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The article explores the concept of the existence of mathematical ideas and whether they existed before the big bang. It presents different perspectives on the nature of mathematics, including the idea that mathematics is eternal and has always existed, the notion that mathematics is a creation of the human mind, and the belief that mathematics exists in an immaterial realm of universal forms. The article also discusses the applicability of mathematical concepts in different contexts, such as Euclidean geometry and curved space. Additionally, the article briefly mentions a survival scenario involving children in the wild and highlights the importance of cultural context in determining a child's ability to survive. [Extracted from the article]

We offer an elementary solution to the shortest distance problem. In addition to being accessible to freshman Calculus students, our approach also provides a bit of extra information, namely a way quantify the "length excess" in terms of how much the path deviates from being a straight line. [ABSTRACT FROM AUTHOR]

Euler's inequality in Euclidean geometry is a famous result relating the circumradius and inradius of a triangle. We provide a new proof of this result using elementary ideas about area and isosceles triangles. [ABSTRACT FROM AUTHOR]

Ptolemy's theorem is a classical result obtained in the late Greek-Roman period, whose first application was to provide computational support to a geocentric cosmological model. This model's most important achievement was that it explained the apparent movement of celestial bodies to a subjective observer on the Earth. What makes Ptolemy's theorem a very interesting case in the history of mathematics is that the Euclidean concept of a Ptolemaic configuration can be investigated in the geometry of general metric spaces, in a situation very similar to the triangle inequality. To complement the historical narrative, in the final part of our paper we introduce a new norm, related to the Euclidean, Chebyshev, and Manhattan norms, and we investigate its properties in relation with other norms, hoping to illustrate how this fundamental configuration traversed Euclidean geometry, complex geometry, and analysis, transformational geometry, to become an interesting classification criterion in metric geometry. [ABSTRACT FROM AUTHOR]

The article focuses on the photographic series by Pygmalion Karatzas which has been inspired by traditional mandalas geometric configurations used in spiritual traditions as a guiding tool for focusing attention and aid meditation.

This work addresses a finite-time visibility-based target tracking problem. The aim of the pursuer is to keep the evader within its field of view, while the evader wants to escape the pursuer's surveillance. Some main contributions of the work are the following: (1) It is proven that minimizing the travel time is equivalent to maximizing the definite integral of the angular speed of the border of the pursuer visibility region. (2) Using Euclidean geometry and set theory methods, the optimality of the player trajectories is demonstrated. (3) It is shown that the motion strategies provided in previous works are not always in Nash equilibrium. [ABSTRACT FROM AUTHOR]

COLUMNS & DEPARTMENTS -> The human brain is both a marvel and a mystery of evolution: Packed into a volume about one-quarter that of an inflated soccer ball, somewhere around 86 billion neurons form networks that enable us to do everything from mindlessly scrolling through Instagram to safely sending people into space. If researchers like Sharpee are right, then to understand the mind, we have to prepare to embrace hyperbolic geometry's tenets - which, when they first appeared, bordered on heresy for the math world. Hungarian mathematician János Bolyai (right) challenged the rules the "Father of Geometry", Euclid (top), had outlined more than 2,000 years earlier. [Extracted from the article]

Graph Neural Networks (GNNs) have demonstrated state-of-the-art performance in a wide variety of analytical tasks. Current GNN approaches focus on learning representations in a Euclidean space, which are effective in capturing non-tree-like structural relations, but they fail to model complex relations in many real-world graphs, such as tree-like hierarchical graph structure. This paper instead proposes to learn representations in both Euclidean and hyperbolic spaces to model these two types of graph geometries. To this end, we introduce a novel approach – Joint hyperbolic and Euclidean g eo m etry c ontrastive graph neural networks (JointGMC). JointGMC is enforced to learn multiple layer-wise optimal combinations of Euclidean and hyperbolic geometries to effectively encode diverse complex graph structures. Further, the performance of most GNNs relies heavily on the availability of large-scale manually labeled data. To mitigate this issue, JointGMC exploits proximity-based self-supervised information in different geometric spaces (i.e., Euclidean, hyperbolic, and Euclidean-hyperbolic interaction spaces) to regularize the (semi-) supervised graph learning. Extensive experimental results on eight real-world graph datasets show that JointGMC outperforms eight state-of-the-art GNN models in diverse graph mining tasks, including node classification, link prediction, and node clustering tasks, demonstrating JointGMC's superior graph representation ability. Code is available at https://github.com/chachatang/jointgmc. [ABSTRACT FROM AUTHOR]

