Paper proves vanishing homology groups for certain hyperbolic groups.
problem Understanding homology groups of specific hyperbolic groups.
method Using twisted Wirtinger presentations to prove homology group vanishing.
result Second homology groups vanish for certain Gromov hyperbolic groups.
We define Wirtinger number for links and prove it equals bridge number.
problem Determining the minimum number of generators for knot groups.
method Defining Wirtinger number and proving it equals bridge number.
result Wirtinger number equals bridge number for links.
Wirtinger number equals virtual bridge number for virtual links.
problem Calculating the virtual bridge number of virtual links.
method Algorithmically computing the minimum number of generators of the link group.
result The Wirtinger number equals the virtual bridge number for virtual links.
A generalization of the affine-geometric Wirtinger inequality for curves to hypersurfaces is given.
Discrete approach proves a new version of Wirtinger's inequality.
problem Classical Wirtinger's inequality for piece-wise functions.
method Theorem of classification of isometries and Fourier series development.
result New elementary proof of Wirtinger's inequality.
Wirtinger curves provide a simplified method to compute fundamental groups of certain algebraic plane curves.
problem Computing the fundamental group of the complement of algebraic plane curves.
method Wirtinger presentation based on the real picture of the curve and application to hypocycloids.
result Wirtinger presentation provides the fundamental group for an infinite subfamily of hypocycloids, relating them to Artin groups.
An algorithm calculates Gabai width for thousands of knots.
problem Calculating Gabai width for many knots.
method Algorithmic definition of Wirtinger width leading to efficient Gabai width bounds.
result Proved Wirtinger width equals Gabai width for knots.
Sharp bounds for curve isoperimetric deficit derived.
problem Finding sharp bounds for the isoperimetric deficit of curves.
method Fourier analysis applied to derive Wirtinger-type inequalities.
result Sharp lower and upper bounds for the isoperimetric deficit proved.
New formula for knot group representations and hyperbolic structures.
problem Understanding representations of knot groups and their geometric implications.
method Direct algebraic formula for geometric parameters of octahedral decompositions.
result Explicit criterion for critical points in Neumann-Zagier--Yokota potential function.
New method calculates bridge indices of spatial graphs using diagram colorings and Wirtinger number.
problem Calculating bridge indices for spatial graphs efficiently.
method Extending Wirtinger number to spatial graphs, implementing Python algorithm, combining algebraic structures and clasping techniques.
result Exact bridge indices for almost unknotted graphs of large bridge index.
Study on weakly G-slim complexes and non-positive immersions for group presentations.
problem Conditions for non-positive immersions in group presentations.
method Investigation of weakly G-slim complexes and their relationship to left-orderable groups.
result Conditions on generalized Wirtinger presentations guaranteeing non-positive immersions for their associated 2-complexes.
Study proves meridional rank conjecture for certain complex links.
problem Determining the minimum number of generators needed for link groups.
method Using Coxeter quotients and Wirtinger numbers, the study provides both lower and upper bounds.
result Proved meridional rank conjecture for specific types of links.
We study the phase retrieval problem, which solves quadratic system of equations, i.e., recovers a vector x∈Rn from its magnitude measurements yi=∣⟨ai,x⟩∣,i=1,...,m. We develop a gradient-like algorithm (referred to as RWF representing reshaped W…
New theory classifies knotted spheres in 4D space.
problem Classifying knotted punctured spheres in 4D space.
method Diagrammatic theory of welded graphs, Tube map extension, Milnor invariants.
result Complete link-homotopy classification of knotted punctured spheres.
We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS^3 built inductively out of BS^1, we prove that the symmetric metrics of certain two-point homogeneous manifolds t…
We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…
New method for blind over-the-air computation without CSI.
problem Over-the-air computation without channel information.
method Wirtinger flow solution with random initialization.
result Statistical optimality and global convergence of the method.
Link's sphere number equals its bridge number.
problem Determining the minimum number of generators for a link's fundamental group.
method Using meridional presentations with embedded two-spheres in fixed diagrams.
result The minimum number of generators equals the bridge number.
For any subvariety of a compact holomorphic symplectic Kaehler manifold, we define the number W(X), which we call Wirtinger number. We show that W(X)≤1, and the equality is reached if and only if the subvariety X⊂M is trianalytic, i. e. compactible with the hyperkaehler structure on M. For a sequence $X_…
Paper introduces danceability index as a new bridge index definition.
problem Defining the bridge index in various mathematical contexts.
method Proves danceability index as equivalent to bridge index, extends to virtual knots.
result Danceability index is a new equivalent definition of the bridge index.
The paper proves existence and uniqueness of slant immersions in complex space forms.
problem Existence and uniqueness of slant immersions in complex space forms.
method Established existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
result Existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
Hadamard Wirtinger Flow recovers sparse signals from fewer measurements.
problem Reconstructing sparse signals from magnitude-only measurements.
method Gradient descent with Hadamard parametrization (HWF).
result A single step of HWF recovers support from k(xmax∗)−2 samples. This research explores complex-valued neural networks and their implementation.
problem The challenges of implementing complex-valued neural networks and their potential for non-complex data.
method Detailed theory and implementation of CVNN, including Wirtinger calculus, complex backpropagation, and modules like complex layers and activation functions. Python implementation using cvnn toolbox.
result Demonstrates the potential of CVNN for non-complex data through simulations.
The paper presents fundamental groups of complements of shadows in 4-balls.
problem Understanding fundamental groups of 4-manifold complements.
method Similar to Wirtinger presentation, focusing on contractible shadows.
result A presentation of fundamental groups of subpolyhedra in 4-balls.
