# A Note on Intuitionistic Fuzzy. Equivalence Relation

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

## Transcript

1 International Mathematical Forum, 5, 2010, no. 67, A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar , Assam, India N. K. Sarma Dept. of Mathematics, Assam University Silchar , Assam, India. Abstract In this paper, some interesting properties of Intuitionistic fuzzy equivalence relation have been discussed. Also Intuitionistic fuzzy equivalence classes have been characterized with the help of (α, β)-cut of Intuitionistic fuzzy relations and finally for any Intuitionistic fuzzy equivalence relation on a finite set, we have given an upper bound of the number of values that the degree of membership and nonmembership can assume. Keywords: Intuitionistic Fuzzy Set, Intuitionistic Fuzzy Relation, Intuitionistic Fuzzy Equivalence Relation, (α, β)-cut of Intuitionistic Fuzzy Sets 1. Introduction The concept of Fuzzy relation on a set was defined by Zadeh [6]. Buhaescu[4] and Bustince[5] discussed some beautiful properties of Intuitionistic fuzzy relation. Banerjee and Basnet [3, 2] introduced and discussed the (α, β)-cut of Intuitionistic Fuzzy Sets and Ideals of a ring. 2. Preliminaries Definition 2.1. Let E be a nonempty set. An Intuitionistic Fuzzy Set (IFS) A of E is an object of the form A = {< x, μ A (x), ν A (x) > x E}, where μ A : E [0, 1] and

2 3302 D. K. Basnet and N. K. Sarma ν A : E [0, 1] define the degree of membership and degree of nonmembership of the element x E respectively and for every x E, 0 μ A (x) + ν A (x) 1. Definition 2.2 If A = {< x, μ A (x), ν A (x) > x E} and B = {< x, μ B (x), ν B (x) > x E} be any two IFS of a set E then A B if and only if for all x E, μ A (x) μ B (x) and ν A (x) ν B (x) A = B if and only if for all x E, μ A (x) = μ B (x) and ν A (x) = ν B (x) A B = {< x, (μ A μ B )(x), (ν A ν B )(x) > x E}, where (μ A μ B )(x) = min { μ A (x), μ B (x)} and (ν A ν B )(x) = max { ν A (x), ν B (x)} A B = {< x, (μ A μ B )(x), (ν A ν B )(x) > x E}, where (μ A μ B )(x) = max{ μ A (x), μ B (x)} and (ν A ν B )(x) = min { ν A (x), ν B (x)} Also we see that a fuzzy set has the form {< x, μ A (x), μ c A(x) > x E}, where μ c A(x) = 1 - μ A (x) 3. Intuitionistic Fuzzy Relation Definition 3.1. Let A be a nonempty set. Then an Intuitionistic fuzzy relation (IF relation) on A is an Intuitionistic fuzzy set {<(x, y), μ A (x, y), ν A (x, y) > (x, y) A A}, where μ A : A A [0, 1] and ν A : A A [0, 1]. Definition 3.2. An IF relation R = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) A A} is said to be reflexive if μ A (x, x) = 1 and ν A (x, x) = 0 for all x A. Also R is said to be symmetric if μ A (x, y) = μ A (y, x) and ν A (x, y) = ν A (y, x) for all x, y A. Definition 3.3. If R 1 = {<(x, y), μ 1 (x, y), ν 1 (x, y) > (x, y) A A} and R 2 = {<(x, y), μ 2 (x, y), ν 2 (x, y) > (x, y) A A} be two IF relations on A then J-composition denoted be R 1 οr 2 is defined by R 1 οr 2 = {<(x, y), (μ 1 ομ 2 )(x, y), (ν 1 ον 2 )(x, y) > (x, y) A A}, where (μ 1 ομ 2 )(x, y)=sup{min{μ 1 (x, z), μ 2 (z, y)}} z A and (ν 1 ον 2 )(x, y)=inf{max{ν 1 (x, z), ν 2 (z, y)}} z A Definition 3.4. An IF relation R on A is called transitive if RοR R. Definition 3.5. An IF relation R on A is called an Intuitionistic fuzzy equivalence relation if R is reflexive, symmetric and transitive. Definition 3.6. For any Intuitionistic fuzzy set A = {< x, μ A (x), ν A (x) > x X} of a set X, we define a (α,β)-cut of A as the crisp subset {x X μ A (x) α, ν A (x) β} of X and it is denoted by C α, β (A). Theorem 3.7. Let R = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) X X} be a relation on a set X. Then A is an IF equivalence relation on X if and only if C α, β (R) is an equivalence relation on X, with 0 α, β 1 and α + β 1 Proof. We have C α, β (R) = {(x, y) X X μ R (x, y) α, ν R (x, y) β}. Since R is an IF equivalence relation so μ A (x, x) = 1 α and ν A (x, x) = 0 β, for all x X and so (x, x) C α, β (R) i.e., C α, β (R) is reflexive.

