Collisions of Extended Bodies in Two Dimensions

Collisions of Extended Bodies in Two Dimensions tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về...

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 Extended Bodies in Two Dimensions Collisions of Extended Bodies in Two Dimensions Bởi: OpenStaxCollege Bowling pins are sent flying and spinning when hit by a bowling ball—angular momentum as well as linear momentum and energy have been imparted to the pins (See [link]) Many collisions involve angular momentum Cars, for example, may spin and collide on ice or a wet surface Baseball pitchers throw curves by putting spin on the baseball A tennis player can put a lot of top spin on the tennis ball which causes it to dive down onto the court once it crosses the net We now take a brief look at what happens when objects that can rotate collide Consider the relatively simple collision shown in [link], in which a disk strikes and adheres to an initially motionless stick nailed at one end to a frictionless surface After the collision, the two rotate about the nail There is an unbalanced external force on the system at the nail This force exerts no torque because its lever arm r is zero Angular momentum is therefore conserved in the collision Kinetic energy is not conserved, because the collision is inelastic It is possible that momentum is not conserved either because the force at the nail may have a component in the direction of the disk’s initial velocity Let us examine a case of rotation in a collision in [link] The bowling ball causes the pins to fly, some of them spinning violently (credit: Tinou Bao, Flickr) 1/9 Collisions of Extended Bodies in Two Dimensions (a) A disk slides toward a motionless stick on a frictionless surface (b) The disk hits the stick at one end and adheres to it, and they rotate together, pivoting around the nail Angular momentum is conserved for this inelastic collision because the surface is frictionless and the unbalanced external force at the nail exerts no torque Rotation in a Collision Suppose the disk in [link] has a mass of 50.0 g and an initial velocity of 30.0 m/s when it strikes the stick that is 1.20 m long and 2.00 kg (a) What is the angular velocity of the two after the collision? (b) What is the kinetic energy before and after the collision? (c) What is the total linear momentum before and after the collision? Strategy for (a) We can answer the first question using conservation of angular momentum as noted Because angular momentum is Iω, we can solve for angular velocity Solution for (a) Conservation of angular momentum states L=L′, where primed quantities stand for conditions after the collision and both momenta are calculated relative to the pivot point The initial angular momentum of the system of stick-disk is that of the disk just before it strikes the stick That is, L = Iω, where I is the moment of inertia of the disk and ω is its angular velocity around the pivot point Now, I = mr2 (taking the disk to be approximately a point mass) and ω = v / r, so that v L = mr2 r = mvr 2/9 Collisions of Extended Bodies in Two Dimensions After the collision, L ′ = I′ω′ It is ω ′ that we wish to find Conservation of angular momentum gives I ′ ω ′ = mvr Rearranging the equation yields ω′ = mvr I′ , where I ′ is the moment of inertia of the stick and disk stuck together, which is the sum of their individual moments of inertia about the nail [link] gives the formula for a rod rotating around one end to be I = Mr2 / Thus, I′ = mr2 + Mr2 ( = m+ M )r2 Entering known values in this equation yields, I′ = (0.0500 kg+0.667 kg)(1.20 m) = 1.032 kg ⋅ m2 The value of I ′ is now entered into the expression for ω ′ , which yields ω′ (0.0500 kg)(30.0 m/s)(1.20 m) = mvr I′ = 1.744 rad/s ≈ 1.74 rad/s = 1.032 kg ⋅ m2 Strategy for (b) The kinetic energy before the collision is the incoming disk’s translational kinetic energy, and after the collision, it is the rotational kinetic energy of the two stuck together Solution for (b) First, we calculate the translational kinetic energy by entering given values for the mass and speed of the incoming disk KE = mv2 = (0.500)(0.0500 kg)(30.0 m/s) = 22.5 J 3/9 Collisions of Extended Bodies in Two Dimensions After the collision, the rotational kinetic energy can be found because we now know the final angular velocity and the final moment of inertia Thus, entering the values into the rotational kinetic energy equation gives KE′ ( ′ ω ′ = (0.5)(1.032 kg ⋅ m2) 1.744 = 2I = 1.57 J rad s ) Strategy for (c) The linear momentum before the collision is that of the disk After the collision, it is the sum of the disk’s momentum and that of the center of mass of the stick Solution of (c) Before the collision, then, linear momentum is p = mv = (0.0500 kg)(30.0 m/s) = 1.50 kg⋅m/s After the collision, the disk and the stick’s center of mass move in the same direction The total linear momentum is that of the disk moving at a new velocity v ′ = rω ′ plus that of the stick’s center of mass, which moves at half this speed because vCM = p ′ = mv ′ + MvCM =mv ′ + Mv′ ( 2r )ω ′ = v′ Thus, Gathering similar terms in the equation yields, ( p′ = m+ M )v ′ M )rω ′ so that ...