(

T
f

)

\sigma (f(N)) \subseteq \sigma ({T_f})

, for all

f
∈
C
(
σ
(
N
)
)

f \in C(\sigma (N))

, and

S
S

has the

W
∗

{W^\ast }

-Spectral Inclusion Property (

W
∗

{W^\ast }

-SIP) if

σ
(
f
(
N
)
)
⊆
σ
(

T
f

)

\sigma (f(N)) \subseteq \sigma ({T_f})

, for all

f
∈

L
∞

(
μ
)

f \in {L^\infty }(\mu )

. It is shown that

S
S

has the

C
∗

{C^\ast }

-SIP if and only if

σ
(
N
)
=
Π
(
S
)

\sigma (N) = \Pi (S)

, the approximate point spectrum of

S
S

. This is equivalent to requiring that

E
(
Δ
)
K

E(\Delta )K

have angle

0
0

with

H
H

, for all nonempty, relatively open

Δ
⊆
σ
(
N
)

\Delta \subseteq \sigma (N)

.

S
S

has the

W
∗

{W^\ast }

-SIP if this angle condition holds for all proper Borel subsets of

σ
(
N
)

\sigma (N)

. If

S
S

is pure and has the

C
∗

{C^\ast }

or

W
∗

{W^\ast }

-SIP, then it is shown that

σ
(
f
(
N
)
)
⊆

σ
e

(

T
f

)

\sigma (f(N)) \subseteq {\sigma _e}({T_f})

, for all appropriate

f
f

.
On computation of eigenvalues for differential equations by Newton’s method
Authors: Nehari Zeev
Published: June 2010

Professor Zeev Nehari, in the last months before his untimely death, had been analyzing and developing a new algorithm for computing eigenvalues of selfadjoint boundary value problems of arbitrary order. Apparently, his main goals were Theorems 3.1 and 3.2 below (which incidentally yield the eigenvalues of the given problem, as well as those of a related problem). Unfortunately, as far as we know, Professor Nehari has not left a proof of the basic Lemma 3.1 which presumably was to be based on

§
2

\S 2

or related results. It seems, however, worthwhile to publish this paper since the results of

§
§
1

\S \S 1

and

2
2

are complete and of independent interest and since the question of the validity of Lemma 3.1 is also of interest. While the presentation in

§
1

\S 1

and

2
2

is in most respects complete, there are a few places where some minor clarifications by selected footnotes were felt to be desirable. Also, Professor Nehari’s original incomplete

§
3

\S 3

has been replaced by a revised and abbreviated version.
The zeros of the second derivative of the reciprocal of an entire function
Authors: Hellerstein Simon , Williamson Jack
Published: June 2010

Let

f
f

be a real entire function of finite order with only real zeros. Assuming that

f
′

f’

has only real zeros, we show that the number of nonreal zeros of

f
f

equals the number of real zeros of

F
F

, where

F
=
1

/

f

F = 1/f

. From this, we show that

F
F

has only real zeros if and only if

f
(
z
)
=
exp
⁡
(
a

z
2

+
b
z
+
c
)

f(z) = \exp (a{z^2} + bz + c)

,

a
⩾
0

a \geqslant 0

, or

f
(
z
)
=

(
A
z
+
B

)
n

f(z) = {(Az + B)^n}

,

A
≠
0

A \ne 0

,

n
n

a positive integer.
A representation-theoretic criterion for local solvability of left invariant differential operators on nilpotent Lie groups
Authors: Corwin Lawrence
Published: June 2010

LetLLbe a left invariant differential operator on the nilpotent Lie groupNN. It is shown that ifπ(L)\pi (L)is invertible for all irreducible representationsπ\piin general position (and if the inverses satisfy some mild technical conditions), thenLLis locally solvable. This result generalizes a theorem ofL{\text {L}}. Rothschild.
Quasisymmetric embeddings in Euclidean spaces
Authors: Väisälä Jussi
Published: June 2010

