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For a graph G = (V , E) , the exponential second Zagreb index is defined as e M 2 (G) = ∑ u v ∈ E e d (u) d (v) , where d (v) is the degree of vertex v ∈ V. Recently, Eliasi (2022) posed a conjecture about (n , m) -graphs with maximum e M 2 when n ≤ m ≤ 2 n − 3. In this paper, we show that this conjecture is true. Furthermore, we determine the graphs with maximum e M 2 among all (n , m) -graphs when 2 n − 3 < m. [ABSTRACT FROM AUTHOR]

In recent years, archaeological research has relied on large datasets, both temporally and geographically, with some archaeologists discussing that there should be a shift towards a more scientific form of conducting archaeological research called "macroarchaeology." Ironically, and contradictorily, this shift towards large-scale research has involved the use of inductive approaches, which means that archaeological material needs to be converted into universal quantitative values. The inductive approaches used by archaeologists today, as argued by Karl Popper, and other authors in recent years, cannot be considered scientific in the strict sense of the word, since there is always a degree of uncertainty in inductive reasoning. This study suggests that archaeological data can be considered as traces of the past, clues that allow us to reconstruct past phenomena. As Carlo Ginzburg's evidential paradigm demonstrates, thinking of the past in terms of traces and clues is much more scientific than appears at first. In addition to traces and clues, a second interpretative procedure can be conducted on data. Based on Ginzburg's conjectural paradigm and discussion on fiction, we can recognize the past as real, while at the same time, conjecture the several ways past agents could have acted otherwise. [ABSTRACT FROM AUTHOR]

Conjectures are made on the text of three passages in Tibullus, Books 1–2: 1.4.26 hastam ... suam for crines ... suos , 2.1.56 membra for bache, 2.4.60 aliis rebus for alias herbas. [ABSTRACT FROM AUTHOR]

• We investigated experimentally the discovery of feedback-learning procedures. • In a dynamic business simulation, participants were incentivized to maximize profits. • Webackward-engineered procedures from recorded actions and information requests. • Anumber of participants discovered an effective heuristic learning procedure. • Implementation errors prevented many participants from realizing its full potential. Domain-general learning rules often enable decision makers to learn from outcome feedback which actions tend to achieve a desired goal. However, in novel and complex environments decision makers must explore how to learn, i.e., acquire procedural knowledge of how to elicit and evaluate outcome feedback that will enable them to navigate toward a desired goal despite the vastness of the set of possible policies. Using a dynamic business simulation, this study investigated: (1) whether and how frequently participants discovered an effective procedure to learn from outcome feedback that allowed them to navigate toward a policy that maximizes long-term business profit (and hence their monetary payoff from the experiment), and (2) whether high monetary incentives affected learning procedures and performance. We found that a number of participants discovered an effective learning procedure and succeeded in approximating the optimal policy. In line with the heuristic method, this learning procedure involved a simplification of the search space and the application of domain-general learning rules to this simplified space. Although the decision histories of about half of the participants feature the key aspect of the effective learning procedure—search among the different steady states of the dynamical system—implementation errors prevented many of the participants from realizing the full potential of the learning procedure. We found no evidence to suggest that high monetary incentives affect the effectiveness of learning. Overall, the study illustrates that a "prepared mind" can discover new, effective learning procedures, although their initial implementation may require substantial refinement. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Udo Riese about the minimal ring of definition for principal series Weil characters of SL 2 (F p) , for p an odd prime. More precisely, we show that the (p + 1) / 2 -dimensional Weil characters can be realized over the ring of integers of Q (ε p) , where ε = (− 1) (p − 1) / 2 , and we provide explicit integral models over these quadratic rings. We do so by studying the Galois action on the integral models of Weil characters recently discovered by Yilong Wang. [ABSTRACT FROM AUTHOR]

Largely disparaged by readers since the eighteenth century but revered before then as a guide to living and dying well, the Tusculan Disputations has throughout its long reception been acknowledged as the most rhetorical of Cicero's dialogues. This essay takes as its point of departure not only this acknowledgment but the principal interlocutor's key comparison between finding the appropriate status or "issue" in a legal case and selecting a circumstantially sensitive strategy when offering consolation for the loss of a loved one. It argues, with the help of Cicero's rhetorical works, that he deploys rhetorical status , with its three questions (conjectural, definitive, and qualitative), to structure the conversation, thereby redressing the perceived failure of Plato's Phaedo to provide adequate guidance for navigating life's vexations, including pain, depression, anxiety, and death. [ABSTRACT FROM AUTHOR]

