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Differential geometry of a kind of helical hypersurface family that depends on six parameters within the seven‐dimensional Euclidean space is explored. The curvatures of these hypersurfaces are determined, and their minimality is examined, along with illustrative examples being provided. Lastly, the Laplace–Beltrami operator for that kind of hypersurfaces is calculated. [ABSTRACT FROM AUTHOR]

Compound flooding, caused by the sequence/co-occurrence of flood drivers (i.e. river discharge and elevated sea level) can lead to devastating consequences for society. Weak and insufficient progress toward sustainable development and disaster risk reduction are likely to exacerbate the catastrophic impacts of these events on vulnerable communities. For this reason, it is indispensable to develop new perspectives on evaluating compound flooding dependence and communicating the associated risks to meet UN Sustainable Development Goals (SDGs) related to climate action, sustainable cities, and sustainable coastal communities. An indispensable first step for studies examining the dependence between these bivariate extremes is plotting the data in the variable space, i.e., visualizing a scatterplot, where each axis represents a variable of interest, then computing a form of correlation between them. This paper introduces the Angles method, based on Euclidean geometry of the so-called " subject space ," for visualizing the dependence structure of compound flooding drivers. Here, we evaluate, for the first time, the utility of this geometric space in computing and visualizing the dependence structure of compound flooding drivers. To assess the effectiveness of this method as a risk communication tool, we conducted a survey with a diverse group of end-users, including academic and non-academic respondents. The survey results provide insights into the perceptions of applicability of the Angles method and highlight its potential as an intuitive alternative to scatterplots in depicting the evolution of dependence in the non-stationary environment. This study emphasizes the importance of innovative visualization techniques in bridging the gap between scientific insights and practical applications, supporting more effective compound flood hazard communication in a warming climate. [ABSTRACT FROM AUTHOR]

Let x be an isometric immersion of a Riemannian n‐manifold M into a Euclidean (n + 1)‐space En+1 which does not pass through the origin of En+1. Then, the tangential part of the position vector field x of x is called the canonical vector field, and the normal part gives rise to a scalar function called the support function. Using the canonical vector field, support function, and mean curvature, we establish three new characterizations of hyperspheres. In addition, we prove that if the energy function of M satisfies the static perfect fluid equation, then M has at most two distinct principal curvatures. As an application, we prove that a complete noncompact hypersurface M is a spherical hypercylinder if the energy function of M satisfies the static perfect fluid equation, and it has exactly two distinct nonsimple principal curvatures. [ABSTRACT FROM AUTHOR]

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2 + 1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras. [ABSTRACT FROM AUTHOR]

Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the 3 n -dimensional configuration space rather than on the 3-dimensional space has seemed uncanny to many, including Schrödinger, Lorentz, and Einstein. Even today, this continues to be seen as a significant issue in the foundations of quantum mechanics. In this article, it will be shown that the wavefunction is, in fact, a genuine object on space. While this may seem surprising, the wavefunction does not possess qualitatively new features that were not previously encountered in objects known from Euclidean geometry and classical physics. The methodology used involves finding equivalent reinterpretations of the wavefunction exclusively in terms of objects from the geometry of space. The result is that we will find the wavefunction to be equivalent to geometric objects on space in the same way as was always the case in geometry and physics. This will be demonstrated to hold true from the perspective of Euclidean geometry, but also within Felix Klein's Erlangen Program, which naturally fits into the classification of quantum particles by the representations of spacetime isometries, as realized by Wigner and Bargmann, adding another layer of confirmation. These results lead to clarifications in the debates about the ontology of the wavefunction. From an empirical perspective, we already take for granted that all quantum experiments take place in space. I suggest that the reason why this works is that they can be interpreted naturally and consistently with the results presented here, showing that the wavefunction is an object on space. [ABSTRACT FROM AUTHOR]