Descartes asserts, leaving it to the reader to persuade himself, that the method he describes suffices to solve any problem solvable by the constructions available in Euclid. He presents it as a movement from the Conics of Apollonius of Perga, a brilliant and difficult triumph of ancient mathematics written in the third century b.c., to Descartes's 1637 Geometry, a decisive stride into modernity. For Apollonius, mathematics was a way of gaining insights into the nature of geometrical forms by envisioning them in the mind; Descartes made mathematics into an activity of manipulation - hence the title of Flaumenhaft's book. David Guaspari Spring 2021 ~ 37 One pillar of the modern world was the project to transform science from a discipline for contemplating nature into a tool for mastering it. [Extracted from the article]

The article discusses the historical significance of Euclid's contributions to mathematics, particularly his development of Euclidean geometry and foundational axioms. Topics include Euclid's role as the "Father of Geometry," his enduring impact on mathematics and logic, and the influence of his work on subsequent generations of mathematicians.

The article presents interview of Professor Hillel Furstenberg who awarded the Abel Prize in mathematics for 2020. Topics discussed include challenging problem in Euclidean geometry: given a triangle, assume two of the angle bisectors have the same length; love of mathematics blended with human kindness; and how prediction theory connects with stationary processes, to look at the past and to associate with it a stationary process.

The author discusses the concept of infinity in mathematics and physics, arguing that infinity prevents the development of unified calculations and theories to define rules of the Universe. A historical overview of infinity theory is presented, including the creation of set theory by mathematician Georg Cantor. Various challenges presented by infinity to physics research are explored, including theories of gravity, the big bang origin of the universe, and the measure problem of the infinite multiverse. Other topics include advocacy against the use of infinity in quantum mechanics by cosmologist Max Tegmark, the development of the atomic clock by physicist David Wineland and infinity-free theories of trigonometry and Euclidean geometry by mathematician Norman Wildberger.

The proposed model takes the GVF vector as a 2D manifold embedded into a 4D Euclidean space. • The proposed model leading to the generalization of the Laplacian operator in Euclidean space to that over manifold. • The two components of the external force are couples with each other to improve the properties of the GVF snake. The GVF snake is effective on converging into concave areas and being insensitive to initialization. However, it fails to converge to areas with deep and narrow concavities or preserve weak edges. In this paper, a novel external force called gradient vector flow over manifold (GVFOM) is proposed for active contours. The proposed model takes the GVF vector as a 2D manifold embedded into a 4D Euclidean space, leading to the generalization of the Laplacian operator in Euclidean Space to that over manifold, and the two components of the external force are couples with each other to improve the properties of the GVF snake. The GVFOM snake has been tested with different kinds of images, and experimental results and comparison manifest that the GVFOM snake has better performance than the GVF and other state-of-the-art snakes on object separation, deep and narrow concavity convergence, weak edge protection, noise robustness while keeping some desirable properties of the GVF model such as initialization insensitivity and large capture range. [ABSTRACT FROM AUTHOR]

A recent preprint abstract discusses the use of non-Euclidean geometric measurements, specifically fractal dimension and lacunarity, to distinguish between different subcomponents of gliomas and determine their IDH and MGMT status. The study found that IDH wildtype gliomas had higher fractality in the enhancing subcomponent compared to IDH mutant gliomas, while IDH mutant gliomas showed higher fractality in the nonenhancing subcomponent. Fractality and lacunarity measures were not sensitive for discriminating MGMT status. These findings suggest that measuring glioma structural heterogeneity using routine MRI scans could provide a noninvasive way to determine the molecular subtype of gliomas and guide clinical management. However, it is important to note that this preprint has not yet undergone peer review. [Extracted from the article]

