New formula simplifies evolution of twist knots and calculates Racah matrices for rectangular representations.
problem Simplifying evolution of twist knots and calculating Racah matrices for rectangular representations.
method Developed a universal formula for triangular evolution matrix B applicable to rectangular representations R=[rs]. Used skew characters and Macdonald polynomials. result Explicit knowledge of twist-family evolution leads to a nearly explicit answer for Racah matrix Sˉ in arbitrary rectangular representation R. Researchers derive exclusive Racah matrix for complex representation with non-trivial multiplicities.
problem Constructing and understanding Racah matrices for complex representations with non-trivial multiplicities.
method Using effective field theory for arborescent knots, they deduced the exclusive Racah matrix for representation R=[3,1] with non-trivial multiplicities. result The exclusive Racah matrix $ar S$ for representation R=[3,1] is operator valued and depends on basis choices in intertwiner spaces. Study reveals hidden structure behind Racah matrices for twisted knots.
problem Understanding non-associativity in representation products of twisted knots.
method Analysis of quantum R-matrices and their eigenvalues to decompose Racah matrices.
result Discovery of pentad structure (Tˉ,Sˉ,S,E,B) associated with universal R-matrix. Method extends factorization to non-rectangular representations, revealing part of the Racah matrix.
problem Factorization of HOMFLY-PT polynomials for non-rectangular representations.
method Extending the differential expansion factorization from rectangular to non-rectangular representations.
result Extracted part of the Racah matrix for non-rectangular representations.
Unified description of adjoint knot polynomials for various knots.
problem Describing adjoint knot polynomials for different types of knots.
method Developing a universal form for quantum dimensions and Racah matrices, extending the eigenvalue conjecture.
result Unified description of adjoint knot polynomials for all arborescent knots.
The paper calculates Racah matrices for up to 3 strands of knots and links.
problem Systematic description of colored knot and link invariants.
method Highest weight method and use of Racah matrices.
result Explicit answers for Racah matrices and colored polynomials for 3-strand knots and links.
The paper calculates HOMFLY polynomials for 3-strand knots using a new method.
problem Systematic description of colored knot polynomials for arbitrary representations.
method Efficient highest-weight method to find inclusive Racah matrices.
result Explicitly found HOMFLY polynomials for 3-strand knots and confirmed conjectures.
The paper calculates R and Racah matrices for SO(5) and finds Kauffman polynomials.
problem Generalizing Reshetikhin-Turaev approach to SO(2n+1) case.
method Provided R and Racah matrices for SO(5) symmetric representation.
result Found Kauffman polynomials for SO(5) symmetric representation.
Quantum Racah matrices for R=[2,2] are fully described and evaluated.
problem Calculating Racah matrices for quantum groups U_q(sl_N).
method Eigenvalue hypothesis for most matrices, highest weight method for degenerate cases.
result Complete Racah matrices for |R| ≤ 4, allowing calculation of HOMFLY polynomials.
A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g+1)-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformatio…
New method extracts Racah matrices from antiparallel double-braid knots.
problem Extract exclusive Racah matrices for arborescent knots.
method Factorization of differential expansion for antiparallel double-braids.
result Found a way to reduce problem to twist knots and provide answers for R=[33]. Paper calculates Racah matrices for 3-strand knots, validating conjectures.
problem Systematic description of colored knot polynomials.
method Highest weight method with Gelfand-Tseitlin tables.
result Explicit Racah matrices and polynomials for 3-strand knots up to 10 crossings.
New findings on knot polynomials for specific representations.
problem Understanding HOMFLY polynomials for twist knots and their representations.
method Differential expansion of HOMFLY polynomials for twist knots and analysis of Racah matrices.
result Deviation of a specific coefficient from skew dimension in R=[333] representation.
New method extracts Racah matrices revealing hidden integrability in knot evolution.
problem Understanding hidden integrability in knot evolution.
method Evolution method to extract Racah matrices from 3-strand mixing matrices.
result Reveals unexpected integrability in the evolution of knots.
