Study eigenvalues of curvature operators to annihilate cobordism invariants.
problem Annihilating rational cobordism invariants on spin manifolds.
method Linear inequalities on curvature operator eigenvalues.
result Curvature conditions stabilize to annihilate invariants.
This paper proves rational eigenvalues for shape operators in certain spaces.
problem Understanding rational properties of shape operators in symmetric spaces.
method Analyzing normal holonomy and shape operators in singular orbits of isotropy representations.
result Shape operators have rational eigenvalues in specific normal holonomy factors.
The paper describes spectra of operators on rational homogeneous varieties.
problem Spectrum of invariant (1,1)-forms and Weitzenböck remainder. method Detailed analysis using index theory and Lie-theoretic data.
result Explicit formula for smallest eigenvalue and new lower bounds.
Proven isoperimetric inequality for Witten-Laplacian eigenvalues.
problem Proving isoperimetric inequality for lower order nonzero Neumann eigenvalues of Witten-Laplacian.
method Analytical proof using Euclidean and hyperbolic spaces.
result Strengthens Szegő-Weinberger inequality and covers Xia-Wang's progress.
The Yamabe invariant is linked to static potentials and eigenvalues.
problem The relationship between Yamabe invariant and static potentials/eigenvalues.
method Analyzes the Yamabe invariant in the context of static potentials and eigenvalues of the Laplacian.
result The Yamabe invariant is closely tied to static potentials and the first eigenvalue of the Laplacian.
We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree d rational curves in Pn. We deduce as special cases algebro-geome…
We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.
We reconsider the (rational) Calogero-Moser system from the point of view of bi-Hamiltonian geometry. By using geometrical tools of the latter, we explicitly construct set(s) of spectral canonical coordinates, that is, complete sets of Darboux coordinates defined by the eigenvalues and the eigenvectors of the Lax matri…
Given a matrix A∈SL(N,Z), form the semidirect product G=ZN⋊AZ where the Z factor acts on ZN by A. Such a G arises naturally as the fundamental group of an N-dimensional torus bundle which fibers over the circle. In this paper we prove that if A has distinct eigenvalues not lying on the…
New examples of hyperbolic 3-manifolds where Seiberg-Witten equations fail.
problem Finding hyperbolic 3-manifolds without irreducible Seiberg-Witten solutions.
method Combining hyperbolic geometry, upper bound for eigenvalues, Selberg trace formula, and precise numerical bounds.
result First examples of hyperbolic 3-manifolds where Seiberg-Witten equations do not admit irreducible solutions.
The study shows conditions for Kähler manifolds to have rational cohomology.
problem Conditions for Kähler manifolds to have rational cohomology.
method Analyzes the Kähler curvature operator and its eigenvalues.
result Compact Kähler manifolds have rational cohomology under certain conditions.
The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior f…
The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres.
problem Conditions for convex hypersurfaces to be rational homology spheres in Riemannian manifolds.
method The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres using vanishing and estimation theorems for Betti numbers.
result Conditions for convex hypersurfaces to be rational homology spheres in Riemannian manifolds.
Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.
problem Prove nullity of biharmonic maps from flat 2-torus to round 2-sphere.
method Use spectral geometry, construct polynomial isomorphism with elliptic curve, determine rational points on spectral curve.
result Nullity of every map in the family is 5, confirming conjecture.
Through simple analytical calculations and numerical simulations, we demonstrate the generic existence of a self-organized macroscopic state in any large multivariate system possessing non-vanishing average correlations between a finite fraction of all pairs of elements. The coexistence of an eigenvalue spectrum predic…
RationalNet improves graph convolutional networks by approximating jump discontinuities more efficiently.
problem Graph convolutional networks struggle with approximating jump discontinuities, leading to oscillations and high computational costs.
method RationalNet uses rational functions to approximate graph signals, avoiding oscillations and reducing computational complexity.
result RationalNet effectively characterizes jump discontinuities, outperforming other methods on both synthetic and real-world graphs.
Study Kähler-Ricci flow on rational homogeneous varieties using algebraic geometry and representation theory.
problem Analyzing the Kähler-Ricci flow on rational homogeneous varieties.
method Combining projective algebraic geometry and representation theory of semisimple Lie groups and Lie algebras.
result Explicit description and computation of solutions and geometric quantities along the flow.
Researchers study eigenvalues on singular Riemannian manifolds, showing how curvature affects Weyl's law.
problem Analyzing eigenvalues of Laplace-Beltrami operator on singular Riemannian manifolds with unbounded geometrical invariants.
method Developed a new quantitative estimate for the remainder of the heat trace and Weyl's function on Riemannian manifolds.
result Constructed singular Riemannian metrics with prescribed non-classical Weyl's law for various slowly varying functions.
We investigate random complex dynamics of rational or polynomial maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in the Riemann sphere due to the automatic coopeartion of many kinds of maps in the system, ev…
New proof for weak mixing in polygonal billiards.
problem Proving weak mixing in polygonal billiards.
method Using Baire category and eigenvalue analysis.
result Billiard flow is weakly mixing for non-rational polygons.
We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume …
We uncover a new anomaly in asset pricing that is linked to the remuneration: the more a company spends on salaries and benefits per employee, the better its stock performs, on average. Moreover, the companies adopting similar remuneration policies share a common risk, which is comparable to that of the value premium. …
Explains eigenvalue and generalized eigenvalue problems with examples.
problem Eigenvalue and generalized eigenvalue problems.
method Introduction and examples from machine learning.
result Solutions to eigenvalue and generalized eigenvalue problems.
