On Pseudo δ-open Fuzzy Sets and Pseudo Fuzzy δ-continuous Functions

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "On Pseudo δ-open Fuzzy Sets and Pseudo Fuzzy δ-continuous Functions"

Transcript

1 Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 29, On Pseudo δ-open Fuzzy Sets and Pseudo Fuzzy δ-continuous Functions A. Deb Ray Department of Mathematics West Bengal State University Berunanpukuria, Malikapur Barasat, North 24 Parganas , India atasi@hotmail.com Pankaj Chettri Department of Mathematics Sikkim Manipal Institute of Technology Majitar, Rangpoo East Sikkim, India pankajct@gmail.com Abstract In this paper, introducing new fuzzy topologies on a fts (X, τ) their interrelations with τ is discussed. The notion of pseudo fuzzy δ-continuous function is also introduced and studied in terms of these new topologies. Mathematics Subject Classification: 03E72, 54A40, 54A99 Keywords: Strong α-level topology, pseudo regular open fuzzy set, pseudo δ-open fuzzy set, ps δ fuzzy topology, ps ro fuzzy topology, pseudo fuzzy δ-continuous function 1 Introduction and Preliminaries In this article, a couple of new fuzzy topologies are introduced with the help of strong α-level topology on a fts (X, τ). In the first section, we discuss their interrelations with the given fuzzy topology τ. Finally, introducing the notion of pseudo fuzzy δ-continuous function we characterize them in terms of these newly developed fuzzy topologies. Let X be a non empty set and I be the closed interval [0, 1]. A fuzzy set μ

2 1404 A. Deb Ray and P. Chettri [7] on X is a function on X into I and the collection of all fuzzy sets on X is denoted by I X. The support of a fuzzy set μ, denoted by suppμ, is the crisp set {x X : μ(x) > 0}. A fuzzy set with a singleton as its support is called a fuzzy point, denoted by x α, and defined as, { α, for z=x x α (z) = 0, otherwise A collection τ I X is called a fuzzy topology [2] on X if (i) 0, 1 τ (ii) μ 1,μ 2,..., μ n τ μ i τ (iii) μ α τ, α Λ (where Λ is an index set) μ α τ Then (X, τ) is called a fuzzy topological space (fts, for short). Let X and Y be two nonempty sets and f : X Y be any function. If A and B are fuzzy sets on X and Y respectively then f(a) and f 1 (B) are respective fuzzy sets on Y and X, given by [7], { sup{a(x) :x f 1 (y)}, iff 1 (y) φ f(a)(y) = 0, otherwise and f 1 (B)(x) =B(f(x)) Let f be a function from a fts X to a fts Y. Then f is fuzzy continuous (fuzzy δ-continuous) iff f 1 (U) is fuzzy open (respectively, fuzzy δ-open ) set in X for each fuzzy open (respectively, fuzzy δ-open ) set U in Y. [5] For each i Λ, if f i : X (Y i,τ i ) are the functions from a set X into fts (Y i,τ i ), then the smallest fuzzy topology on X for which the functions f i,i Λ are fuzzy continuous is called initial fuzzy topology on X generated by the collection of functions {f i,i Λ}. [4] For a fuzzy set μ in X, the set μ α = {x X : μ(x) >α} is called the strong α-level set of X. In a fts (X, τ),w(τ) denotes the collection of all lower semi continuous functions from X into I, i.e w(τ) ={μ : μ 1 (a, 1] τ, a [0, 1]} is a laminated fuzzy topology on X. In a fuzzy topological space (X, τ), for each α I 1 =[0, 1), the collection i α (τ) ={μ 1 (α, 1] : μ τ} is a topology on X and is called strong α-level topology. [4] A fuzzy set μ on a fts (X, τ) is called fuzzy regular open if μ = int(clμ) and its complement is fuzzy regular closed. A fuzzy δ-open set is the union of some fuzzy regular open sets. 2 New fuzzy topologies from old We begin this section with an example showing that in a fts (X, τ) ifμ is fuzzy regular open then μ α need not be regular open in the corresponding topological