We consider quasi-symmetric embeddings

f
:
G
→

R
n

f:G \to {R^n}

,

G
G

open in

R
p

{R^p}

,

p
⩽
n

p \leqslant n

. If

p
=
n

p = n

, quasi-symmetry implies quasi-conformality. The converse is true if

G
G

has a sufficiently smooth boundary. If

p
>
n

p > n

, the Hausdorff dimension of

f
G

fG

is less than

n
n

. If

f
G

fG

has a finite

p
p

-measure,

f
f

preserves the property of being of

p
p

-measure zero. If

p
>
n

p > n

and

n
⩾
3

n \geqslant 3

,

R
n

{R^n}

contains a quasi-symmetric

p
p

-cell which is topologically wild. We also prove auxiliary results on the relations between Hausdorff measure and Čech cohomology.
Random evolution processes with feedback
Authors: Siegrist Kyle
Published: June 2010

A general random evolution Markov process is constructed which switches back and forth at random among a given collection of Markov processes ("modes of evolution") defined on a common evolution state space and indexed by an evolution rule space. Feedback is incorporated by allowing the path of the evolution component to influence the changes in evolution rule. The semigroup of the random evolution process is studied and is used to compare the process with the operator random evolutions of Griego and Hersh. Using deterministic modes of evolution, we generalize the Markov processes constructed by Erickson and by Heath. We also study new random evolution processes constructed from Brownian motions and from regular step processes.
Recognizing the real line
Authors: Gurevich Yuri , Holland W. Charles
Published: June 2010

A certain elementary statement about the group of automorphisms of the real line

R

\mathbf {R}

is sufficient to characterize

R

\mathbf {R}

among homogeneous chains. A similar result holds for the chain of rational numbers.
The diameter of random graphs
Authors: Bollobás Béla
Published: June 2010

Extending some recent theorems of Klee and Larman, we prove rather sharp results about the diameter of a random graph. Among others we show that ifd=d(n)⩾3d = d(n) \geqslant 3andm=m(n)m = m(n)satisfy(log⁡n)/d−3log⁡log⁡n→∞(\log n)/d - 3\,\log \log n \to \infty,2d−1md/nd+1−log⁡n→∞{2^{d - 1}}{m^d}/{n^{d + 1}} - \log n \to \inftyanddd−2md−1/nd−log⁡n→−∞{d^{d - 2}}{m^{d - 1}}/{n^d} - \log n \to - \inftythen almost every graph withnnlabelled vertices andmmedges has diameterdd.
Curvature tensors on almost Hermitian manifolds
Authors: Tricerri Franco , Vanhecke Lieven
Published: June 2010

A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.
Counting divisors with prescribed singularities
Authors: Vainsencher Israel
Published: June 2010

Given a family of divisors

{

D
s

}

\{ {D_s}\}

in a family of smooth varieties

{

Y
s

}

\{ {Y_s}\}

and a sequence of integers

m
1

,
…
,

m
t

{m_1}, \ldots ,{m_t}

, we study the scheme parametrizing the points

(
s
,

y
1

,
…
,

y
t

)

(s,{y_1}, \ldots ,{y_t})

such that

y
i

{y_i}

is a (possibly infinitely near)

m
i

{m_i}

-fold point of

D
s

{D_s}

. We obtain a general formula which yields, as special cases, the formula of de Jonquières and other classical results of Enumerative Geometry. We also study the questions of finiteness and the multiplicities of the solutions.
Hypersingular integrals and parabolic potentials
Authors: Chanillo Sagun
Published: June 2010

In this paper we characterize the potential spaces associated with the heat equation in terms of singular integrals of mixed homogeneity.
Twist maps, coverings and Brouwer’s translation theorem
Authors: Winkelnkemper H. E.
Published: June 2010

We apply the Brouwer Translation Theorem to a class of twist maps of the annulus (which contains

C
1

{C^1}

area preserving maps) to show that, if

h
h

belongs to this class, then a certain set

P

0

{\mathcal {P}_0}

of periodic points of

h
h

cannot be dense. The definition of

P

0

{\mathcal {P}_0}

does not impose any a priori restrictions on the periods of the points of

P

0

{\mathcal {P}_0}

.
Compact groups of homeomorphisms on tree-like continua
Authors: Fugate J. B. , McLean T. B.
Published: June 2010

