# Uniform Convergence of Fourier Series Michael Taylor

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

## Transcript

1 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 2) S N fθ) = 1 fθ ϕ)d N ϕ) dϕ, 3) the last identity by virtue of x N T 1 D N ϕ) = 2N k=0 N k= N e ikϕ = e inϕ 2N = k=0 e ikϕ sinn + 1/2)ϕ, sin ϕ/2 x k N 1 x2n+1 = x, 1 x using e iϕ for x, and multiplying numerator and denominator by e iϕ/2. Using sin N + 1 ) ϕ = cos ϕ 2 2 sin Nϕ + sin ϕ cos Nϕ, 2 we deduce that S N fθ) fθ) = 1 [fθ ϕ) fθ)]d N ϕ) dϕ T 4) = 1 1 g θ ϕ) sin Nϕ dϕ + h θ ϕ) cos Nϕ dϕ, T 1 T 1 5) fθ ϕ) fθ) g θ ϕ) =, h θ ϕ) = fθ ϕ) fθ). Clearly, for N 0, 6) f L 1 T 1 ) = ĥθ±n) = ˆf±N) 0 as N, the convergence to 0 by the Riemann-Lebesgue lemma. Applying the Riemann-Lebesgue lemma to ĝ θ ±N) gives the following. 1

2 2 Proposition 1. Let f L 1 T 1 ). Let K T 1 be compact. Then 7) S N fθ) fθ), uniformly for θ K, provided that 8) {g θ : θ K} is a relatively compact subset of L 1 T 1 ). Proof. The Riemann-Lebesgue lemma plus the compactness hypothesis 8) implies that ĝ θ N) goes to 0 as N, uniformly in θ K. In more detail, take ε > 0. Pick a finite set {θ j : 1 j Mε)} such that, with g j ϕ) = g εj ϕ), 8A) θ K, g j g θ L 1 ε, for some j Mε). The compactness hypothesis 8) guarantees you can do this. The Riemann-Lebesgue lemma says that, for each j {1,..., Mε)}, there exists N j such that 8B) ĝ j N) < ε, N such that N > N j. Now set Ñε) = max{n j : 1 j Mε)}. By 8A) we have, for all θ K, 8C) ) ĝ θ N) min ĝ j N) + ĝ j N) ĝ θ N) j ε + ε, provided N > Ñε). The desired conclusion 7) follows from this, in concert with 4) 6). The following is an important special case. Corollary 2. Let f C ω T 1 ), i.e., 9) fθ ϕ) fθ) Cω ϕ ), θ, ϕ T 1. Assume the modulus of continuity ωt) satisfies 10) Then 7) holds with K = T 1. 0 ωt) t dt <. Proof. We claim the hypotheses 9) 10) imply that 11) g θ is a continuous function of θ with values in L 1 T 1 ).

3 3 Given this, the compactness condition 8) holds, with K = T 1. T 1, θ ν θ 0. We see that So let θ ν, θ 0 12) g θν ϕ) g θ0 ϕ) for all ϕ T 1 \ 0, and that 13) g θν ϕ) C ω ϕ ) ϕ = Hϕ). Hence g θν ϕ) g θ0 ϕ) 0 for all ϕ T 1 \ 0, and 14) g θν ϕ) g θ0 ϕ) 2Hϕ). Now 10) implies H L 1 T 1 ), so the convergence 15) T 1 g θν ϕ) g θ0 ϕ) dϕ 0 follows by the Dominated Convergence Theorem. The following is a version of Riemann localization. Proposition 3. Take f L 1 T 1 ). Assume f = 0 on O, an open subset of T 1, and let K O be compact. Then S N f f uniformly on K. Proof. Take an interval I = ε, ε) so small that 16) θ K, ϕ I = θ ϕ O, so 17) θ K, ϕ I = g θ ϕ) = 0. Then take ψ CT 1 ) such that ψϕ) = 1 for ϕ < ε/2, ψϕ) = 0 for ϕ ε. Then 18) θ K = ψg θ 0 = g θ ϕ) 1 ψϕ) [fθ ϕ) fθ)]. Since 1 ψϕ))/ tanϕ/2) is continuous on T 1, it follows that 19) θ g θ is continuous from K to L 1 T 1 ). Thus 8) holds, and Proposition 3 follows from Proposition 1. Putting together Corollary 2 and Proposition 3 gives the following.

