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Problems that offer multiple revelations during its investigations and that can be approached mathematically in more than one way often involve paper folding and conic sections. These problems can be easily demonstrated with a sheet of wax paper or emulated by the geometer's sketchpad.

In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n), 2 ∤ n, for a ∈ ℤ, ( a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

Abstract  We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) ⃛ ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions. [ABSTRACT FROM AUTHOR]

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

In this note, we slightly generalize Theorem 2 in the paper by M. Tunç and point out that the assumption of Theorem 3 is not sufficient. A misuse of the term 'mean' is also discussed. [ABSTRACT FROM AUTHOR]

This article provides instructions in preparing a paper for publication in the "Journal of Optimization Theory and Applications." Some of these guidelines are the following: 1)submission of manuscripts in triplicate; 2) reference to English as the official language of the journal; 3) inclusion of an abstract of at least 50 to 100 words in each contribution; and, 4) the abstract should be followed by a list of four to five key words identifying the subject.

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections. [ABSTRACT FROM AUTHOR]

In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the "most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the "most likely element". This leads to a conundrum - what is the relationship between these two functions? We show both pointwise and G-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations. [ABSTRACT FROM AUTHOR]

Algebraic topology studies topological spaces with the help of tools from abstract algebra. The main focus of this paper is to show that many concepts from algebraic topology can be conveniently expressed in terms of (normal) factor graphs. As an application, we give an alternative proof of a classical duality result of Kramers and Wannier, which expresses the partition function of the 2-D Ising model at a low temperature in terms of the partition function of the 2-D Ising model at a high temperature. Moreover, we discuss analogous results for the 3-D Ising model and the Potts model. [ABSTRACT FROM AUTHOR]

Creativity is a complex, multi-faceted concept encompassing a variety of related aspects, abilities, properties and behaviours. If we wish to study creativity scientifically, then a tractable and well-articulated model of creativity is required. Such a model would be of great value to researchers investigating the nature of creativity and in particular, those concerned with the evaluation of creative practice. This paper describes a unique approach to developing a suitable model of how creative behaviour emerges that is based on the words people use to describe the concept. Using techniques from the field of statistical natural language processing, we identify a collection of fourteen key components of creativity through an analysis of a corpus of academic papers on the topic. Words are identified which appear significantly often in connection with discussions of the concept. Using a measure of lexical similarity to help cluster these words, a number of distinct themes emerge, which collectively contribute to a comprehensive and multi-perspective model of creativity. The components provide an ontology of creativity: a set of building blocks which can be used to model creative practice in a variety of domains. The components have been employed in two case studies to evaluate the creativity of computational systems and have proven useful in articulating achievements of this work and directions for further research. [ABSTRACT FROM AUTHOR]

We give counterexamples for some statements of Borwein and sufficient conditions for the validity of these. [ABSTRACT FROM AUTHOR]

A comparison of the performance on the final exam for subpopulations of students in College Algebra was used to assess the effectiveness of online homework. Data was collected for two small groups of students, one with traditional paper homework and the other with online homework. The groups of students were further classified by incoming skill level measured by a departmental placement test. The data suggests that online homework was beneficial to the incoming higher-skilled group, whereas, the lower-skilled students did not benefit from online homework. [ABSTRACT FROM PUBLISHER]

We give some remarks on the recent paper [J. Math. Inequal. 13 (4) (2019), 1235– 1244]. Some misleading conclusions of their stability results are carefully discussed and corrected. Moreover, we reestablish their results with a more general assumption and a stronger conclusion. [ABSTRACT FROM AUTHOR]

We give an overview of the main ideas involved in the development of homotopy type theory and the univalent foundations of Mathematics programme. This serves as a background for the research papers published in the special issue. [ABSTRACT FROM AUTHOR]

The article presents abstracts related to mathematics including the equivalent of P to NP-intersect, the application of default logic variants in default logic, and basics of specification.