This paper is concerned with the fixed point sets of certain collections of homeomorphisms on a tree-like continuum. Extending a theorem of P. A. Smith, the authors prove that a periodic homeomorphism has a (nonvoid) continuum as its fixed point set. They then deduce possible periods for homeomorphisms on tree-like continua which satisfy certain decomposability or irreducibility conditions. The main result of the paper is that a compact group of homeomorphisms has a continuum as its fixed point set. This is applied to isometries. The paper concludes with sufficient conditions that a pointwise periodic homeomorphism have a fixed point.
KV-theory of categories
Authors: Weibel Charles A.
Published: June 2010

Quillen has constructed a

K
K

-theory

K

∗

C

{K_{\ast }}C

for nice categories, one of which is the category of projective

R
R

-modules. We construct a theory

K

V

∗

C

K{V_{\ast }}C

for the nice categories parametrized by rings. When applied to projective modules we recover the Karoubi-Villamayor

K
K

-theory

K

V

∗

(
R
)

K{V_{\ast }}(R)

. As an application, we show that the Cartan map from

K

∗

(
R
)

{K_{\ast }}(R)

to

G

∗

(
R
)

{G_{\ast }}(R)

factors through the groups

K

V

∗

(
R
)

K{V_{\ast }}(R)

. We also compute

K

V

∗

K{V_{\ast }}

for the categories of faithful projectives and Azumaya algebras, generalizing results of Bass.
Involutions on Klein spaces 𝑀(𝑝,𝑞)
Authors: Kim Paik Kee
Published: June 2010

The Klein spaces

M
(
p
,

q
)

M(p,\,q)

are defined (up to homeomorphisms) to be the class of closed, orientable, irreducible

3
3

-manifolds with finite fundamental groups, in which a Klein bottle can be embedded. Their fundamental groups act freely on the

3
3

-sphere

S
3

{S^3}

in the natural way. We obtain a complete classification of the PL involutions on Klein spaces

M
(
p
,

q
)

M(p,\,q)

. It can be applied to the study of some transformation group actions on

S
3

{S^3}

and double branched coverings of

S
3

{S^3}

.
Quasicoherent sheaves over affine Hensel schemes
Authors: Greco Silvio , Strano Rosario
Published: June 2010

The following two theorems concerning affine Hensel schemes are proved. Theorem A. Every quasi-coherent sheaf over an affine Hensel scheme is generated by its global sections. Theorem B.

H
p

(
X
,

F
)
=
0

{H^p}(X,\,F) = 0

for all positive

p
p

and all quasi-coherent sheaves

F
F

over an affine Hensel scheme

X
X

.
On nonseparable Banach spaces
Authors: Argyros Spiros A.
Published: June 2010

Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of

L
1

{
−
1
,

1

}
α

{L^1}{\{ - 1,\,1\} ^\alpha }

into

X

∗

{X^{\ast }}

implies, for any infinite cardinal

α
\alpha

, the isomorphic embedding of

l
α
1

l_\alpha ^1

into

X
X

, gets the following form: if

α
=
ω

\alpha = \omega

, has been proved from Pełczynski; if

α
>

ω
+

\alpha > {\omega ^ + }

, the proof is given in this paper; if

α
=

ω
+

\alpha = {\omega ^ + }

, in

Z

.F

.C

.

+

C

.H

.

{\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + {\text {C}}{\text {.H}}{\text {.}}

, an example discovered by Haydon gives a negative answer; if

α
=

ω
+

\alpha = {\omega ^ + }

, in

Z

.F

.C

.

+
⌝

C

.H

.

+

M

.A

.

