Paper develops formulas for shape derivatives in wave scattering.
problem Computing high order shape derivatives for wave scattering is challenging.
method Introduces elegant recurrence formulas using differential forms and Lie derivatives.
result Unified framework for computing high order shape perturbations in scattering problems.
Novel method for shape optimization of non-smooth PDEs.
problem Optimizing shapes governed by non-smooth PDEs.
method Functional variational approach and sensitivity analysis.
result Necessary conditions for locally optimal shapes.
Paper controls shape stability in infinite Riemannian manifolds.
problem Characterizing optimal shapes in infinite-dimensional Riemannian manifolds.
method Uses Riemannian manifold framework and mean curvature analysis.
result Control on shape stability depends only on mean curvature.
This article introduces planar shape signatures derived from homology nerves, which are intersecting 1-cycles in a collection of homology groups endowed with a proximal relator (set of nearness relations) that includes a descriptive proximity. A 1-cycle is a closed, connected path with a zero boundary in a simplicial c…
The article explores derivatives of shapes other than circles and spheres.
problem Understanding derivatives of various shapes.
method First-year calculus approach.
result Derivatives of shapes other than circles and spheres are explored.
The paper improves GP regression for sparse sensor data in structural mode shape reconstruction.
problem Reconstructing full-field structural mode shapes from sparse sensor data.
method Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework.
result The proposed method provides more accurate and reliable mode shapes.
Hierarchical geodesic model for analyzing shapes on manifolds.
problem Analyzing temporal observations on manifold-valued data.
method Adapted functional-based metric for efficiency; variational time discretization of geodesics.
result Performed hypothesis tests and estimated mean trends in longitudinal analysis.
Shape information is of great importance in many applications. For example, the oil-bearing capacity of sand bodies, the subterranean remnants of ancient rivers, is related to their cross-sectional shapes. The analysis of these shapes is therefore of some interest, but current classifications are simplistic and ad hoc.…
The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width, and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differe…
Study uses outer metrics for PDE-constrained shape optimization over diffeomorphism group.
problem Optimizing shapes governed by PDEs over the diffeomorphism group.
method Outer metrics on diffeomorphism group, Riemannian steepest descent method.
result Riemannian approach outperforms other metrics in solving PDE-constrained shape optimization problems.
The paper explores metrics and models for analyzing biological shapes.
problem Analyzing biological shapes using mathematical metrics.
method Review of Riemannian metrics and evolution equations, focusing on diffeomorphic shape analysis.
result Introduction of a new class of metrics involving optimization of a growth tensor.
Optimizes material distribution on surfaces using topological derivatives.
problem Optimal distribution of two materials on smooth submanifolds in Rd. method Topological derivative approach for shape optimization constrained by PDEs.
result Numerical solution of topology optimization problem on surfaces.
New method reconstructs 3D shapes from 2D images using Kendall's shape space.
problem Reconstruct 3D shapes from 2D images, especially for rare specimens.
method Kendall's shape space approach with prior information.
result More robust and plausible shapes compared to previous methods.
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
We introduce a guide to help deep learning practitioners understand and manipulate convolutional neural network architectures. The guide clarifies the relationship between various properties (input shape, kernel shape, zero padding, strides and output shape) of convolutional, pooling and transposed convolutional layers…
Study on shape optimization for specific eigenvalue problems on domains.
problem Shape optimization of eigenvalue problems for fourth order Steklov.
method Asymptotic expansion and sharp upper bound derivation.
result Derivation of eigenvalue spectra and shape optimization conclusions.
Shape-constrained symbolic regression improves model extrapolation with prior knowledge.
problem Improving model extrapolation with prior knowledge in symbolic regression.
method Shape-constrained symbolic regression using evolutionary algorithms with interval arithmetic.
result Models with shape constraints have improved extrapolation but lower accuracy on test sets.
