Geometric Interpretation Research Articles

Solving 𝑆-unit, Mordell, Thue, Thue–Mahler and Generalized Ramanujan–Nagell Equations via the Shimura–Taniyama Conjecture

In the first part we construct algorithms (over Q \mathbb {Q} ) which we apply to solve S S -unit, Mordell, cubic Thue, cubic Thue–Mahler and generalized Ramanujan–Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q \mathbb {Q} with good reduction outside a given finite set of rational primes. The first type of our algorithms uses modular symbols, and the second type combines explicit height bounds with efficient sieves. In particular we construct a refined sieve for S S -unit equations which combines Diophantine approximation techniques of de Weger with new geometric ideas. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we used the resulting data to motivate various conjectures and questions, including Baker’s explicit a b c abc -conjecture and a new conjecture on S S -integral points of any hyperbolic genus one curve over Q \mathbb {Q} . In the second part we establish new results for certain old Diophantine problems (e.g. the difference of squares and cubes) related to Mordell equations, and we prove explicit height bounds for cubic Thue, cubic Thue–Mahler and generalized Ramanujan–Nagell equations. As a byproduct, we obtain here an alternative proof of classical theorems of Baker, Coates and Vinogradov–Sprindžuk. In fact we get refined versions of their theorems, which improve the actual best results in many fundamental cases. We also conduct some effort to work out optimized height bounds for S S -unit and Mordell equations which are used in our algorithms of the first part. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Paršin, Szpiro) combined with the Shimura–Taniyama conjecture, and they all do not use lower bounds for linear forms in (elliptic) logarithms. In the third part we solve the problem of constructing an efficient sieve for the S S -integral points of bounded height on any elliptic curve E E over Q \mathbb {Q} with given Mordell–Weil basis of E ( Q ) E(\mathbb {Q}) . Here we combine a geometric interpretation of the known elliptic logarithm reduction (initiated by Zagier) with several conceptually new ideas. The resulting “elliptic logarithm sieve” is crucial for some of our algorithms of the first part. Moreover, it considerably extends the class of elliptic Diophantine equations which can be solved in practice: To demonstrate this we solved many notoriously difficult equations by combining our sieve with known height bounds based on the theory of logarithmic forms.

Sombor indices of γ-sheet of boron clusters

Regarding Euclidean geometry, introduced the Sombor index and its various types. In graph theory, it is the sum of all pairs of neighbouring vertices. New varieties of Sombor indices are introduced using geometrical interpretation. In this paper, we reviewed recently created Sombor indices for gamma-sheet of boron clusters. In particular, Euclidean geometry introduced the first area-based Sombor index. Perimeters are used to construct the third and fifth Sombor indices, while in terms of angular orientation, the second, fourth, and sixth Sombor indices are determined. Sombor indices can be useful in the analysis of chemical networks because they provide a way to quantify the complexity of the network based on the distances between its constituent molecules. They can be used, for example, to compare the structures of different chemical networks, or to identify regions of high or low connectivity within a network. Degree-based Sombor indices have been used in the analysis of various types of chemical networks, including protein-protein interaction networks, metabolic networks, and gene co-expression networks, among others. They can provide insights into the structure and function of these networks and can be used for tasks such as network clustering, identification of key nodes, and prediction of network behaviour. All the versions of Sombor indices are depend on the degree or valency of bond. First, finding out the degree type of the entire network and then the edge type is also essential to compute the Sombor index, once both types are determined, put the values in the derived formulas, and one can have imported results of Sombor indices.

Manin–Drinfeld cycles and derivatives of $L$-functions

We study algebraic cycles in the moduli space of \operatorname{PGL}_2 -shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner–Drinfeld cycle is the product of the r -th central derivative of an automorphic L -function L(\pi,s) and Waldspurger's toric period integral. When L(\pi,1/2) \neq 0 , this gives a new geometric interpretation for the Taylor series expansion. When L(\pi,1/2) = 0 , the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof sheds new light on the algebraic correspondence introduced by Yun and Zhang, which is the geometric incarnation of “differentiating the L -function”. We realize it as the Lie algebra action of e+f \in \mathfrak{sl}_2 on (\mathbb{Q}_\ell^2)^{\otimes 2d} . The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur–Weyl duality.