We report on an innovative design of algorithmic analysis that supports automatic online assessment of students' exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic activity format for if-then propositions and implemented the activity on a platform that collects and analyzes students' work. Finally, we searched for ways to use variation theory to analyze ninth-grade students' recorded work. We scored and classified data and found correlation between patterns in exploration stages and the conjectures students generated. We demonstrate how automatic identification of mistakes in the early stages is later reflected in the quality of conjectures. [ABSTRACT FROM AUTHOR]

In this article we consider an even entire function of order one as the solution of the classical wave equation in one-dimensional space. We suggest a conjecture that this function has only purely real zeros in the entire complex plane. This is given as a new prospective on a nice connection with number theory and wave equation. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present an interpretation of the novel One Hundred Years of Solitude, by Gabriel García Márquez, based on some of the varied readings generated about it by important Iberoamerican analysts. To achieve this goal, we first took the principles of Hermeneutic Pluralism as a guide which focuses on the abductive work with latent indicial elements in the texts as the key; after that, we abbreviated three readings of the garciamarquiana endeavor. Finally, let us interweave three more readings of the novel, showing the eternal return of violence, which unfolds from the beginning to continue shaping the collective future. [ABSTRACT FROM AUTHOR]

Fuzzy topological topographic mapping ( F T T M ) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. The key to the model is its topological structure that can accommodate electrical or magnetic recorded brain signal. A sequence of FTTM, F T T M n , is an extension of FTTM whereby its form can be arranged in a symmetrical form, i.e., polygon. The special characteristic of F T T M , namely, the homeomorphisms between its components, allows the generation of new F T T M . The generated F T T M s can be represented as pseudo graphs. A pseudo-graph consists of vertices that signify the generated F T T M and edges that connect their incidence components. A graph of pseudo degree zero, G 0 (F T T M n k  ) , however, is a special type of graph where each of the F T T M components differs from its adjacent. A researcher posted a conjecture on G 0 3 (F T T M n 3) in 2014, and it was finally proven in 2021 by researchers who used their novel grid-based method. In this paper, the extended G 0 3 (F T T M n 3) , namely, the conjecture on G 0 4 (F T T M n 4) that was posed in 2018, is narrated and proven using simple mathematical induction. [ABSTRACT FROM AUTHOR]

We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection. In particular, we obtain recurrence relations for them. We also make several conjectures concerning the behaviour of these coefficients, provide numerical evidence supporting these conjectures, and study them theoretically in particular cases. As an auxiliary result of independent interest, we investigate the covariance function of fractional Gaussian noise, prove that it is completely monotone for H > 1 / 2 , and, in particular, monotone, convex, log-convex along with further useful properties. [ABSTRACT FROM AUTHOR]

This paper has two aims: (1) it develops and defends a fully-fledged account of the epistemic normativity of conjecture (2) it goes sharply against orthodoxy, in arguing that conjecture is epistemically more demanding than assertion. According to the view defended here, one's conjecture that p is permissible only if one knows that one has warrant, but not sufficient warrant to believe that p. I argue for my account on three independent grounds: (1) the Bach and Harnish account of the nature of communicative speech acts, (2) the plausible normative relation between assertion and other constatives, and (3) the normativity of belief in conjunction with constatives' epistemic function. [ABSTRACT FROM AUTHOR]

This article presents a new conjecture on Herodotus 1.51.3. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n vertices, m edges having Laplacian eigenvalues μ 1 , μ 2 , ... , μ n − 1 , μ n = 0. The Laplacian energy LE (G) is defined as LE (G) = ∑ i = 1 n | μ i − d ¯ | , where d ¯ = 2 m n is the average degree of G. Radenković and Gutman conjectured that among all trees of order n , the path graph P n has the smallest Laplacian energy. Let n (d) be the family of trees of order n having diameter d. In this paper, we show that Laplacian energy of any tree T ∈ n (4) is greater than the Laplacian energy of P n , thereby proving the conjecture for all trees of diameter 4. We also show the truth of conjecture for all trees with number of non-pendent vertices at most 9 n 2 5 − 2. Further, we give some sufficient conditions for the conjecture to hold for a tree of order n. [ABSTRACT FROM AUTHOR]

