Upper (lower) bounds of the eigenvalues, spread and the open problems for the real symmetric interval matrices

2012 ◽  
Vol 36 (4) ◽  
pp. 413-421 ◽  
Author(s):  
Junliang Wu
Keyword(s):  
Lower Bounds ◽  
Open Problems ◽  
The Real ◽  
Interval Matrices

Mutual Embeddings

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550001
Author(s):  
ILKER NADI BOZKURT ◽  
HAI HUANG ◽  
BRUCE MAGGS ◽  
ANDRÉA RICHA ◽  
MAVERICK WOO
Keyword(s):  
Iterative Methods ◽  
Lower Bounds ◽  
Linear Equations ◽  
Graph Embedding ◽  
Open Problems ◽  
Linear Arrays ◽  
Systems Of Linear Equations

This paper introduces a type of graph embedding called a mutual embedding. A mutual embedding between two n-node graphs [Formula: see text] and [Formula: see text] is an identification of the vertices of V1 and V2, i.e., a bijection [Formula: see text], together with an embedding of G1 into G2 and an embedding of G2 into G1 where in the embedding of G1 into G2, each node u of G1 is mapped to π(u) in G2 and in the embedding of G2 into G1 each node v of G2 is mapped to [Formula: see text] in G1. The identification of vertices in G1 and G2 constrains the two embeddings so that it is not always possible for both to exhibit small congestion and dilation, even if there are traditional one-way embeddings in both directions with small congestion and dilation. Mutual embeddings arise in the context of finding preconditioners for accelerating the convergence of iterative methods for solving systems of linear equations. We present mutual embeddings between several types of graphs such as linear arrays, cycles, trees, and meshes, prove lower bounds on mutual embeddings between several classes of graphs, and present some open problems related to optimal mutual embeddings.

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

scholarly journals On the history of van der Waerden's theorem on arithmetic progressions

2001 ◽  
Vol 32 (4) ◽  
pp. 335-341
Author(s):  
Tom C. Brown ◽  
Jau-Shyong Peter Shiue
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds ◽  
Open Problems ◽  
History Of

In this expository note, we discuss the celebrated theorem known as ``van der Waerden's theorem on arithmetic progressions", the history of work on upper and lower bounds for the function associated with this theorem, a number of generalizations, and some open problems.

3D Camera Tracking for Mixed Reality using Multi-Sensors Technology

2013 ◽  
pp. 2164-2175
Author(s):  
Fakhreddine Ababsa ◽  
Iman Maissa Zendjebil ◽  
Jean-Yves Didier
Keyword(s):  
Mixed Reality ◽  
Research Field ◽  
Sensor Technology ◽  
Camera Tracking ◽  
Open Problems ◽  
The Real ◽  
New Directions ◽  
3D Camera ◽  
Real Objects

The concept of Mixed Reality (MR) aims at completing our perception of the real world, by adding fictitious elements that are not perceptible naturally such as: computer generated images, virtual objects, texts, symbols, graphics, sounds, smells, et cetera. One of the major challenges for efficient Mixed Reality system is to ensure the spatiotemporal coherence of the augmented scene between the virtual and the real objects. The quality of the Real/Virtual registration depends mainly on the accuracy of the 3D camera pose estimation. The goal of this chapter is to provide an overview on the recent multi-sensor fusion approaches used in Mixed Reality systems for the 3D camera tracking. We describe the main sensors used in those approaches and we detail the issues surrounding their use (calibration process, fusion strategies, etc.). We include the description of some Mixed Reality techniques developed these last years and which use multi-sensor technology. Finally, we highlight new directions and open problems in this research field.

scholarly journals New Lower Bounds for the Constants in the Real Polynomial Hardy–Littlewood Inequality

2016 ◽  
Vol 37 (8) ◽  
pp. 927-937 ◽  
Author(s):  
W. Cavalcante ◽  
Daniel Núñez-Alarcón ◽  
D. Pellegrino
Keyword(s):  
Lower Bounds ◽  
Real Polynomial ◽  
The Real