{\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + \urcorner {\text {C}}{\text {.H}}{\text {.}} + {\text {M}}{\text {.A}}{\text {.}}

, is also proved in this paper.
Mathematical theory of single channel systems. Analyticity of scattering matrix
Authors: Sigal I. M.
Published: June 2010

We show that the

S
S

-matrix of a quantum many-body, short-range, single-channel system has a meromorphic continuation whose poles occur at most at the dilation-analytic resonances [28], [24] and at the eigenvalues of the Hamiltonian. In passing, we prove the main spectral theorem (on location of the essential spectrum) and asymptotic completeness for the mentioned class of systems.
Fundamental groups of topological 𝑅-modules
Authors: Bateson Ann
Published: June 2010

The main result of this paper is that if

R
R

is a countable, Noetherian ring, then the underlying abelian group of every

R
R

-module is isomorphic to the fundamental group of some topological

R
R

-module. As a corollary, it is shown that for certain varieties

V
V

(e.g., varieties of finite type) every abelian group in

V
V

is isomorphic to the fundamental group of some arcwise connected topological algebra in

V
V

.
Free Lie subalgebras of the cohomology of local rings
Authors: Avramov Luchezar L.
Published: June 2010

A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra

Ext
A

(
k
,

k
)

{\operatorname {Ext} _A}(k,\,k)

to be a free module over the universal envelope of a free graded Lie subalgebra. It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure. For all rings with

edim
⁡
A
−
depth
⁡
A
⩽
3

\operatorname {edim} A - \operatorname {depth} A \leqslant 3

, and for Gorenstein rings with

edim
⁡
A
−
depth
⁡
A
=
4

\operatorname {edim} A - \operatorname {depth} A = 4

, the following dichotomy is proved: Either

A
A

is a complete intersection, or

Ext
A

(
k
,

k
)

{\operatorname {Ext} _A}(k,\,k)

contains a nonabelian free graded Lie subalgebra.
The minimum modulus of certain small entire functions
Authors: Fenton P. C.
Published: June 2010

Suppose that

f
(
z
)

f(z)

is an entire function satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min Underscript StartAbsoluteValue z EndAbsoluteValue equals r Endscripts StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-slanted-equals upper C left-parenthesis sigma right-parenthesis max Underscript StartAbsoluteValue z EndAbsoluteValue equals r Endscripts StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue comma">

min

|

z

|

=
r

|

f
(
z
)

|

⩽
C
(
σ
)

max

|

z

|

=
r

|

f
(
z
)

|

,

\min \limits _{|z| = r} |f(z)| \leqslant C(\sigma )\max \limits _{|z| = r} |f(z)|,

\]
for

r
⩾

ρ
0

>
0

r \geqslant {\rho _0} > 0

, where

σ
>
0

\sigma > 0

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis sigma right-parenthesis equals product Underscript k equals 1 Overscript normal infinity Endscripts left-brace StartFraction 1 minus exp left-parenthesis minus left-parenthesis 2 k minus 1 right-parenthesis slash 4 sigma right-parenthesis Over 1 plus exp left-parenthesis minus left-parenthesis 2 k minus 1 right-parenthesis slash 4 sigma right-parenthesis EndFraction right-brace squared period">

C
(
σ
)
=

∏

k
=
1

∞

{

1
−
exp
⁡
(
−
(
2
k
−
1
)

/

4
σ
)

1
+
exp
⁡
(
−
(
2
k
−
1
)

/

4
σ
)

}

2

.

C(\sigma ) = \prod \limits _{k = 1}^\infty {{{\left \{ {\frac {{1 - \exp ( - (2k - 1) / 4\sigma )}} {{1 + \exp ( - (2k - 1) / 4\sigma )}}} \right \}}^2}.}

\]
It is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript bar Endscripts Underscript r right-arrow normal infinity Endscripts StartFraction max Underscript StartAbsoluteValue z EndAbsoluteValue equals r Endscripts StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue minus sigma left-parenthesis log r right-parenthesis squared Over log r EndFraction greater-than-or-slanted-equals minus 2 sigma log left-parenthesis max left-parenthesis rho 0 comma StartAbsoluteValue a 1 EndAbsoluteValue right-parenthesis right-parenthesis comma">

lim
_

r
→
∞

max

|

z

|

=
r

|

f
(
z
)

|

−
σ

(
log
⁡
r
)

2

log
⁡
r

⩾
−
2
σ
log
⁡
(
max
(

ρ
0

,

|

a
1

|

)
)
,

\underline {\lim } \limits _{r \to \infty } \frac {{{{\max }_{|z| = r}}|f(z)| - \sigma {{(\log r)}^2}}} {{\log r}} \geqslant - 2\sigma \log (\max ({\rho _0},|{a_1}|)),