Lambert's choice of genre for his Cosmological Letters – the scholarly correspondence – was conscious and well-founded, as it facilitated a certain form of reflection and presentation. Such letters enabled him to integrate more complex reasoning into the presentation of a specific problem than the scholarly dialogues that had been common since Fontenelle. Lambert needed to integrate sophisticated demonstrations from the theories of probability and teleology, as empirical knowledge was insufficient or unproven. With regard to his subject (the order of the comets, satellites, planets and galaxies of the universe), theonomistic and probabilistic cosmology could be convincingly communicated in the form of letters. [ABSTRACT FROM AUTHOR]

This paper concerns the internal structure of reasoning that, basically consisting in conjecturing and refuting, is too often identified with only deducing, abducing and refuting, that is, with just the deductive search of consequences, hypotheses and refutations. With such identification, it is forgotten that in addition to consequences and hypotheses, there is a third class of conjectures, speculations or proper guesses. Speculations are inferentially non-comparable, or orthogonal, with the premise, and generate creativity. It is presented a very simple formal view for the structure of Commonsense Reasoning, a mathematical model allowing to show the importance of speculations, and specially to start with its systematic computational search. [ABSTRACT FROM AUTHOR]

There is a prevailing opinion that conjectural knowledge (zann) cannot be taken as a basis in determining the fundamental theological principles among the theologians. However, from which sources and how to obtain certainty (yaqīn) and which types of knowledge are definitive (qat'ī) have been discussed extensively. Certain and conjectural knowledge meet at a common point in terms of relying on evidence. Conjectural knowledge obtained via reasoning and/or religious scripture that do not express certainty. While conjectural knowledge has been essentially related to the concepts of estimation, doubt, belief and knowledge, by its in term of value, it is stands closer to the concepts of knowledge and belief. Conjectural knowledge contains different degrees of certainty. When it is chosen to be more certain, it is considered as a valid conjecture, when it is disregarded, it is considered as erroneous conjecture. If there is no choice for its certainty, then it is called doubt (shakk). If the information/judgment obtained through reasoning becomes fixed, unchangeable and in a coherence that does not allow it to be otherwise, then it is called definitive (qat'ī) knowledge. To achieve that, theologians generally used methods of argumentations like syllogism, induction, exhaustive investigation and disjunction (al-sabr wa'l-taqsīm) to provide such knowledge about divinity and prophethood. Theologians have used conjectural knowledge, which are based on endoxic (mashhūrāt), already granted (musallamāt) and acceptance of authority (maqbūlāt), in their dialectics to persuade their discussants, rather than to justify the fundamental theological issues of religion. While theologians mostly use rational proofs in theological issues, they also gave significant place to the scriptural narrative that determines the creed. In this context, theologians accepted the narrated evidence as binding and decisive, as long as it is certain in terms of both authenticity and implication. Ultimately, the fundamental theological principles of religion are based on these narrative evidences. Although the theologians do not accept single reports (khabar al-wahid) for their questionable authenticity, they also made inferences regarding subjects of traditions (sam'iyyāt). Acting from this perspective, theologians tried to justify with narrative evidence rather than rational arguments on these issues such as torment in grave, intercession (shafā'ah) and sirât, which are mostly matters of eschatology and the unseen (ghayb). Finally, it has been concluded that the conjectural knowledge is still valid in some secondary theological issues and the assumption that the conjectural knowledge has no deduction value in theology is not absolute. [ABSTRACT FROM AUTHOR]

