Inductive Step Research Articles

DEVELOPMENT OF A MODEL FOR IMPLEMENTING A CASE METHOD FOR INTERACTIVE STUDY PROCESS MANAGEMENT MONITORING

The importance of this research topic lies in the need to gather, store, analyze and disseminate accurate information on the management status of educational institutions to guarantee the provision of quality educational services. This is particularly crucial in the current trend towards digitalizing educational work and utilizing the internet to facilitate practical management tasks. The study aims to construct a case method model to supervise educational process management outcomes. For that purpose, the study scrutinises the implementation methodologies of existing case models to monitor and analyse the situation without human intervention. Also, the investigation entails creating an implementation algorithm and a simulation model of an interactive case method to monitor educational activities. The formalized logical-semantic apparatus is used to address the research problems. The algorithm development and simulation modeling enabled correlation of the obtained answer with the assigned task, automating the results output. Generating answers in the case method involves considering rules based on the base word, answer keywords, and constraints (true/false). These elements are part of the thesaurus specific to the survey domain and are included in the test's base vocabulary. To test an answer against a question, each inductive step is viewed through a logical formula. Logical statements such as conjunction, disjunction, logical negation, implication, and equivalence are represented by formulas. This method enables the evaluation of the respondent's answer based on the nodes within the statement tree integrated into the test. The progression from the initial word to the node-connection generates an automated assessment of individual educational process management standards. This study enables the enhancement of automated monitoring capabilities for the University's educational process results. The research suggests the potential for developing models and algorithms to form question sets and enable individualized assessments based on the respondent's performance. As a consequence, the devised algorithm and simulation model for the interactive case approach are showcased. During the testing, the key words and example words were compared with the responses of the participants and specific results were obtained. Additional case vocabulary terms need to be added to address the limitations encountered during the model testing and the testing period.

Milliken’s Tree Theorem and Its Applications: A Computability-Theoretic Perspective

Milliken’s tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey’s theorem and its many variants and consequences. In this sense, Milliken’s tree theorem is paradigmatic of structural Ramsey theory, which seeks to identify the common combinatorial and logical features of partition results in general. Its investigation in this area has consequently been extensive. Motivated by a question of Dobrinen, we initiate the study of Milliken’s tree theorem from the point of view of computability theory. The goal is to understand how close it is to being algorithmically solvable, and how computationally complex are the constructions needed to prove it. This kind of examination enjoys a long and rich history, and continues to be a highly active endeavor. Applied to combinatorial principles, particularly Ramsey’s theorem, it constitutes one of the most fruitful research programs in computability theory as a whole. The challenge to studying Milliken’s tree theorem using this framework is its unusually intricate proof, and more specifically, the proof of the Halpern-Laüchli theorem, which is a key ingredient. Our advance here stems from a careful analysis of the Halpern-Laüchli theorem which shows that it can be carried out effectively (i.e., that it is computably true). We use this as the basis of a new inductive proof of Milliken’s tree theorem that permits us to gauge its effectivity in turn. The key combinatorial tool we develop for the inductive step is a fast-growing computable function that can be used to obtain a finitary, or localized, version of Milliken’s tree theorem. This enables us to build solutions to the full Milliken’s tree theorem using effective forcing. The principal result of this is a full classification of the computable content of Milliken’s tree theorem in terms of the jump hierarchy, stratified by the size of instance. As usual, this also translates into the parlance of reverse mathematics, yielding a complete understanding of the fragment of second-order arithmetic required to prove Milliken’s tree theorem. We apply our analysis also to several well-known applications of Milliken’s tree theorem, namely Devlin’s theorem, a partition theorem for Rado graphs, and a generalized version of the so-called tree theorem of Chubb, Hirst, and McNicholl. These are all certain kinds of extensions of Ramsey’s theorem for different structures, namely the rational numbers, the Rado graph, and perfect binary trees, respectively. We obtain a number of new results about how these principles relate to Milliken’s tree theorem and to each other, in terms of both their computability-theoretic and combinatorial aspects. In particular, we establish new structural Ramsey-theoretic properties of the Rado graph theorem and the generalized Chubb-Hirst-McNicholl tree theorem using Zucker’s notion of big Ramsey structure.

