Symmetric polynomials and partitions Eugene Mukhin

Size: px
Start display at page:

Download "Symmetric polynomials and partitions Eugene Mukhin"

Transcription

1 Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation w is a one to one map of the set {,..., n} to itself. Thee ae n! pemutations. The poduct of pemutations w w 2 is just the composition of maps. We will wite w x fo x w(),..., x w(n). An invesion in pemutation w is a pai of numbes i < j n, such that w(i) > w(j). A pemutation w is called even o odd if the numbe of invesions is even o odd. The sign of a pemutation w, sgn(w) is if w is odd and sgn(w) = if w is even. Execise: Pove that sgn(w w 2 ) = sgn(w 2 w ) = sgn(w )sgn(w 2 ). Symmetic polynomials ae polynomials which do not change values if some aguments ae switched. Definition: A polynomial p(x) is called symmetic if p(x) = p(w x) fo any pemutation w. Fo example, let n = 3, then a polynomial p(x) = x + x 2 + x 3 is symmetic, say p(3, 5, 2) = p( 5, 2, 3). The polynomial (x) = x + x 2 + x 3 x is not symmetic, (, 2, 3) (2,, 3). Note that p(x) is the sum of all vaiables, no matte how you shuffle the vaiables, but if you pemute the vaiables in, you can also obtain expessions x 2 + x + x 3 x 2, x 3 + x 2 + x x 2 and x 3 + x + x x 2. Execise: Pove that a polynomial p(x) is symmetic if and only if p(x) does not change unde the pemutations of vaiables as an expession..2. Monomial polynomials. Let λ = (λ,..., λ n ). Definition: The monomial symmetic polynomial m λ is the sum of monomial x λ... x λn n and all distinct monomials obtained fom it by a pemutation of vaiables. Fo example, if λ = (2,, ) then m λ = x 2 x 2 x 3 + x x 2 2x 3 + x x 2 x 2 3. The total degee of m λ is i λ i, the degee of m λ in each vaiable x i is λ. In ode to avoid epetitions among m λ we will always assume that λ λ n. A basis is the smallest set of polynomials though which you can expess all the othes. Definition: A set of symmetic polynomials S is called a basis, if ) any symmetic polynomial can be expessed as a sum of polynomials fom S with some coefficients. 2) No polynomial fom S can be expessed as a sum of othe polynomials fom S. Execise: The monomial polynomials {m λ, λ = (λ, λ n 0)} fom a basis.

2 2.3. Patitions. Definition: The vecto λ = (λ,..., λ n ) is called a patition of k if λ λ n 0 and λ = λ +... λ n = k. The numbe k is called length, numbes λ i ae called pats of λ. Patitions can be epesented by pictues called Young diagams (o Fees diagams). The Young diagam of λ consists of n ows of boxes aligned on the left, such that i-th ow is ight on i + -st ow. The length of i-th ow is λ i. The conjugate patition λ is the patition with the Young diagams consisting of columns of lengths λ i. Fo example λ is the numbe of nonzeo pats of λ. If λ = (3, 3, ) then λ = (3, 2, 2). Also λ = λ. Execise: Show that the numbe of patitions of n with odd distinct pats euals to numbe of self conjugated patitions of n (that is patitions λ with the popety λ = λ ). Definition: A patition λ is said to be lage than a patition µ if λ = µ and we have λ µ λ + λ 2 µ + µ 2 λ + λ 2 + λ 3 µ + µ 2 + µ 3... The lagest patition of length k is (k, 0, 0,..., 0). If k n then the smallest patition of length k is (,,...,, 0,..., 0). Execise: Show that λ µ if and only if the Young diagams of λ can be obtained fom Young diagam of µ by aising some boxes fom lowe ows to highe ones. Execise: Find an example of two patitions of 6, none of which is geate then anothe..4. Multiplying monomial polynomials. Let µ + ν be a patition (λ + µ, λ 2 + µ 2,..., λ n + µ n ). Lemma. m λ m µ = m λ+µ + a ν λ,µm ν, a ν λ,µ Z 0. Execise: Poof the lemma. ν<λ+µ 2. Bases 2.. Elementay polynomials. Definition: The elementay polynomials e λ ae defined by the fomulas e λ = e λ e λ2..., e = m λ whee λ = (,...,, 0,..., 0) ( ones).

