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1.
Gaston Casanova 《Advances in Applied Clifford Algebras》1999,9(1):91-94
We define and study here a class of functions probably new defined onC×IR
2 (whereC is a Clifford algebra) with values inC×IR and that we call “hyperbolic analytic functions” because of their analogy with complex analytic functions. 相似文献
2.
E. A. Ramos 《Discrete and Computational Geometry》1996,15(2):147-167
We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR
d
, there arek hyperplanes so that each orthant contains a fraction 1/2
k
of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2
k
−1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2
k−1. We are able to prove that Δ(j, k)=⌈j(2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2
n
withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove
a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5).
This research was supported by the National Science Foundation under Grant CCR-9118874. 相似文献
3.
Let I≥1 be an integer, ω
0=0<ω
1<⋯<ω
I
≤π, and for j=0,…,I, a
j
∈ℂ, a-j=[`(aj)]a_{-j}={\overline{{a_{j}}}}, ω
−j
=−ω
j
, and aj 1 0a_{j}\not=0 if j 1 0j\not=0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form
[(x)\tilde](k) = ?j=-II ajexp(-iwjk) +e(k), k=-2N,?,2N,\tilde{x}(k)= \sum_{j=-I}^I a_j\exp(-i\omega _jk) +\epsilon (k), \quad k=-2N,\ldots,2N, 相似文献
4.
Using the bicomplex numbers
5.
In this second paper, we study the case of substitution tilings of
\mathbb Rd{{\mathbb R}^d} . The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j = 0, . . . , d − 1. We reconstruct the tiling’s equivalence relation in a purely combinatorial way using the AF-relations given by the lower
dimensional substitutions. We define a Bratteli multi-diagram B{{\mathcal B}} which is made of the Bratteli diagrams Bj, j=0, ?d{{\mathcal B}^j, j=0, \ldots d} , of all those substitutions. The set of infinite paths in Bd{{\mathcal B}^d} is identified with the canonical transversal Ξ of the tiling. Any such path has a “border”, which is a set of tails in Bj{{\mathcal B}^j} for some j ≤ d, and this corresponds to a natural notion of border for its associated tiling. We define an étale equivalence relation RB{{\mathcal R}_{\mathcal B}} on B{{\mathcal B}} by saying that two infinite paths are equivalent if they have borders which are tail equivalent in Bj{{\mathcal B}^j} for some j ≤ d. We show that RB{{\mathcal R}_{\mathcal B}} is homeomorphic to the tiling’s equivalence relation RX{{\mathcal R}_\Xi} . 相似文献
6.
Suppose that Ω is a bounded domain with fractal boundary Γ in ${\mathbb R^{n+1}}
7.
Let ξ,ξ
1,ξ
2,… be positive i.i.d. random variables, S=∑
j=1∞
a(j)ξ
j
, where the coefficients a(j)≥0 are such that P(S<∞)=1. We obtain an explicit form of the asymptotics of −ln P(S<x) as x→0 for the following three cases:
8.
Onur Yavuz 《Integral Equations and Operator Theory》2010,68(4):473-485
We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λ
j
with radius r
j
for j = 1, . . . , n from the unit disk. We assume that T is a bounded linear operator on a separable reflexive Banach space whose spectrum contains ∂Ω and does not contain the points
λ1, λ2, . . . , λ
n
, and the operators T and r
j
(T − λ
j
I)−1 are polynomially bounded. Then either T has a nontrivial hyperinvariant subspace or the WOT-closure of the algebra {f(T) : f is a rational function with poles off [`(W)]{\overline\Omega}} is reflexive. 相似文献
9.
Let ξ, ξ1, ξ2, ... be independent identically distributed random variables, and S
n
:=Σ
j=1
n
,ξ
j
, $
\bar S
$
\bar S
:= sup
n≥0
S
n
. If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $
\bar S
$
\bar S
as a → 0 and $
\bar S
$
\bar S
tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:
10.