3 On pseudo δ-open fuzzy sets 1405 space (X, i α (τ)),α I 1 and also μ α may be regular open in (X, i α (τ)),α I 1 inspite of μ not being fuzzy regular open in the fts (X, τ). Example 2.1 Let X be a set with at least two elements. Fix an element y X. { Clearly, τ = {0, 1,A} is a fuzzy topology on X, where A is defined as 0.5, for x = y A(x) = 0.3, otherwise. The fuzzy closed { sets in (X, τ) are 0, 1 and 1 A where 0.5, for x = y (1 A)(x) = 0.7, otherwise. Clearly A 1 A and hence int(cla) =A. i.e A is fuzzy regular open in (X, τ). Now, in the corresponding topological space (X, i α (τ)), α I 1, the X, for α<0.3 open sets are Φ,X and A α where A α = {y}, for 0.3 α<0.5 Φ, for α 0.5. For, 0.3 α<0.5 the closed sets in (X, i α (τ)) are Φ,X and X {y}. It is clear that int(cla α )=X. Hence, A α is not regular open in (X, i α (τ)) for 0.3 α<0.5. Example 2.2 Let X = {x, y, z}. Define fuzzy sets μ, γ and η as follows: μ(a) =0.4, γ(a) =0.55 and η(a) =0.6, a X. If τ = {0, 1,μ,γ,η} then (X, τ) is a fts. The closed fuzzy sets are (1 μ) =η, (1 γ)(a) =0.45, a X and (1 η) =μ. Here, cl(γ) =η and int(clγ) =η. Hence γ is not fuzzy regular open fuzzy set. But, γ α = {x : γ(x) >α}. For α 0.55, γ α =Φ, which is regular open. For α<0.55, γ α = {x, y, z} = X, which is also regular open. Hence for all α I 1,γ α is regular open in(x, i α (τ)). In view of these examples we define the following: Definition 2.1 A fuzzy open set (fuzzy closed set) μ on a fts (X, τ) is said to be pseudo regular open (respectively, pseudo regular closed) fuzzy set if the strong α-level set μ α is regular open (respectively, regular closed) in (X, i α (τ)), α I 1. The following example establishes that pseudo regular closed and pseudo regular open fuzzy sets are not complements of each other. Example 2.3 Let X = {x, y, z, w},τ = {0, 1,μ} where μ is defined as μ(x) = 0.1,μ(y) = 0.2,μ(z) = 0.3,μ(w) = 0.4. Clearly, (X, τ) is a fts. If α =0.3, i α (τ) ={Φ,X,μ α } and μ α = {x X : α<μ(x) 1} = {w}. Closed sets in (X, i α (τ)) are Φ,X and {x, y, z}. Here we shall show that μ is not pseudo regular open but its complement (1 μ) is pseudo regular closed fuzzy set in (X, τ). As the smallest closed set containg μ α is X, cl(μ α )=X.