In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in B ˙ 2 d , r - 3 2 × ( B ˙ 2 d , r - 1 2 ) d when 1 ≤ r < d due to the lack of continuity of the solution. [ABSTRACT FROM AUTHOR]

Š. Schwarz in his paper [SCHWARZ, Š.: Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.: Contribution to the theory of torsion semigroups, Czechoslovak Math. J. 3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory. [ABSTRACT FROM AUTHOR]

The current paper contains the theoretical foundation for the off-the-mainstream model known as Alpha-Beta associative memories (αβ model). This is an unconventional computation model designed to operate as an associative memory, whose main application is the solution of pattern recognition tasks, particularly for pattern recall and pattern classification. Although this model was devised, proposed and created in 2002, it is worth noting that its theoretical support remains unpublished to this day. This is despite the fact that more than a hundred scientific articles have been published with applications, improvements, and new models derived from the αβ model. The present paper includes all the required definitions, and the rigorous mathematical demonstrations of the lemmas and theorems, explaining the operation of the αβ model, as well as the original models it has inspired or that have been derived from it. Also, brief descriptions of 60 selected articles related to the αβ model are presented. These latter works illustrate the competitiveness (and sometimes superiority) of several extensions and models derived from the original αβ model, when compared against some models and paradigms present in the mainstream current scientific literature. [ABSTRACT FROM AUTHOR]

This special issue of the Journal Mathematical Methods in the Applied Sciences on the topic of Modelling, Analysis, and Applications is devoted to the development of mathematical modelling, analysis, and applications, from theoretical and numerical perspectives, involving different applied sciences and engineering. It collects invited contributions, from selected quality full papers submitted to the International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMA2SC2022). The conference took place in Porto, Portugal in 27–29 June 2022 (https://www.isep.ipp.pt/Page/ViewPage/ICMASC). [ABSTRACT FROM AUTHOR]

In this paper, an ancient Chinese algorithm is used to estimate the length constant of a long cooling fin, and an approximate solution formulation is obtained. The obtained results show that this method is a simple but promising method without any requirement for advanced calculus. [ABSTRACT FROM AUTHOR]

The present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information" is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration, It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras. Bibliography: 23 titles. [ABSTRACT FROM AUTHOR]

In the literature, rank-constrained Hermitian nonnegative-definite solutions to the matrix equation AXAH=B have been investigated, under the conditions that B is Hermitian and nonnegative-definite, and the matrix equation is consistent. In this paper, we discuss rank-constrained Hermitian nonnegative-definite least squares solutions to this matrix equation, in which the above conditions may not hold. We derive the rank range and expression of these least squares solutions. Therefore, the results obtained in this paper generalize those in the literature. [ABSTRACT FROM AUTHOR]

Rufus P. Isaacs joined the RAND Corporation4. Santa Monica, California in 1948 and started to develop the theory of dynamic games in the early 1950s. Until winter 1954/55, when Isaacs left the RAND Corporation, he investigated two player. zero-sum dynamic games of the classic pursuit-evasion type. Prior to 1965, Isaacs published his theory only in internal RAND papers and research memoranda. In his first RAND paper (Ref. 1), Isaacs sketched the basic ideas of zero-sum dynamic game theory The ideas already included rudimentary precursors of the maximum principle, dynamic programming, and backward analysis. At the end of 1954 and the beginning of 1955. Isaacs summarized his research in four research memoranda (Refs. 3–6), which ten years later formed the basis of his famous book on Differential Games (Ref. 7). This paper survey's Isaacs' research with an emphasis on the early years of dynamic games. The readers are kindly invited to discuss the author's point of view. Comments and statements sent to the author will be summarized and published later. [ABSTRACT FROM AUTHOR]

