Geometric Lagrangians. for. massive higher-spin fields
|
|
- Κόρη Φιλιππίδης
- 7 χρόνια πριν
- Προβολές:
Transcript
1 Geometric Lagrangians for massive higher-spin fields Dario Francia Chalmers University of Technology - Göteborg Valencia - 3 rd RTN Workshop October, D. F. - in preparation
2 Introduction I: geometry for higher-spin fields? Central object in Maxwell, Yang-Mills (spin 1) and Einstein (spin 2) theories is the curvature : A µ F µ ν, h µν R µ ν, ρ σ. It gives us both dynamical informations and geometrical meaning. Is it possible to find any similar description for higher-spins?
3 Introduction II: massive fields - the Fierz system Free propagation of massive irreps of the Poincaré group can be described by spin-s, massive boson symmetric rank-s tensor ϕ µ1... µ s spin-(s + 1/2), massive fermion symmetric rank-s spinor-tensor ψ µ a 1... µ s satisfying the conditions [Fierz, 1939]: ( m 2 ) ϕ µ1... µ s = 0, (i γ α α m) ψ µ a 1... µ s = 0, α ϕ α µ2... µ s = 0, α ψ α a µ 2... µ s = 0, ϕ α α µ 3... µ s = 0, γ α ψ α a µ 2... µ s = 0. At the Lagrangian level the problem is how to implement the conditions Typically, known solutions { α ϕ α µ2...µ s = 0 ϕ α α µ 3...µ s = 0 { α ψα a µ 2...µ s = 0 γ α ψ α a µ 2... µ s = 0? involve auxiliary fields, are not simple deformations of the massless theory.
4 Introduction III: the examples of spin 1 and spin 2 Spin 1 and spin 2: massive theory as quadratic deformation of the geometric theory: Spin 1 [Proca] L(m = 0) = 1 4 F µνf µν L(m) = 1 4 F µνf µν 1 2 m 2 A µ A µ, ν F µν m 2 A µ = 0, µ ν F µν 0 µ A µ = 0 Spin 2 [Fierz-Pauli] L(m = 0) = 1 2 h µν (R µν 1 2 ηµν R) L(m) = 1 2 h µν {(R µν 1 2 ηµν R) m 2 (h µν η µν h }{{ α α ) } } F ierz P auli mass term µ (R µν 1 2 ηµν R) 0 µ (h µν ν h α α ) = 0 µ h µν ν h α α = 0 Fierz-Pauli constraint
5 Introduction IV: difficulties of the Fronsdal s theory Canonical description of higher-spin gauge fields encoded in the Fronsdal s equation (1978): F ϕ ϕ + 2 ϕ = 0 gauge invariant under δ ϕ = Λ iff Λ ( Λ α α ) 0; Lagrangian description iff ϕ ( ϕ α β α β ) 0. Gauge invariance with a traceless parameter the Einstein tensor does not need to be (and is not) divergenceless {F 1 2 η F } = 1 2 η F, not possible to extend the results of spin 1 and spin 2 [Aragone-Deser-Yang 1987]
6 Introduction V: what do we need to generalise the spin 1 and spin 2 cases? To try and reproduce the geometric construction we need the following: Candidate tensors to play the role of higher-spin curvatures. Candidate Ricci tensors and Dirac tensors, to define the free equations of motion. These have to be consistent with the known result for the non-geometric, constrained theory of Fronsdal. Bianchi identities to be satisfied by the generalised Ricci tensors. Suitable mass deformations, such that the on-shell consequence of the Bianchi identity imply that the system reduces to the Fierz form.
