FNSN 2 - Chapter 11 Searches and limits

Size: px
Start display at page:

Download "FNSN 2 - Chapter 11 Searches and limits"

Transcription

1 FS 2 - Chapter 11 Searhes and limits Paolo Bagnaia last mod. 19-May-17

2 11 Searhes and limits 1. Probability 2. Searhes and limits 3. Limits 4. Maximum likelihood 5. Interpretation of results methods ommonly used for all reent searhes (e.g. LEP, LHC, gravitational waves); not really new (e.g. "Laboratorio di Meania"); but probably forgotten; not a well-organized disussion, beyond the sope of present letures ( referenes + next year). Paolo Bagnaia FS

3 1/2 probability: a new entry in the game modern partile physis makes a large use of (relatively) new sienes : probability and her sister statistis; [already in lassial physis, when measuring an observable, "probability to get x, if true = x* and resolution x"]; [we are sientists, not gamblers, and do OT disuss poker and die here]; [q.m. is intrinsially probabilisti, sine the preditions are distributions, while the experiments produe single/disrete values]; but the sentene "the probability that a theory be true" [e.g. "the p. that the Higgs boson exist and its mass be "] is really new; however, a simple investigation shows that life is intrinsially probabilisti (risk, hane, luk many words whih essentially mean "probability", while experiene, past, use, mean "statistis"). Paolo Bagnaia FS

4 2/2 probability: Kolmogorov axioms Andrei ikolayevih Kolmogorov [Андрей Николаевич Колмогоров] ( ), a Russian (sovieti) mathematiian, in 1933 wrote a fundamental axiomatization of probability; he introdued the spae S of the events (A, B, ) and the event probability as a measure (A) in S. K. axioms are : 1. 0 (A) 1 A S; 2. (S) = 1; 3. A B = Ø (A B) = (A) + (B). Some theorems (easily demonstrated): (Ā) = 1 (A); (A Ā) = 1; (Ø) = 0; A B (A) (B); (A B) = (A) + (B) (A B). A B S Paolo Bagnaia FS

5 1/4 searhes and limits Sometimes, the result of the study is OT the measurement of an observable x : "x = x exp ± x", but, instead, a qualitative "searh" : "the phenomenon Y (does not) exists", or, alternatively : "the phenomenon Y does OT exist in the kinematial range Φ". [statements with "not" apply if effet is not found, and an "exlusion" (a "limit", when Φ is not omplete) is established] In modern experiments, the searhes oupy more than 50% of the published papers, and almost all are negative [but the Higgs searh at LHC, of ourse]. Obviously, a negative result is OT a failure : if any, it is a failure of the theory under test. [but a positive result is muh more pleasant (and rewarding)] A rigorous method, well understood and "easy" to apply, is imperative. This method is a major suess of the LEP era : it uses math, statistis, physis, ommon sense and ommuniation skill. It MUST be in the panoply of eah partile physiist, both theoretiians and experimentalists. use the Higgs searhes at LEP and LHC as a test ase [these letures must remain inside the SM]; after the Higgs disovery, the fous has shifted toward "bsm" searhes, but the method has not hanged. Paolo Bagnaia FS

6 2/4 searhes and limits: definitions In the next slides : L int : integrated luminosity; σ s : ross setion of signal (searhed for); σ b : ross setion of bakground (known); ε s : effiieny for signal (0 1, larger is better); ε b : effiieny for bakground (0 1, smaller is better); s : # expeted events of signal [s = L int ε s σ s ]; b : # expeted events of bakground [b = L int ε b σ b ]; n : # expeted events [n = s + b, or n = b]; : # found events ( flutuates around n with Poisson ( Gauss) statistis; : probability, aording to a given pdf ("probability density funtion"); CL : "onfidene level", a limit (< 1) in the umulative probability; Λ : likelihood funtion for signal+bkgd (Λ s ) or bkgd-only (Λ b ) hypotheses; μ : parameter defining the signal level [n = b + µ s], used for limit definition; p : "p-value", probability to get the same result or another less probable, in the hypothesis of bkgd-only; E[x] : expeted value of the quantity "x". Paolo Bagnaia FS

7 3/4 searhes and limits: verify/falsify A theory (SM, SUSY, ) predits a phenomenon (a partile, a dynami effet), e.g. e +, p, Ω, W ± /Z, H; [in some ases the phenomenon depends on some unknown parameter, e.g. the Higgs boson mass] a new devie (e.g. an aelerator) is potentially able to observe the phenomenon [fully or in a range of the parameters spae still unexplored]; therefore, two possibilities: A. observation: the theory is "verified" (à la Popper) [and the free parameter(s) are measured]; B. non-observation: the theory is "falsified" (à la Popper) [or some subspae in the parameter spae, e.g. an interval in one dimension, is exluded a "limit" is established]; different, less ommon, approah ("model independent") : look for unknown effets, without theoretial guidane, e.g. CP violation, J/ψ. Paolo Bagnaia FS

8 4/4 searhes and limits: blind analysis the key point : usually b s, but ƒ b (b) and ƒ s (s) are very different uts in the event variables (masses, angles), suh that to enhane s wrt b; when is large ( ), statistial flutuations (s.f.) do OT modify the result; usually (not only for impatiene) is small and its s.f. are important; small variations in the seletion ( small hange in the expeted number for s, b) orrespond to large differenes in the results [e.g., if b=0.001, when hanges 0 1, as in the artoon]; a "neutral" analysis is impossible; a posteriori, it is always easy to justify a hange in the uts, whih strongly affets the results; therefore, the only honest proedure onsists in defining the seletion a priori (e.g. by optimizing the expeted visibility on a m event sample); then, the seletion is "blindly" applied to the atual event sample ( "blind analysis"). bakground region signal region whih is the "right" ut? Paolo Bagnaia FS

9 1/9 [in the "good ole times", life was simpler : if the bakground is negligible, the first observation led to the disovery, as for e +, p, Ω, W ± and Z] in most ases, the bakground (reduible or irreduible) is alulable; a disovery is defined as an observation that is inompatible with a +ve statistial flutuation respet to the expeted bakground alone; a limit is established if the observation is inompatible with a ve flutuation respet to the expeted (signal + bakground); both statements are based on a "redutio ad absurdum"; sine all values of in [0, ] are possible, it is ompulsory to predefine a CL to "ut" the pdf; the CL for disovery and exlusion an be different : usually for the disovery striter riteria are required; limits a priori the expeted signal s an be ompared with the flutuation of the bakground (in approximation of large number of events, s b) : n σ = s / b is a figure of merit of the experiment; a posteriori the observed number () is ompared with the expeted bakground (b) or with the sum (s + b). Example. We expet 100 bakground events and 44 signal; we use the "large number" approximation ( n = n) : b = 100, b = b = 10; n = s + b = 144, n = 12. The pre-hosen onfidene level is "3 σ". The disovery orresponds to an observation of > ( ) = 130 events. A limit is established if < ( ) = 108 events. There is no deision if 108 < < 130. The values < 70 and > 180 are "impossible". Paolo Bagnaia FS

