A Possible Solution to the Cosmological Constant Problem By Discrete Space-time Hypothesis
|
|
- Dorothy Gardner
- 6 years ago
- Views:
Transcription
1 A Possible Solution to te Cosological Constant Proble By Discrete Space-tie Hypotesis H.M.Mok Radiation Healt Unit, 3/F., Saiwano Healt Centre, Hong Kong SAR Got, 28 Tai Hong St., Saiwano, Hong Kong, Cina. Abstract Te cosological constant proble is explained by a teory based on te discrete space-tie ypotesis. Te calculated λalue is of te order of [ ] or equialent to about Ω λ =0.. It is in excellent agreeent wit te Type Ia SN obserational data and recent results of BOOMERANG and MAXIMA. Our teory also iplies tat te quantization of te space-tie etric g µν is not necessary since it is not a fundaental field. Te diergence proble of quantu graity is ten of no interest. Cosic inflation is gien out as a consequence of te teory and te unierse is found to be alternatiely doinated by te cosological constant and te ass density at different cosic tie period. Our calculation also sows tat ρ is of siilar order of agnitude as ρ in te present unierse but it is just a coincidence. Tis result supports te antropic principle. PACS nubers: k, Cq, Te international collaboration on te Hig-Z SN Ia obseration, wic was aied at easuring te cosic deceleration and global curature, found tat te unierse is accelerating instead of decelerating. Te obserational results fro bot teas, Perlutter [1] and Scidt [2], indicated tat tere is a non-anising cosological constant ( λ) in 2 2 our unierse. Te alue of Ω ( λc / 3H ) is a few tents of te critical ass density and, Λ 0
2 wen copared wit te Planck scale or electroweak scale, is any order of agnitudes saller tan tat expected in quantu field teory. Recent obserations by te BOOMERANG and MAXIMA also support suc finding [3]. Tis cosological constant proble is one of te ysteries of bot cosology and particle field teory. Viable approac, suc as quintessence, antropic principle and iger diensional brane world solution are under inestigation (Recent concise coents on suc approaces can be found on [4]). Howeer, up to now, tere is still no satisfactory prediction on te λalue using suc attepts. On te oter and, te teoretical estiate of te λ alue by te quantu field teory is based on te spontaneous syetry breaking process proided by te Higgs ecanis []. Altoug suc teory is successful in ost of te particle experients, it is an ad oc ecanis and lacks detail understanding. For exaple, te reason of existence of a scalar field in te acuu is still not known. Te present situation on te λ proble urges us to consider bot probles togeter in a new direction. Altoug it is strange tat tere exists so large discrepancy between te teoretical and experiental λ alue, we ay not be too unfailiar wit suc situation. If we copare te λ proble wit te property of ass density, siilar caracteristic can be found. Due to te atoic structure of atter, ass density is relatiely sall in acroscopic scale (around kg / for coon aterials) but extreely large inside te atoic nucleus (around kg / ). Te case for te cosological constant, also, is ery sall in acroscopic scale but extreely large wen quantu field consideration (i.e. icroscopic scale) as been put in. If te λ proble is analogous to te case of atter density, it indicates tat soe kind of discrete structure ay exist in te acuu. Since te acuu and space-tie are indistinguisable, suc discrete acuu properties ay furter iply tat te space-tie itself is also discrete in nature. Besides, tere is no eidence tat space-tie is soot and continuous in extree icroscopic scale. In quantu graity, space-tie is expected to ae a ery different geoetrical structure in Planck scale, suc as te existence of woroles and space-tie foas. Also, te concept of discrete space-tie is not new. T.D.Lee [6], G.t Hooft [] and oters [8] ad considered suc possibility in resoling te UV/IR diergence proble in quantu graity but tey ad not related it to te proble of te cosological constant at tat tie.
3 If space-tie is really discrete in nature, te cosological constant described by te quantu field teory can be just te situation inside te basic constituents of space-tie and is any order of agnitudes larger tan te acroscopic obserational data. Te quantu field teory description and te cosological obserations of te acuu energy density ay ten bot be correct on its corresponding lengt scales. Furterore, te spontaneous syetry breaking of te scalar field ay be corresponding to te pase transition of suc space-tie condensate like structure. Let us sow te aboe idea in a paradig. If we iagine our 3-D space as a 2-D elastic ebrane, wic is coonly use for te illustration of cosic expansion, te discrete space-tie is corresponding to a ebrane wic is not soot but as its atoic structure (in fact, a pysical ebrane is ade of atos). Te graitational field for te 2-D creatures liing on te ebrane is te properties of deforing te ebrane by ass and general relatiity is ten a kind of continuous teory of elasticity. Tey ay find tat te space is soot in acroscopic scale but as its icroscopic structure. Also, te internal energy of te ebrane (It ay be iewed as te acuu energy by te 2- D creatures.) is extreely large in icroscopic scale (i.e. te nuclear energy or te atoic bonding energy) but sall in acroscopic scale and suc internal energy of is space does not cure te space-tie structure. Tis siple odel gies te properties of our cosological constant. Based on te aboe arguents, we postulate tat : (1) Space-tie is discrete in nature and its fundaental unit is of te order of Planck scale; (2) Te space-tie fors a kind of pase (or say condensate ) wit its constituents; (3) Te scalar field plays bot te role as te order paraeter of suc space-tie pase and te waefunction of its constituents as Cooper pairs in superconductiity [9,10] (We ae to reark tat suc condensate ay not be exactly te sae as in usual understanding but as siilar properties tat is useful to draw analogy between te). Soe iportant consequences can be directly followed fro tese postulates. Firstly, te Higgs particle will be just a kind of excited state of te indiidual space-tie constituent. Secondly, te diergence proble of te quantu graity is of no interest since te space-tie etric g µν is a