Inductive Reasoning Involving Social Kinds

Abstract Most social policies cannot be defended without making inductive inferences. For example, consider certain arguments for racial profiling and affirmative action, respectively. They begin with statistics about crime or socioeconomic indicators. Next, there is an inductive step in which the statistic is projected from the past to the future. Finally, there is a normative step in which a policy is proposed as a response in the service of some goal—for example, to reduce crime or to correct socioeconomic imbalances. In comparison to the normative step, the inductive step of a policy defense may seem trivial. We argue that this is not so. Satisfying the demands of the inductive step is difficult, and doing so has important but underappreciated implications for the normative step. In this paper, we provide an account of induction in social contexts and explore its implications for policy. Our account helps to explain which normative principles we ought to accept, and as a result it can explain why it is acceptable to make inferences involving race in some contexts (e.g., in defense of affirmative action) but not in others (e.g., in defense of racial profiling).

Erratum to “Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants”

This erratum corrects the proof of the main result [1, Thm. 5.2] of [1, Sec. 5]. While this result is correct as stated, its proof is flawed. We adopt the notation of [1, Sec. 5]. The total order |$\prec $| is not a term order, so that [1, Lem. 5.3] loses meaning. In particular, [1, Lem. 5.1] is false because the depth |$d(\sigma )$| is not multiplicative. For |$n = 6$|⁠, the elements |$u, v \in \wedge \{ \Theta _{6}, \Xi _{6}\}$| given by |$u=\xi _{1} \theta _{3}\xi _{3} \theta _{4} \theta _{5} \xi _{5}$| and |$v=\theta _{1} \theta _{2}\xi _{2} \theta _{4} \theta _{6} \xi _{6}$| have the following lattice path representations as in [1, Sec. 5]: We correct the proof of [1, Thm. 5.2] as follows. We shall calculate a Gröbner basis for the ideal |$I_{n} = \langle \delta _{n}\rangle \subset \wedge \{\Theta _{n}, \Xi _{n} \}$| where |$\delta _{n}= \sum _{i=1}^{n} \theta _{i} \xi _{i}$| with respect to the lexicographical term order |$<_{\textrm{lex}}$|⁠. For each Motzkin path |$\sigma $| as in [1, Sec. 5], we define |$j(\sigma )$| to be the |$x$|-coordinate where the depth |$d(\sigma )$| is achieved the 1st time. We have |$d(\sigma )=0$| if and only if |$j(\sigma )=0$|⁠. If |$u, v$| are the Motzkin paths (or monomials) in Example 1, then |$j(u)=5$| and |$j(v)=2$|⁠. Given a Motzkin path |$\sigma =(s_{1}, s_{2},\dots , s_{n})$| and |$i \leq n$|⁠, we define |$k_{i}$| to be the difference between the |$y$|-coordinate of the starting point of |$s_{i}$| and |$d(\sigma )$|⁠. For example, |$k_{1}=-d(\sigma )$| and |$k_{j(\sigma )+1}=0$|⁠. For |$i\leq j(\sigma )$|⁠, we introduce the exterior algebra elements ... ... The initial ideal |$\operatorname{in}_{lex}(\delta _{n}^{k})$| with respect the lexicographical term order contains all monomials |$\sigma $| with depth |$d(\sigma )\leq -k$|⁠. We claim that the leading monomial of |$p(\sigma )\delta _{n}^{-d(\sigma )}$| divides the monomial |$\textrm{wt}(\sigma )$|⁠, and we prove this statement for all |$n$| by induction on |$j(\sigma )$|⁠. The base case |$j(\sigma )=0$| is trivial because all monomials belong to the ideal generated by |$\delta _{n}^{0}=1$|⁠. For the inductive step, we remove the 1st step |$s_{1}$| from |$\sigma $| to get a new path |$\tau =(s_{2}, \dots , s_{n})$| involving only the variables |$\theta _{2}, \dots , \theta _{n}, \xi _{2}, \dots , \xi _{n}$|⁠. Notice that |$p(\sigma )=p_{1}(\sigma )p(\tau )$|⁠. We divide proof in three cases according to the 1st step |$s_{1}$|⁠. Case 1:|$s_{1} = (1,1)$|is an up step. We have |$d(\tau )=d(\sigma )-1$|⁠, |$\textrm{wt}(\sigma )=\textrm{wt}(\tau )$|⁠, and ... Case 2:|$s_{1} = (1,0)$|is a horizontal step. We assume that the horizontal step |$s_{1}$| is labelled with |$\theta $|⁠; the other case is identical. We have |$d(\tau )=d(\sigma )$|⁠, |$\textrm{wt}(\sigma )=\theta _{1} \textrm{wt}(\tau )$|⁠, and ... ... Case 3:|$s_{1} = (1,-1)$|is a down step. We have |$d(\tau )=d(\sigma )+1$|⁠, |$\textrm{wt}(\sigma )=\theta _{1}\xi _{1} \textrm{wt}(\tau )$|⁠, |$p(\sigma )=p(\tau )$|⁠, and ... We conclude that |$\textrm{wt}(\sigma ) \in \textrm{in}_{\textrm{lex}}(\delta _{n}^{-d(\sigma )}) \subseteq \textrm{in}_{\textrm{lex}}(\delta _{n}^{k})$| for all |$k \leq -d(\sigma )$| and the proof is complete. The above theorem substitutes [1, Lem. 5.3]. The 2nd part of the proof of [1, Thm. 5.2] is correct and can be left unchanged. The set |$\{p(\sigma )\delta _{n} \mid \sigma \textnormal{ s.t.} d(\sigma )=-1\}$| is a Gröbner basis for the ideal |$I_{n}= \langle \delta _{n} \rangle $| with respect to the lexicographical term order.

