Verify spiral minimal product structure using Takahashi Theorem.
problem Verify spiral minimal product structure.
method Using Takahashi Theorem with full computational details.
result Verify spiral minimal product structure.
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48 results for “spiral minimal product structure”
Spiral Minimal Productsmath.DG
Paper constructs new minimal submanifolds in spheres by spinning given ones.
problem Creating new minimal submanifolds in spheres from given ones.
method Spin given minimal submanifolds by a curve in a balanced way.
result Generates spiral minimal products forming a two-dimensional family.
The paper discusses new Lagrangian constructions and examples.
problem Exploring new Lagrangian constructions in complex projective spaces.
method Generalized Delaunay construction among minimal Lagrangians.
result Uncountably many new special Lagrangian cones found.
Spirals are not shortest paths in certain sub-Riemannian geometries.
problem Nonminimality of spiral-like curves in sub-Riemannian manifolds.
method Construction of a competing curve to demonstrate non-minimality.
result Spiral-like curves are not length minimizing in sub-Riemannian manifolds.
On spiral minimal surfacesmath.DG
A class of spiral minimal surfaces in E^3 is constructed using a symmetry reduction. The new surfaces are invariant with respect to the composition of rotation and dilatation. The solutions are obtained in closed form %through the Legendre transformation and their asymptotic behaviour is described.
We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a "logarithmic cone", the surfaces are embedded.
Study on visibility properties of spiral sets in higher dimensions.
problem Characterizing density properties of spiral sets.
method Employing visibility concepts from discrete geometry.
result Established conditions for various density properties of spirals.
This article aims to present an elementary analytical solution to the question of the formation of a structure of differentiation of rates of return in a classical gravitation model and in a model of the dynamics of price-wage spirals.
Construction of a spiral with given boundary conditions by inversion of the involute of a circlemath.DG
Method constructs spirals with given tangents and curvatures.
problem Constructing a spiral with specified tangents and curvatures.
method Inversion of the involute of a circle to find the spiral.
result Spiral construction method using linear-fractional map.
Geodesics in Sol geometry described with invariant k and spiral properties.
problem Understanding the geodesic flow in the Sol geometry.
method Self-contained geometric description and analysis of geodesics.
result Characterization of geodesic segments, cut locus, and asymptotic distance growth.
Packaging of Thick Membranes using a Multi-Spiral Folding Approach: Flat and Curved Surfacescond-mat.soft
New multi-spiral approach improves packaging of thick membranes.
problem Deploying thick membranes on curved surfaces efficiently.
method Multi-spiral folding approach with prismatic folding lines.
result Improved deployment performance of thick membranes on curved surfaces.
The Cornu spirals on plane are the curves whose curvatures are linear. Generalized planar cornu spirals and Euler spirals in E^3, the curves whose curvatures are linear are defined in [1,5]. In this study, these curves are presented as the ratio of two rational linear functions. Also here, generalized Euler spirals in …
Conformal geodesics can't spiral in Riemannian manifolds.
problem Existence of spiral conformal geodesics on Riemannian manifolds.
method Analyzing properties of conformal geodesics on Riemannian manifolds.
result No conformal geodesic can become trapped in every neighborhood of a point.
This note is the updated outline of the article "Interpolational properties of planar spiral curves", Fund. and Applied Math., 2001, Vol.7, N.2, 441-463, published in Russian. The main result establishes boundary regions for spiral and piecewise spiral splines, matching given data. The width of such region can serve as…
Wojciech Kamiński disproved a spiral claim for conformal geodesics.
problem Proving that conformal geodesics cannot spiral.
method Provided a counterexample and illustrated proof failure.
result Lemma 4.6 proof was flawed.
Study geodesic curvature of logarithmic spirals on curved surfaces.
problem Understanding geodesic curvature on curved surfaces.
method Computed geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature.
result Asymptotic behavior of geodesic curvature is independent of the ambient surface's curvature.
