A General Note on δ-quasi Monotone and Increasing Sequence

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "A General Note on δ-quasi Monotone and Increasing Sequence"

Transcript

1 International Mathematical Forum, 4, 2009, no. 3, A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in TMIMT, Moradabad, U. P., India Abstract In the present paper, a general theorem concerning ϕ C, α, ρ, γ k summability factors of infinte series, has been proved. Mathematics Subject Classification: 40D15, 40F05, 40G05 Keywords: Absolute Summability, Almost Increasing Sequence, Quasi- Monotone Sequence 1 Introduction A sequence of (b n ) of positive numbers is said to be δ-quasi monotone, if b n 0, b n > 0 ultimately and Δb n δ n, where (δ n ) is a sequence of positive numbers (see[3]). A positive sequence (b n ) is said to be almost increasing if there exists a psitive increasing sequence (c n ) and two positive constants A and B such that Ac n b n Bc n (see[1]). Let (ϕ n ) be a sequence of complex numbers and let a n be a given infinite series with partial sums (s n ). We denote by σn α and tα n the n-th Cesàro means of order α, with α> 1, of the sequence (s n ) and (na n ), respectively, i.e. σ α n = 1 A α n ν=0 n ν s ν (1) where t α n = 1 A α n n ν νa ν, (2) A α n = O (n α ), α > 1, A α 0 =1and A α n =0for n > 0. (3)

2 144 S. K. Saxena The series a n is said to be summable C, α k, k 1 and α> 1 if (see[7]) n k 1 σ α n σn 1 α k <. (4) But since t α n = n ( σ α n σ α n 1) (see[9]) condition (4) can also be written as 1 n tα n k <. (5) The series a n is said to be summable C, α, ρ, γ k, k 1, if (see[8]) n γ(ρk+k 1) k t α n k <, (6) where ρ 0 and γ is a real number. The series a n is said to be summable ϕ C, α k, k 1 and α> 1, if (see [2] and [10]) n k ϕ n t n k <, (7) and it is said to be summable ϕ C, α, ρ, γ k, k 1, ρ 0, γ 1if n γ(ρk+k 1) 2k+1 ϕ n t α n k <. (8) In the special case when γ = 1 and ρ =0,ϕ C, α, ρ, γ k summability is same as ϕ C, α k summability and when ϕ n = n 1 1 k (resp. ϕ n = n β+1 1 k ) ϕ C, α, ρ, γ k summability is same as C, α, ρ, γ k summability. 2 Known Result Mazhar [12] proved the following theorem for C, 1 k summability factors of infinite series. Theorem 2.1 Let λ n 0 as n. Suppose that there exists a sequence of numbers (B n ) such that it is δ-quasi monotone with nδ n log n<, Bn log n is convergent and Δλ n B n for all n. If 1 n t n k = O (log m) as m, (9) Later on Bor and Leindler [4] generalized the above theorem under weaker conditions in the following form for ϕ C, α k summability

3 δ-quasi monotone and increasing sequence 145 Theorem 2.2 Let (X n ) be an almost increasing sequence such that ΔX n = O ( ) X n n and λn 0 as n. Suppose that there exists a sequence of numbers (B n ) such that it is δ-quasi monotone with nx n δ n <, B n X n is convergent and Δλ n B n for all n. If there exists an ε>0 such that the sequence ( n ε k ϕ n k) is non-increasing and if the sequence (ωn), α defined by (see[13]) ωn α = t α n, α =1 max 1 ν n tα ν, 0 <α<1 (10) satisfies the condition n k (ωn α ϕ n ) k = O (X m ) as m, (11) then the series a n λ n is summable ϕ C, α k, k 1, 1 k α 1. 3 Main Result The aim of this paper is to generalize Theorem 2.2 for ϕ C, α, ρ, γ k summability in the following form. Theorem 3.1 Let (X n ) be an almost increasing sequence such that ΔX n = O ( ) X n n as λn 0 as n. Suppose that there exists a sequence of numbers (B n ) such that it is δ-quasi monotone with nx n δ n <, B n X n is convergent ( and Δλ n B n for all n. If there exists an ε>0 such that the sequence n ε k ϕ n k) is non-increasing and if the sequence (ωn α ) defined by (see [13]) as (10) satisfies the condition n γ(ρk+k 1) 2k+1 ( ϕ n ωn) α k = O (X m ) as m, (12) then the series a n λ n is summable ϕ C, α, ρ, γ k,k 1, 1 α 1. k It is also a generalization of Bor and Özarslan [5] We need the following lemmas for the proof of our theorem. Lemma 3.2 [6] If 0 <α 1 and 1 ν n, then n pa p max m pa p. (13) 1 m ν p=0 Lemma 3.3 Under the conditions regarding (λ n ) and (X n ) of the Theorem, we have λ n X n = O (1) as n, (14) p=0