4 1406 A. Deb Ray and P. Chettri So int(clμ α )=X μ α. Hence μ α is not regular open in (X, i α (τ)). This shows that μ is not pseudo regular open in (X, τ). Here, (1 μ)(x) =0.9, (1 μ)(y) =0.8, (1 μ)(z) =0.7, (1 μ)(w) =0.6. (1 μ) α = {x X : α<(1 μ)(x) 1}. For, α 0.6, i α (τ) ={Φ,X} and so cl[int(1 μ) α ]=X. Also for α<0.6, (1 μ) α = X and so cl[int(1 μ) α ]=X, whatever be i α (τ). Hence, in any case (1 μ) α is regular closed in (X, i α (τ)). This shows (1 μ) is pseudo regular closed fuzzy set in (X, τ). Definition 2.2 A fuzzy set μ on a fts (X, τ) is said to be pseudo δ-open (respectively, pseudo δ-closed) fuzzy set if the strong α-level set μ α is δ-open (respectively, δ-closed) in (X, i α (τ)), α I 1. Theorem 2.1 The collection of all pseudo δ-open fuzzy sets on a fts (X, τ) forms a fuzzy topology on X. proof. Straightforward. Definition 2.3 The fuzzy topology as obtained in the above theorem is called pseudo δ-fuzzy topology (in short ps-δ fuzzy topology) on X. The complements of the members of ps-δ fuzzy topology are known as ps δ-closed fuzzy sets. Theorem 2.2 In a fts (X, τ) union of pseudo regular open fuzzy sets is pseudo δ-open. Proof. Let μ = {μ i : i Λ}, where μ i is pseudo regular open fuzzy sets in a fts (X, τ), for each i Λ. Here, μ α i is regular open in (X, i α (τ)), i Λ. As, ( μ i ) α = μ α i is δ-open in (X, i α (τ)), α I 1, μ is pseudo δ-open fuzzy set in (X, τ). The following example shows that the converse of the Theorem( 2.2) is not true in general. i.e Any pseudo δ-open fuzzy set on a fts (X, τ) need not be expressible as union of pseudo regular open fuzzy sets. Example 2.4 Let X = {x, y, z} and the topology τ generated by μ, γ, η where μ(x) = 0.4,μ(y) = 0.4,μ(z) = 0.5, γ(x) = 0.4,γ(y) = 0.6, γ(z) = 0.4 and η(x) = 0.5, η(y) = 0.5, η(z) = 0.6. Consider i α (τ), for each α as follows: Case 1: For α<0.4, μ α = γ α = η α = X and hence i α (τ) ={X, Φ}. Consequently, μ α,γ α and η α are all regular open. Case 2: For α 0.6, μ α = γ α = η α = Φ and hence i α (τ) ={X, Φ}. Consequently, μ α,γ α and η α are all regular open. Case 3: For 0.4 α<0.5, μ α = {z}, γ α = {y}, η α = X and hence i α (τ) = {X, Φ, {y}, {z}, {y, z}}. We observe that int(clμ α )=μ α, int(clγ α )=γ α and int(clη α )=η α, proving all of them to be regular open.

5 On pseudo δ-open fuzzy sets 1407 But (μ γ) α = {y, z} is not regular open as int(cl{y, z}) =X {y, z}. Case 4: For 0.5 α < 0.6, μ α = Φ, γ α = {y}, η α = {z} and hence i α (τ) ={X, Φ, {y}, {z}, {y, z}}. In this case too all μ α,γ α and η α are regular open but (γ η) α = {y, z} is not so. Now we consider a fuzzy set K on X as follows: K(x) =0.4, K(y) =0.6 and X, for α<0.4 K(z) =0.6. Clearly K α = {y, z}, for 0.4 α<0.6 Φ, for α 0.6. X, for α<0.4 Hence, K α μ α γ α, for 0.4 α<0.5 = γ α η α, for 0.5 α<0.6 Φ, for α 0.6. and so, K α is δ-open in (X, i α (τ)), α I 1. Therefore, K is pseudo δ-open fuzzy set in (X, τ). It can be shown easily that K is neither a pseudo regular open fuzzy set in (X, τ) nor is expressible as union of pseudo regular open fuzzy sets in (X, τ). It is clear from the above example that A α = B α, for some α I 1 does not imply A = B. However we have the following Theorem: Theorem 2.3 If A and B are two fuzzy sets in a fts (X, τ) such that A α = B α, α I 1 then A = B. Proof. For α =0,A 0 = B 0 A(x) > 0iffB(x) > 0. Hence A(x) =0iff B(x) = 0. Suppose, y Y is such that A(y) > 0,B(y) > 0 and A(y) B(y). Let A(y) =α 1 and B(y) =α 2. Without any loss of generality let us take α 1 >α 2. Since A(y) =α 1 >α 2, y A α 2, but y/ B α 2 i.e A α2 B α 2, which is a contradiction. Hence, forall y Y, A(y) =B(y) i.e. A = B. Theorem 2.4 If {μ i }be a collection of all pseudo regular open fuzzy sets on a fts (X, τ), then 1. 0, 1 {μ i }. 2. μ 1,μ 2 {μ i } μ 1 μ 2 {μ i }. Proof. 0 and 1 are pseudo regular open fuzzy sets in a fts (X, τ). Hence 0, 1 {μ i }. Let μ 1,μ 2 {μ i }. μ α 1,μ α 2 are regular open in (X, i α (τ)). μ α 1 μα 2 is regular open in (X, i α(τ)). (μ 1 μ 2 ) α = μ α 1 μα 2 is regular open in (X, i α(τ)). μ 1 μ 2 is pseudo regular open fuzzy set in (X, τ). Remark 2.1 In view of Theorem ( 2.4) the collection of all pseudo regular open fuzzy sets on (X, τ) generates a fuzzy topology called pseudo regular

