Volume Of Ball Research Articles

Some inequalities for the volume of the unit ball

Some inequalities for the volume of the unit ball

Effect of Grewia venusta(‘Ururu’) mucilage on the physicochemical and sensory properties of fried cake (‘akara’) prepared from bambara groundnut seed flour

Grewia venusta (‘Ururu’) mucilage was applied in the preparation of fried bambara groundnut cake (‘Akara’) to study its physicochemical and sensory properties. Batches of akara containing 0-2% mucilage were prepared by deep-fat-frying. The physical properties, proximate composition and sensory attributes of the akara were studied. The weight, volume and dimensions of the akara balls increased due to mucilage addition. The air spaces in the akara crumbs increased in number (56-295) with increasing level of mucilage. The akara crumbs became progressively spongy and their texture and flavour received higher scores than the control. The akara with 0% and 0.5% mucilage had the most attractive colour as shown by the panelists’ scores (4.8 and 4.4, respectively). The scores for colour of akara with 1.0% and 1.5% mucilage were not different significantly. The moisture, fibre, fat and ash contents of theakara increased while protein and carbohydrate decreased with increasing mucilage addition. The results suggested that apart from protein and colour, the quality of bambara groundnut akara improved upon addition of mucilage. The preparation of akara from bambara groundnuts with added Grewia venusta mucilage would help to diversify the utilization of the nuts as well as promote the use of a lesser known mucilage in food.

Ends, Fundamental Tones, and Capacities of Minimal Submanifolds Via Extrinsic Comparison Theory

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.

The portion of matrices with real spectrum in the Lie algebra of the real symplectic group

The portion of matrices with real spectrum in a given matrix Lie algebra is the ratio of the volume of the set of matrices with real spectrum in a ball centered at the zero of the algebra to the volume of the entire ball. In this article we calculate the portion of matrices with real spectrum in the Lie algebra of the real symplectic group.

Geometry of locally compact groups of polynomial growth and shape of large balls

We generalize Pansu’s thesis [27] about asymptotic cones of finitely generated nilpotent groups to arbitrary locally compact groups G of polynomial growth. We show that any such G is weakly commensurable to some simply connected solvable Lie group S , the Lie shadow of G and that balls in any reasonable left invariant metric on G admit a well-defined asymptotic shape. By-products include a formula for the asymptotics of the volume of large balls and an application to ergodic theory, namely that the Ergodic Theorem holds for all ball averages. Along the way we also answer negatively a question of Burago and Margulis [7] on asymptotic word metrics and give a geometric proof of some results of Stoll [33] of the rationality of growth series of Heisenberg groups.

Hörmander Functional Calculus for Poisson Estimates

The aim of the article is to show a Hormander spectral multiplier theorem for an operator A whose kernel of the semigroup exp(−zA) satisfies certain Poisson estimates for complex times z. Here exp(−zA) acts on $${L^p(\Omega),\,1 < p < \infty}$$ , where $${\Omega}$$ is a space of homogeneous type with the additional conditions that the volume of balls grows polynomially of exponent d and the measure of annuli is controlled by the corresponding euclidean term. In most of the known Hormander type theorems in the literature, Gaussian bounds and self-adjointness for the semigroup are needed, whereas here the new feature is that the assumptions are the to some extent weaker Poisson bounds, and $${H^\infty}$$ calculus in place of self-adjointness. The order of derivation in our Hormander multiplier result is typically $${\frac{d}{2}}$$ , d being the dimension of the space $${\Omega}$$ . Moreover the functional calculus resulting from our Hormander theorem is shown to be $${R}$$ -bounded. Finally, the result is applied to some examples.

Foliations of a smooth metric measure space by hypersurfaces with constantf-mean curvature

We study smooth codimension-one foliations F of a smooth metric measure space whose leaves have the same constant f -mean curvature. Firstly, we show that all the leaves of F are f -minimal hypersurfaces when either the smooth metric measure space is compact and has nonnegative Bakry‐ Emery Ricci curvature, or the limit of the ratio of the weighted volume of a geodesic ball B and the weighted area of a geodesic sphere @B vanishes. Secondly, we prove that every leaf of F is strongly f -stable. Lastly, we show that there is no complete proper foliation of the Gaussian space whose leaves have the same constant f -mean curvature. In particular, there are no foliations of R nC1 whose leaves are complete proper self-similar solutions for mean curvature flow.

Characterization and mapping of a spotted-leaf genotype, spl Y181 that confers blast susceptibility in rice

A spotted-leaf mutant Y181, produced by 60Coγ-radiation treatment of Zixuan1, was identified in rice. The result of TEM demonstrated the number of starch grains decreased significantly, the distribution and volumes of osmiophilic granules and lipid balls were larger in the leaf cells around the lesion mimics, the chloroplasts in mesophyll cells disappeared and just leaving a few rupture thylakoid membranes in the cytoplasm and some transmutative starch granules in Y181. Compared with the wild type, the content of chlorophyll was decreased significantly, while the H2O2 content, and SOD, POD and CAT activity were obviously increased in mutant. DAB staining and H2O2 treatment further revealed oxidative burst in spotted leaves. Expression of lesion-related gene showed RuBisCo-L, RuBisCo-S, PDI and TPX was down-regulated, while PBZ1 was up-regulated in the leaf cell at the lesion mimics. Disease reaction to Magnaporthe grisea indicated that the resistance of Y181 was reduced. Genetic analysis showed that the mutant was controlled by a recessive nuclear gene, which was delimited to a 390 kb region between CH12-936 and RM7195.

