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…
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48 results for “spin parity”
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.
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 -origamis, calculation of spin parities, and investigation of monodromy groups.
result All minimal -origamis have monodromy groups that are almost always finite simple groups.
Study 3d N=1 vacua from M-theory compactification on Spin(7) space.
problem Quantum corrections in 3d N=1 vacua from M-theory compactification.
method Use Higgs bundles to analyze 3d N=1 vacua and track corrections.
result Topological anomalies are robust and calculable in 3d effective field theory.
Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected c…
Study describes how to realize periods of meromorphic differentials with specific properties.
problem Realizing meromorphic differentials with given zeros, poles, and topological constraints.
method Complete description of period representations for specified conditions on Riemann surfaces.
result A comprehensive method for realizing meromorphic differentials with prescribed characteristics.
Algorithm computes fundamental classes of spin components in moduli space.
problem Computing fundamental classes of spin components in moduli space.
method Reconstruct cycles by boundary restrictions via clutching maps and solve linear equations.
result Algorithm implemented in Sage package for cycle class computation.
Researchers compute the ν-invariant for specific G2-structures on nilmanifolds.
problem Detecting connected components of G2-structure moduli spaces.
method Defined and computed the ν-invariant using Mathai-Quillen currents, harmonic spinors, and η-invariants.
result Determined the parity of harmonic spinor dimensions and deduced ν vanishing on invariant spinors.
Study refines Siegel-Veech constants for abelian differentials.
problem Computing Siegel-Veech constants for abelian differentials.
method Intersection theory and quasimodular forms.
result New identity for Siegel-Veech constants of cylinders.
Study describes how to realize periods of holomorphic differentials with specific properties.
problem Realizing periods of holomorphic differentials with given zeros and invariants.
method Complete description of realizable relative period representations.
result Answers a question posed by Simion Filip about realizing periods of holomorphic differentials.
Invariant obstructs separating coassociative 4-folds.
problem Obstructing the separation of coassociative 4-folds.
method Defining a -valued invariant for coassociative 4-folds with spin structures.
result Invariant provides an obstruction to separating coassociative 4-folds.
Study fermionic theories, their anomalies, and modular transformations.
problem Understanding fermionic theories and their anomalies.
method Use spin-cobordisms, surgeries, and invertible topological quantum field theories.
result Explicit combinatorial expressions for spin-bordism invariants.
Scalar curvature rigidity for products of convex hypersurfaces
problem Rigidity of scalar curvature in products of convex hypersurfaces
method Clifford-linear family index theory
result Scalar curvature inequality implies isometry
Parity functorsmath.GT
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 on based matricesmath.GT
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.
We introduce the 2-colour parity. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gaussian parity, to which it reduces on virtual knots. We show that the 2-c…
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…
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 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.
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.
Parity Calibrationcs.LG
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.
Parity Biquandlesmath.GT
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
Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smootha…
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.
In [3] we constructed the parity-biquandle bracket valued in {\em pictures} (linear combinations of -valent graphs). We gave no example of classical links such that the parity-biquandle bracket of which is not trivial. In the present paper we slightly change the notation of the parity-biquandle bracket and give exam…
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.
Classifies components of strata of k-differentials on Riemann surfaces.
problem Classifying connected components of strata of k-differentials.
method Developed new techniques to study connected components of strata of k-differentials for general k.
result Complete classification of connected components of the strata of quadratic differentials with arbitrary poles.
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 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 …
We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers F_d(1,1) are itself connected torus covers with Veech group SL_2(Z). Using results of Eskin, Masur and Schmoll we calculate …
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 -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.
This study analyzes a non-orientable spacetime model in 1+1D quantum gravity.
problem Analyzing a non-orientable spacetime model in 1+1D quantum gravity.
method Formulated a Jackiw-Teitelboim gravity toy model on the Möbius band, computed Stiefel-Whitney classes, and analyzed the Dirac operator.
result Half-integer momentum quantization, spectral symmetry, vanishing mod-2 index, and η_D(0) = 0 follow.
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 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.
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…
Free Knots and Paritymath.GT
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…
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…
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…