New algorithm recovers signals from noisy measurements with optimal precision.
problem Recovering signals from magnitude-only measurements with arbitrary corruption.
method Robust Wirtinger Flow algorithm for joint signal and corruption estimation.
result Guaranteed linear convergence to optimal precision with optimal sample complexity.
This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x∈Rp from noisy quadratic measurements yj=(aj′x)2+εj, j=1,…,m, with independent sub-exponential noise εj. The goals are to understand the effect of the sparsity of x on the estimation prec…
New method for quandle presentations of surface knots in 4-manifolds.
problem Computing fundamental quandle presentations for surface knots in arbitrary 4-manifolds.
method Wirtinger type presentation of the fundamental quandle for surface links in 4-manifolds.
result Infinitely many pairwise non-local surface knots with specific bridge numbers in certain 4-manifolds.
The paper studies pseudo-hyperbolic structures derived from knot complements.
problem Understanding geometric and combinatorial properties of knot complements.
method Using ideal octahedral decompositions and pseudo-hyperbolic structures, the paper computes complex volumes and cusp shapes of knots and links.
result Concrete formulas for Wirtinger generators and cusp shapes are derived, and explicit solutions are provided for various knots.
New method tackles blind demixing from noisy bilinear measurements.
problem Blind demixing of source signals from noisy bilinear measurements.
method Provably nonconvex demixing via Wirtinger flow, similar to vanilla gradient descent.
result Aggressive step size and computational optimality guarantees without regularization.
New algebraic structures biquasiles defined using dual graph diagrams for knot and link invariants.
problem Defining invariants for oriented knots and links.
method Introducing dual graph diagrams and biquasiles, using combinatorial and algebraic approaches.
result Defined new knot and link invariants using biquasiles.
Free surface-links are shown to be ribbon links.
problem Characterizing surface-links as ribbon links.
method Four proofs are provided, including a stabilization approach.
result Every free surface-link is a ribbon surface-link.
A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. W…
Paper tackles phase retrieval with robust gradient descent for noisy data.
problem Recover signals from magnitude measurements with noise and corruption.
method Robust gradient descent applied to Wirtinger Flow algorithm.
result Improves algorithm's robustness to heavy-tailed noise and adversarial corruption.
A new knot invariant is created using regions and crossings.
problem Creating a new knot invariant for regional and crossing contributions.
method Each planar region and crossing contribute to a generator and relation, respectively, forming a tridle of the link.
result A polynomial invariant can be derived from the presentation matrix of a linear tridle.
A {\em word labeled oriented graph} (WLOG) is an oriented graph G on vertices X={x1,…,xk}, where each oriented edge is labeled by a word in X±1. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of …
New approach to quantify posterior concentration rates using Wasserstein dynamics.
problem Quantifying the speed of posterior distribution concentration in Bayesian statistics.
method Combining local Lipschitz-continuity with dynamic formulation of Wasserstein distance.
result Optimal posterior contraction rates in finite and infinite-dimensional models.
A new deep learning model improves phase retrieval performance.
problem Recovering signals from phaseless measurements.
method Hybrid model-based data-driven deep architecture (Unfolded Phase Retrieval, UPR).
result Significant improvement in phase retrieval performance.
The paper connects complex normalizing flows to Kähler-Ricci flows using geometric and statistical perspectives.
problem Understanding the relationship between complex normalizing flows and Kähler-Ricci flows.
method Develops connections between complex normalizing flows and Kähler-Ricci flows by relating the log determinant to Ricci curvature and using a Bayesian perspective.
result Reconciles the complex normalizing flow and Kähler-Ricci flow, showing they are related under certain conditions.
Study calculates fundamental groups of torus knots using algebraic topology.
problem Calculating the fundamental group of torus knots.
method Algebraic topology and group theory.
result Computed fundamental groups of torus knots.
New groups defined from knot diagrams, invariant under Reidemeister moves.
problem Classical knot groups are not invariant under all Reidemeister moves.
method Define quotient groups based on knot diagrams, invariant under Reidemeister moves.
result New groups include extended knot groups and are invariant under all Reidemeister moves.
This paper solves quadratic systems with sparse or generative priors.
problem Recovering signals from quadratic systems with full-rank matrices.
method Thresholded Wirtinger flow (TWF) and projected gradient descent (PGD) algorithms.
result The proposed methods significantly outperform existing algorithms in signal recovery.
Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R^8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley 4-form. We also…
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
problem Understanding the homotopy types of free racks and quandles.
method Proved analogs of Milnor's theorem for racks and quandles and their pointed variants.
result Identified the homotopy types of free racks and quandles on spaces of generators.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.
problem Finding higher order isoperimetric inequalities for both discrete and smooth curves.
method Unified approach via Fourier analysis of linear operators.
result Unified upper and lower bounds for isoperimetric deficit in smooth curves.
A classical result of H. S. M. Coxeter asserts that a certain quotient B(m,n) of the braid group B(m) on m strands is finite if and only if (m,n) corresponds to the type of one of the five Platonic solids. If k is a knot or virtual knot, one can study similar quotients G(k,n) for the correspond…
The paper explores representations of specific knot groups and their properties.
problem Investigating representations of branched twist spins with a non-trivial center of order 2.
method Analyzes mSL2(Z3)-representations and dihedral group representations of branched twist spins. result Provides sufficient conditions for the existence of mSL2(Z3)-representations and determines the number of dihedral group representations. Paper estimates differences in conditional independence graphs from time-dependent data.
problem Estimating changes in conditional dependencies between two time series with known similar structure.
method Penalized D-trace loss function approach in the frequency domain, using Wirtinger calculus, with convex and non-convex penalties.
result Established sufficient conditions for consistency and graph recovery in high-dimensional settings.