Sampling planar tanglegrams and pairs of disjoint triangulations

A tanglegram consists of two rooted binary trees and a perfect matching between their leaves, and a planar tanglegram is one that admits a layout with no crossings. We show that the problem of generating planar tanglegrams uniformly at random reduces to the corresponding problem for irreducible planar tanglegram layouts, which are known to be in bijection with pairs of disjoint triangulations of a convex polygon. We extend the flip operation on a single triangulation to a flip operation on pairs of disjoint triangulations. Interestingly, the resulting flip graph is both connected and regular, and hence a random walk on this graph converges to the uniform distribution. We also show that the restriction of the flip graph to the pairs with a fixed triangulation in either coordinate is connected, and give diameter bounds that are near optimal. Our results furthermore yield new insight into the flip graph of triangulations of a convex n-gon with a geometric interpretation on the associahedron.

Geometrical and physical interpretation of the Levi-Civita spacetime in terms of the Komar mass density

We revisit the interpretation of the cylindrically symmetric, static vacuum Levi-Civita metric, known in either Weyl, Einstein–Rosen, or Kasner-like coordinates. The Komar mass density of the infinite axis source arises through a suitable compactification procedure. The Komar mass density mu _{K} calculated in Einstein–Rosen coordinates, when employed as the metric parameter, leads to a number of advantages. It eliminates double coverages of the parameter space, vanishes in flat spacetime and when small, it corresponds to the mass density of an infinite string. After a comprehensive analysis of the local and global geometry, we proceed with the physical interpretation of the Levi-Civita spacetime. First we show that the Newtonian gravitational force is attractive and its magnitude increases monotonically with all positive mu _{K}, asymptoting to the inverse of the proper distance in the radial direction. Second, we reveal that the tidal force between nearby geodesics (hence gravity in the Einsteinian sense) attains a maximum at mu _{K}=1/2 and then decreases asymptotically to zero. Hence, from a physical point of view the Komar mass density of the Levi-Civita spacetime encompasses two contributions: Newtonian gravity and acceleration effects. An increase in mu _{K} strengthens Newtonian gravity but also drags the field lines increasingly parallel, eventually transforming Newtonian gravity through the equivalence principle into a pure acceleration field and the Levi-Civita spacetime into a flat Rindler-like spacetime. In a geometric picture the increase of mu _{K} from zero to infty deforms the planar sections of the spacetime into ever deepening funnels, eventually degenerating into cylindrical topology in an appropriately chosen embedding.

Thermodynamic uncertainty relations for steady-state thermodynamics.

A system can be driven out of equilibrium by both time-dependent and nonconservative forces, which gives rise to a decomposition of the dissipation into two nonnegative components, called the excess and housekeeping entropy productions. We derive thermodynamic uncertainty relations for the excess and housekeeping entropy. These can be used as tools to estimate the individual components, which are in general difficult to measure directly. We introduce a decomposition of an arbitrary current into housekeeping and excess parts, which provide lower bounds on the respective entropy production. Furthermore, we also provide a geometric interpretation of the decomposition and show that the uncertainties of the two components are not independent, but rather have to obey a joint uncertainty relation, which also yields a tighter bound on the total entropy production. We apply our results to a paradigmatic example that illustrates the physical interpretation of the components of the current and how to estimate the entropy production.

Noncommutative instantons in diverse dimensions

AbstractThis is a mini-review about generalized instantons of noncommutative gauge theories in dimensions 4, 6 and 8, with emphasis on their realizations in type II string theory, their geometric interpretations, and their applications to the enumerative geometry of non-compact toric varieties.

Through the looking glass, and what algebra found there: historically informed conceptual metaphors of algebraic substitution and Gaussian elimination

Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff & Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks.