3 3 Lemma 2. We have e λ = m λ + µ<λ a λµ m µ. Theefoe {e λ, λ = (n λ λ m 0), m Z 0 } fom a basis of symmetic polynomials in n vaiables. Execise: Poof the lemma. Note that one can expess any symmetic polynomial as a sum of poducts of e i, i = 0,,..., n, whee e 0 =. In the mathematical language e,..., e n ae a set of geneatos of ou ing Powe sum polynomials. Definition: The powe sum polynomials p λ ae defined by the fomulas p λ = p λ p λ2..., p = m λ, whee λ = (, 0,..., 0) Lemma 3. We have p λ = a λ m λ + µ>λ b λµ m m u, b λµ Z 0, whee a λ is a natual numbe. Theefoe {p λ, λ = (λ λ n 0)} fom a basis of symmetic polynomials. Execise: Poof the lemma Complete polynomials. Definition: The complete polynomials h λ ae defined by the fomulas h λ = h λ h λ2..., h = m λ. λ = Lemma 4. Polynomials {h λ, λ = (λ λ n 0)} fom a basis of symmetic polynomials. Execise: Poof the lemma using the elation () below.

4 Schu polynomials. Definition: A Schu function s λ is the sum of function x λ x... x λn i n x i x j with all functions obtained fom it by a pemutation of vaiables. Euivalently, s λ is the antisymmetization of monomial x λ x λ x λn+n n divided by the Vandemond function i<j (x i x j ), ( ) s λ = ( ) sgn(w) x λ w() xλ x λn+n w(n) / i x j ), i<j(x w whee the sum is ove all pemutations of n elements. Execise: Show that s λ is a symmetic polynomial. Lemma 5. i<j s λ = m λ + µ<λ K λµ m µ. Theefoe {s λ, λ = (λ λ n 0)} fom a basis of symmetic polynomials. Execise: Poof the lemma. In fact K λµ ae vey impotant nonnegative integes called Kostka numbes Geneating functions and elations between diffeent bases. We have the geneating functions n n E(t) := e i t i = ( + x i t), H(t) := P (t) := h i t i = p i t i = n n x i t, x i x i t. Note that the fist euality is a vesion of Vieta theoem. We have the elations theefoe H(t)E( t) =, H (t) = P (t)h(t ), ( ) s e i h i = 0, () h = p i h i.

5 5 Execise: Use the elation P (t) = (log H(t)) to show that h = p λ n, z λ = (i m i m i!), z λ λ = whee m i is the numbe of pats of λ eual i. 3. Counting symmetic polynomials 3.. Gaussian binomial coefficients. Definition: The Gaussian binomial coefficient is given by ( ) m = ( m )( m )... ( m + ). ( )( 2 )... ( ) Execise: Pove the following identities ( ) ( m m = ( ) ( ) ( ) m m m = + n n ( ) n ( + i t) = i(i )/2 i n i t = ( ) n + i i ), (2) ( ) m = ( ) m + m, (3) t i, (4) t i. (5) The identity 2 shows that Gaussian binomial coefficients ae genealizations of usual binomial coefficients. The identities 3 ae called Pascal idenitites, the identity 4 is called Newton binomial fomula. Use one of the identities to show that Gaussian binomial coefficient is a polynomial in Main theoem via ecusion elations. Define the counting function of of symmetic polynomials by χ n,k () = a i,k i, a i,k = {λ, λ k, λ = i}. The numbe a i,k counts polynomials of total degee i, such that degee in any vaiable is at most k. Theoem 6. ) ( n + k χ k,n () = k Execise: Pove the theoem using Pascal identity (3)..