10 2/9 limits: problem Problem (based on previous example) : ompute the fator, wrt to previous luminosity, whih allows to avoid the "nodeision" region. Answer : 9/4 = 2.25 Solution : all ƒ the fator; then 100 ƒ ƒ = 144 ƒ ƒ ƒ=9/4; b=225; (s+b)=324, s=99. [ritiism : = 270 is still "no-deision"; find a better solution maybe : 100 ƒ ƒ + 1 = 144 ƒ ƒ but must be integer!] Paolo Bagnaia FS

11 3/9 limits: Poisson statistis In general, the searhes looks for proesses with VERY limited statistis (want to disover asap); therefore the limit ("n large", more preisely n >> n) annot be used (neither its onsequenes, like the Gauss pdf); searhes are learly in the "Poisson regime" : large sample and small probability, suh that the expeted number of events ("suesses") be finite; use the Poisson distribution : m e m ( m) = ; = m; σ = m;! therefore, in a searh, two ases : a. the signal does exist : (b+ s) e (b + s) = b + s; ( b + s) = ;! σ = b+ s; [two possibilities : s known/unknown] b. the signal does OT exist : b e b ( b) = ; = b; σ = b;! the strategy is : use (observed) to distinguish between ase (a) and (b); sine is positive for in [0, ] in both ases, the proedure is to define a CL a priori, and aept the hypothesis (a or b) only if it falls in the predefined interval; modern (LHC) evolution : define a parameter, usually alled "µ" : (b+µ s) e (b + µ s) = b + µ s; ( b + µ s) = ;! σ = b+ µ s; learly, µ = 0 is bkgd only, while µ = 1 means disovery; sometimes results are presented as limits on "µ" [e.g. exlude µ = 0 means "disovery"]. Paolo Bagnaia FS

12 4/9 limits : disovery, exlusion the "rule" on the CL usually aepted by experiments is: experiment, with exatly the expeted number of events "]. DISCOVERY : (b only) , [alled also "5σ" (1) ]; EXCLUSIO : (s+b) ; [alled also "95% CL"]; a priori, the integrated luminosity L int for disovery / exlusion an be omputed : L dis : L int min, suh that 50% of the experiments (2) (i.e. an experiment in 50% of the times) had (b only) "5σ"; (1) this probability orresponds to 5σ for a gaussian pdf only; but the experimentalists use (always) the ut in probability and (sometimes) all it "5σ"; (2) for ombined studies an "experiment" at LEP [LHC] results from the data of all 4 [2] ollaborations; in this ase L int 4(2) L int. L exl : L int min, suh that 50% of the experiments (2) (i.e. an experiment in 50% of the times) had (s+b) "2σ"; B : this rule orresponds to the median ["an experiment, in 50% of the times "], and it is different from the average ["an Paolo Bagnaia FS

13 5/9 limits: flowhart m signal (theory for many values of the parameters θ k, σ, dσ/dosθ, final state partiles, ) m bkgd (σ, dσ/dosθ, final state partiles, ) idential!!! detetor m (response, resolution, failures, ) detetor m ( ) analysis : optimization of uts/seletion to maximize signal visibility (e.g. s/ b) [sometimes the seletion is funtion of free parameters θ k (e.g. m H ) or L int ]. optimal seletion one-way only real data sensitivity disovery θ k??? limit on θ k Paolo Bagnaia FS

14 6/9 limits : Luminosity of disovery, exlusion b b+s ut i assume inreasing luminosity (L int = L dis [L exl ]) and onstant ε s, ε b, σ s, σ b ; assume to start with small L int : the two distributions overlap a lot, no satisfies the system (i.e. the tail above the median is large); when L int inreases, the two distributions are more and more distint (overlap 1/ L int ); The values of L dis and L exl ome from the previous equations; ompute L dis (L exl is similar) : " " = e ( b) i b 7 (5 σ) = ; i! i= ( b s) ( b + s) + " " = e i= i! b = L εσ ; s = L dis B B i ε 0.5; σ dis S S. for a given value of L int, it exists a number of events, suh that the uts at (0.5) in the first (seond) umulative oinide; this value of L int orrespond to L dis ; this is the luminosity when, if the signal exists, 50% of the experiments have (at least) 5σ inompatibility with the hypothesis of bkgd only. Paolo Bagnaia FS

15 7/9 limits : ex. m H (b=0, n=0) n s(m H ) = σ s (m H ) L int ε s [theory + m detetor + analysis] just an example, not an atual plot nothing observed: =n=0 95% CL (=0 n) 1-CL n - ln (1-CL) n % exlusion m limit m H Paolo Bagnaia FS

16 8/9 limits : ex. m H (a priori, b>0) n just an example, not an atual plot s+b = L int [σ s (m H ) ε s + σ b ε b ] B : ε s and ε b may be funtions of m H, or not ( mass independent seletion ). n b = L int σ b ε b 95% CL Σ n j=0 (j n) 1-CL n+ > n expeted 95% CL m limit m H Paolo Bagnaia FS

17 9/9 limits : ex. m H (a posteriori, b>0) n just an example, not an atual plot andidate events (resolution inluded) B the result may be larger or smaller than expetation (m limit is a statistial variable). 95% l m limit m H Paolo Bagnaia FS

18 1/5 maximum likelihood: definition A random variable x follows a "probability density funtion" (pdf) ƒ(x θ k ); the parameters θ k (k = 1,,M), sometimes unknown, define ƒ; assume a repeated measurement ( times) of x : x j ( j = 1,,); define the likelihood Λ and its logarithm ln(λ) (see box). Example : observe deays with (unknown) lifetime τ, measuring the lives t j, j = 1,,. 1 t/ τ 1 j t/ j τ Λ= ƒ(t τ ) = e = e ; τ τ j j= 1 j= 1 1 t/ τ 1 j n( Λ= ) n[ e ] = n( τ ) t j. τ τ Λ= ƒ(x θ ); j= 1 n( Λ= ) n[ƒ(x θ )]. j= 1 j= 1 j= 1 j j k k then look for the value τ*, whih maximizes Λ (or lnλ). τ* is the max.lik. estimate of τ. n( Λ ) 1 = 0 = + 2 tj τ τ* τ* 1 τ * = tj =< t >. j= 1 j= 1 Paolo Bagnaia FS

19 2/5 maximum likelihood: parameter estimate the m.l. method has the following important properties [no dem., see the referenes] : asymptotially onsistent; asymptotially no-bias; the results θ* is asymptotially distributed around θ t, with a variane oinident with the Cramér-Frehet- Rao limit; "invariant" for a hange of parameters, i.e. the m.l. estimate of a quantity, funtion of the parameters, is given by the funtion of the estimates [e.g. (θ 2 )* = (θ*) 2 ]; suh a value is also asymptotially nobias; popular wisdom : "the m.l. method is like a Rolls-Roye: expensive, but the best". B. "asymptotially" means : the onsidered property is valid in the limit meas ; if is finite, the property is OT valid anymore; sometimes the physiists show some "lak of rigor" (say). Paolo Bagnaia FS