6 1408 A. Deb Ray and P. Chettri open fuzzy topology (in short ps ro fuzzy topology) on X. The members of this topology are termed as ps ro open fuzzy sets and their complements as ps ro closed fuzzy sets. Theorem 2.5 In a fts (X, τ), ps ro fuzzy topology is coarser than τ. Proof. Straightforward. Theorem 2.6 In a fts (X, τ), ps ro fuzzy topology is coarser than ps δ fuzzy topology. Proof. Let μ ps ro fuzzy topology. So, μ = γ i i where γ i s are pseudo regular open fuzzy sets on (X, τ). Hence μ α =( γ i) α = γα i i i is δ-open in (X, i α (τ)), α I 1. This shows that μ ps δ fuzzy topology on X. Remark 2.2 In view of Example ( 2.4), in general ps ro fuzzy topology is strictly coarser than ps δ fuzzy topology in a fts (X, τ). 3 Pseudo fuzzy δ-continuous functions Definition 3.1 A function f from a fts X to a fts Y is pseudo fuzzy δ- continuous iff f 1 (U) is pseudo δ-open fuzzy set in X for each pseudo δ-open fuzzy set U in Y. Theorem 3.1 A function f :(X, τ) (Y,σ) is pseudo fuzzy δ-continuous then f : (X, i α (τ)) (Y,i α (σ)), is δ-continuous for each α I 1, where (X, τ), (Y,σ) are fts. Proof. Let v be a δ-open set in i α (σ). As every δ-open set is open, v i α (σ) and so there exist μ σ such that v = μ α.now, f 1 (μ α ) = {x X : f(x) μ α } = {x X : μ(f(x)) >α} = {x X :(μf)(x) >α} = {x X :(f 1 (μ))(x) >α} = {x X : x (f 1 (μ)) α } = (f 1 (μ)) α Consider a fuzzy set ζ on X given by { 1 if μ(x) >α ζ(x) = α otherwise Then ζ β = { μ α Y if β α β < α

7 On pseudo δ-open fuzzy sets 1409 Consequently, ζ β is δ-open for all β I 1. Hence, ζ is a pseudo δ-open on Y. Since, f is pseudo fuzzy δ-continuous, f 1 (ζ) is pseudo δ-open fuzzy set in X. Now, (f 1 (ζ)) α = f 1 (ζ α )=f 1 (μ α )=f 1 (v). Hence f 1 (v) isδ-open set whenever v is so. This proves that f is δ-continuous for each α I 1. Theorem 3.2 A function f :(X, i α (τ)) (Y,i α (σ)), is δ-continuous for each α I 1, where (X, τ), (Y,σ) are fts then f :(X, τ) (Y,σ) is pseudo fuzzy δ-continuous. Proof. Let μ be any fuzzy pseudo δ-open set in (Y,σ). μ α is δ-open in (Y,i α (σ)). By the δ-continuity of f : (X, i α (τ)) (Y,i α (σ)),f 1 (μ α ) = (f 1 (μ)) α is δ-open in (X, i α (τ)). Hence f 1 (μ) is pseudo δ-open fuzzy set in (X, τ), proving f to be pseudo fuzzy δ-continuous. Combining Theorems ( 3.1) and ( 3.2) we get: Theorem 3.3 A function f :(X, τ) (Y,σ) is pseudo fuzzy δ-continuous iff f :(X, i α (τ)) (Y,i α (σ)), is δ-continuous for each α I 1, where (X, τ), (Y,σ) are fts. Theorem 3.4 If a function f :(X, τ) (Y,σ) is pseudo fuzzy δ-continuous then f 1 (μ) is pseudo δ-closed fuzzy set in (X, τ), for all pseudo δ-closed fuzzy set μ in (Y,σ). Proof. Let f :(X, τ) (Y,σ) be pseudo fuzzy δ-continuous. f :(X, i α (τ)) (Y,i α (σ)) is δ-continuous for each α I 1.Nowμ is pseudo δ-closed fuzzy set in (Y,σ). Hence, α I 1, μ α is δ-closed in (Y,i α (σ)), that is (Y μ α )isδ-open in (Y,i α (σ)). Now, f 1 (Y μ α ) = {x X : f(x) / μ α } = {x X : μ(f(x)) α} = {x X :(μf)(x)) α} = X {x X :(μf)(x)) >α} = X {x X :(f 1 (μ))(x) >α} = X (f 1 (μ)) α As, f 1 (Y μ α )isδ-open, (f 1 (μ)) α is δ-closed in (X, i α (τ)), α I 1. Hence f 1 (μ) is pseudo δ-closed fuzzy set in (Y,σ). As a pseudo δ-closed set need not be the complement of a pseudo δ-open set, the converse of the Theorem ( 3.4) may not hold true. However, the following theorem characterizes pseudo fuzzy δ-continuous functions in terms of ps δ- closed fuzzy sets.

