Geometrically interprets frequency in electric circuits.
problem Conventional frequency definition limitations.
method Introduces a multivector definition of frequency.
result Conventional frequency is a special case of the new framework.
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48 results for “geometric interpretations”
The paper proposes geometrizing deep networks to improve deep learning system interpretability.
problem Improving the interpretability of deep learning systems.
method Proposes geometrization of deep networks as a solution.
result Geometrization of deep networks can help understand existing deep learning systems and solve interpretability issues.
The abstract discusses different geometric interpretations of the order of tangency between manifolds.
problem Understanding the order of tangency between manifolds of the same dimension.
method Three geometric interpretations of the order of tangency are provided.
result Different geometric interpretations of the order of tangency between manifolds.
Geometrically interprets virtual knotoids in thickened surfaces.
problem No specific problem stated; focuses on geometric interpretation.
method Geometric interpretation of virtual knotoids as arcs in thickened surfaces.
result Shows virtual knotoid theory as a generalization of classical knotoid theory.
We give a geometric interpretation of Hamilton's matrix Harnack inequality for the Ricci flow as the curvature of a connection on space-time.
Geometrically interprets a duality theorem linking cochain and chain complexes.
problem Understanding a complex duality theorem in geometric terms.
method Introduces a chain isomorphism involving simplicial and cellular complexes.
result Establishes a geometric interpretation of Ranicki duality.
The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpre…
NDM incorporates geometric structure into neural networks for better optimization and interpretability.
problem Efficient and interpretable deep learning architectures.
method NDM is a neural network architecture that explicitly incorporates geometric structure into its design, using a Coordinate Layer, Geometric Layer, and Evolution Layer.
result NDM provides intrinsic regularization, enhancing generalization and robustness.
Researchers interpret SGD using diffusion metrics for clearer geometric understanding.
problem Elusiveness of geometrical significance in stochastic gradient descent.
method Study a deterministic model with geodesics of diffusion metrics.
result Establishes parallel with General Relativity models.
We give a geometric interpretation of all the -th elliptic integrable systems associated to a -symmetric space (in the sense of C.L. Terng). It turns out that we have to introduce the integer defined by m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three cases : the prim…
Geometric interpretation improves VAE performance and robustness.
problem Improving Variational Autoencoder performance and robustness.
method Introducing a geometric perspective on VAEs, sampling from the Riemannian latent space.
result Improved generation and interpolations with competitive or better performance on benchmark datasets.
Geometrically Interpreting Higher Cup Products, and Application to Combinatorial Pin Structureshep-th
Geometrically interprets cup products and defines combinatorial Pin structures.
problem Understanding Steenrod's cup products and their geometric interpretation.
method Constructs vector fields and combinatorial frames to interpret cochain-level formulas.
result Geometrically interprets cup products and defines Pin structures combinatorially.
FiberNet integrates geometry into machine learning for clearer classification.
problem Lack of interpretability in traditional deep learning.
method Reformulates classification as geometric optimization on fiber bundles, introducing learnable Riemannian metrics and variational prototype optimization.
result Clear geometric interpretability and efficiency in classification.
Geometrically interprets integrability of geodesic flow using web theory.
problem Integrability of geodesic flow by quadratic integrals.
method Geometric interpretation through web theory and integrable billiards construction.
result Constructs integrable billiards on surfaces with quadratic geodesic integrals.
Study topological quantum mechanics on orbifolds with geometric interpretation.
problem Quantum mechanical models on symplectic orbifolds.
method Explicit orbifold version of quantum HKR map and exact semi-classical approximation.
result Geometric and quantum field theoretic interpretation of orbifold algebraic index.
Geometrically interprets exact triangles of projectively flat bundles on complex tori.
problem Understanding exact triangles of projectively flat bundles on complex tori.
method Interprets projectively flat bundles geometrically and focuses on intersections of Lagrangian submanifolds.
result Geometric interpretation of exact triangles of projectively flat bundles.
Geometric interpretations and localisation theory for Kane-Mele invariant.
problem Understanding the Kane-Mele invariant in three-dimensional fermionic systems.
method Homotopy theory, geometric interpretations, Mayer-Vietoris Theorem, bundle gerbes.
result Unified cohomological explanation for equivalence between discrete Pfaffian and local geometric computations.
Milnor's triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.
The approach we present is a modification of the Morse theory for unital C*-algebras. We provide tools for the geometric interpretation of noncommutative CW complexes. These objects were introduced and studied in [2],[7] and [14]. Some examples to illustrate these geometric information in practice are given. A classifi…
At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean spa…
Hermann Schwarz, while studying complex analysis, introduced the geometric interpretation for the Poisson kernel in 1890. We shall see here that the geometric interpretation can be useful to develop a new approach to some old classical problems as well as to obtain several new results, mostly related to hyperbolic geom…
Defines contact structures on Heisenberg groups for geometric interpretation.
problem Finding a geometric interpretation for the Yamabe equation on Heisenberg groups.
method Defines contact structures of Heisenberg type, introduces a natural connection, and computes conformal scalar curvature.
result Establishes equivalence between contact Riemannian manifolds and contact structures of Heisenberg type.
We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between correspondin…
Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.
problem Understanding Fock-Goncharov positivity and its geometric implications.
method Geometric interpretation and bending deformations of Fuchsian representations.
result Stabilization of a uniform Finsler quasi-convex disk in the symmetric space.