Summary: We discuss two possible ways to define Nagel points for triangles when using the taxicab distance formula. The two approaches are equivalent in Euclidean geometry since a triangle's extouch points divide the triangle's perimeter in half, but this is not the case with the taxicab distance formula. We will discuss how using extouch points does not always lead to a unique taxicab Nagel point, but using points that split the perimeter does. In fact, we show that the most common proof for the existence of the Euclidean Nagel point and its Barycentric coordinates holds when replacing the Euclidean distance formula with any l p ‐metric . [ABSTRACT FROM AUTHOR]

Let $f\colon M^{2n}\to \mathbb {R}^{2n+p}$ , $2\leq p\leq n-1$ , be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng (2013, Michigan Mathematical Journal 62, 421–441) conjectured that if the codimension is $p\leq 11$ , then, along any connected component of an open dense subset of $M^{2n}$ , the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces in the kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\mathbb {R}^{2n+p}$ of larger dimension than $2n$. This bold conjecture was proved by Dajczer and Gromoll just for codimension 3 and then by Yan and Zheng for codimension 4. In this paper, we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step for a possible classification of the nonholomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Euclidean space is the fabric of the world we live in. Whether and how geometric experience shapes our spatial-temporal representations of the world remained unknown. We deprived male rats of experience with crucial features of Euclidean geometry by rearing them inside spheres, and compared activity of large hippocampal neuronal ensembles during navigation and sleep with that of cuboid cage-reared controls. Sphere-rearing from birth permitted emergence of accurate neuronal ensemble spatial codes and preconfigured and plastic time-compressed neuronal sequences. However, sphere-rearing led to diminished individual place cell tuning, more similar neuronal mapping of different track ends/corners, and impaired pattern separation and plasticity of multiple linear tracks, coupled with reduced preconfigured sleep network repertoires. Subsequent experience with multiple linear environments over four days largely reversed these effects. Thus, early-life experience with Euclidean geometry enriches the hippocampal repertoire of preconfigured neuronal patterns selected toward unique representation and discrimination of multiple linear environments. How geometric experience shapes our representation of the world remains unclear. Here the authors show that early-life experience with Euclidean geometry enriches the hippocampal repertoire of preconfigured neuronal motifs selected for representation of multiple contexts. [ABSTRACT FROM AUTHOR]

For an arbitrary field K and a K-variety V, we introduce the étale-open topology on the set V(K) of K-points of V. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when K is separably closed, real closed, or p-adically closed, respectively. Topological properties of the étale-open topology correspond to algebraic properties of K. For example, the étale-open topology on 픸¹ (K) is not discrete if and only if K is large. As an application, we show that a large stable field is separably closed. [ABSTRACT FROM AUTHOR]

Can we compute what we cannot draw? How must we draw to produce measurable representations, or visual ones? This research inquires into the relationship between mathematics and figurative representation, and more precisely between drawing and computation. The scientific imagery studied here is the representation of the five platonic solids, discussing various representation techniques from classical antiquity to modern times, and their efficacy to help calculate sizes and proportional ratios. Scholars in history of architectural drawing have too often limited their observations to the very few preserved plans and front views dating back to classical antiquity, without enlarging their investigation to other scientific fields that also rely on drawing as a research tool and communication device. Among these other fields stands the mathematical research, especially solid geometry which deals with objects and entities that have shapes that needs to be somehow drawn in 3D to be studied. [ABSTRACT FROM AUTHOR]

This paper aims to establish new upper bounds for the Euclidean operator radius concerning pairs of bounded linear operators in a complex Hilbert space. To achieve this objective, we utilize some Boas-Bellman type inequalities as proof tools. Furthermore, we extend our findings to derive novel upper bounds for the numerical radius of operators in Hilbert spaces. These results contribute to advancing our understanding and analytical capabilities regarding operator properties within the framework of Hilbert spaces. [ABSTRACT FROM AUTHOR]