\]
where

a
1

{a_1}

is the first nonzero zero of

f
f

.
Extensions for AF 𝐶* algebras and dimension groups
Authors: Handelman David
Published: June 2010

Let

A
A

,

C
C

be approximately finite dimensional

(

AF)

C

∗

({\text {AF)}}\,{C^{\ast }}

algebras, with

A
A

nonunital and

C
C

unital; suppose that either (i)

A
A

is the algebra of compact operators, or (ii) both

A
A

,

C
C

are simple. The classification of extensions of

A
A

by

C
C

is studied here, by means of Elliott’s dimension groups. In case (i), the weak Ext group of

C
C

is shown to be

Ext

Z

(

K
0

(
C
)
,

Z

)

{\operatorname {Ext} _{\mathbf {Z}}}({K_0}(C),\,{\mathbf {Z}})

, and the strong Ext group is an extension of a cyclic group by the weak Ext group; conditions under which either Ext group is trivial are determined. In case (ii), there is an unnatural and complicated group structure on the classes of extensions when

A
A

has only finitely many pure finite traces (and somewhat more generally).
Left definite multiparameter eigenvalue problems
Authors: Binding Paul
Published: June 2010

We study the problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis upper T Subscript m Baseline x Subscript m Baseline equals sigma-summation Underscript n equals 1 Overscript k Endscripts lamda Subscript n Baseline upper V Subscript m n Baseline x Subscript m Baseline comma 0 not-equals x Subscript m Baseline element-of upper H Subscript m Baseline comma m equals 1 comma ellipsis comma k comma">

(
∗
)

T
m

x
m

=

∑

n
=
1

k

λ
n

V

m
n

x
m

,

0
≠

x
m

∈

H
m

,

m
=
1
,
…
,
k
,

(\ast )\qquad {T_m}{x_m} = \sum \limits _{n = 1}^k {{\lambda _n}{V_{mn}}{x_m},\qquad 0 \ne } {x_m} \in {H_m},\,m = 1, \ldots ,k,

\]
where

T
m

{T_m}

and

V

m
n

{V_{mn}}

are selfadjoint linear operators on separable Hilbert spaces

H
m

{H_m}

, with

T
m

{T_m}

positive,

T
m

−
1

T_m^{ - 1}

compact and

V

m
n

{V_{mn}}

bounded. We assume “left definiteness” which involves positivity of certain linear combinations of cofactors in the determinant with

(
m
,

n
)

(m,\,n)

th entry

(

x
m

,

V

m
n

x
m

)

({x_m},\,{V_{mn}}{x_m})

. We establish a spectral theory for

(
∗
)

(\ast )

that is in some way simpler and more complete than those hitherto available for this case. In particular, we make use of operators

B
n

=

Δ
n

−
1

Δ
0

{B_n} = \Delta _n^{ - 1}{\Delta _0}

, where the

Δ
n

{\Delta _n}

are determinantal operators on

⊗

m
=
1

k

H
m

\otimes _{m = 1}^k{H_m}

. This complements an established approach to the alternative “right definite” problem (where

Δ
0

{\Delta _0}

is positive) via the operators

Γ
n

=

Δ
0

−
1

Δ
n

{\Gamma _n} = \Delta _0^{ - 1}{\Delta _n}

.
Simple knots in compact, orientable 3-manifolds
Authors: Myers Robert
Published: June 2010

A simple closed curve

J
J

in the interior of a compact, orientable

3
3

-manifold

M
M

is called a simple knot if the closure of the complement of a regular neighborhood of

J
J

in

M
M

is irreducible and boundary-irreducible and contains no non-boundary-parallel, properly embedded, incompressible annuli or tori. In this paper it is shown that every compact, orientable

3
3

-manifold

M
M

such that

∂
M

\partial M

contains no

2
2

-spheres contains a simple knot (and thus, from work of Thurston, a knot whose complement is hyperbolic). This result is used to prove that such a

3
3

-manifold is completely determined by its set

K

(
M
)

\mathcal {K}(M)

of knot groups, i.e, the set of groups

π
1

(
M
−
J
)

