scholarly journals Assouad dimension influences the box and packing dimensions of orthogonal projections

2021 ◽  
Author(s):  
Kenneth Falconer ◽  
Jonathan Fraser ◽  
Pablo Shmerkin
Keyword(s):  
Orthogonal Projections ◽  
Assouad Dimension ◽  
Packing Dimensions

scholarly journals Combinatorial proofs of two theorems of Lutz and Stull

2021 ◽  
pp. 1-12
Author(s):  
TUOMAS ORPONEN
Keyword(s):  
Information Theory ◽  
Euclidean Space ◽  
Pigeonhole Principle ◽  
Algorithmic Information Theory ◽  
Orthogonal Projections ◽  
Information Theoretic ◽  
Secondary Purpose ◽  
Packing Dimensions ◽  
Algorithmic Information ◽  
Projection Theorems

Abstract Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if \[K \subset {\mathbb{R}^n}\] is any set with equal Hausdorff and packing dimensions, then \begin{equation} \[{\dim _{\text{H}}}{\pi _e}(K) = \min \{ {\dim _{\text{H}}}{\text{ }}K{\text{, 1}}\} \] \end{equation} for almost every \[e \in {S^{n - 1}}\] . Here \[{\pi _e}\] stands for orthogonal projection to span ( \[e\] ). The primary purpose of this paper is to present proofs for Lutz and Stull’s projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman’s “potential theoretic” method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to generalise Lutz and Stull’s theorems: the versions in this paper apply to orthogonal projections to m-planes in \[{\mathbb{R}^n}\] , for all \[0 < m < n\] .

scholarly journals Assouad Dimensions and Lower Dimensions of Some Moran Sets

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
JiaQing Xiao
Keyword(s):  
Compression Ratio ◽  
Cantor Sets ◽  
Hausdorff Dimensions ◽  
Low Dimensions ◽  
Assouad Dimension ◽  
Packing Dimensions ◽  
Box Dimensions

We prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions. Subsequently, we consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples in which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition of the smallest compression ratio c ∗ > 0 .

THE AVERAGE FRACTAL DIMENSION AND PROJECTIONS OF MEASURES AND SETS IN Rn

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 747-754 ◽  
Author(s):  
M. ZÄHLE
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Average Density ◽  
Higher Dimension ◽  
Orthogonal Projections ◽  
Analytic Set ◽  
Average Dimension ◽  
Packing Dimensions ◽  
Almost All ◽  
Local Average

In this note we introduce the concept of local average dimension of a measure µ, at x∈ℝn as the unique exponent where the lower average density of µ, at x jumps from zero to infinity. Taking the essential infimum or supremum over x we obtain the lower and upper average dimensions of µ, respectively. The average dimension of an analytic set E is defined as the supremum over the upper average dimensions of all measures supported by E. These average dimensions lie between the corresponding Hausdorff and packing dimensions and the inequalities can be strict. We prove that the local Hausdorff dimensions and the local average dimensions of µ at almost all x are invariant under orthogonal projections onto almost all m- dimensional linear subspaces of higher dimension. The corresponding global results for µ and E (which are known for Hausdorff dimension) follow immediately.

scholarly journals Multifractal dimensions for projections of measures

2021 ◽  
Vol 40 ◽  
pp. 1-15
Author(s):  
Bilel Selmi
Keyword(s):  
Multifractal Analysis ◽  
Probability Measures ◽  
Orthogonal Projections ◽  
Packing Dimensions ◽  
Borel Probability

In this paper, we calculate the multifractal Hausdorff and packing dimensions of Borel probability measures and study their behaviors under orthogonal projections. In particular, we try through these results to improve the main result of M. Dai in \cite{D} about the multifractal analysis of a measure of multifractal exact dimension.

