We consider odd Poisson (odd symplectic) structure on supermanifolds induced by an odd symmetric rank (non-degenerate) contravariant tensor field. We describe the difference between odd Riemannian and odd symplectic structure in terms of the Cartan prolongation of the corresponding Lie algebras, and formulate an an…
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Odd connections on supermanifolds are defined and their properties studied.
problem Defining and understanding odd quasi-connections on supermanifolds.
method Examined odd quasi-connections, defined torsion and curvature, and identified special classes.
result Odd connections on supermanifolds are shown to have torsion and curvature tensors.
Cobordisms of graphs. A sliceness criterion for stably odd knots and related results on cobordismsmath.GT
The paper provides a sliceness criterion for stably odd knots.
problem Sliceness of stably odd knots
method Analyzes cobordisms and free knots to derive a sliceness criterion
result A sliceness criterion for stably odd knots
Proves conditions for odd pretzel knots to be doubly slice.
problem Identifying conditions for odd pretzel knots to be doubly slice.
method Analyzes knots with specific twist parameters and odd integers.
result Identifies all doubly slice odd pretzel knots.
Odd GKM-manifolds with non-negative curvature split cohomology.
problem Understanding cohomology of odd-dimensional GKM-manifolds.
method Proving cohomology splitting for specific manifolds.
result Cohomology splits for GKM manifolds of non-negative curvature.
We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and di…
This paper extends NCFI to odd codimension and computes examples.
problem Extending NCFI to foliations of odd codimension.
method Computing NCFI for various foliated manifolds in both even and odd codimensions.
result NCFI is an invariant of foliations in odd codimension, requiring an odd \(K_1\)-class.
The divergence-like operator on an odd symplectic superspace which acts invariantly on a specially chosen odd vector field is considered. This operator is used to construct an odd invariant semidensity in a geometrically clear way. The formula for this semidensity is similar to the formula of the mean curvature of hype…
The paper solves sliceness for odd free knots.
problem Determining if odd free knots have a spanning disc.
method Using the pairing of knot diagram chords.
result An explicit criterion for sliceness of odd free knots.
We establish several Witten type rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds. We obtain our results by combining the modular method, modular transgression and some careful analysis of odd Chern classes for cocycles in odd -theory. Moreover we discover that in odd dimen…
Adapts scanning algorithm for odd Khovanov homology.
problem Computing odd Khovanov homology efficiently.
method Uses mapping cone construction instead of tensor product.
result Determines odd Khovanov homology of 3-strand torus links.
This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and o…
Extends Landweber-Stong result to torus actions on manifolds of generalized odd type.
problem Signature vanishing for manifolds with torus actions of generalized odd type.
method Extends Landweber-Stong result to torus actions.
result Signature vanishes for manifolds with torus actions of generalized odd type.
ODD Metricsmath.DG
New ODD metrics defined on manifolds with degeneracy conditions.
problem Defining metrics on manifolds with degeneracy conditions.
method Introducing ODD metrics that degenerate on submanifolds while maintaining compatibility.
result ODD metrics satisfy basic properties and induce metric space structures.
Study proposes new methods to convert betting odds into accurate probabilities for sports forecasting.
problem Convert betting odds to accurate outcome probabilities for sports forecasting and market efficiency analysis.
method Proposes two methods: Odds-Only-Equal-Profitability-Confidence (OO-EPC) and Favourite-Longshot-Bias-Adjusted Generalised Linear Model (FL-GLM).
result Proposed methods outperform existing methods in empirical tests and real-world applications.
Odd-dimensional orbifolds' Euler characteristic equals half of their boundary's.
problem Calculating the Euler characteristic of odd-dimensional orbifolds.
method Proved through mathematical analysis of orbifolds and their boundaries.
result The Euler characteristic of an odd-dimensional orbifold is half of its boundary's.
Proves module structure on odd Khovanov homology and applies to ribbon 2-knots.
problem Proving module structure on odd Khovanov homology.
method Geometric description of module structure and link cobordism.
result Combinatorial proof of odd invariant for ribbon 2-knots.
The general Dabrowski-Sitarz-Zalecki type theorem for odd dimensional manifolds with boundary IImath.DG
Proves another theorem for odd dimensional manifolds with boundary.
problem Spectral Einstein functional on odd dimensional manifolds with boundary.
method Proof of a theorem using the Dirac operator.
result Another general Dabrowski-Sitarz-Zalecki type theorem proved.
Paper extends index theorem to odd-dimensional manifolds with even-dimensional boundaries.
problem Index theorem for odd-dimensional manifolds with boundaries.
method Equivariant Toeplitz index theory.
result Established equivariant version of Dai-Zhang's theorem.
On odd Laplace operatorsmath.DG
We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold . We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ``orbit space'' of volume forms. This…
Study odd generalized Einstein metrics on 3D Lie groups.
problem Classify odd generalized Einstein metrics on 3D Lie groups.
method Left-invariant generalized connections, divergence operators, and Ricci tensors.
result Describe all odd generalized Einstein metrics on all 3D Lie groups.
Constructs modular forms and proves divisibility results for odd-dimensional manifolds.
problem Constructing modular forms over specific groups and proving divisibility results.
method SL(2, Z) modular forms and Witten genus in odd dimensions.
result Obtained divisibility results of index of Toeplitz operators on spin and spin^c manifolds.
New supergroups created from odd involutions in supergeometry.
problem Creating new supergroups from odd involutions.
method Constructing Grassmannians by gluing domains with an odd involution.
result Introduced a supergroup associated with odd involutions.
