We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles…
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48 results for “mirror double”
This paper tabulates prime knot projections up to eight double points.
problem Tabulating prime knot projections and their mirror images up to a certain number of double points.
method Systematic flypes and enumeration of tangles with at most four double points, using arrow diagrams.
result Complete table of prime knot projections with their mirror images up to eight double points.
Given a six-dimensional symplectic manifold , a nondegenerate, co-closed four-form introduces a dual symplectic structure independent of via the Hodge duality . We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of nonc…
Double structures and jetsmath.DG
We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to con…
We give an extension of Fox's formula of the Alexander polynomial for double branched covers over the three-sphere. Our formula provides the Reidemeister torsion of a double branched cover along a knot for a non-trivial one dimensional representation by the product of two factors derived from the knot group. One of the…
Study identifies prime strongly positive amphicheiral knots with double symmetry.
problem Characterizing prime strongly positive amphicheiral knots with specific symmetries.
method Examined knots up to 16 crossings, identified prime knots with double symmetry, and presented almost doubly symmetric diagrams.
result Found the first prime strongly positive amphicheiral knot not slice.
Corner contributions to Neumann jump determinants: three model calculations and a BFK conjecturemath.SP
Local corner-factor conjecture for Neumann jump determinants supported by models.
problem Determining the determinant of Neumann jump operator on piecewise curves.
method Formulated conjecture, supported by three model calculations, and discussed connections.
result Support for the conjectural determinant formula \(\Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^Nα_j^{-1/2}\).
Jones polynomial bounds and crossing numbers of knots.
problem Bounding the degree of colored Jones polynomial in terms of crossing number.
method Sharpness of bounds determined for adequate knots; application to satellite knots.
result Sharpness of bounds for adequate knots; determination of crossing numbers for specific knots.
We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsváth-Szabó and Bloom's spectral sequence for the branched double cover of a link in . We prove that there exists a spectral sequence of -modules (where has degree ) which converges to $\widetilde{\m…
A new framework optimizes model transfer across domains with labeled data.
problem Distributional heterogeneity across domains in multi-source learning.
method Conditional Group Distributionally Robust Optimization (CG-DRO) framework with Mirror Prox algorithm and double machine learning.
result Established fast statistical convergence rates and uniformly valid inference for CG-DRO.
Symmetric graphs flow without singularities on their axis.
problem Preserving symmetry in mean curvature flow.
method Weak solution approach, introducing 'vanity', mean curvature flow approximation.
result Singularities occur only on the axis of symmetry.
New framework TVR assesses true DeFi value, revealing substantial double counting.
problem TVL is easily manipulated and inflated in DeFi, leading to unreliable metrics.
method Proposed a new framework TVR to assess true underlying value of DeFi.
result TVR reveals substantial double counting in DeFi, with a gap of $139.87 billion at peak.
Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, Σ(K). We prove, for each integer n > 1, the existence of a spectral sequence from Khovanov's categorification of the reduced n-colored Jones polynomial of the mirror of K to the knot Floer homology of (Σ(K),\widetil…
Mirror flows converge to a limiting flow with a convex potential.
problem Incremental learning in mirror flows
method Rescaled trajectories converge to a limiting mirror flow
result Primal variable minimizes the loss over a time-dependent hypothesis set
New mirror maps improve PMD performance in reinforcement learning.
problem Limited exploration of PMD's full potential due to focus on negative entropy.
method Evolutionary strategies to identify and learn more efficient mirror maps.
result Learned mirror maps outperform negative entropy in various environments.
Ozsvath and Szabo have established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L in S^3 and the Heegaard Floer homology of its double-branched cover. This relationship has since been recast by the authors as a specific instance of a broa…
Study mirror symmetry on manifolds with exceptional holonomy groups.
problem Construct mirrors for Spin(7) and G2 manifolds.
method Apply mirror symmetry to pairs of non-compact manifolds and use CFT analysis.
result Confirm geometric mirror constructions for Spin(7) and G2 manifolds.