7 Plan I. Higher-spin curvatures II. Geometric massive theory I: bosons Ricci tensors, Bianchi identities, mass def ormations. III. Geometric massive theory II: fermions { the example of spin 3/2, mass def ormations. IV. Propagators and uniqueness
8 I. Higher-spin curvatures
9 Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα (but also s = 7/2) Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ = 0. Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
10 Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα (but also s = 7/2) Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ = 0. Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
11 Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα Λ α α 0! Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ 0. (but also s = 7/2) Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
12 Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα Λ α α 0! Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ 0. (but also s = 7/2) (curvatures no constraints Fronsdal s theory is not geometric) Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
13 Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα Λ α α 0! Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ 0. (but also s = 7/2) (curvatures no constraints Fronsdal s theory is not geometric) Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
14 II. Geometric massive theory I: bosons
15 Candidate Ricci tensors (s = 3) Spin-3 curvature: saturate three indices to restore the symmetries of ϕ µ1 µ 2 µ 3 : R ρ1 ρ 2 ρ 3, µ 1 µ 2 µ 3 R, restore dimensions of a relativistic wave operator, introducing non-localities: R 1 R, so defining the simplest candidate Ricci tensor. [D.F. - A. Sagnotti, 2002] This possibility is highly non-unique infinitely many -more singular- ones: 1 R A ϕ (a) 1 R + a 2 R, 2 A ϕ (a) = 0 Meaning? Lagrangian origin (i.e. Bianchi identity)?
16 Geometric Lagrangians for massless bosons Spin s: the most general candidate Ricci tensor A ϕ (a 1,... a k,... ) is such that, for almost all choices of a 1,... a k,... : (CONSISTENCY ) the equation A ϕ = 0 implies the compensator equation F 3 3 α ϕ = 0, with δ α ϕ = Λ Fronsdal form, after partial gauge-fixing. (LAGRANGIAN) it is possible to define an identically divergenceless Einstein tensor E ϕ s.t. L = 1 2 ϕ E ϕ E ϕ = 0 A ϕ = 0,
17 Mass deformation for bosons I Spin 2: R µν 1 2 η µν R m 2 (h µν η µν h ) = 0. M h = h µν η µν h }{{} F ierz P auli mass term, h µ µ h = 0, }{{} F ierz P auli constraint Spin s: General idea: higher traces should appear in the mass term : E ϕ (a 1,... a k,... ) m 2 M ϕ = 0 M ϕ = λ k η k ϕ [k], }{{} generalised F P mass term ϕ ϕ = 0, }{{} F P constraint How?
18 Mass deformation for bosons II: spin 4 µ ϕ ϕ ϕ We look for λ 1, λ 2 s.t. in this way M ϕ = {ϕ + λ 1 η ϕ + λ 2 η 2 ϕ } = µ ϕ + k η µ ϕ ; {E ϕ ({a 1, a 2 }) m 2 M ϕ (λ 1, λ 2 )} = 0 µ ϕ + k η µ ϕ = 0 µ ϕ = 0 µ ϕ = 0 The condition µ ϕ = 0 implies in turn A ϕ ({a 1, a 2 }) = 0 {a 1, a 2 } From the resulting equation (λ 1 = 1, λ 2 = 1) A ϕ ({a 1, a 2 }) m 2 (ϕ η ϕ ϕ ) = 0 ϕ = 0 ϕ = 0 A ϕ ({a 1, a 2 }) = ϕ {a 1, a 2 } the last condition needed to put the system in the Fierz form.
19 Mass deformation for bosons III The general solution is L (m) = 1 2 ϕ {E ϕ ({a 1,... a k... }) m 2 M ϕ }, where, for s = 2n or s = 2n + 1 M ϕ = ϕ η ϕ η 2 ϕ 1 3 η 3 ϕ 1 (2 n 3)!! η n ϕ [n]. The structure of the mass term is to be understood such that M ϕ = µ ϕ + k 1 η µ ϕ k m η [m] µ [m] ϕ +...., so that M ϕ = 0 implies the basic, Fierz-Pauli constraint µ ϕ = ϕ ϕ = 0, together with all its consistency conditions: µ [m] ϕ = 0 m. The massive theory is not unique: The Fierz-Pauli constraint implies A ϕ ({a k}) = 0, ({a k }) Under this condition all traces of ϕ can be shown to vanish on-shell this implies in turn A ϕ ({a k }) = ϕ, ({a k }), and then any Lagrangian equation can be reduced to the Fierz system.