The complex numbers are naturally related to rotations and dilatations in the plane. In this paper we present the function
theory associate to the (universal) Clifford algebra forIR
1,0 [1], the so called hyperbolic numbers [2,3,4], which can be related to Lorentz transformations and dilatations in the two
dimensional Minkowski space-time. After some brief algebraic interpretations (part 1), we present a “Hyperbolic Calculus”
analogous to the “Calculus of one Complex Variable”. The hyperbolic Cauchy-Riemann conditions, hyperbolic derivatives and
hyperbolic integrals are introduced on parts 2 and 3. Then special emphasis is given in parts 4 and 5 to conformal hyperbolic
transformations which preserve the wave equation, and hyperbolic Riemann surfaces which are naturally associated to classical
string motions. 相似文献
11.
Nariya Kawazumi 《Inventiones Mathematicae》1997,131(1):137-149
Let Σ
g,1
be an oriented compact surface of genus g with 1 boundary component, and Γ
g,1
the mapping class group of Σ
g,1
. We define a bigraded series of cohomology classes m
i,j
∈H
2i+j−2
(Γ
g,1
;⋀
j
H
1(Σ
g,1
;ℤ)), 2i+j−2≥1,i,j≥0. When j=0, the class m
i+1,0
is the i-th Morita- Mumford class [Mo][Mu]. It is proved that H
r
(Γ
g,1
;⋀
s
H
1(Σ
g,1
;ℚ)) is generated by m
i,j
's for the case r+s=2 and the case g≥5 and (r,s)=(1,3). Especially the Johnson homomorphism extended to the whole mapping class group by Morita [Mo3] has an implicit representation by the classes m
0,3 and m
0,2
m
1,1 over ℚ.
Oblatum 28-IV-1995 & 8-II-1997 相似文献
12.
Summary. We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx
t
= ∑
j
=0
m
f
j
(x
t
)∘dW
t
j
and dx
t
=∑
j
=0
m
g
j
(x
t
)∘dW
t
j
in ℝ
d
with smooth coefficients satisfying f
j
(0)=g
j
(0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,
13.
Misha Verbitsky 《Mathematische Zeitschrift》2010,264(4):939-957
Let (M, ω) be a Kähler manifold. An integrable function ${\varphi}
14.
We show how B-series may be used to derive in a systematic way the analytical expressions of the high-order stroboscopic averaged
equations that approximate the slow dynamics of highly oscillatory systems. For first-order systems we give explicitly the
form of the averaged systems with O(ej)\mathcal{O}(\epsilon^{j}) errors, j=1,2,3 (2π
ε denotes the period of the fast oscillations). For second-order systems with large O(e-1)\mathcal{O}(\epsilon^{-1}) forces, we give the explicit form of the averaged systems with O(ej)\mathcal{O}(\epsilon^{j}) errors, j=1,2. A variant of the Fermi–Pasta–Ulam model and the inverted Kapitsa pendulum are used as illustrations. For the former
it is shown that our approach establishes the adiabatic invariance of the oscillatory energy. Finally we use B-series to analyze
multiscale numerical integrators that implement the method of averaging. We construct integrators that are able to approximate
not only the simplest, lowest-order averaged equation but also its high-order counterparts. 相似文献
15.
Christian Liedtke 《Mathematische Annalen》2009,343(3):623-637
A non-classical Godeaux surface is a minimal surface of general type with χ = K
2 = 1 but with h
01 ≠ 0. We prove that such surfaces fulfill h
01 = 1 and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall
into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute
their Hodge-, Hodge–Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux
surface in characteristic 5. 相似文献
16.
Sebastian Bogner Bernd Fritzsche Bernd Kirstein 《Complex Analysis and Operator Theory》2007,1(1):55-95
The main theme of this paper is to characterize distinguished subclasses of the matricial Schur class
in terms of Taylor coefficients. Starting point of our investigations is the observation that the Taylor coefficient sequences
of functions from
are exactly the infinite p × q Schur sequences. We draw our attention mainly to the subclass
of
which consists of all p × q Schur functions for which the corresponding Taylor coefficient sequences are nondegenerate p × q Schur sequences. Using an appropriate adaptation of the Schur–Potapov algorithm for functions belonging to
to infinite sequences of complex p × q matrices we obtain an one-to-one correspondence between infinite nondegenerate p × q Schur sequences and the set of all infinite sequences (Ej)j=0∞ of strictly contractive complex p × q matrices. Taking into account the construction of
this gives us an one-to-one correspondence between
and the set of all infinite sequences (Ej)j=0∞ of strictly contractive complex p × q matrices. Hereby, (Ej)j =0∞ is called the sequence of Schur–Potapov parameters (shortly SP-parameters) of f.
