Evenly Convex Sets: Linear Systems Containing Strict Inequalities

2020 ◽  
pp. 1-60
Author(s):  
María D. Fajardo ◽  
Miguel A. Goberna ◽  
Margarita M. L. Rodríguez ◽  
José Vicente-Pérez
Keyword(s):  
Linear Systems ◽  
Convex Sets ◽  
Strict Inequalities

Evenly Convex Polyhedra: Finite Linear Systems Containing Strict Inequalities

2020 ◽  
pp. 61-99
Author(s):  
María D. Fajardo ◽  
Miguel A. Goberna ◽  
Margarita M. L. Rodríguez ◽  
José Vicente-Pérez
Keyword(s):  
Linear Systems ◽  
Convex Polyhedra ◽  
Strict Inequalities ◽  
Finite Linear

scholarly journals On linear systems containing strict inequalities

2003 ◽  
Vol 360 ◽  
pp. 151-171 ◽  
Author(s):  
Miguel A. Goberna ◽  
Valentín Jornet ◽  
Margarita M.L. Rodríguez
Keyword(s):  
Linear Systems ◽  
Strict Inequalities

scholarly journals On Finite Linear Systems Containing Strict Inequalities

2017 ◽  
Vol 173 (1) ◽  
pp. 131-154
Author(s):  
Margarita M. L. Rodríguez ◽  
José Vicente-Pérez
Keyword(s):  
Linear Systems ◽  
Strict Inequalities ◽  
Finite Linear

scholarly journals Analyzing linear systems containing strict inequalities via evenly convex hulls

2006 ◽  
Vol 169 (3) ◽  
pp. 1079-1095 ◽  
Author(s):  
Miguel A. Goberna ◽  
Margarita M.L. Rodrı́guez
Keyword(s):  
Linear Systems ◽  
Convex Hulls ◽  
Strict Inequalities

Reserve of characteristic inclusion for interval linear systems of relations

2021 ◽  
pp. 61-85
Author(s):  
Ирина Александровна Шарая ◽  
Сергей Петрович Шарый
Keyword(s):  
Linear Systems ◽  
Interval Arithmetic ◽  
Quantitative Measure ◽  
Solution Set ◽  
Special Cases ◽  
Interval Linear ◽  
Strict Inequalities ◽  
Interval Linear Systems ◽  
Interval Systems ◽  
Linear Systems Of Equations

В работе рассматриваются интервальные линейные включения Cx ⊆ d в полной интервальной арифметике Каухера. Вводится количественная мера выполнения этого включения, названная “резервом включения”, исследуются ее свойства и приложения. Показано, что резерв включения оказывается полезным инструментом при изучении АЕ-решений и кванторных решений интервальных линейных систем уравнений и неравенств. В частности, использование резерва включения помогает при определении положения точки относительно множества решений, при исследовании пустоты множества решений или его внутренности и т.п In this paper, we consider interval linear inclusions Cx ⊆ d in the Kaucher complete interval arithmetic. These inclusions are important both on their own and because they provide equivalent and useful descriptions for the so-called quantifier solutions and AE-solutions to interval systems of linear algebraic relations of the form Ax σ b , where A is an interval m × n -matrix, x ∈ R , b is an interval m -vector, and σ ∈ {= , ≤ , ≥} . In other words, these are interval systems in which equations and non-strict inequalities can be mixed. Considering the inclusion Cx ⊆ d in the Kaucher complete interval arithmetic allows studing simultaneously and in a uniform way all the different special cases of quantifier solutions and AE-solutions of interval systems of linear relations, as well as using interval analysis methods. A quantitative measure, called the “inclusion reserve”, is introduced to characterize how strong the inclusion Cx ⊆ d is fulfilled. In our work, we investigate its properties and applications. It is shown that the inclusion reserve turns out to be a useful tool in the study of AE-solutions and quantifier solutions of interval linear systems of equations and inequalities. In particular, the use of the inclusion reserve helps to determine the position of a point relative to a solution set, in investigating whether the solution set is empty or not, whether a point is in the interior of the solution set, etc

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

1990 ◽  
Vol 48 (1) ◽  
pp. 520-521
Author(s):  
Neng-Yu Zhang ◽  
Bruce F. McEwen ◽  
Joachim Frank
Keyword(s):  
Function Space ◽  
Convex Sets ◽  
Electron Tomography ◽  
Spinal Ganglion ◽  
Fourier Space ◽  
Missing Information ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Energy Bound ◽  
Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

Well-Posed Linear Systems

2005 ◽  
Author(s):  
Olof Staffans
Keyword(s):  
Linear Systems ◽  
Well Posed

Principles of Linear Systems

1997 ◽  
Author(s):  
Philip E. Sarachik
Keyword(s):  
Linear Systems
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