The paper explores parity in knotoids and virtual knots, proving a conjecture and introducing a new polynomial.
problem Investigating parity in knotoids and its relation to virtual knots.
method Introducing a planar parity bracket polynomial and using the Nikonov/Manturov theorem.
result Minimal diagrams of knot-type knotoids have zero height.
This research classifies Teichmüller curves in genus 2, proving parity conjectures for specific cases.
problem Classifying imprimitive Teichmüller curves in $\M_2$ related to square-tiled surfaces and modular curves.
method Analyzing square-tiled surfaces and their modular curves, proving parity conjectures for specific cases.
result Established the parity conjecture for Wd2[n] in three cases, showing number of components does not depend on d. According to the work of Kontsevich-Zorich, the invariant that classifies non-hyperelliptic connected components of the moduli spaces of Abelian differentials with prescribed singularities,is the parity of the spin structure. We show that for the moduli space of quadratic differentials, the spin structure is constant o…
Counterexample disproves Spencer-Brown's claim about parity-pass algorithm.
problem Disproving Spencer-Brown's claim about the parity-pass algorithm and its relation to edge colorings.
method Provided a counterexample to Spencer-Brown's algorithm on non-polar pentagons.
result The parity-pass algorithm does not necessarily terminate in an extendable edge coloring.
We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads to a possibility of constructing new invariants and proving minimality and non-t…
Researchers solved a number-theoretic hypothesis to determine the spin parity of k-differentials.
problem Determining the spin parity of k-differentials on Riemann surfaces of genus zero and one.
method Proved a number-theoretic hypothesis (Conjecture A.10) by reformulating it in terms of Jacobi symbols and reducing it to a combinatorial identity.
result The spin parity of k-differentials on Riemann surfaces of genus zero and one was completely determined.
This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial. The paper is relatively self-contained and it describes virtual knot theory bot…
We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the prob…
Polynomial-time algorithm solves random parity games with high probability.
problem Solving random parity games efficiently.
method SWCP algorithm based on cycles in subgraphs.
result Polynomial-time solution for large-degree games with high probability.
We establish the volume conjecture for (m,2)-cables of the figure 8 knot, when m is odd. For (m,2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev invariant). There are many cases when the volume conjecture for cables of the fig…
A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension d=2m depends only on the parity of m. Gromov defined several hyperbolization functors which produce an aspherical manifold …
New parity theory for virtual links extends Gaussian parity and yields stronger invariants.
problem Defining a parity theory for a broad class of virtual links.
method Introducing 2-colour parity, extending Gaussian parity, and comparing to Im-Park parity.
result 2-colour parity yields a stronger invariant than Im-Park parity.
Parity functors assign labels to knot diagrams based on crossing parity.
problem Assigning consistent labels to knot diagrams.
method Define parity functors for knot diagrams and surfaces.
result Universal oriented parity functors for free knots and fixed surface knots.
Parity defined for based matrices, a new example of virtual knot parity.
problem Defining parity for a new algebraic structure.
method Introduced parity for based matrices, defined reduced stable parity.
result New example of parity for virtual knots.
In the present paper, we develop the parity theory invented in \cite{ManSb}; we construct new parities for two-component (virtual and free) links. New parities significantly depend on geometrical properties of diagrams; in particular, they are mutation-sensitive. New parities can be used practically in all problems, wh…
Study shows Transformers can generalize to varying task lengths.
problem Understanding when and how Transformers can generalize to different input lengths.
method Proposed a unifying framework and introduced the RASP-Generalization Conjecture.
result Transformers tend to length generalize on tasks if solvable by short RASP programs.
In \cite {FrKn,Sbornik} it was shown that in some knot theories the crucial role is played by {\em parity}, i.e.\ a function on crossings valued in {0,1} and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and …
We identify conditional parity as a general notion of non-discrimination in machine learning. In fact, several recently proposed notions of non-discrimination, including a few counterfactual notions, are instances of conditional parity. We show that conditional parity is amenable to statistical analysis by studying ran…
Universal Gaussian parity proven for 2D knots.
problem Proving universal Gaussian parity for 2D knots.
method Analyzing Gaussian parity on free 2D knots.
result Gaussian parity is universal for 2D knots.