20 III. Geometric massive theory II: fermions
21 The example of spin 3/2 Spin-3/2 massive theory as quadratic deformation of the geometric, Rarita-Schwinger theory: Spin 3/2 L(m = 0) = 1 2 ψ µ ( R µ 1 2 γµ R) + h.c. L(m) = 1 2 ψ µ {( R µ 1 2 γµ R) m (ψ µ γ µ ψ)} + h.c. µ ( R µ 1 2 γµ R) 0 µ (ψ µ γ µ ψ) = 0 µ ψ µ ψ = 0 (fermionic) Fierz-Pauli constraint
22 Mass deformation for fermions In the general case the fermionic analogue of the Fierz-Pauli constraint is µ ψ ψ ψ ψ = 0 Spin 5/2: We look for a mass term M ψ s. t. E ψ m (ψ λ 1 γ ψ λ 2 η ψ ) }{{} = 0, M ψ = µ ψ + ρ 1 γ µ ψ = 0. M ψ The unique solution is L = 1 2 ψ {E ψ m (ψ γ ψ η ψ )} + h.c.. Spin s + 1/2: The same procedure leads to the general solution L = 1 2 ψ {E ψ m (ψ s j=0 1 (2j 1)!! γ η j ψ [j] s i=1 1 (2i 3)!! η i ψ [i] )} + h.c. The generalised Fierz-pauli constraint implies the Fierz system for all geometric Einstein tensors (including those with more singular terms) issue of uniqueness (already at the simplest level).
23 IV. Propagators & uniqueness
24 Massless propagators Are all those solutions really equivalent? Propagator from Lagrangian equation with an external current: E ϕ (a 1,... a k... ) = J ϕ = G J Current exchange J ϕ = J G J consistency conditions on the polarisations flowing: almost all geometric theories give the wrong result, but one. The correct theory has a simple structure: The kinetic tensor has the compensator form A ϕ = F 3 3 γ(ϕ); It satisfies the identities : { A ϕ 1 2 A ϕ 0 A ϕ 0, and the Lagrangian is L = 1 2 ϕ (A ϕ 1 2 η A ϕ + η 2 B ϕ ) ϕ J [ D.F. - J. Mourad - A. Sagnotti, 2007]
25 Massive propagators - spin 4 Consider the Lagrangian L (m) = 1 2 ϕ {E ϕ (a 1, a 2 ) m 2 M ϕ } ϕ J, where J is a conserved current. The on-shell condition A ϕ (a 1, a 2 ) = 0 reduces the equation of motion to A ϕ (a 1, a 2 ) m 2 (ϕ η ϕ ϕ ) = J. where A ϕ (a 1, a 2 ) o.s. = ϕ 2 ϕ ϕ = 0. The whole structure of the propagator is encoded in the coefficients of M ϕ Inverting the equation of motion J ϕ = 1 p 2 m {J J 6 2 D + 3 J J + 3 (D + 1)(D + 3) J J } while the corresponding computation for the massless case gives J ϕ = 1 p 2 {J J 6 D + 2 J J + 3 D(D + 2) J J } thus showing the (generalised) vdvz discontinuity for higher-spin fields.
26 KK reduction and uniqueness How to understand the origin of the Fierz-Pauli mass-term, for s = 2? KK reduction ( m 2 ): R µν 1 2 η µν R (h η h ) +..., How to perform a KK reduction of a non local theory? Maybe not clear ( 1 1 m 2 ), but still it is unambiguously defined the pure massive contribution of the resulting reduction: E ϕ = (ϕ + k 1 η ϕ + k 2 η 2 ϕ +... ) +..., Is it possible to find a geometric theory whose box term encodes the coefficients of the generalised FP mass term? up to spin 4 (at least) it is the one selected by the analysis of the current exchange. Why the mass term works well with all geometric tensors? Not too strange, also true for spin 2: the non-local theory defined by the eom R µν 1 2 η µν R + λ (η 2 ) R m 2 (h η h ), reduces to the Fierz system, and gives the correct current exchange!
