scholarly journals Introduction to logical entropy and its relationship to Shannon entropy

4open ◽  
2022 ◽  
Vol 5 ◽  
pp. 1
Author(s):  
David Ellerman
Keyword(s):  
Shannon Entropy ◽  
Information Age ◽  
Quantitative Measure ◽  
Boolean Logic ◽  
Natural Measure ◽  
Minimum Number ◽  
Concept Of Information ◽  
The Given ◽  
Logical Entropy ◽  
Projective Measurements

We live in the information age. Claude Shannon, as the father of the information age, gave us a theory of communications that quantified an “amount of information,” but, as he pointed out, “no concept of information itself was defined.” Logical entropy provides that definition. Logical entropy is the natural measure of the notion of information based on distinctions, differences, distinguishability, and diversity. It is the (normalized) quantitative measure of the distinctions of a partition on a set-just as the Boole–Laplace logical probability is the normalized quantitative measure of the elements of a subset of a set. And partitions and subsets are mathematically dual concepts – so the logic of partitions is dual in that sense to the usual Boolean logic of subsets, and hence the name “logical entropy.” The logical entropy of a partition has a simple interpretation as the probability that a distinction or dit (elements in different blocks) is obtained in two independent draws from the underlying set. The Shannon entropy is shown to also be based on this notion of information-as-distinctions; it is the average minimum number of binary partitions (bits) that need to be joined to make all the same distinctions of the given partition. Hence all the concepts of simple, joint, conditional, and mutual logical entropy can be transformed into the corresponding concepts of Shannon entropy by a uniform non-linear dit-bit transform. And finally logical entropy linearizes naturally to the corresponding quantum concept. The quantum logical entropy of an observable applied to a state is the probability that two different eigenvalues are obtained in two independent projective measurements of that observable on that state.

scholarly journals AN INTRODUCTION TO LOGICAL ENTROPY AND ITS RELATION TO SHANNON ENTROPY

2013 ◽  
Vol 07 (02) ◽  
pp. 121-145 ◽  
Author(s):  
DAVID ELLERMAN
Keyword(s):  
Shannon Entropy ◽  
Conditional Entropy ◽  
Boolean Logic ◽  
Exclusion Principle ◽  
Counting Measure ◽  
Logical Concept ◽  
Finite Set ◽  
Logical Notion ◽  
Logical Entropy ◽  
Ordered Pairs

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set — just as the usual logical notion of probability based on the Boolean logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g. the inclusion-exclusion principle) — just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition.

scholarly journals Introduction to Quantum Logical Information Theory: Talk

2018 ◽  
Vol 182 ◽  
pp. 02039
Author(s):  
David Ellerman
Keyword(s):  
Information Theory ◽  
Boolean Logic ◽  
Projective Measurement ◽  
Superposition State ◽  
Logical Probability ◽  
Quantitative Version ◽  
Natural Notion ◽  
Logical Entropy ◽  
Logical Information ◽  
Mutual Entropy

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences, and distinguishability, and is formalized using the distinctions (“dits”) of a partition (a pair of points distinguished by the partition). All the definitions of simple, joint, conditional, and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this talk is to outline the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., “qudits” of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates (cohered together in the pure superposition state being measured) that are distinguished by the measurement (decohered in the postmeasurement mixed state). Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.

APPROXIMATING POLYGONS AND SUBDIVISIONS WITH MINIMUM-LINK PATHS

1993 ◽  
Vol 03 (04) ◽  
pp. 383-415 ◽  
Author(s):  
LEONIDAS J. GUIBAS ◽  
JOHN E. HERSHBERGER ◽  
JOSEPH S.B. MITCHELL ◽  
JACK SCOTT SNOEYINK
Keyword(s):  
Greedy Algorithms ◽  
Np Hard ◽  
Basic Approach ◽  
Line Segments ◽  
Minimum Number ◽  
Polygonal Chains ◽  
The Given ◽  
Plane Polygon

We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with. no self-intersections are NP-hard.