20 3/5 maximum likelihood: hypothesis test The likelihood funtion [PDG] is the produt of the pdf for eah event, alulated for the observed values; for searhes, it is the Poisson probability for observing events times the pdf of eah single event (see box); sine there are two hypotheses (b only and b+s), there are two pdf s and therefore two likelihoods; both are funtions of the parameter(s) of the phenomenon under study (m H ); the likelihood ratio is a powerful (atually the most powerful) test between two hypotheses, mutually exlusive; the term -2 ln is there only for onveniene [both for omputing and beause -2ln(Λ) χ 2 for n large]; in the box : Λ =1, C refers to different hannels to be ombined, if any [e.g. H ZZ, γγ]; ƒ s,b are the pdf (usually from m) of the kinematial variables x for event j : s b s ƒ (x ) + b ƒ ( x) given ƒ (x), ƒ (x), ƒ (x) : s b b+ s = s + b ( b + s ) C poisson Λ =Λ = s s(m H) + ; b s = 1 ƒ ( x j ) j = 1 ( b ) C poisson =Λ (m ) = ; = b 1 ƒ( x j ) j = 1 b b H Λ S 2 nq = 2 n = 2( n Λ B n Λ S). ΛB Paolo Bagnaia FS

21 4/5 maximum likelihood: m H at LEP (1) A "simple" appliation of the likelihood ratio for the Higgs mass at LEP2 : "hannel ", =1,,C : one experiment one s one final state [e.g. (L3) ( s = 204 GeV) (e + e HZ bb µ + µ )] (atually C > 100); "m = m H, test mass" : the mass under study ("the hypothesis"), whih must be aepted/rejeted (a sansion in mass, with interval ~ mass resolution); for eah hannel and eah test mass, (in priniple) a different analysis, i.e. a different value of {σ S, σ B, ε S, ε B, L, n},m [L ε S σ S s,m, L ε B σ B b,m, both ƒ(m H )]; for eah hannel and eah test mass, a different set of andidates (= events surviving the uts) (j=1,, ); event j has a set of kinematial variables (4- momenta of partiles, seondary vertees) x j [observables of event j of hannel ]; the m samples (both signal and bkgd) allows to define ƒ S,m(x) and ƒ B,m(x), the pdf for signal and bkgd of all the variables, after uts and fits; define all the appropriate variables (e.g. reonstruted mass m j, seondary vertex probability, define also the total number of andidates for a test mass M m,m ; notie that, generally speaking, all variables are orrelated [e.g. m j = m jm = m j (m H ), beause effiieny, uts and fits do depend on m H ]. Paolo Bagnaia FS

22 5/5 maximum likelihood: m H at LEP (2) and therefore [one again, remember that everything is an impliit funtion of the test mass m H ]. ( + ) + e s b n S B (s b ) n j j Λ S = = = 1 n! j= 1 s + b = e n! (s + b ) = 1 j= 1 n B b ƒ (x j) j= 1 s ƒ (x ) + b ƒ (x ) n S B s ƒ (x j) + b ƒ (x j) ; b n n e b B Λ B = ƒ (x j) = = 1 n! j= 1 b e = = 1 n! ; e (s + b ) n S B s ƒ (x j) + b ƒ (x j) S = 1 n! Λ j= 1 2 nq = 2 n = 2n = b n ΛB e B b ƒ (x j) 1 n! = j= 1 n S s ƒ (x j) = 2 s 2 n 1+ B. = 1 j= 1 bƒ( x j) = 1 Paolo Bagnaia FS

23 1/3 interpretation of results : disovery plot The likelihood is expeted to be larger when the orret pdf is used; this is easily onfirmed by the large n limit [ -2ln(Λ) χ 2 ] : ( Λ Λ ) 2 2 2nQ = 2n χ χ = s b s b -2lnQ b "small" - << 0 "large" = "large" - >> 0 "small" if (s+b) ok if (b) ok ; real data m H the plot is a little artoon of an ideal situation (e.g. Higgs searh at LEP2), that never happened : s+b good separation bad " the ross-setion dereases when m H inreases (TRUE); disovery!!! (FALSE). Paolo Bagnaia FS

24 2/3 interpretation of results: parameter µ put : σ = σ b + µ σ SMs ; plot : horizontal : m H. vertial : µ (= σ obs s / σ SMs ) ; the lines show, with a given L int and analysis, the expeted limit (--), and the atual observed limit ( ), i.e. the µ value exluded at 95% CL; the band ( ) shows the flutuations at ±1σ (±2σ) of the "bkgd only" hypothesis. B : the ase µ 1 has no well-defined physial meaning (= a theory idential to the SM, but with a saled ross setion); if the lines are at µ > 1, the "distane" respet to µ=1 reflets the L int neessary to get the limit in the SM in this hypothetial ase, the region 140 < m H < 170 GeV is exluded at 95% CL, while the expeted limit was GeV (either bad luk or hint of disovery). Paolo Bagnaia FS

25 3/3 interpretation of results: p-value ƒ Q exp (H 0 ) ƒ B (Q H 0 ) Q obs p Q the p-value is the probability to get the same result or another less probable, in the hypothesis of bkgd only. x = statistis (e.g. likelihood ratio); H 0 = "null hypothesis (i.e. bkgd only); i.e. p ƒ(x H 0)dx obs x p small H 0 OT probable disovery!!! vertial : p-value; horizontal : m H. the band ( ) shows the flutuations at 1σ (2σ). B the disovery orresponds to the red line below 5σ (or ), not shown in this fake plot. Paolo Bagnaia FS

26 Referenes 1. lassi textbook : Eadie et al., Statistial methods in experimental physis; 2. modern textbook : Cowan, Statistial data analysis; 3. simple, for experimentalists : Cranmer, Pratial Statistis for the LHC, [arxiv: v1]; (also CER Aademi Training, Feb 2-5, 2009); 4. bayesian : G.D'Agostini, YR CER statistial proedure for Higgs : A.L.Read, J. Phys. G: ul. Part. Phys. 28 (2002) 2693; 6. mass limits : R.Cousins, Am.J.Phys., 63 (5), 398 (1995). 7. [PDG has everything, but very onise] bells are related to dramati events even outside partile physis Paolo Bagnaia FS

27 End End of hapter 11 Paolo Bagnaia FS

Millennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion

Millennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six

More information

Complexity of Regularization RBF Networks

Complexity of Regularization RBF Networks Complexity of Regularization RBF Networks Mark A Kon Department of Mathematis and Statistis Boston University Boston, MA 02215 mkon@buedu Leszek Plaskota Institute of Applied Mathematis University of Warsaw

More information

Heavy Flavour Production at HERA

Heavy Flavour Production at HERA Journal of Physis: onferene Series OPEN AESS Heavy Flavour Prodution at HERA To ite this artile: Ringail Plaakyt and the H1 and Zeus ollaborations 014 J. Phys.: onf. Ser. 556 010 View the artile online

More information

V. Interacting Particles

V. Interacting Particles V. Interating Partiles V.A The Cumulant Expansion The examples studied in the previous setion involve non-interating partiles. It is preisely the lak of interations that renders these problems exatly solvable.

More information

The Laws of Acceleration

The Laws of Acceleration The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the

More information

Maximum Entropy and Exponential Families

Maximum Entropy and Exponential Families Maximum Entropy and Exponential Families April 9, 209 Abstrat The goal of this note is to derive the exponential form of probability distribution from more basi onsiderations, in partiular Entropy. It

More information

arxiv:gr-qc/ v2 6 Feb 2004

arxiv:gr-qc/ v2 6 Feb 2004 Hubble Red Shift and the Anomalous Aeleration of Pioneer 0 and arxiv:gr-q/0402024v2 6 Feb 2004 Kostadin Trenčevski Faulty of Natural Sienes and Mathematis, P.O.Box 62, 000 Skopje, Maedonia Abstrat It this

More information

Methods of evaluating tests

Methods of evaluating tests Methods of evaluating tests Let X,, 1 Xn be i.i.d. Bernoulli( p ). Then 5 j= 1 j ( 5, ) T = X Binomial p. We test 1 H : p vs. 1 1 H : p>. We saw that a LRT is 1 if t k* φ ( x ) =. otherwise (t is the observed

More information

Wave Propagation through Random Media

Wave Propagation through Random Media Chapter 3. Wave Propagation through Random Media 3. Charateristis of Wave Behavior Sound propagation through random media is the entral part of this investigation. This hapter presents a frame of referene

More information

Relativistic Dynamics

Relativistic Dynamics Chapter 7 Relativisti Dynamis 7.1 General Priniples of Dynamis 7.2 Relativisti Ation As stated in Setion A.2, all of dynamis is derived from the priniple of least ation. Thus it is our hore to find a suitable

More information

The Effectiveness of the Linear Hull Effect

The Effectiveness of the Linear Hull Effect The Effetiveness of the Linear Hull Effet S. Murphy Tehnial Report RHUL MA 009 9 6 Otober 009 Department of Mathematis Royal Holloway, University of London Egham, Surrey TW0 0EX, England http://www.rhul.a.uk/mathematis/tehreports

More information

MAC Calculus II Summer All you need to know on partial fractions and more

MAC Calculus II Summer All you need to know on partial fractions and more MC -75-Calulus II Summer 00 ll you need to know on partial frations and more What are partial frations? following forms:.... where, α are onstants. Partial frations are frations of one of the + α, ( +

More information

RESEARCH ON RANDOM FOURIER WAVE-NUMBER SPECTRUM OF FLUCTUATING WIND SPEED

RESEARCH ON RANDOM FOURIER WAVE-NUMBER SPECTRUM OF FLUCTUATING WIND SPEED The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan RESEARCH ON RANDOM FORIER WAVE-NMBER SPECTRM OF FLCTATING WIND SPEED Qi Yan 1, Jie Li 1 Ph D. andidate, Department

More information

Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach

Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach Amerian Journal of heoretial and Applied tatistis 6; 5(-): -8 Published online January 7, 6 (http://www.sienepublishinggroup.om/j/ajtas) doi:.648/j.ajtas.s.65.4 IN: 36-8999 (Print); IN: 36-96 (Online)

More information

Critical Reflections on the Hafele and Keating Experiment

Critical Reflections on the Hafele and Keating Experiment Critial Refletions on the Hafele and Keating Experiment W.Nawrot In 1971 Hafele and Keating performed their famous experiment whih onfirmed the time dilation predited by SRT by use of marosopi loks. As

More information

Computer Science 786S - Statistical Methods in Natural Language Processing and Data Analysis Page 1

Computer Science 786S - Statistical Methods in Natural Language Processing and Data Analysis Page 1 Computer Siene 786S - Statistial Methods in Natural Language Proessing and Data Analysis Page 1 Hypothesis Testing A statistial hypothesis is a statement about the nature of the distribution of a random

More information

Time and Energy, Inertia and Gravity

Time and Energy, Inertia and Gravity Time and Energy, Inertia and Gravity The Relationship between Time, Aeleration, and Veloity and its Affet on Energy, and the Relationship between Inertia and Gravity Copyright 00 Joseph A. Rybzyk Abstrat

More information

Analysis of discretization in the direct simulation Monte Carlo

Analysis of discretization in the direct simulation Monte Carlo PHYSICS OF FLUIDS VOLUME 1, UMBER 1 OCTOBER Analysis of disretization in the diret simulation Monte Carlo iolas G. Hadjionstantinou a) Department of Mehanial Engineering, Massahusetts Institute of Tehnology,

More information

Evaluation of effect of blade internal modes on sensitivity of Advanced LIGO

Evaluation of effect of blade internal modes on sensitivity of Advanced LIGO Evaluation of effet of blade internal modes on sensitivity of Advaned LIGO T0074-00-R Norna A Robertson 5 th Otober 00. Introdution The urrent model used to estimate the isolation ahieved by the quadruple

More information

A Spatiotemporal Approach to Passive Sound Source Localization

A Spatiotemporal Approach to Passive Sound Source Localization A Spatiotemporal Approah Passive Sound Soure Loalization Pasi Pertilä, Mikko Parviainen, Teemu Korhonen and Ari Visa Institute of Signal Proessing Tampere University of Tehnology, P.O.Box 553, FIN-330,

More information

Case I: 2 users In case of 2 users, the probability of error for user 1 was earlier derived to be 2 A1

Case I: 2 users In case of 2 users, the probability of error for user 1 was earlier derived to be 2 A1 MUTLIUSER DETECTION (Letures 9 and 0) 6:33:546 Wireless Communiations Tehnologies Instrutor: Dr. Narayan Mandayam Summary By Shweta Shrivastava (shwetash@winlab.rutgers.edu) bstrat This artile ontinues

More information

Einstein s Three Mistakes in Special Relativity Revealed. Copyright Joseph A. Rybczyk

Einstein s Three Mistakes in Special Relativity Revealed. Copyright Joseph A. Rybczyk Einstein s Three Mistakes in Speial Relativity Revealed Copyright Joseph A. Rybzyk Abstrat When the evidene supported priniples of eletromagneti propagation are properly applied, the derived theory is

