New Jordan Floer homology shows every curve inscribes every isosceles trapezoid.
problem Understanding inscriptions of isosceles trapezoids in Jordan curves.
method Constructing a new Lagrangian Floer homology chain complex.
result Establishes new cases of non-smooth Jordan curves inscribing isosceles trapezoids.
Continuous curves inscribe isosceles trapezoids in complex plane.
problem Proving periodic curves inscribe isosceles trapezoids.
method Lagrangian intersection problem and convergence argument.
result Continuous curves inscribe trapezoids with any similarity type.
Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.
problem Understanding periodic billiard paths in right triangles and orbit closures in strata of Abelian and quadratic differentials.
method Classifying orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials.
result Computed orbit closures and deduced asymptotic number of periodic billiard trajectories in right triangles.
The paper improves estimates of Gaussian curvature for minimal graphs over a unit disk.
problem Estimating Gaussian curvature of minimal graphs over a unit disk.
method Constructing Scherk's type minimal graphs and comparing their curvatures.
result Optimal estimate of Gaussian curvature at the center of the disk.
Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.
problem Finding the minimum of the spectral determinant on isosceles triangles.
method Analyzing the determinant of the Laplacian on Euclidean isosceles triangle envelopes of fixed area.
result Equilateral triangle envelope minimizes the determinant of the Laplacian.
Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon…
Characterizes billiard and quasigeodesic flows in polyhedral convex bodies.
problem Characterizing billiard and quasigeodesic flows in polyhedral convex bodies.
method Alexandrov geometry methods.
result Optimal regularity result for convex bodies: billiard dynamics is continuous if boundary is of class C2,1. The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.
problem Finding extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
method Characterizations obtained under suitable geometric constraints.
result Characterizations of extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
The abstract shows how embeddings inscribe trapezoids or map three points to a line, proving nonexistence of certain maps.
problem Proving the nonexistence of affinely 3-regular maps in infinitely many dimensions.
method Elementary proof using embeddings and nonsingular bilinear maps.
result Recovery of nonexistence results for affinely 3-regular maps without complex algebraic techniques.
Study on Fox's trapezoidal conjecture for specific alternating links.
problem Investigating Fox's trapezoidal conjecture for alternating links.
method Diagrammatic Murasugi sums, Alexander polynomial, and concordance analysis.
result Established inequalities and conditions for the Alexander polynomial and trapezoidal conjecture.
Alternative proof of Alexander polynomial trapezoid conjecture using dimers.
problem Alexander polynomial trapezoid conjecture for special alternating links.
method Dimer model approach to Alexander polynomial.
result Shorter and more accessible proof of Azarpendar, Juhász, and Kálmán's result.
New proof of trapezoidal property for Alexander polynomials of special alternating links.
problem Proving trapezoidal property of Alexander polynomials for special alternating links.
method Analyzing vector configurations from matroids and totally positive matrices.
result Alexander polynomials of special alternating links exhibit log-concavity and trapezoidal properties.
We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.
Optimal Reeb graphs identified for polygon decomposition.
problem Investigating the topological structure of planar polygon decomposition.
method Using oriented Reeb graphs with a marked vertex for height functions.
result Described all possible optimal Reeb graphs for specific polygon configurations.
Characterizes a specific type of alternating knot.
problem Identifying a special class of genus g alternating knots.
method Uses Ozsváth and Szabó's work on alternating knots.
result Shows that if the coefficients of Alexander polynomial satisfy a certain condition, the knot is a specific torus knot.
Study proves Alexander polynomials of certain 4-braid knots satisfy a conjecture and gives formulas for log-concave sequences.
problem Proving the Alexander polynomials of certain 4-braid knots satisfy Fox's Trapezoidal Conjecture.
method Analyzes families of alternating 4-braids and n-braids, providing explicit formulas and verifying log-concavity. result Explicit formulas for signature and first 4 coefficients of Alexander polynomials, showing log-concavity.
Paper introduces clock moves for plane graphs and proves Alexander polynomial properties.
problem Alexander polynomial of plane graphs and unimodality of coefficients.
method Introduces clock moves for plane graphs and develops a spanning tree model of Alexander polynomial.
result Proves unimodal property of Alexander polynomial coefficients and confirms conjectures.
Fox's trapezoidal conjecture for four-strand Turk's head knots is proven.
problem Proving log-concavity of the coefficient sequence of Dn(z) for four-strand Turk's head knots. method Four-block smoothing theorem for products of reciprocal quartics.
result The coefficient sequence of Dn(z) is log-concave. The paper evaluates functions of stable Lévy processes and their extrema efficiently.
problem Efficiently evaluating functions of stable Lévy processes and their extrema.
method Integral representations, conformal acceleration technique, simplified trapezoid rule.
result Efficient numerical procedures for cumulative probability distribution functions (cpdfs) are developed.
New methods for Z-transform inversion and Wiener-Hopf factorization.
problem Efficient numerical inversion of Z-transforms and factorization of functions. method Sinh-deformations of contours, variable changes, and simplified trapezoid rule.
result High precision and speed in evaluating moments and constructing filters.