4 4 Corollary 4. Take f L 1 T 1 ). Let O T 1 be open and assume f O C ω O), with ω satisfying 10). Let K O be compact. Then S N f f uniformly on K. We now produce another strengthening of Corollary 2. Proposition 5. Take f L 1 T 1 ), and let K T 1 be compact. Assume f K CK) and 20) fθ ϕ) fθ) Cω ϕ ), θ K, ϕ T 1, ω is measurable and satisfies 10). Then 7) holds. Proof. Again it suffices to show that 21) θ g θ is continuous from K to L 1 T 1 ). So let θ ν, θ 0 K and θ ν θ 0. We continue to have 13) 14), i.e., 22) g θν ϕ) g θ0 ϕ) 2Hϕ), H L 1 T 1 ). We need a replacement for 12). In fact, we claim that 23) g θν g θ0 in measure, as ν, i.e., if m denotes Lebesgue measure on T 1, then, for each ε > 0, 24) me εν ) 0 as ν, 25) E εν = {ϕ T 1 : g θν ϕ) g θ0 ϕ) > ε}. In fact, 23) follows directly from the continuity of f K, together with the fact that, with f θ ϕ) = fθ ϕ), 26) f θν f θ0 in measure, as ν, itself a consequence of the fact that 27) f θν f θ0 in L 1 -norm, i.e., 28) T 1 f θν ϕ) f θ0 ϕ) dϕ 0, together with Chebechev s inequality, ) 29) m {ϕ : F ϕ) > ε} 1 ε F L 1. It is a variant of the Dominated Convergence Theorem cf. [T], pp ), that 22) 23) imply 15). This completes the proof of Proposition 5. Bringing in an argument used for Riemann localization in Proposition 3, we have the following strengthening of Proposition 5.

5 5 Proposition 6. Take f L 1 T 1 ) and let K T 1 be compact. Assume f K CK). Assume there exists ε > 0 such that 30) fθ ϕ) fθ) < cω ϕ ), θ K, ϕ < ε, ω is measurable on [0, ] and satisfies 10). Then 7) holds. Proof. Take ψ as in 18), and set 31) g θ ϕ) = u θ ϕ) + v θ ϕ), 32) u θ ϕ) = ψϕ) fθ ϕ) fθ), v θ ϕ) = 1 ψϕ) [fθ ϕ) fθ)]. Clearly f L 1 T 1 ) and f K bounded implies {v θ : θ K} is relatively compact in L 1 T 1 ). Meanwhile an analysis parallel to 21) 29) applies here, with g θ replaced by u θ, under the hypotheses given above. Reference [T] M. Taylor, Measure Theory and Integration, AMS, Providence RI, 2006.

### Example 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 περισσότερα

### ST5224: 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 περισσότερα

### 2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

Διαβάστε περισσότερα

### 2 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 περισσότερα

### Every 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 περισσότερα

### Areas and Lengths in Polar Coordinates

Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

### Ordinal 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 περισσότερα

### Areas and Lengths in Polar Coordinates

Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

### 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

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 περισσότερα

### ORDINAL 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 περισσότερα

### Math221: 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 περισσότερα

### 5. Choice under Uncertainty

5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation

Διαβάστε περισσότερα

### Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.

Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:

Διαβάστε περισσότερα

### CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.

Διαβάστε περισσότερα

### The semiclassical Garding inequality

The semiclassical Garding inequality We give a proof of the semiclassical Garding inequality (Theorem 4.1 using as the only black box the Calderon-Vaillancourt Theorem. 1 Anti-Wick quantization For (q,

Διαβάστε περισσότερα

### Fractional 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 περισσότερα

### C.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 περισσότερα

### Solutions 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 περισσότερα

### Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

### Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

Διαβάστε περισσότερα

### Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1

Arithmetical applications of lagrangian interpolation Tanguy Rivoal Institut Fourier CNRS and Université de Grenoble Conference Diophantine and Analytic Problems in Number Theory, The 00th anniversary

Διαβάστε περισσότερα

### Section 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 περισσότερα

### The Pohozaev identity for the fractional Laplacian

The Pohozaev identity for the fractional Laplacian Xavier Ros-Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya (joint work with Joaquim Serra) Xavier Ros-Oton (UPC) The Pohozaev

Διαβάστε περισσότερα

### A Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering

Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix

Διαβάστε περισσότερα

### Phys460.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 περισσότερα

### CHAPTER 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 περισσότερα

### Fourier 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 περισσότερα

### Problem 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 περισσότερα

Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say

Διαβάστε περισσότερα

### Econ 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 περισσότερα

### 4.6 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 περισσότερα

### Intuitionistic Fuzzy Ideals of Near Rings

International Mathematical Forum, Vol. 7, 202, no. 6, 769-776 Intuitionistic Fuzzy Ideals of Near Rings P. K. Sharma P.G. Department of Mathematics D.A.V. College Jalandhar city, Punjab, India pksharma@davjalandhar.com