4 146 S. K. Saxena The statements proof of Lemma 3.3 are proved by Bor and Leindler [4] and hence is omitted. Lemma 3.4 Under the conditions pertaining to (X n ) and (B n ) of the Theorem, we have nb n X n = O (1) (15) nx n ΔB n <. (16) The statements proof of Lemma 3.4 are proved in Theorem 1 and Theorem 2 of Leindler [11] and hence is omitted. Proof of Theorem 3.1 Let (T α n ) be the n-th (C, α), mean of the sequence (na n λ n ). Then, by (2), we have Tn α = 1 A α n Using Abel s transformation, we get Tn α = 1 n 1 Δλ A α ν n p=1 n ννa ν λ ν. n p pa p + λ n A α n n ν νa ν, so that making use of Lemma 1, we have Tn α 1 n 1 Δλ A α ν n p pa p + λ n n p=1 A α n ν νa ν n 1 n 1 A α A α ν ων α Δλ ν + λ n ωn α n = Tn,1 α + T n,2 α, say. Since Tn,1 α + T n, α k ( T 2 k α k T n,1 + α k ) n,1, to complete the proof of the theorem, it is sufficient to show that n γ(ρk+k 1) 2k+1 ϕn Tn,r α k <, for r =1, 2 by (8). Now, when k>1, applying Hölder s inequality with indices k and k, where = 1, we get that 1 k + 1 k n γ(ρk+k 1) 2k+1 ϕn Tn,1 α k n=2 { n 1 n γ(ρk+k 1) 2k+1 (A α n ) k ϕ n k A α ν ωα ν Δλ ν } k

5 δ-quasi monotone and increasing sequence 147 { n 1 } = O (1) n γ(ρk+k 1) 2k+1 n αk ϕ n k ν αk (ων α )k B ν n=2 { n 1 } k 1 B ν = O (1) ν αk (ω α ϕ n k ν )k B ν nγ(ρk+k 1) 2k+1 n=ν+1 nαk m+1 = O (1) ν αk (ων α n ε+γ(ρk+k 1) 2k+1 )k B ν ϕ n=ν+1 n αk+ε n k = O (1) ν αk (ων α ) k B ν ν ε+γ(ρk+k 1) 2k+1 ϕ ν k 1 n=ν+1 n αk+ε = O (1) ν αk (ων α )k B ν ν ε+γ(ρk+k 1) 2k+1 ϕ ν k dx ν x αk+ε = O (1) ν B ν ν γ(ρk+k 1) 2k+1 (ων α ϕ ν ) k = O (1) Δ(ν B ν ) r γ(ρk+k 1) 2k+1 (ωr α ϕ r ) k r=1 + O (1) m B m ν γ(ρk+k 1) 2k+1 (ων α ϕ ν ) k = O (1) (ν +1) ΔB ν B ν X ν + O (1) m B m X m = O (1) ν ΔB ν X ν + O (1) B ν X ν + O (1) m B m X m = O (1) ν ΔB ν X ν + O (1) B ν+1 X ν+1 + O (1) m B m X m = O (1) as m by the virtue of the hypotheses of the Theorem 3.1 and Lemma 3.4. Again, since λ n = O ( 1 X n ) = O (1), by (14) we have n γ(ρk+k 1) 2k+1 ϕn Tn,2 α k = λ n k 1 λ n n γ(ρk+k 1) 2k+1 (ωn α ϕ n ) k = O (1) λ n n γ(ρk+k 1) 2k+1 (ωn α ϕ n ) k

6 148 S. K. Saxena = O (1) Δ λ n ν γ(ρk+k 1) 2k+1 (ων α ϕ ν ) k + O (1) λ m n γ(ρk+k 1) 2k+1 (ωn α ϕ n ) k = O (1) Δ λ n X n + O (1) λ m X m = O (1) B n X n + O (1) λ m X m = O (1) as m, by the virtue of the hypotheses of Theorem 3.1 and Lemma 3.3. Therefore, we get that n γ(ρk+k 1) 2k+1 ϕn Tn,r α k = O (1), as m for r =1, 2. This completes the proof of the Theorem 3.1. References [1] S. Aljančić and D. Arandelovic, O-regular varying functions, Publ. Inst. Math., 22 (1977), [2] M. Balci, Absolute ϕ-summability factors, Comm. Fac. Sci. Univ. Ankara, Ser. A, 29 (1980), [3] R. P. Boas, Quasi positive sequences and trigonometric series, Proc. London Math. Soc. Ser. A, 14 (1965), [4] H. Bor and L. Leindler, A note on δ-quasi-monotone and almost increasing sequences, Mathematical inequalities and Applications, 8(1) (2005), [5] H. Bor and H. S. Özarslan, On the quasi-monotone and almost increasing sequences, Journal of Mathematical Inequalities, 1(4) (2007), [6] L. S. Bosanquit, A mean value theorem, J. London Math. Soc., 16 (1941), [7] T. M. Flett, On an extension of absolute summability and some theorems of Littelwood and Paley, Proc. London Math. Soc., 7 (1957), [8] A. N. Gürkan, On absolute Cesáro summability factors, J. Anal., 7 (1999),

7 δ-quasi monotone and increasing sequence 149 [9] E. Kogbetliantz, Sur la series absolument summability methode des moyennes arithmetiques, Bull. Sci. Math., 49 (1925), [10] L. Leindler, On extension of some theorems of Flett I, Acta. Math. Hungar, 64(3) (1994), [11] L. Leindler, Three theorems connected with δ-quasi monotone sequences and their applications to an integrability theorem, Publ. Math. (Debrecen), 60 (2002), [12] S. M. Mazhar, On a generalized quasi-convex sequence and its applications, Indian J. pure appl. Math. 8 (1977), [13] T. Pati, The summability factors of infinite series, Duke Math. J., 21 (1954), Received: October 11, 2008

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

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

Homomorphism in Intuitionistic Fuzzy Automata

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

2 Composition. Invertible Mappings

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

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

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

Congruence Classes of Invertible Matrices of Order 3 over F 2

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

Example Sheet 3 Solutions

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

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

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

Uniform Convergence of Fourier Series Michael Taylor

Uniform 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 περισσότερα

Homework 3 Solutions

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

Matrices and Determinants

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

A Note on Intuitionistic Fuzzy. Equivalence Relation

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

Statistical Inference I Locally most powerful tests

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

PROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1)

PROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1) GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 995, 535-545 PROPERTIES OF CERTAIN INTEGRAL OPERATORS SHIGEYOSHI OWA Abstract. Two integral operators P α and Q α for analytic functions in the open unit disk

Διαβάστε περισσότερα

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

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

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

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

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 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

Every set of first-order formulas is equivalent to an independent set

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

Other Test Constructions: Likelihood Ratio & Bayes Tests

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

On Inclusion Relation of Absolute Summability

On Inclusion Relation of Absolute Summability It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 53, 2641-2646 O Iclusio Relatio of Absolute Summability Aradhaa Dutt Jauhari A/66 Suresh Sharma Nagar Bareilly UP) Idia-243006 aditya jauhari@rediffmail.com