Estimation of Balance Stability States for Legged Robotic Systems

2016 ◽  
Author(s):  
Carlotta Mummolo ◽  
Luigi Mangialardi ◽  
Joo H. Kim
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Center Of Mass ◽  
Stability Boundary ◽  
Three Dimensional ◽  
Quantitative Measure ◽  
Initial Position ◽  
Robotic Systems ◽  
The Given ◽  
Balance Stability

A three-dimensional criterion is provided for the estimation of balance stability states of legged robotic systems subject to various constraints. A general framework is established to evaluate the balance stability boundary of a given system in the state space of Center of Mass (COM) Cartesian position and velocity. For each assigned COM initial position, an optimization-based iterative algorithm finds the minimum and maximum COM initial velocity that the system can handle along a given direction, such that it maintains the capability to reach a final static equilibrium. The resulting set of velocity extrema constitutes the system’s balance stability boundary, which represents the sufficient condition to estimate falling states versus balanced states, according to the definitions provided herein. The COM state space domain identified with this approach contains all possible balanced states for the given legged system, with respect to the necessary physical, balancing, and design constraints. The balance state estimation is demonstrated for 1- and 2-degrees of freedom planar legged systems in single support. The domain identified by the balance stability boundary can be used as a “map” for the given legged system in which the distance from a given state to the domain boundaries can provide a quantitative measure of balance stability/instability.

scholarly journals Applicability analysis of prediction methods in the system for selection personalized offers by analytical modeling

2021 ◽  
Vol 22 (1) ◽  
pp. 16-22
Author(s):  
Yuriy S. Fedorenko
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Analytical Modeling ◽  
Queuing System ◽  
Software Implementation ◽  
System Response ◽  
System Response Time ◽  
Minimum Number ◽  
The Many ◽  
The Given

The relevance of the work is justified by the frequent occurrence of the need to solve the problems of choosing personalized offers in information systems and the many possible methods of machine learning, among which it is necessary to choose the most suitable one. The purpose of this study is to simulate a system for selecting personalized offers as a queuing system for estimating equipment costs when using each of the methods necessary to service the required part of requests for a given time limit. This solves the problem of assessing the minimum number of servicing devices (backend servers) required to ensure the operation of the system at a given level. The paper shows that the system can be described by a multichannel queuing system without losses. The distribution function of the spent time of the request in the system (the service time plus the waiting time in the queue) is calculated, since in the literature for such systems only the distribution function of the waiting time in the queue is described. Transformations of the expression for the probability of waiting are given, which solve the overflow problem in the software implementation. In the final part, as an example, the system was modeled according to the given parameters, and the minimum number of servicing devices was estimated to ensure a given system response time. Based on the data obtained, it is possible to make a decision on the advisability of using one or another method for predicting the frequency of user clicks or ranking.

scholarly journals Early Termination Clauses for Leasing Contracts with APR Cap

2018 ◽  
Vol 13 (12) ◽  
pp. 290
Author(s):  
Luciano Quattrocchio ◽  
Luisa Tibiletti ◽  
Mariacristina Uberti
Keyword(s):  
Interest Rate ◽  
Early Termination ◽  
Penalty Costs ◽  
Extra Charge ◽  
Minimum Number ◽  
Additional Charges ◽  
The Given

Terminating a leasing contract early may entail the payment of additional charges attributable to penalty and late payment costs. The occurrence of these extra charges push the lease effective Annual Percentage Rate (APR) up. The aim of this note is to discuss the contract early termination extra charge conditions which guarantee the no exceedance of a given APR threshold, whenever the contract expires. In the event the contract provides the lessee the option to terminate the lease prior to the first payment, extra charges shoot APR extraordinarily up and so the penalty costs should be set at zero. In the occurrence that the contract terminates upon the first payment date due to lessee’s exercise of the early termination option, the most severe compliance conditions are those if the termination occurs at the first payment date. If the early termination occurs for lessee’s insolvency, the most severe compliance condition is at correspondence of the first admissible date of the contract redemption for insolvency after a minimum number of unpaid payments. The late payment cost compliance condition requires only that the late payment interest rate not exceeds the given APR cap. Our findings hold whatever the payback amortization is set at the date of entering into the contract. If the French amortization (fixed installments) is used, the compliance constrains turn out looser than those valid for a general amortization.