Derives integral representations for a Lévy process and its extremum, hitting time, with fast evaluation.
problem Efficiently evaluating the joint probability density function of a Lévy process, its supremum, and hitting time.
method Integral representations, Laplace-Fourier transforms, summation by parts, conformal deformation, trapezoid rules, Gaver-Wynn-Rho algorithm.
result Explicit calculations and fast evaluation of the joint cpdf for Lévy processes.
Reconstruct trapezoidal surfaces from point clouds.
problem Reconstructing trapezoidal surfaces from point clouds.
method Kinematic approach: find axis direction, polygonal path, and reconstruct surface.
result Reconstructs trapezoidal surfaces from point clouds.
TENP prunes experts and neurons in Mixture-of-Experts models for efficient deployment.
problem Efficient deployment of large language models constrained by static parameter footprint.
method Structured Trapezoidal ExpertNeuron Pruning (TENP) identifies and retains important experts and neurons.
result DeepSeek model achieves 10% better performance on code generation tasks with 40% expert sparsity.
Path integral method calculates barrier option prices.
problem Barrier option pricing in finance.
method Path integral method applied to trapezoid and square potential barriers.
result Analytical expressions for option pricing derived.
In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. I…
New approach improves computational efficiency of Bass Local Volatility model.
problem Eliminate interpolation and improve computational efficiency in local volatility models.
method Combines local quadratic estimation and lognormal mixture tails for state price densities; uses trapezoidal rule for numerical convolutions.
result Proposed method outperforms traditional numerical methods in option pricing and market case studies.
We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomial…
We address the question of determining the eigenvalues λ_n (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with n nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …
Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the s…
Study proves properties of compact Hermitian surfaces with specific curvature conditions.
problem Characterizing compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature.
method Analyzes surfaces with Gauduchon connections and Lichnerowicz holomorphic sectional curvature.
result Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature are either Kähler or isosceles Hopf surfaces.
The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptio…
Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.
problem Parameterize spectral data of constant mean curvature tori in R^3.
method Use Whitham deformations, blowups, and spectral data analysis.
result Prove the Wente family is parameterized by the bisector of the right angle.
Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location of zeros of the Alexander polynomial of the fibered link complement implying a s…
Study links weaving knots with polynomial coefficients and lattice numbers.
problem Understanding polynomial coefficients of weaving knots and their lattice counterparts.
method Established relationships between Jones and Chebyshev polynomials, and derived explicit formulas for Alexander polynomials.
result Proved coefficients of Jones polynomial are Whitney numbers of Lucas lattices and satisfied Fox's trapezoidal conjecture.
Log-concave coefficient sequences for two-bridge knots proved.
problem Proving log-concavity of Alexander polynomial coefficient sequences for alternating knots.
method Introducing a polynomial Δ(t) associated to Christoffel words and proving its log-concavity. result Strong Fox conjecture for two-bridge knots proved.
We apply multilevel Monte Carlo for option pricing problems using exponential Lévy models with a uniform timestep discretisation to monitor the running maximum required for lookback and barrier options. The numerical results demonstrate the computational efficiency of this approach. We derive estimates of the convergen…
We consider Conway polynomials of two-bridge links as Euler continuant polynomials. As a consequence, we obtain new and elementary proofs of classical Murasugi's 1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a modulo 2 congruence for links, which implies the classical Murasugi's 1971 congruen…
Paper simulates LR fuzzy intervals with interval-valued cores.
problem Generating random fuzzy intervals with interval-valued cores.
method Developed algorithms for simulating LR fuzzy numbers with interval-valued cores.
result Numerically efficient algorithm for simulating fuzzy values.
Hamiltonian method applied to floating barrier options pricing.
problem Pricing of floating barrier options.
method Hamiltonian approach in quantum mechanics applied to barrier options.
result Analytical expressions for pricing kernel and option price derived.
Given a Hermitian manifold (Mn,g), the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call ∇s=(1−2s)∇c+2s∇b the s-Gauduchon connection of M, where ∇c and ∇b are r…
In this paper, we study open complete metric spaces with non-negative curvature. Among other things, we establish an extension of Perelman's soul theorem for possibly singular spaces: "Let X be a complete, non-compact, finite dimensional Alexandrov space with non-negative curvature. Suppose that X has no boundary and h…
Inverse spectral theory reveals shapes from sound.
problem Can the shape of a drum be determined by its sound?
method Inverse isospectral techniques applied to specific shapes.
result The regular n-gon can be uniquely determined by its eigenvalues.
New efficient method for inverse Z-transform reduces complexity significantly.
problem Efficient numerical realization of inverse Z-transform for large n.
method Derives sufficient conditions for new scheme, applies to option pricing.
result Significant reduction in complexity for large n, especially for European options.
The paper derives formulas for option pricing and random walk expectations.
problem Calculating the price of barrier and lookback options.
method Inverse Z-transform, Fourier/Laplace inversion, Wiener-Hopf factorization, and numerical methods.
result Efficient numerical methods for option pricing are developed.
We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this …
We derive a factorization of the Alexander polynomial of the 4-strand Turk's head knot using hypergeometric representations.
problem Deriving a factorization of the Alexander polynomial of the 4-strand Turk's head knot
method Using the reduced Burau representation and multivariable resultant elimination over reciprocal constraints
result Deriving a factorization of the Alexander polynomial in terms of Chebyshev polynomials
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate O(N−1/2).