scholarly journals Induction Models on $\mathbb{N}$

10.29007/kvp3 ◽  
2020 ◽  
Author(s):  
A. Dileep ◽  
Kuldeep S. Meel ◽  
Ammar F. Sabili
Keyword(s):  
Computer Science ◽  
Generating Function ◽  
Generating Functions ◽  
Mathematical Induction ◽  
Base Case ◽  
And Mathematics ◽  
Definition Of ◽  
Induction Model

Mathematical induction is a fundamental tool in computer science and mathematics. Henkin [12] initiated the study of formalization of mathematical induction restricted to the setting when the base case B is set to singleton set containing 0 and a unary generating function S. The usage of mathematical induction often involves wider set of base cases and k−ary generating functions with different structural restrictions. While subsequent studies have shown several Induction Models to be equivalent, there does not exist precise logical characterization of reduction and equivalence among different Induction Models. In this paper, we generalize the definition of Induction Model and demonstrate existence and construction of S for given B and vice versa. We then provide a formal characterization of the reduction among different Induction Models that can allow proofs in one Induction Models to be expressed as proofs in another Induction Models. The notion of reduction allows us to capture equivalence among Induction Models.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

scholarly journals RAMANUJAN CONGRUENCES FOR A CLASS OF ETA QUOTIENTS

2010 ◽  
Vol 06 (04) ◽  
pp. 835-847 ◽  
Author(s):  
JONAH SINICK
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Generating Functions ◽  
Complete Characterization ◽  
Ramanujan Congruences

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 ( mod ℓ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences cN(n) satisfy, where cN(n) is defined by [Formula: see text]. This last result answers a question of H.-C. Chan.

scholarly journals Research of perception of digital world terminology by future teachers of Computer Science and Mathematics

2021 ◽  
Vol 27 (1) ◽  
pp. 160-167
Author(s):  
Tatyana A. Boronenko ◽  
Vera S. Fedotova
Keyword(s):  
Computer Science ◽  
Future Teachers ◽  
Lexical Meaning ◽  
New Words ◽  
Digital World ◽  
And Mathematics ◽  
Definition Of ◽  
Time Form ◽  
Terminology System

The authors of the article consider the issue of how future teachers of Computer Science and Mathematics are going to perceive the terminology of the digital world. The authors note that the terminological system “digital” arises as a result of digitalisation of all spheres of human activity. The results of the classification of digital terminology into thematic groups on the basis of decoding their lexical and semantic meaning are presented. The authors consider the issue of the most productive understanding of digital terminology based on the allocation of thematic groups "Space", "Time". “Form and Matter”, “Consumption”, “Technological Transformations”, “Human in the Digital World”. The authors review scientific works from different fields of science, they select phrases with the adjective “digital”. The authors create assignments to identify understanding of digital terminology. The authors determine special attention to the most accessible and complicated understanding of the lexical meaning of the thematic group, they compare the volumes of thematic groups. The novelty of the research lies in the definition of the lexical and semantic group of digital terminology. Topicality of the research lies in the need to expand the vocabulary of a teacher of Computer Science and Mathematics of digital terminology, to build a digital terminology system. The authors prove that the most capacious thematic group is the group “People in the Digital World”. The authors conclude that an assessment of the current level of mastering digital terminology by a Computer Science and Mathematics teacher is to reveal problems in the lexical interpretation of new words in schoolchildren.

scholarly journals Hereditary Semiorders and Enumeration of Semiorders by Dimension

10.37236/8140 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Mitchel T. Keller ◽  
Stephen J. Young
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Interval Order ◽  
Structural Description ◽  
Interval Orders ◽  
Structural Result ◽  
Forbidden Subposets ◽  
Open Question ◽  
Ascent Sequences

In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2}$ and $\mathbf{1}+\mathbf{3}$. The number of unlabeled semiorders on $n$ points has long been known to be the $n$th Catalan number. However, describing the ascent sequences that correspond to the semiorders under the bijection of Bousquet-Mélou et al. has proved difficult. In this paper, we discuss a major part of the difficulty in this area: the ascent sequence corresponding to a semiorder may have an initial subsequence that corresponds to an interval order that is not a semiorder. We define the hereditary semiorders to be those corresponding to an ascent sequence for which every initial subsequence also corresponds to a semiorder. We provide a structural result that characterizes the hereditary semiorders and use this characterization to determine the ordinary generating function for hereditary semiorders. We also use our characterization of hereditary semiorders and the characterization of semiorders of dimension $3$ given by Rabinovitch to provide a structural description of the semiorders of dimension at most $2$. From this description, we are able to determine the ordinary generating function for the semiorders of dimension at most $2$.