We give a geometric interpretation of the linear trace Harnack inequality for the Ricci flow.
New geometric interpretation explains over-parameterized models and adversarial perturbations.
problem Geometric understanding of over-parameterized regression and adversarial perturbations.
method Alternative geometric interpretation of regression in feature space.
result Adversarial perturbations are a natural feature of biased models due to underlying geometry.
The article interprets Darboux's surfaces using triality in differential geometry.
problem Understanding Darboux's surfaces through triality.
method Geometric interpretation of Darboux's surfaces using triality.
result Darboux's surfaces are related to totally isotropic surfaces in a 6D projective quadric.
The group SL(2) acts on the space of cohomology groups of any hyper-Kahler manifold X. The χ_{y} genus of a hyper-Kahler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the χ_{y} …
The paper explains geometrically why certain mappings have singular points.
problem Understanding singular points in mappings from R^2 to R^3 and higher.
method Analyzing full rank matrices constructed from coefficients of mappings.
result Mappings have only one singular point when ℓ=3 and no singular points when ℓ>3.
Complex frequency generalizes eigenvalues in LTI systems.
problem Characterizing dynamics of signals with complex values.
method Geometric frequency interpretation and transformation analysis.
result Complex frequencies in LTI systems match eigenvalues.
Unified geometric framework for Brownian motion on various manifolds.
problem Modeling Brownian motion on complex Riemannian manifolds.
method Constructing stochastic differential equations with noise and drift terms aligned with Laplace-Beltrami operators.
result Geometrically transparent and mathematically consistent foundation for diffusion processes.
Torsion Energygr-qc
In the present work, torsion energy is defined. Its law of conservation is given. It is shown that this type of energy gives rise to a repulsive force which can be used to interpret supernovae type Ia observations, and consequently the accelerating expansion of the Universe. This interpretation is a pure geometric one …
We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.
Neural networks explained through geometric projections.
problem Understanding the geometric and mathematical underpinnings of neural networks.
method Exploiting connections between integration, Radon transforms, and neural networks.
result Distribution of neural network outputs can be interpreted as nonlinear projections along hypersurfaces.
Model learns disease self-representations for drug repositioning.
problem Drug repositioning for disease treatment.
method Enforces proximity in disease self-representations to preserve human phenome network structure.
result Method outperforms state-of-the-art approaches and produces biologically interpretable disease self-representations.
Abstract geometric structures flow harmonically.
problem Geometric structures on Riemannian manifolds.
method Twistorial interpretation and abstract harmonicity condition.
result Established analytic properties of geometric gradient flow.
Study embeds PC matrices into Grassmannian manifold for geometric interpretation.
problem Understanding algebraic consistency of pairwise comparisons matrices.
method Leverages Plücker coordinates and geometric interpretation of Grassmannian manifold.
result Algebraic consistency condition is equivalent to geometric consistency in .
An analogue of geometric quantization of Poisson algebras obtained by algebraic reduction of symmetries is developed. Interpretation of the obtained results and their application to the problem of commutativity of quantization and reduction are given
Geometrically interprets symplectic structure in 3-manifold triangulations.
problem Understanding symplectic structures in 3-manifold triangulations.
method Geometric interpretation and algorithm construction for symplectic basis.
result Algorithm constructs curves forming a symplectic basis.
GeoTop resolves topological ambiguity in diagnostic imaging using geometric-topological analysis.
problem Topological equivalence between benign and malignant structures in diagnostic images.
method Combines Topological Data Analysis and Lipschitz-Killing Curvatures to resolve ambiguity.
result Achieves 3.6% accuracy improvement and reduces false positives/negatives by 15-18%.
Construct geometric interpretation of Heston model using group quantization.
problem Geometric interpretation of Heston model
method Lifted local Lie groupoid formulation
result Geometric interpretation of Heston pricing operator and Riccati equations
A geometric interpretation of a circle transfer map in cobordism categories.
problem Understanding the circle transfer map in topological contexts.
method Geometric re-interpretation as a morphism of cobordism categories.
result The circle transfer map is homotopic to a composition of functors in cobordism categories.
HLRC offers a new curvature metric for hypergraphs that balances interpretability and efficiency.
problem Challenges in geometric characterization of hypergraphs with higher-order interactions.
method Hypergraph lower Ricci curvature (HLRC) defined in closed form.
result HLRC consistently reveals meaningful higher-order organization in diverse hypergraph datasets.
Notes describe geometric interpretations of cohomology in trisected 4-manifolds.
problem Understanding geometric interpretations of cohomology classes in trisected 4-manifolds.
method Analogy with Hodge theory and sheaf cohomology in algebraic geometry.
result Classes in can be interpreted as -classes.
Geometric interpretation of 2d-4d wall-crossing formulas.
problem Understanding wall-crossing phenomena in coupled 2d-4d systems.
method Deformation theory of holomorphic pairs and relation to scattering diagrams.
result Geometric interpretation of wall-crossing formulas.
A geometrical interpretation of the -structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained
A Geometric Interpretation of the Normal Closure of the Braid Group in the braid group of the torus math.GT
This paper studies the normal closure of braid groups in a torus.
problem Understanding the normal closure of braid groups in a torus.
method Combining results from Birman and Goldberg, the paper explores the geometric interpretation of the normal closure of full braid groups.
result The normal closure of in has a geometric description.
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, …