There are different definitions of axioms, but the one that seems to have general approval is that axioms are statements whose truths are universally accepted but cannot be proven; they are the foundation from which further propositional truths are derived. Previous attempts, led by David Hilbert, to show that all of mathematics can be built into an axiomatic system that is complete and consistent failed when Kurt Gödel proved that there will always be statements which are known to be true but can never be proven within the same axiomatic system. But Gödel and his followers took no account of brain mechanisms that generate and mediate logic. In this largely theoretical paper, but backed by previous experiments and our new ones reported below, we show that in the case of so‐called 'optical illusions', there exists a significant and irreconcilable difference between their visual perception and their description according to Euclidean geometry; when participants are asked to adjust, from an initial randomised state, the perceptual geometric axioms to conform to the Euclidean description, the two never match, although the degree of mismatch varies between individuals. These results provide evidence that perceptual axioms, or statements known to be perceptually true, cannot be described mathematically. Thus, the logic of the visual perceptual system is irreconcilable with the cognitive (mathematical) system and cannot be updated even when knowledge of the difference between the two is available. Hence, no one brain reality is more 'objective' than any other. [ABSTRACT FROM AUTHOR]

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n ≥ 2 points. Some elementary properties of pseudo n-metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n-metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric. [ABSTRACT FROM AUTHOR]

Faced with the mathematical possibility of non-Euclidean geometries, 19th Century geometers were tasked with the problem of determining which among the possible geometries corresponds to that of our space. In this context, the contribution of the Belgian philosopher-mathematician, Joseph Delboeuf, has been unduly neglected. The aim of this essay is to situate Delboeuf's ideas within the context of the philosophies of geometry of his contemporaries, such as Helmholtz, Russell and Poincaré. We elucidate the central thesis, according to which Euclidean geometry is given special status on the basis of the relativity of magnitudes, we uncover its hidden history and show that it is defensible within the context of the philosophies of geometry of the epoch. Through this discussion, we also develop various ideas that have some relevance to present-day methods in gravitational physics and cosmology. [ABSTRACT FROM AUTHOR]

Morphometric analysis is the quantification of shapes, which makes irregular shapes found in nature analyzable and comparable. Cave morphometry has been used for the genetic classification of caves, the digital reconstruction of their conduits, the decoding of their paleoenvironment, and other research purposes. Ratios and indices that have been derived from Euclidean geometry and application of fractal geometry onto karst features and topological parameters are the basic methodologies that have been used for shape quantification. This paper reviews the literature that focuses on methodologies used for morphometric analyses and the applications that these methodologies have found. [ABSTRACT FROM AUTHOR]

Bachmann's absolute geometry provides a common foundation of Euclidean and hyperbolic geometry without any assumptions about order or free mobility. An order structure can be introduced in an additional step either in a synthetic way (by postulating that every line is linearly orderable and admits a partition into sides) or analytically (by the requirement that the field of coordinates is orderable). The class of ordered absolute planes can be divided into two subclasses depending on whether or not the groups of rotations around a point O are cyclically ordered groups. In this paper we study the geometric properties of both kinds of order structures and determine the associated models. Surprisingly, in every ordered absolute plane the groups of rotations around a point O are cyclically orderable, with the only exception of subplanes of Euclidean planes with a negative orthogonality constant. We show that this exceptional case has a deeper reason: These Euclidean planes are subplanes of a Minkowskian plane and inherit the Minkowskian metric. The article concludes with some remarks on the foundations of ordered absolute geometry, which include comments on the well-known axiom systems of Hilbert and Tarski and on the role of the axiom of Pasch. [ABSTRACT FROM AUTHOR]

This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence and a non-trivial inequality concerning triangles. GeoGebra succeeds in proving all these selected examples, while ChatGPT fails in one case. Our thesis is that both GeoGebra and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery can cooperate in order to obtain sound and—more relevant—interesting results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct relations and characterizations between the pedal curves and surfaces and to find components of the vector of α(t) by the means of the pedal on the surface M. Also, the formula of pedal curves of a curve α is generalized in n‐dimensional Euclidean space En. Some special results are obtained within the scope of pedal curves and given with obtained characterization of the pedal curve in n‐dimensional space En. [ABSTRACT FROM AUTHOR]