The article talks about imitation (taqlid) in religious beliefs : must the beliefs, which man adopts, be based on the basis of reasoning and inference, or that imitation is permissible in these beliefs? Is it necessary to have certainty in religious beliefs, or conjecture is sufficient? If assuming that certainty is necessary, then, what is meant by certainty? Is it the dogmatic belief that conforms to reality, or every dogmatic thing, whether it conforms to reality or not? Is there a correlation between the opinion that sees the permissibility of imitation in beliefs and the other which sees that conjecture is sufficient (concerning beliefs)? In this study, we discuss the limits of this topic whether or not it covers everything related to beliefs. And through the topics discussed in the study, it can be concluded that the essential point is the necessity of searching for religious beliefs and getting to certainty from certain evidence. However, if ordinary people put aside the getting to certainty from certain evidence --even though they are required to get to it-- due to their lack of understanding, it would be sufficient for them to be considered faithful when they feel certain of the true beliefs that they obtain through imitation (taqlid) or simple innate evidence. [ABSTRACT FROM AUTHOR]

Proximity π and remoteness ρ are respectively the minimum and the maximum, over the vertices of a connected graph, of the average distance from a vertex to all others. The distance eigenvalues of a connected graph G , denoted by ∂ 1 ≥ ∂ 2 ≥ ⋯ ≥ ∂ n , are those of its distance matrix. In this paper, we prove π + ∂ 3 > 0 for any graph with a diameter at least 3. This result leads to relations between ∂ 3 and several distance invariants of a graph such as remoteness, diameter, radius, average eccentricity and average distance. In particular, it confirms ρ + ∂ 3 > 0 for any graph with a diameter at least 3 conjectured by Aouchiche and Hansen (2016). [ABSTRACT FROM AUTHOR]

Nicholas of Cusa (1401-11 August 1464) assumes essentially the Erigenian and Alanian thesis of the incognoscibility of God and the quiddity of things. He develops and deepens the awareness of the superiority of negation about the affirmation in divinis which had enabled Medieval Neoplatonism to develop a semantic theory of language in clear break with the Aristotelian triptych which establishes the need of a correspondence between thing, thought and sign: according to Nicholas of Cusa, the essence of language is in the production of a meaning that can be entirely independent from the possibility of apprehending its object. The present research intends to show that the thesis of divine incognoscibility is intimately linked to another thesis of an epistemological order that radically decrees that all affirmative proposition is a conjecture about reality and that our mind is unable to adequately reach the truth. There is no certain human knowledge: the quiddity of things which determines their true essences is inaccessible. [ABSTRACT FROM AUTHOR]

The environment impacts human health in profound ways, yet few theories define the form of the relationship between human physiology and the environment. It is conjectured that such complex systems cannot interact directly, but rather their interaction requires the formation of an intermediary "interface." This position contrasts with current epidemiological constructs of causation, which implicitly assume that two complex systems transfer information directly while remaining separate entities. Further, it is contended that dynamic, process‐based interfaces incorporate components from all the interacting systems but exhibit operational independence. This property has many consequences, the foremost being that characteristics of the interface cannot be fully resolved by only studying the systems involved in the interaction. The interface itself must be the subject of inquiry. Without refocusing the attention on biodynamic interfaces, how the environment impacts health cannot be discerned. Also see the video abstract here https://youtu.be/XeyjeZeyo4o. [ABSTRACT FROM AUTHOR]

We show that no cubic graphs of order 26 have crossing number larger than 9, which proves a conjecture of Ed Pegg Jr and Geoffrey Exoo that the smallest cubic graphs with crossing number 11 have 28 vertices. This result is achieved by first eliminating all girth 3 graphs from consideration, and then using the recently developed QuickCross heuristic to find drawings with few crossings for each remaining graph. We provide a minimal example of a cubic graph on 28 vertices with crossing number 10, and also exhibit for the first time a cubic graph on 30 vertices with crossing number 12, which we conjecture is minimal. [ABSTRACT FROM AUTHOR]

This paper describes outcomes from a project conducted at a small US university where 23 secondary school mathematics teachers participated in the study. Based on sketches created in a dynamic geometry software (DGS) and written responses to an assignment related to developing the trapezoid area formula, we explored how teachers used DGS when visualizing and verifying geometrical conjectures. We found that teachers used a combination of four strategies—eye-balling, measurement, constructions, and written statements—in their DGS sketches. Some teachers used DGS as paint software and their DGS files did not pass the dragging test. Teachers' misconceptions became obvious while they were visualizing conjectures in the software, which were mediated by the skill for use of technology. Also, some used shortcuts that disagreed with conjectures and distorted their solutions, thus employing practices that could be damaging in teaching situations. These results and the methodological approach used in this study will be of interest to mathematics teacher educators, facilitators of professional development programs, and mathematics teachers. [ABSTRACT FROM AUTHOR]