8 1410 A. Deb Ray and P. Chettri Theorem 3.5 A function f :(X, τ) (Y,σ) is pseudo fuzzy δ-continuous iff f 1 (μ) is ps δ-closed fuzzy set in a fts (X, τ), where μ is ps-δ-closed fuzzy set in a fts (Y,σ). Proof. Let f :(X, τ) (Y,σ) be pseudo fuzzy δ-continuous. Then f :(X, i α (τ)) (Y,i α (σ)) is δ-continuous for each α I 1. Let μ be ps δ-closed fuzzy set in a fts (Y,σ). 1 μ is pseudo δ-open fuzzy set in (Y,σ). (1 μ) α is δ-open and so f 1 ((1 μ) α )=(f 1 (1 μ)) α is δ-open fuzzy set in (X, i α (τ)), α I 1. This shows that f 1 (1 μ) is pseudo δ-open fuzzy set in (X, τ). Now, (1 f 1 (1 μ))(x) = 1 f 1 (1 μ)(x) = 1 (1 μ)(f(x)) = μ(f(x)) = f 1 (μ)(x). Hence, f 1 (μ) isps δ-closed fuzzy set in (X, τ). Conversely, Let μ be any pseudo δ-open and so (1 μ) isps δ-closed fuzzy set in (Y,σ). As, f 1 (1 μ) isps δ-closed, 1 f 1 (1 μ) is pseudo δ-open fuzzy set in (X, i α (τ)), α I 1. Again, f 1 (μ) =1 f 1 (1 μ),f 1 (μ) is pseudo δ-open fuzzy set in (X, τ). Hence f is pseudo fuzzy δ-continuous. References [1] Azad, K. K : On fuzzy semi-continuity,fuzzy almost continuity and weaklycontinuity, J. Math. Anal. Appl., 82(1981), [2] Chang, C. L : Fuzzy topological spaces, J. Math. Anal. Appl., 24(1968), [3] Lowen, R : Initial and final Fuzzy Topologies and the Fuzzy Tychonoff theorem, J. Math. Anal. Appl., 58(1977), [4] Lowen, R : Fuzzy Topological Spaces and Fuzzy Compactness, J. Math. Anal. Appl.,56(1976), [5] Noiri, T : On δ-continuous functions, Journal of Korean Mathematical Society., 16(1980), [6] Velicko, N : H-closed topological spaces,amer. Soc. Transl., 78(2)(1968), [7] Zadeh, L.A : Fuzzy Sets, Information and Control., 8(1965),

9 On pseudo δ-open fuzzy sets 1411 Received: November, 2009

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

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

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

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 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

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

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

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

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

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

Operation Approaches on α-γ-open Sets in Topological Spaces

Operation Approaches on α-γ-open Sets in Topological Spaces Int. Journal of Math. Analysis, Vol. 7, 2013, no. 10, 491-498 Operation Approaches on α-γ-open Sets in Topological Spaces N. Kalaivani Department of Mathematics VelTech HighTec Dr.Rangarajan Dr.Sakunthala

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Intuitionistic Supra Gradation of Openness

Intuitionistic Supra Gradation of Openness Applied Mathematics & Information Sciences 2(3) (2008), 291-307 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. Intuitionistic Supra Gradation of Openness A. M. Zahran 1, S. E.