Geodesics spiral around compact subsets in CAT(0) spaces.
problem Understanding geodesic spiraling in CAT(0) spaces.
method Logarithm law-type result for geodesics in quotients of rank one CAT(0) spaces.
result Proved logarithm law for geodesic spiraling in certain CAT(0) spaces.
Around Vogt's theoremmath.DG
Vogt's theorem, concerning boundary angles of a convex arc with monotonic curvature (spiral arc), is taken as a starting point to establish basic properties of spirals. The theorem is expanded by removing requirements of convexity and curvature continuity; the cases of inflection and multiple windings are considered. P…
We derive formulas for Alexander polynomials of spiral knots.
problem Understanding spiral knots, a braid-theoretic generalization of torus knots.
method Recursive formula for Alexander polynomials, genus formula.
result Simple genus formula for spiral knots.
We consider a simple model of rational agents competing in a single product market described by simple linear demand curve. Contrary to accepted economic theory, the agents' production levels synchronise in the absence of conscious collusion, leading to a downward spiraling of market total production until the monopoly…
Researchers found spiraling conformal geodesics in 3D space.
problem Existence of spiraling conformal geodesics in Euclidean signature.
method Constructed an example in 3D Euclidean signature, answering a question posed by Friedrich and Tod.
result Found an example of spiraling conformal geodesics in 3D space, not real analytic.
Unwinding spiralsmath.GT
We show that there is no bi-Lipschitz homeomorphism of that maps a spiral with a sub-exponential decay of winding radii to an unwinded arc. This result is sharp as shows an example of a logarithmic spiral.
This paper characterizes spiral Delone sets in higher dimensions.
problem To extend the theory of spiral sets to higher dimensions and show the existence of spiral Delone sets.
method Characterized Delone property of spiral sets in terms of packing and covering conditions for spherical sequences.
result Construction of explicit spiral Delone sets in \(\mathbb{R}^n\) for all \(n \ge 2\).
Geodesics spiral around Reeb orbits in 3D contact manifolds.
problem Understanding geodesics in sub-Riemannian geometry.
method Normal form along Reeb orbits due to Melrose.
result Sub-Riemannian geodesics spiral around Reeb orbits in both phase and configuration spaces.
Improves latent space structure for better data representation.
problem Limited ability of conventional priors to encode data manifold structure.
method Introduces an Encoded Prior Sliced Wasserstein AutoEncoder with iterative training and geodesic interpolation.
result Learned manifold encoding preserves topological and geometric properties of data.
The article completes the research of two-point G Hermite interpolation problem with spirals by inversion of conics. A simple algorithm is proposed to construct a family of 4th degree rational spirals, matching given G Hermite data. A possibility to reduce the degree to cubic is discussed.
A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including thos…
Cone structures over minimal products can't be calibrated smoothly.
problem Calibrating cones over minimal products with smooth calibrations.
method Extending a key result from [Zha26], showing obstruction.
result Cone structures over minimal products cannot be calibrated by smooth calibrations.
In this study, some characterizations of Euler spirals in E_1^{3} have been presented by using their main property that their curvatures are linear. Moreover, discussing some properties of Bertrand curves and helices, the relationship between these special curves in E_1^{3} have been investigated with different theorem…
What are the possible shapes of various things and why? For instance, when a closed wire or a frame is dipped into a soap solution and is raised up from the solution, the surface spanning the wire is a soap film. What are the possible shapes of soap films and why? Or, for instance, why is DNA like a double spiral stair…
The paper finds that circles and logarithmic spirals are the only constant-speed ramps for a specific force field.
problem Determining planar curves for constant-speed motion under specific force conditions.
method Analyzing the motion of a particle under friction and a central force field.
result Every solution to the constant-speed motion problem approaches either a circle or a logarithmic spiral.
Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lin…
Minimal surfaces with uniform curvature (or area) bounds have been well understood and the regularity theory is complete, yet essentially nothing was known without such bounds. We discuss here the theory of embedded (i.e., without self-intersections) minimal surfaces in Euclidean 3-space without a priori bounds. The st…
The study examines minimal surfaces in Riemannian products of surfaces.
problem Exploring geometric and topological restrictions on minimal surfaces in Riemannian products of surfaces.
method Analyzes totally geodesic surfaces and minimal 2-spheres, 2-tori, and 2-spheres in Riemannian products of surfaces with constant curvature.
result Generically, a totally geodesic surface in a Riemannian product is either a slice or a product of geodesics. Minimal 2-spheres and 2-tori have specific properties under certain curvature conditions.
This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small necks can be obtained b…
The paper studies Einstein-type structures in warped product manifolds.
problem Characterizing Einstein-type structures in warped product manifolds.
method Analyzing conditions for minimal, totally umbilical, and geodesic immersions.
result Characterization of rotational hypersurfaces in .
Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.
problem Proving Doyle conjecture for hexagonal lattice circle packings.
method Using Liouville theorem of discrete harmonic functions based on logarithmic radii ratio observation.
result Proves rigidity of Doyle spirals in hexagonal lattice circle packings with bounded radii ratios.
A spiral unibike track emerges from a mathematical construction.
problem Finding a unibike curve with a spiral shape.
method Starting with a polar square root curve, iteratively applying a differential equation to create a spiral unibike track.
result A spiral unibike curve is found with a precision error less than 10^-7.
A method for interpreting SVMs using polynomial kernels, revealing model complexity.
problem Interpreting SVMs built with truncated orthogonal polynomial kernels.
method Orthogonal Representation Contribution Analysis (ORCA) with normalized Orthogonal Kernel Contribution (OKC) indices.
result The method reveals structural aspects of model complexity not captured by predictive accuracy.
We investigate the approximate j-dimensionality of the singularity sets of minimal surfaces prescribed by Simon. This leads to the clasification of 8 variations of approximately j-dimensional surfacs in terms of dimension and locally finite Hausdorff measure. We show that the singularity sets must either be well behave…
Given a negatively curved geodesic metric space , we study the statistical asymptotic penetration behavior of (locally) geodesic lines of in small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of . We prove Khintchine-type and logarithme law-type results for the s…
A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential at the pole. We introduce the space of such measured foliations, and prove that f…
Proposes a new method to estimate Bayesian neural network depth.
problem Estimating the depth of Bayesian neural networks.
method Uses a discrete truncated normal distribution to learn depth mean and variance, inferring posterior distributions by minimizing variational free energy.
result Improves test accuracy and reduces posterior depth variance on the spiral dataset.
We present a confidence-based single-layer feed-forward learning algorithm SPIRAL (Spike Regularized Adaptive Learning) relying on an encoding of activation spikes. We adaptively update a weight vector relying on confidence estimates and activation offsets relative to previous activity. We regularize updates proportion…
On existence, uniqueness and scalability of adversarial robustness measures for AI classifiersstat.ML
Mathematical conditions and practical computations for adversarial robustness measures are established.
problem Existence, uniqueness, and scalability of adversarial robustness measures for AI classifiers.
method Formulated and proven mathematical conditions for existence, uniqueness, and explicit analytical computation of minimal adversarial paths and distances. Practical computation demonstrated on various AI tools and synthetic benchmarks.
result Explicit mathematical conditions and practical computations for adversarial robustness measures are established.
The spiral index of knotsmath.GT
Two new invariants that are closely related to Milnor's curvature-torsion invariant are introduced. The first, the spiral index of a knot, captures the minimum number of maxima among all knot projections that are free of inflection points. This invariant is closely related to both the bridge and braid index of the knot…
From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are giv…
Symplectic manifolds derived from Torelli groups, showing complex structure invariants without holomorphic structures.
problem Deriving symplectic structures from Torelli groups without holomorphic structures.
method Symplectization of mapping tori derived from Torelli group representatives.
result Symplectic manifolds with complex structure invariants but no holomorphic structures.