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1.
For a self‐affine tile in $\mathbf {R}^2$ generated by an expanding matrix $A\in M_2(\mathbf {Z})$ and an integral consecutive collinear digit set ${\mathcal D}$, Leung and Lau [Trans. Amer. Math. Soc. 359 , 3337–3355 (2007).] provided a necessary and sufficient algebraic condition for it to be disklike. They also characterized the neighborhood structure of all disklike tiles in terms of the algebraic data A and ${\mathcal D}$. In this paper, we completely characterize the neighborhood structure of those non‐disklike tiles. While disklike tiles can only have either six or eight edge or vertex neighbors, non‐disklike tiles have much richer neighborhood structure. In particular, other than a finite set, a Cantor set, or a set containing a nontrivial continuum, neighbors can intersect in a union of a Cantor set and a countable set. 相似文献
2.
3.
Given a function ψ in
the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions
In this paper we prove that the set of functions generating affine systems that are a Riesz basis of
${\cal L}^2({\Bbb R}^d)$ is dense in We also prove that a stronger result is true for affine systems that are a frame of
In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze
the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems,
that are compactly supported in frequency, are dense in the unit sphere of
with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this
Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems. 相似文献
4.
In this paper,the boundedness for the multilinear commutators of Bochner-Riesz operator is considered.We prove that the multilinear commutators generated by Bochner-Riesz operator and Lipschitz function are bounded from Lp(Rn)into ∧˙(β-np)(Rn)and from Lnβ(Rn)into BMO(Rn). 相似文献
5.
The purpose of this paper is to present a general stochastic calculus
approach to insider trading. We consider a market driven by a standard Brownian
motion $B(t)$ on a filtered probability space $\displaystyle
(\Omega,\F,\left\{\F\right\}_{t\geq 0},P)$ where the coefficients are
adapted to a filtration ${\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$,
with $\F_t\subset\G_t$ for all $t\in [0,T]$, $T>0$ being a fixed terminal time.
By
an {\it insider} in this market we
mean a person who has access to a filtration (information)
$\displaystyle{\Bbb H}=\left\{\H_t\right\}_{0\leq t\leq T}$ which is strictly
bigger than the filtration
$\displaystyle{\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$.
In this context an insider strategy is represented by an
$\H_t$-adapted process
$\phi(t)$ and we interpret all anticipating integrals as
the forward integral defined in
[23] and [25].
We consider an optimal portfolio problem with
general utility for an insider with access to a general information
$\H_t \supset\G_t$ and show that if
an optimal insider portfolio $\pi^*(t)$ of this problem exists, then
$B(t)$ is an $\H_t$-semimartingale, i.e. the enlargement
of filtration property holds. This is a converse of previously
known results in this field.
Moreover, if $\pi^*$ exists
we obtain an explicit expression in terms of $\pi^*$ for the
semimartingale decomposition of $B(t)$ with respect to $\H_t$.
This is a generalization
of results in [16], [20] and [2]. 相似文献
6.
Let $\Omega \subset \Bbb{R}^2$ denote a bounded domain whose boundary
$\partial \Omega$ is Lipschitz and contains a segment $\Gamma_0$ representing
the austenite-twinned martensite interface. We prove
$$\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla
u(x,y))dxdy=0}$$ for any elastic energy density $\varphi : \Bbb{R}^2
\rightarrow [0,\infty)$ such that $\varphi(0,\pm 1)=0$. Here
${\cal W}(\Omega)$ consists of all Lipschitz functions $u$ with
$u=0$ on $\Gamma_0$ and $|u_y|=1$ a.e. Apart from the trivial case
$\Gamma_0 \subset \reel \times \{a\},~a\in \Bbb{R}$, this result is
obtained through the construction of suitable minimizing sequences
which differ substantially for vertical and non-vertical
segments. 相似文献
7.
Let (M,g) be a compact Riemannian manifold of dimension 3, and let ? denote the collection of all embedded surfaces homeomorphic to \input amssym ${\Bbb R}{ \Bbb P}^2$ . We study the infimum of the areas of all surfaces in ?. This quantity is related to the systole of (M,g). It makes sense whenever ? is nonempty. In this paper, we give an upper bound for this quantity in terms of the minimum of the scalar curvature of (M,g). Moreover, we show that equality holds if and only if (M,g) is isometric to \input amssym ${\Bbb R}{ \Bbb P}^3$ up to scaling. The proof uses the formula for the second variation of area and Hamilton's Ricci flow. © 2010 Wiley Periodicals, Inc. 相似文献
8.
Franki Dillen Johan Fastenakels Joeri Van der Veken Luc Vrancken 《Monatshefte für Mathematik》2007,152(2):89-96
In this article we study surfaces in
for which the unit normal makes a constant angle with the
-direction. We give a complete classification for surfaces satisfying this simple geometric condition. 相似文献
9.
Given a collection S of subsets of some set
and
the set cover problem is to find the smallest subcollection
that covers
that is,
where
denotes
We assume of course that S covers
While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually
Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance
exist, under a certain general condition: that for a random subset
and nondecreasing function f(·), there is a decomposition of the complement
into an expected at most f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|))
can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that
are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family
of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects,
and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in
and for guarding an x-monotone polygonal chain. 相似文献
10.