New parities defined on virtual knots linked to crossing indices.
problem Defining parities on virtual knots.
method Connecting parities to invariant cycles on arcs and quasi-indices on crossings.
result New series of parities on virtual knots defined.
New examples show non-trivial parity-biquandle bracket.
problem Constructing non-trivial parity-biquandle bracket examples.
method Slightly changed notation and constructed examples of knots and links.
result Minimality theorem: graphs appear as link invariants.
This paper tackles fair Bayes-optimal classifiers under predictive parity, proving their limitations and proposing a new algorithm.
problem Ensuring fair Bayes-optimal classifiers under predictive parity, especially when group performance levels vary widely.
method Proving the limitations of fair Bayes-optimal classifiers under predictive parity and proposing a new adaptive thresholding algorithm, FairBayes-DPP.
result Fair Bayes-optimal classifiers under predictive parity may not hold if group performance levels vary widely, leading to within-group unfairness.
Study of origamis in minimal stratum with single cylinders, calculating spin parities and monodromy groups.
problem Understanding the structure and properties of origamis in the minimal stratum of moduli space.
method Construction and analysis of minimal [1,1]-origamis, calculation of spin parities, and investigation of monodromy groups. result All minimal [1,1]-origamis have monodromy groups that are almost always finite simple groups. Parity calibration aims to predict increase-decrease events, not values.
problem Forecasting future increase-decrease events rather than exact values.
method Online binary calibration method to achieve parity calibration.
result Online binary calibration achieves parity calibration effectively.
We use crossing parity to construct a generalization of biquandles for virtual knots which we call Parity Biquandles. These structures include all biquandles as a standard example referred to as the even parity biquandle. Additionally, we find all Parity Biquandles arising from the Alexander Biquandle and Quaternionic …
Diversified risk parity strategies outperform equally-weighted portfolios in various asset universes.
problem Finding optimal portfolio allocations that balance risk and reward.
method Integrates various reward-risk measures and generic allocation rules into diversified risk parity.
result Diversified reward-risk parity strategies exhibit higher average returns, Sharpe ratios, and Calmar ratios compared to equally-weighted risk portfolios.
Functorial maps and weak parities are equivalent descriptions of rules of substitution virtual crossings for classical in diagrams of a knot in a way compatible with Reidemeister moves. We introduce the notion of maximal weak parity and describe it for knots in a given closed oriented surface. This weak parity defines …
Introduce a two-variable parity polynomial for virtual knotoids
problem Define a polynomial invariant for virtual knotoids
method Based on the parity of classical crossings
result Can distinguish pairs not distinguished by odd writhe and affine index polynomial
Paper solves k-sparse parity problem with sign SGD, matching SQ lower bound.
problem Solving k-sparse parity problems efficiently.
method Sign stochastic gradient descent on neural networks.
result Matches Statistical Query lower bound for solving k-sparse parity problems.
Neural networks struggle with learning fixed parities.
problem Difficulty of learning fixed parities with neural networks.
method Using perturbed gradient descent on one-hidden-layer ReLU networks.
result Training neural networks on fixed parities fails to produce meaningful results.
Study shows physical drift affects put-call parity enforcement, not just option payoffs.
problem Inconsistency between quoted put-call parity and actual market behavior.
method Examined SPX and RUT index options, used drift-preserving GBM term to improve fit.
result Physical drift enters the enforcement of risk-neutral parity, not just option payoffs.
Study inequalities between knot invariants and compute new bounds.
problem Understanding inequalities and bounds between knot invariants.
method Directed graph of inequalities, database construction, systematic transitivity criterion.
result 139 new exact values for unknotting number and doubly slice genus.