{\pi _1}(M - J)

as

J
J

ranges over all the simple closed curves in

M
M

. In addition, it is proven that a closed

3
3

-manifold

M
M

is homeomorphic to

S
3

{S^3}

if and only if every simple closed curve in

M
M

shrinks to a point inside a connected sum of graph manifolds and

3
3

-cells.
Operator-self-similar processes in a finite-dimensional space
Authors: Hudson William N. , Mason J. David
Published: June 2010

A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar.
On critical point theory for indefinite functionals in the presence of symmetries
Authors: Benci Vieri
Published: June 2010

We consider functionals which are not bounded from above or from below even modulo compact perturbations, and which exhibit certain symmetries with respect to the action of a compact Lie group. We develop a method which permits us to prove the existence of multiple critical points for such functionals. The proofs are carried out directly in an infinite dimensional Hilbert space, and they are based on minimax arguments. The applications given here are to Hamiltonian systems of ordinary differential equations where the existence of multiple time-periodic solutions is established for several classes of Hamiltonians. Symmetry properties of these Hamiltonians such as time translation invariancy or evenness are exploited.
2-sided embeddings of projective planes into 3-manifolds
Authors: Ochiai Mitsuyuki
Published: June 2010

Let

M
M

be a nonorientable closed

3
3

-manifold which admits a

2
2

-sided embedding of a projective plane. Then we first prove the following theorem: If

M
M

has a Heegaard splitting of genus two, then

M
M

is homeomorphic to

P

2

×

S

1

{P^{2}}\times {S^{1}}

. Next, let

M
M

be a nonorientable

3
3

-manifold whose fundamental group is abelian. We verify that if

M
M

has a Heegaard splitting of genus two, then

M
M

is either the nonorientable

2
2

-sphere bundle over the circle or

P

2

×

S

1

{P^{2}}\times {S^{1}}

.
An elementary integral solution operator for the Cauchy-Riemann equations on pseudoconvex domains in 𝐶ⁿ
Authors: Range R. Michael
Published: June 2010

An integral representation formula for

(
0
,
q
)

(0,q)

forms is constructed on a strictly pseudoconvex domain

D
D

in

C

n

\mathbf {C}^n

by using only the local geometry of the boundary of

D
D

. By combining this representation with elementary results about compact operators in Banach spaces, one obtains the solution of the Levi problem and, more importantly, an integral solution operator for

∂
¯

\bar {\partial }

on

D
D

. The construction does not need any a priori knowledge of the solvability of

∂
¯

\bar {\partial }

and thus allows us to establish fundamental global results by a direct and elementary method.
On the absence of positive eigenvalues of Schrödinger operators with long range potentials
Authors: Kalf Hubert , Krishna Kumar V.
Published: June 2010

In this paper we consider the problem of obtaining upper bounds for the positive bound states associated with the Schrödinger operators with long range potentials. We have extended the size of the class of long range potentials for which one can establish the nonexistence of positive eigenvalues, improving upon the recent results of G. B. Khosrovshahi, H. A. Levine and L. E. Payne (Trans. Amer. Math. Soc. 253 (1979), 211-228).
Monotone decompositions of 𝜃-continua
Authors: Grace E. E.
Published: June 2010

A

θ
\theta

-continuum (

θ
n

{\theta _n}

-continuum) is a compact, connected, metric space that is not separated into infinitely many (more than

n
n

) components by any subcontinuum. The following results are among those proved. The first generalizes earlier joint work with E. J. Vought for

θ
n

{\theta _n}

-continua, and the second generalizes earlier work by Vought for

θ
1

{\theta _1}

-continua. A

θ
\theta

-continuum

X
X

admits a monotone, upper semicontinuous decomposition

D

\mathcal {D}

such that the elements of

D

\mathcal {D}

have void interiors and the quotient space

X

/

D

X/\mathcal {D}

is a finite graph, if and only if, for each nowhere dense subcontinuum

H
H

of

X
X

, the continuum

T
(
H
)
=
{
x
∈
X

|

T(H) = \{x \in X|

if

K
K

is a subcontinuum of

X
X

and

x
x

is in the interior of

K
K

, then

K
∩
H
≠
∅
}

K \cap H \ne \emptyset \}

is nowhere dense. Also, if

X
X

satisfies this condition, then

X
X

is in fact a

θ
n

{\theta _n}

-continuum, for some natural number

n
n

, and, for each natural number

m
m

,

X
X

is a

θ
m

{\theta _m}

-continuum, if and only if

X

/

D

X/\mathcal {D}

is a

θ
m

{\theta _m}

-continuum.
On the triangulation of stratified sets and singular varieties
Authors: Johnson F. E. A.
Published: June 2010

