This paper develops a cohomological hierarchy for bistable visual paradoxes.
problem Understanding the hierarchy of visual paradoxes built from bistable elements.
method Develops a cohomological hierarchy using Z2 coefficients and a discrete Stokes theorem. result Reveals a hierarchy of paradox classes from H0 through H2, refined at each degree by the relative/absolute distinction. Bistable structures associated with non-linear deformation behavior, exemplified by the Venus flytrap and slap bracelet, can switch between different functional shapes upon actuation. Despite numerous efforts in modeling such large deformation behavior of shells, the roles of mechanical and nonlinear geometric effects …
Novel bistable structures made from four-bar linkages, proving existence and construction.
problem Existence and construction of bistable mechanical structures composed of four-bar linkages.
method Geometric construction starting from infinitesimally flexible quad nets, applying Whiteley de-averaging.
result Construction of bistable structures from well-known quad nets, allowing control of geometric parameters.
A deep learning model organizes RNA graphs to reveal folding patterns and properties.
problem Organizing and understanding the complex folding patterns of RNA secondary structures.
method Geometric scattering autoencoder (GSAE) network for learning graph embeddings.
result GSAE accurately reflects bistable RNA structures and can sample new folding trajectories.
Study shows bifurcating price dynamics in ASME with traders.
problem Understanding price dynamics in artificial stock markets.
method Agent-based model of endogenous traders interacting through a LOB.
result Bistability in price equilibria: zero-price and persistent positive-price states.
Following findings by Ormerod and Mounfield, Wright rises the problem whether a power or an exponential law describes the distribution of occurrences of economic recession periods. In order to clarify the controversy a different set of GDP data is hereby examined. The conclusion about a power law distribution of recess…
Improved noise estimation in latent neural SDEs enhances model accuracy.
problem Latent neural SDEs underestimate noise, limiting their stochastic dynamics modeling.
method Explicit additional noise regularization in the loss function.
result Model accurately captures diffusion component of stochastic time series data.
I prove the bistability of linear evolution equations x′=A(t)x in a Banach space E, where the operator-valued function A is of the form A(t)=f′(t)G(t,f(t)) for a binary operator-valued function G and a scalar function f. The constant that bounds the solutions of the equation is computed explicitly; it i…
We consider a system of diffusion processes that interact through their empirical mean and have a stabilizing force acting on each of them, corresponding to a bistable potential. There are three parameters that characterize the system: the strength of the intrinsic stabilization, the strength of the external random per…
Noise can stabilize systemic risk models with uncertain robustness.
problem Understanding systemic risk in financial systems with uncertain parameters.
method Analyzing a mean-field model of systemic risk with uncertain coefficients and noise.
result Noise can induce stability in systemic risk models, contrary to intuition.
We consider minimal surfaces M which are complete, embedded and have finite total curvature in R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation Δu+f(u)=0inR3. Here f=−W′ with W bistable and balanced, for instance W(u)=41(1−u2)2. We assume that …
New method finds basins of attraction without needing system models.
problem Determining basins of attraction (BoA) for nonlinear systems without prior knowledge.
method Hybrid Active Learning (HAL) method combining AST, AL, and DBS.
result Efficiently finds and labels boundary of BoA without model knowledge.
We formulate and analyze a multi-agent model for the evolution of individual and systemic risk in which the local agents interact with each other through a central agent who, in turn, is influenced by the mean field of the local agents. The central agent is stabilized by a bistable potential, the only stabilizing force…
The paper classifies reversible elements in Seifert-fibered spaces and braid groups.
problem Classifying reversible elements in Seifert-fibered spaces and braid groups.
method Classification of reversible elements in Fuchsian groups, application to Seifert-fibered groups, and analysis of 3-torsion elements.
result Classification and analysis of reversible and 3-torsion elements in Seifert-fibered spaces and braid groups.
Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.
problem Classifying reciprocal elements in Hecke groups.
method Classifying and parametrizing reciprocal classes in Hecke groups Γp for p≥3. result Generalizes Sarnak's result on reciprocal elements in the modular group.
We know that any element A of the group SO(3) can be represented as A = A1 A2 A1', where A1, A1' are elements of SO1(2)={A is an element of SO(3) | Ae1=e1}, and SO2(2)={A is an element of SO(3) | Ae2=e2} . This fact is known as Euler's angle. When this situation, a matrix A is called the generator. In the present paper…
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serr…
In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 e…
The paper classifies reversible and strongly reversible elements in Hermitian isometry groups.
problem Classifying reversible and strongly reversible elements in Hermitian isometry groups.
method Classification through group theory and algebraic manipulation.
result New classification of strongly reversible elements in Sp(n).
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental…
New findings on generating mapping class groups using pseudo-Anosov elements.
problem Generating mapping class groups using specific types of elements.
method Proving the generation of mapping class groups by pseudo-Anosov elements and conjugate reducible but not periodic elements.
result The mapping class group can be generated by two conjugate pseudo-Anosov elements with arbitrarily large dilatations for surfaces of genus greater than or equal to nine.
This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are…
Classifies reversible and strongly reversible elements in quaternionic groups.
problem Classifying reversible and strongly reversible elements in quaternionic groups.
method Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
result Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
Persistent elements are ubiquitous in knot groups, especially for hyperbolic knots.
problem Identifying persistent elements in knot groups under Dehn fillings.
method Combining techniques from knot theory and hyperbolic geometry, including Dehn fillings and automorphisms.
result Persistent elements are structurally pervasive in knot groups, not just rare exceptions.