Διαβάστε περισσότερα

ST5224: Advanced Statistical Theory II

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

Second Order Partial Differential Equations

Second 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 περισσότερα

EE512: Error Control Coding

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

Section 8.3 Trigonometric Equations

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

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

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

On a four-dimensional hyperbolic manifold with finite volume

On 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 περισσότερα

Homomorphism of Intuitionistic Fuzzy Groups

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

Intuitionistic Fuzzy Ideals of Near Rings

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

DIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS

DIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS GANIT J. Bangladesh Math. oc. IN 606-694) 0) -7 DIRECT PRODUCT AND WREATH PRODUCT OF TRANFORMATION EMIGROUP ubrata Majumdar, * Kalyan Kumar Dey and Mohd. Altab Hossain Department of Mathematics University

Διαβάστε περισσότερα

Fractional Colorings and Zykov Products of graphs

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

A summation formula ramified with hypergeometric function and involving recurrence relation

A summation formula ramified with hypergeometric function and involving recurrence relation South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin

Διαβάστε περισσότερα

5. Choice under Uncertainty

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

SOME PROPERTIES OF FUZZY REAL NUMBERS

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

1. Introduction and Preliminaries.

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

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R + Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b

Διαβάστε περισσότερα

Reminders: linear functions

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

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS

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

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

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

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

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

n=2 In the present paper, we introduce and investigate the following two more generalized

n=2 In the present paper, we introduce and investigate the following two more generalized MATEMATIQKI VESNIK 59 (007), 65 73 UDK 517.54 originalni nauqni rad research paper SOME SUBCLASSES OF CLOSE-TO-CONVEX AND QUASI-CONVEX FUNCTIONS Zhi-Gang Wang Abstract. In the present paper, the author

Διαβάστε περισσότερα

Commutative Monoids in Intuitionistic Fuzzy Sets

Commutative Monoids in Intuitionistic Fuzzy Sets Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,

Διαβάστε περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

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

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER

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

Concrete Mathematics Exercises from 30 September 2016

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

Cyclic or elementary abelian Covers of K 4

Cyclic or elementary abelian Covers of K 4 Cyclic or elementary abelian Covers of K 4 Yan-Quan Feng Mathematics, Beijing Jiaotong University Beijing 100044, P.R. China Summer School, Rogla, Slovenian 2011-06 Outline 1 Question 2 Main results 3

Διαβάστε περισσότερα

( y) Partial Differential Equations

( 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 περισσότερα

Lecture 21: Properties and robustness of LSE

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

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

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

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

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

Generating Set of the Complete Semigroups of Binary Relations

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

Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, RIMS

Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, RIMS Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, 2017 @ RIMS Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem Introduction PP: The set of all positive definite operators

Διαβάστε περισσότερα

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

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

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

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

derivation of the Laplacian from rectangular to spherical coordinates

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

Fractional Calculus of a Class of Univalent Functions With Negative Coefficients Defined By Hadamard Product With Rafid -Operator

Fractional Calculus of a Class of Univalent Functions With Negative Coefficients Defined By Hadamard Product With Rafid -Operator EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 4, No. 2, 2, 62-73 ISSN 37-5543 www.ejpam.com Fractional Calculus of a Class of Univalent Functions With Negative Coefficients Defined By Hadamard

Διαβάστε περισσότερα

Lecture 34 Bootstrap confidence intervals

Lecture 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 περισσότερα

C.S. 430 Assignment 6, Sample Solutions

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

Subclass of Univalent Functions with Negative Coefficients and Starlike with Respect to Symmetric and Conjugate Points

Subclass of Univalent Functions with Negative Coefficients and Starlike with Respect to Symmetric and Conjugate Points Applied Mathematical Sciences, Vol. 2, 2008, no. 35, 1739-1748 Subclass of Univalent Functions with Negative Coefficients and Starlike with Respect to Symmetric and Conjugate Points S. M. Khairnar and

Διαβάστε περισσότερα

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

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

Tridiagonal matrices. Gérard MEURANT. October, 2008

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

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

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

12. Radon-Nikodym Theorem

12. Radon-Nikodym Theorem 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

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.

