Polynomial processes have the property that expectations of polynomial functions (of degree , say) of the future state of the process conditional on the current state are given by polynomials (of degree ) of the current state. Here we explore the application of polynomial processes in the context of structur…
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48 results for “dynamic polynomial degree”
Bayesian optimisation for expensive experiments with shape prior.
problem Expensive experiments with time-varying control variables.
method Developed a novel Bayesian optimisation framework using Bernstein polynomial basis and dynamic polynomial degree adjustment.
result Demonstrated effectiveness on polymer fibre design and learning rate optimisation.
We extend the exploration regarding dynamical approach of macroeconomic variables by tackling systematically expenditure using Statistical Physics models (for the first time to the best of our knowledge). Also, using polynomial distribution which characterizes the behavior of dynamical systems in certain situations, we…
Polynomial distribution can be applied to dynamical systems in certain situations. Macroeconomic systems characterized by economic variables such as income and wealth can be modelled similarly using polynomials. We extend our previous work to data regarding income from a more diversified pool of countries, which contai…
The study computes trace fields and minimal polynomials for specific knots and links.
problem Computing trace fields and minimal polynomials for specific knots and links.
method Using factorization theorems for sparse polynomials.
result Results depend on the degrees of the trace fields over Q being sufficiently large.
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.
The paper defines a path metric on a stable component of polynomial families.
problem Understanding the geometry of polynomial families with parabolic relations.
method Constructing a positive semi-definite pressure form on a bounded stable component of the moduli space.
result The pressure form defines a path metric on the stable component.
The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees th…
New relations for vertex polynomial in graphs of any degree.
problem Understanding vertex polynomial in graphs of varying degrees.
method Proved local relations for digons, triangles, quadrilaterals, and pentagons.
result Established new relations for vertex polynomial in graphs of arbitrary degree.
Survey on using low-degree polynomials to assess statistical tasks complexity.
problem Understanding the complexity of statistical tasks using polynomial functions.
method Applying low-degree polynomials to measure the complexity of statistical tasks, including detection, recovery, and estimation.
result Low-degree polynomials provide a framework to predict and explain statistical-computational tradeoffs.
We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developin…
Polynomial neural networks explore thresholds for maximum expressiveness.
problem Understanding the limits of polynomial neural networks' expressiveness.
method Introducing activation degree threshold to measure network expressiveness and proving its existence and upper bounds.
result Polynomial neural networks with equi-width architectures achieve the maximum expressiveness.
Low-degree method fails to predict robust subspace recovery problem.
problem Predicting computational tractability of robust subspace recovery problem.
method Low-degree polynomial framework, anti-concentration properties.
result Low-degree method fails to predict computational tractability of robust subspace recovery problem even up to high degree.
New findings on computational limits for estimating hidden structures.
problem Estimating hidden structures in noisy data.
method Use of low-degree polynomials as a restricted model of computation.
result Established low-degree hardness of recovery problems for easy detection problems.
Sharp upper bound for quasi polynomial degree of manifold configuration spaces.
problem Determining the exact degree of quasi-polynomial homology groups of configuration spaces.
method Analyzing extremal homology groups of unordered configuration spaces of manifolds.
result The upper bound for the degree of quasi-polynomials is sharp for every manifold.
New moving average adapts weight dynamically based on polynomial and wavefunction.
problem Lagging traditional moving averages in adjusting to changes in data.
method Develops a moving average with weight as a polynomial of a wavefunction from an eigenproblem.
result Immediate 'switch' without lag, adapting to changes in data.
Local high-degree polynomial integrals of geodesic flows and the generalized hodograph methodmath.DS
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
NGRC shows numerical instabilities with short lags and high-degree polynomials.
problem Numerical instabilities in NGRC feature matrix.
method Combining numerical linear algebra and dynamical systems theory, we study feature matrix conditioning. We evaluate different numerical algorithms for solving the regularized least-squares problem.
result SVD-based training achieves accurate forecasts without regularization, preferable for short lags and high-degree polynomials.
New findings show that common optimization algorithms struggle with random problems.
problem Finding near-optimal solutions to random optimization problems.
method Low-degree polynomials, Boolean circuits, and Langevin dynamics.
result These algorithms fail to produce nearly optimal solutions with high probability.
Alexander polynomial degree correlates with knot defect, proving conjecture for defect zero.
problem Characterizing knot polynomials and their defects.
method Analyzing differential expansions and degree in of Alexander polynomials.
result Proved Alexander polynomial degree correlates with knot defect, especially for defect zero.
Homology groups of spaces of nonsingular polynomial embeddings of degrees are calculated. A general algebraic technique of such calculations for spaces of polynomial knots of arbitrary degrees is described.
We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many cases.
We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how the Jones slopes and the linear term behave under cabling. As an application we ve…
Stability in the cohomology of the space of complex irreducible polynomials in several variablesmath.AG
The cohomology of complex irreducible polynomials stabilizes with degree or variables.
problem Understanding the cohomology of complex irreducible polynomials.
method Proving homological stability in cohomology as degree or variables increase.
result Cohomology stabilizes with respect to both degree and number of variables.
Study disproves conjecture about low-degree polynomials in hypothesis testing.
problem Conjecture about limitations of polynomial-time algorithms in hypothesis testing.
method Used counterexamples to refute the conjecture and modified the conjecture to rule out the counterexample.
result Disproved conjecture about limitations of low-degree polynomials in hypothesis testing.
This note gives a proof that the -polynomial of any nontrivial knot in has nontrivial -degree.
Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potent…
The basin of infinity of a polynomial map $f : {\bf C} \arrow {\bf C}$ carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface . As diverges in the moduli space of polynomials, the surface collapses along its foliation to yield a metrized simplicial t…
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.
The paper explores how low-degree polynomials can detect shuffled linear regression models.
problem Detecting multivariate shuffled linear regression models from independent Gaussian random matrices.
method Investigates the effectiveness of low-degree polynomial algorithms for distinguishing the model from independent Gaussian random matrices.
result Establishes a phase transition phenomenon in the performance of low-degree polynomial algorithms for distinguishing the model.
Estimates piecewise polynomials and bounded variation functions using optimal decision trees.
problem Estimating piecewise smooth functions in general dimensions.
method Dyadic CART and Optimal Regression Tree (ORT) estimators for piecewise polynomials and bounded variation functions.
result Oracle inequalities and risk bounds for ORT estimators, demonstrating adaptivity and optimality.
The alternating knots, links and twists projected on the sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond to the stable and unstable manifolds connecting the saddles. Each face is then …
Psyquandle Coloring Quiversmath.GT
Enhances psyquandle invariants for singular and pseudoknots.
problem Counting invariants for singular knots and pseudoknots.
method Uses quivers to extend in-degree polynomial invariants.
result Obtains biquandle coloring quivers and in-degree polynomial invariants.
The monotonicity of the Franz-Parisi potential is equivalent with Low-degree MMSE lower boundsmath.ST
New work shows FP potential monotonicity equals low-degree polynomial estimators limits.
problem Establishing a precise mathematical relationship between statistical physics and polynomial estimators limits.
method Analyzing Gaussian additive models (GAMs) to show FP potential monotonicity equals low-degree polynomial estimators limits.
result For a broad family of Gaussian additive models, the power of low-degree polynomials is equivalent to the monotonicity of the annealed FP potential.
We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [KP10] and [BKP14]. We also give a sharp…
In this paper, we investigate twist sequences for Kauffman finite-type invariants and Goussarov-Polyak-Viro finite-type invariants. It is shown that one obtains a Kauffman or GPV type of degree if and only if an invariant is a polynomial of degree on every twist lattice of the right form. The main resul…
The study characterizes and constructs polynomial harmonic morphisms on spheres.
problem Characterizing and constructing polynomial harmonic morphisms on spheres.
method Characterization and construction of polynomial harmonic morphisms using eigenfamilies.
result Strong restrictions and classification of polynomial harmonic morphisms in low dimensions.
New bound on Jones polynomial for specific positive links.
problem Finding bounds on the Jones polynomial for positive links.
method Using previous results on positive fibered links, we found a new bound for a specific family of positive links.
result We provided a bound on the maximum degree of the Jones polynomial for positive links with a specific coefficient.
Two new polynomial invariants for long virtual knots.
problem Defining new polynomial invariants for long virtual knots.
method Introducing and , establishing properties, and showing realizability.
result First derivatives of and at define finite type invariants of degree three.
Enhanced invariant for linkoids using quivers.
problem Counting invariants for linkoids.
method Use of quivers to generalize in-degree polynomial invariant.
result Introduced in-degree quiver polynomial matrix as a new invariant.
Cappell-Shaneson polynomialsmath.GT
The paper defines and classifies Cappell-Shaneson polynomials.
problem Characterizing Cappell-Shaneson polynomials.
method Algebraic conditions on polynomials, reduction modulo primes, and construction of infinite series.
result Complete lists of Cappell-Shaneson polynomials of degrees 4 and 5, and several infinite series of degree 6.
New bounds on HOMFLY polynomial for homogeneous links.
problem Bounding the minimum -degree of HOMFLY polynomial for homogeneous links.
method Proved a slice version of Cromwell's inequality and a related conjecture.
result New bounds on the minimum -degree of HOMFLY polynomial for homogeneous links.
New upper bound on Jones polynomial for fibered positive links.
problem Classifying positive and non-positive knots of crossing number ≤ 12.
method Proved a new upper bound on the maximum degree of Jones polynomial for fibered positive knots.
result Maximum degree of Jones polynomial for fibered positive knots is at most four times the minimum degree.
New knots with specific properties have identical polynomial values.
problem Identifying knots with matching polynomial values after braiding.
method Constructing infinitely many hyperbolic knots and analyzing their braided satellites.
result Mutually distinct hyperbolic knots have identical HOMFLY polynomial values up to given -degrees.
In-degree quiver polynomials for surface-links computed.
problem Computing in-degree quiver polynomials for surface-links.
method Defined using a quandle and set of endomorphisms, computed for surface-links with ch-index up to 10.
result Example computations for surface-links with ch-index up to 10.
Study robustness of polynomial neural networks using algebraic geometry.
problem Certify robustness radius of polynomial neural networks.
method Metric algebraic geometry, Euclidean distance degree, symbolic elimination, homotopy-continuation methods.
result Found decision boundaries with lower ED degree than generic cubic hypersurfaces.
We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.
We give a criterion to detect whether the derivatives of the HOMFLY polynomial at a point is a Vassiliev invariant or not. In particular, for a complex number b we show that the derivative P_K^{(m,n)}(b,0)=d^m/da^m d^n/dx^n P_K(a,x)|(a, x) = (b, 0) of the HOMFLY polynomial of a knot K at (b,0) is a Vassiliev invariant …