Box and packing dimensions of projections and dimension profiles

2001 ◽  
Vol 130 (1) ◽  
pp. 135-160 ◽  
Author(s):  
J. D. HOWROYD
Keyword(s):  
Packing Dimension ◽  
Dimension Theory ◽  
Box Dimension ◽  
Orthogonal Projections ◽  
Packing Dimensions ◽  
Almost All ◽  
Box Dimensions

For E a subset of ℝn and s ∈ [0, n] we define upper and lower box dimension profiles, B-dimsE and B-dimsE respectively, that are closely related to the box dimensions of the orthogonal projections of E onto subspaces of ℝn. In particular, the projection of E onto almost all m-dimensional subspaces has upper box dimension B-dimmE and lower box dimension B-dimmE. By defining a packing type measure with respect to s-dimensional kernels we are able to establish the connection to an analogous packing dimension theory.

scholarly journals Further steps on the reconstruction of convex polyominoes from orthogonal projections

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Andrea Frosini ◽  
Simone Rinaldi ◽  
Lama Tarsissi ◽  
Laurent Vuillon
Keyword(s):  
Discrete Geometry ◽  
Convex Subset ◽  
Convex Sets ◽  
A Priori ◽  
Orthogonal Projections ◽  
Reconstruction Process ◽  
Combinatorial Properties ◽  
The Family ◽  
Discrete Counterpart ◽  
Interior Part

AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

scholarly journals Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization

2021 ◽  
Vol 51 (3) ◽  
Author(s):  
Maurice A. de Gosson
Keyword(s):  
Uncertainty Principle ◽  
Convex Geometry ◽  
Geometric Approach ◽  
Point Of View ◽  
Reconstruction Problem ◽  
Orthogonal Projections ◽  
Topological Version ◽  
Mahler Conjecture ◽  
Quantum Indeterminacy ◽  
Dual Quantum

AbstractWe define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

scholarly journals Nuclear Magnetic Resonance Metabolomics with Double Pulsed-Field-Gradient Echo and Automatized Solvent Suppression Spectroscopy for Multivariate Data Matrix Applied in Novel Wine and Juice Discriminant Analysis

Molecules ◽  
2021 ◽  
Vol 26 (14) ◽  
pp. 4146
Author(s):  
José Enrique Herbert-Pucheta ◽  
José Daniel Lozada-Ramírez ◽  
Ana E. Ortega-Regules ◽  
Luis Ricardo Hernández ◽  
Cecilia Anaya de Parrodi
Keyword(s):  
Statistical Analysis ◽  
Discriminant Analysis ◽  
Field Gradient ◽  
Multivariate Statistical Analysis ◽  
Data Matrix ◽  
Cabernet Sauvignon ◽  
Gradient Echo ◽  
Multivariate Statistical ◽  
Orthogonal Projections ◽  
Pulsed Field

The quality of foods has led researchers to use various analytical methods to determine the amounts of principal food constituents; some of them are the NMR techniques with a multivariate statistical analysis (NMR-MSA). The present work introduces a set of NMR-MSA novelties. First, the use of a double pulsed-field-gradient echo (DPFGE) experiment with a refocusing band-selective uniform response pure-phase selective pulse for the selective excitation of a 5–10-ppm range of wine samples reveals novel broad 1H resonances. Second, an NMR-MSA foodomics approach to discriminate between wine samples produced from the same Cabernet Sauvignon variety fermented with different yeast strains proposed for large-scale alcohol reductions. Third a comparative study between a nonsupervised Principal Component Analysis (PCA), supervised standard partial (PLS-DA), and sparse (sPLS-DA) least squares discriminant analysis, as well as orthogonal projections to a latent structures discriminant analysis (OPLS-DA), for obtaining holistic fingerprints. The MSA discriminated between different Cabernet Sauvignon fermentation schemes and juice varieties (apple, apricot, and orange) or juice authentications (puree, nectar, concentrated, and commercial juice fruit drinks). The new pulse sequence DPFGE demonstrated an enhanced sensitivity in the aromatic zone of wine samples, allowing a better application of different unsupervised and supervised multivariate statistical analysis approaches.

Sign in / Sign up

Export Citation Format

Share Document