In this paper, finite element methods for the problems of wave propagation in a fluid-saturated porous medium are discussed. The medium is composed of a porous elastic solid (soil, rock, etc.) saturated by a compressible viscous fluid (oil, water, etc.), and the fluid may flow relatively to the solid. Biot's lowfrequency dynamic equations are chosen to describe the problems mentioned above, with stress-given boundary conditions, ABGs (Absorbing Boundary Conditions) on artificial boundaries and conditions on interfaces between the fluid-saturated porous medium and elastic solids. In the paper, a new kind of discrete ABCs is presented, and a discrete-time Galerkin method are utilized for obtaining approximate solutions. The numerical results show that they both are effective. Two dilatational waves (fast wave P1 and slow wave P2) and one rotational wave (S wave) are clearly visible in the figures of computational results, which coincide with theoretical analysis very well. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the effect of diffusion, which is ubiquitous in nature, on the immune system using a reaction-diffusion model in order to understand the dynamical behavior of complex patterns and control the dynamics of different patterns. Through control theory and linear stability analysis of local equilibrium, we obtain the optimal condition under which the system loses stability and a Turing pattern occurs. By combining mathematical analysis and numerical simulation, we show the possible patterns and how these patterns evolve. In addition, we establish a bridge between the complex patterns and the biological mechanism using the results from a previous study in Nature Cell Biology. The results in this paper can help us better understand the biological significance of the immune system. [ABSTRACT FROM AUTHOR]

The article discusses about developing expressive mathematical language in young children. Topics discussed include that teachers can help develop children's expressive mathematical language by giving them opportunities to speak and write as they use hands-on materials alongside, discusses types of mathematical writing including exploratory writing, argumentative writing and mathematically creative writing; and it also includes samples in which teachers notice the math in what a child is doing.

The purpose of this paper is twofold. The first goal is to provide a simple and constructive proof of Korn inequalities in half-space with weighted norms. The proof leads to explicit values of the constants. The second objective is to use these inequalities to show that the linear elasticity system in half-space admits a coercive variational formulation. This formulation corresponds to the physical case in which the solution is evanescent at infinity. [ABSTRACT FROM AUTHOR]

In this paper, a new class of Hermite-Hadamard type integral inequalities using a strong type of convexity, known as n-polynomial exponential type s-convex function, is studied. This class is established by utilizing the Hölder's inequality, which has several applications in optimization theory. Some existing results of the literature are obtained from newly explored consequences. Finally, some novel limits for specific means of positive real numbers are shown as applications. [ABSTRACT FROM AUTHOR]

Mathematics is the most logical and unbiased scientific field there is. What is true in mathematics today will forever stay true, and mathematical arguments will always clearly indicate which side "wins". These are some of the characteristics of mathematics that I have liked very much since I was young. In this article, I will try to demonstrate the beauty of mathematics and show you that it appears in many situations in our daily lives--including places we might not expect. Then, I will give you a taste of two studies I conducted in a mathematical field called combinatorics, and I will explain how they are related to everyday problems. One of these problems is how to perform accurate mathematical analyses of large streams of data that we cannot save in a computer's memory. Finally, I will try to give you an idea of my life as a mathematician--the challenges and difficulties, alongside the immense enjoyment and satisfaction I derive from them. Join me to an extended dive into my work on the wonders of combinatorics. [ABSTRACT FROM AUTHOR]

Among several number triangles that exist in mathematics, Pascal's triangle is the most fascinating and prominent structure possessing numerous properties. In this paper, we will introduce a number triangle consisting of positive integers arranged in ascending to descending order so that the entries in each row would be mirror images with respect to its middle number. Using this simple number triangle, we have proved six amusing results which interestingly depend only as functions of its centred numbers. Moreover some of the concepts and results discussed in this paper have connection with Tamil literary work and triangular numbers. [ABSTRACT FROM AUTHOR]

This paper deals with undiscounted infinite stage two-person zero-sum stochastic games with finite state and action spaces. It was recently shown that such games possess a value. But in general there are no optimal strategies. We prove that for each player there exists a nonempty set of easy initial states, i.e. starting states for which the player possesses an optimal stationary strategy. This result is proved with the aid of facts derived by Bewley and Kohlberg for the limit discount equation for stochastic games. [ABSTRACT FROM AUTHOR]