3 A note on intuitionistic fuzzy equivalence relation 3303 Next let (x, y) C α, β (R), then μ R (x, y) α, ν R (x, y) β. But R is IF equivalence so μ R (y, x) = μ R (x, y) α and ν R (y, x) = ν R (x, y) β and hence (y, x) C α, β (R). Therefore C α, β (R) is symmetric. Finally let (x, y) C α, β (R) and (y, z) C α, β (R), then μ R (x, y) α, ν R (x, y) β and μ R (y, z) α, ν R (y, z) β min {μ R (x, y), μ R (y, z)} α and max{ν R (x, y), ν R (y, z)} β max{ min {μ R (x, y), μ R (y, z)}} α and min{max{ν R (x, y), ν R (y, z)}} β y y (μ R ομ R )(x, z) α and (ν R ον R )(x, z) β But R is IF equivalence relation so μ R (x, z) (μ R ομ R )(x, z) α and ν R (x, z) (ν R ον R )(x, z) β (x, z) C α, β (R), which shows that C α, β (R) is transitive. Conversely suppose that C α, β (R) is an equivalence relation on X. Taking α = 1 and β = 0 we get C 1, 0 (R) is equivalence and so a reflexive relation and so (x, x) C 1, 0 (R), for all x X. Thus μ R (x, x) 1, ν R (x, x) 0 and consequently μ R (x, x) = 1 and ν R (x, x) = 0. Therefore the IF relation R is reflexive. For any x, y X, let μ R (x, y) = α and ν R (x, y) = β. Then α + β 1 and so by hypothesis C α, β (R) is an equivalence and hence symmetric relation on X. Also (x, y) C α, β (R), so by symmetry (y, x) C α, β (R). Therefore μ R (y, x) α = μ R (x, y) and ν R (y, x) β = ν R (x, y). Similarly if μ R (y, x) = δ and ν R (y, x) = σ then (x, y) C δ, σ (R) and similarly as above we get μ R (x, y) δ = μ R (y, x) and ν R (x, y) σ = ν R (y, x) and hence μ R (x, y) = μ R (y, x) and ν R (x, y) = ν R (y, x). So the IF relation R is symmetric. Finally let x, y, z X and min{μ R (x, z), μ R (z, y)} = α and max{ ν R (x, z), ν R (z, y)} = β, then0 α, β 1 and α + β 1. Therefore C α, β (R) is an equivalence relation on X. Since μ R (x, y) α, μ R (z, y) α and ν R (x, z) β, ν R (z, y) β, so (x, z) C α, β (R) and (z, y) C α, β (R). As C α, β (R) is equivalence relation so by transitivity (x, y) C α, β (R).Therefore μ R (x, y) α and ν R (x, y) β μ R (x, y) α = min{μ R (x, z), μ R (z, y)} and ν R (x, y) β = max{ν R (x, z), ν R (z, y)}, z μ R (x, y) Sup{ min{μ R (x, z), μ R (z, y)}} z and ν R (x, y) Inf{max{ ν R (x, z), ν R (z, y)}} z μ R (x, y) (μ R ομ R )(x, y) and ν R (x, y) (ν R ον R )(x, y) μ R μ R ομ R and ν R ν R ον R, Which shows that the IF relation R is transitive and hence it is an IF equivalence relation. Definition 3.8. Let R = {<(x, y), μ R (x, y), ν R (x, y) > (x, y) X X} be an IF equivalence on a set X. Let a be any element of X. Then the IFS defined by ar = {< x, (aμ R )(x), (aν R )(x) > x X}, where (aμ R )(x) = μ R (a, x), (aν R )(x) = ν R (a, x) x X, is called an IF equivalence class of a with respect to R. Theorem 3.9. Let R = {<(x, y), μ R (x, y), ν R (x, y) > (x, y) X X} be an IF

4 3304 D. K. Basnet and N. K. Sarma equivalence relation on a set X. Let a be any element of X. Then for 0 α, β 1 and α + β 1, C α, β (ar) = [a], the equivalence class of a with respect to the equivalence relation C α, β (R) in X. Proof. We have [a] = {x X (a, x) C α, β (R)} = {x X μ R (a, x) α and ν R (a, x) β} = {x X (aμ R )(x) α and (aν R )(x) β} = C α, β (ar). With the help of this result we are now giving an alternative proof of a well known result of equivalence relation as follows. Theorem Let R = {<(x, y), μ R (x, y), ν R (x, y) > (x, y) X X} be an IF equivalence relation on a set X. Then [a] = [b] if and only if (a, b) C α, β (R), where [a], [b] are equivalence classes of a and b with respect to the equivalence relation C α, β (R) in X for 0 α, β 1 and α + β 1 Proof. Let [a] = [b] then by theorem 3.9, C α, β (ar) = C α, β (br) {x X (aμ R )(x) α, (aν R )(x) β} = {x X (bμ R )(x) α, (bν R )(x) β} As the two sets on both sides of the equality are nonempty let x C α, β (ar) = C α, β (br) (aμ R )(x) α, (aν R )(x) β and (bμ R )(x) α, (bν R )(x) β μ R (a, x) α, ν R (a, x) β and μ R (b, x) α, ν R (b, x) β min{μ R (a, x), μ R (x, b)} α and max{ν R (a, x), ν R (x, b)} β sup{ min{μ R (a, x), μ R (x, b)}} α and inf{max{ν R (a, x), ν R (x, b)}} β x x (μ R ομ R )(a, b) α and (ν R ον R )(a, b) β μ R (a, b) (μ R ομ R )(a, b) α and ν R (a, b) (ν R ον R )(a, b) β (a, b) C α, β (R) Conversely let (a, b) C α, β (R) μ R (a, b) α and ν R (a, b) β... (i) Let x C α, β (ar), then (aμ R )(x) α, (aν R )(x) β μ R (a, x) α, ν R (a, x) β min{μ R (b, a), μ R (a, x)} α and max{ν R (b, a), ν R (a, x)} β, using (i) sup{min{μ R (b, a), μ R (a, x)}} α and inf{max{ν R (b, a), ν R (a, x)}} β a a (μ R ομ R )(b, x) α and (ν R ον R )(b, x) β μ R (b, x) (μ R ομ R )(b, x) α and ν R (b, x) (ν R ον R )(b, x) β (bμ R )(x) α and (bν R )(x) β x C α, β (br) C α, β (ar) C α, β (br) Similarly we can show that C α, β (br) C α, β (ar) Hence C α, β (ar) = C α, β (br) i.e., [a] = [b] and this completes the proof. Theorem The intersection of two IF equivalence relations on a set is again an IF equivalence relation on the set. Proof. Let A = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) X X} and B = {<(x, y), μ B (x, y), ν B (x, y) > (x, y) X X} be two IF equivalence relations on a set X. Now for any 0 α, β 1 and α + β 1. We have C α, β (A B) = C α, β (A)

