arXiv research

We relate the total curvature and the isoperimetric deficit of a curve $γ$ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $γ$. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent.

Study inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.

Quantitative estimates for $Q$-curvature near minimizing metrics on Riemannian manifolds.

The paper proves reverse inequalities in various geometric settings using curvature radius data.

Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.

The study proves stability of quermassintegral inequalities in hyperbolic space.

The study optimizes cell membranes' shapes based on curvature and proves existence of minimizers.

In the class of smoothly embedded surfaces of sphere type we prove that the isoperimetric deficit can be controlled by the Willmore deficit.

Sharp bounds for curve isoperimetric deficit derived.

In his paper "Shapes of Polyhedra and Triangulations of the Sphere", Thurston found that the set of shapes of convex polyhedra with prescribed cone-deficits has a complex hyperbolic structure. Inspired by his work, this paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits. We s…

Deep neural networks map brain lesions to deficits for better brain function understanding.

The paper improves inequalities for nearly spherical sets using quermassintegrals.

New deficit functions link elliptic and parabolic inequalities, proving log Sobolev.

Study analyzes household capital risk and poverty trapping, deriving a new function for capital deficit distribution.

The paper analyzes risk measures and optimal reserve allocation strategies.

We prove that finite perimeter subsets of $\mathbb{R}^{n+1}$ with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Colbois co…

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical interpretation for the case when equality holds.

We prove a quantitative version of Obata's Theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the genera…

We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the si…

This study analyzes public debts and deficits between European countries. The statistical evidence here seems in general to reveal that sovereign debts and government deficits of countries within European Monetary Unification-in average- are getting worse than countries outside European Monetary Unification, in particu…

Study mass and center of mass in flat 3-manifolds, proving existence of foliations.

Study compares nonsmooth spaces with integrable Ricci bounds.

We construct knot invariants on the basis of ascribing Euclidean geometric values to a triangulation of sphere S^3 where the knot lies. The main new feature of this construction compared to the author's earlier papers on manifold invariants is that now nonzero "deficit angles" (in the terminology of Regge calculus) can…

The accurate diagnosis and assessment of neurodegenerative disease and traumatic brain injuries (TBI) remain open challenges. Both cause cognitive and functional deficits due to focal axonal swellings (FAS), but it is difficult to deliver a prognosis due to our limited ability to assess damaged neurons at a cellular le…

Successful implementation of California's Renewable Portfolio Standard (RPS) mandating 33 percent renewable energy generation by 2020 requires inclusion of a robust strategy to mitigate increased risk of energy deficits (blackouts) due to short time-scale (sub 1 hour) intermittencies in renewable energy sources. Of the…

A sharp quantitative polygonal isoperimetric inequality is obtained.

The paper proves stability for scalar-flat metrics on manifolds with boundary.

Study stability of rigid motions and Möbius transformations on spheres, proving new rigidity estimates.

Kuwert and Schätzle showed in 2001 that the Willmore flow converges to a standard round sphere, if the initial energy is small. In this situation, we prove stability estimates for the barycenter and the quadratic moment of the surface. Moreover, in codimension one we obtain stability bounds for the enclosed volume and …

One of the main issues affecting the Italian NHS is the healthcare deficit: according to current agreements between the Italian State and its Regions, public funding of regional NHS is now limited to the amount of regional deficit and is subject to previous assessment of strict adherence to constraint on regional healt…

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the mani…

Optimally stabilizes Möbius group maps in spheres across dimensions.

A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational O(2) symmetry in a subspace orthogonal to a singular surface $Σ$ so that the surfa…

We postulates, and then show experimentally, that liquidity deficit is the driving force of the markets. In the first part of the paper a kinematic of liquidity deficit is developed. The calculus-like approach, which is based on Radon--Nikodym derivatives and their generalization, allows us to calculate important chara…

We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and s…

New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.

Paper finds eigenvalue bounds for hyperbolic space domains.

Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.

MRI identifies chronic symptoms in mTBI patients.

Early and accurate identification of parkinsonian syndromes (PS) involving presynaptic degeneration from non-degenerative variants such as Scans Without Evidence of Dopaminergic Deficit (SWEDD) and tremor disorders, is important for effective patient management as the course, therapy and prognosis differ substantially …

The paper develops a model for sovereign debt dynamics with explicit maturity structure.

The aim of the present article is to offer a strictly mathematical, statistical treatment of the current account balances in EU and in the Eurozone. Based on Eurostat data, an overview of the total and annual balances is first made for different collections among the EU countries. Then, using the Mathematica technical …

Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.

Given a simple closed plane curve $Γ$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4πF\leq π|F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $Γ$. In this note we improve this inequality finding strictly posi…

Similar to humans and animals, deep artificial neural networks exhibit critical periods during which a temporary stimulus deficit can impair the development of a skill. The extent of the impairment depends on the onset and length of the deficit window, as in animal models, and on the size of the neural network. Deficit…

We give conditions on a knot on which the Morton-Franks-Williams inequality is not sharp. As applications, we show infinitely many examples of knots where the inequality is not sharp and also prove (by giving examples) that the deficit of the inequality can be arbitrarily large.

Polyhedral surfaces can be broken down into parallelograms.