As an application of mathematics to engineering problems, this paper formulates a simple optimal stopping problem to decide the opening time of harvesting farmed fishery resources that maximizes an economic objective function. A sufficient condition for unique existence of the internal optimal opening time is provided and its concrete mathematical analysis is carried out. Comparative statics of the optimal opening time clearly reveals its dependence on the parameters of the farming environment. The problem is finally applied to analyzing management of a commercially important fishery resource in Japan. [ABSTRACT FROM AUTHOR]

Grey wolf optimization (GWO) algorithm is a new emerging algorithm that is based on the social hierarchy of grey wolves as well as their hunting and cooperation strategies. Introduced in 2014, this algorithm has been used by a large number of researchers and designers, such that the number of citations to the original paper exceeded many other algorithms. In a recent study by Niu et al., one of the main drawbacks of this algorithm for optimizing real‐world problems was introduced. In summary, they showed that GWO's performance degrades as the optimal solution of the problem diverges from 0. In this paper, by introducing a straightforward modification to the original GWO algorithm, that is, neglecting its social hierarchy, the authors were able to largely eliminate this defect and open a new perspective for future use of this algorithm. The efficiency of the proposed method was validated by applying it to benchmark and real‐world engineering problems. [ABSTRACT FROM AUTHOR]

Based on the existence of pullback attractors for stochastic suspension bridge in [7], in the paper, we further investigate the upper semicontinuity of pullback attractors for the problem. [ABSTRACT FROM AUTHOR]

In this paper we propose an approach to introduce a concept of bounded index in an infinite-dimensional space. Our object of investigation is the space ℓ∞ equipped with the norm ∥x∥∞ = sup{|xn|: n ∈ N}. We consider entire curves from C to ℓ∞ and prove a proposition indicating connection between of the l-index boundedness of every component of the curve and the l-index boundedness of the curve. Moreover, we obtain sufficient conditions of the l-index boundedness of entire curves in the space. They describe local behavior of norm of derivatives of the entire curves on the discs. Also, there is posed a problem on necessary conditions of the l-index boundedness of entire curves in infinite-dimensional spaces. [ABSTRACT FROM AUTHOR]

Let T and S be two self-adjoint positive invertible operators of a complex Hilbert space H. In this paper, we investigate some inequalities involving the quadratic forms of the weighted arithmetic and harmonic means of T and S. Application for operator entropies is also discussed. [ABSTRACT FROM AUTHOR]

The augmented Zagreb index (AZI) of a graph G is defined as AZI (G) =∑uv∊E(G) (dudv/du+dv-2)3 where E(G) and du denote set of edges of G and degree of the vertex u in G respectively. In this paper we establish some general results and bounds of AZI for certain unicyclic graphs and their corresponding chemical representation. We also obtain some results pertaining to AZI of certain trees. [ABSTRACT FROM AUTHOR]

In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of v, namely v∈R. These modified inequalities are used to establish corresponding operator inequalities on a Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper, we study varieties and characterize convolution operators on the algebras related to the new structures of KPC-hypergroups, which are a generalization of the classical ones. [ABSTRACT FROM AUTHOR]

The Justified Plan Graph (JPG) technique was developed in the late 1970s and refined in the following two decades as a means of undertaking qualitative and quantitative research into the spatial structure or permeability of buildings. Famously used by Space Syntax researchers to uncover the social logic of architectural types, the technique remains an important, if not widely understood, approach to the analysis of the built environment. This paper uses the JPG method to undertake a three-stage analysis of the early houses of Pritzker Prize winning architect Glenn Murcutt; the stages are visual analysis, mathematical analysis and theoretical analysis. Through this process the paper offers a rare application of the JPG method to multiple works by the same architect and demonstrates the construction of a series of 'inequality genotypes', a partial 'statistical genotype' and a partial 'statistical archetype' for these houses. Instead of seeking to uncover the social structure of Murcutt's housing, the paper analyses the architect's distinctive approach to ordering space within otherwise simple volumes or forms. The ultimate purpose of this analysis is to offer an alternative space-based, rather than form-based, insight into this architect's work. [ABSTRACT FROM AUTHOR]

In this article, the author discusses the concept of individuality and collaboration in mathematics. The author reflects on the achievement of Andrew Wiles, a world known mathematicians, in which Wiles solved the problem of Fermat's last theorem without collaborating with other mathematicians. The author also shares some of the collaborative works in the development of mathematics including the Bers Mafia and the collaboration of Paul Erd—os and I. M. Gel'fand.

