1,895 research outputs found
Computing Garsia Entropy for Bernoulli Convolutions with Algebraic Parameters
We introduce a parameter space containing all algebraic integers
that are not Pisot or Salem numbers, and a sequence of
increasing piecewise continuous function on this parameter space which gives a
lower bound for the Garsia entropy of the Bernoulli convolution .
This allows us to show that for all
with representations in certain open regions of the parameter space.Comment: 21 pages, 2 figures, 5 table
Golden gaskets: variations on the Sierpi\'nski sieve
We consider the iterated function systems (IFSs) that consist of three
general similitudes in the plane with centres at three non-collinear points,
and with a common contraction factor \la\in(0,1).
As is well known, for \la=1/2 the invariant set, \S_\la, is a fractal
called the Sierpi\'nski sieve, and for \la<1/2 it is also a fractal. Our goal
is to study \S_\la for this IFS for 1/2<\la<2/3, i.e., when there are
"overlaps" in \S_\la as well as "holes". In this introductory paper we show
that despite the overlaps (i.e., the Open Set Condition breaking down
completely), the attractor can still be a totally self-similar fractal,
although this happens only for a very special family of algebraic \la's
(so-called "multinacci numbers"). We evaluate \dim_H(\S_\la) for these
special values by showing that \S_\la is essentially the attractor for an
infinite IFS which does satisfy the Open Set Condition. We also show that the
set of points in the attractor with a unique ``address'' is self-similar, and
compute its dimension.
For ``non-multinacci'' values of \la we show that if \la is close to 2/3,
then \S_\la has a nonempty interior and that if \la<1/\sqrt{3} then \S_\la$
has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of
the model in question.Comment: 27 pages, 10 figure
Application of two-sided approximations method to solution of first boundary value problem for one-dimensional nonlinear heat conductivity equation
The first boundary value problem for a one-dimensional nonlinear heat equation is considered, where the heat conductivity coefficient and the power function of heat sources have a power-law dependence on temperature. For a numerical analysis of this problem, it is proposed to use the method of two-sided approximations based on the method of Green’s functions. After replacing the unknown function, the boundary value problem is reduced to the Hammerstein integral equation, which is considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a single positive solution of the problem and the conditions for two-sided convergence of successive approximations to it are obtained. The developed method is programmatically implemented and researched in solving test problems. The results of the computational experiment are illustrated by graphical and tabular information. The conducted experiments confirmed the efficiency and effectiveness of the developed method that allowed recommending its practical use for solving problems of system analysis and mathematical modeling of nonlinear processes
Electrostatic deposition of graphene in a gaseous environment: A deterministic route to synthesize rolled graphenes?
The synthesis of single-wall carbon nanotubes (SWCNTs) of desired diameters
and chiralities is critical to the design of nanoscale electronic devices with
desired properties.1-6 The existing methods are based on self-assembly, 7-16
therefore lacking the control over their diameters and chiralities. The present
work reports a direct route to roll graphene. Specifically, we found that the
electrostatic deposition of graphene yielded: (i) flat graphene layers under
high vacuum (10-7 Torr), (ii) completely scrolled graphene under hydrogen
atmosphere, (iii) partially scrolled graphene under nitrogen atmosphere, and
(iv) no scrolling for helium atmospheres. Our study shows that the application
of the electrostatic field facilitates the rolling of graphene sheets exposed
to appropriate gases and allows the rolling of any size graphene. The technique
proposed here, in conjunction with a technique that produces graphene
nanoribbons (GNRs) of uniform widths, will have significant impact on the
development of carbon nanotube based devices. Furthermore, the present
technique may be applied to obtain tubes/scrolls of other layered materials
Realizing quantum gates with optically-addressable Yb ion qudits
The use of multilevel information carriers, also known as qudits, is a
promising path for exploring scalability of quantum computing devices. In this
work, we present a proof-of-principle realization of a quantum processor that
uses optically-addressed Yb ion qudits in a linear trap. The rich
level structure of Yb ions makes it possible to use the Zeeman
sublevels of the quadrupole clock transition at 435.5 nm for efficient and
robust qudit encoding. We demonstrate the realization of the universal set of
gates consisting of single-qudit rotations and two-qudit entangling operation
with a two-ququart system, which is formally equivalent to a universal
gate-based four-qubit processor. Our results paves a way towards further
studies of more efficient implementations of quantum algorithms with
trapped-ion-based processors.Comment: 9 pages, 3 figure
Intersections of homogeneous Cantor sets and beta-expansions
Let be the -part homogeneous Cantor set with
. Any string with
such that is called a code of . Let
be the set of having a unique code,
and let be the set of which make the intersection a
self-similar set. We characterize the set in a
geometrical and algebraical way, and give a sufficient and necessary condition
for . Using techniques from beta-expansions, we
show that there is a critical point , which is a
transcendental number, such that has positive
Hausdorff dimension if , and contains countably
infinite many elements if . Moreover, there exists a
second critical point
such that
has positive Hausdorff dimension if
, and contains countably infinite many elements if
.Comment: 23 pages, 4 figure
Cantor type functions in non-integer bases
Cantor's ternary function is generalized to arbitrary base-change functions
in non-integer bases. Some of them share the curious properties of Cantor's
function, while others behave quite differently
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