Collisions of Point Masses in Two Dimensions

Annals of Mathematics Cover times for Brownian motionand random walks in two dimensions By Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni Annals of Mathematics, 160 (2004), 433–464 Cover times for Brownian motion and random walks in two dimensions By Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni* Abstract Let T (x, ε) denote the first hitting time of the disc of radius ε centered at x for Brownian motion on the two dimensional torus T 2 . We prove that sup x∈ T 2 T (x, ε)/|log ε| 2 → 2/π as ε → 0. The same applies to Brownian mo- tion on any smooth, compact connected, two-dimensional, Riemannian mani- fold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z 2 n is asymptotic to 4n 2 (log n) 2 /π. De- termining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied nonrigorously in the physics literature. We also es- tablish a conjecture, due to Kesten and R´ev´esz, that describes the asymptotics for the number of steps needed by simple random walk in Z 2 to cover the disc of radius n. 1. Introduction In this paper, we introduce a unified method for analyzing cover times for random walks and Brownian motion in two dimensions, and resolve several open problems in this area. 1.1. Covering the discrete torus. The time it takes a random walk to cover a finite graph is a parameter that has been studied intensively by probabilists, combinatorialists and computer scientists, due to its intrinsic appeal and its applications to designing universal traversal sequences [5], [10], [11], testing graph connectivity [5], [19], and protocol testing [24]; see [2] for an introduction *The research of A. Dembo was partially supported by NSF grant #DMS-0072331. The research of Y. Peres was partially supported by NSF grant #DMS-9803597. The research of J. Rosen was supported, in part, by grants from the NSF and from PSC-CUNY. The research of all authors was supported, in part, by a US-Israel BSF grant. 434 AMIR DEMBO, YUVAL PERES, JAY ROSEN, AND OFER ZEITOUNI to cover times. Aldous and Fill [4, Chap. 7] consider the cover time for random walk on the discrete d-dimensional torus Z d n = Z d /nZ d , and write: ‘‘Perhaps surprisingly, the case d =2turns out to be the hardest of all explicit graphs for the purpose of estimating cover times.” The problem of determining the expected cover time T n for Z 2 n was posed informally by Wilf [29] who called it “the white screen problem” and wrote “Any mathematician will want to know how long, on the average, it takes until each pixel is visited.” (see also [4, p. 1]). In 1989, Aldous [1] conjectured that T n /(n log n) 2 → 4/π. He noted that the upper bound T n /(n log n) 2 ≤ 4/π + o(1) was easy, and pointed out the dif- ficulty of obtaining a corresponding lower bound. A lower bound of the correct order of magnitude was obtained by Zuckerman [30], and in 1991, Aldous [3] showed that T n /E(T n ) → 1 in probability. The best lower bound prior to the present work is due to Lawler [21], who showed that lim inf E(T n )/(n log n) 2 ≥ 2/π. Our main result in the discrete setting, is the proof of Aldous’s conjecture: Theorem 1.1. If T n denotes the time it takes for the simple random walk in Z 2 n to cover Z 2 n completely, then lim n→∞ T n (n log n) 2 = 4 π in probability.(1.1) The main interest in this result is not the value of the constant, but rather that establishing a limit theorem, with Collisions of Point Masses in Two Dimensions Collisions of Point Masses in Two Dimensions Bởi: OpenStaxCollege In the previous two sections, we considered only one-dimensional collisions; during such collisions, the incoming and outgoing velocities are all along the same line But what about collisions, such as those between billiard balls, in which objects scatter to the side? These are two-dimensional collisions, and we shall see that their study is an extension of the one-dimensional analysis already presented The approach taken (similar to the approach in discussing two-dimensional kinematics and dynamics) is to choose a convenient coordinate system and resolve the motion into components along perpendicular axes Resolving the motion yields a pair of one-dimensional problems to be solved simultaneously One complication arising in two-dimensional collisions is that the objects might rotate before or after their collision For example, if two ice skaters hook arms as they pass by one another, they will spin in circles We will not consider such rotation until later, and so for now we arrange things so that no rotation is possible To avoid rotation, we consider only the scattering of point masses—that is, structureless particles that cannot rotate or spin We start by assuming that Fnet = 0, so that momentum p is conserved The simplest collision is one in which one of the particles is initially at rest (See [link].) The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in [link] Because momentum is conserved, the components of momentum along the x- and y-axes (px and py) will also be conserved, but with the chosen coordinate system, py is initially zero and px is the momentum of the incoming particle Both facts simplify the analysis (Even with the simplifying assumptions of point masses, one particle initially at rest, and a convenient coordinate system, we still gain new insights into nature from the analysis of two-dimensional collisions.) 1/10 Collisions of Point Masses in Two Dimensions A two-dimensional collision with the coordinate system chosen so that m2 is initially at rest and v1 is parallel to the x -axis This coordinate system is sometimes called the laboratory coordinate system, because many scattering experiments have a target that is stationary in the laboratory, while particles are scattered from it to determine the particles that make-up the target and how they are bound together The particles may not be observed directly, but their initial and final velocities are Along the x-axis, the equation for conservation of momentum is p1x + p2x = children in msubsup + children in msubsup Where the subscripts denote the particles and axes and the primes denote the situation after the collision In terms of masses and velocities, this equation is m1v1x + m2v2x = m12 children in msubsup + m22 children in msubsup But because particle is initially at rest, this equation becomes m1v1x = m12 children in msubsup + m22 children in msubsup The components of the velocities along the x-axis have the form v cos θ Because particle initially moves along the x-axis, we find v1x = v1 Conservation of momentum along the x-axis gives the following equation: m1v1 = m12 children in msubsup cos θ1 + m22 children in msubsup cos θ2, where θ1 and θ2 are as shown in [link] Conservation of Momentum along the x-axis m1v1 = m12 children in msubsup cos θ1 + m22 children in msubsup cos θ2 2/10 Collisions of Point Masses in Two Dimensions Along the y-axis, the equation for conservation of momentum is p1y + p2y = children in msubsup + children in msubsup or m1v1y + m2v2y = m12 children in msubsup + m22 children in msubsup But v1y is zero, because particle initially moves along the x-axis Because particle is initially at rest, v2y is also zero The equation for conservation of momentum along the y -axis becomes = m12 children in msubsup + m22 children in msubsup The components of the velocities along the y-axis have the form v sin θ Thus, conservation of momentum along the y-axis gives the following equation: = m12 children in msubsup sin θ1 + m22 children in msubsup sin θ2 Conservation of Momentum along the y-axis = m12 children in msubsup sin θ1 + m22 children in msubsup sin θ2 The equations of conservation of momentum along the x-axis and y-axis are very useful in analyzing two-dimensional collisions of particles, where one is originally stationary (a common laboratory situation) But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level Determining the Final Velocity of an Unseen Object from the Scattering of Another Object Suppose the following experiment is performed A 0.250-kg object (m1) is slid on a frictionless surface into a dark room, where it strikes an initially stationary object with mass of 0.400 kg (m2) The 0.250-kg object emerges from the room at ...© JAPI • VOL. 54 • JULY 2006 www.japi.org 539 Original Article Health Problems and Disability of Elderly Individuals in Two Population Groups from Same Geographical Location GK Medhi*, NC Hazarika**, PK Borah*, J Mahanta*** Abstract Objective : To compare morbidity, disability (ADL-IADL disability) along with behavioral and biological correlates of diseases and disability of two elderly population groups (tea garden workers and urban dwellers) living in same geographical location. Methods : Two hundred and ninety three and 230 elderly from urban setting and tea garden respectively aged > 60 years were included in the study. Subjects were physical examined and activity of daily living instrumental activity of daily living (ADL-IADL) was assessed. Diagnosis of diseases was made on the basis of clinical evaluation, diagnosis and/or treatment of diseases done earlier elsewhere, available investigation reports, and electrocardiography. Hypertension was defined according to JNC-VI classification. BMI (weight/height 2 ) was calculated. Logistic regression analysis was performed to see the impact of important background characteristics on non-communicable diseases (NCD) and disability. Results : Hypertension (urban - 68% and tea garden - 81.4%), musculoskeletal diseases (urban - 62.5% and teagarden - 67.5%), COPD and other respiratory problems (urban - 30.4% and tea garden - 32.2%), cataract (urban 40.3% and tea garden - 33%), gastro-intestinal problems (urban - 13% and tea garden - 6.5%) were more commonly observed health problems among community dwellings elderly across both the groups. However in contrast to urban group, serious NCDs like Ischaemic Heart Disease (IHD), diabetes were yet to emerge as health problems among tea garden dwellers. Infectious morbidities, undernutrition and disability (ADL-IADL disability) were more pronounced among tea garden dwellers. Utilization of health service by tea garden elderly was very low in comparison to the urban elderly. Both tea garden men and women had very high rates of risk factors like use of non-smoked tobacco and consumption of alcohol. On the other hand, smoking and obesity was more common in urban group. Most morbidities and disabilities were associated with identifiable risk factors, such as obesity, tobacco (smoked and non-smoked) and alcohol consumption. Educational status was also found to be an important determinant of diseases and disability of elderly population. Age showed a J-shaped relationship with disability and morbidity. Sex difference in health status was also detected. Conclusion : This study highlights the physical dimension of health problems of elderly individuals. Social circumstances and health risk behaviours play important role in the variation of health and functional status between the two groups. Life-style modification is warranted to prevent onset of chronic diseases. To improve quality of life, rectification of poor health status through affordable health service for disease screening and better management of illness, nutritional improvement and greater health awareness are necessary particularly among low socio-economic group. Low-cost intervention like cataract surgery could make a difference in the quality of life of elderly Indian. © ahead for health care in Annals of Mathematics Cauchy transforms of point masses: The logarithmic derivative of polynomials By J. M. Anderson and V. Ya. Eiderman* Annals of Mathematics, 163 (2006), 1057–1076 Cauchy transforms of point masses: The logarithmic derivative of polynomials By J. M. Anderson and V. Ya. Eiderman* 1. Introduction For a polynomial Q N (z)= N  k=1 (z −z k ) of degree N, possibly with repeated roots, the logarithmic derivative is given by Q  N (z) Q(z) = N  k=1 1 z −z k . For fixed P>0 we define sets Z(Q N ,P) and X(Q N ,P)by Z(Q N ,P)=  z : z ∈ C,      N  k=1 1 z −z k      >P  , X(Q N ,P)=  z : z ∈ C, N  k=1 1 |z −z k | >P  . (1.1) Clearly Z(Q N ,P) ⊂X(Q N ,P). Let D(z,r) denote the disk {ζ : ζ ∈ C, |ζ − z| <r}. In [2] it was shown that X(Q N ,P) is contained in a set of disks D(w j ,r j ) with centres w j and radii r j such that  j r j < 2N P (1 + log N), *Research supported in part by the Russian Foundation of Basic Research (Grant no. 05-01-01021) and by the Royal Society short term study visit Programme no. 16241. The second author thanks University College, London for its kind hospitality during the preparation of this work. The first author was supported by the Leverhulme Trust (U.K.). 1058 J. M. ANDERSON AND V. YA. EIDERMAN or, as we prefer to state it, M(X(Q N ,P)) < 2N P (1 + log N).(1.2) Here M denotes 1-dimensional Hausdorff content defined by M(A) = inf  j r j , where the infimum is taken over all coverings of a bounded set A by disks with radii r j . The question of the sharpness of the bound in (1.2) was left open in [2]. We prove – Theorem 2.3 below – that the estimate (1.2) for X is essentially best possible. Obviously, (1.2) implies the same estimate for M(Z(Q N ,P)). It was sug- gested in [2] that in this case the (1+ log N) term could be omitted at the cost of multiplying by a constant. The above suggestion means that in the passage from the sum of moduli to the modulus of the sum in (1.1) essential cancella- tion should take place. As a contribution towards this end the authors showed that any straight line L intersects Z(Q N ,P) in a set F P of linear measure less than 2eP −1 N. Further information about the complement of F P under certain conditions on {z k } is obtained in [1]. Clearly we may assume that N>1 and we do so in what follows, for ease of notation. However, it was shown in [3] that there is an absolute positive constant c such that for all N  3 one can find a polynomial Q N of degree N for which the projection Π of Z(Q N ,P) onto the real axis has measure greater than c P N(log N) 1 2 (log log N) − 1 2 ,N 3.(1.3) Throughout this paper c will denote an absolute positive constant, not neces- sarily the same at each occurrence. Marstand suggested in [3] that the best result for M (Z(Q N ,P)) would be obtained by omitting the log log -term in (1.3). It is the object of this paper to show that this is indeed the case and that the corresponding result is then, apart from a constant best possible (The- orems 2.1 and 2.2 below). Thus the cancellation mentioned above does indeed occur but in general it is not as “strong” as was suggested in [2]. 2. Results We prove Theorem 2.1. Let z k ,1 k  N , N>1, be given points in C. There is an absolute constant c such that for every P>0 there exists a set of disks D j = D(w j ,r j ) so that      N  k=1 1 z −z k      <P, z∈ C\  j D j (2.1) CAUCHY TRANSFORMS OF POINT MASSES 1059 and  j r j < c P N(log N) 1 2 . In other words M(Z(Q N ,P)) < c P N(log [...]... flow inversion Here the main expenses of separation are caused by liquid and gas flow circulation and heating (cooling) The physico-chemical and engineering bases of production of the isotopes of the elements mentioned above in counter-flow columns are considered in this book The theory of isotope separation in such columns is sufficiently explained in several monographs So, in chapter 1 only information... –1– Theory of Isotope Separation in CounterCurrent Columns: Review 1.1 SEPARATION FACTOR Isotope separation in two-phase systems is based on the thermodynamic isotope effect (TDIE), the value of which is conventionally determined by the separation factor of a binary isotopic mixture, α, representing the ratio of the relative concentration of isotopes in two different substances or phases in equilibrium:... hydride phases of palladium and inter-metallic compounds, as well as by phase isotope exchange in sorption systems (first of all, with zeolites) At present, no less important are problems of separation of isotopes of the other biogenic elements such as carbon, nitrogen, and oxygen Heavy stable isotopes of these elements, 13C, 15 