5 A note on intuitionistic fuzzy equivalence relation 3305 C α, β (B) (by theorem 3.6 of [3] ). By theorem 3.7, C α, β (A) and C α, β (B) are equivalence relations on X and so being intersection of two equivalence relations C α, β (A B) is also an equivalence relation on X and again by theorem 3.7, A B is an IF equivalence relation on X. Note However the union of two IF equivalence relation on a set is not necessarily an IF equivalence relation on the set as shown by the following example: Let x = {a, b, c} and A = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) X X} and B = {<(x, y), μ B (x, y), ν B (x, y) > (x, y) X X} be two IFS on X, where μ A (a, a) = μ A (b, b) = μ A (c, c) = 1, μ A (a, b) = μ A (b, a) = μ A (a, c) = μ A (c, a) = 0.2 μ A (b, c) = μ A (c, b) = 0.7, ν A (a, a) = ν A (b, b) = ν A (c, c) = 0, ν A (a, b) = ν A (b, a) = ν A (a, c) = ν A (c, a) = 0.6 ν A (b, c) = ν A (c, b) = 0.1 and μ B (a, a) = μ B (b, b) = μ B (c, c) = 1, μ B (a, b) = μ B (b, a) = μ B (b, c) = μ B (c, b) = 0.4 μ B (a, c) = μ B (c, a) = 0.6, ν B (a, a) = ν B (b, b) = ν B (c, c) = 0, ν B (a, b) = ν B (b, a) = ν B (b, c) = ν B (c, b) = 0.5 ν B (a, c) = ν B (c, a) = 0.3 Then it is easy to check that A and B are IF equivalence relation on X. Now A B = {<(x, y), (μ A μ B ) (x, y), (ν A ν B )(x, y) > (x, y) X X} and it is not transitive as shown below: {(μ A μ B ) ο (μ A μ B )}(a, b) = Sup{ min{(μ A μ B )(a, a), (μ A μ B )(a, b)}, min{(μ A μ B )(a, b), (μ A μ B )(b, b)}, min{(μ A μ B )(a, c), (μ A μ B )(c, b)}} = Sup{ min{1, 0.4}, min{0.4, 1}, min{0.6, 0.7}} = = max{μ A (a, b), μ B (a, b)} = (μ A μ B )(a, b) This shows that A B is not an IF equivalence relation on X. Before proceeding further we observe the followings: Let A = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) X X} be an IF equivalence relation on a set X = {a, b, c}. So μ A (a, a) = μ A (b, b) = μ A (c, c) = 1, μ A (a, b) = μ A (b, a), μ A (a, c) = μ A (c, a) and μ A (b, c) = μ A (c, b). Without loss of generality let μ A (a, b) μ A (b, c) μ A (c, a). By symmetry and transitivity we have μ A (a, b) (μ A ομ A )(a, b) = Sup{ min{μ A (a, a), μ A (a, b)}, min{μ A (a, b), μ A (b, b)}, min{μ A (a, c), μ A (c, b)}} min{μ A (a, c), μ A (c, b)} = μ A (c, b) = μ A (b, c) Similarly μ A (b, c) (μ A ομ A )(b, c) = Sup{min{μ A (b, a), μ A (a, c)}, min{μ A (b, b), μ A (b, c)}, min{μ A (b, c), μ A (c, c)}} min{μ A (b, a), μ A (a, c)} = μ A (b, a) = μ A (a, b) Therefore μ A (a, b) = μ A (b, c). This shows that #(Imμ A ) 3. Similarly we can show #(Imν A ) 3. Next let X = {a, b, c, d} and A = {<(x, y), μ A (x, y), ν A (x, y)> (x, y) X X} be an IF equivalence relation on X. We have μ A (a, a)=μ A (b, b) = μ A (c, c) = μ A (d, d) = 1, μ A (a, b) = μ A (b, a), μ A (a, c) = μ A (c, a), μ A (a, d) = μ A (d, a), μ A (b, c) = μ A (c, b), μ A (b, d) = μ A (d, b), μ A (c, d) = μ A (d, c). Without loss of generality let μ A (a, b) μ A (a, c) μ A (a, d) μ A (b, c) μ A (b, d) μ A (c, d).

