scholarly journals Geometry, Coordinatization and Cardinality of the Rational Numbers from Physical Perspective

2021 ◽  
Vol 2090 (1) ◽  
pp. 012037
Author(s):  
Kaushik Ghosh
Keyword(s):  
Line Segment ◽  
Statistical Physics ◽  
Rational Numbers ◽  
Partition Functions ◽  
Actual Process ◽  
Real Numbers ◽  
Hausdorff Topology ◽  
Straight Line ◽  
Definition Of ◽  
Positive Integers

Abstract In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

The definition of measure in function space

1950 ◽  
Vol 46 (1) ◽  
pp. 19-27
Author(s):  
H. R. Pitt
Keyword(s):  
Function Space ◽  
Real Variable ◽  
Fundamental Result ◽  
Real Numbers ◽  
Real Functions ◽  
Definition Of ◽  
Image Position ◽  
The Right ◽  
Positive Integers

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differenceswith respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfyingis measurable for any realbi, tiand has measure F(t; b).

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case

Algorithms ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 56
Author(s):  
Gokarna Sharma ◽  
Ramachandran Vaidyanathan ◽  
Jerry L. Trahan
Keyword(s):  
Mobile Robots ◽  
Line Segment ◽  
Autonomous Robots ◽  
Straight Line Segment ◽  
Autonomous Mobile Robots ◽  
Asymptotically Optimal ◽  
Time Translation ◽  
Straight Line ◽  
A New Technique ◽  
Colored Lights

We consider the distributed setting of N autonomous mobile robots that operate in Look-Compute-Move (LCM) cycles and use colored lights (the robots with lights model). We assume obstructed visibility where a robot cannot see another robot if a third robot is positioned between them on the straight line segment connecting them. In this paper, we consider the problem of positioning N autonomous robots on a plane so that every robot is visible to all others (this is called the Complete Visibility problem). This problem is fundamental, as it provides a basis to solve many other problems under obstructed visibility. In this paper, we provide the first, asymptotically optimal, O(1) time, O(1) color algorithm for Complete Visibility in the asynchronous setting. This significantly improves on an O(N)-time translation of the existing O(1) time, O(1) color semi-synchronous algorithm to the asynchronous setting. The proposed algorithm is collision-free, i.e., robots do not share positions, and their paths do not cross. We also introduce a new technique for moving robots in an asynchronous setting that may be of independent interest, called Beacon-Directed Curve Positioning.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

scholarly journals Automatic contraction of unstructured tensor networks

2020 ◽  
Vol 8 (1) ◽  
Author(s):  
Adam Jermyn
Keyword(s):  
Partition Function ◽  
Statistical Physics ◽  
Discrete Models ◽  
Correlation Structure ◽  
Central Problem ◽  
Partition Functions ◽  
Lattice Systems ◽  
Tensor Networks ◽  
Tensor Network

The evaluation of partition functions is a central problem in statistical physics. For lattice systems and other discrete models the partition function may be expressed as the contraction of a tensor network. Unfortunately computing such contractions is difficult, and many methods to make this tractable require periodic or otherwise structured networks. Here I present a new algorithm for contracting unstructured tensor networks. This method makes no assumptions about the structure of the network and performs well in both structured and unstructured cases so long as the correlation structure is local.

Jungian Analytical Method as a Process for Transformative Catechesis

Horizons ◽  
2008 ◽  
Vol 35 (1) ◽  
pp. 72-93
Author(s):  
Robert Brancatelli
Keyword(s):  
Depth Psychology ◽  
Analytical Method ◽  
Jungian Psychology ◽  
Actual Process ◽  
Paschal Mystery ◽  
Psychological Transformation ◽  
New Type ◽  
Definition Of

This paper elaborates a theory of catechesis that is concerned with the psychological transformation of adult Christians. It offers a definition of this new type of catechesis as well as a comparison with experiential catechesis. It then presents a process for transformative catechesis based on the analytical method of Jungian depth psychology. This process includes anamnesis, interpretation, discernment, and ritual commitment, with the ultimate aim of helping adults identify and experience the paschal mystery in their own lives. It begins by examining the suitability of Jungian psychology for a catechetical process, presents the actual process, and then explores the theological implications of Jungian-based catechesis for those working in ministry.

Signal Recovery by Linear Estimation

2019 ◽  
pp. 260-385
Author(s):  
John Stillwell
Keyword(s):  
Human Activities ◽  
Mathematical Concept ◽  
Rational Numbers ◽  
Precise Definition ◽  
Computable Analysis ◽  
Signal Recovery ◽  
Linear Estimation ◽  
Definition Of ◽  
Computable Sequence ◽  
Informal Description

This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.

RADISHCHEV PLOT IN MULTIVARIATE ANALYSIS OF THE PROBLEM OF MULTIPLE TARGETS GROUP PURSUIT

2021 ◽  
pp. 22-31
Author(s):  
A. A. Dubanov
Keyword(s):  
Multivariate Analysis ◽  
Line Segment ◽  
Iterative Process ◽  
Straight Line Segment ◽  
Line Of Sight ◽  
Radius Of Curvature ◽  
Time Step ◽  
Minimum Radius ◽  
Group Pursuit ◽  
Straight Line

This article discusses a kinematic model of the problem of group pursuit of a set of goals. The article discusses a variant of the model when all goals are achieved simultaneously. And also the possibility is considered when the achievement of goals occurs at the appointed time. In this model, the direction of the speeds by the pursuer can be arbitrary, in contrast to the method of parallel approach. In the method of parallel approach, the velocity vectors of the pursuer and the target are directed to a point on the Apollonius circle. The proposed pursuit model is based on the fact that the pursuer tries to follow the predicted trajectory of movement. The predicted trajectory of movement is built at each moment of time. This path is a compound curve that respects curvature constraints. A compound curve consists of a circular arc and a straight line segment. The pursuer's velocity vector applied to the point where the pursuer is located touches the given circle. The straight line segment passes through the target point and touches the specified circle. The radius of the circle in the model is taken equal to the minimum radius of curvature of the trajectory. The resulting compound line serves as an analogue of the line of sight in the parallel approach method. The iterative process of calculating the points of the pursuer’s trajectory is that the next point of position is the point of intersection of the circle centered at the current point of the pursuer’s position, with the line of sight corresponding to the point of the next position of the target. The radius of such a circle is equal to the product of the speed of the pursuer and the time interval corresponding to the time step of the iterative process. The time to reach the goal of each pursuer is a dependence on the speed of movement and the minimum radius of curvature of the trajectory. Multivariate analysis of the moduli of velocities and minimum radii of curvature of the trajectories of each of the pursuers for the simultaneous achievement of their goals i based on the methods of multidimensional descriptive geometry. To do this, the projection planes are entered on the Radishchev diagram: the radius of curvature of the trajectory and speed, the radius of curvature of the trajectory and the time to reach the goal. On the first plane, the projection builds a one-parameter set of level lines corresponding to the range of velocities. In the second graph, corresponding to a given range of speeds, functions of the dependence of the time to reach the target on the radius of curvature. The preset time for reaching the target and the preset value of the speed of the pursuer are the optimizing factors. This method of constructing the trajectories of pursuers to achieve a variety of goals at given time values may be in demand by the developers of autonomous unmanned aerial vehicles.

The EXPRESS-I Language

1994 ◽  
Author(s):  
Douglas Schenck ◽  
Peter Wilson
Keyword(s):  
Information Modeling ◽  
Real Numbers ◽  
Geometric Point ◽  
Computer Automation ◽  
Definition Of ◽  
Data Elements ◽  
Specific Implementation ◽  
And Behavior ◽  
Specification Techniques ◽  
Test Suites

Now we turn to the question: ‘Once I have created an abstract declaration in EXPRESS, what would an instance of that thing look like?’ EXPRESS-I allows you to create instances of EXPRESS things that have values in place of references to datatypes. The main reason for doing this is to study some realistic examples of things that otherwise might be difficult to understand. After all, it is one thing to describe a tree and quite another to actually see one. Some of the design goals of EXPRESS-I are based on these requirements: • Major information modeling projects are large and complex. Managing them without appropriate tools based on formal languages and methods is a risky proposition. Informal specification techniques eliminate the possibility of employing computer automation in checking for inconsistencies in presentation or specification. • The language should focus on the display of the realization of the properties of entities, which are the things of interest. The definition of entities is in terms of data and behavior. Data represents the properties by which an entity is realized and behavior is represented by constraints. • The language should seek to avoid, as far as possible, specific implementation views. That is, EXPRESS-I models do not suggest the structure of databases, object bases, or of information bases in general. • The language should provide a means for displaying small populated models of EXPRESS schemas as examples for design reviews. • The language should provide a means for supporting the specification of test suites for information model processors. EXPRESS-I represents entity instances in terms of the values of its attributes (attributes are the traits or characteristics considered important for use and understanding). These values have a representation which might be considered simple (an integer value) or something more complex (an entity value). A geometric point might be defined in terms of three real numbers named x, y and z, and the actual values associated with those attributes might be 1.0, 2.5 and 7.9. The EXPRESS-I instance language provides a means of displaying instantiations of EXPRESS data elements. The language is designed principally for human readability and for ease of generating EXPRESS-I element instances from definitions in an EXPRESS schema.

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