4 collectie effect of te space-tie constituents (like te strain tensor of elasticity of a ebrane) and is not a fundaental field itself. General Relatiity is also not a fundaental field teory but is just a collectie description. Using te language of te paradig described aboe, tere is no need to insist in quantizing te wae propagation of te strain tensor on a 2-D ebrane since te atoic lattice ibration is te one tat needed to be quantized (ponons). Back to our case, it is te field of te space-tie constituents (or say te scalar field) and its lattice like ibration energy tat need to be quantized. Terefore, te proble of quantu graity is reduced to te quantization of te ibration energy of te space-tie constituents and in soe sense we can say tat te quantization of Higgs field is already part of a quantu graity teory! It sees tat te concept in condensed atter pysics is useful in building te quantized odel of te space-tie structure. If we take te for of te scalar field potential as V = V g (1) µ φ φ + ( φ φ) were µ 2 > 0, g > 0 and assuing tat V 0 (Tey as its usual eaning as in te 0 = electroweak teory. SM Higgs is assued ere for siplicity and also to aoid unnecessary coplexity). Te energy density at broken syetry ρ (wic will be sown to be different fro te acroscopic acuu energy density in cosological obseration) of te scalar field is gien by 4 ρ = [], were is te electroweak ass scale. Since te sape of te potential around te iniu φ 0 deterines te ass of te quantu particle and, in our case, it is equal to te Higgs ass, te energy density ρ expression can ten be interpreted as te contribution of te scalar field of ass wit nuber density icroscopic acuu energy density wit alue 3. Te negatie sign of ρ akes it beaes as te 4 (We use te sign conention tat +e acuu energy density corresponding to e noral energy density.). Also, because te scalar field is te waefunction of te space-tie constituents, ρ acts as a kind of
5 binding energy (or internal energy) of te constituents and, as entioned aboe, suc binding energy will not cure te space-tie since it beaes as te internal energy of te space-tie structure. If pase transition appens on te space-tie condensate, tis binding energy can be released as positie energy and is iportant in early unierse eolution tat will be discussed later. Due to te energy density constituents and wit te alue equals to ρ, pressure will be created on eac space-tie P = ρ = (2) 4 Te sign on te RHS of equation (2) sows tat te pressure is negatie (i.e. beae as a stretcing force on indiidual space-tie constituent but as a binding force between te space-tie constituents.). Tis alue is wat we expect fro quantu field teory []. Howeer, as discussed aboe, te space-tie is postulated to be discrete and its fundaental unit is of te order of Planck scale wit estiated density of about 3. Terefore, te acroscopic acuu energy density sould be weiged by a factor of ( M 3 / ) to aerage it out on an ideally soot acroscopic space scale (just as aerage out te nuclear ass density to get te acroscopic ass density of atter by considering te atoic spacing), were M is te Planck ass. Te acroscopic acuu energy density ρ ν will becoe 3 4 ρ = = (3) 3 M M Te cosological constant is ten equal to λ = κρ ~ (4) M
6 (Tis expression was first appeared in te autor s 1999 e-print paper [11] in te attepts on te explanation of te cosological constant proble by te electroweak and Planck scale.). If we put to about 100 GeV, te alue of λ is λ ~ 10 [ GeV] = 10 [ ] () (we use GeV ~ 10 ). If we use a diensionless Hubble constant = 0. 6 (recent obserational results gie te Hubble constant in te range [12] ), our calculated cosological constant will be equialent to about 0.ρc ( ρc = 3H 2 0 / 8πG is te critical ass density wen λ = 0 ). It is in excellent agreeent wit te Type Ia SN obseration data. Besides te explanation of te present λ alue, tis teory as iportant iplications on te early deelopent of te unierse. We can first put equation (4) into a ore general for as λ = κρ ~ T (6) M were T is te VEV ass scale of te scalar field in different transition stage in te early unierse. Te λ alue for te 3 transition stages, te Planck stage, GUT stage and te electroweak stage were ten equal to 2 M, GUT / M and / M respectiely. If we beliee tat te ass energy density (including radiation and atter) of te unierse is cae fro te cange of te acroscopic acuu energy density, we expect tat at te oent just after te GUT and te electroweak transition te atter density were as and 3 GUT / M respectiely. One of te point reains uncertain and cannot be gien by tis teory is te ass energy density at Planck tie. If it was coparable to te acuu energy at tat tie, te λ effect would doinated te ass energy effect at about M s (estiated by Friedann odel) and te unierse would be fast cooled to GUT transition
7 teperature by te accelerated expansion due to te cosological constant. Tis ade te GUT transition stages cae earlier tan in te ot Friedann unierse (at 3 10 s ). λ = GUT After GUT transition, te ass energy density would be equal to / M 4 M and. Te unierse was ten doinated by ass and continuous cooling by te usual Friedann expansion. Assuing tat te unierse was radiation doinated and te relation between te cosic teperature and tie was as T 1/ 2 t (we neglect te cosological constant effect before it doinated te expansion for siplicity), we expect tat te unierse would be doinated by te cosological constant at te tie around s (Tis is estiated by te calculation tat te ass energy density is greater tan 42 acuu energy density by 28 order of agnitude at around 10 s and tis corresponding to te cange of teperature of about order of agnitude. Terefore, te λ would be doinated at s = 10 s.). Te unierse would ten be fast cooled by te accelerated expansion in a sort tie. At te tie just before te electroweak transition, te λ alue was ten about 40 order of agnitude larger tan te ass energy density and 91 order of agnitude larger tan te present λ alue. Suc uge cosological constant effect igt cause an extree large expansion of te unierse in tat period. Tis is wat we coonly called te inflation! Altoug it is also drien by a uge λ alue, te acceleration process is not due to te false acuu as oter inflation odels [9] but is an intrinsic property of te space-tie condensate. As in te aboe estiated tie of transition, we ae not consider te additional acceleration effect by te λ before it becoe doinated. It is ten reasonable to expect tat te inflation also occurred at around te tie order s. One can find tat 1/ 2 t ~ 1 λ at tat tie. Te inflation would be ended at te copletion of te electroweak transition. Tis eans tat it also cae earlier tan te expected tie of about s in te ot Friedann unierse. As in te usual inflation odel, te inflation also occurred in te period between te GUT and electroweak transition but te cosic tie was different.
