Volume Of Ball Research Articles

Anosov magnetic flows, critical values and topological entropy

We study the magnetic flow determined by a smooth Riemannian metric g and a closed 2-form Ω on a closed manifold M. If the lift of Ω to the universal cover M̃ is exact, we can define a critical value c(g,Ω) in the sense of Mañé (1997 Bol. Soc. Bras. Mat. 28 141-53) for the lift of the flow to M̃. We have c(g,Ω)<∞ if and only the lift of Ω has a bounded primitive. This critical value can be expressed in terms of an isoperimetric constant defined by (g,Ω), which coincides with Cheeger's isoperimetric constant when M is an oriented surface and Ω is the area form of g. When the magnetic flow of (g,Ω) is Anosov on the unit tangent bundle SM, we show that 1/2>c(g,Ω) and any closed bounded form in M̃ of degree ⩾2 has a bounded primitive.Next we consider the one-parameter family of magnetic flows on SM associated with the pair (g,λΩ) for λ⩾0, where Ω is such that its lift to M̃ has a bounded primitive.We introduce a volume entropy hv(λ) defined as the exponential growth rate of the average volume of certain balls. We show thathv(λ)⩽htop(λ), where htop(λ) isthe topological entropy of the magnetic flow of (g,λΩ) on SM andthat equality holds if the magnetic flow of (g,λΩ) is Anosov on SM.If λ1⩽λ2 and the magnetic flows for (g,λ1Ω)and (g,λ2Ω) are both Anosov on SM, then hv(λ1)⩾hv(λ2).We construct an example of a Riemannian metricof negative curvature on a closed oriented surface of higher genussuch that if ϕλ is the magnetic flow associated withthe area form with intensity λ, then there are values of the parameter0<λ1<λ2 with the property that ϕλ1has conjugate points and ϕλ2 is Anosov. Variations of this example show that it is also possible to exit andreenter the set of Anosov magnetic flows arbitrarily many timesalong the one-parameter family. Moreover, we can startwith a Riemannian metric with conjugate points and endup with an Anosov magnetic flow for some λ>0.Finally we have a version of the example (in which Ω is no longer the area form) such thatthe topological entropy of ϕλ1 is greater than the topological entropy of the geodesic flow, which in turn is greater than thetopological entropy of ϕλ2.

A note on a theorem of Borwein, Borwein, Fee and Girgensohn

A simple application of a theorem of (1) shows that the product of the volume of the unit ball of n by the volume of the unit ball of its dual space is increasing for 1 p 2.

High-frequency estimates for the Neumann scattering phasein non-smooth obstacle scattering

In this paper the high-frequency asymptotics of the Neumannscattering phase s(λ) in acoustic obstacle scattering isconsidered.Under certain conditions it is proved that if the boundary∂Γ of an obstacle Γ has finite upperδ-Minkowski content µ*(δ,∂Γ), thenthere is a positive constant Cn,δ, depending only onn and δ, such thatandas λ→∞, where wn is the volume of theunit ball in ℝn.

Inequalities for the Volume of the Unit Ball in Rn

Let Ωn=πn/2/Γ(1+n/2) be the volume of the unit ball in Rn. We determine the best possible constants a, b, A, B, α, and β such that the inequalitiesaΩn/(n+1)n+1≤Ωn≤bΩn/(n+1)n+1,n+A/2π≤Ωn−1/Ωn≤n+B/2π,and1+1/nα≤Ω2n/Ωn−1Ωn+1≤1+1/nβare valid for all integers n≥1. Our results refine and complement inequalities proved by G. D. Anderson et al., K. H. Borgwardt, and D. A. Klain and G.-C. Rota.

Friction and wear properties of carbon-ion implanted titanium nitride films

TiN films were deposited by ion plating on high speed tool steel substrates, and implanted by carbon ions with fluences up to 5×10 17 ions/cm 2 and with energies ranging from 50 to 150 keV. Friction and wear tests were carried out by a pin-on-disk tribometer with stainless steel balls as a counter material. The surface layers of TiN films modified by the carbon-ion implantation were characterized by X-ray diffraction (XRD) for crystal structure identification and X-ray photoelectron spectroscopy (XPS) for chemical composition analysis. The carbon-ion implantation reduced the friction coefficient of the TiN films against the stainless steel balls, and also the wear volume of the steel balls. The duration of the low friction coefficient was extended with increasing carbon dose. Adhesion of the counter material could be prevented by the carbon-ion implantation into TiN film, leading to a drastic decrease of the friction coefficient. The tribological properties of the carbon-implanted TiN films can be controlled by carbon dose and ion energy: wear rate and frictional behavior of the TiN coating can be reduced.