Διαβάστε περισσότερα

ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II)

ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) Commun. Korean Math. Soc. 25 (2010), No. 3, pp. 457 475 DOI 10.4134/CKMS.2010.25.3.457 ON FUZZY BITOPOLOGICAL SPACES IN ŠOSTAK S SENSE (II) Ahmed Abd El-Kader Ramadan, Salah El-Deen Abbas, and Ahmed Aref

Διαβάστε περισσότερα

Homomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PS-algebras

Homomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PS-algebras Annals of Pure and Applied athematics Vol. 8, No. 1, 2014, 93-104 ISSN: 2279-087X (P), 2279-0888(online) Published on 11 November 2014 www.researchmathsci.org Annals of Homomorphism and Cartesian Product

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

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

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

Fuzzifying Tritopological Spaces

Fuzzifying Tritopological Spaces International Mathematical Forum, Vol., 08, no. 9, 7-6 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/imf.08.88 On α-continuity and α-openness in Fuzzifying Tritopological Spaces Barah M. Sulaiman

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

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

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

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

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

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

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

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

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

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

Partial Differential Equations in Biology The boundary element method. March 26, 2013

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

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

The Simply Typed Lambda Calculus

The 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 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

A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings

A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 1 (2011), pp. 61-71 International Research Publication House http://www.irphouse.com A Note on Characterization

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

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

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

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

Lecture 2. Soundness and completeness of propositional logic

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

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

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

Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα

Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα [ 1 ] Πανεπιστήµιο Κύπρου Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα Νίκος Στυλιανόπουλος, Πανεπιστήµιο Κύπρου Λευκωσία, εκέµβριος 2009 [ 2 ] Πανεπιστήµιο Κύπρου Πόσο σηµαντική είναι η απόδειξη

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

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

«Έντυπο και ψηφιακό βιβλίο στη σύγχρονη εποχή: τάσεις στην παγκόσμια βιομηχανία».

«Έντυπο και ψηφιακό βιβλίο στη σύγχρονη εποχή: τάσεις στην παγκόσμια βιομηχανία». ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΙΟΝΙΩΝ ΝΗΣΩΝ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑΣ Ταχ. Δ/νση : ΑΤΕΙ Ιονίων Νήσων- Λεωφόρος Αντώνη Τρίτση Αργοστόλι- Κεφαλληνίας, Ελλάδα 28100, +30

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

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

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

A General Note on δ-quasi Monotone and Increasing Sequence

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

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

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

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

Knaster-Reichbach Theorem for 2 κ

Knaster-Reichbach Theorem for 2 κ Knaster-Reichbach Theorem for 2 κ Micha l Korch February 9, 2018 In the recent years the theory of the generalized Cantor and Baire spaces was extensively developed (see, e.g. [1], [2], [6], [4] and many

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

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

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

GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)

GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ) IFSCOM016 1 Proceeding Book No. 1 pp. 155-161 (016) ISBN: 978-975-6900-54-3 SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)

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

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

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

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 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 περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006

ΚΥΠΡΙΑΚΗ 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 περισσότερα

On density of old sets in Prikry type extensions.

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

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

ΚΥΠΡΙΑΚΗ 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 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

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

«Χρήσεις γης, αξίες γης και κυκλοφοριακές ρυθμίσεις στο Δήμο Χαλκιδέων. Η μεταξύ τους σχέση και εξέλιξη.»

«Χρήσεις γης, αξίες γης και κυκλοφοριακές ρυθμίσεις στο Δήμο Χαλκιδέων. Η μεταξύ τους σχέση και εξέλιξη.» ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΑΓΡΟΝΟΜΩΝ ΚΑΙ ΤΟΠΟΓΡΑΦΩΝ ΜΗΧΑΝΙΚΩΝ ΤΟΜΕΑΣ ΓΕΩΓΡΑΦΙΑΣ ΚΑΙ ΠΕΡΙΦΕΡΕΙΑΚΟΥ ΣΧΕΔΙΑΣΜΟΥ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ: «Χρήσεις γης, αξίες γης και κυκλοφοριακές ρυθμίσεις στο Δήμο Χαλκιδέων.