We study equations over torsion-free groups in terms of their `t-shape' (the occurences of the variable t in the equation). A t-shape is good if any equation with that shape has a solution. It is an outstanding conjecture that all t-shapes are good. In [Klyachko's methods and the solution of equations over torsion-free…
Reward shaping is one of the most effective methods to tackle the crucial yet challenging problem of credit assignment in Reinforcement Learning (RL). However, designing shaping functions usually requires much expert knowledge and hand-engineering, and the difficulties are further exacerbated given multiple similar tas…
The paper learns pose variations within shape populations using constrained mixtures of factor analyzers.
problem Learning pose variations within a shape population with articulated parts and relative rotations.
method Formulated as mixtures of factor analyzers, segmentation by component posterior probabilities, and constraints on factor loading matrices for rotation matrices.
result Automatic learning of pose variations from shape populations, resulting in smooth and realistic animations.
Inverse curvature flows shape star-shaped hypersurfaces into spheres.
problem Evolution of star-shaped hypersurfaces inside a convex cone.
method Inverse curvature flows, convexity of the cone, gradient and Hölder estimates.
result Hypersurfaces converge to a round sphere as time goes to infinity.
Coupled entropy corrects flaws in Tsallis entropy for complex systems.
problem Misinterpretation of generalized temperature and entropy.
method Derived from generalized Pareto and Student's t distributions.
result Provides balanced measure of uncertainty for complex systems.
Optimal trading strategy with predictor and costs, derived equations and shape.
problem Optimal trading strategy in presence of price predictor, costs, and risk control.
method Path-integral method to derive equations for band edges, solved explicitly for Ornstein-Uhlenbeck predictor.
result Explicit equations and shape of the optimal band strategy derived and analyzed.
The paper characterizes law-invariant star-shaped risk measures.
problem Understanding and characterizing law-invariant star-shaped risk measures.
method Developed characterizations for positively homogeneous and star-shaped functionals, derived Kusuoka-type representations, and offered representations of general law-invariant star-shaped functionals.
result Characterizations of law-invariant star-shaped functionals, including their connections to Value-at-Risk and Expected Shortfall.
Kernel-based tests for shape constraints in finance.
problem Enforcing shape relations on latent functions in financial econometrics.
method Kernel-based nonparametric framework for mean-variance optimization.
result Established statistical properties and a joint Wald-type statistic for testing shape constraints.
In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to s…
A new method for analyzing shapes using FDA techniques.
problem Statistical shape analysis of deformed contours.
method Functional Data Analysis (FDA) with basis expansion and principal component analysis.
result Successfully identifies deformation parameters and captures contour distributions.
The shape equation and linking conditions for a vesicle with two-phase domains are derived. We refine the conjecture on the general neck condition for the limit shape of a budding vesicle proposed by Jülicher and Lipowsky [Phys. Rev. Lett. \textbf{70}, 2964 (1993); Phys. Rev. E \textbf{53}, 2670 (1996)], and then we us…
The paper uses 3D shapes to reveal sundial design adjustments based on latitude.
problem Identifying sundial design adjustments based on installation location.
method Shape analysis in a high-dimensional space, regression in shape space.
result Sundial design adjustments were latitude-dependent.
The projective shape of a configuration of k points or "landmarks" in RP(d) consists of the information that is invariant under projective transformations and hence is reconstructable from uncalibrated camera views. Mathematically, the space of projective shapes for these k landmarks can be described as the quotient sp…
The paper derives inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
problem Establishing inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
method Using the Gauss equation and hypotheses for cosymplectic and nearly cosymplectic manifolds, the paper derives inequalities for the norm of the second fundamental form and the shape operator.
result The contact warped product submanifolds in cosymplectic manifolds exhibit a geometric property called D1-minimality, leading to an optimal general inequality. Study eigenvalues for special curvature equations on star-shaped surfaces.
problem Eigenvalue problem for prescribed curvature equations on star-shaped, k-convex hypersurfaces.
method Established existence of a unique eigenvalue and hypersurface through uniform estimates in p for Lp-type equations.
result Existence of a unique eigenvalue and associated hypersurface under certain conditions.