Communicated by Daniel Alpay.
Submitted: August 17, 2006; Accepted: September 13, 2006 相似文献
17.
For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT
α=Σ
j=1
∞
a
j
T
j
, where {a
j
} are the coefficients in the power series expansion (1-t)α=1-Σ
j=1
∞
a
j
t
j
in the open unit disk, which satisfya
j
>0 anda
j
>0 and Σ
j=1
∞
a
j
=1. The operator calculus justifies the notation(I−T)
α
:=I−T
α
(e.g., (I−T
1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for
T if there is anx∈X such that(I−T)
α
x=y, i.e.,y is a coboundary forT
α
. The fractional Poisson equation forT is the Poisson equation forT
α
. We show that if(I−T)X is not closed, then(I−T)
α
X strictly contains(I−T)X (but has the same closure).
ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ
k=1
∞
T
k
y/k
1-α converges in norm, and conclude that lim
n
‖(1/n
1-α)Σ
k=1
n
T
k
y‖=0 for suchy.
For a Dunford-Schwartz operatorT onL
1 of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T)
α
L
1 for some 0<α<1, then the one-sided Hilbert transform Σ
k=1
∞
T
k
f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T)
α
L
p
with α>1−1/p=1/q, then Σ
k=1
∞
T
k
f/k
1/p
converges a.e., and thus (1/n
1/p
) Σ
k=1
n
T
k
f converges a.e. to zero. Whenf∈(I−T)
1/q
L
p
(the case α=1/q), we prove that (1/n
1/p
(logn)1/q
)Σ
k=1
n
T
k
f converges a.e. to zero. 相似文献
18.
We investigate a certain well-established generalization of the Davenport constant. For j a positive integer (the case j = 1, is the classical one) and a finite Abelian group (G, +, 0), the invariant D
j
(G) is defined as the smallest ℓ such that each sequence over G of length at least ℓ has j disjoint non-empty zero-sum subsequences. We investigate these quantities for elementary 2-groups of large rank (relative
to j). Using tools from coding theory, we give fairly precise estimates for these quantities. We use our results to give improved
bounds for the classical Davenport constant of certain groups. 相似文献
19.
Djordje Mili?evi? 《Geometric And Functional Analysis》2011,21(6):1375-1418
We prove that, on a distinguished class of arithmetic hyperbolic 3-manifolds, there is a sequence of L
2-normalized high-energy Hecke–Maass eigenforms fj{\phi_{j}} which achieve values as large as l1/4+o(1)j{\lambda^{1/4+o(1)}_{j}}, where ( D+lj ) fj = 0{( \Delta+\lambda_{j} ) \phi_{j} = 0}. Arithmetic hyperbolic 3-manifolds on which this exceptional behavior is exhibited are, up to commensurability, precisely
those containing immersed totally geodesic surfaces. We adapt the method of resonators and connect values of eigenfunctions
to the global geometry of the manifold by employing the pre-trace formula and twists by Hecke correspondences. Automorphic
representations corresponding to forms appearing with highest weights in the optimized spectral averages are characterized
both in terms of base change lifts and in terms of theta lifts from GSp2. 相似文献
20.
We study a necessary and sufficient condition for Jacobi integrals of weight
-r+\fracj2-r+\frac{j}{2}, r∈ℤ≥0, and index ℳ(j) on ℋ×ℂ
j
to have a dual Jacobi form of weight
r+\fracj2+2r+\frac{j}{2}+2 and index ℳ(j). Such a meromorphic Jacobi integral with a dual Jacobi form is called a mock Jacobi form whose concept was first introduced
by Zagier in Séminaire Bourbaki, 60éme année, 2006–2007, N° 986. In fact, we show the map Lr+1M(j)L^{r+1}_{\mathcal{M}^{(j)}} from the space of mock Jacobi forms to that of Jacobi forms is surjective by constructing the corresponding inverse image
via Eichler integral of vector valued modular forms which are coming from the theta decomposition of Jacobi forms. We discuss
Lerch sums as a typical example. 相似文献
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