相关论文: Fractal uncertainty principle for discrete Cantor …

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

The purpose of this paper is to study the fractal phenomena in large data sets and the associated questions of dimension reduction. We examine situations where the classical Principal Component Analysis is not effective in identifying the…

We construct $\delta$-regular sets with $\delta\ge \frac12$ for which the analog of the Bourgain--Dyatlov Fractal Uncertainty Principle fails for the Walsh--Fourier transform.

Quantum gravity has been baffling the theoretical physicist for decades now: both for its mathematical obscurity and phenomenological testing. Nevertheless, the new era of precision cosmology presents a promising avenue to test the effects…

We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption…

The diffraction spectrum of coherent waves scattered from fractal supports is calculated exactly. The fractals considered are of the class generated iteratively by successive dilations and translations, and include generalizations of the…

We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence…

Using the discrete-scale invariance theory, we show that the coupling constants of fundamental forces, the atomic masses and energies, and the elementary particle masses, obey to the fractal properties.

Chebotar\"ev proved that every minor of a discrete Fourier matrix of prime order is nonzero. We prove a generalization of this result that includes analogues for discrete cosine and discrete sine matrices as special cases. We establish…

We construct meta-intransitive systems of independent random variables of any finite order from basic tuple of random variables which generalize intransitive dice. Under this construction, the equality of some linear functional is…

A partition is finitary if all its members are finite. For a set $A$, $\mathscr{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{ZF}$ (without the axiom of choice) that there exist an infinite…

Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular…

The derivation of the Heisenberg Uncertainty Principle (HUP) from the Uncertainty Theorem of Fourier Transform theory demonstrates that the HUP arises from the dependency of momentum on wave number that exists at the quantum level. It also…

The union-closed sets conjecture (sometimes referred to as Frankl's conjecture) states that every finite, nontrivial union-closed family of sets has an element that is in at least half of its members. Although the conjecture is known to be…

We establish a fractal transference principle for continued fraction expansions over the field of Laurent series. Let $S$ be an infinite subset of the set of all polynomials over a finite field of $q$ elements of positive degree with growth…

The paper is devoted to the development of control procedures with a guide for conflict-controlled dynamical systems described by ordinary fractional differential equations with the Caputo derivative of an order $\alpha \in (0, 1).$ For the…

For open systems derived from quadratic total Hamiltonians, we derive a dynamic fluctuation-dissipation (FD) inequality valid for any total initial state and without regard to the sign of the dissipation. With the added constraint that this…

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of…

In this article, we provide a simple and systematic way to represent general (inhomogeneous) fractals that may look different at different scales and places. By using set-valued compression maps, we express these general fractals as…

Generalized uncertainty principle and breakdown of the spacetime continuum certainly represent two important results derived of various approaches related to quantum gravity and black hole physics near the well-known Planck scale. The…