PROFIL KEMAMPUAN LITERASI DIGITAL SISWA SEKOLAH DASAR DALAM PENGGUNAAN MEDIA PEMBELAJARAN JARAK JAUH

The COVID-19 pandemic has changed the online learning process from offline to online. One of the references for measuring digital literacy levels is to directly observe the learning process that occurs using platforms that support distance learning. The purpose of this study was to analyze the digital literacy skills of elementary school students in the use of distance learning media and analyze the understanding of digital media of elementary school students in the use of distance learning media. The method used is qualitative with a phenomenological design. Data collection techniques using interview techniques, field observations, and documentation. The data validity technique uses source and technique triangulation. At the same time, the data analysis technique uses an inductive step. The results showed that the students of SD N 1 Brajan had a good level of digital literacy; students knew the benefits of digital media used as learning media, such as understanding the help of the media features used for practical benefits. Meanwhile, in applying digital media, students can also use it as a learning medium. This is because students are used to using these media in everyday life. In contrast, some students have difficulty applying digital media because they have the reason that the cellphones they use are still alternating with their parents, so the infrequent use of digital media will affect students' digital literacy skills. Teachers can make efforts to improve students' digital skills by providing learning media that can provide additional learning in the use of learning media that are foreign to students.

Making Sense of Normalization

Abstract This article examines current practices of normalization of names in Norse philology and computational linguistics that to a large extent build on deductive reasoning and external authoritative sources such as grammars, dictionaries and gazetteers. Instead, a survey of manuscript evidence and quantification of name forms at several levels of abstraction is proposed as an alternative inductive principle of normalization. A case study of name-form distributions in a dataset of 6,633 spatial attestations in East Norse literature from the Norse World resource serves as a point of departure for a discussion of the advantages and disadvantages of the approach. The comparison between attestations linked to the five most frequent place-names in Old Swedish and Old Danish shows the existence of typical spellings. However, there are still examples of norm negotiations and competitive distributions. Thus, the first inductive step of normalization can be complemented by further processing based on correspondences between phonology and spelling. Finally, stratified normalization of place-names pioneered by Norse World is seen as more versatile compared to traditional methods; the approach has a potential to facilitate both more nuanced philological and linguistic research as well as the further development of named-entity recognition tools.