6 3306 D. K. Basnet and N. K. Sarma Now by transitivity min{μ A (a, c), μ A (c, b)} μ A (a, b) μ A (a, c) μ A (a, b), which gives μ A (a, b) = μ A (a, c). Similarly we can show that μ A (a, b) = μ A (a, d), μ A (b, c) = μ A (b, d). Therefore #(Imμ A ) 4. Similarly we can show that #(Imν A ) 4. The above observations actually lead to the following theorem. Theorem Let X = {a 1, a 2, a 3,..., a n } be a set having n elements and A = {<(x, y), μ A (x, y), ν A (x, y)> (x, y) X X} be an IF equivalence relation on X. Then #(Imμ A ) n and #(Imν A ) n. Proof. We shall prove the first result only and the second will follow similarly. We prove the result by induction on n. For n = 1 the result is trivial [for n = 3 and n = 4 the result is true by the above observations]. Suppose the result is true for any set with n 1 elements. Now X ={a 1, a 2, a 3,..., a n-1 } {a n }. When restricted to Y = {a 1, a 2, a 3,..., a n-1 }, A is also an IF equivalence relation on this set. By induction hypothesis #(Imμ A ) n 1 for the set Y with n 1 elements. For the IF equivalence relation A in X, in addition to these elements in Imμ A there can be only the following elements: μ A (a 1, a n ), μ A (a 2, a n ), μ A (a 3, a n ),..., μ A (a n-1, a n ) and μ A (a n, a n ). Out of these, μ A (a n, a n ) = 1 = μ A (a 1, a 1 ) = μ A (a 2, a 2 ) =... = μ A (a n-1, a n-1 ). Without loss of generality let μ A (a 1, a n ) μ A (a 2, a n ) μ A (a 3, a n )... μ A (a n-1, a n ). By transitivity min{μ A (a 1, a n ), μ A (a n, a 2 )} μ A (a 1, a 2 ) μ A (a 1, a n ) μ A (a 1, a 2 ) Also min{μ A (a 1, a 2 ), μ A (a 2, a n )} μ A (a 1, a n ) μ A (a 1, a 2 ) μ A (a 1, a n ) [as μ A (a 1, a n ) μ A (a 2, a n )] Therefore μ A (a 1, a n ) = μ A (a 1, a 2 ). Similarly, min{μ A (a 2, a n ), μ A (a n, a 3 )} μ A (a 2, a 3 ) μ A (a 2, a n ) μ A (a 2, a 3 ) Also min{μ A (a 2, a 3 ), μ A (a 3, a n )} μ A (a 2, a n ) μ A (a 2, a 3 ) μ A (a 2, a n ) [as μ A (a 2, a n ) μ A (a 3, a n )] Therefore μ A (a 2, a n ) = μ A (a 2, a 3 ). Proceeding in this way we can show that μ A (a 3, a n ) = μ A (a 3, a 4 ), μ A (a 4, a n ) = μ A (a 4, a 5 ),..., μ A (a n-2, a n ) = μ A (a n-2, a n-1 ) Thus out of the n elements μ A (a 1, a n ), μ A (a 2, a n ), μ A (a 3, a n ),..., μ A (a n-1, a n ) and μ A (a n, a n ) all the elements except μ A (a n-1, a n ) coincide with some of the n 1 elements of Imμ A restricted to Y = {a 1, a 2, a 3,..., a n-1 }. Hence Imμ A on X can have a maximum of n elements. This completes the proof. References [1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), no. 1, [2] B. Banerjee and D. K. Basnet, Intuitionistic Fuzzy Subrings and Ideals, J. of Fuzzy Mathematics, Vol. 11, No. 1, 2003, [3] D. K. Basnet, (α, β)-cut of Intuitionistic fuzzy ideals (submitted)

7 A note on intuitionistic fuzzy equivalence relation 3307 [4] T. T. Buhaescu, Some observationson Intuitionistic fuzzy relations, Itimerat Seminar on Functional Equations, [5] H. Bustince, Conjuntos Intuicionistas e Intervalo valorados Difusos: Propiedades y Construccion Relatciones Intuicionistas Fuzzy. Thesis, Univerrsidad Publica de Navarra,(1994). [6] L. A. Zadeh, Similarity relations and fuzzy orderings, Information Sci. 3 (1971), Received: July, 2010

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Section 9.2 Polar Equations and Graphs

180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify

Διαβάστε περισσότερα

### 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

### 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

### 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

### 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

### ( ) 2 and compare to M.

Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8

Διαβάστε περισσότερα

### SOLVING CUBICS AND QUARTICS BY RADICALS

SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with

Διαβάστε περισσότερα

### 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

### 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

### Bounding Nonsplitting Enumeration Degrees

Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,

Διαβάστε περισσότερα

### Solutions to Selected Homework Problems 1.26 Claim: α : S S which is 1-1 but not onto β : S S which is onto but not 1-1. y z = z y y, z S.

