Geometric Interpretation Research Articles

Contrast gain control is a reparameterization of a population response curve.

Neurons in primary visual cortex (area V1) adapt in varying degrees to the average contrast of the environment, suggesting that the representation of visual stimuli may interact with the state of cortical gain control in complex ways. To investigate this possibility, we measured and analyzed the responses of neural populations in mouse V1 to visual stimuli as a function of contrast in different environments, each characterized by a unique distribution of contrast values. Our findings reveal that, for a fixed stimulus, the population response can be described by a vector function r(gec), where the gain ge is a decreasing function of the mean contrast of the environment. Thus, gain control can be viewed as a reparameterization of a population response curve, which is invariant across environments. Different stimuli are mapped to distinct curves, all originating from a common origin, corresponding to a zero-contrast response. Altogether, our findings provide a straightforward, geometric interpretation of contrast gain control at the population level and show that changes in gain are well-matched among members of a population.

Projected state ensemble of a generic model of many-body quantum chaos

Abstract The projected ensemble is based on the study of the quantum state of a subsystem A conditioned on projective measurements in its complement. Recent studies have observed that a more refined measure of the thermalization of a chaotic quantum system can be defined on the basis of convergence of the projected ensemble to a quantum state design, i.e. a system thermalizes when it becomes indistinguishable, up to the kth moment, from a Haar ensemble of uniformly distributed pure states. Here we consider a random unitary circuit with the brick-wall geometry and analyze its convergence to the Haar ensemble through the frame potential and its mapping to a statistical mechanical problem. This approach allows us to highlight a geometric interpretation of the frame potential based on the existence of a fluctuating membrane, similar to those appearing in the study of entanglement entropies. At large local Hilbert space dimension q, we find that all moments converge simultaneously with a time scaling linearly in the size of region A, a feature previously observed in dual unitary models. However, based on the geometric interpretation, we argue that the scaling at finite q on the basis of rare membrane fluctuations, finding the logarithmic scaling of design times t k = O ( log ⁡ k ) . Our results are supported with numerical simulations performed at q = 2.

A note on Weyl gauge symmetry in gravity

Abstract A scale invariant theory of gravity, containing at most two derivatives, requires, in addition to the Riemannian metric, a scalar field and (or) a gauge field. The gauge field is usually used to construct the affine connection of Weyl geometry. In this note, we incorporate both the gauge field and the scalar field to build a generalised scale invariant Weyl affine connection. The Ricci tensor and the Ricci scalar made out of this generalised Weyl affine connection contain, naturally, kinetic terms for the scalar field and the gauge field. This provides a geometric interpretation for these terms. It is also shown that scale invariance in the presence of a cosmological constant and mass terms is not completely lost. It becomes a duality transformation relating various fields.

Mathematical model for predicting the yield of apple trees on rootstock 62-396

Traditionally, forecasting apple tree yields was carried out on the basis of statistics of already obtained long-term yield dynamics or based on generative formations formed on the tree. The purpose of the work is to identify the most significant soil parameters for the formation of the apple tree varieties yield on the dwarf rootstock 62-396 and to build a mathematical model for predicting fruit yield in the conditions of the Central Chernozem region of the Russian Federation. Field research was carried out in 2004-2022 in industrial apple tree plantations at the age of full fruiting in the Lipetsk (two farms) and Tambov (eight farms) regions. Seven horticultural soil types have been studied. Based on statistical analysis, a mathematical model of apple tree productivity was obtained in the form of an analytical dependence on three factors. A geometric interpretation of the regression model is given using the Mathcad symbolic mathematics system in the form of response surfaces and the corresponding level lines. Practical recommendations are given for choosing rational parameters within selected intervals of factor variation.