27 Comments & Conclusions Foregoing locality, linear dynamics of higher-spin gauge fields in geometric fashion; infinitely many formulations are indeed available, at the free level. relationship with a parallel, local, unconstrained formulation, or analysis of the propagator, shows that there is one preferred geometric theory. [ D.F. - A. Sagnotti, 05, 06] Generalisation of the Fierz-Pauli mass term, for bosons and fermions, involving all (γ )traces of the field. Description of massive theory, for spin greater than two, that does not involve auxiliary fields (and no dimension-dependent coefficients). Mass term unique Issue of uniqueness: Massive theory degenerate memory of the correct theory, by KK analysis? The van Dam - Veltman - Zakharov discontinuity is shown to be present for any spin.
6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραHigher Derivative Gravity Theories
Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραSpace-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραSymmetric Stress-Energy Tensor
Chapter 3 Symmetric Stress-Energy ensor We noticed that Noether s conserved currents are arbitrary up to the addition of a divergence-less field. Exploiting this freedom the canonical stress-energy tensor
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότερα= {{D α, D α }, D α }. = [D α, 4iσ µ α α D α µ ] = 4iσ µ α α [Dα, D α ] µ.
PHY 396 T: SUSY Solutions for problem set #1. Problem 2(a): First of all, [D α, D 2 D α D α ] = {D α, D α }D α D α {D α, D α } = {D α, D α }D α + D α {D α, D α } (S.1) = {{D α, D α }, D α }. Second, {D
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραThe challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραCHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραLecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραOverview. Transition Semantics. Configurations and the transition relation. Executions and computation
Overview Transition Semantics Configurations and the transition relation Executions and computation Inference rules for small-step structural operational semantics for the simple imperative language Transition
Διαβάστε περισσότεραAppendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Διαβάστε περισσότεραFrom the finite to the transfinite: Λµ-terms and streams
From the finite to the transfinite: Λµ-terms and streams WIR 2014 Fanny He f.he@bath.ac.uk Alexis Saurin alexis.saurin@pps.univ-paris-diderot.fr 12 July 2014 The Λµ-calculus Syntax of Λµ t ::= x λx.t (t)u
Διαβάστε περισσότεραΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης
Α.Τ.Ε.Ι. ΙΟΝΙΩΝ ΝΗΣΩΝ ΠΑΡΑΡΤΗΜΑ ΑΡΓΟΣΤΟΛΙΟΥ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ «Η διαμόρφωση επικοινωνιακής στρατηγικής (και των τακτικών ενεργειών) για την ενδυνάμωση της εταιρικής
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραΣυστήματα Διαχείρισης Βάσεων Δεδομένων
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo
Διαβάστε περισσότεραLecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραFrom Fierz-Pauli to Einstein-Hilbert
From Fierz-Pauli to Einstein-Hilbert Gravity as a special relativistic field theory Bert Janssen Universidad de Granada & CAFPE A pedagogical review References: R. Feynman et al, The Feynman Lectures on
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραDerivation of Optical-Bloch Equations
Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be
Διαβάστε περισσότεραTutorial problem set 6,
GENERAL RELATIVITY Tutorial problem set 6, 01.11.2013. SOLUTIONS PROBLEM 1 Killing vectors. a Show that the commutator of two Killing vectors is a Killing vector. Show that a linear combination with constant
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραBounding Nonsplitting Enumeration Degrees
Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,
Διαβάστε περισσότεραOn a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ
Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 1: Elements of Syntactic Structure Το περιεχόμενο του μαθήματος διατίθεται με άδεια
Διαβάστε περισσότεραMA 342N Assignment 1 Due 24 February 2016
M 342N ssignment Due 24 February 206 Id: 342N-s206-.m4,v. 206/02/5 2:25:36 john Exp john. Suppose that q, in addition to satisfying the assumptions from lecture, is an even function. Prove that η(λ = 0,
Διαβάστε περισσότεραThe kinetic and potential energies as T = 1 2. (m i η2 i k(η i+1 η i ) 2 ). (3) The Hooke s law F = Y ξ, (6) with a discrete analog
Lecture 12: Introduction to Analytical Mechanics of Continuous Systems Lagrangian Density for Continuous Systems The kinetic and potential energies as T = 1 2 i η2 i (1 and V = 1 2 i+1 η i 2, i (2 where
Διαβάστε περισσότερα3+1 Splitting of the Generalized Harmonic Equations
3+1 Splitting of the Generalized Harmonic Equations David Brown North Carolina State University EGM June 2011 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότερα8.324 Relativistic Quantum Field Theory II
8.324 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall 200 Lecture 2 3: GENERAL ASPECTS OF QUANTUM ELECTRODYNAMICS 3.: RENORMALIZED LAGRANGIAN Consider the Lagrangian
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραChapter 3: Ordinal Numbers
Chapter 3: Ordinal Numbers There are two kinds of number.. Ordinal numbers (0th), st, 2nd, 3rd, 4th, 5th,..., ω, ω +,... ω2, ω2+,... ω 2... answers to the question What position is... in a sequence? What
Διαβάστε περισσότεραSection 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Διαβάστε περισσότεραDr. D. Dinev, Department of Structural Mechanics, UACEG
Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents
Διαβάστε περισσότεραGeneral 2 2 PT -Symmetric Matrices and Jordan Blocks 1
General 2 2 PT -Symmetric Matrices and Jordan Blocks 1 Qing-hai Wang National University of Singapore Quantum Physics with Non-Hermitian Operators Max-Planck-Institut für Physik komplexer Systeme Dresden,
Διαβάστε περισσότεραLecture 21: Properties and robustness of LSE
Lecture 21: Properties and robustness of LSE BLUE: Robustness of LSE against normality We now study properties of l τ β and σ 2 under assumption A2, i.e., without the normality assumption on ε. From Theorem
Διαβάστε περισσότεραVariational Wavefunction for the Helium Atom
Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραForced Pendulum Numerical approach
Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Διαβάστε περισσότεραSection 8.2 Graphs of Polar Equations
Section 8. Graphs of Polar Equations Graphing Polar Equations The graph of a polar equation r = f(θ), or more generally F(r,θ) = 0, consists of all points P that have at least one polar representation
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραDiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραAbout these lecture notes. Simply Typed λ-calculus. Types
About these lecture notes Simply Typed λ-calculus Akim Demaille akim@lrde.epita.fr EPITA École Pour l Informatique et les Techniques Avancées Many of these slides are largely inspired from Andrew D. Ker
Διαβάστε περισσότεραOn the Galois Group of Linear Difference-Differential Equations
On the Galois Group of Linear Difference-Differential Equations Ruyong Feng KLMM, Chinese Academy of Sciences, China Ruyong Feng (KLMM, CAS) Galois Group 1 / 19 Contents 1 Basic Notations and Concepts
Διαβάστε περισσότεραAssalamu `alaikum wr. wb.
LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump
Διαβάστε περισσότεραSymmetries and Feynman Rules for Ramond Sector in WZW-type Superstring Field Theories
Symmetries and Feynman Rules for Ramond Sector in WZW-type Superstring Field Theories Hiroshi Kunitomo YITP 215/3/5 @Nara Women s U. Nara arxiv:1412.5281, 153.xxxx Y TP YUKAWA INSTITUTE FOR THEORETICAL
Διαβάστε περισσότερα