502 research outputs found
inverse orbit generating functions almost always have natural boundaries
The function sends to resp. according
as is odd, resp. even, where . The map
sends integers to integers, and for let mean that is in the forward orbit of under iteration of
We consider the generating functions which are holomorphic in the unit disk. We give
sufficient conditions on for the functions have the unit
circle as a natural boundary to analytic continuation. For the
function these conditions hold for all to show that
has the unit circle as a natural boundary except possibly for and . The Conjecture is equivalent to the assertion that
is a rational function of for the remaining values .Comment: 15 page
Area Inequalities for Embedded Disks Spanning Unknotted Curves
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric
inequality that bounds the area of an embedded disk spanning the curve in terms
of two parameters: the length L of the curve and the thickness r (maximal
radius of an embedded tubular neighborhood) of the curve. For fixed length, the
expression giving the upper bound on the area grows exponentially in 1/r^2. In
the direction of lower bounds, we give a sequence of length one curves with r
approaching 0 for which the area of any spanning disk is bounded from below by
a function that grows exponentially with 1/r. In particular, given any constant
A, there is a smooth, unknotted length one curve for which the area of a
smallest embedded spanning disk is greater than A.Comment: 31 pages, 5 figure
On Lebesgue measure of integral self-affine sets
Let be an expanding integer matrix and be a finite subset
of . The self-affine set is the unique compact set satisfying
the equality . We present an effective algorithm to
compute the Lebesgue measure of the self-affine set , the measure of
intersection for , and the measure of intersection of
self-affine sets for different sets .Comment: 5 pages, 1 figur
All functions g:N-->N which have a single-fold Diophantine representation are dominated by a limit-computable function f:N\{0}-->N which is implemented in MuPAD and whose computability is an open problem
Let E_n={x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For any
integer n \geq 2214, we define a system T \subseteq E_n which has a unique
integer solution (a_1,...,a_n). We prove that the numbers a_1,...,a_n are
positive and max(a_1,...,a_n)>2^(2^n). For a positive integer n, let f(n)
denote the smallest non-negative integer b such that for each system S
\subseteq E_n with a unique solution in non-negative integers x_1,...,x_n, this
solution belongs to [0,b]^n. We prove that if a function g:N-->N has a
single-fold Diophantine representation, then f dominates g. We present a MuPAD
code which takes as input a positive integer n, performs an infinite loop,
returns a non-negative integer on each iteration, and returns f(n) on each
sufficiently high iteration.Comment: 17 pages, Theorem 3 added. arXiv admin note: substantial text overlap
with arXiv:1309.2605. text overlap with arXiv:1404.5975, arXiv:1310.536
Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives
We consider the problem of finding commuting self-adjoint extensions of the
partial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domain
C_c^\infty(\Omega) where the self-adjointness is defined relative to
L^2(\Omega), and \Omega is a given open subset of R^d. The measure on \Omega is
Lebesgue measure on R^d restricted to \Omega. The problem originates with I.E.
Segal and B. Fuglede, and is difficult in general. In this paper, we provide a
representation-theoretic answer in the special case when \Omega=I\times\Omega_2
and I is an open interval. We then apply the results to the case when \Omega is
a d-cube, I^d, and we describe possible subsets \Lambda of R^d such that
{e^(i2\pi\lambda \dot x) restricted to I^d:\lambda\in\Lambda} is an orthonormal
basis in L^2(I^d).Comment: LaTeX2e amsart class, 18 pages, 2 figures; PACS numbers 02.20.Km,
02.30.Nw, 02.30.Tb, 02.60.-x, 03.65.-w, 03.65.Bz, 03.65.Db, 61.12.Bt,
61.44.B
Tiling groupoids and Bratteli diagrams
Let T be an aperiodic and repetitive tiling of R^d with finite local
complexity. Let O be its tiling space with canonical transversal X. The tiling
equivalence relation R_X is the set of pairs of tilings in X which are
translates of each others, with a certain (etale) topology. In this paper R_X
is reconstructed as a generalized "tail equivalence" on a Bratteli diagram,
with its standard AF-relation as a subequivalence relation.
Using a generalization of the Anderson-Putnam complex, O is identified with
the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is
built from this sequence, and its set of infinite paths dB is homeomorphic to
X. The diagram B is endowed with a horizontal structure: additional edges that
encode the adjacencies of patches in T. This allows to define an etale
equivalence relation R_B on dB which is homeomorphic to R_X, and contains the
AF-relation of "tail equivalence".Comment: 34 pages, 4 figure
Data analysis of gravitational-wave signals from spinning neutron stars. IV. An all-sky search
We develop a set of data analysis tools for a realistic all-sky search for
continuous gravitational-wave signals. The methods that we present apply to
data from both the resonant bar detectors that are currently in operation and
the laser interferometric detectors that are in the final stages of
construction and commissioning. We show that with our techniques we shall be
able to perform an all-sky 2-day long coherent search of the narrow-band data
from the resonant bar EXPLORER with no loss of signals with the dimensionless
amplitude greater than .Comment: REVTeX, 26 pages, 1 figure, submitted to Phys. Rev.
Sets Uniquely Determined by Projections on Axes I. Continuous Case
This paper studies sets S in Rn which are uniquely reconstructible from their hyperplane integral projections Pi(xi ;S) = ∬ . . . ∫ΧS ( {x1, . . . ,xi, . . . ,xn) dx1 . . . dxi - 1 dxi + 1 . . .dxn onto the n coordinate axes of Rn. It is shown that any additive set S = {x = (x1, . . .,xn) : ∑i = 1n fi(xi)≧0}, where each fi(xi) is a bounded measurable function, is uniquely reconstructible. In particular, balls are uniquely reconstructible. It is shown that in R2 all uniquely reconstructible sets are additive. For n≧3, Kemperman has shown that there are uniquely reconstructible sets in Rn of bounded measure that are not additive. It is also noted for n≧3 that neither of the properties of being additive and being a set of uniqueness is closed under monotone pointwise limits.
A necessary condition for S to be a set of uniqueness is that S contain no bad configuration. A bad configuration is two finite sets of points T1 in Int(S) and T2 in Int(Sc), where Sc=Rn - S, such that T1 and T2 have the same number of points in any hyperplane xi = c for 1≦ i ≦n, and all c ∈ R2. We show that this necessary condition is sufficient for uniqueness for open sets S in R2.
The results show that prior information about a density f in R2 to be reconstructed in tomography (namely if f is known to have only values 0 and 1) can sometimes reduce the problem of reconstructing f to knowing only two projections of f. Thus even meager prior information can in principle be of enormous value in tomography
Boundaries of Disk-like Self-affine Tiles
Let be a disk-like self-affine tile generated by an
integral expanding matrix and a consecutive collinear digit set , and let be the characteristic polynomial of . In the
paper, we identify the boundary with a sofic system by
constructing a neighbor graph and derive equivalent conditions for the pair
to be a number system. Moreover, by using the graph-directed
construction and a device of pseudo-norm , we find the generalized
Hausdorff dimension where
is the spectral radius of certain contact matrix . Especially,
when is a similarity, we obtain the standard Hausdorff dimension where is the largest positive zero of
the cubic polynomial , which is simpler than
the known result.Comment: 26 pages, 11 figure
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