We introduce a kind of "perturbation" for the Li-Keiper coefficients around the Koebe function (the K function) and establish a closed system of Equations for the Li-Keiper coefficients. We then check the correctness of some of the many possible solutions offered by the system, related to the discrete derivative of order n of a function. We also report a numerical finding which support our stability conjecture, i.e., that the tiny part λtiny(n) (the fluctuations around the trend) are bounded in absolute values by γ·n, where γ is the Euler-Mascheroni constant. [ABSTRACT FROM AUTHOR]

In this note, we report on an implementation of discovery-oriented problems in courses on Real Analysis and Differential Equations. We explain a type of task design that gives students the opportunity to conjecture, refute and prove. What is new is that the complexity in our problems is limited and thus the tasks can also be used in homework assignments. In addition to several concrete examples, we also discuss feedback and assessment outcomes of our students. [ABSTRACT FROM AUTHOR]

In the paper, the authors firstly present a concise proof for complete monotonicity of a function involving a difference between the logarithmic and psi functions, secondly compute completely monotonic degree of the above-mentioned function, and finally pose several conjectures on completely monotonic degrees of remainders and their derivatives for the asymptotic formula of the logarithm of the classical Euler gamma function. [ABSTRACT FROM AUTHOR]

The Alon–Tarsi conjecture states that if n is even, then the sum of the signs of the Latin squares of order n is non-zero (Alon and Tarsi, 1992). The conjecture has been proven in the cases n = p + 1 (Drisko, 1997), and n = p − 1 (Glynn, 2010), where p is an odd prime. This paper is intended to be a concise and largely self-contained account of these results, along with streamlined, and in some cases, original proofs that should be readily accessible to a mathematician with a background in combinatorics. We also discuss the relation between the Alon–Tarsi conjecture and Rota's basis conjecture (Huang and Rota, 1994), and present some related problems, such as Zappa's extension of the Alon–Tarsi conjecture (Zappa, 1997), and Drisko's proof of the extended conjecture for n = p (Drisko, 1998). [ABSTRACT FROM AUTHOR]

Highlights • Clearly and obviously express inference, conjecture, assumption, self-evidence. • Inferences and assumptions function as indirect requests for information. • Conjectures have a challenging function. • Self-evidence is a post-evidential meaning of clearly and obviously. • Clearly and obviously perform textual functions. Abstract The aim of this paper is to examine the types of evidential meaning expressed by clearly and obviously in receiver-directed talk, and to identify their interpersonal functions. It focuses on their uses in utterances containing the second person pronoun subject, excerpted from the British National Corpus. This study demonstrates that clearly and obviously are used to express inference, conjecture, assumption and self-evidence. The four notions can be organised on a cline, each showing a different degree of (inter)subjectivity, which is why this study argues that (inter)subjectification is responsible not only for their development from adverbs of manner into evidential adverbs (which is generally agreed on), but also for the different types of evidential meaning they express. It also argues that self-evidence is a post-evidential meaning of the two adverbs. This paper demonstrates that inferences, conjectures and assumptions function as strategies for eliciting a response from the addressee. Inferences and assumptions serve as prompts; conjectures have a challenging function. Inferences are also used to provide justification for the speech act performed by the speaker, while assumptions prepare the ground for the speaker's talk. Self-evidence is connected with the interpersonal functionality of concurrence and with textual functions performed by the two adverbs. [ABSTRACT FROM AUTHOR]

For y∊ℝ, a sequence x = (xn)∊ℓ, and a non-negative regular matrix A, Bartoszewicz et. al., in 2015, defined the notion of the A-density δA(y) of the indices of those xn that are close to y. Their main result states that if the set of limit points of (xn) is countable and density δA(y) exists for any y∊R where A is a non-negative regular matrix, then limn→∞(Ax)n = P y2R δA(y) Σ y: In this note we first show that the result can be extended to a more general class of matrices and then consider a conjecture which naturally arises from our investigations. [ABSTRACT FROM AUTHOR]