6 Main theoem via h k. Note that LHS of (5) is eual to H(t) whee x i ae substituted with i. Execise: Pove Theoem 6 by compaing monomials in h k with n + vaiables and patitions λ contibuting to a i. 4. Appendix 4.. Eule Identity. The Eule function is defined by the fomula ϕ(t) = ( t i ). The coefficient of t k in function /ϕ(t) euals the numbe of all patitions of k. Execise: Use geneating functions to pove that the numbe of patitions of n with odd pats is eual to the numbe of patitions of n with uneual pats. Lemma 7. (Eule identity) ϕ(t) = n= ( ) i t (3n2 n)/2. The numbes (3n 2 n)/2 ae called pentagon numbes. Compae to numbes n, tiangula numbes n(n + )/2, suae numbes n 2. Execise: Pove Eule identity by constucting a map fom patitions consisting of odd numbe of uneual pats to patitions consisting of even numbe of uneual pats Roges Ramanudjan Identities. Lemma = + n(n+) 8 ( )( 2 )... ( n ), 4 6 = + n2 9 ( )( 2 )... ( n ). Hee in LHS of the fist identity we have powes of which have emaindes 2 o 3 mod 5 and in LHS of the second identity the powes have emaindes o 4 mod 5. Execise: Refomulate Roges-Ramanudjan identities in the language of patitions A challenge. Let N k be a numbe of diffeent figues obtained by a putting two Young diagams of patitions λ, µ, such that λ + µ = k on top of each othe. Fo example, N 0 = N =, N 2 = 3, N 4 = 5, N 5 = 0, N 6 = 6. n= n= CHALLENGE. Compute the function N(t) = N it i. At the moment I know the answe but I do not know an elementay poof of it.

Saturated and weakly saturated hypergraphs

Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 6-7 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B

The Binomial Distribution

The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

Week 3-4: Permutations and Combinations

Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

Model Question Paper Mathematics Class XII

Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

The LCOE is defined as the energy price (\$ per unit of energy output) for which the Net Present Value of the investment is zero.

Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

Semipartial (Part) and Partial Correlation

Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

Chapter 4: Matrix Norms

EE448/58 Vesion.0 John Stensby Chate 4: Matix Noms The analysis of matix-based algoithms often equies use of matix noms. These algoithms need a way to quantify the "size" of a matix o the "distance" between

Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

On Some Functions Involving the lcm and gcd of Integer Tuples

SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91-100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

Chapter 30: Magnetic Fields Due to Currents

d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.1-1 T) i to ue a lage cuent flowing though a wie.

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

Integer sequences from walks in graphs

otes on umbe Theoy and Discete Mathematics Vol. 9, 3, o. 3, 78 84 Intege seuences fom walks in gahs Enesto Estada, and José A. de la Peña Deatment of Mathematics and Statistics, Univesity of Stathclyde

Voltage ( = Electric Potential )

V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

Separation probabilities for products of permutations

Sepaation pobabilities fo poducts of pemutations Olivie Benadi, Rosena R. X. Du, Alejando H. Moales and Richad P. Stanley Mach 1, 2012 Abstact We study the mixing popeties of pemutations obtained as a

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

CHAPTER 10 Aggregate Demand I

CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

Closed-form expressions for integrals of traveling wave reductions of integrable lattice equations

Closed-fom expessions fo integals of taveling wave eductions of integable lattice equations Dinh T Tan, Pete H van de Kamp, G R W Quispel Depatment of Mathematics, La Tobe Univesity, Victoia 386, Austalia

12.1. FÖRSTER RESONANCE ENERGY TRANSFER

ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1-1 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

An Introduction to Omega

An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

Lab M4: The Torsional Pendulum and Moment of Inertia

M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the

A r. (Can you see that this just gives the formula we had above?)

24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

Do Vibrations Make Sound?

Do Vibations Make Sound? Gade 1: Sound Pobe Aligned with National Standads oveview Students will lean about sound and vibations. This activity will allow students to see and hea how vibations do in fact

Problem Set # 9 Solutions

Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

Seshadri constants and surfaces of minimal degree

Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth

Fast FPT-algorithms for cleaning grids

Fast FPT-algoithms fo cleaning gids Josep Diaz Dimitios M. Thilikos Abstact We conside the poblem that given a gaph G and a paamete k asks whethe the edit distance of G and a ectangula gid is at most k.

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

Supplementary Material for EpiDiff

Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module

Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

Converting knowledge Into Practice

Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

FI3300 Corporate Finance

Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multi-peiod multi-cf time-value-of-money poblems: Geneal case Pepetuity

Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

Questions for Review. By buying bonds This period you save s, next period you get s(1+r)

MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

Strength Analysis and Optimization Design about the key parts of the Robot

Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Pint): 2320-9356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.25-29 Stength Analysis and Optimization Design

Chapter 2. Electrostatics

Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.

