Aguiar and Ardila defined the Hopf monoid GP of generalized permutahedra and showed that it contains many submonoids that correspond to combinatorial objects. They also give a basic polynomial invariant of generalized permutahedra, which then specializes to the submonoids. We define the Hopf monoid of directed graphs a…
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48 results for “basic polynomials”
We generalize the index polynomial invariant to the case of virtual tangles. Three polynomial invariants result from this generalization; we give a brief overview of their definition and some basic properties.
We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyl's First Fundamental Theorems, provides a characterization of the recently discovered Clifford foliations in terms of basic polynomials. This link a…
The study provides a formula for the volume of leaf spaces of certain foliations.
problem Calculating the volume of leaf spaces for singular Riemannian foliations.
method Proved a version of Weyl's Law for the basic spectrum of closed singular Riemannian foliations.
result Explicit formula for the volume of leaf spaces in terms of basic polynomials.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
Polynomial knotsmath.GT
A polynomial knot is a smooth embedding whose components are polynomials. The case is of particular interest. It is both an object of real algebraic geometry as well as being an open ended topological knot. This paper contains basic results for these knots as well as many examples.
The interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to a par…
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.
We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.
We call an Ising model tractable when it is possible to compute its partition function value (statistical inference) in polynomial time. The tractability also implies an ability to sample configurations of this model in polynomial time. The notion of tractability extends the basic case of planar zero-field Ising models…
The paper constructs quantum invariants for knotoid diagrams.
problem Quantum invariants for knotoid diagrams in .
method Decompose Morse knotoid diagrams into basic elementary diagrams, each associated with a matrix solving the quantum Yang-Baxter equation. Define quantum state sum models to recover various polynomials.
result Recover and define new polynomials for Morse knotoids.
Paper constructs a new approach to extract Affine Index Polynomial from Sawollek Polynomial.
problem Examining relationships between Affine Index Polynomial and Sawollek Polynomial.
method New approach to extract Affine Index Polynomial from Sawollek Polynomial.
result Constructs a concise proof of Mellor's Theorem.
This paper studies how knots combine using Alexander Polynomials.
problem How knots combine and their determinants behave.
method Basic knot theory, Alexander Polynomials, and composition techniques.
result Generalized solution for knot determinants in compositions.
We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletio…
The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the -polynomial and a fixed quantum invariant (such as the Jones polynomial) under filling. Our c…
Defines a new knot invariant and studies its properties.
problem Understanding the structure of knot invariants.
method Defining and analyzing the twisted Alexander vanishing order.
result Characterizes knots with a zero-twisted Alexander polynomial.
In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition…
Algorithm learns polynomials in Gaussian inputs with reduced sample complexity.
problem Learning polynomials of few relevant dimensions in high-dimensional data.
method Filtered PCA for warm start, geodesic SGD for accuracy.
result Sample complexity roughly , runtime .
This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we describe basic properties of such polynomials, show that they satisfy an analogue of t…
The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormalmath.SG
Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.
problem Understanding HOMFLYPT polynomials and their geometric origins.
method Defining worldsheet skein module and D-module, considering skein valued open curve counts.
result Worldsheet skein D-module for Hopf link conormal is generated by three operator polynomials.
The abstract conjectures a link between knot homologies and quiver partition functions.
problem Understanding the relationship between knot homologies and quiver partition functions.
method Interpreting quiver nodes as holomorphic curves with boundary on the knot conormal, and studying recursion relations.
result Generalized quiver partition functions are related to knot homologies.
Study Betti and Hodge numbers of solvmanifolds from integer polynomials.
problem Computing Betti and Hodge numbers of solvmanifolds constructed from integer polynomials.
method Analyzing de Rham and Dolbeault cohomology of solvmanifolds under algebraic conditions.
result Explicit generating polynomials for Hodge numbers in quasi full rank case.
Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
Let be a polynomial of degree with a Cremer point and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets . The \emph{red dwarf} are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a cont…
Study active learning of PTFs with derivative access.
problem Active learning of polynomial threshold functions (PTFs).
method Algorithm for active learning degree- univariate PTFs with derivative access.
result Computational efficient algorithm for active learning degree- univariate PTFs.
We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result the s…
We derive a formula expanding the bracket with respect to a natural deformation parameter. The expansion is in terms of a two-variable polynomial algebra of diagram resolutions generated by basic operations involving the Goldman bracket. A functorial characterization of this algebra is given. Differentiability properti…
Study describes moduli spaces of flat bundles on Sasakian manifolds.
problem Understanding moduli spaces of flat bundles on Sasakian manifolds.
method Shows moduli space of simple flat bundles is a union of spaces with fixed basic structures.
result Detailed description of non-abelian Hodge correspondence on compact Sasakian manifolds.
Introduced coloring-allowed invariants of planar knotoids with the coloring number.
problem Construction of polynomial invariants of knotoids with signs of crossings.
method Defined coloring-allowed invariants of planar knotoids with the coloring number.
result Discussed the 4-phases functions of coloring-allowed invariants.
We prove a Slice Theorem around closed leaves in a singular Riemannian foliation, and we use it to study the -algebra of smooth basic functions, generalizing to the inhomogeneous setting a number of results by G.~Schwarz. In particular, in the infinitesimal case we show that this algebra is generated by a fin…
Virtual Knot Theorymath.GT
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss…
Algorithm learns arbitrary ReLU neurons under Gaussian inputs.
problem Learn an arbitrary ReLU activation over Gaussian marginals.
method Statistical Query (SQ) algorithm that outputs a ReLU activation achieving loss.
result First constant factor approximation for arbitrary bias in polynomial time.
Paper uses ensemble learning for more accurate power flow modeling.
problem Improving accuracy and efficiency of power flow modeling.
method Applies polynomial regression and ensemble learning (GB, bagging) to create a more accurate linear power flow model.
result Data-driven model outperforms traditional methods in accuracy and speed.
New examples of isoparametric foliations on are provided.
problem Isoparametric foliations on of codimension 1.
method Construction of new examples and classification of bi-homogeneous polynomials.
result Isoparametric foliations can be realized as level sets of specific bi-homogeneous polynomials.
Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from the `cubic' point of view. Finite type invariants of knots and homology 3-spheres fit perfectly into this conception. In …
PolyLUT uses polynomials to reduce FPGA latency.
problem Reducing latency in FPGA-based neural network inference.
method Training neural networks using multivariate polynomials as basic building blocks.
result Achieved significant latency and area improvements.
Flat coordinates for Frobenius manifolds defined on the orbit space of a Coxeter group W are specified through a certain system of generators of W-invariant polynomials. In this note, starting from basic invariants proposed by M.Mehta, we calculate flat coordinates for the exceptional groups of type E_7 and E_8, leadin…
The paper explores polynomial functions with bounded Hess^+ complements and their properties.
problem Understanding the properties of functions with bounded Hess^+ complements.
method Detailed analysis of polynomial functions and their Hess^+ complements.
result Polynomial functions with bounded Hess^+ complements have specific properties like connectedness and convexity.
Infinitesimal holomorphic realizations for the Schrödinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The squeezed states for the unitary irreducible representation of the Jacobi group are introd…
This study extends verifiable learning to boosted tree ensembles, enabling efficient security verification.
problem Efficiently verifying the robustness of boosted tree ensembles against norm-based attackers.
method Formal verification of robustness for large-spread boosted tree ensembles, considering -norm and pseudo-polynomial time for -norm verification.
result Polynomial time verification for -norm attackers, NP-hard for other norms, and pseudo-polynomial time for -norm verification.
We define a Khovanov homotopy type for colored links and quantum spin networks and derive some of its basic properties. In the case of -colored B-adequate links, we show a stabilization of the homotopy types as the coloring , generalizing the tail behavior of the colored Jones …
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.
Explains basic knot and link theory for students.
problem Understanding simple invariants of knots and links.
method Defining and explaining linking number, Arf-Casson invariants, Alexander-Conway polynomials, and skein relations.
result Introduction of stronger invariants and the Vassiliev-Kontsevich theorem.
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulate…
We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L), where M is a closed oriented 3-manifold and L is an oriented link in M, by a tria…
A compact oriented 4-manifold is defined to be of ``superconformal simple type'' if certain polynomials in the basic classes (constructed using the Seiberg-Witten invariants) vanish identically. We show that all known 4-manifolds of are of superconformal simple type, and that the numerical invariants of 4-man…
Flat coordinates found for algebraic Frobenius manifolds in low dimensions.
problem Understanding algebraic Frobenius manifolds in small dimensions.
method Using reflection representations of finite Coxeter groups, finding flat coordinates of the Frobenius metric.
result Explicit relations between flat coordinates of the Frobenius metric and intersection form for most known examples up to dimension 4.
An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence…