arXiv research

Abstract: Unknown status of Jacobian Conjecture, proof has a gap.

Notes a flaw in a proof about embedding graphs.

The article explores the fundamental gap in Bakry-Emery geometry.

Proves effective slope gaps for lattice surfaces.

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…

Probabilistic method proves gap estimates on sphere.

Unpublished results of S Straus and W Browder state that two notions of homotopy equivalence for manifolds with smooth group actions - isovariant and equivariant - often coincide under a condition called the Gap Hypothesis; the proofs use deep results in geometric topology. This paper analyzes the difference between th…

In this note, we prove the magnetic spectral gap-labelling conjecture as stated in [arXiv:1508.01064], in all dimensions, for principal solenoidal tori.

This paper has been withdrawn due to a gap in the proof of the main result.

Proves gap rigidity theorem for Hermitian symmetric spaces.

In this note, we prove an ${L^{\frac{n}{2}}}$-energy gap result for Yang-Mills connections on a principal $G$-bundle over a compact manifold without using Lojasiewicz-Simon gradient inequality (arXiv:1502.00668).

Estimates spectral gap for Brownian motion on sticky-reflecting domains.

The paper was withdrawn due to a gap in the proof of Lemma 3.

There is a gap in the proof of the main theorem in the article [ShCh13a] on optimal bounds for the Morse lemma in Gromov-hyperbolic spaces. We correct this gap, showing that the main theorem of [ShCh13a] is correct. We also describe a computer certification of this result.

In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely wh…

This paper has been withdrawn by the author due to a serious gap in the proof of the main theorem.

Simplified proofs for splitting homotopy idempotents.

The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.

This note fixes a small gap in Kerckhoff's proof that the limit set of the handlebody set has measure zero.

New estimates show spectral gap stability in RCD spaces, close to Beta distribution.

This paper studies dynamic stochastic optimization problems parametrized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of solutions and the absence of a duality gap. Our proof uses extended dynamic programm…

Study proves rigidity and gap theorems for specific metrics.

Game-theoretic analysis of mining gaps in blockchain systems.

In this note we fill a gap in the proof of the main theorem (Theorem 1.2) of our paper 'Surfaces in 4-manifolds', Math. Res. Letters 4 (1997), 907-914.

A gap in the proof of the finsler "unicorn" conjecture in the paper "Regular Landsberg metrics are always Berwald" by Z. I. Szabo is pointed out

Random hyperbolic surfaces have a spectral gap that approaches 1/4 as genus grows.

New method for spectral and Bergman kernels under local spectral gap condition.

Study improves understanding of why agentic theorem provers succeed.

Every smooth fiber bundle admits a complete (Ehresmann) connection. This result appears in several references, with a proof on which we have found a gap, that does not seem possible to remedy. In this note we provide a definite proof for this fact, explain the problem with the previous one, and illustrate with examples…

Paper provides a rigorous proof of the index theorem for economists.

UCB algorithm's arm-sampling behavior is revealed, leading to new insights and proofs.

In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which gives both a sharp lower bound for the spectral gap of a Schröinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We…

The trace set of a Fuchsian group $Γ$ ist the set of length of closed geodesics in the surface $Γ\backslash \mathbb{H}$. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C property actually characterizes arithm…

Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.

Corrected proof for 3D harmonic manifolds with minimal horospheres.

Negative curvature restricts the gap between the first and second eigenvalues of convex domains.

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature est…

Proves polynomial error rate for equidistribution of unipotent flows.

New proof found for Khovanov's assertion about Frobenius algebra twists.

Skeletal signatures were introduced in [J W Anderson and A Wootton, A Lower Bound for the Number of Group Actions on a Compact Riemann Surface, Algebr. Geom. Topol. 12 (2012) 19--35.] as a tool to describe the space of all signatures with which a group can act on a surface of genus $σ\geq 2$. In the present paper we pr…

This note provides a novel, simple analysis of the method of conjugate gradients for the minimization of convex quadratic functions. In contrast with standard arguments, our proof is entirely self-contained and does not rely on the existence of Chebyshev polynomials. Another advantage of our development is that it clar…

Researchers find spectral gaps in quantum flag manifolds using twisted operators.

New curvature measure defined for graphs, with bounds on diameter and spectral gap.

Let $G=*_λG_λ$ be a free product of torsion-free groups, and let $g\in[G,G]$ be any element not conjugate into a $G_λ$. Then scl$_G(g)\ge1/2$. This generalizes, and gives a new proof of a theorem of Duncan-Howie.

Paper withdrawn because of a gap in the proof of Proposition 3 of Thomas Schick: "Integrality of L2-Betti numbers", Math. Ann. 317, 727-750 (arXiv.org/abs/math.gt/0001101). Most results of the withdrawn paper were based on this proposition.

The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.

Random surfaces have a strong spectral gap with polynomial rate.

Original abstract: "We construct periodic solutions of nonlinear wave equations using analytic continuation. The construction applies in particular to Einstein equations, leading to infinite-dimensional families of time-periodic solutions of the vacuum, or of the Einstein-Maxwell-dilaton-scalar fields-Yang-Mills-Higgs-…