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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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…
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…
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…
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…
Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around R⊂C. The Fourier transform techniques reduces calculation of probability distributions and option prices to evaluation of integrals whose i…
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.
Study Alexander polynomials of special alternating links and generalize Fox's conjecture.
problem Distinguish special alternating links up to isotopy using polynomial invariants.
method Combinatorial and discrete geometric properties of Alexander polynomials of special alternating links.
result Generalized Alexander polynomials of special alternating links can be expressed in terms of volumes of root polytopes of unimodular matrices.
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 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 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.
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). Efficiently price VIX options using multilevel Monte Carlo in rough Bergomi model.
problem Pricing VIX options in a rough Bergomi model with high computational complexity.
method Combining rectangle discretization, Cholesky sampling, and multilevel Monte Carlo.
result Reduced computational complexity to O(ε−2log2(ε)) and asymptotically optimal O(ε−2). New method for efficient pricing of double barrier options in Lévy models.
problem Difficulties in accurately and quickly calculating prices of double barrier options in jump models.
method GWR-SINH method based on Gaver-Wynn-Rho acceleration applied to Bromwich integral.
result Accurate and fast calculations of prices of double barrier options in jump models achieved.
New method improves inference for discrete diffusion models, achieving better quality and efficiency.
problem High dimensionality of discrete diffusion models causes inference challenges.
method Developed high-order numerical inference schemes for discrete diffusion models.
result Second-order accuracy of the θ-Trapezoidal method in KL divergence. A non-parametric method for evaluation of the aggregate loss distribution (ALD) by combining and numerically inverting the empirical characteristic functions (CFs) is presented and illustrated. This approach to evaluate ALD is based on purely non-parametric considerations, i.e., based on the empirical CFs of frequency …
Fast method developed for pricing barrier options and joint Lévy process distributions.
problem Accurate pricing of barrier options and joint distributions in Lévy models.
method Dual space calculations, Wiener-Hopf factorization, sinh-deformations, Gaver-Wynn Rho acceleration.
result Achieves precision of 10−15 in seconds and 10−9−10−8 in fractions of a second. We consider a two-factor model for the valuation of a non callable defaultable bond which pays coupons at certain given dates. The model under consideration is the Jump to Default Constant Elasticity of Variance (JDCEV) model. The JDCEV model is an improvement of the reduced form approach, which unifies credit and equi…
In this thesis, we consider the knot energy "integral Menger curvature" which is the triple integral over the inverse of the classic circumradius of three distinct points on the given knot to the power p∈[2,∞). We prove the existence of the first variation for a subset of a certain fractional Sobolev space if…
Efficiently calculates privacy guarantees for 2020 Census data.
problem Evaluate privacy guarantees for 2020 U.S. Census data releases.
method Sieve-accelerated quadrature method to evaluate tail probabilities of high-dimensional convolutions.
result Achieves 1,824-fold speedup over prior methods while maintaining error tolerances.