In the present paper, we compare the geometric–arithmetic index G A and the chromatic number χ of a connected graph with given order. We prove, among other results, an upper bound on the ratio G A ∕ χ . We also prove lower bounds on the chromatic number in terms of geometric–arithmetic index and number of vertices of a connected graph. The results obtained for the chromatic number χ are extended to the clique number ω . [ABSTRACT FROM AUTHOR]

An algebraic condition for the singularity of certain Toeplitz matrix pencils is derived which involves only the principal minors of the constant parts of the pencils. This leads to an algebraic conjecture which is equivalent to the so-called Toeplitz pencil conjecture. [ABSTRACT FROM AUTHOR]

This article explores the inscriptions and material metamorphoses of Henry VII in Horace Walpole’s ‘paper fabric’, a reversible world of writing, collecting, and book-making. InAnecdotes of Painting in England(1762), Walpole celebrates the funerary monument of Henry VII by Pietro Torrigiano at Westminster Abbey. InHistoric Doubts on the Life and Reign of King Richard the Third(1768), conjecture and speculation become methodological prompts to unveil the textual and architectural discontinuities of history. Walpole’s next historical experiment consists in placing a bust of Henry VII in the agonies of death in the Star Chamber at his house at Strawberry Hill in Twickenham. The bust’s importance is captured by its reappearance propped on top of a frontispiece and its dissemination in other reproductions in extra-illustrated copies ofA Description of the Villa of Mr. Horace Walpole, Youngest Son of Sir Robert Walpole Earl of Orford, at Strawberry-Hill near Twickenham, Middlesex …(1784). A dramatic representation of the bust in John Carter’s extra-illustrated copy ofA Description, later engraved in hisSpecimens of the Ancient Sculpture and Painting now remaining in this Kingdom(1780–94), shows the alternative trajectories of Henry VII from Westminster Abbey to Strawberry Hill, from Walpole’s cosmopolitan collection of curiosities to Carter’s paper collection of national gothic specimens. [ABSTRACT FROM AUTHOR]

An often-ignored Humean contribution to Scottish Enlightenment is 'conjectural history', an eighteenth-century historical genre that attempted to trace the origins and development of particular institutions from prehistory to modernity. But conjectural methodology prevented histories from establishing any facts. What was then its point? I propose a way to justify Hume's practice of conjectural history by appealing to his scattered comments on historical explanation. Conjectural histories explain the origin of modern institutions by offering the rationale that must have caused their emergence and stability and that justifies their current approbation, providing the reader with means for understanding and criticism. [ABSTRACT FROM AUTHOR]

Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight 1. Strongly equistable graphs are graphs such that for every [ABSTRACT FROM AUTHOR]

Based on a presentation of the production ofnouvellesin Hispanic America and its current study, in which the permanent academic and critical interest in Juan Carlos Onetti'sLos adioses(1954) stands, this article raises and discusses the procedures used by the author to achieve a narrative that by its nature ofnouvellerefuses the explicit in exchange for the suggestion. Among such procedures are the character of the narrator, the ellipsis, and a language of conjecture and doubt. With all of those procedures Onetti builds an open and multiple text in which his major emphasis during the writing process lies in the development of the meaning of the text, instead of just narrating the story, as proposed by Henry James in various prefaces to hisnouvelles. [ABSTRACT FROM AUTHOR]

How can we teach inquiry? In this paper, I offer practical techniques for teaching inquiry effectively using activities built from routine textbook exercises with minimal advanced preparation, including rephrasing exercises as questions, creating activities that inspire students to make conjectures, and asking for counterexamples to reasonable, but false, conjectures. [ABSTRACT FROM AUTHOR]

Alessandro Ferrara’s The Democratic Horizon: Hyperpluralism and the Renewal of Political Liberalism poses an important challenge to recent defenders of ‘realism’ in political theory and shows that a renewal of Rawlsian ideal theory is possible. Ferrara focuses on the contemporary condition of ‘hyperpluralism’, in which every comprehensive worldview and religion has to admit the equal validity of at least one other conception, and claims that only a ‘pluralist justification of pluralism’ can lead to a genuine revival of the democratic horizon. Naming such a project of democracy the ‘multivariate’ polity, he uses Rawls’ method of conjecture to show how such a justification strategy is possible. I argue that the multivariate polity may attain pluralism but not stability and may fail in securing democratic respect among equal citizens for each other’s point of view. [ABSTRACT FROM AUTHOR]