J. Jakubík noted in [JAKUBÍK, J.: Modular Lattice of Locally Finite Length, Acta Sci. Math. 37 (1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving sublattices isomorphic to S7. In the present paper we extend the scope of Šik's theorem to the class of upper semimodular, upper continuous and strongly atomic lattices. Moreover, we show that corresponding result of Jakubík from [JAKUBÍK, J.: Modular Lattice of Locally Finite Length, Acta Sci. Math. 37 (1975), 79–82] cannot be strengthened is analogous way. [ABSTRACT FROM AUTHOR]

The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory, numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s Theorem. This paper is primarily concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains. When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises and V.I. Romanovsky. [ABSTRACT FROM AUTHOR]

This is a reply to Ramon Ferrer-I-Cancho's paper in this issue "Some Word Order Biases from Limited Brain Resources: A Mathematical Approach." In this reply, I challenge the Euclidean distance model proposed in that paper by proposing a simple alternative model based on linear ordering. [ABSTRACT FROM AUTHOR]

This paper will consider the range of British participation in mathematics internationally during the nineteenth century through an analysis of British mathematical contributions to scientific journals outside of Britain. Viewing scientific papers contained in journals as significant indicators of research, we consider scientists who authored or read and responded in print to papers in a given area within a given group of journals to constitute a publication community. The extent of publication by British mathematicians in these journals can help characterize the role of foreign publication in nineteenth-century British mathematics. Moreover, the isolation of educational, societal, and personal circumstances which motivated British mathematicians to present their work to foreign journals highlights limited but concentrated groups of mathematicians committed to developing and strengthening international mathematical ties with Britain. [ABSTRACT FROM AUTHOR]

Discussions of mathematical problem-solving and heuristic reasoning have typically examined how proofs that are already known might be found. This approach has at least three problems: first, provers engaged in discovering proofs for themselves cannot have this perspective; second, if a proof is difficult, formulaic strategies quickly run out; third, beginning with a proof already in-hand separates reasoning about a proof from the actual circumstances in which such reasoning occurs. As an alternative approach to the study of mathematical reasoning, this paper presents a detailed descriptive account of the work of finding a specific proof, including the shifting of perspectives, the wrong paths, the mistakes and the outright errors. Even the appearance of a sketched diagram or of a course of mathematical writing can suggest unanticipated possibilities for finding a proof. This material is used to illustrate the paper's central claim – that the ways that provers go about working on proofs provide the context for continuing that work and for discovering the reasoning that a particular proof is then seen to require. [ABSTRACT FROM AUTHOR]

The paper reports some new solutions for heat transfer through extended surfaces or fins using the symbolic algebra package Maple 8, which is widely available. The four specific problems chosen for the present study are: (a) a rectangular convection fin with the heat transfer coefficient varying either linearly or exponentially with the distance from the base, (b) a truncated conical spine with convection at both ends, (c) a heat-generating annular fin with a constant base heat flux and an adiabatic tip, and (d) a convection fin array made of a rectangular fin and two triangular fins. Each problem is formulated in a manner that makes its solution novel and distinct from what is available in the literature. Solutions are provided in symbolic forms. Using the numerical and graphical capabilities of Maple, the results are presented in the form of numerical data as well as graphical displays. The paper demonstrates that Maple provides an effective and convenient tool for the analysis of extended surface heat transfer problems that otherwise demand tedious algebraic manipulations. [ABSTRACT FROM AUTHOR]