Solutions to Selected Homework Problems 1.26 Claim: α : S S which is 1-1 but not onto β : S S which is onto but not 1-1. Proof. ( ) Since α is 1-1, β : S S such that β α = id S. Since β α = id S is onto,

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

### 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

### 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

### Μηχανική Μάθηση Hypothesis Testing

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider

Διαβάστε περισσότερα

### Example of the Baum-Welch Algorithm

Example of the Baum-Welch Algorithm Larry Moss Q520, Spring 2008 1 Our corpus c We start with a very simple corpus. We take the set Y of unanalyzed words to be {ABBA, BAB}, and c to be given by c(abba)

Διαβάστε περισσότερα

### Second Order RLC Filters

ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

### Strukturalna poprawność argumentu.

Strukturalna poprawność argumentu. Marcin Selinger Uniwersytet Wrocławski Katedra Logiki i Metodologii Nauk marcisel@uni.wroc.pl Table of contents: 1. Definition of argument and further notions. 2. Operations

Διαβάστε περισσότερα

### Risk! " #\$%&'() *!'+,'''## -. / # \$

Risk! " #\$%&'(!'+,'''## -. / 0! " # \$ +/ #%&''&(+(( &'',\$ #-&''&\$ #(./0&'',\$( ( (! #( &''/\$ #\$ 3 #4&'',\$ #- &'',\$ #5&''6(&''&7&'',\$ / ( /8 9 :&' " 4; < # \$ 3 " ( #\$ = = #\$ #\$ ( 3 - > # \$ 3 = = " 3 3, 6?3

Διαβάστε περισσότερα

### 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

### 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

### UNIT - I LINEAR ALGEBRA. , such that αν V satisfying following condition

UNIT - I LINEAR ALGEBRA Definition Vector Space : A non-empty set V is said to be vector space over the field F. If V is an abelian group under addition and if for every α, β F, ν, ν 2 V, such that αν

Διαβάστε περισσότερα

### A Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics

A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions

Διαβάστε περισσότερα

### 2. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν.

Experiental Copetition: 14 July 011 Proble Page 1 of. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν. Ένα μικρό σωματίδιο μάζας (μπάλα) βρίσκεται σε σταθερή απόσταση z από το πάνω μέρος ενός

Διαβάστε περισσότερα

### The Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality

The Probabilistic Method - Probabilistic Techniques Lecture 7: The Janson Inequality Sotiris Nikoletseas Associate Professor Computer Engineering and Informatics Department 2014-2015 Sotiris Nikoletseas,

Διαβάστε περισσότερα

### ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011

Διάρκεια Διαγωνισμού: 3 ώρες Απαντήστε όλες τις ερωτήσεις Μέγιστο Βάρος (20 Μονάδες) Δίνεται ένα σύνολο από N σφαιρίδια τα οποία δεν έχουν όλα το ίδιο βάρος μεταξύ τους και ένα κουτί που αντέχει μέχρι

Διαβάστε περισσότερα

### ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ

ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΕΠΛ 342 Βάσεις εδοµένων ιδάσκων: Γ. Σαµάρας 5η σειρά ασκήσεων: Συναρτησιακές Εξαρτήσεις και Κανονικοποίηση. Λύσεις Μέρος Α. Συναρτησιακές Εξαρτήσεις 1. Αποδείξτε

Διαβάστε περισσότερα

### MATRICES

MARICES 1. Matrix: he arrangement of numbers or letters in the horizontal and vertical lines so that each horizontal line contains same number of elements and each vertical row contains the same numbers

Διαβάστε περισσότερα

### THE GENERAL INDUCTIVE ARGUMENT FOR MEASURE ANALYSES WITH ADDITIVE ORDINAL ALGEBRAS

THE GENERAL INDUCTIVE ARGUMENT FOR MEASURE ANALYSES WITH ADDITIVE ORDINAL ALGEBRAS STEFAN BOLD, BENEDIKT LÖWE In [BoLö ] we gave a survey of measure analyses under AD, discussed the general theory of order

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 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a

Διαβάστε περισσότερα

### Math 6 SL Probability Distributions Practice Test Mark Scheme

Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry

Διαβάστε περισσότερα

### Ψηφιακή Επεξεργασία Εικόνας

ΠΑΝΕΠΙΣΤΗΜΙΟ ΙΩΑΝΝΙΝΩΝ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΪΚΑ ΜΑΘΗΜΑΤΑ Ψηφιακή Επεξεργασία Εικόνας Μετασχηματισμοί έντασης και χωρικό φιλτράρισμα Διδάσκων : Αναπληρωτής Καθηγητής Νίκου Χριστόφορος Άδειες Χρήσης Το παρόν εκπαιδευτικό

Διαβάστε περισσότερα

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

Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 29, 1403-1411 On Pseudo δ-open Fuzzy Sets and Pseudo Fuzzy δ-continuous Functions A. Deb Ray Department of Mathematics West Bengal State University Berunanpukuria,

Διαβάστε περισσότερα

### Trigonometric Formula Sheet

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

Διαβάστε περισσότερα

### Σχέσεις, Ιδιότητες, Κλειστότητες

Σχέσεις, Ιδιότητες, Κλειστότητες Ορέστης Τελέλης telelis@unipi.gr Τµήµα Ψηφιακών Συστηµάτων, Πανεπιστήµιο Πειραιώς Ο. Τελέλης Πανεπιστήµιο Πειραιώς Σχέσεις 1 / 26 Εισαγωγή & Ορισµοί ιµελής Σχέση R από