A new Approach to Hyper Dual Split Quaternions with Different Polar Representation

Hamilton first introduced quaternions in 1843 as a way to represent rotations in three dimensional space, and since then, they have become the important tool in many fields. One advantage of quaternions over other methods of representing rotations is their ability to avoid the problem of gimbal lock, which can occur when using Euler angles. Quaternions also have a relatively simple algebraic structure and can be efficiently implemented in computer algorithms. In recent years, quaternions have been used in the development of virtual reality systems and computer games, where they are used to represent orientations of objects in three-dimensional space. They have also been applied in robotics, control theory, and signal processing. Overall, quaternions have become the valuable tools in many areas of mathematics and engineering, and their usage continue to expand. Sangwine and Bihan introduced a quaternion polar representation that draws inspiration from the Cayley-Dickson form. In their formulation, they express quaternions using a complex modulus and argument. The Cayley-Dickson construction is a mathematical procedure that extends the concept of complex numbers to higher dimensions, paving the way for the development of quaternions. On the other hand, the complex argument represents the direction or orientation of the quaternion in a manner analogous to the argument of a complex number. This approach provides a concise and insightful way to represent quaternions, offering a geometric interpretation that aligns with the principles of complex analysis.

Defining geometric gauge theory to accommodate particles, continua, and fields

Gauge theory underpins the quantum field theories of the standard model, and in a previous paper was shown via a geometric approach to describe classical electromagnetism in a form which approximates quantum electrodynamics. Here we formalize and generalize the notion of a geometric gauge theory, then apply this framework to classical physical models, including an improved Lagrangian for matter field electromagnetism. We find a remarkably consistent series of actions, with straightforward limits under which each previous one may be obtained. Ancillary benefits include a gauge-independent Galilean Lagrangian, a geometric interpretation for the unusual metric dependence of four-momentum, a modern treatment of the effects of worldline variation on the four-current, a gauge theory of gravity which includes a matter field, and consistent units for matter field electromagnetism.

Flexural wave propagation in canonical quasicrystalline-generated waveguides

We investigate the propagation of harmonic flexural waves in periodic two-phase phononic multi-supported continuous beams whose elementary cells are designed according to the quasicrystalline standard Fibonacci substitution rule. The resulting dynamic frequency spectra are studied with the aid of a trace-map formalism which provides a geometrical interpretation of the recursive rule governing traces of the relevant transmission matrices: the traces of three consecutive elementary cells can be represented as a point on the surface defined by an invariant function of the square root of the circular frequency, and the recursivity implies the description of a discrete orbit on the surface. In analogy with the companion axial problem, we show that, for specific layouts of the elementary cell (the canonical configurations), the orbits are almost periodic. Likewise, for the same layouts, the stop-/pass-band diagrams along the frequency domain are almost periodic. Several periodic orbits exist and each corresponds to a self-similar portion of the dynamic spectra whose scaling law can be investigated by linearising the trace map in the neighbourhood of the orbit. The obtained results provide a new piece of theory to better understand the dynamic behaviour of two-phase flexural periodic waveguides whose elementary cell is obtained from quasicrystalline generation rules.

To justification of concrete strength criterion at biaxial compression

Introduction. Numerous experimental data from Russian and foreign researchers indicate that the classical plasticity hypotheses do not take into account the different resistance to uniaxial tension and compression, the influence of the spherical tensor. At the same time, experiments show that the ultimate resistance depends on the type of stress state, and hydrostatic pressure increases the strength and plasticity of solids. Aim. To establish the dependence of the influence of the second component of stresses during biaxial compression of concrete on the parameters of the complete diagrams of deformation of the material σbR and εbR it is necessary to describe these diagrams, and to construct a closed curve on the plane of the main stresses (concrete strength criterion). Materials and methods. Based on the experimental materials of foreign and domestic researchers, including the experiments of the authors of the article, methods of mechanics of deformed solids, limit curves and a closed curve on the plane of the main stresses in the form of a chain line forming the smallest area strength surface in the form of a catenoid are proposed. Results. The article provides an analysis of the known strength criteria from the point of view of their geometric interpretation in the stress space. It is shown that these studies relate mainly to metals and metal structures, and for the design of reinforced concrete and steel-concrete structures in a complex stress state, it is necessary to develop an appropriate criterion for the strength of concrete. Conclusions. As a result, the proposed limit curves and the surface (material strength criterion) on the plane of main stresses in the form of a catenoid accurately reflect the behavior of concrete under conditions of uniform and uneven flat stress state, and the equation of the surface in the form of a catenoid is a generalization of the equations of limit curves for each of the three types of flat stress state. At the same time, there is no overestimation of strength in the "compression – compression" area in this case.

Geometric interpretation of efficient weight vectors

Pairwise comparison matrices (PCMs) are frequently used in different multicriteria decision making problems. A weight vector is said to be efficient if no other weight vector is at least as good in estimating the elements of the PCM, and strictly better in at least one position. Understanding the efficient weight vectors is crucial to determine the appropriate weight calculation technique for a given problem. In this paper we study the set of efficient weight vectors for three and four dimensions (alternatives) from a geometric viewpoint, which is a complementary to the algebraic approach used in the literature. Besides providing well-interpretable demonstrations, we also draw attention to the particular role of weight vectors calculated from spanning trees. Weight vectors corresponding to line graphs are vertices of the (polyhedral, but usually nonconvex) set of efficient weight vectors, while weight vectors corresponding to other spanning trees are also on the boundary.

Stated SL(n$n$)‐skein modules and algebras

AbstractWe develop a theory of stated SL()‐skein modules, , of 3‐manifolds marked with intervals in their boundaries. These skein modules, generalizing stated SL(2)‐modules of the first author, stated SL(3)‐modules of Higgins', and SU(n)‐skein modules of the second author, consist of linear combinations of framed, oriented graphs, called ‐webs, with ends in , considered up to skein relations of the ‐Reshetikhin–Turaev functor on tangles, involving coupons representing the anti‐symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3‐manifold along a disk resulting in a 3‐manifold yields a homomorphism for all . That result allows to analyze the skein modules of 3‐manifolds through the skein modules of their pieces. The theory of stated skein modules is particularly rich for thickened surfaces , in whose case, is an algebra, denoted by . One of the main results of this paper asserts that the skein algebra of the ideal bigon is isomorphic with and it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on . (In particular, the coproduct is given by a splitting homomorphism.) We show that for surfaces with boundary every splitting homomorphism is injective and that is a free module with a basis induced from the Kashiwara–Lusztig canonical bases. Additionally, we show that a splitting of a thickened bigon near a marking defines a right ‐comodule structure on , or dually, a left ‐module structure. Furthermore, we show that the skein algebra of surfaces glued along two sides of a triangle is isomorphic with the braided tensor product of Majid. These results allow for geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. Building upon the above results, we prove that the factorization homology with coefficients in the category of representations of is equivalent to the category of left modules over for surfaces with . We also establish isomorphisms of our skein algebras with the quantum moduli spaces of Alekseev–Schomerus and with the internal algebras of the skein categories for these surfaces and .

Dissecting polytopes: Landau singularities and asymptotic expansions in 2 → 2 scattering

Parametric representations of Feynman integrals have a key property: many, frequently all, of the Landau singularities appear as endpoint divergences. This leads to a geometric interpretation of the singularities as faces of Newton polytopes, which facilitates algorithmic evaluation by sector decomposition and asymptotic expansion by the method of regions. Here we identify cases where some singularities appear instead as pinches in parametric space for general kinematics, and we then extend the applicability of sector decomposition and the method of regions algorithms to such integrals, by dissecting the Newton polytope on the singular locus. We focus on 2 → 2 massless scattering, where we show that pinches in parameter space occur starting from three loops in particular nonplanar graphs due to cancellation between terms of opposite sign in the second Symanzik polynomial. While the affected integrals cannot be evaluated by standard sector decomposition, we show how they can be computed by first linearising the graph polynomial and then splitting the integration domain at the singularity, so as to turn it into an endpoint divergence. Furthermore, we demonstrate that obtaining the correct asymptotic expansion of such integrals by the method of regions requires the introduction of new regions, which can be systematically identified as facets of the dissected polytope. In certain instances, these hidden regions exclusively govern the leading power behaviour of the integral. In momentum space, we find that in the on-shell expansion for wide-angle scattering the new regions are characterised by having two or more connected hard subgraphs, while in the Regge limit they are characterised by Glauber modes.

