Fractal balls

To the best of our knowledge, the analysis of densely folded media has not deserved special attention. The stress and strain analysis of this type of structures involves considerable difficulties concerning very strong non-linear effects. This paper presents a theory that could be classified as a geometric theory of folded media, in the sense that it ultimately leads to a kind of geometric constitutive law, or, in other words, a law that establishes the relationship between the geometry of the folded media and other variables such as the confinement capacity and the plastic strain energy. The discussion presented here is restricted to the particular case of compact balls produced by crushing together very thin plates or sheets. It is shown that both the geometry of the folded sheet and the plastic work density can be used as self-similarity tests. These criteria are equivalent for the case of thin plates or sheets made of the same material and with the same thickness. For the general case, the geometry of the folded sheet is not valid anymore as similarity criterion but there are strong arguments in favor of the plastic work density as a general criterion. If self-similarity is obtained for a ball set resulting from crumpling thin plates or sheets, it is possible to define two variables, the packing capacity and the slenderness ratio, that are related according to a power law. That is, the balls have a fractal representation. The power law scaling is derived from the mass conservation principle. The theory is claimed to be valid provided that certain assumptions referring to the geometry and material properties are satisfied. The results have shown that the theory is coherent and worthwhile of experimental validation. Some applications are suggested. A possible challenging investigation is related to the optimal geometry of biological membranes.     

Rolling balls and octonions

In this semi-expository paper we disclose hidden symmetries of a classical nonholonomic kinematic model and try to explain the geometric meaning of the basic invariants of vector distributions. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 258, pp. 17–27.     

Networks and closed balls

Neural networks calledtangent networks are constructed by explicit reference to the geometry of a set, and then blended intocascades which approximate characteristic functions of closed balls. In this way some known results about approximation by single hidden layer neural networks are re-proved in a very constructive and geometrical fashion.     

Monomorphism Operator and Perpendicular Operator

For a quiver Q, a k-algebra A, and an additive full subcategory 𝒳 of A-mod, the monomorphism category Mon(Q, 𝒳) is introduced. The main result says that if T is an A-module such that there is an exact sequence 0 → T _m  → … → T ₀ → D(A _A ) → 0 with each T _i  ∈ add(T), then Mon(Q, ^⊥ T) =^⊥(kQ ? _k T); and if T is cotilting, then kQ ? _k T is a unique cotilting Λ-module, up to multiplicities of indecomposable direct summands, such that Mon(Q, ^⊥ T) =^⊥(kQ ? _k T).

As applications, the category of the Gorenstein-projective (kQ ? _k A)-modules is characterized as Mon(Q, 𝒢𝒫(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q, 𝒳) can be described; and a sufficient and necessary condition for Mon(Q, A) being of finite type is given.     

Essentially Normal Operator + Compact Operator = Strongly Irreducible Operator

It is shown that given an essentially normal operator with connected spectrum, there exists a compact operator such that is strongly irreducible.

     

Characterization of balls by Riesz-potentials

For a bounded convex domain and consider the unit- density Riesz-potential . We show in this paper that u  =  const. on ∂G if and only if G is a ball. This result corresponds to a theorem of L.E. Fraenkel, where the ball is characterized by the Newtonian-potential (α = 2) of unit density being constant on ∂G. In the case α = N the kernel |x − y| ^α-N is replaced by  − log|x − y| and a similar characterization of balls is given. The proof relies on a recent variant of the moving plane method which is suitable for Green-function representations of solutions of (pseudo-)differential equations of higher-order.      

Variational inequalities over Euclidean balls

This paper investigates solution stability of parametric variational inequalities over Euclidean balls in finite dimensional spaces. We provide exact formulas for computing required coderivatives of the normal cone mappings to Euclidean balls via the initial data. On the basis of these formulas, we establish necessary and sufficient conditions for Lipschitzian stability of the solution maps of the aforementioned variational inequalities.     

Pucci eigenvalues on geodesic balls

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng’s eigenvalue comparison theorem for the Laplace–Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng’s bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng’s upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principal Pucci eigenvalues of an \({O(n)}\)-invariant hypersurface immersed in \({{\mathbb{R}}^{n+1}}\) with one smooth boundary component are smaller than the eigenvalues of an \({n}\)-dimensional Euclidean ball with the same boundary.     

Weak Operator Topology, Operator Ranges and Operator Equations via Kolmogorov Widths

Let K be an absolutely convex infinite-dimensional compact in a Banach space χ. The set of all bounded linear operators T on χ satisfying TK ⊃ K is denoted by G(K). Our starting point is the study of the closure WG(K) of G(K) in the weak operator topology. We prove that WG(K) contains the algebra of all operators leaving [`(lin(K))]\overline{{\rm lin}(K)} invariant. More precise results are obtained in terms of the Kolmogorov n-widths of the compact K. The obtained results are used in the study of operator ranges and operator equations.     

Covering an ellipsoid with equal balls

The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, our goal is to find the minimum number of unit balls needed to cover this ellipsoid. A tight asymptotic bound on the logarithm of this number is obtained.     

Linear balls and the multiplicity conjecture

A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley–Reisner ring has a linear resolution. It turns out that the Stanley–Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley–Reisner rings satisfy the multiplicity conjecture will be presented.     

From approximate balls to approximate ellipses

A ball spans a set of n points when none of the points lie outside it. In Zarrabi-Zadeh and Chan (Proceedings of the 18th Canadian conference on computational geometry (CCCG’06), pp 139–142, 2006) proposed an algorithm to compute an approximate spanning ball in the streaming model of computation, and showed that the radius of the approximate ball is within 3/2 of the minimum. Spurred by this, in this paper we consider the 2-dimensional extension of this result: computation of spanning ellipses. The ball algorithm is simple to the point of being trivial, but the extension of the algorithm to ellipses is non-trivial. Surprisingly, the area of the approximate ellipse computed by this approach is not within a constant factor of the minimum and we provide an elegant proof of this. We have implemented this algorithm, and experiments with a variety of inputs, except for a very pathological one, show that it can nevertheless serve as a good heuristic for computing an approximate ellipse.     

The Gaussian measure of shifted balls

Summary Let be a centered Gaussian measure on a Hilbert spaceH and let be the centered ball of radiusR>0. ForaH and , we give the exact asymptotics of (B _R(t)+t·a) ast. Also, upper and lower bounds are given when is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates.Supported in part by NSF grant number DMS-9024961