8 After te electroweak transition, te ass energy becae 3 GUT / M and λ = / M. Te cosological constant was dropped to te present alue, te inflation ten stopped and te unierse becae atter doinated again. Wen te atter density of te unierse continuously decreased by te expansion of te unierse, te λ alue becoes doinate again as te present cosological obseration [1,2]. If te radiation doinated unierse ended at around s and enter te atter doinate stage, te dilution factor for te ass energy density due to te cosic expansion fro s contribute about GUT was about / M Te additional factor in te atter stage up to now Tat eans te estiated present ass energy density is = 10 / M = 10λ. It is pretty close to te obseration tat te ass energy density about te sae order as te acuu energy density in present unierse. Te estiation is a bit large and ay due to uncertainty in estiating te ending tie of te electroweak transition. It is because only alf an order of agnitude cange can cause suc deiation. If we beliee tat tere is no furter pase transition of te space-tie condensate in future, our unierse will ten doinate by λ alue foreer. Up to now, we know tat te unierse eolution stages starting fro Big Bang igt be alternatiely doinated by cosological constant and atter density in different transition stages and te cosological constant will win te process finally. Also, fro te ass energy density estiation aboe, it sees tat ρ ~ ρ in te present unierse ay be just a coincidence, not due to underlying pysical teory. Tis result supports te antropic principle []. Fro te aboe teory, we find tat te cosological constant proble can be resoled by postulating tat space-tie is discrete in nature wit its pase transition properties described by te scalar field. Te cosological constant calculated by our teory is in excellent agreeent wit te Type Ia SN obseration data. Since te λ alue is dependent on te VEV of te scalar field and terefore it is closely related to te pase transition of te space-tie condensate. Te eolution of te unierse including te Big Bang igt be a series of pase transition of tis condensate. Our teory autoatically gies out te inflation process in te early unierse but te stage of acceleration is
9 different fro oter odels of inflation. It explains te cosological constant proble and te inflation ecanis togeter in a single siple teory. Te eolution of te unierse is found to be alternatiely doinated by te cosological constant and te ass density at different transition stages. Our calculation sows tat ρ ~ ρ in te present unierse. It is ten just a coincidence, not due to underlying pysical teory. Proided tat no furter pase transition will occur, our teory predicts tat te unierse will be doinated by te λ alue foreer. One ay find tat all te aboe results is not acieed by fine tuning paraeters but follows autoatically fro our postulates. In our teory, te scalar field also becoe ore pysical tan just an unknown acuu potential but is te pase paraeter and waefunction of te space-tie condensate. Te diergence proble of quantu graity can be autoatically soled because te space-tie itself is not a fundaental field but a collectie effect of a ore fundaental Higgs process. Te aboe arguents sows tat te space-tie structure can be a ore coplicated structure ten just a continuous ateatical space so tat an eolution on te space-tie concept is terefore necessary. Reference [1] S.Perlutter et al., Astropys. J. 1, 6 (1999) [2] B.P.Scidt et al., Astropys. J 0, 46 (1998) [3] J.R.Bond et al., astro-p/ [4] S.Weinberg, astro-p/ [] S.Weinberg, Re. Mod. Pys (1989) [6] T.D.Lee, Pys. Lett. 122B, 21 (1983)
10 [] G.t Hooft, Recent Deelopent in Graitation, edited by M.Ley & S.Denser, (Plenu, New York, 199); Under te Spell of te Gauge Principle (World Scientific, 1994) [8] L.Bobelli et al, Pys. Re. Lett. 9, 21 (198) [9] A.Linde, Particle Pysics and Inflationary Cosology (Harwood Acadeic, GbH, 1990) [10] I.Dynikoa et al, Graitation and Cosology 4, 0 (1998) [11] H.M.Mok, ttp://publis.aps.org/eprint/grateway/eplist/aps1999aug1_001 [12] W.L.Freedan, astro-p/001236; B.R.Parodi et al., astro-p/
Derivative at a point
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Derivative at a point Wat you need to know already: Te concept of liit and basic etods for coputing liits. Wat you can
More informationSF Chemical Kinetics.
SF Cheical Kinetics. Lecture 5. Microscopic theory of cheical reaction inetics. Microscopic theories of cheical reaction inetics. basic ai is to calculate the rate constant for a cheical reaction fro first
More informationKinetic Molecular Theory of Ideal Gases
Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions.
More information1 Proving the Fundamental Theorem of Statistical Learning
THEORETICAL MACHINE LEARNING COS 5 LECTURE #7 APRIL 5, 6 LECTURER: ELAD HAZAN NAME: FERMI MA ANDDANIEL SUO oving te Fundaental Teore of Statistical Learning In tis section, we prove te following: Teore.