The proportions in the size of the avian egg albumen, yolk, and shell are crucial for understanding bird survival and reproductive success because their relationships with volume and surface area can affect ecological and life history strategies. Prior studies have focused on the relationship between the albumen and the yolk, but little is known about the scaling relationship between eggshell mass and shape and the mass of the albumen and the yolk. Toward this end, 691 eggs of six precocial species were examined, and their 2-D egg profiles were photographed and digitized. The explicit Preston equation, which assumes bilateral symmetrical geometry, was used to fit the 2-D egg profiles and to calculate surface areas and volumes based on the hypothesis that eggs can be treated as solids of profile revolution. The scaling relationships of eggshell mass (Ms), albumen mass (Ma), and yolk mass (My), as well as the surface area (S), volume (V), and total mass (Mt) were determined. The explicit Preston equation was validated in describing the 2-D egg profiles. The scaling exponents of Ma vs. Ms, My vs. Ms , and My vs. Ma were smaller than unity, indicating that increases in Ma and My fail to keep pace with increases in Ms , and that increases in My fail to keep pace with increases in Ma. Therefore, increases in unit nutrient contents (i.e. the yolk) involve disproportionately larger increases in eggshell mass and disproportionately larger increases in albumen mass. The data also revealed a 2/3-power scaling relationship between S and V for each species, that is, the simple Euclidean geometry is obeyed. These findings help to inform our understanding of avian egg construction and reveal evolutionary interspecific trends in the scaling of egg shape, volume, mass, and mass allocation. [ABSTRACT FROM AUTHOR]

The combined use of origami and dynamic geometry software has recently appeared in mathematics education to enrich students' geometric thinking. The objective of this research is to study the roles played by the interaction of two artifacts, paper folding and GeoGebra, in a construction-proving problem as well as its generalization in the Euclidean geometry context. For this, we designed and implemented two mathematical tasks with 52 secondary education students (15–16 years old, 10th grade) during the COVID-19 emergency lockdown period in Italy. The tasks involved four phases: constructing, exploring, conjecturing, and proving. This article presents an epistemic analysis of the tasks and a cognitive analysis of the answers given by one of the students. The theoretical tools of the onto-semiotic approach supported these analyses. Cognitive analysis allows us to confront the intended meanings of the task and the meanings actually employed by a student, thus drawing specific conclusions about the roles of such artifacts in written arguments and give an interpretation of their combined use in mathematics education. [ABSTRACT FROM AUTHOR]

While there are well-known synthetic methods in the literature for finding the image of a point under circular inversion in l 2 -normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in creating a synthetic construction of the circular inversion in l 1 -normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the l 1 -norm. [ABSTRACT FROM AUTHOR]

Copyright of Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales is the property of Academia Colombiana de Ciencias Exactas, Fisicas y Naturales and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. "Classroom" is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. This article explores generalizations of the Pythagorean theorem using classical Euclidean plane geometry tools. It introduces innovative extensions that shed new light on the theorem's geometric relationships. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a solution to three conjectures from Euclidean geometry concerning the location of the Mittenpunkt. The first two are solved without dependence on computer technology and with only a moderate amount of calculations. They were initially tackled by heavy calculations using computer algebra systems. [ABSTRACT FROM AUTHOR]

We consider and characterize closed convex subsets of the Euclidean space which are simultaneously Motzkin decomposable and generalized Minkowski or, shortly, MdgM sets. We also prove the existence of suitably defined fixed points for, possibly multivalued, functions defined on MdgM sets along with the existence of classical fixed points for some single valued self functions of MdgM sets. The first mentioned type of existence results are based on Kakutani fixed point theorem, and the second type are based both on the Brouwer fixed point theorem and the Banach contraction principle. [ABSTRACT FROM AUTHOR]