Διαβάστε περισσότερα

Solutions to Exercise Sheet 5

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

Τμήμα Ψηφιακών Συστημάτων. Διπλωματική Εργασία

Τμήμα Ψηφιακών Συστημάτων. Διπλωματική Εργασία Π Α Ν Ε Π Ι Σ Τ Η Μ Ι Ο Π Ε Ι Ρ Α Ι Ω Σ Τμήμα Ψηφιακών Συστημάτων Διπλωματική Εργασία ΜΕΛΕΤΗ ΠΕΡΙΠΤΩΣΗΣ: Η ΣΥΜΒΟΛΗ ΤΟΥ SCRATCH ΣΤΗ ΔΙΔΑΣΚΑΛΙΑ ΤΟΥ ΠΡΟΓΡΑΜΜΑΤΙΣΜΟΥ ΣΤΗ Β /ΘΜΙΑ ΕΚΠΑΙΔΕΥΣΗ ΦΟΥΝΤΟΥΛΑΚΗ ΜΑΡΙΑ

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

Διπλωματική εργασία: Τα κίνητρα των παιδιών προσχολικής ηλικίας για μάθηση

Διπλωματική εργασία: Τα κίνητρα των παιδιών προσχολικής ηλικίας για μάθηση ΤΜΗΜΑ ΕΠΙΣΤΗΜΩΝ ΤΗΣ ΕΚΠΑΙΔΕΥΣΗΣ ΚΑΙ ΤΗΣ ΑΓΩΓΗΣ ΣΤΗΝ ΠΡΟΣΧΟΛΙΚΗ ΗΛΙΚΙΑ ΜΕΤΑΠΤΥΧΙΑΚΟ ΠΡΟΓΡΑΜΜΑ ΣΠΟΥΔΩΝ ΚΑΤΕΥΘΥΝΣΗ: ΚΟΙΝΩΝΙΚΗ ΘΕΩΡΙΑ, ΠΟΛΙΤΙΚΗ ΚΑΙ ΠΡΑΚΤΙΚΕΣ ΣΤΗΝ ΕΚΠΑΙΔΕΥΣΗ Διπλωματική εργασία: Τα κίνητρα

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

6.3 Forecasting ARMA processes

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

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

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

ETOΣ 16o - AP. ΦΥΛΛΟΥ 61 - ΑΠΡΙΛΙΟΣ-ΜΑΪΟΣ-ΙΟΥΝΙΟΣ 2010 - AIOΛOY 100 AΘHNA, T.θ. 4043 - T.K. 102 10 - (AΘHNA) ΕMAIL: lynistaina@gmail.

ETOΣ 16o - AP. ΦΥΛΛΟΥ 61 - ΑΠΡΙΛΙΟΣ-ΜΑΪΟΣ-ΙΟΥΝΙΟΣ 2010 - AIOΛOY 100 AΘHNA, T.θ. 4043 - T.K. 102 10 - (AΘHNA) ΕMAIL: lynistaina@gmail. Αιόλου 100 682 κωδ. αρ. 4666 ΛΥΝΙΣΤΙΑΝΙΚΗ ΦΩΝΗ Όργανο επικοινωνίας Συνδέσμου Αποδήμων Λυνιστιάνων Ολυμπίας ETOΣ 16o - AP. ΦΥΛΛΟΥ 61 - ΑΠΡΙΛΙΟΣ-ΜΑΪΟΣ-ΙΟΥΝΙΟΣ 2010 - AIOΛOY 100 AΘHNA, T.θ. 4043 - T.K. 102

Διαβάστε περισσότερα

The challenges of non-stable predicates

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

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 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

Oscillatory integrals

Oscillatory integrals Oscilltory integrls Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto August, 0 Oscilltory integrls Suppose tht Φ C R d ), ψ DR d ), nd tht Φ is rel-vlued. I : 0, ) C by Iλ)

Διαβάστε περισσότερα

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

Main source: Discrete-time systems and computer control by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a

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

Η ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ

Η ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΝΟΜΙΚΗ ΣΧΟΛΗ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΤΟΜΕΑΣ ΙΣΤΟΡΙΑΣ ΦΙΛΟΣΟΦΙΑΣ ΚΑΙ ΚΟΙΝΩΝΙΟΛΟΓΙΑΣ ΤΟΥ ΔΙΚΑΙΟΥ Διπλωματική εργασία στο μάθημα «ΚΟΙΝΩΝΙΟΛΟΓΙΑ ΤΟΥ ΔΙΚΑΙΟΥ»

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

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

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