More informationTutorial 2 (Solution) 1. An electron is confined to a one-dimensional, infinitely deep potential energy well of width L = 100 pm.
Seester 007/008 SMS0 Modern Pysics Tutorial Tutorial (). An electron is confined to a one-diensional, infinitely deep potential energy well of widt L 00 p. a) Wat is te least energy te electron can ave?
More informationc hc h c h. Chapter Since E n L 2 in Eq. 39-4, we see that if L is doubled, then E 1 becomes (2.6 ev)(2) 2 = 0.65 ev.
Capter 39 Since n L in q 39-4, we see tat if L is doubled, ten becoes (6 ev)() = 065 ev We first note tat since = 666 0 34 J s and c = 998 0 8 /s, 34 8 c6 66 0 J sc 998 0 / s c 40eV n 9 9 60 0 J / ev 0
More informationChapter 10 Light- Reflectiion & Refraction
Capter 0 Ligt- Relectiion & Reraction Intext Questions On Page 68 Question : Deine te principal ocus o a concae irror. Principal ocus o te concae irror: A point on principal axis on wic parallel ligt rays
More informationqwertyuiopasdfghjklzxcvbnmqwerty uiopasdfghjklzxcvbnmqwertyuiopasd fghjklzxcvbnmqwertyuiopasdfghjklzx cvbnmqwertyuiopasdfghjklzxcvbnmq
qwertyuiopasdfgjklzxcbnmqwerty uiopasdfgjklzxcbnmqwertyuiopasd fgjklzxcbnmqwertyuiopasdfgjklzx cbnmqwertyuiopasdfgjklzxcbnmq Projectile Motion Quick concepts regarding Projectile Motion wertyuiopasdfgjklzxcbnmqwertyui
More informationNeural Networks Trained with the EEM Algorithm: Tuning the Smoothing Parameter
eural etworks Trained wit te EEM Algorit: Tuning te Sooting Paraeter JORGE M. SATOS,2, JOAQUIM MARQUES DE SÁ AD LUÍS A. ALEXADRE 3 Intituto de Engenaria Bioédica, Porto, Portugal 2 Instituto Superior de
More informationlecture 35: Linear Multistep Mehods: Truncation Error
88 lecture 5: Linear Multistep Meods: Truncation Error 5.5 Linear ultistep etods One-step etods construct an approxiate solution x k+ x(t k+ ) using only one previous approxiation, x k. Tis approac enoys
More informationKinetic Molecular Theory of. IGL is a purely empirical law - solely the
Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions.
More informationThe Schrödinger Equation and the Scale Principle
Te Scrödinger Equation and te Scale Princile RODOLFO A. FRINO Jul 014 Electronics Engineer Degree fro te National Universit of Mar del Plata - Argentina rodolfo_frino@aoo.co.ar Earlier tis ear (Ma) I wrote
More informationKey Terms Electric Potential electrical potential energy per unit charge (JC -1 )
Chapter Seenteen: Electric Potential and Electric Energy Key Ter Electric Potential electrical potential energy per unit charge (JC -1 ) Page 1 of Electrical Potential Difference between two points is
More informationHORIZONTAL MOTION WITH RESISTANCE
DOING PHYSICS WITH MATLAB MECHANICS HORIZONTAL MOTION WITH RESISTANCE Ian Cooper School of Physics, Uniersity of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS ec_fr_b. This script
More informationarxiv:hep-ph/ v2 5 Dec 1995
Theral Inflation and the Moduli Proble David H. Lyth School of Physics and Cheistry, Lancaster University, Lancaster, LA1 4YB, U.K. Ewan D. Stewart Research Center for the Early Universe, School of Science,
More information4. DEVIATIONS FROM HOMOGENEITY: THE PECULIAR VELOCITY FIELD 4.1 INTRODUCTION
Section 4 4. DEVIATIONS FROM HOMOGENEITY: THE PECULIAR VELOCITY FIELD 4.1 INTRODUCTION In addition to the effects on the propagation of light rays and the graitational lensing effects (Sect. ), the cosological
More informationLAB #3: ELECTROSTATIC FIELD COMPUTATION
ECE 306 Revised: 1-6-00 LAB #3: ELECTROSTATIC FIELD COMPUTATION Purpose During tis lab you will investigate te ways in wic te electrostatic field can be teoretically predicted. Bot analytic and nuerical
More informationEN40: Dynamics and Vibrations. Midterm Examination Tuesday March
EN4: Dynaics and ibrations Midter Exaination Tuesday Marc 4 14 Scool of Engineering Brown University NAME: General Instructions No collaboration of any kind is peritted on tis exaination. You ay bring
More informationLecture 6. Announcements. Conservation Laws: The Most Powerful Laws of Physics. Conservation Laws Why they are so powerful
Conseration Laws: The Most Powerful Laws of Physics Potential Energy gh Moentu p = + +. Energy E = PE + KE +. Kinetic Energy / Announceents Mon., Sept. : Second Law of Therodynaics Gie out Hoework 4 Wed.,
More information[ ] ( t) ( ω ω. I. Part 3 Transitions and relation to observables. Left off at transition probability for absorbance: choose ω = ω mk
I. Part Transitions and relation to obserables I- Left off at transition probability for absorbance: coose k P ( t) k ( ) ( sin Ε 4 µ k ( [ k k ) t ) ] last ter is band sape f (t, f πt δ ( for t Long tie:
More informationThe accelerated expansion of the universe is explained by quantum field theory.
The accelerated expansion of the universe is explained by quantu field theory. Abstract. Forulas describing interactions, in fact, use the liiting speed of inforation transfer, and not the speed of light.
More informationA NEW ELECTROSTATIC FIELD GEOMETRY. Jerry E. Bayles
INTRODUCTION A NEW ELECTROSTATIC FIELD GEOMETRY by Jerry E Bayles The purpose of this paper is to present the electrostatic field in geoetrical ters siilar to that of the electrogravitational equation
More informationOne Dimensional Collisions
One Diensional Collisions These notes will discuss a few different cases of collisions in one diension, arying the relatie ass of the objects and considering particular cases of who s oing. Along the way,
More informationTHE ESSENCE OF QUANTUM MECHANICS
THE ESSENCE OF QUANTUM MECHANICS Capter belongs to te "Teory of Spae" written by Dariusz Stanisław Sobolewski. Http: www.tsengines.o ttp: www.teoryofspae.info E-ail: info@tsengines.o All rigts resered.
More informationOn seismic landslide hazard assessment
Yang, J. (27). Géotecnique 57, No. 8, 77 713 doi: 1.168/geot.27.57.8.77 TECHNICAL NOTE On seismic landslide azard assessment J. YANG* KEYWORDS: eartquakes; landslides; slopes INTRODUCTION Seismic landslides
More information13 Harmonic oscillator revisited: Dirac s approach and introduction to Second Quantization
3 Haronic oscillator revisited: Dirac s approach and introduction to Second Quantization. Dirac cae up with a ore elegant way to solve the haronic oscillator proble. We will now study this approach. The
More informationA = h w (1) Error Analysis Physics 141
Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.
More information5.1 The derivative or the gradient of a curve. Definition and finding the gradient from first principles
Capter 5: Dierentiation In tis capter, we will study: 51 e derivative or te gradient o a curve Deinition and inding te gradient ro irst principles 5 Forulas or derivatives 5 e equation o te tangent line
More informationThe product of force and displacement ( in the direction of force ), during which the force is acting, is defined as work.
5 WORK, ENERGY ND POWER Page 5. Work The product of force and displaceent ( in the direction of force ), during which the force is acting, is defined as work. When N force is applied on a particle and
More informationJournal of Modern Physics, 2011, 2, doi: /jmp Published Online November 2011 (http://www.scirp.
Journal of Modern Physics, 11,, 1331-1347 doi:1.436/jp.11.11165 Published Online Noveber 11 (http://www.scirp.org/journal/jp) Transforation of the Angular Power Spectru of the Cosic Microwave Background
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015
Lecture : Transition State Teory. tkins & DePaula: 7.6-7.7 University o Wasinton Departent o Ceistry Ceistry 453 Winter Quarter 05. ctivated Kinetics Kinetic rate uations are overned by several principles.
More informationThe Characteristic Planet
The Characteristic Planet Brano Zivla, bzivla@gail.co Abstract: I have calculated a relation significant for planets fro a logical starting point that a whole and its parts are ianently depandant on each
More informationA KERNEL APPROACH TO ESTIMATING THE DENSITY OF A CONDITIONAL EXPECTATION. Samuel G. Steckley Shane G. Henderson
Proceedings of te 3 Winter Siulation Conference S Cick P J Sáncez D Ferrin and D J Morrice eds A KERNEL APPROACH TO ESTIMATING THE DENSITY OF A CONDITIONAL EXPECTATION Sauel G Steckley Sane G Henderson
More informationChapter 6 1-D Continuous Groups
Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores:
More informationThe derivative function
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative
More information12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015
18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring 2015 12 Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tie-independent) elastic systes.
More informationThe anisotropy of the cosmic background radiation due to mass clustering in a closed universe
Astron. Astrophys. 9, 79 7 (997) ASTRONOMY AND ASTROPHYSICS The anisotropy of the cosic background radiation due to ass clustering in a closed universe Daing Chen, Guoxuan Song, and Yougen Shen, Shanghai
More informationProblem Set 7: Potential Energy and Conservation of Energy AP Physics C Supplementary Problems
Proble Set 7: Potential Energy and Conservation of Energy AP Pysics C Suppleentary Probles 1. Approxiately 5.5 x 10 6 kg of water drops 50 over Niagara Falls every second. (a) Calculate te aount of potential
More informationENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 2 LINEAR IMPULSE AND MOMENTUM
ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D5 TUTORIAL LINEAR IMPULSE AND MOMENTUM On copletion of this ttorial yo shold be able to do the following. State Newton s laws of otion. Define linear
More informationNUMERICAL MODELLING OF THE TYRE/ROAD CONTACT
NUMERICAL MODELLING OF THE TYRE/ROAD CONTACT PACS REFERENCE: 43.5.LJ Krister Larsson Departent of Applied Acoustics Chalers University of Technology SE-412 96 Sweden Tel: +46 ()31 772 22 Fax: +46 ()31
More informationOptical Properties of Plasmas of High-Z Elements
Forschungszentru Karlsruhe Techni und Uwelt Wissenschaftlishe Berichte FZK Optical Properties of Plasas of High-Z Eleents V.Tolach 1, G.Miloshevsy 1, H.Würz Project Kernfusion 1 Heat and Mass Transfer
More informationESTIMATION OF THE VISCOELASTIC PARAMETERS OF LAMINATED COMPOSITES. PART I. ANALYTICAL CONSIDERATIONS
nd International Conerence danced Coposite aterials Engineering COT 8 9 October 8, Braso, Roania ESTITION OF THE VISCOELSTIC RETERS OF LINTED COOSITES. RT I. NLYTICL CONSIDERTIONS.Katouzian, S. Vlase,.V.Guian
More informationSupplementary Materials: Proofs and Technical Details for Parsimonious Tensor Response Regression Lexin Li and Xin Zhang
Suppleentary Materials: Proofs and Tecnical Details for Parsionious Tensor Response Regression Lexin Li and Xin Zang A Soe preliinary results We will apply te following two results repeatedly. For a positive
More informationNUCLEAR THERMAL-HYDRAULIC FUNDAMENTALS
NUCLEAR THERMAL-HYDRAULIC FUNDAMENTALS Dr. J. Micael Doster Departent of Nuclear Engineering Nort Carolina State University Raleig, NC Copyrigted POER CYCLES Te analysis of Terodynaic Cycles is based alost
More informationReading from Young & Freedman: For this topic, read the introduction to chapter 25 and sections 25.1 to 25.3 & 25.6.