Prescribing growth type of complete Riemannian manifolds of bounded geometry

We describe certain properties of growth types of nondecreasing sequences. We build a complete, connected Riemannian surface of bounded geometry and of a given growth type provided that the type satisfies some natural conditions. 0. Introduction. Growth of leaves plays an important role in the study of topology and dynamics of foliations. The existence of leaves with nonexponential or polynomial growth has some influence on the structure of foliations. Constructing leaves with neither exponential nor polynomial growth can be pretty difficult (see [CC] and references there). The space of all growth types is very rich and contains many types which cannot be compared with polynomial, fractional or exponential ones. In this article, we show that any growth type ξ (not greater than the exponential one and satisfying simple conditions described in Sections 2, 3) can be realized by the volumes of balls on a suitable complete Riemannian manifold of bounded geometry. We believe that in the near future we will be able to apply our construction to obtain leaves of a given growth type on some compact foliated manifolds. 1. Growth types. In this section we recall the notion of the growth type of nondecreasing functions and of complete, connected Riemannian manifolds (compare [HH], [E]). Let I be the set of nonnegative nondecreasing functions on N: I = {f : N→ R+ : f(n) ≤ f(n+ 1) for all n ∈ N}. Define a preorder in I. Let f, h ∈ I. We say that h dominates f (and write f h) if for some A ∈ R+ and B ∈ N, f(n) ≤ Ah(Bn) for any n ∈ N. 2000 Mathematics Subject Classification: Primary 53C99; Secondary 37C85.

A Strong Law for the Largest Nearest-Neighbour Link between Random Points

Suppose that X1, X2, X3, … are independent random points in Rd with common density f, having compact support Ω with smooth boundary ∂Ω, with f[mid ]Ω continuous. Let Rni, k denote the distance from Xi to its kth nearest neighbour amongst the first n points, and let Mn, k = maxi[les ]nRni, k. Let θ denote the volume of the unit ball. Then as n → ∞,formula hereIf instead the points lie in a compact smooth d-dimensional Riemannian manifold K, then nθMdn, k/ log n → (minKf)−1, almost surely.

Isoperimetric Inequalities for Extrinsic Balls in Minimal Submanifolds and their Applications

S.-Y. Cheng, P. Li and S.-T. Yau proved comparison theorems for the volume of extrinsic balls in minimal submanifolds of space forms. These results were extended by S. Markvorsen for minimal submanifolds of a riemannian manifold with just an upper bound on the sectional curvature. In the paper an isoperimetric inequality for extrinsic balls in minimal submanifolds of a riemannian manifold N with sectional curvatures bounded from above by a non-positive constant is found. As a corollary of this result an alternative proof is obtained of the comparison for the volume of extrinsic balls stated by the preceding authors, but now the equality case is characterized when the upper bound for the sectional curvatures of the ambient manifold is strictly negative. Finally, when the sectional curvatures of N are bounded from above for any constant (positive or negative), it is proved that the ∞-isoperimetric quotient of the extrinsic balls is bounded from below by the mean curvature of the geodesic spheres in the m-dimensional real space forms.

On the Hilbert kernel density estimate

Abstract Let X be an R d -valued random variable with unknown density f. Let X1,…,Xn be i.i.d. random variables drawn from f. We study the pointwise convergence of a new class of density estimates, of which the most striking member is the Hilbert kernel estimate 1 V d n log n ∑ i=1 n 1 ||x−X i || d , where Vd is the volume of the unit ball in R d . This is particularly interesting as this density estimate is basically of the format of the kernel estimate (except for the log n factor in front) and the kernel estimate does not have a smoothing parameter.

A Strong Law for the Longest Edge of the Minimal Spanning Tree

Suppose $X_1, X_2, X_3,\ldots$ are independent random points in $\mathbf{R}^d,d\geq 2$, with common density $f$, having connected compact support $\Omega$ with smooth boundary $\partial\Omega$, with $f|_{\Omega}$ continuous. Let $M_{n}$ denote the smallest $r$ such that the union of balls of diameter $r$ centered at the first $n$ points is connected. Let $\theta$ denote the volume of the unit ball. Then as $n\to\infty$, $$n\theta M^d_n/\log n \to \max\Big(\big(\min\limits_{\Omega}f\big)^{-1},2(1 - 1/d)\big(\min\limits_{\partial\Omega}f\big)^{-1}\Big),\quad\text{a.s.}$$

On manifolds with non-negative Ricci curvature and Sobolev inequalities

— Let M be a complete n-dimensional Riemanian manifold with non-negative Ricci curvature in which one of the Sobolev inequalities (∫ |f |dv )1/p ≤ C(∫ |∇f |qdv)1/q, f ∈ C∞ 0 (M), 1 ≤ q 1. Moreover, for q > 1, the equality in (1) is attained by the functions (λ + |x|q/(q−1))1−(n/q), λ > 0, where |x| is the Euclidean length of the vector x in IR. We are actually interested here in the geometry of those manifolds M for which one of the Sobolev inequalities (1) is satisfied with the best constant C = K(n, q) of IR. The result of this note is the following theorem. Theorem. Let M be a complete n-dimensional Riemannian manifold with nonnegative Ricci curvature. If one of the Sobolev inequalities (1) is satisfied with C = K(n, q), then M is isometric to IR. The particular case q = 1 (p = n/(n − 1)) is of course well-known. In this case indeed, the Sobolev inequality is equivalent to the isoperimetric inequality ( voln(Ω) )(n−1)/n ≤ K(n, 1)voln−1(∂Ω) where ∂Ω is the boundary of a smooth bounded open set Ω in M . If we let V (x0, s) = V (s) be the volume of the geodesic ball B(x0, s) = B(s) with center x0 and radius s in M , we have d ds voln ( B(s) ) = voln−1 ( ∂B(s) ) . Hence, setting Ω = B(s) in the isoperimetric inequality, we get V (s)(n−1)/n ≤ K(n, 1)V ′(s) for all s. Integrating yields V (s) ≥ (nK(n, 1))−nsn, and since K(n, 1) = n−1ω n , for every s, (2) V (s) ≥ V0(s) where V0(s) = ωns is the volume of the Euclidean ball of radius s in IR. IfM has nonnegative Ricci curvature, by Bishop’s comparison theorem (cf. e.g. [Ch]) V (s) ≤ V0(s) for every s, and by (2) and the case of equality,M is isometric to IR. The main interest of the Theorem therefore lies in the case q > 1. As usual, the classical value q = 2 (and p = 2n/(n − 2)) is of particular interest (see below). It should be noticed that known results already imply that the scalar curvature of M is zero in this case (cf. [He], Prop. 4.10). Proof of the Theorem. It is inspired by the technique developed in the recent work [B-L] where a sharp bound on the diameter of a compact Riemannian manifold satisfying a Sobolev inequality is obtained, extending the classical Myers theorem. We thus assume that the Sobolev inequality (1) is satisfied with C = K(n, q) for some q > 1. Recall first that the extremal functions of this inequality in IR are the functions (λ + |x|q)1−(n/q), λ > 0, where q′ = q/(q − 1). Let now x0 be a fixed point in M and let θ > 1. Set f = θ−1d(·, x0) where d is the distance function on M . The idea is then to apply the Sobolev inequality (1), with C = K(n, q), to (λ+ f ′ )1−(n/q), for every λ > 0 to deduce a differential inequality whose solutions may be compared to the extremal Euclidean case. Set, for every λ > 0, F (λ) = 1 n− 1 ∫ 1 (λ+ fq)n−1 dv. Note first that F is well defined and continuously differentiable in λ. Indeed, by Fubini’s theorem, for every λ > 0,

Small Scale Limit Theorems for the Intersection Local Times of Brownian Motion

In this paper we contribute to the investigation of the fractal nature of the intersection local time measure on the intersection of independent Brownian paths. We particularly point out the difference in the small scale behaviour of the intersection local times in three-dimensional space and in the plane by studying almost sure limit theorems motivated by the notion of average densities introduced by Bedford and Fisher. We show that in 3-space the intersection local time measure of two paths has an average density of order two with respect to the gauge function $\varphi(r)=r$, but in the plane, for the intersection local time measure of p Brownian paths, the average density of order two fails to converge. The average density of order three, however, exists for the gauge function $\varphi_p(r)=r^2[\log(1/r)]^p$. We also prove refined versions of the above results, which describe more precisely the fluctuations of the volume of small balls around these gauge functions by identifying the density distributions, or lacunarity distributions, of the intersection local times.

Influence of mass and volume of ruminal contents on voluntary intake and digesta passage of a forage diet in steers.

To assess the influence of volume and mass of ruminal contents on voluntary intake and related variables, five ruminally cannulated steers (550 kg) were fed a low-quality forage diet (43.1% ADF, 8.1% CP) in a 5 x 5 Latin square experiment. Mass and volume of ruminal contents were altered by adding varying numbers and weights of filled tennis balls (6.7-cm diameter) to the rumen immediately before the initiation of each experimental period. Treatments consisted of 0 balls (control), 50 balls with a 1.1 specific gravity (SG), 100 balls with a 1.1 SG, 50 balls with a 1.3 SG, and 100 balls with a SG of 1.3. The total volume of balls was 7.25 and 14.5 L for 50 and 100 balls, respectively. The total weight of balls was 8.5 and 17 kg for 50 and 100 balls with a 1.1 SG and 10.75 and 21.5 kg for 50 and 100 balls with a 1.3 SG, respectively. Daily DMI was 8.3, 7.3, 7.0, 6.5, and 6.0 kg for control; 50, 1.1 SG; 50, 1.3 SG; 100, 1.1 SG; and 100, 1.3 SG, respectively. Addition of balls to the rumen reduced (P < .01) DMI. Increasing the number (P < .01) and SG (P <. 01) of the balls decreased DMI further. However, digestibilities of DM, NDF, ADF, and CP were not influenced by treatment. Increasing the number of balls in the rumen increased (P < .05) rate of passage of digesta from the rumen, but increasing SG of the balls did not alter rate of passage. There was a treatment x hour interaction (P < .05) in the proportion of ruminal digesta with a functional specific gravity (FSG) less than 1.1, which decreased with time after feeding for the control but increased with time after feeding for other treatments. Ruminal passage rate of inert particles added in the rumen of different SG (1.1 and 1.3) and length (1 and 3 mm) decreased (P < .05) as SG of the balls increased. Mean fecal particle size was greater for those treatments with the heavier balls. Both the number and SG of balls (P < .10) influenced total VFA, and total concentrations were greater for the control and for the 1.1 SG than for the 1.3 SG treatments.

The subindependence of coordinate slabs inl p n balls

It is proved that if the probabilityP is normalised Lebesgue measure on one of thelpn balls in Rn, then for any sequencet1, t2, …, tnof positive numbers, the coordinate slabs {|xi|≤ti} are subindependent, namely,\(P\left( {\mathop \cap \limits_1^n \{ \left| {x_i } \right| \leqslant t_i \} } \right) \leqslant \prod\limits_1^n {P(\{ \left| {x_i } \right| \leqslant t_i \} )} \). A consequence of this result is that the proportion of the volume of thel1n ball which is inside the cube[−1, t]n is less than or equal tofn(t)=(1−(1−t)n)n. It turns out that this estimate is remarkably accurate over most of the range of values oft. A reverse inequality, demonstrating this, is the second major result of the article.

Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality for a 2-dimensional minimal surface in the n-dimensional space form of nonpositive constant curvature κ under the assumptions that lies in the exterior of a convex domain and contains a subset Γ which is contained in and along which meets perpendicularly and that is connected, or more generally radially-connected from a point in Γ. Also we obtain an optimal version of linear isoperimetric inequalities for minimal submanifolds in a simply connected Riemannian manifolds with sectional curvatures bounded above by a nonpositive number. Moreover, we show the monotonicity property for the volume of a geodesic ball in such minimal submanifolds. We emphasize that in all the results of this paper minimal submanifolds need not be area minimizing or even stable.

The disappearing liquid

A one-liter (Pyrex) volumetric flask is modified by replacing the neck with a narrower tube (wide enough to permit insertion of a magnetic stirring bar) about 35 cm long, and adding a septum adapter to the side of the base. The procedure is as follows:1. A magnetic stirring bar and 500 mL of 95% ethanol (which can be colored with methylene blue to make it more visible) are added to the flask.2. Water which has been similarly colored is added to a bulb to which are attached a tube and hypodermic needle. Air is removed from the tubing and the needle is inserted into the magic flask via the septum. The water, being of higher density than the alcohol, will form a layer under the alcohol without mixing if care is taken.3. When the alcohol has reached the top of the neck the needle is gently removed.4. The flask is placed on a magnetic stirrer.5. When the magnetic stirrer is turned on, the water and alcohol mix; a decrease in volume occurs and the liquid level in the neck drops sharply - from 15 to 25 cm, depending on the care taken in filling the flask.It is possible to perform the experiment without modifying the volumetric flask. This can be done by adding 500 - 600 mL of 95% ethanol to the volumetric flask, in which a stirring bar has been placed. Then:1. As above.2. Slowly add water (also colored) to the bottom of the flask by means of a funnel attached by a rubber tube to a long glass tube (a 10-mL pipet would be suitable). This can take about 15 min.3. When the volumetric flask has been filled to the top, the flow is stopped with a pinch clamp and the tube is gently removed.4. The void left at the neck of the flask is completely filled with ethanol.5. As above. The decrease in height is 8-10 cm.Two questions have to be answered: (i) Where did the liquid in the neck go? (ii) Why do bubbles form on mixing?The answer to the first question lies in the partial molal volumes of alcohol and water, or the fact that the density of the mixture is greater than the average density of the two pure liquids. This can be demonstrated by using a 2-foot cylinder (4 in. i.d.) half filled with 1/4-in. styrofoam balls, on which are layered 1-in. balls to fill the cylinder to the top. Mixing the balls results in a clear decrease in the total volume of balls in the cylinder.The answer to the second question is that the solubility of air in the two liquids is significantly different, and on mixing the air is displaced owing to a change in its activity.Caution: Ethanol is extremely flammable. Care must be taken to ensure that no sparks or flames are near the demonstration.

The volume of geodesic balls and tubes about totally geodesic submanifolds in compact symmetric spaces

Let M be a compact Riemannian symmetric space. We give an analytical expression for the area and volume functions of geodesic balls in M and for the area and volume functions of tubes around some totally geodesic submanifolds P of M. We plot the graphs of these functions for some compact irreducible Riemannian symmetric spaces of rank two.

Nonlinear parabolic problems on manifolds, and a nonexistence result for the noncompact Yamabe problem

We study the Cauchy problem for the semilinear parabolic equations Δ u − R u + u p − u t = 0 \Delta u - R u + u^{p} - u_{t} =0 on M n × ( 0 , ∞ ) \mathbf {M}^{n} \times (0, \infty ) with initial value u 0 ≥ 0 u_{0} \ge 0 , where M n \mathbf {M}^{n} is a Riemannian manifold including the ones with nonnegative Ricci curvature. In the Euclidean case and when R = 0 R=0 , it is well known that 1 + 2 n 1+ \frac {2}{n} is the critical exponent, i.e., if p &gt; 1 + 2 n p &gt; 1 + \frac {2}{n} and u 0 u_{0} is smaller than a small Gaussian, then the Cauchy problem has global positive solutions, and if p &gt; 1 + 2 n p&gt;1+\frac {2}{n} , then all positive solutions blow up in finite time. In this paper, we show that on certain Riemannian manifolds, the above equation with certain conditions on R R also has a critical exponent. More importantly, we reveal an explicit relation between the size of the critical exponent and geometric properties such as the growth rate of geodesic balls. To achieve the results we introduce a new estimate for related heat kernels. As an application, we show that the well-known noncompact Yamabe problem (of prescribing constant positive scalar curvature) on a manifold with nonnegative Ricci curvature cannot be solved if the existing scalar curvature decays “too fast” and the volume of geodesic balls does not increase “fast enough”. We also find some complete manifolds with positive scalar curvature, which are conformal to complete manifolds with positive constant and with zero scalar curvatures. This is a new phenomenon which does not happen in the compact case.

State of residual stress induced by cyclic rolling contact loading

The accumulation of plastic microdeformation during cyclic stressing under rolling contact loading conditions results in a complex state of residual stress in the subsurface volume of deep groove ball bearing inner rings. The three-dimensional state of residual stress was analysed experimentally for the first time, employing a combination of methods based on X-ray diffraction. The residual stress field exhibited components of internal stress which were maximal close to the depth at which stresses induced by the external load were at a maximum. From this position the values of the principal components of residual stress decreased with steep gradients towards the surface and to a larger depth. Compressive residual stresses existed in the circumferential and axial directions while in the surface normal direction a tensile residual stress was present. The trends observed by the diffraction analysis were in qualitative agreement with measurements of ring distortion performed upon sectioning.

State of residual stress induced by cyclic rolling contact loading

The accumulation of plastic microdeformation during cyclic stressing under rolling contact loading conditions results in a complex state of residual stress in the subsurface volume of deep groove ball bearing inner rings. The three-dimensional state of residual stress was analysed experimentally for the first time, employing a combination of methods based on X-ray diffraction. The residual stress field exhibited components of internal stress which were maximal close to the depth at which stresses induced by the external load were at a maximum. From this position the values of the principal components of residual stress decreased with steep gradients towards the surface and to a larger depth. Compressive residual stresses existed in the circumferential and axial directions while in the surface normal direction a tensile residual stress was present. The trends observed by the diffraction analysis were in qualitative agreement with measurements of ring distortion performed upon sectioning.