Διαβάστε περισσότερα

### de Rham Theorem May 10, 2016

de Rham Theorem May 10, 2016 Stokes formula and the integration morphism: Let M = σ Σ σ be a smooth triangulated manifold. Fact: Stokes formula σ ω = σ dω holds, e.g. for simplices. It can be used to define

Διαβάστε περισσότερα

### MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed

Διαβάστε περισσότερα

### Other 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 περισσότερα

### b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.

Διαβάστε περισσότερα

### 6.3 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 περισσότερα

### F A S C I C U L I M A T H E M A T I C I

F A S C I C U L I M A T H E M A T I C I Nr 46 2011 C. Carpintero, N. Rajesh and E. Rosas ON A CLASS OF (γ, γ )-PREOPEN SETS IN A TOPOLOGICAL SPACE Abstract. In this paper we have introduced the concept

Διαβάστε περισσότερα

### SCHOOL 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 περισσότερα

### Inverse 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 περισσότερα

### 3.4 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 περισσότερα

### derivation 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 περισσότερα

### On density of old sets in Prikry type extensions.

On density of old sets in Prikry type extensions. Moti Gitik December 31, 2015 Abstract Every set of ordinals of cardinality κ in a Prikry extension with a measure over κ contains an old set of arbitrary

Διαβάστε περισσότερα

### SOME PROPERTIES OF FUZZY REAL NUMBERS

Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical

Διαβάστε περισσότερα

### Sequent Calculi for the Modal µ-calculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008

Sequent Calculi for the Modal µ-calculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008 Introduction Koz: Axiomatisation for the modal µ-calculus over K Axioms: All classical

Διαβάστε περισσότερα

### Finite 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 περισσότερα

### 1. Introduction and Preliminaries.

Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We

Διαβάστε περισσότερα

### Some new generalized topologies via hereditary classes. Key Words:hereditary generalized topological space, A κ(h,µ)-sets, κµ -topology.

Bol. Soc. Paran. Mat. (3s.) v. 30 2 (2012): 71 77. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v30i2.13793 Some new generalized topologies via hereditary

Διαβάστε περισσότερα

### Lecture 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 περισσότερα

### A Note on Intuitionistic Fuzzy. Equivalence Relation

International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com

Διαβάστε περισσότερα

### Reminders: 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 περισσότερα

### EE512: 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 περισσότερα

### Chapter 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 περισσότερα

### Pg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is

Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =

Διαβάστε περισσότερα

### Bounding 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 περισσότερα

### Iterated trilinear fourier integrals with arbitrary symbols

Cornell University ICM 04, Satellite Conference in Harmonic Analysis, Chosun University, Gwangju, Korea August 6, 04 Motivation the Coifman-Meyer theorem with classical paraproduct(979) B(f, f )(x) :=

Διαβάστε περισσότερα

### 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 περισσότερα

### Congruence Classes of Invertible Matrices of Order 3 over F 2

International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

### HOMEWORK 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 περισσότερα

### Απόκριση 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

### Section 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 περισσότερα

### Matrices 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 περισσότερα

### Lecture 15 - Root System Axiomatics

Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the

Διαβάστε περισσότερα

### Lecture 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 περισσότερα

### Boundedness of Some Pseudodifferential Operators on Bessel-Sobolev Space 1

M a t h e m a t i c a B a l k a n i c a New Series Vol. 2, 26, Fasc. 3-4 Boundedness of Some Pseudodifferential Operators on Bessel-Sobolev Space 1 Miloud Assal a, Douadi Drihem b, Madani Moussai b Presented

Διαβάστε περισσότερα

### GAUGES OF BAIRE CLASS ONE FUNCTIONS

GAUGES OF BAIRE CLASS ONE FUNCTIONS ZULIJANTO ATOK, WEE-KEE TANG, AND DONGSHENG ZHAO Abstract. Let K be a compact metric space and f : K R be a bounded Baire class one function. We proved that for any

Διαβάστε περισσότερα

### SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals

Διαβάστε περισσότερα

### A General Note on δ-quasi Monotone and Increasing Sequence

International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in

Διαβάστε περισσότερα

### Lecture 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 περισσότερα

### Lecture 13 - Root Space Decomposition II

Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).