2. 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 περισσότερα

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

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

Approximation of distance between locations on earth given by latitude and longitude

Approximation 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 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 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

The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients

The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients Robert Gardner Brett Shields Department of Mathematics and Statistics Department of Mathematics and Statistics

Διαβάστε περισσότερα

A study on generalized absolute summability factors for a triangular matrix

A study on generalized absolute summability factors for a triangular matrix Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 doi: 0.376/proc.20.2.06 Available olie at www.eap.ee/proceedigs A study o geeralized absolute suability factors for a triagular atrix Ere Savaş

Διαβάστε περισσότερα

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

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

Finite Field Problems: Solutions

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

Roman Witu la 1. Let ξ = exp(i2π/5). Then, the following formulas hold true [6]:

Roman Witu la 1. Let ξ = exp(i2π/5). Then, the following formulas hold true [6]: Novi Sad J. Math. Vol. 43 No. 1 013 9- δ-fibonacci NUMBERS PART II Roman Witu la 1 Abstract. This is a continuation of paper [6]. We study fundamental properties applications of the so called δ-fibonacci

Διαβάστε περισσότερα

On a Subclass of k-uniformly Convex Functions with Negative Coefficients

On a Subclass of k-uniformly Convex Functions with Negative Coefficients International Mathematical Forum, 1, 2006, no. 34, 1677-1689 On a Subclass of k-uniformly Convex Functions with Negative Coefficients T. N. SHANMUGAM Department of Mathematics Anna University, Chennai-600

Διαβάστε περισσότερα

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

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

Problem Set 3: Solutions

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

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

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

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

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

The Fekete Szegö Theorem for a Subclass of Quasi-Convex Functions

The Fekete Szegö Theorem for a Subclass of Quasi-Convex Functions Pure Mathematical Sciences, Vol. 1, 01, no. 4, 187-196 The Fekete Szegö Theorem for a Subclass of Quasi-Convex Functions Goh Jiun Shyan School of Science and Technology Universiti Malaysia Sabah Jalan

Διαβάστε περισσότερα

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

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

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

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

New bounds for spherical two-distance sets and equiangular lines

New bounds for spherical two-distance sets and equiangular lines New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a

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)

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

Math221: HW# 1 solutions

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

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter

Διαβάστε περισσότερα

Quadratic Expressions

Quadratic Expressions Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots

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. 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

Homework 8 Model Solution Section

Homework 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 περισσότερα

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

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

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

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

Section 7.6 Double and Half Angle Formulas

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

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 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

Strain gauge and rosettes

Strain 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 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

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

Chapter 3: Ordinal Numbers

Chapter 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 περισσότερα

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Jesse 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

ON INTEGRAL MEANS FOR FRACTIONAL CALCULUS OPERATORS OF MULTIVALENT FUNCTIONS. S. Sümer Eker 1, H. Özlem Güney 2, Shigeyoshi Owa 3

ON INTEGRAL MEANS FOR FRACTIONAL CALCULUS OPERATORS OF MULTIVALENT FUNCTIONS. S. Sümer Eker 1, H. Özlem Güney 2, Shigeyoshi Owa 3 ON INTEGRAL MEANS FOR FRACTIONAL CALCULUS OPERATORS OF MULTIVALENT FUNCTIONS S. Sümer Eker 1, H. Özlem Güney 2, Shigeyoshi Owa 3 Dedicated to Professor Megumi Saigo, on the occasion of his 7th birthday

Διαβάστε περισσότερα

The k-α-exponential Function

The k-α-exponential Function Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,

Διαβάστε περισσότερα

Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp

Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp.115-126. α, β, γ ORTHOGONALITY ABDALLA TALLAFHA Abstract. Orthogonality in inner product spaces can be expresed using the notion of norms.

Διαβάστε περισσότερα

The k-bessel Function of the First Kind

The k-bessel Function of the First Kind International Mathematical Forum, Vol. 7, 01, no. 38, 1859-186 The k-bessel Function of the First Kin Luis Guillermo Romero, Gustavo Abel Dorrego an Ruben Alejanro Cerutti Faculty of Exact Sciences National

Διαβάστε περισσότερα

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

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

Areas and Lengths in Polar Coordinates

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

F19MC2 Solutions 9 Complex Analysis

F19MC2 Solutions 9 Complex Analysis F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at

Διαβάστε περισσότερα