Wrinkles form on a thin sheet bonded to a sphere, revealing energy and length scale behaviors.
problem Understanding the wrinkle formation on a thin sheet bonded to a sphere.
method Analyzing the energy of the system with the sheet's thickness as a small parameter, determining leading and next-order behaviors.
result The wrinkling pattern varies with radius, with the number of wrinkles being approximately integer multiples of the sheet thickness.
Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.
problem Understanding complex wrinkling patterns in thin elastic hyperbolic surfaces.
method Non-Euclidean plate theory and investigation of branch points.
result Branch points are natural defects in hyperbolic sheets, influencing their morphology robustly.
Paper develops a framework for hyperbolic Monge-Ampère equation on strips, proving well-posedness and stability.
problem Addressing the rigidity-flexibility dichotomy for wrinkled patterns in thin elastic sheets.
method Develops hodograph transformation and parametrix-corrector decomposition to handle corner singularities and prove well-posedness.
result Proves existence and uniqueness of hodograph weak solutions and derives energy estimates for stability.
New geometric theory explains nonuniform origami responses.
problem Understanding nonuniform responses in origami sheets.
method Purely geometric continuum theory capturing nonuniform, nonlinear response.
result Three modes govern nonuniform response, varying smoothly across the sheet.
This paper explores parallels between minimal surfaces and Einstein manifolds.
problem Understanding Einstein manifolds, which are less studied.
method Synthesizes parallels between minimal surfaces and Einstein four-manifolds.
result Certain Einstein four-manifolds admit a minimal immersion into a higher-dimensional sphere.
Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.
problem Analyzing the Willmore energy of surfaces with curvature concentration.
method Using isoperimetric inequalities and framed loops, derive new lower bounds for the bending energy.
result Optimal blowup rates of the Willmore energy when curvature is concentrated.
The elastic energy functional of a thin elastic rod or sheet is generalized to the case of an M-dimensional manifold in N-dimensional space. We derive potentials for the stress field and curvatures and find the generalized von Karman equations for a manifold in elastic equilibrium. We perform a scaling analysis of an M…
A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms …
In the Minkowski space-time, a world hyper-sheet is a timelike hypersurface consisting of a one-parameter family of spacelike submanifolds. Recently, Bousso and Randall introduced the notion of caustics of world hyper-sheets in order to define the notion of holographic domains in space-time. Here, we give a mathematica…
Paper tackles score following in full-page sheet music images.
problem Score following in unprocessed sheet music images.
method Directly predicts score positions from audio and image input.
result Outperforms state-of-the-art methods in alignment precision.
Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces established.
problem Establishing curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces.
method Pointwise curvature estimate and sheeting theorem for weakly stable CMC hypersurfaces.
result Effective version of the compactness theorem for weakly stable CMC hypersurfaces.
A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds in Lorentz-Minkowski space. In this paper we investigate differential geometry of world sheets in Lorentz-Minkowski space as an application of the theory of big wave fronts.
The paper examines rigidity of thin domains under specific boundary conditions.
problem Linear geometric rigidity of shallow thin domains with zero Dirichlet boundary conditions.
method Analyzes two scaling regimes for ε in (h, √h] and (√h, 1), proving rigidity formulas.
result Rigidity does not depend on curvature in the small parameter regime ε ∈ (h, √h].
New system tracks musical performances in raw sheet images without preprocessing.
problem Lack of direct score position estimation in raw sheet images.
method Proposes an Audio-Conditioned U-Net architecture.
result Direct score position estimation in entire unprocessed sheet images.
Study compares empirical systemic risk with balance sheet risk in interbank networks.
problem Disentangling balance sheet risk from network effects in systemic risk.
method Generalised DebtRank dynamics and maximum-entropy approach to compare observed and expected systemic risk.
result Systemic risk levels are compatible but differ significantly during turbulent times.
The paper analyzes how open-end fund sales affect prices and returns.
problem How open-end fund sales impact prices and returns.
method Continuous-time market-clearing model to derive expected-return restrictions.
result Forced-sale pressure predicts actual fund selling and positive returns.
Study of bubble-sheet singularity in mean curvature flow.
problem Analyzing a specific type of singularity in mean curvature flow.
method Deriving an asymptotic profile for a neighborhood of the singularity.
result An asymptotic profile for a neighborhood of the singularity is derived.
If a tangle, K, in the 3-ball has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.
We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants …
New proof shows most thin knots satisfy Cabling Conjecture.
problem Cabling Conjecture for thin knots using Heegaard Floer homology.
method Heegaard Floer homology and immersed curves techniques.
result Almost all thin knots satisfy the Cabling Conjecture.
We produce embeddings of knots in thin position that admit compressible thin levels. We also find the bridge number of tangle sums where each tangle is high distance.
The study investigates how branch points affect the shape and mechanics of hyperbolic surfaces.
problem Understanding the role of branch points in the shape and mechanics of hyperbolic surfaces.
method Developed a discrete differential geometric (DDG) approach to study deformations of hyperbolic objects with distributed branch points.
result Branch points influence the overall morphology of hyperbolic surfaces without concentrating energy, leading to sub-exponential growth in maximum curvature.
The paper finds many thin subgroups isomorphic to Gromov-Piatetski-Shapiro lattices.
problem Understanding thin subgroups in special linear groups.
method Constructing and embedding non-arithmetic hyperbolic manifolds into SL(n+1)(R).
result Non-arithmetic lattices in SO(n,1) can be embedded into SL(n+1)(R) as thin subgroups.
Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, Φ(c), such that if K is a nontrivial knot in the three-sphere with a diagram with c crossings and a particularly s…
The automatic segmentation of human knee cartilage from 3D MR images is a useful yet challenging task due to the thin sheet structure of the cartilage with diffuse boundaries and inhomogeneous intensities. In this paper, we present an iterative multi-class learning method to segment the femoral, tibial and patellar car…
We describe a natural decomposition of a normal complex surface singularity (X,0) into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…
Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S3, k) from a thin position of k. In this article, we will prove that if a thin position of k induces a thin decomposition of double branched cover of (S3,k) by Howards and…
Abby Thompson proved that if a link K is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin…
A new XVA strategy rooted in balance sheet perspective improves equity process for bank shareholders.
problem Counterparty risk valuation adjustments (XVAs) in financial derivatives.
method Develops a cost-of-capital XVA strategy in a balance sheet perspective, solving explicitly in static setup and dynamically in trade context.
result Ensures a submartingale equity process corresponding to a target hurdle rate on capital at risk.
New method characterizes thin links via Conway spheres and tangle decompositions.
problem Characterize thin links without relying on specific knot invariants.
method Developed a relative version of thinness for tangles and used it to characterize thinness via tangle decompositions along Conway spheres.
result Characterized thin links via Conway spheres and tangle decompositions.
Regularized Stein thinning improves MCMC output approximations.
problem Pathologies in Stein thinning leading to poor approximations.
method Theoretical analysis and regularization to improve KSD.
result Regularized Stein thinning alleviates pathologies and improves efficiency.
The paper studies eigenvalues in negatively curved spaces, showing their relationship to the thick-thin decomposition.
problem Investigating the relationship between eigenvalues and the geometric structure of negatively curved spaces.
method Analyzing the eigenvalues of the Laplacian on the thick part of a manifold's thick-thin decomposition.
result Eigenvalues in the thick part of a negatively curved manifold are significantly smaller than those in the entire manifold.
Thin groups found in specific lattices.
problem Embedding right-angled Coxeter groups in arithmetic lattices.
method Using Agol's unpublished argument, embedding in indefinite orthogonal groups.
result Irreducible right-angled Coxeter groups embed as thin subgroups.
KSD Thinning uses KSD to thin MCMC samples efficiently.
problem Efficiently representing posterior distributions in Bayesian inference.
method KSD Thinning: retains only samples exceeding a KSD threshold.
result Established convergence and complexity tradeoffs for KSD Thinning.
Data thinning splits observations into independent parts for convolution-closed distributions.
problem Validation of unsupervised learning results in settings with limited data.
method Data thinning, splitting observations into independent parts following the same distribution.
result Data thinning provides an attractive alternative to cross-validation in settings with limited sample splitting.
Kernel thinning compresses distributions more effectively than i.i.d. sampling or standard thinning.
problem Efficiently compressing distributions for better sampling and integration accuracy.
method Introduces kernel thinning, a procedure that compresses an n-point approximation of a distribution into a sqrt(n)-point approximation with comparable integration error.
result Kernel thinning achieves a maximum discrepancy in integration error of O_d(n^(-1/2) sqrt(log n)) in probability for compactly supported distributions and O_d(n^(-1/2) (log n)^(d+1/2) sqrt(log log n)) for sub-exponential distributions.
The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…
Study shows (p,q)-cables of non-trivial knots are not thin.
problem Proving (p,q)-cables of non-trivial knots are not thin. method Bordered Floer theory of Lipshitz-Ozsváth-Thurston and Zemke's theorem.
result Proves (p,q)-cables of non-trivial knots are not thin. Generalizes data thinning for various distributions.
problem Thinning random variables without losing information.
method Relaxes summation requirement to sufficiency.
result Generalizes thinning to more distributions.
New method reduces summary points for datasets while maintaining quality.
problem Thinning datasets to reduce summary points while maintaining quality.
method Low-rank analysis of sub-Gaussian thinning.
result Guarantees high-quality compression for any distribution and kernel.
Proposes efficient training method for deep thin networks.
problem Deploying deep learning models with accuracy and compactness.
method Three-stage method: widen, warm up, fine tune.
result Deep thin networks trained with method outperform standard deep networks.
Wu has shown that if a link or a knot L in S3 in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that L⊂S3 is prime, then the thin sphere of lowest width also does not have any vertical c…
In this fact sheet we give some preliminary research results on the Bayesian Decision Theory. This theory has been under construction for the past two years. But what started as an intuitive enough idea, now seems to have the makings of something more fundamental.
Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
problem Understanding the structure of arithmetic locally symmetric spaces.
method Analyzing thin parts and deducing asymptotic results on Betti numbers.
result Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
We give a method for searching for thin positions of a given link.
New thin subgroups found in special linear groups via bending techniques.
problem Finding thin subgroups of lattices in special linear groups.
method Techniques from convex projective geometry.
result Infinitely many non-commensurable lattices with thin subgroups.
Model shows how banks' hidden-to-maturity accounting can mask run risk and lead to financial instability.
problem Run risk and hidden-to-maturity accounting in banking systems.
method Balance sheet model and optimization problem to assess run risk and resilience.
result Held-to-maturity accounting can mask revaluation losses and increase run risk.
We show that every thin position for a connected sum of small knots is obtained in an obvious way: place each summand in thin position so that no two summands intersect the same level surface, then connect the lowest minimum of each summand to the highest maximum of the adjacent summand below.