Visualizing the sum of squares induction proof

We provide a visual proof for the inductive step of the sum of squares formula proof.

Full-program induction: verifying array programs sans loop invariants

Arrays are commonly used in a variety of software to store and process data in loops. Automatically proving safety properties of such programs that manipulate arrays is challenging. We present a novel verification technique, called full-program induction, for proving (a sub-class of) quantified as well as quantifier-free properties of programs manipulating arrays of parametric size N. Instead of inducting over individual loops, our technique inducts over the entire program (possibly containing multiple loops) directly via the program parameter N. The technique performs non-trivial transformations of the given program and pre-conditions during the inductive step. The transformations assist in effectively reducing the assertion checking problem by transforming a program with multiple loops to a program which has fewer and simpler loops or is loop free. Significantly, full-program induction does not require generation or use of loop-specific invariants. To assess the efficacy of our technique, we have developed a prototype tool called Vajra. We demonstrate the performance of Vajra vis-a-vis several state-of-the-art tools on a large set of array manipulating benchmarks from the international software verification competition (SV-COMP) and on several programs inspired by algebraic functions that perform polynomial computations.

A polynomial‐time algorithm for simple undirected graph isomorphism

A polynomial‐time algorithm for simple undirected graph isomorphism

Contractible Edges and Peripheral Cycles in 3-Connected Graphs

Peripheral cycles (induced non-separating cycles) in a general 3-connected graph are analogous to the faces of a polyhedron. Using the works of various authors, this paper explores the distribution of contractible edges in 3-connected graphs as needed to prove a major result originally by Tutte: each edge in a 3-connected graph is part of at least 2 peripheral cycles that share only the edge and its end vertices. A complete, alternative proof of this theorem is provided. The inductive step is generalized into a new independent lemma, which states that each edge in a 3-connected graph with a non-adjacent contractible edge has at least as many peripheral cycles as in the contracted one.

Using the Maple computer algebra system to study mathematical induction

Mathematical induction is one of the main methods of the proof that students study and use it in their studies in higher education. This important method, in addition to mathematics, is widely used in computer science and a number of other related disciplines. However, even if the principles of proof using the method of induction are taught, understood and mastered by students well, many students have great difficulty with algebraic transformations necessary for proving, for example, finding the inductive step. The use of the Maple computer algebra system allows students to overcome this obstacle and, more attention and effort to pay to understanding the concept of proofs using the method of mathematical induction. In addition, students can prove more complex algebraic statements, and their activities can be of a pronounced research nature.

Aubert duals of discrete series: the first inductive step

Aubert duals of discrete series: the first inductive step

Introduction to Discrete Mathematics and Fourier Analysis. A Practical Guide for third Year Undergraduate Students.

This book covers a quick introduction to high school mathematics, arithmetic and geometric series, infinite geometric series, mathematical induction, binomial theorem, factorials, permutations and combinations, determinants, systems of linear equations, boolean algebra, orthogonal functions, Legendre, Hermite, and Laguerre polynomials and orthogonal properties. For example, mathematical induction is a proof based on logical assumptions of equations that involve integers of one number that shows the proof for the next number. It is based on the base step and the inductive step. The base step is to prove the integers based on a formula. The inductive step involves statement of one integer number that shows the proof of the statement for the next integer. The determinant is a scalar and it is found by multiplying the two elements of the principal diagonal and subtracting them from the multiplication of the elements of the other diagonal.

Explicit asymptotic expansions for tame supercuspidal characters

We combine the ideas of a Harish-Chandra–Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim–Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim–Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. The only additional information needed in the inductive step is a fourth root of unity, which we expect to be useful in proving stability and endoscopic-transfer identities.

