Geometric Notions Research Articles

Bohr and Rogosinski inequalities for operator valued holomorphic functions

For any complex Banach space X and each p∈[1,∞), we introduce the p-Bohr radius of order N(∈N) is R˜p,N(X) defined byR˜p,N(X)=sup⁡{r≥0:∑k=0N‖xk‖prpk≤‖f‖H∞(D,X)p}, where f(z)=∑k=0∞xkzk∈H∞(D,X). Here D={z∈C:|z|<1} denotes the unit disk. We also introduce the following geometric notion of p-uniformly C-convexity of order N for a complex Banach space X for some N∈N. For p∈[2,∞), a complex Banach space X is called p-uniformly C-convex of order N if there exists a constant λ>0 such that(0.1)(‖x0‖p+λ‖x1‖p+λ2‖x2‖p+⋯+λN‖xN‖p)1/p≤maxθ∈[0,2π)⁡‖x0+∑k=1Neiθxk‖ for all x0, x1, …, xN ∈X. We denote Ap,N(X), the supremum of all such constants λ satisfying (0.1). We obtain the lower and upper bounds of R˜p,N(X) in terms of Ap,N(X). In this paper, for p∈[2,∞) and each N∈N, we prove that a complex Banach space X is p-uniformly C-convex of order N if, and only if, the p-Bohr radius of order NR˜p,N(X)>0. We also study the p-Bohr radius of order N for the Lebesgue spaces Lq(μ) for 1≤p<q<∞ or 1≤q≤p<2. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk D into B(H), where B(H) denotes the space of all bounded linear operator on a complex Hilbert space H.

The robustness of the generalized Gini index

In this paper, we introduce a map varPhi , which we call zonoid map, from the space of all non-negative, finite Borel measures on {mathbb {R}}^n with finite first moment to the space of zonoids of {mathbb {R}}^n. This map, connecting Borel measure theory with zonoids theory, allows to slightly generalize the Gini volume introduced, in the context of Industrial Economics, by Dosi (J Ind Econ 4:875–907, 2016). This volume, based on the geometric notion of zonoid, is introduced as a measure of heterogeneity among firms in an industry and it turned out to be a quite interesting index as it is a multidimensional generalization of the well-known and broadly used Gini index. By exploiting the mathematical context offered by our definition, we prove the continuity of the map varPhi which, in turn, allows to prove the validity of a SLLN-type theorem for our generalized Gini index and, hence, for the Gini volume. Both results, the continuity of varPhi and the SLLN theorem, are particularly useful when dealing with a huge amount of multidimensional data.

Community detection in networks by dynamical optimal transport formulation

Detecting communities in networks is important in various domains of applications. While a variety of methods exist to perform this task, recent efforts propose Optimal Transport (OT) principles combined with the geometric notion of Ollivier–Ricci curvature to classify nodes into groups by rigorously comparing the information encoded into nodes’ neighborhoods. We present an OT-based approach that exploits recent advances in OT theory to allow tuning between different transportation regimes. This allows for better control of the information shared between nodes’ neighborhoods. As a result, our model can flexibly capture different types of network structures and thus increase performance accuracy in recovering communities, compared to standard OT-based formulations. We test the performance of our algorithm on both synthetic and real networks, achieving a comparable or better performance than other OT-based methods in the former case, while finding communities that better represent node metadata in real data. This pushes further our understanding of geometric approaches in their ability to capture patterns in complex networks.

Étale Covers and Fundamental Groups of Schematic Finite Spaces

We introduce the category of finite étale covers of an arbitrary schematic space X and show that, equipped with an appropriate natural fiber functor, it is a Galois Category. This allows us to define the étale fundamental group of schematic spaces. If X is a finite model of a scheme S, we show that the resulting Galois theory on X coincides with the classical theory of finite étale covers on S, and therefore, we recover the classical étale fundamental group introduced by Grothendieck. To prove these results, it is crucial to find a suitable geometric notion of connectedness for schematic spaces and also to study their geometric points. We achieve these goals by means of the strong cohomological constraints enjoyed by schematic spaces.