LOWER BOUNDS FOR STREETS AND GENERALIZED STREETS

2001 ◽  
Vol 11 (04) ◽  
pp. 401-421 ◽  
Author(s):  
ALEJANDRO LÓPEZ-ORTIZ ◽  
SVEN SCHUIERER
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real Line ◽  
Upper Bound ◽  
Competitive Ratio ◽  
Search Strategy ◽  
The Real ◽  
Simple Polygons ◽  
On Line ◽  
Special Classes

We present lower bounds for on-line searching problems in two special classes of simple polygons called streets and generalized streets. In streets we assume that the location of the target is known to the robot in advance and prove a lower bound of [Formula: see text] on the competitive ratio of any deterministic search strategy—which can be shown to be tight. For generalized streets we show that if the location of the target is not known, then there is a class of orthogonal generalized streets for which the competitive ratio of any search strategy is at least [Formula: see text] in the L2-metric—again matching the competitive ratio of the best known algorithm. We also show that if the location of the target is known, then the competitive ratio for searching in generalized streets in the L1-metric is at least 9 which is tight as well. The former result is based on a lower bound on the average competitive ratio of searching on the real line if an upper bound of D to the target is given. We show that in this case the average competitive ratio is at least 9-O(1/ log D).

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals ГРАНИЦЫ КРАМЕРА-РАО И БХАТТАЧАРИЯ ДЛЯ ОЦЕНКИ ПОТЕНЦИАЛЬНОЙ ТОЧНОСТИ СУБПИКСЕЛЬНОГО СОВМЕЩЕНИЯ ИЗОБРАЖЕНИЙ

2018 ◽  
pp. 34-40
Author(s):  
Виталий Анатольевич Душепа
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Monte Carlo Methods ◽  
Parameter Estimates ◽  
Registration Accuracy ◽  
The Real ◽  
Registration Algorithm ◽  
Subpixel Accuracy ◽  
Bhattacharyya Bounds ◽  
Cramer Rao Bound

The subject matter of the article is theoretical lower bounds of parameter estimates applied to the problem of image co-registration. The goal is to study and compare the Cramer-Rao and Bhattacharyya bounds. The tasks to be solved are: to formulate algorithms for calculating the Cramer-Rao and Bhattacharyya bounds for estimating the subpixel shifts of two images; using the Monte Carlo methods to compare the calculated bounds with the results of the real registration algorithm. The methods used are computer simulation; Monte Carlo methods. Monte Carlo experiments were used both for calculating theoretical bounds (partial derivatives were estimated using numerical differentiation formulas) and for calculating the accuracy of the real algorithm. The subpixel accuracy of the registration algorithm was achieved by the intensity interpolation method, in this case, the problem of image coordinates determination was considered as an optimization problem solved by a numerical method. The following results were obtained. It is experimentally confirmed that, when calculating the lower bounds of the registration accuracy in the traditional formulation (when we do not take into account the errors of interpolation used to achieve subpixel accuracy), the Bhattacharyya bound always passes above the Cramer-Rao bound, that corresponds to the theory. However, although the Bhattacharyya bound provides more accurate estimates than the Cramer-Rao bound, its values at high signal-to-noise ratios are still too optimistic for registration accuracy in real situations. Both artificially modeled images (based on the fractal Brownian motion model) and fragments of real satellite images were used in computational experiments. Conclusions. The scientific novelty of the results obtained is that this work, in many respects following the research carried out by other authors, in contrast to them considers the maximum likelihood function taking into account the possibility of linear brightness transformation between two images, which is the most frequently used model in practice. However, the possibility of rotating two images in this article is not taken into account

Sharp Localized Inequalities for Fourier Multipliers

2014 ◽  
Vol 66 (6) ◽  
pp. 1358-1381
Author(s):  
Adam Osękowski
Keyword(s):  
Lower Bounds ◽  
Lévy Processes ◽  
Levy Processes ◽  
Fourier Multipliers ◽  
Probabilistic Methods ◽  
Independent Interest ◽  
Probability Measures ◽  
Sharp Bounds ◽  
The Real

AbstractIn this paper we study sharp localized Lq → Lp estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on 2×2 matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling–Ahlfors operator.

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