Διαβάστε περισσότερα

### Αλγόριθμοι και πολυπλοκότητα NP-Completeness (2)

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Αλγόριθμοι και πολυπλοκότητα NP-Completeness (2) Ιωάννης Τόλλης Τμήμα Επιστήμης Υπολογιστών NP-Completeness (2) x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 12 22 32 11 13 21

Διαβάστε περισσότερα

### ECON 381 SC ASSIGNMENT 2

ECON 8 SC ASSIGNMENT 2 JOHN HILLAS UNIVERSITY OF AUCKLAND Problem Consider a consmer with wealth w who consmes two goods which we shall call goods and 2 Let the amont of good l that the consmer consmes

Διαβάστε περισσότερα

### Διπλωματική Εργασία του φοιτητή του Τμήματος Ηλεκτρολόγων Μηχανικών και Τεχνολογίας Υπολογιστών της Πολυτεχνικής Σχολής του Πανεπιστημίου Πατρών

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑΣ ΥΠΟΛΟΓΙΣΤΩΝ ΤΟΜΕΑΣ:ΗΛΕΚΤΡΟΝΙΚΗΣ ΚΑΙ ΥΠΟΛΟΓΙΣΤΩΝ ΕΡΓΑΣΤΗΡΙΟ ΗΛΕΚΤΡΟΝΙΚΩΝ ΕΦΑΡΜΟΓΩΝ Διπλωματική Εργασία του φοιτητή του Τμήματος Ηλεκτρολόγων

Διαβάστε περισσότερα

### Συστήματα Διαχείρισης Βάσεων Δεδομένων

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo

Διαβάστε περισσότερα

### Potential Dividers. 46 minutes. 46 marks. Page 1 of 11

Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and

Διαβάστε περισσότερα

### Derivation of Optical-Bloch Equations

Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be

Διαβάστε περισσότερα

### ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης

Α.Τ.Ε.Ι. ΙΟΝΙΩΝ ΝΗΣΩΝ ΠΑΡΑΡΤΗΜΑ ΑΡΓΟΣΤΟΛΙΟΥ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ «Η διαμόρφωση επικοινωνιακής στρατηγικής (και των τακτικών ενεργειών) για την ενδυνάμωση της εταιρικής

Διαβάστε περισσότερα

### Right Rear Door. Let's now finish the door hinge saga with the right rear door

Right Rear Door Let's now finish the door hinge saga with the right rear door You may have been already guessed my steps, so there is not much to describe in detail. Old upper one file:///c /Documents

Διαβάστε περισσότερα

### EE101: Resonance in RLC circuits

EE11: Resonance in RLC circuits M. B. Patil mbatil@ee.iitb.ac.in www.ee.iitb.ac.in/~sequel Deartment of Electrical Engineering Indian Institute of Technology Bombay I V R V L V C I = I m = R + jωl + 1/jωC

Διαβάστε περισσότερα

### If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2

Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

### TMA4115 Matematikk 3

TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet

Διαβάστε περισσότερα

### «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

### If we restrict the domain of y = sin x to [ π 2, π 2

Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

### Testing for Indeterminacy: An Application to U.S. Monetary Policy. Technical Appendix

Testing for Indeterminacy: An Application to U.S. Monetary Policy Technical Appendix Thomas A. Lubik Department of Economics Johns Hopkins University Frank Schorfheide Department of Economics University

Διαβάστε περισσότερα

### Partial Trace and Partial Transpose

Partial Trace and Partial Transpose by José Luis Gómez-Muñoz http://homepage.cem.itesm.mx/lgomez/quantum/ jose.luis.gomez@itesm.mx This document is based on suggestions by Anirban Das Introduction This

Διαβάστε περισσότερα

### Homework 3 Solutions

Homework 3 Solutions Differential Topology (math876 - Spring2006 Søren Kold Hansen Problem 1: Exercise 3.2 p. 246 in [MT]. Let {ɛ 1,..., ɛ n } be the basis of Alt 1 (R n dual to the standard basis {e 1,...,

Διαβάστε περισσότερα

### Συντακτικές λειτουργίες

2 Συντακτικές λειτουργίες (Syntactic functions) A. Πτώσεις και συντακτικές λειτουργίες (Cases and syntactic functions) The subject can be identified by asking ποιος (who) or τι (what) the sentence is about.

Διαβάστε περισσότερα

### ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΓΕΩΤΕΧΝΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΚΑΙ ΔΙΑΧΕΙΡΙΣΗΣ ΠΕΡΙΒΑΛΛΟΝΤΟΣ. Πτυχιακή εργασία

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΓΕΩΤΕΧΝΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΚΑΙ ΔΙΑΧΕΙΡΙΣΗΣ ΠΕΡΙΒΑΛΛΟΝΤΟΣ Πτυχιακή εργασία ΑΝΑΛΥΣΗ ΚΟΣΤΟΥΣ-ΟΦΕΛΟΥΣ ΓΙΑ ΤΗ ΔΙΕΙΣΔΥΣΗ ΤΩΝ ΑΝΑΝΕΩΣΙΜΩΝ ΠΗΓΩΝ ΕΝΕΡΓΕΙΑΣ ΣΤΗΝ ΚΥΠΡΟ ΜΕΧΡΙ ΤΟ 2030

Διαβάστε περισσότερα

### A Lambda Model Characterizing Computational Behaviours of Terms

A Lambda Model Characterizing Computational Behaviours of Terms joint paper with Silvia Ghilezan RPC 01, Sendai, October 26, 2001 1 Plan of the talk normalization properties inverse limit model Stone dualities

Διαβάστε περισσότερα

### Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής

Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής Να γραφεί πρόγραμμα το οποίο δέχεται ως είσοδο μια ακολουθία S από n (n 40) ακέραιους αριθμούς και επιστρέφει ως έξοδο δύο ακολουθίες από θετικούς ακέραιους

Διαβάστε περισσότερα

### MATHEMATICS. 1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81

1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81 We know that KA = A If A is n th Order 3AB =3 3 A. B = 27 1 3 = 81 3 2. If A= 2 1 0 0 2 1 then

Διαβάστε περισσότερα

### Online Appendix to. When Do Times of Increasing Uncertainty Call for Centralized Harmonization in International Policy Coordination?