Low-dimensional organization of global brain states of reduced consciousness

Brain states are frequently represented using a unidimensional scale measuring the richness of subjective experience (level of consciousness). This description assumes a mapping between the high-dimensional space of whole-brain configurations and the trajectories of brain states associated with changes in consciousness, yet this mapping and its properties remain unclear. We combine whole-brain modeling, data augmentation, and deep learning for dimensionality reduction to determine a mapping representing states of consciousness in a low-dimensional space, where distances parallel similarities between states. An orderly trajectory from wakefulness to patients with brain injury is revealed in a latent space whose coordinates represent metrics related to functional modularity and structure-function coupling, increasing alongside loss of consciousness. Finally, we investigate the effects of model perturbations, providing geometrical interpretation for the stability and reversibility of states. We conclude that conscious awareness depends on functional patterns encoded as a low-dimensional trajectory within the vast space of brain configurations.

The Spinor Expressions of Mannheim Curves in Euclidean 3-Space

In this paper, the spinor formulations of Mannheim curve pair are investigated. First of all, two spinors matching to Mannheim curve pair are given and by considering the relationships between the Frenet frames of Mannheim curve pair, the relationship between two spinors matching to this curve pair are gotten. Therefore, a geometric interpretations of spinors are obtained using the Mannheim curve pair and considering Mannheim curve as helix the spinor formulations of Mannheim curve pair are given. Moreover, the spinor formulations are also obtained for the curvatures of the Mannheim curve pair. Consequently, an example of these spinors is obtained. Therefore, it is thought that this study will make an important contribution to the mathematical analysis and geometric interpretation of spinors, which have many uses in physics.

Spinor Representations of Positional Adapted Frame in the Euclidean 3-Space

The main goal of this study is to bring together the spinors, which have a major place in several disciplines from mathematics to physics, and Positional Adapted Frame (PAF) which is a new type frame that attracts the attention of many researchers. In accordance with this purpose, we introduce the spinor representations for the trajectories endowed with PAF in the Euclidean 3-space $\mathbb{E}^3$, and construct the spinor equations of PAF vectors. Then, we find the relations between spinor representations of PAF and Serret-Frenet frame. Also we give some results and present some geometric interpretations with respect to this relationship. Moreover, we present an illustrative numerical example in order to support the given theorems and results.

Cancellation of spurious poles in N=4 SYM: Physical and geometric

This paper shows that not only do the codimension one spurious poles of NkMHV tree level diagrams in N=4 SYM theory cancel in the tree level amplitude as expected, but their vanishing loci have a geometric interpretation that is tightly connected to their representation in the positive Grassmannians. In general, given a positroid variety, Σ, and a minimal matrix representation of it in terms of independent variable valued matrices, MV, one can define a polynomial, R(V) that is uniquely defined by the Grassmann necklace, I, of the positroid cell. The vanishing locus of R(V) lies on the boundary of the positive variety Σ‾∖Σ, but not all boundaries intersect the vanishing loci of a factor of R(V). We use this to show that the codimension one spurious poles of N=4 SYM, represented in twistor space, cancel in the tree level amplitude.

On the Symmetries of a Kaehler Manifold

The natural symmetries of Riemannian manifolds are described by the symmetries of its Riemann curvature tensor. In that sense, the most symmetric manifolds are the constant sectional curvature ones. Its natural generalizations are locally symmetric manifolds, semisymmetric manifolds, and pseudosymmetric manifolds. The analogous generalizations of constant holomorphic sectional curvature Kaehler manifolds are locally symmetric Kaehler manifolds, semisymmetric Kaehler manifolds, and holomorphically pseudosymmetric Kaehler manifolds. Do they differ in some way from their Riemannian analogues? Yes, we prove they can all be characterized only in terms of holomorphic planes. Furthermore, the concept of holomorphically pseudosymmetric Kaehler manifold is different from the classical notion of pseudosymmetric Riemannian manifold proposed by Deszcz. We study some relations between both definitions of pseudosymmetry and the so called double sectional curvatures in the sense of Deszcz. We also present a geometric interpretation of the complex Tachibana tensor and a new characterization of constant holomorphic sectional curvature Kaehler manifolds.

On geometrical interpretation of alignment phenomenon

The observed alignment of spots in the X-ray films in cosmic ray emulsion experiments is analyzed and interpreted in the framework of geometrical approach. It is shown that the high degree of alignment can appear partly due to the selection procedure of most energetic particles itself and the threshold on the energy deposition together with the transverse momentum conservation.