N, 17O, and 18O, are indispensable when studying metabolic processes in humans... determined in terms of the maximum extraction degree (Γm) representing an extraction degree for infinite number of theoretical separation plates (NTP), and relative withdrawal θ If a separation column is infinitely high, isotopic concentrations at its upper end will be interrelated by the separation factor α (see Figure 1.4a), i.e the operating line lower point (with xF ordinate) will be resting on... the greatest interest are the CHEX reactions in gas–liquid systems A distinguishing feature of the kinetics study in such systems is that, unlike systems with a solid phase, the surface of phase contact here is not strictly fixed Moreover, to eliminate the influence of diffusion processes in the contacting phases on chemical kinetics, it is necessary to intensively mix the phases, which is generally... isotope separation is given in chapter 5; nitrogen in chapter 6; and oxygen in chapter 7 In each chapter the thermodynamic isotope effects in two-phase systems are considered: the mass exchange, the main methods of heavy stable isotope enrichment by rectification and chemical exchange, production of light isotopes of carbon, nitrogen, and oxygen, and perspective processes of separation of these isotopes. .. addend that is mainly responsible for the departure of the scale-up factor (SF), allowing for HTU (HETP) increase in packed columns of greater diameters, from one; that is, such an approach assumes the absence of influences of the real structure of flows in the separation column on hIE In the above examination we did not dwell on the βIE value calculations and on possible influence of βIE on the diffusion... is used in subsequent chapters, is given Besides, in chapter 1 the hydrodynamic features of small packing, used as contact devices in columns for isotope separation RESEARC H Open Access Common coupled coincidence and coupled fixed point results in two generalized metric spaces Wasfi Shatanawi 1* , Mujahid Abbas 2 and Talat Nazir 2 * Correspondence: swasfi@hu.edu. jo 1 Department of Mathematics, The Hashemite University, Zarqa 13115, Jordan Full list of author information is available at the end of the article Abstract In this article, we prove the existence of common coupled coincidence and coupled fixed point of generalized contractive type mappings in the context of two generalized metric spaces. These results generalize several comparable results from the current literature. We also provide illustrative examples in support of our new results. 2000 MSC: 47H10. Keywords: coupled coincidence point, common coupled fixed point, weakly compa- tible maps, generalized metric space 1 Introduction and preliminaries The study of common fixed points of mappings satisfying certain contractive condi- tions has been at the center of rigorous research activity [1-5]. Mustafa and Sims [4] generalized the concept of a metric space and call it a generalized metric space. Based on the notion of generalized metric spaces, Mustafa et al. [5-9] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [10] initiated the study of common fixed point theory in generalized metric spaces (see also [11]). Saadati et al. [12] proved some fixed p oint results for contractive mappings in partially ordered G-metric spaces. Abbas et al. [13] obtained some periodic point results in generalized metric spaces. Shatanawi [14] obtained some fixed point results for contractive mappings satisfying F-maps in G-metric spaces (see also [15]). Bhashkar and Lakshmikantham [16] introdu ced the concept of a coupled fixed point of a mapping F : X × X ® X (a nonempty set) and established some coupled fixed point theorems in partially ordered complete metric spaces. Later, Lakshmikantham and Ćirić [3] proved coupled coincidence and coupled common fixed point results for nonlinear mappings F : X × X ® X and g : X ® X satisfying certain contractive condi- tions in partially ordered complete metric spaces. Recently, Abbas et a l. [17] obtained some coupled common fixed point results in two generalized metric spaces. Choudh- uryandMaity[18]alsoprovedtheexistence of coupled fixed points in generalized metric spaces. Recently, Aydi et al. [19] generalized the results of Choudhury and Maity [18]. For other works on G-metric spaces, we refer the reader to [20,21]. The aim of this article is t o prove some common coupled coincidence and coupled fixed points results for mappings defined on a set equipped with two generalized Shatanawi et al. Fixed Point Theory and Applications 2011, 2011:80 http://www.fixedpointtheoryandapplications.com/content/2011/1/80 © 2011 Shatanawi et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.o rg/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly ... 0.7485 ) Thus, children in msubsup = 0.886 m/s 4/10 Collisions of Point Masses in Two Dimensions Discussion It is instructive to calculate the internal kinetic energy of this two- object system before... = v ′ 12 sin2 θ1 + v ′ 22 sin2 θ2 + 2v ′ 1v ′ sin θ1sin θ2 Add these two equations and simplify: 9/10 Collisions of Point Masses in Two Dimensions v12 = v ′ 12 + v ′ 22 + 21 children in msup1... kinetic energy of the helium nucleus? 8/10 Collisions of Point Masses in Two Dimensions (a) 5.36 × 105 m/s at −29.5º (b) 7.52 × 10 −13 J Professional Application Two cars collide at an icy intersection