Online Appendix to When o Times of Increasing Uncertainty Call for Centralized Harmonization in International Policy Coordination? Andrzej Baniak Peter Grajzl epartment of Economics, Central European University,

Διαβάστε περισσότερα

### Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών. ΗΥ-570: Στατιστική Επεξεργασία Σήµατος. ιδάσκων : Α. Μουχτάρης. εύτερη Σειρά Ασκήσεων.

Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 2015 ιδάσκων : Α. Μουχτάρης εύτερη Σειρά Ασκήσεων Λύσεις Ασκηση 1. 1. Consder the gven expresson for R 1/2 : R 1/2

Διαβάστε περισσότερα

### ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΕΠΑΝΑΣΧΕΔΙΑΣΜΟΣ ΓΡΑΜΜΗΣ ΣΥΝΑΡΜΟΛΟΓΗΣΗΣ ΜΕ ΧΡΗΣΗ ΕΡΓΑΛΕΙΩΝ ΛΙΤΗΣ ΠΑΡΑΓΩΓΗΣ REDESIGNING AN ASSEMBLY LINE WITH LEAN PRODUCTION TOOLS

ΔΙΑΤΜΗΜΑΤΙΚΟ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗ ΔΙΟΙΚΗΣΗ ΤΩΝ ΕΠΙΧΕΙΡΗΣΕΩΝ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΕΠΑΝΑΣΧΕΔΙΑΣΜΟΣ ΓΡΑΜΜΗΣ ΣΥΝΑΡΜΟΛΟΓΗΣΗΣ ΜΕ ΧΡΗΣΗ ΕΡΓΑΛΕΙΩΝ ΛΙΤΗΣ ΠΑΡΑΓΩΓΗΣ REDESIGNING AN ASSEMBLY LINE WITH

Διαβάστε περισσότερα

### ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ

ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΛΕΝΑ ΦΛΟΚΑ Επίκουρος Καθηγήτρια Τµήµα Φυσικής, Τοµέας Φυσικής Περιβάλλοντος- Μετεωρολογίας ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Πληθυσµός Σύνολο ατόµων ή αντικειµένων στα οποία αναφέρονται

Διαβάστε περισσότερα

### ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007

Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Όλοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι του 10000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Αν κάπου κάνετε κάποιες υποθέσεις

Διαβάστε περισσότερα

### HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? What is the 50 th percentile for the cigarette histogram?

HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? The point on the horizontal axis such that of the area under the histogram lies to the left of that point (and to the right) What

Διαβάστε περισσότερα

### Test Data Management in Practice

Problems, Concepts, and the Swisscom Test Data Organizer Do you have issues with your legal and compliance department because test environments contain sensitive data outsourcing partners must not see?

Διαβάστε περισσότερα

### EPL 603 TOPICS IN SOFTWARE ENGINEERING. Lab 5: Component Adaptation Environment (COPE)

EPL 603 TOPICS IN SOFTWARE ENGINEERING Lab 5: Component Adaptation Environment (COPE) Performing Static Analysis 1 Class Name: The fully qualified name of the specific class Type: The type of the class

Διαβάστε περισσότερα

### Θεωρία Πληροφορίας και Κωδίκων

Θεωρία Πληροφορίας και Κωδίκων Δρ. Νικόλαος Κολοκοτρώνης Λέκτορας Πανεπιστήμιο Πελοποννήσου Τμήμα Επιστήμης και Τεχνολογίας Υπολογιστών Τέρμα Οδού Καραϊσκάκη, 22100 Τρίπολη E mail: nkolok@uop.gr Web: http://www.uop.gr/~nkolok/

Διαβάστε περισσότερα

### On geodesic mappings of Riemannian spaces with cyclic Ricci tensor

Annales Mathematicae et Informaticae 43 (2014) pp. 13 17 http://ami.ektf.hu On geodesic mappings of Riemannian spaces with cyclic Ricci tensor Sándor Bácsó a, Robert Tornai a, Zoltán Horváth b a University

Διαβάστε περισσότερα

### Δίκτυα Δακτυλίου. Token Ring - Polling

Δίκτυα Δακτυλίου Token Ring - Polling Όλοι οι κόμβοι είναι τοποθετημένοι σε ένα δακτύλιο. Εκπέμπει μόνο ο κόμβος ο οποίος έχει τη σκυτάλη (token). The token consists of a number of octets in a specific

Διαβάστε περισσότερα

### ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ

ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ Γηπισκαηηθή Δξγαζία ηνπ Φνηηεηή ηνπ ηκήκαηνο Ζιεθηξνιόγσλ Μεραληθώλ θαη Σερλνινγίαο Ζιεθηξνληθώλ