© 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 Dynamin-like protein-dependent formation of Woronin bodies in Saccharomyces cerevisiae upon heterologous expression of a single protein Christian Wu ¨ rtz, Wolfgang Schliebs, Ralf Erdmann and Hanspeter Rottensteiner Institut fu ¨ r Physiologische Chemie, Ruhr-Universita ¨ t Bochum, Germany The HEX1 protein of Neurospora crassa, identified by Jedd and Chua [1] and Tenney et al. [2], is the major component of a class of microbodies limited to euasco- mycetes and some deuteromycetes, the so-called Woro- nin body [3,4]. Because of the syncytial growth of filamentous fungi, wounding of hyphae can lead to a severe loss of cytoplasm and subcellular organelles, if the plasma membrane or a nearby septum is not rap- idly sealed. For this reason, the Woronin body is pres- ent in filamentous euascomycetes and plugs septal pores immediately after cells have been damaged [1,2]. In addition to septal pore sealing in cases of injury, Woronin bodies have also been described as being required for efficient pathogenesis, survival during nitrogen starvation [5] and conidiation [6] in various fungi. Although it is more than 140 years since the dis- covery of this very specialized organelle [4], our knowl- edge of the biogenesis of Woronin bodies remains incomplete. Electron microscopy studies provided the first evidence that Woronin bodies are derived from other microbodies [7]. These findings have been extended by reports showing that Woronin body formation is initiated in the vicinity of glyoxysomes and may proceed through fission from them [8], and by the demonstration that PEX14 is a key player in the biogenesis of both glyoxysomes and Woronin bodies [9]. Furthermore, the presence of a C-terminal canonical peroxisomal targeting signal type 1 (PTS1) is required for the proper topogenesis of HEX1 [9] and allows HEX1 to be imported into peroxisomes upon heterologous expression in yeast [1]. Peroxisomal Keywords filamentous fungi; Neurospora crassa; peroxisome; protein import; yeast Correspondence H. Rottensteiner, Institut fu ¨ r Physiologische Chemie, Abt. Systembiochemie, Ruhr- Universita ¨ t Bochum, D-44780 Bochum, Germany Fax: +49 234 321 4266 Tel: +49 234 322 7046 E-mail: hanspeter.rottensteiner@rub.de (Received 22 January 2008, revised 27 March 2008, accepted 2 April 2008) doi:10.1111/j.1742-4658.2008.06430.x Filamentous ascomycetes harbor Woronin bodies and glyoxysomes, two types of microbodies, within one cell at the same time. The dominant pro- tein of the Neurospora crassa Woronin body, HEX1, forms a hexagonal core crystal via oligomerization and evidence has accumulated that Woro- nin bodies bud off from glyoxysomes. We analyzed whether HEX1 is suffi- cient to induce Woronin body formation upon heterologous expression in Saccharomyces cerevisiae, an organism devoid of this specialized organelle. In wild-type strain BY4742, initial import of HEX1 into existing peroxi- somes enabled the formation of organelles with a hexagonal crystal. The observed structures mimicked the shape of genuine Woronin bodies, but exhibited a lower density and were significantly larger. Double-immuno- fluorescence analysis revealed that hexagonal HEX1 structures only occa- sionally co-localized with peroxisomal marker proteins, indicating that the Woronin-body-like structures are well separated from peroxisomes. In cells lacking Vps1p and Dnm1p, dynamin-like proteins required for the division of peroxisomes, the Woronin-body-like organelles remained attached to peroxisomes. The data indicate that Woronin [...]... 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 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 ... [link] in which the disk originally spins clockwise at 1000 rpm and has a radius of 1.50 cm 7/9 Collisions of Extended Bodies in Two Dimensions Twin skaters approach one another as shown in [link]... kinetic energy by entering given values for the mass and speed of the incoming disk KE = mv2 = (0.500)(0.0500 kg)(30.0 m/s) = 22.5 J 3/9 Collisions of Extended Bodies in Two Dimensions After the... Repeat [link] in which the stick is free to have translational motion as well as rotational motion 8/9 Collisions of Extended Bodies in Two Dimensions (a) 1.70 rad/s (b) Initial KE = 22.5 J, final