We show that every compact stratified set in the sense of Thom can be triangulated as a simplicial complex. The proof uses that author’s description of a stratified set as the geometric realisation of a certain type of diagram of smooth fibre bundles and smooth imbeddings, and the triangulability of smooth fibre bundles. As a consequence, we obtain proofs of the classical triangulation theorems for analytic and subanalytic sets, and a correct proof of Yang’s theorem that the orbit space of a smooth compact transformation group is triangulable.
Nonlinear mappings that are globally equivalent to a projection
Authors: Plastock Roy
Published: June 2010

The Rank theorem gives conditions for a nonlinear Fredholm map of positive index to be locally equivalent to a projection. In this paper we wish to find conditions which guarantee that such a map is globally equivalent to a projection. The problem is approached through the method of line lifting. This requires the existence of a locally Lipschitz right inverse,

F
↓

(
x
)

{F^ \downarrow }(x)

, to the derivative map

F
′

(
x
)

{F^\prime }(x)

and a global solution to the differential equation

P
′

(
t
)
=

F
↓

(
P
(
t
)
)
(
y
−

y
0

)

{P^\prime }(t) = {F^ \downarrow }(P(t))(y - {y_0})

. Both these problems are solved and the generalized Hadamard-Levy criterion
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline inf Underscript StartAbsoluteValue x EndAbsoluteValue greater-than s Endscripts left-parenthesis 1 slash StartAbsoluteValue upper F Superscript down-arrow Baseline left-parenthesis x right-parenthesis EndAbsoluteValue right-parenthesis d s equals normal infinity">

∫
0
∞

inf

|

x

|

>
s

(

1

/

|

F
↓

(
x
)

|

)

d
s
=
∞

\int _0^\infty {\inf \limits _{|x| > s} \left ({1/|{F^ \downarrow }(x)|} \right )\,ds = \infty }

\]
is shown to be sufficient for

F
F

to be globally equivalent to a projection map (Theorem 3.2). The relation to fiber bundle mappings is explored in §4.
On generalized Peano derivatives
Authors: Lee Cheng Ming
Published: June 2010

A function

F
F

is said to have a generalized

n
n

th Peano derivative at

x
x

if

F
F

is continuous in a neighborhood of

x
x

and if there exists a positive integer

k
k

such that a

k
k

th primitive of

F
F

in the neighborhood has the

(
k
+
n
)

(k + n)

th Peano derivative at

x
x

; and in this case this

(
k
+
n
)

(k + n)

th Peano derivative at

x
x

is proved to be independent of the integer

k
k

and the

k
k

th primitives, and is called the generalized

n
n

th Peano derivative of

F
F

at

x
x

which is denoted as

F

[
n
]

(
x
)

{F_{[n]}}(x)

. If

F

[
n
]

(
x
)

{F_{[n]}}(x)

exists and is finite for all

x
x

in an interval, then it is shown that

F

[
n
]

{F_{[n]}}

shares many interesting properties that are known for the ordinary Peano derivatives. Using the generalized Peano derivatives, a notion called absolute generalized Peano derivative is studied. It is proved that on a compact interval, the absolute generalized Peano derivatives are just the generalized Peano derivatives. In particular, Laczkovich’s absolute (ordinary) Peano derivatives are generalized Peano derivatives.
Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution
Authors: Duistermaat J. J.
Published: June 2010