Study rigid classes on hyperkahler manifolds, showing general ones are rigid.
problem Characterize rigid classes on compact hyperkahler manifolds.
method Analyze eigenvectors of hyperbolic automorphisms and use BBF form.
result General parabolic classes on hyperkahler manifolds are rigid.
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
The paper compares Steklov and Laplacian eigenvalues on graphs.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on graphs.
method Analyzing eigenvalues and discussing rigidity.
result Obtained Lichnerowicz-type estimates and combinatorial estimates for Steklov eigenvalues.
Paper finds bounds for Steklov eigenvalues on manifolds.
problem Eigenvalue bounds for Steklov eigenvalues on manifolds.
method Eigenvalue comparison theorems and bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem.
result Established bounds for Steklov eigenvalues and Wentzell eigenvalues.
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature (3,2). method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree ≤5 in the quadric. Rational knots and links in solid torus characterized by continued fractions.
problem Characterizing rational knots and links in solid torus.
method Using rational tangles and continued fractions, and generalizing to skein module invariants.
result Rational links in solid torus fully characterized by rational tangles and continued fractions.
New method for simplifying knots with specific properties.
problem Understanding knots with a specific unknotting number.
method Derive and apply the Montesinos trick for proper rational tangle replacement.
result Prove that knots with proper rational unknotting number one are prime and classify certain types.
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
We give an example of a subgroup of SL(2,C) which is a strictly ascending HNN extension of a non-abelian finitely generated free group F. In particular, we exhibit a free group F in SL(2,C) of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir. The main ingre…
We note that a rational 3-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational 3-tangle diagrams up to isotopy. However, there is no perfect classification about rational 3-tangle diagrams such as the classification of rational 2-tangle diagrams cor…
New rational band moves simplify knot classification.
problem Classifying knots using rational moves.
method Introduced oriented rational band moves and proved their effectiveness.
result Knots that can be unlinkified by rational moves are rationally slice.
Classifies 3-manifolds bounding rational homology balls.
problem Identifying 3-manifolds that bound rational homology balls.
method Used constraints from Donaldson's diagonalization theorem and Heegaard Floer correction terms.
result Complete classification of spherical 3-manifolds bounding rational homology balls.
Study rational homology spheres with two legs, classifying those bounding rational balls.
problem Classifying Seifert rational homology spheres with two complementary legs.
method Complete classification of Seifert manifolds with 3 exceptional fibers and two complementary legs.
result Classification of Seifert manifolds bounding rational homology balls.
Classifies surgeries on torus knots and cables that bound rational homology balls.
problem Which surgeries on torus knots and cables bound rational homology balls?
method Classification based on integral surgeries and rational numbers q/p for cables.
result Set of rational numbers q/p for cables of a given knot K is bounded.
Jones polynomial coincidences explored for rational knots.
problem Identifying coincidences in Jones polynomial of rational knots.
method Moves on continued fraction expansion of rational knots, conjectured to generate all coincidences.
result Conjectured moves are sufficient to generate all Jones rational coincidences.
Estimates eigenvalue of p-Laplacian on curved spaces.
problem Estimating eigenvalues of p-Laplacian on curved spaces.
method Integral curvature conditions to estimate eigenvalues.
result Various estimates of the first eigenvalue.
The paper explores inequalities between eigenvalues on Riemannian manifolds.
problem Investigating relationships between eigenvalues on Riemannian manifolds.
method Constructing gradient estimates for a first eigenfunction to derive inequalities.
result Obtained some relationships between weighted p-Laplacian first eigenvalues. The study improves bounds for Laplace eigenvalues in Kaehler manifolds.
problem Improving bounds for Laplace eigenvalues in Kaehler manifolds.
method Generalizing classical inequalities to higher eigenvalues and applying to analytic varieties.
result Proves inequalities for Laplace eigenvalues of Kaehler manifolds and analytic varieties.
Lower bounds on rational slice genus using Heegaard Floer invariants.
problem Measuring complexity of homology classes in 4-manifolds.
method Introducing rational slice genus, bounding Heegaard Floer τ invariants, using satellite links and closed braids.
result Lower bounds on rational slice genus in terms of Heegaard Floer τ invariants.
New rational homology balls found in specific knot 2-handlebodies.
problem Finding rational homology balls in 2-handlebodies.
method Proving existence and properties of rational homology balls through smooth embeddings and rational blow-ups.
result Rationally blown-up 4-manifolds cannot be obtained by ordinary blow-ups under certain conditions.
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove in…
The article explores toric spaces of regular polyhedra, highlighting rational and non-rational cases.
problem Exploring toric spaces associated with regular convex polyhedra.
method Symplectic and complex toric spaces associated with five regular convex polyhedra.
result The regular dodecahedron and icosahedron cannot be treated via standard toric geometry.
The paper connects Steklov eigenvalues and Laplacian eigenvalues on Riemannian manifolds.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on Riemannian manifolds.
method Using sectional curvature conditions, the paper proves mutual control between Steklov and Laplacian eigenvalues.
result Derives a Weyl-type upper bound for Steklov eigenvalues.
Study connects taffy pulling, fractions, and rational tangles.
problem Understanding the relationship between taffy pulling, fractions, and rational tangles.
method Developed a taffy analogue for Conway's characterization of rational tangles and gave a geometric connection.
result Direct geometric connection between rational tangles and taffy pulls.