Jean W. Rioux's book, "Thomas Aquinas' Mathematical Realism," explores Aquinas' philosophy of mathematics and its relevance to contemporary problems in the field. Rioux contrasts Aquinas' views with those of Aristotle, highlighting their complementarity. Aquinas' philosophy emphasizes the conditions that mathematical objects must fulfill to be considered legitimate, rather than simply questioning their existence. Rioux also examines Aquinas' views on the construction of mathematical objects and their relationship to proof construction. The book provides insights into the nature of mathematics and its various aspects, making it a valuable resource for scholars in medieval philosophy and the philosophy of mathematics. [Extracted from the article]

We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $\mathbb {F}$ of characteristic zero as a normalised $\mathbb {F}$ -linear functional on $\mathbb {F}[\alpha _1, \alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $\mathrm {SO}(2,\mathbb {F})$ -invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers $$ \begin{align*} S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!} \end{align*} $$ an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha _1^{2m}\alpha _2^{2n}$. [ABSTRACT FROM AUTHOR]

A cyclic antipodal pair of a circle is a pair of points that are the intersection of the circle with the diameter of the circle. In this article, a recent proof of Ptolemy's Theorem—simultaneously in both (i) Euclidean geometry and (ii) the relativistic model of hyperbolic geometry (also known as the Klein model)—motivates the study of four cyclic antipodal pairs of a circle, ordered arbitrarily counterclockwise. The translation of results from Euclidean geometry into hyperbolic geometry is obtained by means of hyperbolic trigonometry, called gyrotrigonometry, to which Einstein addition gives rise. Identities that extend the Pythagorean identity in both Euclidean and hyperbolic geometry are obtained. [ABSTRACT FROM AUTHOR]

The article presents the author's experience in a philosophy and metaphysics seminar taught by Professor Héctor Massa. Two forms of practicing metaphysics were addressed: testimonial and conceptual, using passages from the works of Berdiáyev and Kant. The influence of Kant on 20th-century analytical philosophy and its importance in the philosophy of science were discussed. Different philosophers were also mentioned, and the relationship between Euclidean geometry and spatial intuition was explored from Kant's perspective. Additionally, strategies for defending a non-conceptualist reading of Kant were presented, and Sellars' stance on perceptual experiences and their relation to empirical knowledge was discussed. [Extracted from the article]

We present an axiom system for the plane Euclidean geometry of ruler and dividers constructions, expressed in Tarski’s language, with points as the only variables and with the predicates of betweenness and equidistance as the only primitive notions. All axioms are statements about at most 5 points, and it is known that no axiom system with all axioms statements about at most 4 points exists in that language, so the axiom system presented here can be called the simplest possible one. This corrects an error in Pambuccian (Math Chronicle 18:63–74, 1989) and shows that the Pasch axiom can be replaced by three statements, a 3-variable one, a 4-variable one, and a 5-variable one in the context of plane Euclidean geometry over Pythagorean ordered fields. Viewed from the vantage point of the weak elementary Euclidean Pasch-free geometry, the Pasch axiom turns out to be equivalent to both a universal 5-variable sentence and to a positive 5-variable sentence. [ABSTRACT FROM AUTHOR]

Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in n-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi [3]. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper. [ABSTRACT FROM AUTHOR]

The 21st-century mathematics classrooms should equip learners with well-grounded knowledge and thinking skills pertaining to geometry. However, Euclidean geometry remains one of the challenging, if not the most difficult topic for many learners. As a result, the purpose of this article is to interpret Grade 12 learners' learning difficulties in Euclidean geometry. We use Van Hiele's geometric thinking model and Hoffer's skills to argue an interpretation of learning difficulties in Euclidean geometry as a focal point towards creating effective teaching and learning of this important topic. This explanatory sequential mixed-methods approach involved 60 Grade 12 learners who wrote a geometry test and completed a questionnaire based on Van Hiele's geometric thinking levels. In addition, semistructured interviews were conducted with a sample of 12 learners and four educators to investigate their views about geometry learning difficulties. The findings of the study revealed that learners had poor conceptualisation of properties of shapes, visualisation skills, circle theorems and geometry terminology, resulting in them experiencing learning difficulties. The recommendations are that, during instruction learners should be given the opportunity to manipulate real geometric objects to enhance their visualisation and visual thinking skills. In addition, we recommend that educators should teach level-specific geometry vocabulary to enable learners to understand concepts at different Van Hiele's levels. Furthermore, we recommend that educators should use constructivist teaching approaches that encourage learners' conceptual understanding instead of traditional methods that promote rote memorisation of geometric facts. Educators should develop learners' broad knowledge of geometry to overcome geometry-related errors and misconceptions. [ABSTRACT FROM AUTHOR]