We describe mirror symmetry on higher dimensional tori, paying special attention to the behaviour of D-branes under mirror symmetry. To find the mirror D-branes the description of mirror symmetry on D-branes due to Ooguri, Oz en Yin is used. This method allows us to deal with the coisotropic D-branes recently introduce…
Derives Mirror Descent from gradient flow on a Riemannian manifold.
problem No specific problem stated; focuses on derivation.
method Derives Mirror Descent from gradient flow on a Riemannian manifold with a natural discretization.
result Generalizes Mirror Descent to non-Hessian metrics.
Motivated by Strominger-Yau-Zaslow's mirror symmetry proposal and Kontsevich's homological mirror symmetry conjecture, we study mirror phenomena (in A-model) of certain results from Donaldson-Thomas theory for Calabi-Yau 4-folds.
Study homological mirror symmetry for Hirzebruch surfaces using Morse homotopy.
problem Homological mirror symmetry for Hirzebruch surfaces .
method Using Strominger-Yau-Zaslow construction and Morse homotopy.
result Homological mirror symmetry holds for Hirzebruch surfaces .
This paper deforms complex tori and their mirrors using gerbes.
problem Deforming complex tori and their mirror partners.
method Using flat gerbes to deform complex tori and their mirrors, constructing holomorphic line bundles over deformed objects.
result Deformed complex tori and their mirrors can be studied using flat gerbes.
Constructs mirror pairs for solvmanifolds using Lie groups.
problem Finding mirror pairs for non-Kaehler solvmanifolds.
method Left-invariant affine structures on Lie groups.
result Explicitly finds SYZ mirror symmetric partners for all known compact 6D solvmanifolds.
Homological mirror symmetry for toric Fano surfaces using Morse homotopy.
problem Establishing homological mirror symmetry for toric Fano surfaces.
method Applying SYZ construction and using Morse homotopy of the moment polytope.
result Homological mirror symmetry achieved for toric Fano surfaces.
New analysis shows GMD can converge linearly under PL-like conditions.
problem Establishing linear convergence for generalized mirror descent.
method PL-based analysis for time-dependent mirrors, Taylor-series approach for stochastic GMD.
result Linear convergence of stochastic GMD under PL-like conditions.
Equitable Mirrorsastro-ph.IM
In this article we explore some finer properties of equi-areal mirrors and introduce techniques for developing new mirror surfaces that simultaneously minimize angular and areal distortion.
Mirror flow optimizes separable data problems, converging to a maximum margin classifier.
problem Optimizing classification problems with separable data using mirror flow.
method Examine mirror flow on linearly separable classification problems, focusing on the horizon function of the mirror potential.
result Mirror flow converges to a maximum margin classifier for separable data under certain conditions.
Study connects mirror symmetry invariants to K-stability for toric manifolds.
problem Relating invariants from mirror symmetry to K-stability for toric polarized manifolds.
method Analyzes expansions involving base loci of linear systems from Landau-Ginzburg potentials.
result Shows Z-stability naturally arises from mirror symmetry considerations.
Reparameterizes mirror descent as gradient descent for efficient sparse learning.
problem Efficiently training small sparse networks with mirror descent.
method Develops a framework to convert mirror descent updates into gradient descent updates on different parameters.
result Mirror descent can be reparameterized as gradient descent on modified parameters, facilitating standard backpropagation.
Topological Mirror Symmetrymath.AG
This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special La…
Paper develops efficient algorithms for robust optimization across multiple groups.
problem Minimizing maximal empirical risk across distinct groups in robust optimization.
method Develops ALEG and ALEM algorithms for two-level finite-sum convex-concave minimax optimization.
result Achieves ε-accuracy with complexity O(m√(nlnm/ε)) and outperforms state-of-the-art methods.
NGMs create mirrored features to assess neural network feature importance.
problem Lack of feature relevance information in DNNs limits their applicability.
method Structured perturbation and kernel-based conditional dependence measure for feature importance evaluation.
result Controls feature selection error rate and maintains high selection power with correlated features.