The Kostant convexity theorem for real flag manifolds is generalized to a Hamiltonian framework. More precisely, it is proved that if

f
f

is the momentum mapping for a Hamiltonian torus action on a symplectic manifold

M
M

and

Q
Q

is the fixed point set of an antisymplectic involution of

M
M

leaving

f
f

invariant, then

f
(
Q
)
=
f
(
M
)
=

f(Q) = f(M) =

a convex polytope. Also it is proved that the coordinate functions of

f
f

are tight, using "half-turn" involutions of

Q
Q

.
Brownian motion and a generalised little Picard’s theorem
Authors: Kendall Wilfrid S.
Published: June 2010

Goldberg, Ishihara, and Petridis have proved a generalised little Picard’s theorem for harmonic maps; if a harmonic map of bounded dilatation maps euclidean space, for example, into a space of negative sectional curvatures bounded away from zero then that map is constant. In this paper a probabilistic proof is given of a variation on this result, requiring in addition that the image space has curvatures bounded below. The method involves comparing asymptotic properties of Brownian motion with the asymptotic behaviour of its image under such a map.
On the 𝜓-mixing condition for stationary random sequences
Authors: Bradley Richard C.
Published: June 2010

For strictly stationary sequences of random variables two mixing conditions are studied which together form the

ψ
\psi

-mixing condition. For the dependence coefficients associated with these two mixing conditions this article gives results on the possible limiting values and possible rates of convergence to these limits.
On the singular structure of three-dimensional, area-minimizing surfaces
Authors: Morgan Frank
Published: June 2010

A sufficient condition is given for the union of two three-dimensional planes through the origin in

R

n

{{\mathbf {R}}^n}

to be area-minimizing. The condition is in terms of the three angles

0
⩽

γ
1

⩽

γ
2

⩽

γ
3

0 \leqslant {\gamma _1} \leqslant {\gamma _2} \leqslant {\gamma _3}

which characterize the geometric relationship between the planes. If

γ
3

⩽

γ
1

+

γ
2

{\gamma _3} \leqslant {\gamma _1} + {\gamma _2}

, the union of the planes is area-minimizing.
𝑆𝐿(2,𝐶) actions on compact Kaehler manifolds
Authors: Carrell James B. , Sommese Andrew John
Published: June 2010

Whenever

G
=
S
L
(
2
,
C
)

G = SL(2,C)

acts holomorphically on a compact Kaehler manifold

X
X

, the maximal torus

T
T

of

G
G

has fixed points. Consequently,

X
X

has associated Bialynicki-Birula plus and minus decompositions. In this paper we study the interplay between the Bialynicki-Birula decompositions and the

G
G

-action. A representative result is that the Borel subgroup of upper (resp. lower) triangular matrices in

G
G

preserves the plus (resp. minus) decomposition and that each cell in the plus (resp. minus) decomposition fibres

G
G

-equivariantly over a component of

X
T

{X^T}

. We give some applications; e.g. we classify all compact Kaehler manifolds

X
X

admitting a

G
G

-action with no three dimensional orbits. In particular we show that if

X
X

is projective and has no three dimensional orbit, and if

Pic
(
X
)
≅

Z

\text {Pic}(X) \cong {\mathbf {Z}}

, then

X
=
C

P

n

X = C{{\mathbf {P}}^n}

. We also show that if

X
X

admits a holomorphic vector field with unirational zero set, and if

Aut
0

⁡
(
X
)

\operatorname {Aut}_0(X)

is reductive, then

X
X

is unirational.
Steenrod and Dyer-Lashof operations on 𝐵𝑈
Authors: Lance Timothy
Published: June 2010

This paper describes a simple, fast algorithm for the computation of Steenrod and Dyer-Lashof operations on

B
U

BU

. The calculations are carried out in

H

∗

(
B
U
,

Z

(
p
)

)

H^{\ast }(BU,{{\mathbf {Z}}_{(p)}})

and

H

∗

(
B
U
,

Z

(
p
)

)

{H_{\ast }}(BU,{{\mathbf {Z}}_{(p)}})

where

p
p

local lifts are determined by the values on primitives and Cartan formulas. This algorithm also provides a description of Steenrod and Dyer-Lashof operations on the fiber of any

H
H

map (or infinite loop map)

B
U
→
B
U

BU \to BU

, and applications to the classifying spaces of surgery which arise in this fashion will appear shortly.