With the advancements in quantum error correction technique, quantum low density parity check (QLDPC) codes have emerged as a promising area in the field of quantum error correction. This paper focuses on the requirement of QLDPC codes based on points excluding the origin, and lines that do not pass through the origin of Euclidean Geometry. It also explores QLDPC codes based on all the points and lines. Finally, a series of simulation analyses are presented. [ABSTRACT FROM AUTHOR]

This paper establishes the mapping properties of the fractional integral operators on the grand Morrey spaces and the grand Hardy-Morrey spaces defined on the Euclidean spaces. We obtain our results by refining the Rubio de Francia extrapolation method as the existing extrapolation method cannot be directly applied to the grand Morrey spaces. This method also yields the mapping properties of nonlinear operators. In particular, we establish the Sobolev embedding, the Poincaré inequality and the mapping properties of the fractional geometric maximal functions on the grand Morrey spaces. [ABSTRACT FROM AUTHOR]

This paper goes into the challenges faced in the teaching of geometry, emphasizing its foundational principles. It investigates an alternate viewpoint by relating geometric principles to verses from the Holy Quran, implying that geometric conceptions have a spiritual and intellectual dimension. The historical contributions of Islam to geometric sciences are also examined, emphasizing the confluence between religion and geometry. The historical context of geometry in Islamic education is also explored, with an emphasis on the substantial contributions of some Muslim scholars to the topic between the 9th and 15th centuries. The literature overview presents much research on geometry education, including inquiry-based techniques, academic talent profiles, and the impact of various teaching methods on student achievement. Despite the variety of teaching methods, obstacles such as curriculum issues, teacher training, and student attitudes continue. In addressing the complexity of geometry teaching, the methodologies section highlights the significance of appropriate research design. The traditional teaching style and activity-based teaching/learning are addressed as two opposing methods. The latter is praised for its ability to foster innovative learning experiences. The results and discussion section critically assesses the “Foundations of Geometry” curriculum at top universities, identifying issues that need to be revised to line with contemporary expectations. The obstacles to teaching geometry are examined, including students’ apathy and lack of prior knowledge, and solutions such as real-world examples, continual professional development, and activity-based teaching approaches are proposed. Finally, the article proposes a comprehensive reevaluation of geometry education that takes historical, religious, and current perspectives into account. It emphasizes the need for dynamic teaching methods, technology integration, and a revamped curriculum to make geometry more accessible and entertaining for students.

We study whether Pareto-optimal stable matchings can be reached via pairwise swaps in one-to-one matching markets with initial assignments. We consider housing markets, marriage markets, and roommate markets as well as three different notions of swap rationality. Our main results are as follows. While it can be efficiently determined whether a Pareto-optimal stable matching can be reached when defining swaps via blocking pairs, checking whether this is the case for all such sequences is computationally intractable. When defining swaps such that all involved agents need to be better off, even deciding whether a Pareto-optimal stable matching can be reached via some sequence is intractable. This confirms and extends a conjecture made by Damamme, Beynier, Chevaleyre, and Maudet (2015) who have shown that convergence to a Pareto-optimal matching is guaranteed in housing markets with single-peaked preferences. We prove that in marriage and roommate markets, single-peakedness is not sufficient for this to hold, but the stronger restriction of one-dimensional Euclidean preferences is. [ABSTRACT FROM AUTHOR]

