arXiv research

Paper proves a noncompact version of Gromov's band-width estimate.

Extends width estimates to family case using index theory.

Study on Gehring link problem and width of bands in curved manifolds.

Develops connections between operator K-theory and positive scalar curvature.

The Rosenberg index vanishes if a manifold admits a wide Riemannian band or cube-like domain.

The paper explores index theory for Dirac operators to understand scalar curvature properties.

The paper explores rigidity theorems for spectral curvature bounds in 3-manifolds.

Study shows upper limit for torical band width with spectral curvature bounds.

Establish optimal Lipschitz lower bounds for functions on manifolds with negative curvature, revealing interplay between width, boundary area, and topology.

Estimates bandwidth for CMC initial data sets.

The study refines known counterexamples in 4D to satisfy certain inequalities.

Inspired by Gromov's work on 'Metric inequalities with scalar curvature' we establish band width inequalities for Riemannian bands of the form $(V=M\times[0,1],g)$, where $M^{n-1}$ is a closed manifold. We introduce a new class of orientable manifolds we call filling enlargeable and prove: If $M$ is filling enlargeable…

Proves a quantitative index theorem for positive scalar curvature metrics.

The paper proves rigidity for certain product spaces and bounds for band widths.

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrtσ$, where $C_n = \…

Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.

Proves special alternating knots cannot be decomposed as non-trivial band sums.

The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.

Proves principles and estimates for initial data sets in Einstein equations.

Counterexamples found for knot conjectures.

Develops a robust hedging valuation adjustment measure for dynamic hedging under liquidity-demand stress.

Knots from a specific band sum have similar homologies but are distinct.

Paper develops a robust HVA measure for dynamic hedging under liquidity stress.

This work uses sampling theory to analyze smoothness and error bounds of finite neural networks.

We prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimal systolic in- equality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric…

New basis confirms Thurston's conjecture and reveals knot configurations.

We present various examples of cosmetic bandings on knots and links, that is, bandings on knots and links leaving their types unchanged. As a byproduct, we give a hyperbolic knot which admits exotic chirally cosmetic surgeries yielding hyperbolic manifolds. This gives a counterexample to a conjecture raised by Bleiler,…

The paper constructs optimal confidence bands for kernel gradient flow estimators.

Proves conjecture about sphere widths under rotational symmetry.

The paper connects Chebyshev polynomials and Gram determinants on Möbius bands.

It has been conjectured that the geometric invariant of knots in 3-space called the width is nearly additive. That is, letting w(K) in N denote the width of a knot K in S^3, the conjecture is that w(K # K') = w(K) + w(K') - 2. We give an example of a knot K_1 so that for K_2 any 2-bridge knot, it appears that w(K_1 # K…

We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary comp…

We show that the torus knot $T_{4,9}$ bounds a smooth Möbius band in the $4$-ball, giving a counterexample to Batson's non-orientable analogue of Milnor's conjecture on the smooth slice genera of torus knots.

Proves the index of a Möbius band in 4D ball equals 5.

Wide networks with polynomial activations have proven asymptotic behavior.

Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.

We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional $I(u) = \int |\nabla u|^2 + V(u)$, where $V(u)$ is the characteristic function of the interval $(-1,1)$. This functional is a close relative of the scalar Ginzburg-Landau functional $J(u) = \int |\nabla u|^…

We compute the non-orientable 4-ball genus for a new family of torus knots.

UTOPIA aggregates multiple prediction intervals efficiently.

In conventional chemisorption model, the d-band center theory (augmented sometimes with the upper edge of d-band for imporved accuarcy) plays a central role in predicting adsorption energies and catalytic activity as a function of d-band center of the solid surfaces, but it requires density functional calculations that…

The paper confirms a conjecture about convex bodies and their properties.

The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have $tr(K_1 \# K_2) = \max\{tr(K_1),tr(K_2)\}$, confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any…

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surfa…

This paper gives mathematical models for flat knotted ribbons, and makes specific conjectures for the least length of ribbon (for a given width) needed to tie the trefoil knot and the figure eight knot. The first conjecture states that (for width one) the least length of ribbon needed to tie an open-ended trefoil knot …

Study a relative aspherical conjecture and prove 3-manifold obstruction to positive scalar curvature.

We show that two open books in a given closed, oriented three-manifold admit isotopic stabilizations, where the stabilization is made by successive plumbings with Hopf bands, if and only if their associated plane fields are homologous. Since this condition is automatically fulfilled in an integral homology sphere, the …

In the paper we prove the conjecture by Alexander Zupan that $w(K) \geqslant n^2w(J)$ where w denote the width and $K$ and $J$ are satellite knot and its companion with winding number $n$. Also we proved that for satellite knot with braid pattern, the equality holds.

We found an infinite family of counterexamples to Batson's conjecture.