Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chor…
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48 results for “polynomial expansions”
Computes Jones polynomial for specific knots.
problem Computing Jones polynomial for double twist knots.
method Using cyclotomic expansion and Kauffman bracket skein theory.
result Answers a question about Jones polynomial for specific knots.
We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion o…
Develops AMITE for analyzing neural network nonlinearities.
problem Addressing difficulties in verification, explainability, and security in neural network analysis.
method Analytically modified integral transform expansion (AMITE) for neural network nonlinearities.
result First to provide six mutually exclusive desired expansion properties.
In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus correctio…
In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern--Simons invaraint and Reidemeister torsion.
A new method builds sparse polynomial chaos expansions for models with dependent inputs.
problem Quantifying uncertainty in models with dependent inputs.
method Data-driven approach to construct orthonormal polynomials recursively based on input correlations.
result Reduces the number of observations and improves numerical stability and computational efficiency.
Colored knot polynomials possess a peculiar Z-expansion in certain combinations of differentials, which depends on the representation. The coefficients of this expansion are functions of the three variables (A,q,t) and can be considered as new distinguished coordinates on the space of knot polynomials, analogous to the…
Enhances polynomial chaos models with uncertainty intervals.
problem Uncertainty quantification in surrogate models.
method Jackknife-based conformal prediction integrated into polynomial chaos expansions.
result Produces accurate predictive intervals for low-accuracy models.
Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Asymptotic expansion formula for colored Jones polynomial using twist knots.
result Obtained asymptotic expansion formulas for twist knots at specified roots of unity.
Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Saddle point method applied to colored Jones polynomial.
result Asymptotic expansion formula for twist knots at given root of unity.
Solves generalized twisted rabbit problems for higher degree polynomials.
problem When a quadratic polynomial is twisted by a cyclic subgroup, what polynomial is equivalent?
method Uses -adic expansion instead of 4-adic for higher degree polynomials.
result Provides a solution that depends on the -adic expansion of the power of the mapping class element.
Derives a series expansion for Asian option pricing with polynomial jump-diffusion moments.
problem Pricing Asian options with polynomial jump-diffusion processes.
method Uses Hermite polynomials and moments of the underlying process for closed-form computation.
result Explicit computation of Greeks and accurate series expansion for Asian options.
In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivia…
We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum repr…
We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any kno…
Estimates hybrid dynamical systems with polynomial expansions and Markovian switching.
problem Identifying hybrid dynamical systems with nonlinear autoregressive exogenous (NARX) components and Markovian switching.
method Probabilistic framework using Expectation Maximization for parameter estimation, including submodel coefficients, hidden state values, and transition probabilities. Disentangles mode classification and NARX regression tasks. Uses soft-labels and coordinate descent approach for parameter fitting.
result Demonstrated on a SMNARX problem with three nonlinear sub-models, achieving parsimonious models through l1-norm bridge estimation and hard-thresholding.
New link polynomials linked to cluster theory.
problem Connecting link polynomials to cluster theory.
method Introducing new link polynomials and their expansion over perfect matchings.
result Bracket polynomials of certain links can be realized as specializations of cluster variables.
Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.
problem Proving the Kontsevich-Witten tau-function formula.
method Directly shows Q-polynomial expansion satisfies Virasoro constraints.
result Direct proof of the formula without matrix model.
Twisting cubic rabbitsmath.DS
Solves infinite family of cubic polynomial problems.
problem Infinite family of twisted polynomial problems.
method Using Dehn twists and 9-adic expansions.
result Result of twisting depends on 9-adic expansion.
Analyzes the differential expansion of knot polynomials, focusing on its applicability and modifications.
problem Understanding the differential expansion of colored knot polynomials, especially for non-trivial knots and those with defects.
method Examines the current status of differential expansion, analyzes its applicability to non-trivial knots, and introduces a new transformation.
result A new transformation that converts to standard -factors and allows for the calculation of .
We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accura…
The paper calculates super Weil-Petersson volumes for large genus.
problem Calculating super Weil-Petersson volumes for large genus.
method Analyzes super intersection numbers, proves coefficients are polynomials, and provides an algorithm to compute them.
result Proves existence of a complete asymptotic expansion of super Weil-Petersson volumes.
Study proposes a method to construct copulas using corrected Hermite polynomial expansion for estimating foreign exchange volatility.
problem Estimating cross foreign exchange volatility with complex correlation structures.
method Applying corrections to the finite sum of multivariate Hermite polynomial expansions to construct copulas.
result The proposed copula method accurately reproduces the volatility smile of cross currency pairs.
Paper finds coefficients of Catalan states using Θ_A-state expansion.
problem Finding coefficients of Catalan states of lattice crossings.
method Uses Θ_A-state expansion to express coefficients as a linear combination of other states.
result Shows that coefficients can be found using Θ_A-state expansion.
The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the -symbols…
The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. We descr…
The paper connects ADO polynomials to Vassiliev invariants for knots.
problem Connecting ADO polynomials to Vassiliev invariants for knots.
method Exploiting the colored Jones polynomials and their decomposition as Vassiliev invariants, the authors transpose this to ADO polynomials.
result A unique computable expansion of ADO polynomials as Vassiliev invariants.
Jones rational coincidencesmath.GT
Jones polynomial coincidences explored for rational knots.
problem Identifying coincidences in Jones polynomial of rational knots.
method Moves on continued fraction expansion of rational knots, conjectured to generate all coincidences.
result Conjectured moves are sufficient to generate all Jones rational coincidences.
We obtain a complete time expansion of the pull-back operator generated by a real analytic flow of real analytic automorphisms acting on analytic tensor sections of a manifold. Our expansion is given in terms of multiple Lie derivatives. Motivated by this expansion, we provide a rather simple and explicit estimate for …
We formulate a conjecture (already proven by A. Kricker) about the structure of Kontsevich integral of a knot. We describe its value in terms of the generating functions for the numbers of external edges attached to closed 3-valent diagrams. We conjecture that these functions are rational functions of the exponentials …
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative re…
Bayesian adaptive PCE method improves surrogate modeling and sensitivity analysis.
problem Lack of fully Bayesian PCE methods in statistics.
method Developed a novel fully Bayesian adaptive PCE method with R implementation.
result Bayesian adaptive PCE provides competitive performance for various UQ tasks.
In many applications (in particular information systems, such as pattern recognition, machine learning, cheminformatics, bioinformatics to name but a few) the assessment of uncertainty is essential - i.e., the estimation of the underlying probability distribution function. More often than not, the form of this function…
New formulas derived for lattice crossing coefficients, improving computation efficiency.
problem Computing coefficients of Catalan states in lattice crossings.
method Using plucking polynomial and Θ_A-state expansion, deriving new properties and formulas.
result Coefficients of Catalan states factor under specific conditions, leading to more efficient computation.
New polynomial for checkerboard-colorable 4-valent virtual graphs.
problem No specific problem stated; focuses on a new polynomial.
method Euler circuit expansion to assign polynomial to graphs.
result New combinatorial formulation of Kauffman-Jones polynomial.
Empirical analysis of many colored knot polynomials, made possible by recent computational advances in Chern-Simons theory, reveals their stability: for any given negative N and any given knot the set of coefficients of the polynomial in r-th symmetric representation does not change with r, if it is large enough. This …
This work reconstructs knot invariants from Alexander polynomials, proving consistency with known theorems.
problem Reconstructing knot invariants from Alexander polynomials.
method Quantization, deformation, and rewriting of Alexander polynomials.
result Derives new formulae for colored superpolynomials and proves consistency with Melvin-Morton-Rozansky theorem.
We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial.
In this paper, we computed the first three coefficients of the asymptotic expansion of Zelditch. We also proved that in general, the -th coefficient is a polynomial of the curvature and its derivative of weight .
Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.
problem Solving special Lagrangian equations near infinity with specific conditions.
method Modified Kelvin transforms to characterize remainders in asymptotic expansions.
result Remainders in asymptotic expansions are characterized by a single smooth function in even dimensions and in odd dimensions.
New derivation of knot invariants from universal invariant.
problem Deriving knot invariants from universal invariant.
method Using Hopf algebra and a Mathematica implementation to compute .
result Derivation of large-color expansion from universal invariant.
New formula recovers degree of colored Jones polynomials for pretzel knots.
problem Determining the degree of colored Jones polynomials for specific knots.
method Alternate expansion of the colored Jones polynomial for pretzel links, focusing on 3-tangle knots.
result Determined the degrees of the colored Jones polynomials for a new family of 3-tangle pretzel knots.
Approximates discounted moments for financial products using polynomial expansions.
problem Approximating discounted moments of stochastic processes for financial applications.
method High-order power series expansion of the infinitesimal generator.
result Error decreases to around 10 to 100 times machine precision for higher orders.
A new method for creating simpler models from complex ones.
problem Creating accurate approximations of complex models at reduced costs.
method Sequential adaptive surrogate modeling based on locally spectral expansions.
result Stochastic spectral embedding (SSE) shows good approximation capabilities and scalability.
We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity. Our conjecture claims that the asymptotic expansion of the colored Jones polynomial is a the formal wave function of an integrable sys…
We collect statistics which consist of the coefficients in the expansion of the generating polynomials that count the Kauffman states associated with certain classes of pretzel knots having n tangles, of r half-twists respectively.
Polynomial Chaos Expansion improves operator learning for PDEs.
problem Approximating mappings between infinite-dimensional functional spaces.
method Polynomial Chaos Expansion (PCE) for operator learning.
result PCE achieves strong performance in operator learning and uncertainty quantification.