Mutually beneficial cooperation is a common part of economic systems as firms in partial cooperation with others can often make a higher sustainable profit. Though cooperative games were popular in 1950s, recent interest in noncooperative games is prevalent despite the fact that cooperative bargaining seems to be more useful in economic and political applications. In this paper we assume that the strategy space and time are inseparable with respect to a contract. Furthermore, it is assumed that each firm's strategy polygon is a geodesic polygon which changes its shape every point of time with the stubbornness strategy surface of firm's executive board follow a Gaussian free field. This gives us more flexibility to deal with generalized geodesic cooperative games which is the main contribution of this paper. Under this environment we show that the strategy spacetime is a dynamic curved Liouville-like 2-brane quantum gravity surface under asymmetric information and that traditional Euclidean geometry fails to give a proper feedback Nash equilibrium. Cooperation occurs when two firms' strategies fall into each other's influence curvature in this strategy spacetime. Small firms in an economy dominated by large firms are subject to the influence of large firms. We determine an optimal feedback semicooperation of the small firm in this case using a Liouville-Feynman path integral method. [ABSTRACT FROM AUTHOR]

The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proofs has always been a grand challenge to the interdisciplinary field of mathematics and artificial intelligence. This is the fourth article in a series of our works, in our previous work, we established a geometric formalized system known as FormalGeo. Moreover, we annotated approximately 7000 geometric problems, forming the FormalGeo7k dataset. Despite the fact that FGPS (Formal Geometry Problem Solver) can achieve interpretable algebraic equation solving and human-like deductive reasoning, it often experiences timeouts due to the complexity of the search strategy. In this paper, we introduced FGeo-TP (theorem predictor), which utilizes the language model to predict the theorem sequences for solving geometry problems. The encoder and decoder components in the transformer architecture naturally establish a mapping between the sequences and embedding vectors, exhibiting inherent symmetry. We compare the effectiveness of various transformer architectures, such as BART or T5, in theorem prediction, and implement pruning in the search process of FGPS, thereby improving its performance when solving geometry problems. Our results demonstrate a significant increase in the problem-solving rate of the language model-enhanced FGeo-TP on the FormalGeo7k dataset, rising from 39.7% to 80.86%. Furthermore, FGeo-TP exhibits notable reductions in solution times and search steps across problems of varying difficulty levels. [ABSTRACT FROM AUTHOR]

At first glance, thermodynamic properties of gravity with asymptotically AdS conditions and those with box boundary conditions, where the spatial section of the boundary is a sphere of finite radius, appear similar. Both exhibit a similar phase structure and Hawking-Page phase transition. However, when we introduce a U(1) gauge field to the system, discrepancies in thermodynamic properties between the two cases has been reported in [7] (JHEP 11 (2016) 041). In this paper, by accepting the assumption that all Euclidean saddles contribute to the partition function, I found that these discrepancies are resolved due to the contribution from the "bag of gold (BG)," which is the class of Euclidean geometries whose area of bolt is bigger than that of the boundary. As a result, the Hawking-Page phase structure is restored, with the unexpected properties that the upper bound of thermodynamic entropy is always larger than the boundary area divided by 4G when the chemical potential is non-zero, and that such high entropy states are realized at sufficiently high temperature. [ABSTRACT FROM AUTHOR]

An axiom system for Euclidean geometry with Euclid’s version of the parallel postulate, in which the order axioms are introduced in terms of the separation a line introduces in the plane, as pioneered by Sperner (Math Ann 121:107–130, 1949), in which the compass can be used only to transport segments, which lacks the Pasch axiom, is shown to imply the Pasch axiom due to the very form in which Euclid chose to express his fifth postulate. This shows, as first noted without proof by Salvatore di Noi in that same year 1949, that Euclid did not need the Pasch axiom. [ABSTRACT FROM AUTHOR]