485 research outputs found
Discrepancy bounds for normal numbers generated by necklaces in arbitrary base
Mordechay B. Levin has constructed a number which is normal in base
2, and such that the sequence
has very small discrepancy . Indeed we have . This construction technique of Levin was
generalized by Becher and Carton, who generated normal numbers via perfect
nested necklaces, and they showed that for these normal numbers the same upper
discrepancy estimate holds as for the special example of Levin. In this paper
now we derive an upper discrepancy bound for so-called semi-perfect nested
necklaces and show that for the Levin's normal number in arbitrary prime base
this upper bound for the discrepancy is best possible, i.e., with for infinitely many . This result
generalizes a previous result where we ensured for the special example of Levin
for the base , that is best
possible in . So far it is known by a celebrated result of Schmidt that for
any sequence in , with for infinitely
many . So there is a gap of a factor in the question, what is the
best order for the discrepancy in that can be achieved for a normal number.
Our result for Levin's normal number in any prime base on the one hand might
support the guess that is the best order in
that can be achieved by a normal number, while generalizing the class of known
normal numbers by introducing e.g. semi-perfect necklaces on the other hand
might help for the search of normal numbers that satisfy smaller discrepancy
bounds in than .Comment: 29 page
Insertion in constructed normal numbers
Defined by Borel, a real number is normal to an integer base b ≥ 2 if in its base-b expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider the problem of insertion in constructed base-b normal expansions to obtain normality to base (b + 1).Fil: Becher, Veronica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentin
The Pseudo-Pascal Triangle of Maximum Deng Entropy
PPascal triangle (known as Yang Hui Triangle in Chinese) is an important model in mathematics while the entropy has been heavily studied in physics or as uncertainty measure in information science. How to construct the the connection between Pascal triangle and uncertainty measure is an interesting topic. One of the most used entropy, Tasllis entropy, has been modelled with Pascal triangle. But the relationship of the other entropy functions with Pascal triangle is still an open issue. Dempster-Shafer evidence theory takes the advantage to deal with uncertainty than probability theory since the probability distribution is generalized as basic probability assignment, which is more efficient to model and handle uncertain information. Given a basic probability assignment, its corresponding uncertainty measure can be determined by Deng entropy, which is the generalization of Shannon entropy. In this paper, a Pseudo-Pascal triangle based the maximum Deng entropy is constructed. Similar to the Pascal triangle modelling of Tasllis entropy, this work provides the a possible way of Deng entropy in physics and information theory
Systems of iterative functional equations : theory and applications
Tese de doutoramento, Matemática (Análise Matemática), Universidade de Lisboa, Faculdade de Ciências, 2015We formulate a general theoretical framework for systems of iterative functional equations between general spaces. We find general necessary conditions for the existence of solutions such as compatibility conditions (essential hypotheses to ensure problems are well-defined). For topological spaces we characterize continuity of solutions; for metric spaces we find sufficient conditions for existence and uniqueness. For a number of systems we construct explicit formulae for the solution, including affine and other general non-linear cases. We provide an extended list of examples. We construct, as a particular case, an explicit formula for the fractal interpolation functions with variable parameters. Conjugacy equations arise from the problem of identifying dynamical systems from the topological point of view. When conjugacies exist they cannot, in general, be expected to be smooth. We show that even in the simplest cases, e.g. piecewise affine maps, solutions of functional equations arising from conjugacy problems may have exotic properties. We provide a general construction for finding solutions, including an explicit formula showing how, in certain cases, a solution can be constructively determined. We establish combinatorial properties of the dynamics of piecewise increasing, continuous, expanding maps of the interval such as description/enumeration of periodic and pre-periodic points and length of pre-periodic itineraries. We include a relation between the dynamics of a family of circle maps and the properties of combinatorial objects such as necklaces and words. We provide some examples. We show the relevance of this for the representation of rational numbers. There are many possible proofs of Fermat's little theorem. We exemplify those using necklaces and dynamical systems. Both methods lead to generalizations. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps. The representation of numbers plays an important role in much of this work. Starting from the classical base p representation we present other type of representation of numbers: signed base p representation, Q-representation and finite base p representation of rationals. There is an extended p representation that generalizes some of the listed representations. We consider the concept of bold play in gambling, where the game has a unique win pay-off. The probability that a gambler reaches his goal using the bold play strategy is the solution of a functional equation. We compare with the timid play strategy and extend to the game with multiple pay-offs.Centro de Matemática e Aplicações Fundamentais da ULisboa; Fundação da Faculdade de Ciências da ULisbo
The calculus of multivectors on noncommutative jet spaces
The Leibniz rule for derivations is invariant under cyclic permutations of
co-multiples within the arguments of derivations. We explore the implications
of this principle: in effect, we construct a class of noncommutative bundles in
which the sheaves of algebras of walks along a tesselated affine manifold form
the base, whereas the fibres are free associative algebras or, at a later
stage, such algebras quotients over the linear relation of equivalence under
cyclic shifts. The calculus of variations is developed on the infinite jet
spaces over such noncommutative bundles.
In the frames of such field-theoretic extension of the Kontsevich formal
noncommutative symplectic (super)geometry, we prove the main properties of the
Batalin--Vilkovisky Laplacian and Schouten bracket. We show as by-product that
the structures which arise in the classical variational Poisson geometry of
infinite-dimensional integrable systems do actually not refer to the graded
commutativity assumption.Comment: Talks given at Mathematics seminar (IHES, 25.11.2016) and Oberseminar
(MPIM Bonn, 2.02.2017), 23 figures, 60 page
Two's Company, Three's a Crowd:Consensus-Halving for a Constant Number of Agents
We consider the -Consensus-Halving problem, in which a set of
heterogeneous agents aim at dividing a continuous resource into two (not
necessarily contiguous) portions that all of them simultaneously consider to be
of approximately the same value (up to ). This problem was
recently shown to be PPA-complete, for agents and cuts, even for very
simple valuation functions. In a quest to understand the root of the complexity
of the problem, we consider the setting where there is only a constant number
of agents, and we consider both the computational complexity and the query
complexity of the problem.
For agents with monotone valuation functions, we show a dichotomy: for two
agents the problem is polynomial-time solvable, whereas for three or more
agents it becomes PPA-complete. Similarly, we show that for two monotone agents
the problem can be solved with polynomially-many queries, whereas for three or
more agents, we provide exponential query complexity lower bounds. These
results are enabled via an interesting connection to a monotone Borsuk-Ulam
problem, which may be of independent interest. For agents with general
valuations, we show that the problem is PPA-complete and admits exponential
query complexity lower bounds, even for two agents
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