Volume of Geodesic Balls in Finsler Manifolds of Hyperbolic Type

Let be a compact Finsler manifold of hyperbolic type, and be its universal Finslerian covering. In this paper we show that the growth function of the volume of geodesic balls of is of purely exponential type.

The Volume of the Unit n-Ball

SummaryTwo simple proofs are given for the fact that the volume of the unit ball in n-dimensional Euclidean space approaches 0 as n approaches ∞. (Some authors use the term “unit sphere” for what is here called the unit ball.) One argument involves covering the unit ball by simplices. The other argument involves covering the unit ball by rectangular solids.

Inequalities for the Volume of the Unit Ball in $${\mathbb{R}^{n}}$$ R n

The volume of the unit ball in \({\mathbb{R}^{n}}\) is defined by $$\Omega_{n} = \frac{\pi^{n/2}}{\Gamma(n/2+1)},\qquad n = 1,2,3,\ldots,$$ where Γ denotes the classical gamma function of Euler. In several recently published papers numerous authors studied properties of Ωn. In particular, various inequalities involving Ωn are given in the literature. In this paper, we continue the work on this subject and offer new inequalities. More precisely, we offer sharp upper and lower bounds for $$\frac{\Omega_{n}^{2}}{\Omega_{n-1} \Omega_{n+1}},\quad\frac{\Omega_{n}}{\Omega_{n-1}+\Omega_{n+1}} \quad {\rm and} \quad\Omega_{n}.$$

MEAN VALUE TYPE INEQUALITIES FOR QUASINEARLY SUBHARMONIC FUNCTIONS

AbstractThe mean value inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalise subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the characterisation of non-negative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalised mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the radius of this ball.

The tetrahedral property and a new Gromov–Hausdorff compactness theorem

We present the Tetrahedral Compactness Theorem, which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov–Hausdorff sense to a countably Hm rectifiable metric space of the same dimension. The tetrahedral property depends only on distances between points in spheres; yet we show it provides a lower bound on the volumes of balls. The proof is based upon intrinsic flat convergence and a new notion called the sliced filling volume of a ball.

On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry

Addressing a question of Gromov, we give a rate in Pansu’s theorem about the convergence to the asymptotic cone of a finitely generated nilpotent group equipped with a left-invariant word metric rescaled by a factor. We obtain a convergence rate (measured in the Gromov–Hausdorff metric) offor nilpotent groups of classandfor nilpotent groups of class 2. We also show that the latter result is sharp, and we make a connection between this sharpness and the presence of so-called abnormal geodesics in the asymptotic cone. As a corollary, we get an error term of the formfor the volume of Cayley balls of a general nilpotent group of classr. We also state a number of related conjectural statements.

Calculation of the Volume of Multidimensional Cones and Balls Using Cavalieri's Principle

Formulas for the volume of multidimensional cones and balls are deduced based on Cavalieri's principle and basic properties of the Lebesgue measure (without using integration).

Polynomial growth of the volume of balls for zero-entropy geodesic systems

The aim of this paper is to state and prove a polynomial analogue of the classical Manning inequality, relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim, we use a numerical conjugacy invariant for dynamical systems, the polynomial entropy. It is infinite when the topological entropy is positive. We first prove that the growth rate of the volume of balls is bounded above by means of the polynomial entropy of the geodesic flow. For the flat torus this inequality becomes an equality. We then study explicitely the example of the torus of revolution (which is a case of strict inequality). We give an exact asymptotic equivalent of the growth rate of volume of balls.

Monotonicity of functions connected with the gamma function and the volume of the unit ball

In the present study, we prove the monotonicity of two functions involving the gamma function Γ(x) and relating to the n-dimensional volume of the unit ball 𝔹 n in ℝ n .

A sharp bound for the area of minimal surfaces in the unit ball

Let Σ be a k-dimensional minimal surface in the unit ball B n which meets the boundary ∂ B n orthogonally. We show that the area of Σ is bounded from below by the volume of the unit ball in $${\mathbb{R}^k}$$ .

Monotonicity and logarithmic convexity relating to the volume of the unit ball

Let $\Omega_n$ stand for the volume of the unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$. In the present paper, we prove that the sequence $\Omega_{n}^{1/(n\ln n)}$ is logarithmically convex and that the sequence $\frac{\Omega_{n}^{1/(n\ln n)}}{\Omega_{n+1}^{1/[(n+1)\ln(n+1)]}}$ is strictly decreasing for $n\ge2$. In addition, some monotonic and concave properties of several functions relating to $\Omega_{n}$ are extended and generalized.

Complete monotonicity of a function involving the ratio of gamma functions and applications

In the paper, necessary and sucient conditions are presented for a function involving a ratio of gamma functions to be logarithmically com- pletely monotonic. This extends and generalizes the main result of Guo and Qi (Taiwanese J. Math. 7 (2003), no. 2, 239-247) and others. As applications, several inequalities involving the volume of the unit ball in R n are derived, which refine, generalize and extend some known inequalities.