For each $n>2$ we construct a convex body
$K\subset {\Bbb R}^3$ and a finite family ${\cal F}$ of disjoint translates
of $K$ such that any $n-1$ members ${\cal F}$ admit a line transversal, but
${\cal F}$ has no line transversal. 相似文献
11.
We consider well‐posedness of the aggregation equation ∂tu + div(uv) = 0, v = −▿K * u with initial data in \input amssym ${\cal P}_2 {\rm (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ in dimensions 2 and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x|α, α > 2 − d, and prove local well‐posedness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ for sufficiently large p < ps. In the special case of K(x) = |x|, the exponent ps = d/(d = 1) is sharp for local well‐posedness in that solutions can instantaneously concentrate mass for initial data in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ with p < ps. We also give an Osgood condition on the potential K(x) that guarantees global existence and uniqueness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ . © 2010 Wiley Periodicals, Inc. 相似文献
12.
Balint Farkas Mate Matolcsi Peter Mora 《Journal of Fourier Analysis and Applications》2006,12(5):483-494
Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde’s Spectral Set
Conjecture in both directions. Namely, in
Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a nonspectral tile.
In search of lower dimensional nonspectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias
and Wang [14]. In particular, we prove here that the USC and the "tile → spectral" direction of Fuglede’s conjecture are equivalent
in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabó [13] for the existence
of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples
of tiles which have no spectra. We overcome these difficulties by invoking some ideas of Révész and Farkas [2], and obtain
nonspectral tiles in
. Fuglede’s conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The one-dimensional case
is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1]. 相似文献
13.
Peter Hornung 《纯数学与应用数学通讯》2011,64(3):367-441
Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C∞, and let gij = δij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C3 away from a certain singular set Σ and C∞ away from a larger singular set Σ ∪ Σ0. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ0. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc. 相似文献
14.
We study the problem of best approximations of a vector
by rational vectors of a lattice
whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend
some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem
which generalize and improve former results. 相似文献
15.
Let
be a univariate, separable polynomial of degree n with roots x
1,…,x
n
in some algebraic closure
of the ground field
. It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all
such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on
by permutation of the coordinates and each
defines a relation P − P(x
1,…,x
n
) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an
H-basis of G-invariant relations if and only if the algebra of coinvariants
has dimension ‖G‖ over
. To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite
group G has dimension ‖G‖ if and only if G = Σ
n
acting via the tautological permutation representation. 相似文献
16.
《复变函数与椭圆型方程》2012,57(3):271-276
Let $ \Pi_{n,M} $ be the class of all polynomials $ p(z) = \sum _{0}^{n} a_{k}z^{k} $ of degree n which have all their zeros on the unit circle $ |z| = 1$ , and satisfy $ M = \max _{|z| = 1}|\,p(z)| $ . Let $ \mu _{k,n} = \sup _{p\in \Pi _{n,M}} |a_{k}| $ . Saff and Sheil-Small asked for the value of $\overline {\lim }_{n\rightarrow \infty }\mu _{k,n} $ . We find an equivalence between this problem and the Krzyz problem on the coefficients of bounded non-vanishing functions. As a result we compute $$ \overline {\lim }_{n\rightarrow \infty }\mu _{k,n} = {{M} \over {e}}\quad {\rm for}\ k = 1,2,3,4,5.$$ We also obtain some bounds for polynomials with zeros on the unit circle. These are related to a problem of Hayman. 相似文献
17.
Let
denote the linear space over
spanned by
. Define the (real) inner product
, where V satisfies: (i) V is real analytic on
; (ii)
; and (iii)
. Orthogonalisation of the (ordered) base
with respect to
yields the even degree and odd degree orthonormal Laurent polynomials
, and
. Define the even degree and odd degree monic orthogonal Laurent polynomials:
and
. Asymptotics in the double-scaling limit
such that
of
(in the entire complex plane),
, and
(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a
matrix Riemann-Hilbert problem on
, and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further
developed in [2],[3]. 相似文献
18.
Given a finite subset
of an additive group
such as
or
, we are interested in efficient covering of
by translates of
, and efficient packing of translates of
in
. A set
provides a covering if the translates
with
cover
(i.e., their union is
), and the covering will be efficient if
has small density in
. On the other hand, a set
will provide a packing if the translated sets
with
are mutually disjoint, and the packing is efficient if
has large density.
In the present part (I) we will derive some facts on these concepts when
, and give estimates for the minimal covering densities and maximal packing densities of finite sets
. In part (II) we will again deal with
, and study the behaviour of such densities under linear transformations. In part (III) we will turn to
.
Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395,
USA
The first author was partially supported by NSF DMS 0074531. 相似文献
19.
设■该文主要讨论了上述奇异积分算子在广义的调幅空间上的有界性,其中粗糙核Ω∈L~1(S~(n-2))h(y)为有界的径向函数,而γ(y)是满足一定条件的超曲面. 相似文献
20.
R. G. Novikov 《Selecta Mathematica, New Series》1997,3(2):245-302
We consider the Dirac-ZS-AKNS system (1) where (the space of functions with n derivatives in L
1), (2) We consider for (1) the transition matrix and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case the scattering matrix We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions,
of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with , the following formulas are valid: (3) and, in addition, for the case of the Dirac system (4) where denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac
system under condition (2) with
III. As applications of the results mentioned above, we show that 1) for any real-valued initial data , the Cauchy problem for the sh-Gordon equation has a unique solution such that and for any t > 0, 2) in addition, for , for such a solution the following formula is valid: where
denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results. 相似文献