Διαβάστε περισσότερα

### Exercises to Statistics of Material Fatigue No. 5

Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can

Διαβάστε περισσότερα

### w o = R 1 p. (1) R = p =. = 1

Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:

Διαβάστε περισσότερα

### Local Approximation with Kernels

Local Approximation with Kernels Thomas Hangelbroek University of Hawaii at Manoa 5th International Conference Approximation Theory, 26 work supported by: NSF DMS-43726 A cubic spline example Consider

Διαβάστε περισσότερα

### The circle theorem and related theorems for Gauss-type quadrature rules

OP.circle p. / The circle theorem and related theorems for Gauss-type quadrature rules Walter Gautschi wxg@cs.purdue.edu Purdue University OP.circle p. 2/ Web Site http : //www.cs.purdue.edu/ archives/22/wxg/codes

Διαβάστε περισσότερα

### Online Appendix I. 1 1+r ]}, Bψ = {ψ : Y E A S S}, B W = +(1 s)[1 m (1,0) (b, e, a, ψ (0,a ) (e, a, s); q, ψ, W )]}, (29) exp( U(d,a ) (i, x; q)

Online Appendix I Appendix D Additional Existence Proofs Denote B q = {q : A E A S [0, +r ]}, Bψ = {ψ : Y E A S S}, B W = {W : I E A S R}. I slightly abuse the notation by defining B q (L q ) the subset

Διαβάστε περισσότερα

### SOLUTIONS 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 περισσότερα

### Tridiagonal 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 περισσότερα

### Homework 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 περισσότερα

### Chapter 2. Ordinals, well-founded relations.

Chapter 2. Ordinals, well-founded relations. 2.1. Well-founded Relations. We start with some definitions and rapidly reach the notion of a well-ordered set. Definition. For any X and any binary relation

Διαβάστε περισσότερα

### Homomorphism of Intuitionistic Fuzzy Groups

International Mathematical Forum, Vol. 6, 20, no. 64, 369-378 Homomorphism o Intuitionistic Fuzz Groups P. K. Sharma Department o Mathematics, D..V. College Jalandhar Cit, Punjab, India pksharma@davjalandhar.com

Διαβάστε περισσότερα

### ω 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 περισσότερα

### 557: MATHEMATICAL STATISTICS II RESULTS FROM CLASSICAL HYPOTHESIS TESTING

Most Powerful Tests 557: MATHEMATICAL STATISTICS II RESULTS FROM CLASSICAL HYPOTHESIS TESTING To construct and assess the quality of a statistical test, we consider the power function β(θ). Consider a

Διαβάστε περισσότερα

### Homomorphism in Intuitionistic Fuzzy Automata

International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic

Διαβάστε περισσότερα

### Generating Set of the Complete Semigroups of Binary Relations

Applied Mathematics 06 7 98-07 Published Online January 06 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/036/am067009 Generating Set of the Complete Semigroups of Binary Relations Yasha iasamidze

Διαβάστε περισσότερα

### The 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 περισσότερα

### Notes on the Open Economy

Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.

Διαβάστε περισσότερα

### Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions

International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for

Διαβάστε περισσότερα

### STAT200C: Hypothesis Testing

STAT200C: Hypothesis Testing Zhaoxia Yu Spring 2017 Some Definitions A hypothesis is a statement about a population parameter. The two complementary hypotheses in a hypothesis testing are the null hypothesis

Διαβάστε περισσότερα

### Partial 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 περισσότερα

### Integrals in cylindrical, spherical coordinates (Sect. 15.7)

Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.

Διαβάστε περισσότερα

### Solution Series 9. i=1 x i and i=1 x i.

Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x

Διαβάστε περισσότερα

### Lifting Entry (continued)

ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu

Διαβάστε περισσότερα

### Trigonometric Formula Sheet

Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ

Διαβάστε περισσότερα

### THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano

235 Kragujevac J. Math. 30 (2007) 235 242. THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS Daniel A. Romano Department of Mathematics and Informatics, Banja Luka University, Mladena Stojanovića

Διαβάστε περισσότερα

### ANSWERSHEET (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 περισσότερα

### Concrete 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 περισσότερα

### Online Appendix to Measuring the Bullwhip Effect: Discrepancy and Alignment between Information and Material Flows

Online Appendix to Measuring the Bullwhip Effect: Discrepancy and Alignment between Information and Material Flows i Chen Wei uo Kevin Shang S.C. Johnson Graduate School of Management, Cornell University,

Διαβάστε περισσότερα

### Lecture 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 περισσότερα

### Logsine integrals. Notes by G.J.O. Jameson. log sin θ dθ = π log 2,

Logsine integrals Notes by G.J.O. Jameson The basic logsine integrals are: log sin θ dθ = log( sin θ) dθ = log cos θ dθ = π log, () log( cos θ) dθ =. () The equivalence of () and () is obvious. To prove

Διαβάστε περισσότερα

### 1 String with massive end-points

1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε

Διαβάστε περισσότερα