However, the obviousness of gravitational attraction hides the underlying assumptions embedded in the force model of gravity. RUNNER-UP Writers' Award XXXI: Space "I know that I am mortal by nature, and ephemeral; but when I trace at my pleasure the windings to and fro of the heavenly bodies, I no longer touch the earth with my feet: I stand in the presence of Zeus himself and take my fill of ambrosia", exclaimed Ptolemy in the second century CE. It took the intellectual prowess of Albert Einstein to rectify that error and link the physics of gravity to the curved geometry of our universe. [Extracted from the article]

Abstract Important in pairwise comparisons methods (PC) concepts of reciprocal matrices, consistency conditions and priority vectors of consistent PC matrices are investigated in a more general framework where mappings take values in a not necessarily abelian group. A problem about Pauli's matrices is posed. Notions of left and right priority mappings with respect to a group operation of consistent mappings are introduced. Hou's fixed point theorems on consistent matrices are generalized to theorems on consistent mappings with values in an arbitrarily fixed lattice-ordered group. Relevant to topology basic properties of Koczkodaj's metric which induces Koczkodaj's inconsistency index in PC are studied. It is shown in ZF that Koczkodaj's metric is Cantor complete and not totally bounded, it is equivalent with the Euclidean metric on the set of positive real numbers. A version of Heine–Borel Theorem for Koczkodaj's metric is obtained and briefly applied to the compact bornology of R + n. [ABSTRACT FROM AUTHOR]

Understanding fracture in cementitious materials caused by the deposition and growth of corrosion products requires scale-bridging approaches due to the large length-scale difference between the micro-pores, where deposition occurs, and the structure, where deterioration manifests. Cementitious materials bear a highly heterogeneous micro-structure owing to the fractal nature of micro-pores. Simultaneously, a corrosion-driven fracture is a multi-physics problem involving ionic diffusion, chemical reactions, and stress development. This multi-scale and multi-physical character makes scale-bridging studies computationally costly, often leading to the use of simplified fractal porous media, which has important consequences for the quantitative interpretation of the results. Recent advances in homogenization approaches using Fast-Fourier-Transform (FFT) based methods have raised interest due to their ease of implementation and low computational cost. This paper presents an FFT-based framework for solving corrosion-driven fractures within fractal porous media. We demonstrate the effectiveness of the Fourier-based spectral method in resolving the multiple corrosion-driven mechanisms such as ionic diffusion, stress development, and damage within a fractal porous microstructure. Based on the presented methodology, we analyse the impact of simplifying fractal porous media with simple Euclidean geometry on corrosion-driven fracture. Our results demonstrate the importance of preserving both the porosity and fractal nature of pores for precise and reliable modelling of corrosion-driven failure mechanisms. [ABSTRACT FROM AUTHOR]

The article offers a preview of the pill-shape Capsule side table from Slash Objects.

A novel scale space approach, SphereSiZer, is proposed for exploring structure in spherical data, that is, directional data on the unit sphere of the three-dimensional Euclidean space. The method finds statistically significant gradients of the smooths of the probability density function underlying the observed data. Bootstrap is used to establish significance and inference is summarized with planar maps of contour plots of smooths of the data, overlaid with arrows that indicate the directions and magnitudes of the significant gradients. An effective way to explore such maps is a movie where each frame corresponds to a fixed level of smoothing, that is, a particular spatial scale on the sphere. The SphereSiZer is demonstrated using simulated data as well as two real-data examples. The first example examines the distribution of infant head normal vector directions. The presence of local maxima in the normal vector distribution may indicate head deformity, such as severe flatness or asymmetry. The second example considers the distribution of earthquakes in the Northern Hemisphere. [ABSTRACT FROM AUTHOR]

Ongoing research addressing shape retrieval render highly acute feature descriptors that are computationally intensive. Hence, a simple approach through a novel tessellated version of the Tetrakis square tiling for acute feature characterization aiding shape retrieval is contributed in this paper. The proposed descriptor, named Triangulated Feature Descriptor (TFD), performs feature characterization and abstraction by fusing hybrid geometrical concepts. The mechanism initially tiles the image into square regions that are later tessellated into right-angled triangles. The right-angled triangular neighbors interact locally using the trigonometric identities to produce an angle-based feature map representing an image. The attained feature maps are locally segmented to finally produce a shape histogram. Since, the intended descriptor is modeled based on intrinsic angular interactions between the local neighbors the resulting descriptor is highly invariant to diverse affine transformations. This characteristic is examined by performing rigorous experiments on MPEG-7, TARI-1000, Articulated and PHOS datasets. Competitive analysis reveals a consistent retrieval rate greater than 85% achieved by TFD with increased performance scores. [ABSTRACT FROM AUTHOR]