ABOUT THE PROBLEM OF CLASSIFICATION OF THE CONCEPTS OF INFORMATICS STUDIED AT SECONDARY SCHOOL

2018 ◽  
pp. 4-7
Author(s):  
S. I. Zenko
Keyword(s):  
Secondary School ◽  
Foreign Language ◽  
Computer Science ◽  
Parts Of Speech ◽  
The Third ◽  
The Cross ◽  
Definition Of

The article raises the problem of classification of the concepts of computer science and informatics studied at secondary school. The efficiency of creation of techniques of training of pupils in these concepts depends on its solution. The author proposes to consider classifications of the concepts of school informatics from four positions: on the cross-subject basis, the content lines of the educational subject "Informatics", the logical and structural interrelations and interactions of the studied concepts, the etymology of foreign-language and translated words in the definition of the concepts of informatics. As a result of the first classification general and special concepts are allocated; the second classification — inter-content and intra-content concepts; the third classification — stable (steady), expanding, key and auxiliary concepts; the fourth classification — concepts-nouns, conceptsverbs, concepts-adjectives and concepts — combinations of parts of speech.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

Creating and Capturing Value through Crowdsourcing

2018 ◽  
Keyword(s):  
Information Systems ◽  
Systems Biology ◽  
Computer Science ◽  
Open Innovation ◽  
Product Innovation ◽  
Collective Intelligence ◽  
Innovation Management ◽  
Private And Public ◽  
Definition Of ◽  
The Uk

Examples of the value that can be created and captured through crowdsourcing go back to at least 1714, when the UK used crowdsourcing to solve the Longitude Problem, obtaining a solution that would enable the UK to become the dominant maritime force of its time. Today, Wikipedia uses crowds to provide entries for the world’s largest and free encyclopedia. Partly fueled by the value that can be created and captured through crowdsourcing, interest in researching the phenomenon has been remarkable. For example, the Best Paper Awards in 2012 for a record-setting three journals—the Academy of Management Review, Journal of Product Innovation Management, and Academy of Management Perspectives—were about crowdsourcing. In spite of the interest in crowdsourcing—or perhaps because of it—research on the phenomenon has been conducted in different research silos within the fields of management (from strategy to finance to operations to information systems), biology, communications, computer science, economics, political science, among others. In these silos, crowdsourcing takes names such as broadcast search, innovation tournaments, crowdfunding, community innovation, distributed innovation, collective intelligence, open source, crowdpower, and even open innovation. The book aims to assemble papers from as many of these silos as possible since the ultimate potential of crowdsourcing research is likely to be attained only by bridging them. The papers provide a systematic overview of the research on crowdsourcing from different fields based on a more encompassing definition of the concept, its difference for innovation, and its value for both the private and public sectors.

scholarly journals A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1161
Author(s):  
Hari Mohan Srivastava ◽  
Sama Arjika
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Additional Parameter ◽  
Physical Sciences ◽  
General Family ◽  
Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

scholarly journals Experimental Evaluation of Shear Behavior of Stone Masonry Wall

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2313
Author(s):  
Maria Luisa Beconcini ◽  
Pietro Croce ◽  
Paolo Formichi ◽  
Filippo Landi ◽  
Benedetta Puccini
Keyword(s):  
Shear Behavior ◽  
Stone Masonry ◽  
Masonry Walls ◽  
Mechanical Parameters ◽  
In Situ Tests ◽  
Capacity Curves ◽  
Historical Structures ◽  
Definition Of

The evaluation of the shear behavior of masonry walls is a first fundamental step for the assessment of existing masonry structures in seismic zones. However, due to the complexity of modelling experimental behavior and the wide variety of masonry types characterizing historical structures, the definition of masonry’s mechanical behavior is still a critical issue. Since the possibility to perform in situ tests is very limited and often conflicting with the needs of preservation, the characterization of shear masonry behavior is generally based on reference values of mechanical properties provided in modern structural codes for recurrent masonry categories. In the paper, a combined test procedure for the experimental characterization of masonry mechanical parameters and the assessment of the shear behavior of masonry walls is presented together with the experimental results obtained on three stone masonry walls. The procedure consists of a combination of three different in situ tests to be performed on the investigated wall. First, a single flat jack test is executed to derive the normal compressive stress acting on the wall. Then a double flat jack test is carried out to estimate the elastic modulus. Finally, the proposed shear test is performed to derive the capacity curve and to estimate the shear modulus and the shear strength. The first results obtained in the experimental campaign carried out by the authors confirm the capability of the proposed methodology to assess the masonry mechanical parameters, reducing the uncertainty affecting the definition of capacity curves of walls and consequently the evaluation of seismic vulnerability of the investigated buildings.

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