Spherical winding and helicity

In ideal magnetohydrodynamics, magnetic helicity is a conserved dynamical quantity and a topological invariant closely related to Gauss linking numbers. However, for open magnetic fields with non-zero boundary components, the latter geometrical interpretation is complicated by the fact that helicity varies with non-unique choices of a field’s vector potential or gauge. Evaluated in a particular gauge called the winding gauge, open-field helicity in Cartesian slab domains has been shown to be the average flux-weighted pairwise winding numbers of field lines, a measure constructed solely from field configurations that manifest its topological origin. In this paper, we derive the spherical analogue of the winding gauge and the corresponding winding interpretation of helicity, in which we formally define the concept of spherical winding of curves. Using a series of examples, we demonstrate novel properties of spherical winding and the validity of spherical winding helicity. We further argue for the canonical status of the winding gauge choice among all vector potentials for magnetic helicity by exhibiting equivalences between local coordinate changes and gauge transformations.

Wald-Zoupas prescription with soft anomalies

We show that the Wald-Zoupas prescription for gravitational charges is valid in the presence of anomalies and field-dependent diffeomorphism, but only if these are related to one another in a specific way. The geometric interpretation of the allowed anomalies is exposed looking at the example of BMS symmetries: They correspond to soft terms in the charges. We determine if the Wald-Zoupas prescription coincides with an improved Noether charge. The necessary condition is a certain differential equation, and when it is satisfied, the boundary Lagrangian of the resulting improved Noether charge contains in general a non-trivial corner term that can be identified a priori from a condition of anomaly-freeness. Our results explain why the Wald-Zoupas prescription works in spite of the anomalous behaviour of BMS transformations, and should be helpful to relate different branches of the literature on surface charges.

Analysis of (sub-)Riemannian PDE-G-CNNs

Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalizes G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously (1) reduce network complexity, (2) increase classification performance, and (3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper, we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry.

Markov chain Monte Carlo methods for graph refinement in spinfoam cosmology

We study the behavior of the Lorentzian Engle-Pereira-Rovelli-Livine spinfoam amplitude with homogeneous boundary data, under a graph refinement going from five to twenty boundary tetrahedra. This can be interpreted as a wave function of the Universe, for which we compute boundary geometrical operators, correlation functions, and entanglement entropy. The numerical calculation is made possible by adapting the Metropolis-Hastings algorithm, along with recently developed computational methods appropriate for the deep quantum regime. We confirm that the transition amplitudes are stable against such refinement. We find that the average boundary geometry does not change, but the new degrees of freedom correct the quantum fluctuations of the boundary and the correlations between spatial patches. The expectation values are compatible with their geometrical interpretation and the correlations between neighboring patches decay when computed across different spinfoam vertices.

Geometry of Relevant Implication II

This note extends earlier results on geometrical interpretations of the logic KR to prove some additional results, including a simple undecidability proof for the four-variable fragment of KR.

Qualitative stability analysis of cosmological parameters in f(T, B) gravity

We analyze the cosmological solutions of f(T, B) gravity using dynamical system analysis where T is the torsion scalar and B be the boundary term scalar. In our work, we assume three specific cosmological models. For first model, we consider f(T,B)=f_{0}(B^{k}+T^{m}), where k and m are constants. For second model, we consider f(T,B)=f_{0}T B, for third model, we consider f(T,B)=alpha T^{2}. We generate an autonomous system of differential equations for each models by introducing new dimensionless variables. To solve this system of equations, we use dynamical system analysis. We also investigate the critical points and their natures, stability conditions and their behaviors of Universe expansion. For first and second models, we get two stable critical points, while for third model we get one stable critical point. The phase plots of this system are analyzed in detail and study their geometrical interpretations also. For these three models, we evaluated density parameters such as Omega _{r}, Omega _{m}, Omega _{Lambda } and omega _{eff} and deceleration parameter (q) and find their suitable range of the parameter lambda for stability. For first model, we get omega _{eff}=-0.833,-0.166 and for second model, we get omega _{eff}=-frac{1}{3}. This shows that both the models are in quintessence phase. For third model we get accelerated expansion of the Universe. Further, we compare the values of EoS parameter and deceleration parameter with the observational values.