What the argument from evil should, but cannot, be

AbstractMichael Tooley has developed a sophisticated evidential version of the argument from evil that aims to circumvent sceptical theist responses. Evidential arguments from evil depend on the plausibility of inductive inferences from premises about our inability to see morally sufficient reasons for God to permit evils to conclusions about there being no morally sufficient reasons for God to permit evils. Tooley's defence of this inductive step depends on the idea that the existence of unknown rightmaking properties is no more likely, a priori, than the existence of unknown wrongmaking properties. I argue that Tooley's argument begs the question against the theist, and, in doing so, commits an analogue of the base rate fallacy. I conclude with some reflections on what a successful argument from evil would have to establish.

102.22 Proof without Words: sums of sums of factorials is one less than a factorial: inductive step

102.22 Proof without Words: sums of sums of factorials is one less than a factorial: inductive step

Han's conjecture and Hochschild homology for null-square projective algebras

Let $\mathcal H$ be the class of algebras verifying Han's conjecture. In this paper we analyse two types of algebras with the aim of providing an inductive step towards the proof of this conjecture. Firstly we show that if an algebra $\Lambda$ is triangular with respect to a system of non necessarily primitive idempotents, and if the algebras at the idempotents belong to $\mathcal H$, then $\Lambda$ is in $\mathcal H$. Secondly we consider a $2\times 2$ matrix algebra, with two algebras on the diagonal, two projective bimodules in the corners, and zero corner products. They are not triangular with respect to the system of the two diagonal idempotents. However, the analogous result holds, namely if both algebras on the diagonal belong to $\mathcal H$, then the algebra itself is in $\mathcal H$.

On birational boundedness of Fano fibrations

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of $\epsilon$-Fano type. Furthermore, we show birational boundedness of $3$-folds of $\epsilon$-Fano type.

Are Model Organisms Theoretical Models?

Abstract This article compares the epistemic roles of theoretical models and model organisms in science, and specifically the role of non-human animal models in biomedicine. Much of the previous literature on this topic shares an assumption that animal models and theoretical models have a broadly similar epistemic role—that of indirect representation of a target through the study of a surrogate system. Recently, Levy and Currie (2015) have argued that model organism research and theoretical modelling differ in the justification of model-to-target inferences, such that a unified account based on the widely accepted idea of modelling as indirect representation does not similarly apply to both. I defend a similar conclusion, but argue that the distinction between animal models and theoretical models does not always track a difference in the justification of model-to-target inferences. Case studies of the use of animal models in biomedicine are presented to illustrate this. However, Levy and Currie’s point can be argued for in a different way. I argue for the following distinction. Model organisms (and other concrete models) function as surrogate sources of evidence, from which results are transferred to their targets by empirical extrapolation. By contrast, theoretical modelling does not involve such an inductive step. Rather, theoretical models are used for drawing conclusions from what is already known or assumed about the target system. Codifying assumptions about the causal structure of the target in external representational media (e.g. equations, graphs) allows one to apply explicit inferential rules to reach conclusions that could not be reached with unaided cognition alone (cf. Kuorikoski and Ylikoski 2015).

Limits of quotients of polynomial functions in three variables

A method for computing limits of quotients of real analytic functions in two variables was developed in [4]. In this article we generalize the results obtained in that paper to the case of quotients q = f ( x, y, z )/ g ( x, y, z ) of polynomial functions in three variables with rational coefficients. The main idea consists in examining the behavior of the function q along certain real variety X ( q ) (the discriminant variety associated to q ). The original problem is then solved by reducing to the case of functions of two variables. The inductive step is provided by the key fact that any algebraic curve is birationally equivalent to a plane curve. Our main result is summarized in Theorem 2. In Section 4 we describe an effective method for computing such limits. We provide a high level description of an algorithm that generalizes the one developed in [4], now available in Maple as the &lt;code&gt;limit/multi&lt;/code&gt; command.