We revisit our construction of mirror symmetries for compactifications of Type II superstrings on twisted connected sum manifolds. For a given manifold, we discuss evidence for the existence of mirror symmetries of two kinds: one is an autoequivalence for a given Type II superstring on a mirror pair of $G_2…
The paper discusses a solution to homological mirror symmetry for complex tori, especially when the matrix is singular.
problem Homological mirror symmetry for complex tori, particularly when the matrix is singular.
method Proposes a new approach to define a mirror partner for complex tori of dimension when the matrix is singular.
result Proposes a method to avoid the problem of defining a mirror partner for complex tori of higher dimensions when the matrix is singular.
Find first (0,2) mirror symmetry examples on Hopf surfaces.
problem Find (0,2) mirror symmetry on compact non-Kähler manifolds.
method Use Borisov's approach with vertex algebras and chiral de Rham complex. Study Killing spinors on quadratic Lie algebras and embeddings of superconformal vertex algebras.
result Construct first (0,2) mirror pairs of Hopf surfaces.
New target-based TD learning algorithms improve deep Q-learning convergence.
problem Improving convergence of deep Q-learning algorithms.
method Introducing averaging TD, double TD, and periodic TD algorithms.
result Established asymptotic convergence analyses for averaging TD and double TD, and finite sample analysis for periodic TD.
Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot…
New algorithm reduces optimization complexity in adaptive mirror descent.
problem Optimizing complex, non-smooth, non-convex functions efficiently.
method SVRAMD: Variance Reduced Adaptive Mirror Descent.
result Variance reduction accelerates convergence in adaptive mirror descent.
Mirror descent algorithm recovers low-rank matrices in matrix sensing.
problem Matrix sensing with low-rank matrices under certain conditions.
method Discrete-time mirror descent applied to empirical risk with Bregman divergence analysis.
result Mirror descent converges to a matrix minimizing a specific nuclear norm-related quantity.
Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithmstat.CO
New algorithm improves sampling from constrained spaces.
problem Sampling from constrained spaces efficiently.
method Metropolis-adjusted Mirror Langevin algorithm.
result Unbiased sampling with improved mixing time.
Develops parameter-free online mirror descent for optimal dynamic regret.
problem Optimal online linear optimization in unbounded domains.
method Modified online mirror descent framework for parameter-free algorithms.
result First unconstrained online linear optimization achieving optimal dynamic regret.
Study shows Calabi-Yau manifolds are non-hyperbolic via mirror symmetry.
problem Proving non-hyperbolicity of Calabi-Yau manifolds.
method Using mirror symmetry, entire curves are found on Calabi-Yau manifolds.
result Calabi-Yau manifolds are Kobayashi non-hyperbolic.
We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on Calabi-Yau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class…
In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow c…
Researchers match complex affine structures in mirror constructions.
problem Matching complex affine structures in SYZ fibrations of Del Pezzo surfaces.
method Floer-theoretical gluing method to construct mirrors using immersed Lagrangians.
result The constructed mirror agrees with Carl-Pomperla-Siebert's mirror.
Continuous-time mirror descent solves sparse phase retrieval efficiently.
problem Recovering sparse signals from magnitude-only measurements.
method Continuous-time mirror descent applied to unconstrained empirical risk minimization problem.
result Mirror descent recovers -sparse vectors with minimum non-zero entry order of from Gaussian measurements.
Paper explains scattering diagrams' role in mirror symmetry.
problem Reconstruction problem in mirror symmetry.
method Introduction of scattering diagrams and their role in SYZ and HMS conjectures.
result Scattering diagrams help in understanding mirror symmetry.
Mirror descent linked to information ratio via Bayesian regret bounds.
problem Understanding stability in mirror descent and its relation to information ratio.
method Developed a connection between mirror descent and information ratio using Bayesian regret bounds.
result Mirror descent with suitable estimators and distributions achieves bounds similar to information-directed sampling.