The classical notion of rigidity (that a formation of agents in $\mathbb {R}^2$ or $\mathbb {R}^3$ is rigidly constrained by interagent distances up to rigid-body transformations of space) is inherently dependent on the nature of Euclidean space and the nature of distance measurements. In this paper, we present a generalized formulation of rigidity, where agent states may lie in heterogeneous and non-Euclidean state spaces with arbitrary differentiable measurement constraints. A key aspect of our approach is to recognize the crucial role that the symmetry action of rigid-body transformations plays in classical rigidity theory. We consider a general symmetry given by a Lie-group action on a heterogeneous state space and define global rigidity of a formation to be the case where the interagent measurements fully constrain the agent locations up to the invariance encoded by the group action. In this framework, we develop general definitions of local rigidity and infinitesimal rigidity and introduce a new notion of robust rigidity that we believe will be important for control applications. To motivate the development, we show how the proposed theory can be applied to generalizations of the established problems of network localization and formation control. The provided results are directly applicable to networks of robotic vehicles involving a mixture of bearing and distance sensors, as well as cases where a collection of ground, submersible, and aerial vehicles operate in a single formation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability analysis for continuous-time positive singular systems with time-varying delays. First, an auxiliary system is introduced to establish the positivity condition for singular time-delay systems. Based on the positivity condition, the stability criterion is obtained for the positive singular systems with constant delays. By analyzing the monotonic property of the system trajectory, we extend the stability condition to the cases with time-varying delays. A numerical simulation is employed to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

This paper presents a novel non-linear extension of the Random Projection method based on the degree-2 homogeneous polynomial kernel. Our algorithm is able to implicitly map data points to the high-dimensional feature space of that kernel and from there perform a Random Projection to an Euclidean space of the desired dimensionality. Pairwise distances between data points in the kernel feature space are approximately preserved in the resulting representation. As opposed to previous kernelized Random Projection versions, our method is data-independent and preserves much of the computational simplicity of the original algorithm. This is achieved by focusing on a specific kernel function, what allowed us to analyze the effect of its associated feature mapping in the distribution of the Random Projection hyperplanes. Finally, we present empirical evidence that the proposed method outperforms alternative approaches in terms of pairwise distance preservation, while being significantly more efficient. Also, we show how our method can be used to approximate the accuracy of non-linear classifiers with efficient linear classifiers in some datasets. [ABSTRACT FROM AUTHOR]

This paper studies stability- and symmetry-preserving $H^2$ optimal model reduction problems of linear systems, which include linear gradient systems as a special case. The problem is formulated as a nonlinear optimization problem on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. To solve the problem by using the trust-region method, the gradient and Hessian of the objective function are derived. Furthermore, it is shown that if we restrict our systems to gradient systems, the gradient and Hessian can be obtained more efficiently. More concretely, by symmetry, we can reduce linear matrix equations to be solved. In addition, by a simple example, we show that the solutions to our problem and a similar problem in some literature works are not unique, and the solution sets of both problems do not contain each other in general. Also, it is revealed that the attained optimal values do not coincide. Numerical experiments show that the proposed method gives a reduced system with the same structure with the original system although the balanced truncation method does not. [ABSTRACT FROM PUBLISHER]

This paper offers an understanding of the contents of the first book of spherics, whoever wrote it-- a rational reconstruction of what it may once have said that goes beyond the historical evidence, which is the second-century BCE Spherics of Theodosios. The reconstruction is done by bringing to the fore what is glossed over and adding the missing conclusion. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerically robust blind equalization scheme to mitigate cochannel and intersymbol interferences and separate signals from a convolutive mixture in multiple-input multiple-output (MIMO) communication systems. The robustness of the proposed scheme is achieved through a particular dual-mode operation applied to a constant-modulus-based space-time algorithm that corrects phase rotations. In the first mode of operation, the consistency in the nonlinear estimates of the transmitted signals is ensured. In the second mode, the Euclidean norms of the equalizer coefficient vectors decrease along the iterations, which allows the algorithm to return to the first mode of operation, where the estimation of the transmitted signals really occurs. In particular, we apply this scheme to a multimodulus version of the multiuser Shalvi-Weinstein algorithm (MU-SWA). By means of a deterministic stability analysis assuming the persistence of excitation condition, we show that the dual-mode MU-SWA does not diverge. [ABSTRACT FROM AUTHOR]

Our lead article for this issue is a timely exploration of mathematics and epidemiology. And Yves Nievergelt rounds out this issue's articles by considering the problem of round-off error in computer algebra systems. Everyone smiles the first time they notice that the standard Graph HT ht formula for the circumference of a circle is the derivative of the standard area formula Graph HT ht . [Extracted from the article]

He uses the occasions to bring up various topics in mathematics itself, such as puzzles, the birthday paradox, math mosaics, and cryptology. Author Chartier surveys instances of use of analytics in various sports, with a light touch. There are many books about mathematics and art; this one distinguishes itself as an "unorthodox geometry textbook", with exercises and fun art projects. [Extracted from the article]

This paper investigates the robust H\infty observer-based control (OBC) for linear time-invariant disturbed uncertain fractional-order systems (DU-FOS). First, the existence conditions for robust H\infty OBC are given. Then, based on the H\infty -norm analysis using the generalized Kalman–Yakubovich–Popov lemma for FOS, and following the fractional derivative order $\alpha$ , new sufficient linear matrix inequalities (LMIs) conditions are obtained to ensure the stability of the estimation errors and the stabilization of the DU-FOS simultaneously. All observer matrices gains and control laws can be computed by solving a unique LMI condition in one step. Numerical simulation is given to illustrate the validity of the proposed method. [ABSTRACT FROM PUBLISHER]

We consider the problem of obtaining state dependent switching strategies for switching linear time-invariant systems which satisfy prespecified dwell-time constraints. Continuous-time strategies which yield exponential stability and a guaranteed level of performance are presented. We also consider discrete-time strategies in which the sampling time is independent of the dwell-times. A numerical example is included to validate the developed theory. [ABSTRACT FROM AUTHOR]

The article discusses the computer program Digital Amati developed by Harry Mairson which allows luthiers to construct and adapt patterns quickly and easily based on François Denis's principles which showed how the old Italians used Euclidean geometry to design their instruments.

The numerical simulation of electromagnetic scattering problem by cylinders of arbitrary cross-section made of homogeneous and anisotropic material is carried out. The anisotropic materials are characterized by symmetric and positive definite magnetic permeability tensors. An efficient and powerful meshfree strong form technique based on the particular solutions of anisotropic radial basis functions (A-RBFs) is extended to deal with anisotropic problem directly. The A-RBFs are constructed by introducing a special norm associated to the appearing anisotropic tensor and replacing it with the standard Euclidean norm in RBFs. The particular solutions for the anisotropic Laplace operator are approximated explicitly by using the anisotropic radial basis functions. Because of material discontinuity at the interface of anisotropic and free regions, two different sets of basis functions, the approximate particular solutions of the anisotropic generalized multiquadric (A-GMQ) and the standard generalized multiquadric (GMQ) functions are employed to approximate the total electric field inside and the scattered field outside the anisotropic scatterer respectively. Some numerical simulations are implemented to confirm the accuracy and efficiency of the method for anisotropic scatterer with arbitrary cross-sections. [ABSTRACT FROM AUTHOR]

In some applications, measurement errors and multiple repeated measurements often lead to a set-based classification task where objects are represented with a set of samples, and the traditional support vector machines (SVMs) do not work in these settings. To deal with this problem, we construct a new classifier called support function machine (SFM) in this work. First, sets in d -dimensional Euclidean space R d are mapped into an infinite-dimensional Banach space C ( S ) (whose elements are functions) via support functions, and then set-based classification in R d is converted into function-based classification in C ( S ). Second, we define the hyperplane via the Riesz representation theorem in Banach space, and discuss the Hausdorff distance of hyperpalnes and maximum margin principle (MMP) in C ( S ). Based on MMP, we construct an optimal problem and discuss some of its properties. Thereafter, we establish an SFM to solve set-based classification. Experiments about water quality evaluation and set-valued data classifications show the superiority of SFM. [ABSTRACT FROM AUTHOR]