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of ℓ-form-valued k-forms on triangulations. result Generalizes existing finite element methods for various tensor fields.
The paper improves convergence rates of curvature approximations using Regge elements.
problem Improving convergence rates of curvature approximations using Regge elements.
method Investigates the interplay between polynomial degree of curvature lifting and metric tensor degree in Regge finite element space.
result Higher convergence rates are achieved by reducing the polynomial degree of curvature lifting and using linear Regge elements.
New group with non-loxodromic Morse element found.
problem Finding non-loxodromic Morse elements in groups.
method Small-cancellation techniques to construct a Morse local-to-global group.
result Found an infinite-order Morse element that is not loxodromic.
The paper solves a complex option pricing model using finite elements.
problem Risk-Adjusted Pricing Methodology (RAPM) Black-Scholes model with transaction costs.
method Spatial finite element models based on P1 and/or P2 elements, combined with a Crank-Nicolson-type temporal scheme.
result Results compare favorably with finite difference methods in the literature.
Paper introduces Deep Sets for Symmetric Elements (DSS) layers for learning sets of symmetric elements.
problem Learning sets of symmetric elements is underexplored.
method Characterized equivariant layers, showed DSS layers are universal approximators, and demonstrated their effectiveness.
result DSS layers improve set-learning architectures across various data types.
The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
The paper classifies 3-manifold groups with specific torsion elements.
problem Classifying 3-manifold groups with generalized torsion elements of order two.
method Analyzing the fundamental groups of 3-manifolds and their conjugates.
result 3-manifold groups with generalized torsion elements of order two have been classified.
New examples of hyperbolic links with generalized torsion elements found.
problem Finding generalized torsion elements in the fundamental groups of hyperbolic links.
method Analyzing the Weeks manifold, figure-eight sister manifold, and Whitehead sister link to identify generalized torsion elements.
result First examples of hyperbolic links with link groups admitting generalized torsion elements.
Proves mapping class group generated by two torsion elements for certain surfaces.
problem Generating mapping class group with two torsion elements.
method Analyzes surfaces of different genera and orders, proving generation by two elements of specific orders.
result Mapping class group generated by two torsion elements for g≥6 and other genera. Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
Paper solves convertible bond valuation using finite elements with penalty method.
problem Valuation of convertible bonds under penalty TF model.
method Solves TF system of equations using P1 and P2 finite elements with penalty method.
result Numerical solutions compare favorably with finite difference method.
Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.
problem Tackles constructing Poisson structures on gauge orbits of Maurer-Cartan elements.
method Constructs Poisson structures on gauge orbits of Maurer-Cartan elements of dgla L, associating a compatible Batalin-Vilkovisky algebra to each MC element.
result MCP structures yield a notion of hamiltonian flow of MC elements and define Lie algebroids on gauge orbits.
Loxodromic elements are pseudo-Anosov on specific graphs.
problem Characterizing loxodromic elements in specific groups.
method Analyzing subgroups acting on multiarc and curve graphs, and the handlebody group on disk graphs.
result Loxodromic elements are pseudo-Anosov on witness graphs.
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
problem Approximating Levi-Civita connection and curvature on 2D manifolds with finite elements.
method Using Regge finite elements, piecewise polynomial symmetric (0,2)-tensor fields, and distributional sense for non-regular tensors.
result Distributional quantities converge to their smooth counterparts under refinement of triangulation.
We show that the mapping class group of a closed oriented surface of genus at least three is generated by 3 elements of order 3 and by 4 elements of order 4. Note that the mapping class group cannot be generated by finitely many torsion elements of same order if genus is equal to one or two.
An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: deci…
Characterizes stably elliptic elements in Lie groups and their properties.
problem Understanding stably elliptic elements in Lie groups and their geometric and algebraic properties.
method Characterization through fixed point algebra and Weyl group action; relates to maximal invariant cones and compactness of order intervals.
result Connected components of stably elliptic elements can be described using Weyl group action on a compactly embedded Cartan subalgebra.
Positive Thompson links are proven for oriented subgroup elements.
problem Proving properties of Thompson links with positive elements.
method Analyzing elements of the oriented subgroup of the Thompson group.
result Positive oriented Thompson links are established.
The paper classifies and decomposes quaternionic projective transformations.
problem Classifying and decomposing elements of the projective linear group PSL(3,H). method Algebraic characterization of dynamical types using reversibility, decomposition of elements into simple elements.
result Offered a complete classification for elements of SL(3,R). Let Sg be the closed oriented surface of genus g and let Mod(Sg) be the mapping class group. When the genus is at least 3, Mod(Sg) can be generated by torsion elements. We prove the follow results. For g≥4, Mod(Sg) can be generated by 4 torsion elements. Three generators are invo…
New loxodromic elements found in infinite-type surfaces.
problem Finding loxodromic elements in infinite-type surfaces.
method Adapting Thurston-Veech construction for infinite-type surfaces.
result Infinitely many loxodromic elements produced without leaving finite-type subsurfaces invariant.
We have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the ind…
The paper constructs hyperbolic elements in multiple spaces.
problem Constructing hyperbolic elements in multiple Gromov-hyperbolic spaces.
method Explicit construction under minimal conditions.
result Set of simultaneously hyperbolic elements has strictly positive density.