PHY10 Electricity Topic 6 (Lectures 9 & 10) Electric Current and Resistance n this topic, we will cover: 1) Current in a conductor ) Resistivity 3) Resistance 4) Oh s Law 5) The Drude Model of conduction
More informationTheoretical Astrophysics and Cosmology Master Degree in Astronomy and Erasmus-Mundus A.A. 2016/17 Alberto Franceschini Cosmology Course
Theoretical Astrophysics and Cosology Master Degree in Astronoy and Erasus-Mundus A.A. 16/17 Alberto Franceschini Cosology Course Hoogeneous Friedan Universe.1 PROGRAMME FOR THE COSMOLOGY COURSE. The Hoogeneous
More informationMEASUREMENTS OF POLARIMETRIC PARAMETERS AT LOW SIGNAL-TO-NOISE RATIOS
P9. MEAUEMENT OF POLAIMETIC PAAMETE AT LOW IGNAL-TO-NOIE ATIO V. M. Melniko and D.. rnic Cooperatie Institute for Mesoscale Meteorological tudies, Uniersity of Oklaoa and NOAA/National eere tors Laboratory,
More informationMechanics Research Communications
Mecanics Researc Counications 38 2011) 393 398 Contents lists available at ScienceDirect Mecanics Researc Counications j o ur nal oep age: www.elsevier.co/locate/ecresco On te aronic vibrations in linear
More informationA Different Derivation of the Calogero Conjecture
1 Abstract: A Different Derivation of the Calogero Conjecture Ioannis Iraklis Haranas Physics and Astronoy Departent York University 314 A Petrie Science Building Toronto Ontario CANADA E ail: ioannis@yorku.ca
More informationPhysics 11 HW #7 Solutions
hysics HW #7 Solutions Chapter 7: Focus On Concepts: 2, 6, 0, 3 robles: 8, 7, 2, 22, 32, 53, 56, 57 Focus On Concepts 7-2 (d) Moentu is a ector quantity that has a agnitude and a direction. The agnitudes
More informationPhysics Teach Yourself Series Topic 15: Wavelike nature of matter (Unit 4)
Pysics Teac Yourself Series Topic 15: Wavelie nature of atter (Unit 4) A: Level 14, 474 Flinders Street Melbourne VIC 3000 T: 1300 134 518 W: tss.co.au E: info@tss.co.au TSSM 2017 Page 1 of 8 Contents
More informationPHY 171. Lecture 14. (February 16, 2012)
PHY 171 Lecture 14 (February 16, 212) In the last lecture, we looked at a quantitative connection between acroscopic and icroscopic quantities by deriving an expression for pressure based on the assuptions
More informationma x = -bv x + F rod.
Notes on Dynaical Systes Dynaics is the study of change. The priary ingredients of a dynaical syste are its state and its rule of change (also soeties called the dynaic). Dynaical systes can be continuous
More informationA KERNEL APPROACH TO ESTIMATING THE DENSITY OF A CONDITIONAL EXPECTATION. Samuel G. Steckley Shane G. Henderson
Proceedings of te 3 Winter Siulation Conference S Cick P J Sáncez D Ferrin and D J Morrice eds A KERNEL APPROACH TO ESTIMATING THE DENSITY OF A CONDITIONAL EXPECTATION Sauel G Steckley Sane G Henderson
More informationDetermining Limits of Thermal NDT of Thick Graphite/Epoxy Composites
ECNDT 006 - We.3.8.1 Deterining Liits of Teral NDT of Tick Grapite/Epoy Coposites Vladiir VAVILOV Institute of Introscopy Tosk Russia Abstract. Te known approac to inspecting tin coposites by using infrared
More information1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point
MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note
More informationEXPERIMENTAL INVESTIGATION OF TANGENTIAL CONTACT STIFFNESS AND EQUIVALENT DAMPING
Proceedings in Manufacturing Systes, Volue 7, Issue, ISSN 7-9 EXPERIMENTL INVESTIGTION OF TNGENTIL CONTCT STIFFNESS ND EQUIVLENT DMPING Iuliana PISCN,*, Tierry JNSSENS, Farid L-BENDER, Cristina PUPĂZĂ
More informationChapter 1: Basics of Vibrations for Simple Mechanical Systems
Chapter 1: Basics of Vibrations for Siple Mechanical Systes Introduction: The fundaentals of Sound and Vibrations are part of the broader field of echanics, with strong connections to classical echanics,
More informationScattering and bound states
Chapter Scattering and bound states In this chapter we give a review of quantu-echanical scattering theory. We focus on the relation between the scattering aplitude of a potential and its bound states
More informationAn Approximate Model for the Theoretical Prediction of the Velocity Increase in the Intermediate Ballistics Period
An Approxiate Model for the Theoretical Prediction of the Velocity... 77 Central European Journal of Energetic Materials, 205, 2(), 77-88 ISSN 2353-843 An Approxiate Model for the Theoretical Prediction
More informationThe Verlet Algorithm for Molecular Dynamics Simulations
Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical
More informationRationality Problems of the Principles of Equivalence and General Relativity
Rationality Probles of the Principles of Equivalence and General Relativity Mei Xiaochun (Departent of Physics, Fuzhou University, E-ail: xc1@163.co Tel:86-591-8761414) (N.7-B, South Building, Zhongfu
More informationWe consider the abelian Higgs model with the Lagrangian:
(c) 06 Roanian Journal of Physics (for accepted papers only) BOUT THE MSS CORRECTIONS IN N BELIN HIGGS MODEL 3 4 5 RENT JOR National Institute of Physics and Nuclear Engineering PO Box MG-6, Bucharest-Magurele,
More informationExample A1: Preparation of a Calibration Standard
Suary Goal A calibration standard is prepared fro a high purity etal (cadiu) with a concentration of ca.1000 g l -1. Measureent procedure The surface of the high purity etal is cleaned to reove any etal-oxide
More informationEdge Detection Based on the Newton Interpolation s Fractional Differentiation
Te International Arab Journal of Information Tecnology, Vol. 11, No. 3, May 014 3 Edge Detection Based on te Newton Interpolation s Fractional Differentiation Caobang Gao 1,, Jiliu Zou, 3, and Weiua Zang
More informationHomotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media
Hootopy analysis of D unsteady, nonlinear groundwater flow troug porous edia Autor Song, Hao, Tao, Longbin Publised 7 Journal Title Journal of Coastal Researc Copyrigt Stateent 7 CERF. Te attaced file
More informationP (t) = P (t = 0) + F t Conclusion: If we wait long enough, the velocity of an electron will diverge, which is obviously impossible and wrong.
4 Phys520.nb 2 Drude theory ~ Chapter in textbook 2.. The relaxation tie approxiation Here we treat electrons as a free ideal gas (classical) 2... Totally ignore interactions/scatterings Under a static
More informationNumerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
More informationKINETIC THEORY. Contents
KINETIC THEORY This brief paper on inetic theory deals with three topics: the hypotheses on which the theory is founded, the calculation of pressure and absolute teperature of an ideal gas and the principal
More informationSubmanifold density estimation
Subanifold density estiation Arkadas Ozakin Georgia Tec Researc Institute Georgia Insitute of Tecnology arkadas.ozakin@gtri.gatec.edu Alexander Gray College of Coputing Georgia Institute of Tecnology agray@cc.gatec.edu
More informationWhy gravity is not an entropic force
Wy gravity is not an entropic force San Gao Unit for History and Pilosopy of Science & Centre for Time, SOPHI, University of Sydney Email: sgao7319@uni.sydney.edu.au Te remarkable connections between gravity
More informationField Mass Generation and Control. Chapter 6. The famous two slit experiment proved that a particle can exist as a wave and yet
111 Field Mass Generation and Control Chapter 6 The faous two slit experient proved that a particle can exist as a wave and yet still exhibit particle characteristics when the wavefunction is altered by
More informationUfuk Demirci* and Feza Kerestecioglu**
1 INDIRECT ADAPTIVE CONTROL OF MISSILES Ufuk Deirci* and Feza Kerestecioglu** *Turkish Navy Guided Missile Test Station, Beykoz, Istanbul, TURKEY **Departent of Electrical and Electronics Engineering,
More informationSolutions to the problems in Chapter 6 and 7
Solutions to the probles in Chapter 6 and 7 6.3 Pressure of a Feri gas at zero teperature The nuber of electrons N and the internal energy U, inthevoluev,are N = V D(ε)f(ε)dε, U = V εd(ε)f(ε)dε, () The
More informationTransverse waves. Waves. Wave motion. Electromagnetic Spectrum EM waves are transverse.
Transerse waes Physics Enhanceent Prograe for Gifted Students The Hong Kong Acadey for Gifted Education and, HKBU Waes. Mechanical waes e.g. water waes, sound waes, seisic waes, strings in usical instruents.
More informationCurrent Developments in the Field of Shock Calibration
XVIII IMEKO WORLD CONGRESS Metrology for a Sustainale Developent Septeer, 17, 6, Rio de Janeiro, Brazil Current Developents in te Field of Sock Caliration T. Bruns 1, A. Link, C. Elster 3 1 Pysikalisc-Tecnisce
More informationClassical systems in equilibrium
35 Classical systes in equilibriu Ideal gas Distinguishable particles Here we assue that every particle can be labeled by an index i... and distinguished fro any other particle by its label if not by any
More informationPhase space in classical physics
Pase space in classical pysics Quantum mecanically, we can actually COU te number of microstates consistent wit a given macrostate, specified (for example) by te total energy. In general, eac microstate
More informationStationary Gaussian Markov processes as limits of stationary autoregressive time series
Stationary Gaussian Markov processes as liits of stationary autoregressive tie series Pilip A. rnst 1,, Lawrence D. Brown 2,, Larry Sepp 3,, Robert L. Wolpert 4, Abstract We consider te class, C p, of
More informationName: Partner(s): Date: Angular Momentum
Nae: Partner(s): Date: Angular Moentu 1. Purpose: In this lab, you will use the principle of conservation of angular oentu to easure the oent of inertia of various objects. Additionally, you develop a
More informationMomentum, p. Crash! Collisions (L8) Momentum is conserved. Football provides many collision examples to think about!