A nonlinear controlled system in Euclidean space, which is specified in a finite time interval, is considered. The approach problem of a controlled system with a target set in a finite time interval is studied. An algorithm for an approximate calculation of the solution of the problem is presented. An illustrative example of a “flywheel–pendulum” controlled mechanical system is considered. [ABSTRACT FROM AUTHOR]

Jeremiah Bartz wraps up our article offerings this month with a discourse on so-called "triangular triples", meaning a triple of triangular numbers where the largest is the sum of the other two. In this case, the grazing area is bounded by a series of elliptical arcs, creating a region Brawner refers to as an "elliptagon." James Brawner gets us started with a tour-de-force of calculus, geometry, and trigonometry. [Extracted from the article]

In a previous work we provided the explicit form of the nonlinear PDEs, subjected to the appropriate boundary conditions, which have to be satisfied by transport coefficients for systems out of Onsager's region. Since the proposed PDEs are obtained without neglecting any term present in the balance equations (i.e., the mass, momentum, and energy balance equations), we propose them as a good candidate for describing also transport in thermodynamic systems in turbulent regime. As a special case, we derive the nonlinear PDEs for transport coefficients when the thermodynamic system is subjected to two thermodynamic forces. In this case, the obtained PDE is, in thermodynamical field theory (TFT), analogous to Liouville's equation in Riemannian (or pseudo-Riemannian) geometry. The validity of our model is tested by analysing a concrete example where Onsager's relations manifestly disagree with experience: transport in Tokamak-plasmas. More specifically, we compute the electron mass and energy losses in turbulent FTU (Frascati Tokamak Upgrade)-plasmas. We show the agreement between the theoretical predictions and experimental observations. This approach allows to predict the values of the Bohm and the gyro-Bohm coefficients. To the best of our knowledge, it is the first time that such coefficients have been evaluated analytically. The aim of this series of works is to apply our approach to the Divertor Tokamak Test facility (DTT), to be built in Italy, and to ITER. [ABSTRACT FROM AUTHOR]

As noise is omnipresent, real-world quantities measured by scientists and engineers are commonly obtained in the form of statistical distributions. In turn, perhaps the most compact representation of a given statistical distribution is via the mean-variance approach: the mean manifesting the distribution’s ‘typical’ value, and the variance manifesting the magnitude of the distribution’s fluctuations about its mean. The mean-variance approach is based on an underlying Euclidean-geometry perspective. So very often real-world quantities of interest are non-negative sizes, and their measurements yield statistical size distributions. In this paper, and in the context of size distributions, we present an alternative to the Euclidean-based mean-variance approach: a mean-equality approach that is based on an underlying socioeconomic perspective. We establish two equality indices that score, on a unit-interval scale, the intrinsic ‘egalitarianism’ of size distributions: (i) the poverty equality index which is particularly sensitive to the existence of very small “poor” sizes; (ii) the riches equality index which is particularly sensitive to the existence of very large “rich” sizes. These equality indices, their properties, their computation, their application, and their connections to the mean-variance approach – are explored and described comprehensively. [ABSTRACT FROM AUTHOR]

The aim of this work is to provide a relationship between the relative multifractal spectra of orthogonal projections of a measure μ in Euclidean space and those of μ . As an application we study the relative multifractal analysis of the projections of a measure. [ABSTRACT FROM AUTHOR]

Jensen-type [Jensen-Shannon (JS) and Jensen-Tsallis] kernels were first proposed by Martins et al. (2009). These kernels are based on JS divergences that originated in the information theory. In this paper, we extend the Jensen-type kernels on probability measures to define positive-definite kernels on Euclidean space. We show that the special cases of these kernels include dot-product kernels. Since Jensen-type divergences are multidistribution divergences, we propose their multipoint variants, and study spectral clustering and kernel methods based on these. We also provide experimental studies on benchmark image database and gene expression database that show the benefits of the proposed kernels compared with the existing kernels. The experiments on clustering also demonstrate the use of constructing multipoint similarities. [ABSTRACT FROM AUTHOR]