Parity mappings from the chords of a Gauss diagram to the integers is defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and virtual knots. One family consists of degree n Vassiliev invariants.
We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. These invariants are determined by pairs consisting of a biquandle 2-cocycle φ^0 and a map φ^1 with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide …
Counterfactual fairness not equivalent to demographic parity, finds study.
problem Equivalence between causal and probabilistic concepts in fairness metrics.
method Close examination of recent claim about counterfactual fairness.
result Counterfactual fairness is not equivalent to demographic parity.
Transformers solve parity problems efficiently with step-by-step reasoning.
problem Training transformers to solve complex, recursive problems like parity.
method Training a one-layer transformer to solve k-parity, incorporating intermediate parities into the loss function, and using teacher forcing or augmented data. result Transformers can learn parity in one gradient update with intermediate supervision or self-consistency checks.
2-dimensional knots and links are studied in the article. The notion of parity is introduced via techniques similar to the ones used by the second named author in 1-dimensional case. By using parity new invariants are constructed and known invariants are refined.
New tools study curvature measures of convex bodies, revealing structured spaces.
problem Investigate translation invariant curvature measures of convex bodies.
method Introduce new tools to study curvature measures, proving conjectures about their structure.
result Space of curvature measures has length at most 2 as a representation of the general linear group in degrees 0 and n-2.
Study calculates quantum hyperbolic invariants for figure-eight knot complement, finding it either 0 or half the volume.
problem Computing quantum hyperbolic invariants for knot complements.
method Computed the real part of the semi-classical limit of quantum hyperbolic invariants of the figure-eight knot complement.
result The real part is rigid and either 0 or half the hyperbolic volume of the knot complement.
New causal analysis reconciles predictive and statistical fairness.
problem Mutual exclusivity of predictive and statistical fairness notions.
method Derive a new causal decomposition formula for fairness measures.
result Predictive and statistical fairness are complementary, not mutually exclusive.
Proposes a framework to create fair IDRs by enforcing demographic parity constraints.
problem Discrimination in IDRs trained on biased data.
method Incorporates DP and CDP constraints into IDR estimation.
result Theoretically optimal IDRs can be efficiently obtained through perturbations.
BriarPatches obscure sensitive attributes to achieve demographic parity.
problem Achieving demographic parity in model predictions.
method Pixel-space interventions that obscure sensitive attributes from classifier representations.
result BriarPatches push downstream predictors towards demographic parity.
New method controls bias in training data for fair outcomes.
problem Ensuring equal treatment between different groups in machine learning.
method Contrastive information estimation to control mutual information between representations and protected attributes.
result Our method provides strong theoretical guarantees on the parity of any downstream algorithm.
The article develops a model for skewness risk in risk parity portfolios.
problem Managing skewness risk in asset allocation models.
method Modeling asset returns with skewness and jumps, deriving analytical formulas for risk contributions.
result Skewness-based risk parity portfolios outperform volatility-based portfolios in managing jump risks.
We mathematically compare four competing definitions of group-level nondiscrimination: demographic parity, equalized odds, predictive parity, and calibration. Using the theoretical framework of Friedler et al., we study the properties of each definition under various worldviews, which are assumptions about how, if at a…
We investigate an application of crossing parity for the bracket expansion of the Jones polynomial for virtual knots. In addition we consider an application of parity for the arrow polynomial as well as for the categorifications of both polynomials. We present a number of examples found through our calculations. We pro…
Defines new invariant for virtual knots.
problem Determining unknottability of virtual knots.
method Parity virtual Alexander polynomial.
result Many virtual knots cannot be unknotted by odd crossing changes.
Incorrect parity-based descriptions of realizable Gauss diagrams found, but bipartite graphs provide a valid approach.
problem Incorrect descriptions of realizable Gauss diagrams using parity conditions.
method Used bipartite graphs to describe realizable Gauss diagrams.
result Realizable Gauss diagrams can be accurately described using bipartite graphs.