Canonical lifts in multisymplectic De Donder–Weyl Hamiltonian field theories

We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder–Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called natural Noether symmetries present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether’s theorem. The Klein–Gordon field, the Polyakov bosonic string, and Einstein–Cartan gravity in 3+1 dimensions are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known Virasoro constraint.

Pricing a product with network effects for sale to a fixed population of customers

AbstractWe consider a deterministic dynamic pricing problem for a product that exhibits network effects and that is sold to a fixed heterogeneous population of customers. We begin by introducing a demand model wherein those customers are arrayed over two‐dimensional space according to a bivariate probability distribution. Each customer's location in space provides a description of that customer's intrinsic value for the product as well as the extent to which the customer is influenced by the network effect. In the pricing problem, as sales accumulate over time, the set of customers who have already purchased the product grows, while the set of customers who have not yet purchased the product shrinks. The total customer population remains fixed. Those who have not yet purchased constitute the remaining population of potential buyers of the product. As time moves forward, the mix of customers that remain as potential buyers evolves endogenously. The demand model yields a geometric interpretation of the remaining population of potential buyers, and gives rise to a dynamic program with states that are sets in two‐dimensional space. It is not practical to solve the dynamic pricing problem to optimality, so we present bounds and comparative statics results that help us identify tractable heuristics and obtain rigorous performance guarantees. In numerical experiments, we find that fixed‐price policies may perform poorly, especially when the network effect is strong or the time horizon is long. We also introduce a stochastic version of the problem that uses a spatial Poisson process to describe the customers, and we develop and analyze a heuristic approach for that formulation.

Contrast gain control is a reparameterization of a population response curve.

Neurons in primary visual cortex (area V1) adapt in different degrees to the average contrast of the environment, suggesting that the representation of visual stimuli may interact with the state of cortical gain control in complex ways. To investigate this possibility, we measured and analyzed the responses of neural populations to visual stimuli as a function of contrast in different environments, each characterized by a unique distribution of contrast. Our findings reveal that, for a given stimulus, the population response can be described by a vector function , where the gain is a decreasing function of the mean contrast of the environment. Thus, gain control can be viewed as a reparameterization of a population response curve, which is invariant across environments. Different stimuli are mapped to distinct curves, all originating from a common origin, corresponding to a zero-contrast response. Altogether, our findings provide a straightforward, geometric interpretation of contrast gain control at the population level and show that changes in gain are well coordinated among members of a neural population.

Impacts of thermally stratified medium on transient convective heat transfer between co‐axial horizontal fixed pipes: Applications of the thermal stratification

AbstractThermal stratification improves coaxial pipe systems’ efficiency and stability. Thermal stratification enables accurate temperature maintenance, reduces thermal stress, optimizes heat transmission performance, and minimizes usage of energy to guarantee the system's long‐term performance. The main aim of the current study is to investigate the impacts of thermal stratification on buoyancy force flow and thermal transmission between coaxial fixed pipes. In the present research, the applications of thermally stratified medium on transient convective heat transfer between two coaxial fixed pipes are studied. A two‐dimensional mathematical formulation in terms of mutually nonlinear partial differential equations is used to analyze the unsteady flow and temperature field between the co‐axial pipes, when the internal pipe is uniformly heated and the outer wall of the external pipe is placed at infinity from the surface of the inner fixed pipe. Flow is assumed along the axial direction of the internal pipe and stationary boundary condition is assumed at the surface of the inner pipe. The coupled equations of the simulated model are solved numerically by applying the Implicit Finite Difference Technique. The computed outcomes in the form of geometrical interpretation are highlighted by using the technically advanced software TECHPLOT‐360. Comprehensive detail of the obtained results for the non‐dimensional parameters included in the flow formulation is predicted for steady state velocity, temperature distribution, time‐dependent surface shearness and time‐dependent energy shearness in results and discussion section of the manuscript. The emphasis is placed on the thermal stratification parameter in the above mentioned chief quantities. From the obtained results, it is predicted that the fluid flow pattern and thermal distribution are both reduced for rising values of the thermal stratification parameter S = 0.001, 0.03, 0.05, and 0.07. Minimum flow and thermal profile are observed at S = 0.07. Further, the amplitude of the time‐dependent surface shearness is uniformly distributed throughout the medium and the amplitude of the time‐dependent energy shearness is reduced effectively for S = 1.0, 5.0, and 10.0.