Collisions (L8) Crash! collisions can be ery coplicated two objects bang into each other and exert strong forces oer short tie interals fortunately, een though we usually do not know the details of the
More informationFokker-Planck Description of Conductance-Based Integrate-and-Fire Neuronal Networks
Fokker-Planck Description of Conductance-Based Integrate-and-Fire Neuronal Networks Gregor Koačič, Louis Tao, Aaditya V. Rangan, 3 and Daid Cai 3,4 Matheatical Sciences Departent, Rensselaer Polytechnic
More informationCoping with Friction for Non-penetrating Rigid Body Simulation
SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 Coping with Friction for Non-penetrating Rigid ody Siulation Daid araff Progra of Coputer Graphics Cornell Uniersity Ithaca, NY 14853
More informationNumerical Solution for Non-Stationary Heat Equation in Cooling of Computer Radiator System
(JZS) Journal of Zankoy Sulaiani, 9, 1(1) Part A (97-1) A119 Nuerical Solution for Non-Stationary Heat Equation in Cooling of Coputer Radiator Syste Aree A. Maad*, Faraidun K. Haa Sal**, and Najadin W.
More informationSome consequences of a Universal Tension arising from Dark Energy for structures from Atomic Nuclei to Galaxy Clusters
unning Head: Universal Tension fro DE Article Type: Original esearch Soe consequences of a Universal Tension arising fro Dark Energy for structures fro Atoic Nuclei to Galaxy Clusters C Sivara Indian Institute
More informationChapter 2 Ising Model for Ferromagnetism
Capter Ising Model for Ferromagnetism Abstract Tis capter presents te Ising model for ferromagnetism, wic is a standard simple model of a pase transition. Using te approximation of mean-field teory, te
More informationA nonstandard cubic equation
MATH-Jan-05-0 A nonstandard cubic euation J S Markoitch PO Box West Brattleboro, VT 050 Dated: January, 05 A nonstandard cubic euation is shown to hae an unusually econoical solution, this solution incorporates
More informationProc. of the IEEE/OES Seventh Working Conference on Current Measurement Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES Belinda Lipa Codar Ocean Sensors 15 La Sandra Way, Portola Valley, CA 98 blipa@pogo.co
More informationSOLUTIONS. PROBLEM 1. The Hamiltonian of the particle in the gravitational field can be written as, x 0, + U(x), U(x) =
SOLUTIONS PROBLEM 1. The Hailtonian of the particle in the gravitational field can be written as { Ĥ = ˆp2, x 0, + U(x), U(x) = (1) 2 gx, x > 0. The siplest estiate coes fro the uncertainty relation. If
More informationDoppler Effect (Text 1.3)
Doppler Effet (et 1.3) Consider a light soure as a soure sending out a tik eery 1/ν and these tiks are traeling forward with speed. tik tik tik tik Doppler Effet (et 1.3) Case 1. Obserer oing transersely.
More information2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all
Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either
More informationEstimation for the Parameters of the Exponentiated Exponential Distribution Using a Median Ranked Set Sampling
Journal of Modern Applied Statistical Metods Volue 14 Issue 1 Article 19 5-1-015 Estiation for te Paraeters of te Exponentiated Exponential Distribution Using a Median Ranked Set Sapling Monjed H. Sau
More informationGeneviève BELANGÉR, Kristjan KANNIKE, Alexander PUKHOV and Martti RAIDAL
Geneviève BELANGÉR,, Alexander PUKHOV and Martti RAIDAL cuola Normale uperiore di Pisa E-mail: belanger@lapt.cnrs.fr, kristjan.kannike@cern.c, pukov@lapt.cnrs.fr, martti.raidal@cern.c calar dark matter
More informationA Reconsideration of Matter Waves
A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,
More informatione = n 1 ( ) 3 [ m 3] = n [ m 3] n
Magnetospheric Physics - Hoework Solutions, /7/4 7. Plasa definition Can a plasa be aintained at teperatures of T e K Hint: Calculate the density liit using the plasa paraeter and explain your result).
More informationUNCERTAINTIES IN THE APPLICATION OF ATMOSPHERIC AND ALTITUDE CORRECTIONS AS RECOMMENDED IN IEC STANDARDS
Paper Published on the16th International Syposiu on High Voltage Engineering, Cape Town, South Africa, 2009 UNCERTAINTIES IN THE APPLICATION OF ATMOSPHERIC AND ALTITUDE CORRECTIONS AS RECOMMENDED IN IEC
More information12.1 Static Magnetic Field in the Presence of Magnetic Materials
Electroagnetics P1-1 Lesson 1 Magnetostatics in Materials 1.1 Static Magnetic Field in the Presence o Magnetic Materials Concept o induced agnetic dipoles An aterial has an icroscopic agnetic dipoles (i.e.,
More informationMolecular interactions in beams
Molecular interactions in beas notable advanceent in the experiental study of interolecular forces has coe fro the developent of olecular beas, which consist of a narrow bea of particles, all having the
More informationWYSE Academic Challenge Sectional Physics 2006 Solution Set
WYSE Acadeic Challenge Sectional Physics 6 Solution Set. Correct answer: d. Using Newton s nd Law: r r F 6.N a.kg 6./s.. Correct answer: c. 6. sin θ 98. 3. Correct answer: b. o 37.8 98. N 6. N Using Newton
More information3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
More information