Left atrium shape has been shown to be an independent predictor of recurrence after atrial fibrillation (AF) ablation. Shape-based representation is imperative to such an estimation process, where correspondence-based representation offers the most flexibility and ease-of-computation for population-level shape statisti…
New method uses Riemannian geometry to describe molecular shapes.
problem Predicting drug-like molecules using shape similarity.
method Riemannian geometry applied to molecular surfaces.
result RGMolSA method captures molecular shape effectively.
New framework classifies high-dimensional shapes using ray intersections, establishing data requirements.
problem Classifying high-dimensional shapes in real-world data.
method Ray-based classification (RBC) framework using intersections of one-dimensional representations (rays) with shape boundaries.
result Established bounds on the number of rays necessary for shape classification, defined by key angular metrics.
Introduces Star-Shaped DDPMs for non-Gaussian distributions.
problem Difficulties in defining DDPMs for non-Gaussian distributions.
method Star-shaped diffusion process, duality with specific Markovian diffusions, efficient algorithms.
result SS-DDPMs can model distributions like Beta, von Mises-Fisher, Dirichlet, Wishart.
Paper tackles hard shape constraints in kernel machines.
problem Enforcing shape requirements in a hard fashion is challenging.
method Tightened second-order cone constrained reformulation for kernel machines.
result Performance guarantees and efficiency demonstrated in various applications.
The paper explores risk measures and arbitrage in financial markets.
problem Quantifying and managing risk in financial markets.
method Introduces new risk measure axioms and characterizes arbitrage conditions.
result Derives the consistent price interval for financial contracts.
New method uses Riemannian geometry to quantify molecular shapes.
problem Quantifying molecular similarity for drug discovery.
method Riemannian geometry and Kähler quantization (KQMolSA).
result KQMolSA method compares well to existing shape similarity methods.
A novel 3D shape registration method using spectral graph embedding and probabilistic matching.
problem Challenges in 3D shape analysis and registration, especially with large variability.
method Combining spectral graph matching with Laplacian embedding for large graphs, using commute-time embedding and PCA.
result A method to register shapes with different samplings and isometric deformations.
We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a n…
New method estimates shape distance in neural representations with limited data.
problem Measuring geometric similarity between high-dimensional network representations.
method Method-of-moments estimator with tunable bias-variance tradeoff.
result New estimator achieves lower bias than standard methods in high-dimensional settings.
New Hopf algebras help classify 4D shapes.
problem Classifying 4D shapes up to deformations.
method Developed non-factorizable ribbon Hopf algebras.
result Some derived invariants are boundary-dependent.
New findings connect shaped and unshaped neural networks using differential equations.
problem Understanding the behavior of neural networks with different activation scaling methods.
method Deriving differential equation-based asymptotic characterizations for shaped and unshaped neural networks.
result Two types of unshaped networks converge to the same infinite-depth-and-width limit at initialization.
We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in Rn+1 to become star-shaped. As a consequence, we demonstrate that any connected surface moving by the flow which is not initially a topological sphere develops a singularity or self-intersection …
The paper analyzes extreme risk measures with limited distributional information.
problem Investigating risk measures under partial knowledge of distribution moments and shape.
method Employing probability inequalities and modified Schwarz inequality to derive bounds on distortion risk measures.
result Unified framework for calculating best- and worst-case scenarios of distortion risk measures.
We analyze a monetary system of random money transfer on the basis of double entry bookkeeping. Without boundary conditions, we do not reach a price equilibrium and violate text-book formulas of economists quantity theory (MV=PQ). To match the resulting quantity of money with the model assumption of a constant price, w…
Unified framework for hard affine SDP constraints in vRKHSs.
problem Incorporating shape constraints into predictive models for rich function classes.
method Unified convex optimization framework using second-order cone tightening.
result Unified and modular approach for handling multiple shape constraints.