More information

Likelihood-confidence intervals for quantiles in Extreme Value Distributions

Likelihood-confidence intervals for quantiles in Extreme Value Distributions Likelihood-onfidene intervals for quantiles in Extreme Value Distributions A. Bolívar, E. Díaz-Franés, J. Ortega, and E. Vilhis. Centro de Investigaión en Matemátias; A.P. 42, Guanajuato, Gto. 36; Méxio

More information

12.1 Events at the same proper distance from some event

12.1 Events at the same proper distance from some event Chapter 1 Uniform Aeleration 1.1 Events at the same proper distane from some event Consider the set of events that are at a fixed proper distane from some event. Loating the origin of spae-time at this

More information

Relativity in Classical Physics

Relativity in Classical Physics Relativity in Classial Physis Main Points Introdution Galilean (Newtonian) Relativity Relativity & Eletromagnetism Mihelson-Morley Experiment Introdution The theory of relativity deals with the study of

More information

UPPER-TRUNCATED POWER LAW DISTRIBUTIONS

UPPER-TRUNCATED POWER LAW DISTRIBUTIONS Fratals, Vol. 9, No. (00) 09 World Sientifi Publishing Company UPPER-TRUNCATED POWER LAW DISTRIBUTIONS STEPHEN M. BURROUGHS and SARAH F. TEBBENS College of Marine Siene, University of South Florida, St.

More information

Measuring & Inducing Neural Activity Using Extracellular Fields I: Inverse systems approach

Measuring & Inducing Neural Activity Using Extracellular Fields I: Inverse systems approach Measuring & Induing Neural Ativity Using Extraellular Fields I: Inverse systems approah Keith Dillon Department of Eletrial and Computer Engineering University of California San Diego 9500 Gilman Dr. La

More information

Properties of Quarks

Properties of Quarks PHY04 Partile Physis 9 Dr C N Booth Properties of Quarks In the earlier part of this ourse, we have disussed three families of leptons but prinipally onentrated on one doublet of quarks, the u and d. We

More information

3 Tidal systems modelling: ASMITA model

3 Tidal systems modelling: ASMITA model 3 Tidal systems modelling: ASMITA model 3.1 Introdution For many pratial appliations, simulation and predition of oastal behaviour (morphologial development of shorefae, beahes and dunes) at a ertain level

More information

18.05 Problem Set 6, Spring 2014 Solutions

18.05 Problem Set 6, Spring 2014 Solutions 8.5 Problem Set 6, Spring 4 Solutions Problem. pts.) a) Throughout this problem we will let x be the data of 4 heads out of 5 tosses. We have 4/5 =.56. Computing the likelihoods: 5 5 px H )=.5) 5 px H

More information

Chapter 26 Lecture Notes

Chapter 26 Lecture Notes Chapter 26 Leture Notes Physis 2424 - Strauss Formulas: t = t0 1 v L = L0 1 v m = m0 1 v E = m 0 2 + KE = m 2 KE = m 2 -m 0 2 mv 0 p= mv = 1 v E 2 = p 2 2 + m 2 0 4 v + u u = 2 1 + vu There were two revolutions

More information

DIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS

DIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS CHAPTER 4 DIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS 4.1 INTRODUCTION Around the world, environmental and ost onsiousness are foring utilities to install

More information

The Second Postulate of Euclid and the Hyperbolic Geometry

The Second Postulate of Euclid and the Hyperbolic Geometry 1 The Seond Postulate of Eulid and the Hyperboli Geometry Yuriy N. Zayko Department of Applied Informatis, Faulty of Publi Administration, Russian Presidential Aademy of National Eonomy and Publi Administration,

More information

The Hanging Chain. John McCuan. January 19, 2006

The Hanging Chain. John McCuan. January 19, 2006 The Hanging Chain John MCuan January 19, 2006 1 Introdution We onsider a hain of length L attahed to two points (a, u a and (b, u b in the plane. It is assumed that the hain hangs in the plane under a

More information

Danielle Maddix AA238 Final Project December 9, 2016

Danielle Maddix AA238 Final Project December 9, 2016 Struture and Parameter Learning in Bayesian Networks with Appliations to Prediting Breast Caner Tumor Malignany in a Lower Dimension Feature Spae Danielle Maddix AA238 Final Projet Deember 9, 2016 Abstrat

More information

Supplementary information for: All-optical signal processing using dynamic Brillouin gratings

Supplementary information for: All-optical signal processing using dynamic Brillouin gratings Supplementary information for: All-optial signal proessing using dynami Brillouin gratings Maro Santagiustina, Sanghoon Chin 2, Niolay Primerov 2, Leonora Ursini, Lu Thévena 2 Department of Information

More information

A simple expression for radial distribution functions of pure fluids and mixtures

A simple expression for radial distribution functions of pure fluids and mixtures A simple expression for radial distribution funtions of pure fluids and mixtures Enrio Matteoli a) Istituto di Chimia Quantistia ed Energetia Moleolare, CNR, Via Risorgimento, 35, 56126 Pisa, Italy G.

More information

Mean Activity Coefficients of Peroxodisulfates in Saturated Solutions of the Conversion System 2NH 4. H 2 O at 20 C and 30 C

Mean Activity Coefficients of Peroxodisulfates in Saturated Solutions of the Conversion System 2NH 4. H 2 O at 20 C and 30 C Mean Ativity Coeffiients of Peroxodisulfates in Saturated Solutions of the Conversion System NH 4 Na S O 8 H O at 0 C and 0 C Jan Balej Heřmanova 5, 170 00 Prague 7, Czeh Republi balejan@seznam.z Abstrat:

More information

The experimental plan of displacement- and frequency-noise free laser interferometer

The experimental plan of displacement- and frequency-noise free laser interferometer 7th Edoardo Amaldi Conferene on Gravitational Waves (Amaldi7) Journal of Physis: Conferene Series 122 (2008) 012022 The experimental plan of displaement- and frequeny-noise free laser interferometer K

More information

Gluing Potential Energy Surfaces with Rare Event Simulations

Gluing Potential Energy Surfaces with Rare Event Simulations This is an open aess artile published under an ACS AuthorChoie Liense, whih permits opying and redistribution of the artile or any adaptations for non-ommerial purposes. pubs.as.org/jctc Gluing Potential

More information

Calibration of Piping Assessment Models in the Netherlands

Calibration of Piping Assessment Models in the Netherlands ISGSR 2011 - Vogt, Shuppener, Straub & Bräu (eds) - 2011 Bundesanstalt für Wasserbau ISBN 978-3-939230-01-4 Calibration of Piping Assessment Models in the Netherlands J. Lopez de la Cruz & E.O.F. Calle

More information

Physical Laws, Absolutes, Relative Absolutes and Relativistic Time Phenomena

Physical Laws, Absolutes, Relative Absolutes and Relativistic Time Phenomena Page 1 of 10 Physial Laws, Absolutes, Relative Absolutes and Relativisti Time Phenomena Antonio Ruggeri modexp@iafria.om Sine in the field of knowledge we deal with absolutes, there are absolute laws that

More information

Quantum Mechanics: Wheeler: Physics 6210

Quantum Mechanics: Wheeler: Physics 6210 Quantum Mehanis: Wheeler: Physis 60 Problems some modified from Sakurai, hapter. W. S..: The Pauli matries, σ i, are a triple of matries, σ, σ i = σ, σ, σ 3 given by σ = σ = σ 3 = i i Let stand for the

More information

arxiv:physics/ v1 [physics.class-ph] 8 Aug 2003

arxiv:physics/ v1 [physics.class-ph] 8 Aug 2003 arxiv:physis/0308036v1 [physis.lass-ph] 8 Aug 003 On the meaning of Lorentz ovariane Lszl E. Szab Theoretial Physis Researh Group of the Hungarian Aademy of Sienes Department of History and Philosophy

More information

23.1 Tuning controllers, in the large view Quoting from Section 16.7:

23.1 Tuning controllers, in the large view Quoting from Section 16.7: Lesson 23. Tuning a real ontroller - modeling, proess identifiation, fine tuning 23.0 Context We have learned to view proesses as dynami systems, taking are to identify their input, intermediate, and output

More information

Four-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field

Four-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field Four-dimensional equation of motion for visous ompressible substane with regard to the aeleration field, pressure field and dissipation field Sergey G. Fedosin PO box 6488, Sviazeva str. -79, Perm, Russia

More information

UTC. Engineering 329. Proportional Controller Design. Speed System. John Beverly. Green Team. John Beverly Keith Skiles John Barker.

UTC. Engineering 329. Proportional Controller Design. Speed System. John Beverly. Green Team. John Beverly Keith Skiles John Barker. UTC Engineering 329 Proportional Controller Design for Speed System By John Beverly Green Team John Beverly Keith Skiles John Barker 24 Mar 2006 Introdution This experiment is intended test the variable

More information

Lecture 3 - Lorentz Transformations

Lecture 3 - Lorentz Transformations Leture - Lorentz Transformations A Puzzle... Example A ruler is positioned perpendiular to a wall. A stik of length L flies by at speed v. It travels in front of the ruler, so that it obsures part of the

More information

QUANTUM MECHANICS II PHYS 517. Solutions to Problem Set # 1

QUANTUM MECHANICS II PHYS 517. Solutions to Problem Set # 1 QUANTUM MECHANICS II PHYS 57 Solutions to Problem Set #. The hamiltonian for a lassial harmoni osillator an be written in many different forms, suh as use ω = k/m H = p m + kx H = P + Q hω a. Find a anonial

More information

Theory. Coupled Rooms

Theory. Coupled Rooms Theory of Coupled Rooms For: nternal only Report No.: R/50/TCR Prepared by:. N. taey B.., MO Otober 00 .00 Objet.. The objet of this doument is present the theory alulations to estimate the reverberant

More information

Relativity fundamentals explained well (I hope) Walter F. Smith, Haverford College

Relativity fundamentals explained well (I hope) Walter F. Smith, Haverford College Relativity fundamentals explained well (I hope) Walter F. Smith, Haverford College 3-14-06 1 Propagation of waves through a medium As you ll reall from last semester, when the speed of sound is measured

More information

Chapter 8 Hypothesis Testing

Chapter 8 Hypothesis Testing Leture 5 for BST 63: Statistial Theory II Kui Zhang, Spring Chapter 8 Hypothesis Testing Setion 8 Introdution Definition 8 A hypothesis is a statement about a population parameter Definition 8 The two

More information

Supplementary Materials

Supplementary Materials Supplementary Materials Neural population partitioning and a onurrent brain-mahine interfae for sequential motor funtion Maryam M. Shanehi, Rollin C. Hu, Marissa Powers, Gregory W. Wornell, Emery N. Brown

More information

u x u t Internal Waves

u x u t Internal Waves Internal Waves We now examine internal waves for the ase in whih there are two distint layers and in whih the lower layer is at rest. This is an approximation of the ase in whih the upper layer is muh

More information

9 Geophysics and Radio-Astronomy: VLBI VeryLongBaseInterferometry

9 Geophysics and Radio-Astronomy: VLBI VeryLongBaseInterferometry 9 Geophysis and Radio-Astronomy: VLBI VeryLongBaseInterferometry VLBI is an interferometry tehnique used in radio astronomy, in whih two or more signals, oming from the same astronomial objet, are reeived

More information

Wavetech, LLC. Ultrafast Pulses and GVD. John O Hara Created: Dec. 6, 2013

Wavetech, LLC. Ultrafast Pulses and GVD. John O Hara Created: Dec. 6, 2013 Ultrafast Pulses and GVD John O Hara Created: De. 6, 3 Introdution This doument overs the basi onepts of group veloity dispersion (GVD) and ultrafast pulse propagation in an optial fiber. Neessarily, it

More information

The Complete Energy Translations in the Detailed. Decay Process of Baryonic Sub-Atomic Particles. P.G.Bass.

The Complete Energy Translations in the Detailed. Decay Process of Baryonic Sub-Atomic Particles. P.G.Bass. The Complete Energy Translations in the Detailed Deay Proess of Baryoni Su-Atomi Partiles. [4] P.G.Bass. PGBass P12 Version 1..3 www.relativitydomains.om August 218 Astrat. This is the final paper on the

More information

The Unified Geometrical Theory of Fields and Particles

The Unified Geometrical Theory of Fields and Particles Applied Mathematis, 014, 5, 347-351 Published Online February 014 (http://www.sirp.org/journal/am) http://dx.doi.org/10.436/am.014.53036 The Unified Geometrial Theory of Fields and Partiles Amagh Nduka

More information

Charmed Hadron Production in Polarized pp Collisions

Charmed Hadron Production in Polarized pp Collisions Charmed Hadron Prodution in Polarized pp Collisions oshiyuki Morii Λ and Kazumasa Ohkuma Λ Division of Sienes for Natural Environment, Faulty of Human Development, Kobe University, Nada, Kobe 657-851,

More information

Breakdown of the Special Theory of Relativity as Proven by Synchronization of Clocks

Breakdown of the Special Theory of Relativity as Proven by Synchronization of Clocks Breakdown of the Speial Theory of Relativity as Proven by Synhronization of Cloks Koshun Suto Koshun_suto19@mbr.nifty.om Abstrat In this paper, a hypothetial preferred frame of referene is presumed, and

More information

Weighted K-Nearest Neighbor Revisited

Weighted K-Nearest Neighbor Revisited Weighted -Nearest Neighbor Revisited M. Biego University of Verona Verona, Italy Email: manuele.biego@univr.it M. Loog Delft University of Tehnology Delft, The Netherlands Email: m.loog@tudelft.nl Abstrat

