A Note on Intuitionistic Fuzzy. Equivalence Relation


 Ειδοθεα Θεοδωρίδης
 2 χρόνια πριν
 Προβολές:
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 Jcomposition 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 n1 } {a n }. When restricted to Y = {a 1, a 2, a 3,..., a n1 }, 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 n1, 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 n1, a n1 ). 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 n1, 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 n2, a n ) = μ A (a n2, a n1 ) 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 n1, a n ) and μ A (a n, a n ) all the elements except μ A (a n1, a n ) coincide with some of the n 1 elements of Imμ A restricted to Y = {a 1, a 2, a 3,..., a n1 }. 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
Homomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 22489940 Volume 3, Number 1 (2013), pp. 3945 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Διαβάστε περισσότεραIntuitionistic Fuzzy Ideals of Near Rings
International Mathematical Forum, Vol. 7, 202, no. 6, 769776 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 περισσότεραCommutative 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 περισσότεραHomomorphism of Intuitionistic Fuzzy Groups
International Mathematical Forum, Vol. 6, 20, no. 64, 369378 Homomorphism o Intuitionistic Fuzz Groups P. K. Sharma Department o Mathematics, D..V. College Jalandhar Cit, Punjab, India pksharma@davjalandhar.com
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραEvery set of firstorder formulas is equivalent to an independent set
Every set of firstorder formulas is equivalent to an independent set May 6, 2008 Abstract A set of firstorder formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραGÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)
IFSCOM016 1 Proceeding Book No. 1 pp. 155161 (016) ISBN: 9789756900543 SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότερα1. 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 περισσότεραCoefficient Inequalities for a New Subclass of Kuniformly Convex Functions
International Journal of Computational Science and Mathematics. ISSN 097489 Volume, Number (00), pp. 6775 International Research Publication House http://www.irphouse.com Coefficient Inequalities for
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular SturmLiouville. ii Singular SturmLiouville mixed boundary conditions. iii Not SturmLiouville ODE is not in SturmLiouville form. iv Regular SturmLiouville note
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραChapter 3: Ordinal Numbers
Chapter 3: Ordinal Numbers There are two kinds of number.. Ordinal numbers (0th), st, 2nd, 3rd, 4th, 5th,..., ω, ω +,... ω2, ω2+,... ω 2... answers to the question What position is... in a sequence? What
Διαβάστε περισσότεραA Note on Characterization of Intuitionistic Fuzzy Ideals in Γ NearRings
International Journal of Computational Science and Mathematics. ISSN 09743189 Volume 3, Number 1 (2011), pp. 6171 International Research Publication House http://www.irphouse.com A Note on Characterization
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239246 HIKARI Ltd, www.mhikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραSOME INTUITIONISTIC FUZZY MODAL OPERATORS OVER INTUITIONISTIC FUZZY IDEALS AND GROUPS
IFSCOM016 1 Proceeding Book No. 1 pp. 8490 (016) ISBN: 9789756900543 SOME INTUITIONISTIC FUZZY MODAL OPERATORS OVER INTUITIONISTIC FUZZY IDEALS AND GROUPS SINEM TARSUSLU(YILMAZ), GÖKHAN ÇUVALCIOĞLU,
Διαβάστε περισσότεραSequent Calculi for the Modal µcalculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008
Sequent Calculi for the Modal µcalculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008 Introduction Koz: Axiomatisation for the modal µcalculus over K Axioms: All classical
Διαβάστε περισσότεραMINIMAL 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 περισσότεραHomomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PSalgebras
Annals of Pure and Applied athematics Vol. 8, No. 1, 2014, 93104 ISSN: 2279087X (P), 22790888(online) Published on 11 November 2014 www.researchmathsci.org Annals of Homomorphism and Cartesian Product
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(αβ) cos α cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β sin α cos(αβ cos α cos β sin α NOTE: cos(αβ cos α cos β cos(αβ cos α cos β Proof of cos(αβ cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραSOME PROPERTIES OF FUZZY REAL NUMBERS
Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 2127 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical
Διαβάστε περισσότερα2. 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 20050308 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Διαβάστε περισσότεραSome 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 ISSN21751188 on line ISSN00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v30i2.13793 Some new generalized topologies via hereditary
Διαβάστε περισσότεραStatistical 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 onesided
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests SideNote: So far we have seen a few approaches for creating tests such as NeymanPearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00 Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραNowherezero flows Let be a digraph, Abelian group. A Γcirculation in is a mapping : such that, where, and : tail in X, head in
Nowherezero flows Let be a digraph, Abelian group. A Γcirculation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowherezero Γflow is a Γcirculation such that
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a nontrivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normalorder
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 0303 :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραF 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 περισσότεραDIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS
GANIT J. Bangladesh Math. oc. IN 606694) 0) 7 DIRECT PRODUCT AND WREATH PRODUCT OF TRANFORMATION EMIGROUP ubrata Majumdar, * Kalyan Kumar Dey and Mohd. Altab Hossain Department of Mathematics University
Διαβάστε περισσότεραk 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 περισσότερα12. RadonNikodym Theorem
Tutorial 12: RadonNikodym Theorem 1 12. RadonNikodym 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 περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραMath 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 περισσότεραGenerating Set of the Complete Semigroups of Binary Relations
Applied Mathematics 06 7 9807 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 περισσότεραSolution 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 περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραThe operators over the generalized intuitionistic fuzzy sets
Int. J. Nonlinear Anal. Appl. 8 (2017) No. 1, 1121 ISSN: 20086822 (electronic) http://dx.doi.org/10.22075/ijnaa.2017.11099.1542 The operators over the generalized intuitionistic fuzzy sets Ezzatallah
Διαβάστε περισσότεραOn a fourdimensional 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 fourdimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραLecture 15  Root System Axiomatics
Lecture 15  Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the
Διαβάστε περισσότεραPhys460.nb Solution for the tdependent 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 περισσότεραPractice 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 περισσότεραSection 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Διαβάστε περισσότεραAreas 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 περισσότεραSpaceTime Symmetries
Chapter SpaceTime Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
Διαβάστε περισσότεραChapter 2. Ordinals, wellfounded relations.