Let q = 1, and q be the largest value of ( k + 1)( n − q) for all integers 1 ≤ k ≤ n − 1 with n ≥ 2. The sequence Q = { q, q, q,...} is called Levine-O'Sullivan sequence. In this paper, we use the combinational and analysis skill and the mathematical induction to study the asymptotic properties of q, and give an interesting asymptotic formula for it. This solved a conjecture proposed by Professor Chen Yonggao. [ABSTRACT FROM AUTHOR]

For a function f from ω to ω , a topological space X satisfies ⋃ f (Γ , Γ) if for each sequence (U n : n ∈ ω , U n has no finite subcover) of elements of Γ, select for each n a finite subset F n ⊆ U n such that | F n | ≤ f (n) for all n and { ⋃ F n : n ∈ ω } is an element of Γ, where Γ denotes the family of all open γ -covers of X. In this paper, we prove the following results. (1) Assume the Continuum Hypothesis. There is a set of real numbers that satisfies ⋃ fin (Γ , Γ) and S 1 (Γ , O) but not ⋃ id (Γ , Γ) , where id is the identity function from ω to ω. (2) Assume b = c. There is a set of real numbers that satisfies ⋃ id (Γ , Γ) but not ⋃ k (Γ , Γ) for all natural numbers k ≥ 1. (3) Assume the Continuum Hypothesis. For each natural number k ≥ 2 , there is a set of real numbers that satisfies ⋃ k + 1 (Γ , Γ) but not ⋃ k (Γ , Γ). These results answer an open problem proposed by Zdomskyy and a conjecture proposed by Tsaban. [ABSTRACT FROM AUTHOR]

This paper treats subgroup Nash equilibriums which concept is given as an extension of Nash equilibrium of a strategic game with non-partitional information, and addresses the problem how to reach the equilibrium by communication through messages according to network among players. A subgroup Nash equilibrium of a strategic game consists of (1) a subset S of players, (2) independent mixed strategies for each member of S together with (3) the conjecture of the actions for the other players outside S provided that each member of S maximizes his/her expected payoff according to the product of all mixed strategies for S and the conjecture about other players' actions. Suppose that the players have a reflexive and transitive informationwith a common prior distribution, and that each player in a subgroup S predicts the other players' actions as the posterior of the others' actions given his/her information. He/she communicates privately his/her belief about the other players' actions through messages to the recipient in S according to the communication network in S.We show that in the pre-play communication according to the revision process of their predictions about the other players' actions, their future predictions converges to a subgroup Nash equilibrium of the game in the long run. [ABSTRACT FROM AUTHOR]

The Randić index R ( G ) of a graph G is defined by R ( G ) = ∑ u v 1 d ( u ) d ( v ) , where d ( u ) is the degree of a vertex u and the summation extends over all edges u v of G . The eccentricity ϵ ( v ) of a vertex v is the maximum distance from it to any other vertex and the average eccentricity ϵ ̄ ( G ) of graph G is the mean value of eccentricities of all vertices of G . There are relations between the Randić index and the average eccentricity of connected graphs conjectured by a computer program called AGX: for any connected graph G on n ≥ 14 vertices, both lower bounds of R ( G ) + ϵ ̄ ( G ) and R ( G ) ⋅ ϵ ̄ ( G ) are achieved only by a star. In this paper, we show that both conjectures are true. [ABSTRACT FROM AUTHOR]

Conjecture is a knot theoretical equivalent form of the Kervaire conjecture. We show that Conjecture is true for all the pretzel knots of the form where , and are odd positive integers. [ABSTRACT FROM AUTHOR]

This paper discusses the coincidence condition of two Bézier curves of the same degree, which is necessary for a curve/curve intersection algorithm. It proves that two Bézier curves of the same degree are coincident with each other if and only if they have a coincident control polygon or they can be reparameterized into the same non-reparameterizable Bézier curve of a lower degree. A simple method is also provided for detecting the reparameterizable case where a Bézier curve can be reparameterized into a non-reparameterizable one of a lower degree. [ABSTRACT FROM AUTHOR]