Multistability and anomalies in oscillator models of lossy power grids

The analysis of dissipatively coupled oscillators is challenging and highly relevant in power grids. Standard mathematical methods are not applicable, due to the lack of network symmetry induced by dissipative couplings. Here we demonstrate a close correspondence between stable synchronous states in dissipatively coupled oscillators, and the winding partition of their state space, a geometric notion induced by the network topology. Leveraging this winding partition, we accompany this article with an algorithms to compute all synchronous solutions of complex networks of dissipatively coupled oscillators. These geometric and computational tools allow us to identify anomalous behaviors of lossy networked systems. Counterintuitively, we show that loop flows and dissipation can increase the system’s transfer capacity, and that dissipation can promote multistability. We apply our geometric framework to compute power flows on the IEEE RTS-96 test system, where we identify two high voltage solutions with distinct loop flows.

Pseudoholomorphic curves relative to a normal crossings symplectic divisor: compactification

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than the relative compactification and there is a projection map from the latter onto the former. The latter is constructed via expanded degenerations of the target. Our construction does not need any modification of (or any extra structure on) the target. Unlike the classical moduli spaces of stable maps, these log moduli spaces are often virtually singular. We describe an explicit toric model for the normal cone (i.e. the space of gluing parameters) to each stratum in terms of the defining combinatorial data of that stratum. In [FT2], we introduce a natural set up for studying the deformation theory of log (and relative) curves and obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. In a forthcoming paper, we will prove a gluing theorem for smoothing log curves in the normal direction to each stratum. With some modifications to the theory of Kuranishi spaces, the latter will allow us to construct a virtual fundamental class for every such log moduli space and define relative GW invariants without any restriction.

Indirect Distance Measurement and Geometric Work in the Construction of Trigonometric Notions

Background: Among the multiple phenomena reported around the teaching and learning of trigonometry is the exclusive use as technical tools that trigonometric ratios receive. This use allows and promotes the construction of limited meanings and the dissociation of school trigonometry from geometric notions and procedures. Objective: Given this problem, we studied the germinal construction of trigonometric notions in a historical setting to identify meaningful uses of mathematical knowledge to enrich those that inhabit the school and its associated meanings. Design: We conducted this study from socio-epistemological theory and through a particular configuration of content analysis. Setting and participants: Being a documentary cut study, we did not have participants stricto sensu . Data collection and analysis: The historical study focused on the mathematical preliminaries of Ptolemy’s Almagest , a work that the literature points out as the oldest evidence of the birth of trigonometry. Results: Among the results, we highlight the indirect measurement of distances as the use that allowed the initial construction of trigonometric notions and the synergy of three uses of geometric knowledge - which we call geometric work - as fundamental to this process. Conclusions: We conclude by stressing the importance of creating concrete didactic situations that explore the operation of these results in teaching environments and the need for historical studies around other points in the construction of trigonometric notions.

Stability of entropic optimal transport and Schrödinger bridges

We establish the stability of solutions to the entropically regularized optimal transport problem with respect to the marginals and the cost function. The result is based on the geometric notion of cyclical invariance and inspired by the use of c-cyclical monotonicity in classical optimal transport. As a consequence of stability, we obtain the wellposedness of the solution in this geometric sense, even when all transports have infinite cost. More generally, our results apply to a class of static Schrödinger bridge problems including entropic optimal transport.

τ-tilting finiteness of non-distributive algebras and their module varieties

We treat the τ-tilting finiteness of minimal representation-infinite algebras and particularly the non-distributive ones. Building upon the new results of Bongartz, we fully determine which algebras in this family are τ-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the minimal representation-infinite biserial algebras. Consequently, we obtain nontrivial explicit conditions for τ-tilting infiniteness of a large family of algebras. This also produces concrete families of “minimal τ-tilting infinite algebras”, recently studied in our work.We further use our results to establish a conjectural connection between the τ-tilting theory and two geometric notions (the dense orbit property and Schur-representation finiteness) introduced by Chindris, Kinser and Weyman while studying module varieties. We verify the conjectures for the algebras studied in this note: For the minimal representation-infinite algebras which are non-distributive or biserial, if Λ has the dense orbit property, then Λ is τ-tilting finite. Moreover, we prove that such an algebra is Schur-representation-finite if and only if it is τ-tilting finite. This gives a categorical interpretation of Schur-representation-finiteness over this family of algebras.

Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.

Calculus of Orthogonal Projectors

It is possible to express all geometric notions connected with closed linear subspaces in terms of algebraic properties of the orthoprojectors onto these linear spaces. In this paper, sucient conditions for the calculus of a family of orthoprojectors in B(H) have been given with meaningful consideration of the sum, the product and dierence of orthoprojectors to be a projector. This has been done by giving the algebraic formulations of orthogonality for the sum, product and dierence. From the paper, it is observed that there is a natural one-to-one correspondence between the set of all closed linear subspaces of a Hilbert space H and the set of all orthoprojectors on H. This paper will help in the study of vector space with many diverse applications such as orthogonal polynomials, QR decomposition of projectors and Gram-Schmidt orthogonalization.

Skew-symmetric hom-algebroids and their representation theory

In this paper, we introduce skew-symmetric hom-algebroids and linear almost Poisson structures on dual hom-bundles. In addition, we show that these two geometric notions are related to each other and present several interesting examples of them. Moreover, we study the representation theory of skew-symmetric hom-algebroids and their dual spaces. Finally, we give the notion of phase spaces of skew-symmetric algebroids.

Bulk area law for boundary entanglement in spin network states: Entropy corrections and horizon-like regions from volume correlations

For quantum gravity states associated to open spin network graphs, we study how the entanglement entropy of the boundary degrees of freedom (spins on open edges) is affected by the bulk data, specifically by its combinatorial structure and by the quantum correlations among intertwiner degrees of freedom. For a specific assignment of bulk edge spins and slightly entangled intertwiners, we recover the Ryu-Takayanagi formula (with a properly (discrete) geometric notion of area, thanks to the underlying quantum gravity formalism) and its corrections due to the entanglement entropy of the bulk state. We also show that the presence of a region with highly entangled intertwiners deforms the minimal-area surface, which is then prevented from entering that region when the entanglement entropy of the latter exceeds a certain bound. This entanglement-based mechanism leads thus to the formation of a black hole-like region in the bulk.

Violations at the Reference Point of Discontinuity: Limitations of Prospect Theory and an Alternative Model of Risk Choices

The tilted S-shaped utility function proposed in Prospect Theory (PT) relied fundamentally on the geometrical notion that there is a discontinuity between gains and losses, and that individual preferences change relative to a reference point. This results in PT having three distinct parameters; concavity, convexity and the reference point represented as a disjoint between the concavity and convexity sections of the curve. The objective of this paper is to examine the geometrical violations of PT at the zero point of reference. This qualitative study adopted a theoretical review of PT and Markowitz’s triply inflected value function concept to unravel methodological assumptions which were not fully addressed by either PT or cumulative PT. Our findings suggest a need to account for continuity and to resolve this violation of PT at the reference point. In so doing, an alternative preference transition theory, was proposed as a solution that includes a phase change space to cojoin these three separate parameters into one continuous nonlinear model. This novel conceptual model adds new knowledge of risk and uncertainty in decision making. Through a better understanding of an individual’s reference point in decision making behaviour, we add to contemporary debate by complementing empirical studies and harmonizing research in this field. Doi: 10.28991/ESJ-2022-06-01-03 Full Text: PDF

Deformations of Lagrangian submanifolds in log-symplectic manifolds

This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that the gauge equivalence relation of the DGLA corresponds with the geometric notion of equivalence by Hamiltonian isotopies. We discuss the corresponding moduli space, and we prove a rigidity statement for the more flexible equivalence relation by Poisson isotopies.

