Orthogonal Projection of an Infinite Round Cone in Real Hilbert Space

We fully characterize orthogonal projections of infinite right circular (round) cones in real Hilbert spaces. Another interpretation is that, given two vectors in a real Hilbert space, we establish the optimal estimate on the angle between the orthogonal projections of the two vectors. The estimate depends on the angle between the two vectors and the position of only one of the two vectors. Our results also make a contributions to Cauchy-Bunyakovsky-Schwarz type inequalities

LpL_{p} inequalities for the growth of polynomials with restricted zeros

summary:Let $P(z)=\sum _{\nu =0}^{n}a_{\nu }z^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $|z|1$, Boas and Rahman proved $$\left\Vert P(Rz)\right\Vert _{p}\le \big (\left\Vert R^{n}+z\right\Vert_{p}/\left\Vert1+z\right\Vert_{p}\big )\left\Vert P\right\Vert _{p}.$$ In this paper, we improve the above inequality for $0\le p 1$ is also given

Inequalities

Inequalities appear in various fields of natural science and engineering. Classical inequalities are still being improved and/or generalized by many researchers. That is, inequalities have been actively studied by mathematicians. In this book, we selected the papers that were published as the Special Issue ‘’Inequalities’’ in the journal Mathematics (MDPI publisher). They were ordered by similar topics for readers’ convenience and to give new and interesting results in mathematical inequalities, such as the improvements in famous inequalities, the results of Frame theory, the coefficient inequalities of functions, and the kind of convex functions used for Hermite–Hadamard inequalities. The editor believes that the contents of this book will be useful to study the latest results for researchers of this field

Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate

In 1995, Nadirashvili and subsequently Ashbaugh and Benguria proved the Rayleigh Conjecture concerning the first eigenvalue of the bilaplacian with clamped boundary conditions in dimension $2$ and $3$. Since then, the conjecture has remained open in dimension $d>3$. In this document, we contribute in answering the conjecture under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti's comparison principle, made possible after a fine study of the geometry of the eigenfunction's nodal domains.Comment: 28 pages, 3 figure

Quantifying Shape of Star-Like Objects Using Shape Curves and A New Compactness Measure

Shape is an important indicator of the physical and chemical behavior of natural and engineered particulate materials (e.g., sediment, sand, rock, volcanic ash). It directly or indirectly affects numerous microscopic and macroscopic geologic, environmental and engineering processes. Due to the complex, highly irregular shapes found in particulate materials, there is a perennial need for quantitative shape descriptions. We developed a new characterization method (shape curve analysis) and a new quantitative measure (compactness, not the topological mathematical definition) by applying a fundamental principle that the geometric anisotropy of an object is a unique signature of its internal spatial distribution of matter. We show that this method is applicable to “star-like” particles, a broad mathematical definition of shape fulfilled by most natural and engineered particulate materials. This new method and measure are designed to be mathematically intermediate between simple parameters like sphericity and full 3D shape descriptions. For a “star-like” object discretized as a polyhedron made of surface planar elements, each shape curve describes the distribution of elemental surface area or volume. Using several thousand regular and highly irregular 3-D shape representations, built from model or real particles, we demonstrate that shape curves accurately encode geometric anisotropy by mapping surface area and volume information onto a pair of dimensionless 2-D curves. Each shape curve produces an intrinsic property (length of shape curve) that is used to describe a new definition of compactness, a property shown to be independent of translation, rotation, and scale. Compactness exhibits unique values for distinct shapes and is insensitive to changes in measurement resolution and noise. With increasing ability to rapidly capture digital representations of highly irregular 3-D shapes, this work provides a new quantitative shape measure for direct comparison of shape across classes of particulate materials