The paper tabulates knot polynomials for a specific class of knots.
problem Computing knot polynomials for arborescent knots efficiently.
method Family approach and Feynman diagram technique with auxiliary matrix model field theory.
result New tables of colored knot polynomials for arborescent knots.
Basing on evaluation of the Racah coefficients for SU_q(3) (which supported the earlier conjecture of their universal form) we derive explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list the answ…
KNTZ trick simplifies knot polynomial calculations for twist knots.
problem Completing the structure of differential expansion for twist knots.
method Converting arborescent evolution matrix into triangular form.
result Conjecture for triangular matrix B in non-rectangular case. Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this …
New method for calculating colored HOMFLY-PT polynomials for links with different symmetric representations.
problem Calculating colored HOMFLY-PT polynomials for links with arbitrary symmetric representations.
method Using quantum Racah coefficients (6j-symbols) of Uq(sl2) to simplify the evaluation. result Multi-colored link polynomials H[r1],[r2] for a specific link L7a3 are successfully evaluated. With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new struc…
Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371 corresponds to the case m=2, and our generalizations sheds additional light on …
After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra {\cal G}, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra Σ is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming u…
This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in var…
Researchers extend differential expansion to links using special framing.
problem Extending differential expansion from knots to links.
method Use of special framing and recent achievements in 6j-symbols. result Differential expansions for Whitehead and Borromean rings differ from previous findings.
The eigenvalue conjecture is supported for colored Alexander polynomials.
problem Supporting the eigenvalue conjecture for colored Alexander polynomials.
method Connecting Alexander polynomials and eigenvalues of braid group generators.
result Support for the eigenvalue conjecture for i>2, where direct evaluation is difficult.
Factorization of DE coefficients is violated in antiparallel triple pretzels, but described elegantly.
problem Understanding the origins of factorization in double braids and its extension to antiparallel triple pretzels.
method Defect-preserving deformation from trefoil to antiparallel triple pretzels, analysis of DE coefficients.
result Factorization of DE coefficients is violated but described by an elegant formula for symmetric representations.
DeepTMR reorders matrices without prior knowledge of structural patterns.
problem Matrix reordering without prior structural knowledge.
method DeepTMR uses a neural network to automatically extract features and reorder matrices.
result Trained network produces denoised mean matrix for visualization.
Paper speeds up matrix multiplication on Intel PIII using SIMD.
problem Efficiently multiplying large matrices for faster algorithm performance.
method Implemented matrix-matrix multiply using Intel Pentium SIMD architecture.
result Average performance 2.09 times faster than public domain routines.
The paper constructs Goeritz matrices from Dehn colorings.
problem Constructing Goeritz matrices from Dehn colorings.
method Purely algebraic construction of Goeritz matrices from Dehn coloring matrices for prime knot diagrams.
result A new method to construct Goeritz matrices from Dehn colorings.
New matrix reveals cluster info in sparse directed graphs.
problem Analyzing cluster information in directed graphs.
method Proposed complex non-backtracking matrix integrating Hermitian adjacency matrix and non-backtracking matrix properties.
result The complex non-backtracking matrix holds cluster information, especially for sparse directed graphs.
The CN matrix of a pure braid projection is characterized and applied.
problem Understanding the structure of CN matrices for braid projections.
method Discussion and characterization of patterns and specific matrices.
result Characterization of CN matrix of a pure 6-braid projection and related matrices.
Characterizes the OU matrix for up to 5 strands in braids.
problem Understanding the structure of braid diagrams through their matrices.
method Characterization of the OU matrix for up to 5 strands in braids.
result Standard form of the OU matrix for general braids of up to 5 strands is given and characterized.
Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.
problem Classifying SL(n) covariant matrix-valued valuations on Lp-spaces.
method Established a complete classification for continuous and SL(n) covariant matrix-valued valuations on Lp(Rn,|x|2dx), eliminating matrix symmetry assumption.
result Unique characterization of such valuations by the moment matrix in n>2, rotation matrix in 2D.