Noncyclic contractions and relatively nonexpansive mappings in strictly convex fuzzy metric spaces

&lt;abstract&gt;&lt;p&gt;A concept of fuzzy projection operator is introduced and use to investigate the non-emptiness of the fuzzy proximal pairs. We then consider the classes of noncyclic contractions and noncyclic relatively nonexpansive mappings and survey the existence of best proximity pairs for such mappings. In the case that the considered mapping is noncyclic relatively nonexpansive, we need a geometric notion of fuzzy proximal normal structure defined on a nonempty and convex pair in a convex fuzzy metric space. We also prove that every nonempty, compact and convex pair of subsets of a strictly convex fuzzy metric space has the fuzzy proximal normal structure.&lt;/p&gt;&lt;/abstract&gt;

Finite Blaschke products and the golden ratio

Geometric properties of finite Blaschke products have been intensively studied by many different aspects. In this paper, our aim is to study geometric properties of finite Blaschke products related to the golden ratio $\alpha =\frac{1+\sqrt{5}}{2}$. Mainly, we focus on the relationships between the zeros of canonical finite Blaschke products of lower degree and the golden ratio. We show that the geometric notions such as "golden triangle, "golden ellipse" and "golden rectangle" are closely related to the geometry of finite Blaschke products.

Sector search strategies for odor trail tracking

Ants, mice, and dogs often use surface-bound scent trails to establish navigation routes or to find food and mates, yet their tracking strategies remain poorly understood. Chemotaxis-based strategies cannot explain casting, a characteristic sequence of wide oscillations with increasing amplitude performed upon sustained loss of contact with the trail. We propose that tracking animals have an intrinsic, geometric notion of continuity, allowing them to exploit past contacts with the trail to form an estimate of where it is headed. This estimate and its uncertainty form an angular sector, and the emergent search patterns resemble a "sector search." Reinforcement learning agents trained to execute a sector search recapitulate the various phases of experimentally observed tracking behavior. We use ideas from polymer physics to formulate a statistical description of trails and show that search geometry imposes basic limits on how quickly animals can track trails. By formulating trail tracking as a Bellman-type sequential optimization problem, we quantify the geometric elements of optimal sector search strategy, effectively explaining why and when casting is necessary. We propose a set of experiments to infer how tracking animals acquire, integrate, and respond to past information on the tracked trail. More generally, we define navigational strategies relevant for animals and biomimetic robots and formulate trail tracking as a behavioral paradigm for learning, memory, and planning.

Measuring the derivative using surfaces

Abstract Students need a robust understanding of the derivative for upper-division mathematics and science courses, including thinking about derivatives as ratios of small changes in multivariable and vector contexts. In Raising Calculus to the Surface activities, multivariable calculus students collaboratively discover properties of derivatives by using tangible tools to solve context-rich problems. In this paper, we present examples of student reasoning about derivatives during the first of a sequence of three Raising Calculus activities. In this sequence, students work collaboratively on dry-erasable surfaces (tangible graphs of functions of two variables) using an inclinometer, a tool that can measure derivatives in any direction on the surfaces, to invent procedures to determine derivatives in any direction. Since students are not given algebraic expressions for the underlying functions, they must coordinate conceptual and geometric notions of derivatives, building on their understandings from introductory differential calculus. We discuss examples of student reasoning that demonstrate how the activity supports student realization of the need to define a path, attend to direction and the orientation of the coordinate axes and recognize covariation between related quantities. This first activity enables students to initially recognize the ratio of small changes approach to derivatives. We briefly describe how students utilize the ratio of small changes approach in subsequent activities as they measure partial derivatives on surfaces (using an inclinometer) and on contour maps.

The Generalized Covering Radii of Linear Codes

Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of linear codes– the generalized covering radius. The generalized covering-radius hierarchy of a linear code characterizes the trade-off between storage amount, latency, and access complexity, in such database systems. Several equivalent definitions are provided, showing this as a combinatorial, geometric, and algebraic notion. We derive bounds on the code parameters in relation with the generalized covering radii, study the effect of simple code operations, and describe a connection with generalized Hamming weights.