This article describes the process that was undertaken to find an important relationship between the Goldbach Conjecture and Dirichlet theorem. Perhaps it could be suppressed the first part of this article, but I do not know made in order to show the historical process that he ran to get the result. It is hoped that the results presented actually show the effort made to attain an test that lasted many years in achieved. [ABSTRACT FROM AUTHOR]

Communication about prediction is complex in a number of ways. First, language is by nature recursive — language is an indicator of meaning as well as a force that shapes meaning. Second, the same language used to communicate prediction in uncertain environments is used for other purposes. In this article, we describe how the recursive nature of language impacted the choices we made in a cross-sectional longitudinal study aimed at gaining insight into children’s language repertoires relating to conjecture. We then explore some Grade 6 students’ communication about prediction to develop insight into their meaning and meaning-making with prediction language. From this we raise questions about interpreting data from such contexts. Finally, we discuss implications for educators. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n ≥ 3 . The remoteness ρ = ρ ( G ) is the maximum, over all vertices, of the average distance from a vertex to all others. The radius r = r ( G ) is the minimum, over all vertices, of the eccentricity of a vertex. Aouchiche and Hansen (2011) conjectured that ρ − r ≥ 3 − n 4 if n is odd and ρ − r ≥ 2 n − n 2 4 ( n − 1 ) if n is even. In this paper, we confirm this conjecture. In addition, we completely characterize extremal graphs attaining the lower bound. [ABSTRACT FROM AUTHOR]

This article presents a classification of conjectures on the text of the New Testament. It focusses on the types of arguments used by conjectural critics. The argumentation for a conjecture basically comprises (1) the perception of a problem (or problems) in the transmitted text and (2) the suggestion of a cause (or causes) for the supposed scribal change. Type (or types) of perceived problems and of supposed causes are classified,and illustrated with a range of important conjectures. [ABSTRACT FROM AUTHOR]

As educational systems around the world attempt to reform their mathematics programs to increase students' opportunities to engage in processes central to the practice of mathematics such as proof, it is important to understand how this mathematical act is portrayed in national curriculum documents that drive that change. This study examined the presence of reasoningand- proving (RP) in Ireland's national reform-oriented secondary syllabi for junior cycle (ages 12-15) and senior cycle (ages 15-18) students. The analyses reveal that there were no differences among direct and indirect RP learning outcomes within each syllabus, but statistically significant differences did exist across syllabi in these categories. Students were provided with statistically different opportunities to engage in pattern identification, conjecture formulation, and argument construction in both syllabi. There were significantly fewer opportunities to engage in conjecture formulation for junior cycle students and significantly more opportunities to construct arguments for senior cycle students. There were no instances of proof as falsification across both syllabi, but students were given similar opportunities to experience proof as explanation, verification, and generation of new knowledge. Across both syllabi there were statistically significantly more RP learning outcomes that were divorced from content than those that were connected to content. The results as well as the implications of these results for the design of national curriculum documents are discussed. [ABSTRACT FROM AUTHOR]

The construction of the bound for the intensity of the environmental fluctuations, based on the Lyapunov function (LF), is one of the important problem in dynamical system. But the method of identifying proper LF is really very difficult and some time might be impossible, as it involves complete blind search procedure which is arbitrary and ad hoc. In addition, the random fluctuations or in statistical terminology the error terms involved in the above experiments are completely unknown to us. In usual practice, researchers assume the simple independent structure of the error terms. But for the sake of completeness of the model as well as for the realistic environmental set-up (where the error terms or the equations are dependent), the study of more complicated and generalized structure of the random environmental fluctuations is demanding. In this paper, we propose a probabilistic bound for the intensity of the environmental fluctuations. The proposed method has several key advantages – (a) it is simple and can be used as a handling tool or a working formula for the experimental biologists, (b) can be used regardless the situation where the identification of proper LF is either difficult or not possible, (c) it can be applied for a system of any dimensions, and moreover (d) applicable under more general set-up of error terms. [ABSTRACT FROM AUTHOR]

In the paper, two sharp inequalities for bounding the psi function ψ and the harmonic numbers H n are established respectively, some known results are improved and some remarks are given. [ABSTRACT FROM AUTHOR]