New method for matrix completion under complex missing data patterns.
problem Matrix completion with complex missing data patterns.
method Estimate the probability matrix of observation via low-rank matrix estimation and use inverse probabilities weighting to complete the target matrix.
result Optimal asymptotic convergence rates for observation probabilities and target matrix estimation.
The paper tackles matrix completion under nonlinear distortions.
problem Matrix completion with nonlinear distortions.
method Alternates between low-rank matrix estimation and monotonic function estimation.
result Empirical results show the method's competitiveness.
Unified approach for robust low rank matrix estimation with adversaries.
problem Robust low rank matrix estimation in the presence of adversaries.
method Unified approach combining Huber loss and nuclear norm penalization.
result Sharp estimation error bounds for matrix compressed sensing and completion.
New methods improve recommendation accuracy for users and items with few ratings.
problem Skewed distribution and low ratings affect recommendation accuracy.
method Four matrix completion-based approaches: FARP, TMF, TMF + Dropout, IFWMF.
result Improved prediction accuracy for users and items with few ratings.
A hierarchical Gaussian prior model improves low-rank matrix completion.
problem Low-rank matrix completion with improved structure exploitation.
method Hierarchical Gaussian prior model with GAMP embedded variational Bayesian inference.
result The proposed method outperforms state-of-the-art matrix completion methods.
New NMF algorithm uses Toeplitz matrix for facial recognition.
problem Facial recognition performance improvement.
method Proposes TNMF algorithm with Toeplitz penalty for NMF.
result TNMF outperforms ZNMF and other constrained NMF algorithms.
A new algorithm speeds up matrix operations in Neural Networks.
problem Time-consuming matrix operations in Neural Networks.
method An algorithm that increases the degree of parallelism of matrix multiplication.
result The algorithm speeds up several matrix operations in Neural Networks.
Selective sampling improves matrix completion with known structure.
problem Reconstructing a low-rank matrix with incomplete data.
method Designing observation sets based on matrix structure and selective sampling.
result Improved reconstruction accuracy with selective sampling.
Bayesian HMF integrates multiple datasets for in/out-of-matrix prediction.
problem Data integration across different entity types and sparsity levels.
method Bayesian hybrid matrix factorisation model combining multiple methods.
result Consistently better in-matrix and out-of-matrix predictions compared to state-of-the-art methods.
A new method for matrix completion identifies low-rank submatrices.
problem Matrix completion for non-low-rank matrices.
method Targeted framework: extract low-rank submatrices, complete separately.
result Significantly smaller reconstruction errors than classical methods.
A new matrix factorization method that approximates data without requiring nonnegativity or convexity.
problem Approximating data matrices without the constraints of nonnegativity or convexity.
method A multi-objective optimization problem finds conical combinations of templates that approximate a given data matrix.
result The method allows for approximation of data sets without the usual constraints of nonnegativity or convexity.
The paper defines the OU matrix for braid diagrams and finds determinant relationships.
problem Understanding the layeredness of braid diagrams.
method Defining the OU matrix and analyzing its determinant for layered braid diagrams.
result The determinant of the OU matrix for layered braid diagrams is the product of the determinants of the layers.
New method improves robust low-rank matrix completion for computer vision.
problem Robust low-rank matrix completion for partially observed data.
method Formulated as a nonsmooth Riemannian optimization problem over Grassmann manifold, solved with an alternating manifold proximal gradient continuation method.
result Demonstrated advantages over existing approaches in background extraction from surveillance videos.
NIMFA is a Python library for nonnegative matrix factorization.
problem Efficiently factorizing nonnegative matrices for various applications.
method Unified interface, state-of-the-art methods, initialization approaches, quality scoring, supports dense and sparse matrices.
result Unified and efficient implementation of nonnegative matrix factorization methods.
Link colorings linked to Goeritz matrix.
problem Understanding link colorings and their relation to the Goeritz matrix.
method Exploring the relationship between link colorings and the Goeritz matrix.
result Established a connection between link colorings and the Goeritz matrix.