Geometric genuine multipartite entanglement for four-qubit systems

Xie and Eberly introduced a genuine multipartite entanglement (GME) measure ‘concurrence fill’ (Xie and Eberly, 2021) for three-party systems. It is defined as the area of a triangle whose side lengths represent squared concurrence in each bi-partition. However, it has been recently shown that concurrence fill is not monotonic under LOCC, hence not a faithful measure of entanglement. Though it is not a faithful entanglement measure, it encapsulates an elegant geometric interpretation of bipartite squared concurrences. There have been a few attempts to generalize GME quantifier to four-party settings and beyond. However, some of them are not faithful, and others simply lack an elegant geometric interpretation. The recent proposal from Xie et al.. constructs a concurrence tetrahedron, whose volume gives the amount of GME for four-party systems; with generalization to more than four parties being the hypervolume of the simplex structure in that dimension. Here, we show by construction that to capture all aspects of multipartite entanglement, one does not need a more complex structure, and the four-party entanglement can be demonstrated using 2D geometry only. The subadditivity together with the Araki-Lieb inequality of linear entropy is used to construct a direct extension of the geometric GME quantifier to four-party systems resulting in quadrilateral geometry. Our quantifier can be geometrically interpreted as a combination of three quadrilaterals whose sides result from the concurrence in one-to-three bi-partition, and diagonal as concurrence in two-to-two bipartition.

On the charge algebra of causal diamonds in three dimensional gravity

Covariant phase space methods are applied to the analysis of a causal diamond in 2+1-dimensional pure Einstein gravity. It is found that the reduced phase space is parametrized by a family of charges with a dual geometrical interpretation: they are geometric observables on the corner of the diamond, and they generate diffeomorphisms. The Poisson brackets among them close into an algebra. Knowledge of the corner charges therefore permits reconstruction of the diamond geometry, which realizes a form of local holography. The results are contrasted with the literature, and the path to a quantum description of spacetime geometry is discussed.

On Vafa–Witten equations over Kähler manifolds

Abstract In this paper, we study the analytic properties of solutions to the Vafa–Witten equation over a compact Kähler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the C ∗ \mathbb{C}^{*} invariant locus of the moduli space is shown to behave similarly to the Hermitian Yang–Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes’ results on rank two solutions over Kähler surfaces together with a new complex geometric interpretation. The moduli space of SU ⁢ ( 2 ) \mathsf{SU}(2) monopoles and some related examples are also discussed in the final section.

A geometric interpretation of the Peano Axioms as a didactic strategy for teaching and learning natural numbers

Peano’s axioms are a set of propositions that allow a formal definition of the set of natural numbers. In these axioms, Peano rescues the essence of counting objects and materializes it by assuming the existence of a set, of a function in this set (which he calls successor) and an axiom of induction. In this work a geometrical representation and interpretation of each of these axioms is made, which allows understanding the independence and necessity of each one of them, which facilitates giving the conceptual rigor of them by making use of mathematical language. Finally, using Euclidean geometry, a geometric construction of a system of natural numbers on a straight line is made.

Geometric BV for twisted Courant sigma models and the BRST power finesse

We study twisted Courant sigma models, a class of topological field theories arising from the coupling of 3D 0-/2-form BF theory and Chern-Simons theory and containing a 4-form Wess-Zumino term. They are examples of theories featuring a nonlinearly open gauge algebra, where products of field equations appear in the commutator of gauge transformations, and they are reducible gauge systems. We determine the solution to the master equation using a technique, the BRST power finesse, that combines aspects of the AKSZ construction (which applies to the untwisted model) and the general BV-BRST formalism. This allows for a geometric interpretation of the BV coefficients in the interaction terms of the master action in terms of an induced generalised connection on a 4-form twisted (pre-)Courant algebroid, its Gualtieri torsion and the basic curvature tensor. It also produces a frame independent formulation of the model. We show, moreover, that the gauge fixed action is the sum of the classical one and a BRST commutator, as expected from a Schwarz type topological field theory.