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 “spiralling structure”
Analyzes wage-price spiral and stagflation dynamics in economic models.
problem Understanding the formation of wage-price spirals and stagflation.
method Analytical solution to classical gravitation and price-wage spiral models.
result Elementary solution to rates of return differentiation in economic models.
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.
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.
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.
Unwinding spiralsmath.GT
No bi-Lipschitz homeomorphism can unwind spirals with sub-exponential winding radii.
problem Unwinding spirals with sub-exponential winding radii.
method Analyzing bi-Lipschitz homeomorphisms of R^2.
result No bi-Lipschitz homeomorphism exists for spirals with sub-exponential winding radii.
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.
Quadratic differentials induce spiralling foliations on Riemann surfaces.
problem Understanding the structure of foliations induced by quadratic differentials.
method Introduced a space of measured foliations and used harmonic maps to real trees.
result Any measured foliation is realized by a quadratic differential with second order poles at marked points.
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.
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.
Study equiangular surfaces in 3D, extending plane spirals.
problem Understanding 3D surfaces with constant normal-vector angles.
method Investigates three-dimensional extensions of equiangular spirals.
result Identifies self-similar structures in sea shell geometry.
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.
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…
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.
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…
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.
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.
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.
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…
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.
SPIRAL uses spikes to adaptively update weights, improving robustness and reducing overfitting.
problem Improving robustness and reducing overfitting in learning algorithms.
method Adaptive weight updates based on confidence estimates and activation offsets, regularized by spike rates.
result SPIRAL is more robust and less prone to overfitting compared to averaged perceptron and AROW.
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.
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.
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…
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 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…
Modeling stablecoins reveals deleveraging spirals and attacks.
problem Stablecoin markets' liquidity issues during crises.
method Developed a model of stable assets, including non-custodial stablecoins, and analyzed dynamics and liquidity.
result Stablecoin markets face deleveraging feedback effects causing illiquidity during crises.
Method approximates planar curves with circular arcs of equal length.
problem Approximating planar curves with circular arcs of equal length.
method Proposed by I.Kh. Sabitov and A.V. Slovesnov, extended with new inequalities and computer modeling.
result Derived inequalities for the length of a convex spiral arc with prescribed Hermite data.
A nice limaçon-like spiralmath.DG
A limaçon-like curve, allowing 2π-transition with monotone curvature between concentric curvature elements, is presented. The curve is 4th degree algebraic, 4th degree rational, and shares other common features with Pascal's limaçon.
New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.
problem Designing aesthetic shapes in equiaffine geometry.
method Introducing a new symmetry (ESA) to characterize planar curves.
result The new class of curves includes the quadratic curve and logarithmic spiral.
Deep learning models predict generalization gaps without specific task or architecture.
problem Predicting when deep learning works across different tasks and architectures.
method Created a dataset of 13,500 neural networks trained on various spiral datasets and parameters. Used this dataset to train predictors for generalization gaps.
result DNNs and RNNs outperform linear models in predicting generalization gaps, with RNNs achieving .
Bounding shears in ideal triangulations on hyperbolic surfaces.
problem Bounding shears in ideal triangulations on hyperbolic surfaces.
method Showing an ideal triangulation with bounded shear parameters on hyperbolic surfaces.
result An upper bound on shear parameters depends logarithmically on the surface's topology.
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…
Model explains deleveraging risks in non-custodial stablecoins.
problem Deleveraging risks in non-custodial stablecoins during market crises.
method Developed a stochastic model incorporating speculators' profit optimization and collateral liquidation costs.
result Identified deflationary deleveraging spirals and higher price variance in unstable domains.
Mitigates DeFi liquidations with reversible call options.
problem Systemic failures in DeFi due to liquidations.
method Introduces reversible call options to prevent liquidations.
result Reduces liquidated collateral by 89.82% in simulations.
In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces could be thought as the bi-dimensional analogue of the generalized helices. Some pi…
Minimal products of spherical immersions are studied with geometric and dynamical properties.
problem Minimal products of spherical immersions and their geometric properties.
method Profile flow, Liouville integrability, phase map analysis, Routh completion, primitive order analysis.
result Minimal products have a scalar Sturm form plus two nonnegative squares in their Hessian.