Displacement, Velocity And Acceleration

Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,

Chapter 4: Fluid Kinematics

Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

Efficient Redundancy Techniques for Latency Reduction in Cloud Systems

Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo

Exam 3: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

Explicit, analytical solution of scaling quantum graphs. Abstract

Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 06459-0155, USA E-mail: ydabaghian@wesleyan.edu (Januay 6, 2003)

PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary

PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC

9.3 Surface Area of Pyramids

Page 1 of 9 9.3 Suface Aea of Pyamids and Cones Goa Find the suface aeas of pyamids and cones. Key Wods pyamid height of a pyamid sant height of a pyamid cone height of a cone sant height of a cone The

Capital Investment and Liquidity Management with collateralized debt.

TSE 54 Novembe 14 Capital Investment and Liquidity Management with collatealized debt. Ewan Piee, Stéphane Villeneuve and Xavie Wain 7 Capital Investment and Liquidity Management with collatealized debt.

Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation

Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,

CHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL

CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The

Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

Financing Terms in the EOQ Model

Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

Channel selection in e-commerce age: A strategic analysis of co-op advertising models

Jounal of Industial Engineeing and Management JIEM, 013 6(1):89-103 Online ISSN: 013-0953 Pint ISSN: 013-843 http://dx.doi.og/10.396/jiem.664 Channel selection in e-commece age: A stategic analysis of

Quantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w

1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a

CRRC-1 Method #1: Standard Practice for Measuring Solar Reflectance of a Flat, Opaque, and Heterogeneous Surface Using a Portable Solar Reflectometer

CRRC- Method #: Standad Pactice fo Measuing Sola Reflectance of a Flat, Opaque, and Heteogeneous Suface Using a Potable Sola Reflectomete Scope This standad pactice coves a technique fo estimating the

Comparing Availability of Various Rack Power Redundancy Configurations

Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance the availability

Research on Risk Assessment of the Transformer Based on Life Cycle Cost

ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,

How To Write A Theory Of The Concept Of The Mind In A Quey

Jounal of Atificial Intelligence Reseach 31 (2008) 157-204 Submitted 06/07; published 01/08 Conjunctive Quey Answeing fo the Desciption Logic SHIQ Bite Glimm Ian Hoocks Oxfod Univesity Computing Laboatoy,

Chapter 4: Fluid Kinematics

4-1 Lagangian g and Euleian Desciptions 4-2 Fundamentals of Flow Visualization 4-3 Kinematic Desciption 4-4 Reynolds Tanspot Theoem (RTT) 4-1 Lagangian and Euleian Desciptions (1) Lagangian desciption

PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

Liquidity and Insurance for the Unemployed

Liquidity and Insuance fo the Unemployed Robet Shime Univesity of Chicago and NBER shime@uchicago.edu Iván Wening MIT, NBER and UTDT iwening@mit.edu Fist Daft: July 15, 2003 This Vesion: Septembe 22, 2005

Physics HSC Course Stage 6. Space. Part 1: Earth s gravitational field

Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe

Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

Chris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment

Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability

When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.

Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

Radians mc-ty-adians-2009-1 Atschoolweusuallyleantomeasueanangleindegees. Howeve,theeaeothewaysof measuinganangle. Onethatweaegoingtohavealookatheeismeasuinganglesinunits called adians. In many scientific

Thank you for participating in Teach It First!

Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident

NURBS Drawing Week 5, Lecture 10

CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

MATHEMATICAL SIMULATION OF MASS SPECTRUM

MATHEMATICA SIMUATION OF MASS SPECTUM.Beánek, J.Knížek, Z. Pulpán 3, M. Hubálek 4, V. Novák Univesity of South Bohemia, Ceske Budejovice, Chales Univesity, Hadec Kalove, 3 Univesity of Hadec Kalove, Hadec

Liquidity and Insurance for the Unemployed*

Fedeal Reseve Bank of Minneapolis Reseach Depatment Staff Repot 366 Decembe 2005 Liquidity and Insuance fo the Unemployed* Robet Shime Univesity of Chicago and National Bueau of Economic Reseach Iván Wening

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita

Approximation Algorithms for Data Management in Networks

Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute

Financial Derivatives for Computer Network Capacity Markets with Quality-of-Service Guarantees

Financial Deivatives fo Compute Netwok Capacity Makets with Quality-of-Sevice Guaantees Pette Pettesson pp@kth.se Febuay 2003 SICS Technical Repot T2003:03 Keywods Netwoking and Intenet Achitectue. Abstact

How To Find The Optimal Stategy For Buying Life Insuance

Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications

JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat