Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.
problem Understanding quasi-cluster algebras on non-orientable surfaces.
method Developed matrix formulae and proved skein relations for quasi-cluster variables.
result Laurent expansion and skein relations for quasi-cluster variables on non-orientable surfaces.
Matrix formulas for super Teichmüller spaces generalize previous work and yield super λ-lengths.
problem Calculating super λ-lengths on bordered surfaces with marked points.
method Using holonomy matrices of elements in the supergroup OSp(1|2) to compute super λ-lengths in decorated super Teichmüller spaces.
result Matrix formulas for arcs on bordered surfaces yield super λ-lengths in Penner-Zeitlin's decorated super Teichmüller space.
Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
Harer-Zagier formulas generalized to knot matrix models.
problem Understanding knot polynomials through matrix models.
method Defined knot matrix models and extracted averages.
result Harer-Zagier formulas factorize for torus knots but not for others.
Derives adjoint formulas for matrix operations and applies them to specific cases.
problem Computing adjoints for matrix operations and specific matrix types.
method Derives adjoint formulas for matrix operations and applies them to specific cases.
result Closed-form expressions for adjoints in specific matrix types.
Matrix formulas for knot invariants derived from Tait graphs.
problem Computing knot invariants for alternating links.
method Squarefree matrix extraction from Tait graph vertices.
result Explicit formulas for CWRk for k≥4. We present a formula for the trace of any symmetric power of a n×n matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and n−2 polynomial functions defined recursively.
The paper proves a formula for complex Monge-Ampère equations on manifolds.
problem Proving a mean value formula for complex Monge-Ampère equations.
method Using subharmonic Hermitian matrix valued functions and Liouville type theorems.
result Obtained a Liouville theorem for complex Monge-Ampère equations.
This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -- with or without self-intersections -- we associate an element of (the fractio…
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…
New Hessian estimates for heat equations on manifolds.
problem Estimating Hessian matrices for heat-type equations on Riemannian manifolds.
method Using Bismut-Stroock Hessian formula, with explicit coefficients and delay/growth rate functions.
result Novel backward weak Harnack inequality and precise pointwise Hessian estimates for eigenfunctions.
In this article we give an explicit description of the representation matrix of a Heisenberg type action constructed by Blanchet, Habegger, Masbaum and Vogel. We give the matrix in terms of a ribbon graph and its admissible colorings. We show that components of the representation matrix satisfies the {\it external edge…
Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.
problem Proving the Kontsevich-Witten tau-function formula.
method Directly shows Q-polynomial expansion satisfies Virasoro constraints.
result Direct proof of the formula without matrix model.
Two formulas for Chern classes of tensor products of vector bundles are presented.
problem Calculating Chern classes for tensor products of vector bundles.
method Two formulas using matrices and polynomials to compute Chern classes.
result Determinantal formulas for Chern classes of tensor products.
We study the heat kernel asymptotics for the Laplace type differential operators on vector bundles over Riemannian manifolds. In particular this includes the case of the Laplacians acting on differential p-forms. We extend our results obtained earlier for the scalar Laplacian and present closed formulas for all heat in…
Gradient descent solves rank-one matrix estimation problem with detailed time evolution analysis.
problem Estimating a rank-one symmetric matrix corrupted by noise.
method Gradient descent on a sphere, using local versions of the semi-circle law.
result Explicit formulas for the time evolution of the estimator and cost function, revealing phase transitions.
On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group O(n) by using only Euclidean coordinates …
Paper finds formulas for mutual information and MMSE in matrix tensor product problems.
problem High-dimensional inference problems involving matrix tensor products.
method Single-letter formulas for mutual information and MMSE, using new techniques.
result Analytical formulas describe leading order terms in mutual information and MMSE.
Simple matrix formulas for Grassmannian curvatures.
problem Modeling Grassmannian for curvature calculations.
method Symmetric orthogonal matrices and standard matrix operations.
result Explicit, simple formulas for various curvatures.
This handbook simplifies Grassmann manifold geometry for matrix-based algorithms.
problem Modeling linear subspaces in various applications.
method Expository work on Grassmann manifold geometry, including new algorithms and formulas.
result Improved understanding and computational tools for the Grassmann manifold.
In this article we give a unified treatment of the construction of all possible Weitzenböck formulas for all irreducible, non--symmetric holonomy groups. The resulting classification is two--fold, we construct explicitly a basis of the space of Weitzenböck formulas on the one hand and characterize Weitzenböck formulas …
Formula proves symmetry breaking operators for differential forms.
problem Symmetry breaking operators between differential forms on spheres and their hyperplanes.
method Explicit residue formula for meromorphic continuation of operators.
result Simple construction of symmetry breaking operators and determination of zeros.
This paper sets fundamental limits for rank-one matrix estimation with varying noise levels.
problem Estimating a rank-one matrix from Gaussian observations with different noise levels across blocks.
method Novel reduction from heterogeneous noise to homogeneous noise, proving asymptotic error bounds.
result Asymptotically exact formulas for minimum mean-squared error in estimating rank-one matrix and factors.
In Part I (arXiv:1209.2045) we computed the Stokes data, though not the "connection matrix", for the smooth solutions of the tt*-Toda equations whose existence we established by p.d.e. methods. Here we give an alternative proof of the existence of some of these solutions by solving a Riemann-Hilbert problem. In the pro…
New method solves inverse Ising problem without iteration.
problem Solving inverse Ising problems accurately and non-iteratively.
method Tree-reweighted approximation for analytic solution.
result Optimal interaction matrix derived analytically.
New proof for knot state-sum formula using bijection between states.
problem Proving a knot state-sum formula for colored Jones polynomial.
method Established bijection between states on arc-graph and bichromatic digraph, used flow property of R-matrix.
result Two state models are essentially the same, extending formula to links.
Estimates the probability of a random symmetric tensor being close to rank-one.
problem Estimating the probability of a random symmetric tensor being close to rank-one.
method Using Weyl's tube formula and techniques from Random Matrix theory, we study metric invariants of the real Veronese variety.
result Explicit formula for the reach and curvature coefficients of the real Veronese variety with respect to the Bombieri-Weyl metric.
A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the correspondi…
Researchers extend knot theory formulas to non-rectangular cases.
problem Applying universal-matrix precursor formulas to non-rectangular knot representations.
method Reformulated previously known formulas for simplest non-rectangular representations [r,1].
result Demonstrated drastic simplification of formulas after reformulation.
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. Proves estimates for Kähler-Ricci flow solutions.
problem Positive solutions to Kähler-Ricci flow.
method Matrix Li-Yau-Hamilton estimates coupled with flow.
result Monotonicity formula derived.
Abstract: Determinants and formulas for operators on various spaces.
problem Determinants and formulas for operators on different algebras and spaces.
method Use of Poincaré type determinants, invariant operators, and full matrix-symbols.
result Explicit formulas for determinants of elliptic operators and periodic pseudo-differential operators.
This paper studies HOMFLY polynomials of specific and infinite classes of knots.
problem Computing HOMFLY polynomials in general is difficult; this paper examines specific cases.
method Examined two specific knots and a general infinite class of knots.
result Observed apparent patterns in the polynomials of specific knots and conjectured properties of the general class.
Formula for computing cross-moments of polynomial processes.
problem Computing cross-moments of polynomial jump-diffusion dynamics.
method Explicit formula based on linear combinations of exponentials of the generator matrix.
result Closed and compact formulations for correlators, useful in financial pricing.
We prove an adiabatic decomposition formula of the zeta-determinant of the Laplace type operator with respect to Dirichlet boundary condition. We allow the non-invertible tangential operator. As a result, our adiabatic decomposition formula involves the scattering matrix over the manifold with cylindrical end. We also …
We show that integration over a G-manifold M can be reduced to integration over a minimal section Σ with respect to an induced weighted measure and integration over a homogeneous space G/N. We relate our formula to integration formulae for polar actions and calculate some weight functions. In case of a compact …
This talk is a report on joint work with A. Vaintrob [arXiv:math.CO/0109104 and math.GT/0111102]. It is organised as follows. We begin by recalling how the classical Matrix-Tree Theorem relates two different expressions for the lowest degree coefficient of the Alexander-Conway polynomial of a link. We then state our fo…
We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta al…
Proves conjecture about integer sums of torus knot torsions.
problem Integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for torus knots.
method Introduced Verlinde numbers from modular S-matrix, proved integrality through recursion formulas.
result Proven integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for all torus knots and non-negative g.
The paper defines and calculates fourth fundamental form and i-th curvatures for hypersurfaces in 4D Euclidean space.
problem Calculating curvatures for hypersurfaces in 4D Euclidean space.
method Defining fourth fundamental form and i-th curvatures for hypersurfaces, calculating them on rotational hypersurface, and studying hypersurfaces satisfying a specific differential equation.
result Fourth fundamental form and i-th curvatures are defined and calculated for hypersurfaces in 4D Euclidean space.
A1GM method improves efficiency in reconstructing missing data using KL divergence.
problem Efficiently reconstructing missing data in matrices.
method Fast non-gradient-based rank-1 NMF using KL divergence.
result A1GM outperforms gradient methods in efficiency with competitive reconstruction errors.
Paper derives local Plücker formulas for special orthogonal groups.
problem Deriving Plücker formulas for special orthogonal groups.
method Reduction to classical A_n case.
result Local Plücker formulas for special orthogonal groups derived.
New formulas derived for Jones polynomial of rational links.
problem Calculating the Jones polynomial of rational links.
method Colored Brylawski's tensor product formula for Tutte polynomials, finite automaton for crossing signs.
result Generalization of existing formulas for rational links.
New method improves covariance estimation for weighted samples.
problem Improving covariance estimation for weighted sample data.
method Asymptotic non-linear shrinkage formulas for covariance and precision matrix estimators of weighted sample covariances.
result Asymptotic non-linear shrinkage formulas for covariance and precision matrix estimators of weighted sample covariances.
Study O(p,q)-invariant valuations and derive Crofton formulas.
problem Classify and describe O(p,q)-invariant valuations. method Explicit formulas for signatures, functorial properties, and Crofton distributions.
result Every O(p,q)-invariant valuation is given by an invariant Crofton formula. After defining convex near-polygons, a formula enumerating the number of triangulations of such configurations is derived in terms of edge-polynomials. The paper describes also a transfer-matrix approach for computing quantities related to triangulations.
Solves PCR with fewer calls to ridge regression.
problem Principal component regression (PCR) with high accuracy.
method Reduces PCR to ridge regression calls and develops stable recurrence for matrix Chebyshev polynomials.
result Achieves PCR with multiplicative accuracy up to 1+γ using fewer calls.