More information

CHAPTER 26 The Special Theory of Relativity

CHAPTER 26 The Special Theory of Relativity CHAPTER 6 The Speial Theory of Relativity Units Galilean-Newtonian Relativity Postulates of the Speial Theory of Relativity Simultaneity Time Dilation and the Twin Paradox Length Contration Four-Dimensional

More information

A Motion Paradox from Einstein s Relativity of Simultaneity

A Motion Paradox from Einstein s Relativity of Simultaneity Motion Paradox from Einstein s Relativity of Simultaneity Espen Gaarder Haug Norwegian University of Life Sienes November 5, 7 bstrat We are desribing a new and potentially important paradox related to

More information

A Queueing Model for Call Blending in Call Centers

A Queueing Model for Call Blending in Call Centers A Queueing Model for Call Blending in Call Centers Sandjai Bhulai and Ger Koole Vrije Universiteit Amsterdam Faulty of Sienes De Boelelaan 1081a 1081 HV Amsterdam The Netherlands E-mail: {sbhulai, koole}@s.vu.nl

More information

22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E')

22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') 22.54 Neutron Interations and Appliations (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') Referenes -- J. R. Lamarsh, Introdution to Nulear Reator Theory (Addison-Wesley, Reading, 1966),

More information

Relative Maxima and Minima sections 4.3

Relative Maxima and Minima sections 4.3 Relative Maxima and Minima setions 4.3 Definition. By a ritial point of a funtion f we mean a point x 0 in the domain at whih either the derivative is zero or it does not exists. So, geometrially, one

More information

To work algebraically with exponential functions, we need to use the laws of exponents. You should

To work algebraically with exponential functions, we need to use the laws of exponents. You should Prealulus: Exponential and Logisti Funtions Conepts: Exponential Funtions, the base e, logisti funtions, properties. Laws of Exponents memorize these laws. To work algebraially with exponential funtions,

More information

2 The Bayesian Perspective of Distributions Viewed as Information

2 The Bayesian Perspective of Distributions Viewed as Information A PRIMER ON BAYESIAN INFERENCE For the next few assignments, we are going to fous on the Bayesian way of thinking and learn how a Bayesian approahes the problem of statistial modeling and inferene. The

More information

Hankel Optimal Model Order Reduction 1

Hankel Optimal Model Order Reduction 1 Massahusetts Institute of Tehnology Department of Eletrial Engineering and Computer Siene 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Hankel Optimal Model Order Redution 1 This leture overs both

More information

The gravitational phenomena without the curved spacetime

The gravitational phenomena without the curved spacetime The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime,

More information

Indian Institute of Technology Bombay. Department of Electrical Engineering. EE 325 Probability and Random Processes Lecture Notes 3 July 28, 2014

Indian Institute of Technology Bombay. Department of Electrical Engineering. EE 325 Probability and Random Processes Lecture Notes 3 July 28, 2014 Indian Institute of Tehnology Bombay Department of Eletrial Engineering Handout 5 EE 325 Probability and Random Proesses Leture Notes 3 July 28, 2014 1 Axiomati Probability We have learned some paradoxes

More information

Armenian Theory of Special Relativity (Illustrated) Robert Nazaryan 1 and Haik Nazaryan 2

Armenian Theory of Special Relativity (Illustrated) Robert Nazaryan 1 and Haik Nazaryan 2 29606 Robert Nazaryan Haik Nazaryan/ Elixir Nulear & Radiation Phys. 78 (205) 29606-2967 Available online at www.elixirpublishers.om (Elixir International Journal) Nulear Radiation Physis Elixir Nulear

More information

EE 321 Project Spring 2018

EE 321 Project Spring 2018 EE 21 Projet Spring 2018 This ourse projet is intended to be an individual effort projet. The student is required to omplete the work individually, without help from anyone else. (The student may, however,

More information

CONDITIONAL CONFIDENCE INTERVAL FOR THE SCALE PARAMETER OF A WEIBULL DISTRIBUTION. Smail Mahdi

CONDITIONAL CONFIDENCE INTERVAL FOR THE SCALE PARAMETER OF A WEIBULL DISTRIBUTION. Smail Mahdi Serdia Math. J. 30 (2004), 55 70 CONDITIONAL CONFIDENCE INTERVAL FOR THE SCALE PARAMETER OF A WEIBULL DISTRIBUTION Smail Mahdi Communiated by N. M. Yanev Abstrat. A two-sided onditional onfidene interval

More information

A model for measurement of the states in a coupled-dot qubit

A model for measurement of the states in a coupled-dot qubit A model for measurement of the states in a oupled-dot qubit H B Sun and H M Wiseman Centre for Quantum Computer Tehnology Centre for Quantum Dynamis Griffith University Brisbane 4 QLD Australia E-mail:

More information

Non-Markovian study of the relativistic magnetic-dipole spontaneous emission process of hydrogen-like atoms

Non-Markovian study of the relativistic magnetic-dipole spontaneous emission process of hydrogen-like atoms NSTTUTE OF PHYSCS PUBLSHNG JOURNAL OF PHYSCS B: ATOMC, MOLECULAR AND OPTCAL PHYSCS J. Phys. B: At. Mol. Opt. Phys. 39 ) 7 85 doi:.88/953-75/39/8/ Non-Markovian study of the relativisti magneti-dipole spontaneous

More information

INTRO VIDEOS. LESSON 9.5: The Doppler Effect

INTRO VIDEOS. LESSON 9.5: The Doppler Effect DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS INTRO VIDEOS Big Bang Theory of the Doppler Effet Doppler Effet LESSON 9.5: The Doppler Effet 1. Essential Idea: The Doppler Effet desribes the phenomenon

More information

Modes are solutions, of Maxwell s equation applied to a specific device.

Modes are solutions, of Maxwell s equation applied to a specific device. Mirowave Integrated Ciruits Prof. Jayanta Mukherjee Department of Eletrial Engineering Indian Institute of Tehnology, Bombay Mod 01, Le 06 Mirowave omponents Welome to another module of this NPTEL mok

More information

22.01 Fall 2015, Problem Set 6 (Normal Version Solutions)

22.01 Fall 2015, Problem Set 6 (Normal Version Solutions) .0 Fall 05, Problem Set 6 (Normal Version Solutions) Due: November, :59PM on Stellar November 4, 05 Complete all the assigned problems, and do make sure to show your intermediate work. Please upload your

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 16 Aug 2004

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 16 Aug 2004 Computational omplexity and fundamental limitations to fermioni quantum Monte Carlo simulations arxiv:ond-mat/0408370v1 [ond-mat.stat-meh] 16 Aug 2004 Matthias Troyer, 1 Uwe-Jens Wiese 2 1 Theoretishe

More information

Studies of b-hadron production and CKM matrix elements using semileptonic decays

Studies of b-hadron production and CKM matrix elements using semileptonic decays BEACH 218 Penihe, June 17-23, 218 Studies of b-hadron prodution and CKM matrix elements using semileptoni deays Olaf Steinkamp on behalf of the LHCb ollaboration Physik-Institut der Universität Zürih Winterthurerstrasse

More information

LOGISTIC REGRESSION IN DEPRESSION CLASSIFICATION

LOGISTIC REGRESSION IN DEPRESSION CLASSIFICATION LOGISIC REGRESSIO I DEPRESSIO CLASSIFICAIO J. Kual,. V. ran, M. Bareš KSE, FJFI, CVU v Praze PCP, CS, 3LF UK v Praze Abstrat Well nown logisti regression and the other binary response models an be used

More information

Nuclear Shell Structure Evolution Theory

Nuclear Shell Structure Evolution Theory Nulear Shell Struture Evolution Theory Zhengda Wang (1) Xiaobin Wang () Xiaodong Zhang () Xiaohun Wang () (1) Institute of Modern physis Chinese Aademy of SienesLan Zhou P. R. China 70000 () Seagate Tehnology

More information

19 Lecture 19: Cosmic Microwave Background Radiation

19 Lecture 19: Cosmic Microwave Background Radiation PHYS 652: Astrophysis 97 19 Leture 19: Cosmi Mirowave Bakground Radiation Observe the void its emptiness emits a pure light. Chuang-tzu The Big Piture: Today we are disussing the osmi mirowave bakground

More information

arxiv:gr-qc/ v7 14 Dec 2003

arxiv:gr-qc/ v7 14 Dec 2003 Propagation of light in non-inertial referene frames Vesselin Petkov Siene College, Conordia University 1455 De Maisonneuve Boulevard West Montreal, Quebe, Canada H3G 1M8 vpetkov@alor.onordia.a arxiv:gr-q/9909081v7

More information

Final Review. A Puzzle... Special Relativity. Direction of the Force. Moving at the Speed of Light

Final Review. A Puzzle... Special Relativity. Direction of the Force. Moving at the Speed of Light Final Review A Puzzle... Diretion of the Fore A point harge q is loated a fixed height h above an infinite horizontal onduting plane. Another point harge q is loated a height z (with z > h) above the plane.

More information

QCLAS Sensor for Purity Monitoring in Medical Gas Supply Lines

QCLAS Sensor for Purity Monitoring in Medical Gas Supply Lines DOI.56/sensoren6/P3. QLAS Sensor for Purity Monitoring in Medial Gas Supply Lines Henrik Zimmermann, Mathias Wiese, Alessandro Ragnoni neoplas ontrol GmbH, Walther-Rathenau-Str. 49a, 7489 Greifswald, Germany

More information

Process engineers are often faced with the task of

Process engineers are often faced with the task of Fluids and Solids Handling Eliminate Iteration from Flow Problems John D. Barry Middough, In. This artile introdues a novel approah to solving flow and pipe-sizing problems based on two new dimensionless

More information

Particle-wave symmetry in Quantum Mechanics And Special Relativity Theory

Particle-wave symmetry in Quantum Mechanics And Special Relativity Theory Partile-wave symmetry in Quantum Mehanis And Speial Relativity Theory Author one: XiaoLin Li,Chongqing,China,hidebrain@hotmail.om Corresponding author: XiaoLin Li, Chongqing,China,hidebrain@hotmail.om

More information

Generalized Dimensional Analysis

Generalized Dimensional Analysis #HUTP-92/A036 7/92 Generalized Dimensional Analysis arxiv:hep-ph/9207278v1 31 Jul 1992 Howard Georgi Lyman Laboratory of Physis Harvard University Cambridge, MA 02138 Abstrat I desribe a version of so-alled

More information

4 Puck s action plane fracture criteria

4 Puck s action plane fracture criteria 4 Puk s ation plane frature riteria 4. Fiber frature riteria Fiber frature is primarily aused by a stressing σ whih ats parallel to the fibers. For (σ, σ, τ )-ombinations the use of a simple maximum stress

More information

1 Josephson Effect. dx + f f 3 = 0 (1)

1 Josephson Effect. dx + f f 3 = 0 (1) Josephson Effet In 96 Brian Josephson, then a year old graduate student, made a remarkable predition that two superondutors separated by a thin insulating barrier should give rise to a spontaneous zero

More information

EECS 120 Signals & Systems University of California, Berkeley: Fall 2005 Gastpar November 16, Solutions to Exam 2

EECS 120 Signals & Systems University of California, Berkeley: Fall 2005 Gastpar November 16, Solutions to Exam 2 EECS 0 Signals & Systems University of California, Berkeley: Fall 005 Gastpar November 6, 005 Solutions to Exam Last name First name SID You have hour and 45 minutes to omplete this exam. he exam is losed-book

More information

Name Solutions to Test 1 September 23, 2016

Name Solutions to Test 1 September 23, 2016 Name Solutions to Test 1 September 3, 016 This test onsists of three parts. Please note that in parts II and III, you an skip one question of those offered. Possibly useful formulas: F qequb x xvt E Evpx

More information

SUGRA Models at the LHC

SUGRA Models at the LHC SUGRA odels at the LHC Bhaskar Dutta and Teruki Kamon Texas A& University PASCOS 9, DESY, Germany 9 th July 9 SUGRA odels at the LHC SUSY at the LHC D (or l + l -, τ+τ High P T jet [mass differene is large]

More information

Tests of fit for symmetric variance gamma distributions

Tests of fit for symmetric variance gamma distributions Tests of fit for symmetri variane gamma distributions Fragiadakis Kostas UADPhilEon, National and Kapodistrian University of Athens, 4 Euripidou Street, 05 59 Athens, Greee. Keywords: Variane Gamma Distribution,

More information

Where as discussed previously we interpret solutions to this partial differential equation in the weak sense: b

Where as discussed previously we interpret solutions to this partial differential equation in the weak sense: b Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential

More information

Sensor management for PRF selection in the track-before-detect context

Sensor management for PRF selection in the track-before-detect context Sensor management for PRF seletion in the tra-before-detet ontext Fotios Katsilieris, Yvo Boers, and Hans Driessen Thales Nederland B.V. Haasbergerstraat 49, 7554 PA Hengelo, the Netherlands Email: {Fotios.Katsilieris,

More information

Development of Fuzzy Extreme Value Theory. Populations

Development of Fuzzy Extreme Value Theory. Populations Applied Mathematial Sienes, Vol. 6, 0, no. 7, 58 5834 Development of Fuzzy Extreme Value Theory Control Charts Using α -uts for Sewed Populations Rungsarit Intaramo Department of Mathematis, Faulty of

More information

When p = 1, the solution is indeterminate, but we get the correct answer in the limit.

When p = 1, the solution is indeterminate, but we get the correct answer in the limit. The Mathematia Journal Gambler s Ruin and First Passage Time Jan Vrbik We investigate the lassial problem of a gambler repeatedly betting $1 on the flip of a potentially biased oin until he either loses

More information