Extends odd Khovanov bracket to link cobordisms and proves functoriality up to sign.
problem Proving functoriality of odd Khovanov homology up to sign.
method Extending odd Khovanov bracket to link cobordisms and proving functoriality up to sign.
result Functoriality of odd Khovanov homology up to sign.
The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^{odd}(4) the only GL^+(2,R) orbit closures are closed orbits, the Prym locus Q(3,-1^3), and H^{odd}(4). Together with work of Matheus-Wright, this implies that there are only f…
We observe that an anti-symplectic manifold locally always admits a parity structure. The parity structure can be viewed as a complex-like structure on the manifold. This induces an odd metric and its Levi-Civita connection, and thereby a new notion of an odd Kaehler geometry. Oversimplified, just to capture the idea, …
An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of …
Odd -theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential -theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We chara…
Short note solves Khovanov homology conjecture with odd torsion examples.
problem Khovanov homology conjecture and odd torsion in even homology.
method Introduction of specific knots and links with odd torsion.
result First known examples of knots and links with odd torsion in even Khovanov homology (order 9, 27, 81, 25).
Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
New categories help understand knot algebra.
problem Understanding the action of on knot homology.
method Introducing and equating two new supercategories.
result Equivalence of categories for .
Odd Pfaffian formsmath.DG
Defines odd Pfaffian form for odd-dimensional manifolds, proving Chern-Gauss-Bonnet formula.
problem Proving Chern-Gauss-Bonnet formula for incomplete edge singularities.
method Defining odd Pfaffian form through curvature tensor, proving formula for various metrics.
result Proves intrinsic Chern-Gauss-Bonnet formula for edge singularities and complete manifolds.
No contact Anosov diffeomorphisms exist on odd-dimensional manifolds.
problem Existence of contact Anosov diffeomorphisms on odd-dimensional manifolds.
method Analysis of Anosov diffeomorphisms and invariant contact structures.
result No invariant contact structure exists for Anosov diffeomorphisms on odd-dimensional manifolds.
We develop isometry and inversion formulas for the Segal--Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.
For odd-dimensional spheres, there's always a second short geodesic.
problem Finding the second shortest closed geodesic on odd-dimensional spheres.
method Analyzing non-reversible Finsler metrics on spheres of odd dimension.
result There is a second closed geodesic with Morse index ≤ 4(m+2)(m-1)+2.
Proves a generalized table theorem for odd Euler characteristic surfaces.
problem Proving a generalized table theorem for surfaces with odd Euler characteristic.
method Using the square peg problem for smooth curves, the result is generalized to real valued functions on Riemannian surfaces with odd Euler characteristic.
result Proves the table conjecture for even functions on the two sphere.
A new method constructs odd characteristic classes in cyclic homology.
problem Constructing odd characteristic classes in equivariant loop space homology.
method Constructing a Chern character in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex.
result Induces a well-defined group homomorphism from theory to the odd homology group.
A pretzel knot is called if all its twist parameters are odd, and if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5-stranded pretzel knots are . We d…
Defines odd Khovanov homology via categorification of q-Schur algebra.
problem Constructing odd Khovanov homology via representation theory.
method Supercategorification of q-Schur algebra, odd foams, tensor product on chain complexes.
result Odd Khovanov homology defined via categorification.
Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.
problem Characterizing chirally cosmetic surgeries on specific knot types.
method Recent methods of Ichihara, Ito, and Saito applied to genus 2 and 3 alternating odd pretzel knots.
result Most genus 2 and 3 alternating odd pretzel knots do not admit chirally cosmetic surgeries.
Odd primes act on alternating knots via flypes.
problem Understanding odd order group actions on alternating knots.
method Investigation of flypes between reduced alternating diagrams.
result Odd prime actions on alternating knots are isotopic through a single flype.
Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.
problem Calculating volumes of specific strata of quadratic differentials.
method Expressed volumes as a sum over stable graphs, with coefficients as intersection numbers of psi classes with combinatorial classes.
result Formula for volumes of odd strata of quadratic differentials.
Odd-dimensional manifolds with infinitely many different geometries of positive Ricci curvaturemath.DG
The study finds infinitely many different geometries for odd-dimensional manifolds with positive Ricci curvature.
problem Finding different geometries for odd-dimensional manifolds with positive Ricci curvature.
method Examining closed manifolds in odd dimensions .
result Large classes of manifolds admit infinitely many different geometries of positive Ricci curvature.
New method uses geometric mean to avoid non-collapsibility in case-control studies.
problem Non-collapsibility of odds ratio under outcome-dependent sampling.
method Proposes geometric mean aggregation to avoid non-collapsibility and provides estimation and inference methods.
result Geometric odds ratio is collapsible under outcome-dependent sampling.
The Law of Total Oddsmath.PR
The law of total probability may be deployed in binary classification exercises to estimate the unconditional class probabilities if the class proportions in the training set are not representative of the population class proportions. We argue that this is not a conceptually sound approach and suggest an alternative ba…
We generalize categorified Jones-Wenzl projectors in odd Khovanov homology.
problem Categorify Jones-Wenzl projectors for odd Khovanov homology.
method Develop grading multicategories to replace grading categories, proving existence and uniqueness of categorified projectors.
result Existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology.
Study spectral flow for Callias operators to prove obstructions to positive scalar curvature.
problem Prove obstructions to positive scalar curvature on manifolds.
method Use spectral flow and odd K-cowaist to derive obstructions.
result Infinite odd K-cowaist is an obstruction to the existence of PSC metrics.
Supergeneralization of $\DC P(N)$ provided by even and odd Kählerian structures from Hamiltonian reduction are construct.Operator which used in Batalin-- Vilkovisky quantization formalism and mechanics which are bi-Hamiltonian under corresponding even and odd Poisson brackets are considered.