Chapter 2. Ordinals, wellfounded relations. 2.1. Wellfounded Relations. We start with some definitions and rapidly reach the notion of a wellordered set. Definition. For any X and any binary relation
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.355.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.55.3. This is our third
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, 1, 1, 1) p 0 = p 0 p = p i = p i p μ p μ = p 0 p 0 + p i p i = E c 2  p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations.   
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 20121231 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραOperation Approaches on αγopen Sets in Topological Spaces
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 10, 491498 Operation Approaches on αγopen Sets in Topological Spaces N. Kalaivani Department of Mathematics VelTech HighTec Dr.Rangarajan Dr.Sakunthala
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραNew bounds for spherical twodistance sets and equiangular lines
New bounds for spherical twodistance sets and equiangular lines Michigan State University Oct 831, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a
Διαβάστε περισσότεραTHE 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 περισσότεραω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Διαβάστε περισσότεραMINIMAL INTUITIONISTIC GENERAL LFUZZY AUTOMATA
italian journal of pure applied mathematics n. 35 2015 (155 186) 155 MINIMAL INTUITIONISTIC GENERAL LUZZY AUTOMATA M. Shamsizadeh M.M. Zahedi Department of Mathematics Kerman Graduate University of Advanced
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραAreas 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 περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε 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 περισσότεραF19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Διαβάστε περισσότεραSOLVING 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 περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS ShiahSen Wang The graphs are prepared by ChienLun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότεραBounding Nonsplitting Enumeration Degrees
Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2permitting for the enumeration degrees. Till now,
Διαβάστε περισσότεραCHARACTERIZATION OF BIPOLAR FUZZY IDEALS IN ORDERED GAMMA SEMIGROUPS
JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 23034866, ISSN (o) 23034947 www.imvibl.org /JOURNALS / JOURNAL Vol. 8(2018), 141156 DOI: 10.7251/JIMVI1801141C Former BULLETIN OF
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραA TwoSided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering
Electronic Companion A TwoSie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix
Διαβάστε περισσότεραMyhill Nerode Theorem for Fuzzy Automata (Minmax Composition)
Intern. J. Fuzzy Mathematical Archive Vol. 3, 2013, 5867 ISSN: 2320 3242 (P), 2320 3250 (online) Published on 30 December 2013 www.researchmathsci.org International Journal of Myhill Nerode Theorem for
Διαβάστε περισσότεραA General Note on δquasi Monotone and Increasing Sequence
International Mathematical Forum, 4, 2009, no. 3, 143149 A General Note on δquasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in
Διαβάστε περισσότερα9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Διαβάστε περισσότεραJordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp
Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp.115126. α, β, γ ORTHOGONALITY ABDALLA TALLAFHA Abstract. Orthogonality in inner product spaces can be expresed using the notion of norms.
Διαβάστε περισσότεραIntuitionistic Supra Gradation of Openness
Applied Mathematics & Information Sciences 2(3) (2008), 291307 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. Intuitionistic Supra Gradation of Openness A. M. Zahran 1, S. E.
Διαβάστε περισσότεραA new modal operator over intuitionistic fuzzy sets
1 st Int. Workshop on IFSs, Mersin, 14 Nov. 2014 Notes on Intuitionistic Fuzzy Sets ISSN 1310 4926 Vol. 20, 2014, No. 5, 1 8 A new modal operator over intuitionistic fuzzy sets Krassimir Atanassov 1, Gökhan
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραLecture 21: Properties and robustness of LSE
Lecture 21: Properties and robustness of LSE BLUE: Robustness of LSE against normality We now study properties of l τ β and σ 2 under assumption A2, i.e., without the normality assumption on ε. From Theorem
Διαβάστε περισσότεραENGR 691/692 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework 1: Bayesian Decision Theory (solutions) Due: September 13
ENGR 69/69 Section 66 (Fall 06): Machine Learning Assigned: August 30 Homework : Bayesian Decision Theory (solutions) Due: Septemer 3 Prolem : ( pts) Let the conditional densities for a twocategory onedimensional
Διαβάστε περισσότεραNew Operations over Interval Valued Intuitionistic Hesitant Fuzzy Set
Mathematics and Statistics (): 67 04 DOI: 0.89/ms.04.000 http://www.hrpub.org New Operations over Interval Valued Intuitionistic Hesitant Fuzzy Set Said Broumi * Florentin Smarandache Faculty of Arts
Διαβάστε περισσότεραSubclass of Univalent Functions with Negative Coefficients and Starlike with Respect to Symmetric and Conjugate Points
Applied Mathematical Sciences, Vol. 2, 2008, no. 35, 17391748 Subclass of Univalent Functions with Negative Coefficients and Starlike with Respect to Symmetric and Conjugate Points S. M. Khairnar and
Διαβάστε περισσότερα