Διαβάστε περισσότερα

### MATRIX INVERSE EIGENVALUE PROBLEM

English NUMERICAL MATHEMATICS Vol.14, No.2 Series A Journal of Chinese Universities May 2005 A STABILITY ANALYSIS OF THE (k) JACOBI MATRIX INVERSE EIGENVALUE PROBLEM Hou Wenyuan ( ΛΠ) Jiang Erxiong( Ξ)

Διαβάστε περισσότερα

### AME SAMPLE REPORT James R. Cole, Ph.D. Neuropsychology

Setting the Standard since 1977 Quality and Timely Reports Med-Legal Evaluations Newton s Pyramid of Success AME SAMPLE REPORT Locations: Oakland & Sacramento SCHEDULING DEPARTMENT Ph: 510-208-4700 Fax:

Διαβάστε περισσότερα

### ΤΟ ΜΟΝΤΕΛΟ Οι Υποθέσεις Η Απλή Περίπτωση για λi = μi 25 = Η Γενική Περίπτωση για λi μi..35

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΤΟΜΕΑΣ ΣΤΑΤΙΣΤΙΚΗΣ ΚΑΙ ΕΠΙΧΕΙΡΗΣΙΑΚΗΣ ΕΡΕΥΝΑΣ ΑΝΑΛΥΣΗ ΤΩΝ ΣΥΣΧΕΤΙΣΕΩΝ ΧΡΕΟΚΟΠΙΑΣ ΚΑΙ ΤΩΝ

Διαβάστε περισσότερα

### Bessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!

Bessel functions The Bessel function J ν (z of the first kind of order ν is defined by J ν (z ( (z/ν ν Γ(ν + F ν + ; z 4 ( k k ( Γ(ν + k + k! For ν this is a solution of the Bessel differential equation

Διαβάστε περισσότερα

### 1. Ηλεκτρικό μαύρο κουτί: Αισθητήρας μετατόπισης με βάση τη χωρητικότητα

IPHO_42_2011_EXP1.DO Experimental ompetition: 14 July 2011 Problem 1 Page 1 of 5 1. Ηλεκτρικό μαύρο κουτί: Αισθητήρας μετατόπισης με βάση τη χωρητικότητα Για ένα πυκνωτή χωρητικότητας ο οποίος είναι μέρος

Διαβάστε περισσότερα

### SPECIAL FUNCTIONS and POLYNOMIALS

SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195

Διαβάστε περισσότερα

### ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 10η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 0η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Best Response Curves Used to solve for equilibria in games

Διαβάστε περισσότερα

### 1. A fully continuous 20-payment years, 30-year term life insurance of 2000 is issued to (35). You are given n A 1

Chapter 7: Exercises 1. A fully continuous 20-payment years, 30-year term life insurance of 2000 is issued to (35). You are given n A 1 35+n:30 n a 35+n:20 n 0 0.068727 11.395336 10 0.097101 7.351745 25

Διαβάστε περισσότερα

### Démographie spatiale/spatial Demography

ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ Démographie spatiale/spatial Demography Session 1: Introduction to spatial demography Basic concepts Michail Agorastakis Department of Planning & Regional Development Άδειες Χρήσης

Διαβάστε περισσότερα

### Generalized Double Star Closed Sets in Interior Minimal Spaces

IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 4 Ver. II (Jul - Aug. 2015), PP 56-61 www.iosrjournals.org Generalized Double Star Closed Sets in Interior Minimal

Διαβάστε περισσότερα

### n r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1)

8 Higher Derivative of the Product of Two Fuctios 8. Leibiz Rule about the Higher Order Differetiatio Theorem 8.. (Leibiz) Whe fuctios f ad g f g are times differetiable, the followig epressio holds. r

Διαβάστε περισσότερα

### ΑΝΩΤΑΤΗ ΣΧΟΛΗ ΠΑΙ ΑΓΩΓΙΚΗΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΚΗΣ ΕΚΠΑΙ ΕΥΣΗΣ ΠΑΡΑΔΟΤΕΟ ΕΠΙΣΤΗΜΟΝΙΚΗ ΕΡΓΑΣΙΑ ΣΕ ΔΙΕΘΝΕΣ ΕΠΙΣΤΗΜΟΝΙΚΟ ΠΕΡΙΟΔΙΚΟ

ΑΝΩΤΑΤΗ ΣΧΟΛΗ ΠΑΙ ΑΓΩΓΙΚΗΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΚΗΣ ΕΚΠΑΙ ΕΥΣΗΣ (Α.Σ.ΠΑΙ.Τ.Ε.) «Αρχιμήδης ΙΙΙ Ενίσχυση Ερευνητικών ομάδων στην Α.Σ.ΠΑΙ.Τ.Ε.» Υποέργο: 8 Τίτλος: «Εκκεντρότητες αντισεισμικού σχεδιασμού ασύμμετρων

Διαβάστε περισσότερα

### Physical DB Design. B-Trees Index files can become quite large for large main files Indices on index files are possible.

B-Trees Index files can become quite large for large main files Indices on index files are possible 3 rd -level index 2 nd -level index 1 st -level index Main file 1 The 1 st -level index consists of pairs

Διαβάστε περισσότερα