scholarly journals The culture of security developed through education

2021 ◽  
Author(s):  
Ludmila Curos ◽  
Keyword(s):  
National Security ◽  
Personal Development ◽  
Information Age ◽  
State Level ◽  
Educational Institutions ◽  
Security Education ◽  
Beliefs And Values ◽  
The Individual ◽  
The Given ◽  
Strong Culture

The article argues for the need to develop a culture of security, which must be a benchmark of conduct, morals, education and training; to promote the ensemble of information, attitudes, beliefs and values through which the individual relates them to the political, military and economic system, which characterizes the level of ensuring security to each individual and to society as a whole. The need to develop a culture of security and to educate the population in this spirit derives from the diversification of threats to national security and how they are spread. Educating and promoting a strong culture of security will prevent the transformation of vulnerabilities into threats to national security. Due to the fact that the citizens are at the basis of the development of the culture of security, education in this field brings a considerable contribution in the proper promotion of the given culture at all levels of education, based on the principle of lifelong learning. The actors concerned with security education use several means of training through concrete actions, such as: meetings with civil society representatives, round tables, conferences, debates or various partnerships with academic or research institutions, scientific communication sessions, meetings with pupils and students in educational institutions of all levels, trainings with active young people, introduction of course units on the development of the culture of security, with all its components, from healthy living and personal development to work protection, civil protection, education for society, etc. Therefore, the culture of security is vital in the information age we are going through, in which we all need guidelines at both personal and state level. A citizen with a high culture of security is the natural and invaluable partner for the community he belongs to and for the nation.

scholarly journals The complete list of prime knots whose flat plumbing basket numbers are 6 or less

2015 ◽  
Vol 24 (07) ◽  
pp. 1550042
Author(s):  
Yoon-Ho Choi ◽  
Yun Ki Chung ◽  
Dongseok Kim
Keyword(s):  
Complete Classification ◽  
Classification Theorem ◽  
Complete List ◽  
Sequential Presentation ◽  
Minimum Number ◽  
The Given

Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surfaces can be presented by a sequential presentation known as flat plumbing basket code first found by Furihata, Hirasawa and Kobayashi. The minimum number of flat plumbings to obtain flat plumbing basket surfaces of a link is defined to be the flat plumbing basket number of the given link. In this paper, we use these sequential presentations to find the complete classification theorem of prime knots whose flat plumbing basket number is 6 or less. As applications, this result improves the work of Hirose and Nakashima which finds the flat plumbing basket number of prime knots up to nine crossings.

scholarly journals Methods for simulating nutritional requirement and response studies with all organisms to increase research efficiency

2017 ◽  
Vol 117 (3) ◽  
pp. 473-477
Author(s):  
Dmitry Vedenov ◽  
Rashed A. Alhotan ◽  
Runlian Wang ◽  
Gene M. Pesti
Keyword(s):  
Response Curve ◽  
Theoretical Perspective ◽  
Nutrient Level ◽  
Nutritional Requirement ◽  
Microsoft Excel ◽  
Response Models ◽  
Nutrient Levels ◽  
Limiting Nutrient ◽  
Minimum Number ◽  
The Given

AbstractNutritional requirements and responses of all organisms are estimated using various models representing the response to different dietary levels of the nutrient in question. To help nutritionists design experiments for estimating responses and requirements, we developed a simulation workbook using Microsoft Excel. The objective of the present study was to demonstrate the influence of different numbers of nutrient levels, ranges of nutrient levels and replications per nutrient level on the estimates of requirements based on common nutritional response models. The user provides estimates of the shape of the response curve, requirements and other parameters and observation to observation variation. The Excel workbook then produces 1–1000 randomly simulated responses based on the given response curve and estimates the standard errors of the requirement (and other parameters) from different models as an indication of the expected power of the experiment. Interpretations are based on the assumption that the smaller the standard error of the requirement, the more powerful the experiment. The user can see the potential effects of using one or more subjects, different nutrient levels, etc., on the expected outcome of future experiments. From a theoretical perspective, each organism should have some enzyme-catalysed reaction whose rate is limited by the availability of some limiting nutrient. The response to the limiting nutrient should therefore be similar to enzyme kinetics. In conclusion, the workbook eliminates some of the guesswork involved in designing experiments and determining the minimum number of subjects needed to achieve desired outcomes.

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