scholarly journals Mass Transference Principle: From Balls to Arbitrary Shapes

2020 ◽  
Author(s):  
Henna Koivusalo ◽  
Michał Rams
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Arbitrary Shape ◽  
Strong Result ◽  
Transference Principle ◽  
Arbitrary Shapes

Abstract The mass transference principle, proved by Beresnevich and Velani in 2006, is a strong result that gives lower bounds for the Hausdorff dimension of limsup sets of balls. We present a version for limsup sets of open sets of arbitrary shape.

Experimental force reconstruction on plates of arbitrary shape using neural networks

2021 ◽  
Vol 263 (3) ◽  
pp. 3407-3416
Author(s):  
Tyler Dare
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Inverse Problem ◽  
Artificial Neural Network ◽  
Rectangular Plate ◽  
Arbitrary Shape ◽  
Impulsive Force ◽  
Force Reconstruction ◽  
Output Space ◽  
Arbitrary Shapes

Measuring the forces that excite a structure into vibration is an important tool in modeling the system and investigating ways to reduce the vibration. However, determining the forces that have been applied to a vibrating structure can be a challenging inverse problem, even when the structure is instrumented with a large number of sensors. Previously, an artificial neural network was developed to identify the location of an impulsive force on a rectangular plate. In this research, the techniques were extended to plates of arbitrary shape. The principal challenge of arbitrary shapes is that some combinations of network outputs (x- and y-coordinates) are invalid. For example, for a plate with a hole in the middle, the network should not output that the force was applied in the center of the hole. Different methods of accommodating arbitrary shapes were investigated, including output space quantization and selecting the closest valid region.

scholarly journals Approximation properties of -expansions II

2017 ◽  
Vol 38 (5) ◽  
pp. 1627-1641
Author(s):  
SIMON BAKER
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Measure Zero ◽  
Hausdorff Measures ◽  
Approximation Properties ◽  
Real Numbers ◽  
Positive Real ◽  
Transference Principle ◽  
Significant Step ◽  
Full Lebesgue Measure

Let $\unicode[STIX]{x1D6FD}\in (1,2)$ be a real number. For a function $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$, define $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ to be the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $(\unicode[STIX]{x1D716}_{i})_{i=1}^{n}\in \{0,1\}^{n}$ satisfying $0\leq x-\sum _{i=1}^{n}(\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D6FD}^{i})\leq \unicode[STIX]{x1D6F9}(n)$. In Baker [Approximation properties of $\unicode[STIX]{x1D6FD}$-expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$, the condition $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)=\infty$ implies that $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers $(\unicode[STIX]{x1D714}_{n})_{n=1}^{\infty }$ satisfying $\lim _{n\rightarrow \infty }\unicode[STIX]{x1D714}_{n}=\infty$, for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1.497,\ldots ,2)$, the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714}_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. We also study the case where $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)<\infty$ in which the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$.

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

Lower bounds for the Hausdorff dimension of attractors

1995 ◽  
Vol 7 (3) ◽  
pp. 457-469 ◽  
Author(s):  
Michael Y. Li ◽  
James S. Muldowney
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds

JARNÍK’S THEOREM WITHOUT THE MONOTONICITY ON THE APPROXIMATING FUNCTION

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950044
Author(s):  
CHAO MA ◽  
SHAOHUA ZHANG
Keyword(s):  
Hausdorff Dimension ◽  
Short Proof ◽  
Main Tool ◽  
Hausdorff Measures ◽  
Transference Principle ◽  
Negative Function

Let [Formula: see text] be a non-negative function such that [Formula: see text] as [Formula: see text]. The well-known Jarník–Besicovtich theorem concerns the Hausdorff dimension of the set of [Formula: see text]- approximable numbers. In this paper, we give an alternative but short proof of the Jarník–Besicovitch theorem for approximating functions with no monotonicity. The main tool is the appropriate usage of the mass transference principle of Beresnevich–Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2) 164(3) (2006) 971–992].

Lower Bounds to the Nth Eigenvalue of the Helmholz Equation Over Three-Dimensional Regions of Arbitrary Shape

1974 ◽  
Vol 41 (3) ◽  
pp. 819-820
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Helmholtz Equation ◽  
Lower Bounds ◽  
Arbitrary Shape ◽  
Three Dimensional ◽  
Helmholz Equation

A method is presented to compute a lower bound to the nth eigenvalue of the Helmholtz equation over three-dimensional regions. The shape of the regions is arbitrary and only their volume need be known.

scholarly journals Dimension estimates for sets of uniformly badly approximable systems of linear forms

2015 ◽  
Vol 11 (07) ◽  
pp. 2037-2054 ◽  
Author(s):  
Ryan Broderick ◽  
Dmitry Kleinbock
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Higher Dimensions ◽  
Upper And Lower Bounds ◽  
Homogeneous Dynamics ◽  
Linear Forms ◽  
One Dimensional ◽  
Approximation Constant ◽  
The One ◽  
Badly Approximable

The set of badly approximable m × n matrices is known to have Hausdorff dimension mn. Each such matrix comes with its own approximation constant c, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed c. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions. The lower bounds are established via the technique of Schmidt games, while for the upper bound we use homogeneous dynamics methods, namely exponential mixing of flows on the space of lattices.

scholarly journals Dynamics of Newton's functions of Barna's polynomials

2002 ◽  
Vol 29 (2) ◽  
pp. 79-84
Author(s):  
Piyapong Niamsup
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Real Roots ◽  
Exceptional Sets ◽  
Real Polynomials

We define Barna's polynomials as real polynomials with all real roots of which at least four are distinct. In this paper, we study the dynamics of Newton's functions of such polynomials. We also give the upper and lower bounds of the Hausdorff dimension of exceptional sets of these Newton's functions.

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