Spectrum of (, q)-fuzzy Prime h-ideals of a Hemiring
|
|
- Lenard Sutton
- 6 years ago
- Views:
Transcription
1 World Appled Scences Journal 17 (12): , 2012 ISSN IDOSI Publcatons, 2012 Spectrum of (, q)-fuzzy Prme h-deals of a Hemrng 1 M. Shabr and 2 T. Mahmood 1 Department of Mathematcs, Quad--Azam Unversty, Islamabad, Pakstan 2 Department of Mathematcs and Statstcs, Internatonal Islamc Unversty, Islamabad, Pakstan Abstract: In ths paper we defne (, q)-fuzzy prme h-deal, (, q)-fuzzy semprme h deal. We also construct spectrum of (, q)-fuzzy prme h-deals n two dfferent ways. In the last secton we also dscuss mlpcaton-based (, q)-fuzzy prme h-deals mathematcs subject classfcaton: 16Y60 08A72 03G25 03E72 Key words: (, q)-fuzzy prme h-deal (, q)-fuzzy semprme h-deal spectrum of (, q)- fuzzy prme h-deal INTRODUCTION Semrngs, ntroduced by Vandver [1] n 1934., appears n natural way for studyng optmzaton theory, graph theory, theory of dscrete event dynamcal systems, matrces, determnants, generalzed fuzzy computaton, theory of automata, formal languages theory, codng theory, analyss of computer programmes. Hemrngs, whch are semrngs wth commutatve addton and absorbng addtve dentty, are used n some applcatons to the theory of automata, the theory of formal languages and n computer scences [2-5]. Ideals of semrngs play an mportant role n the structure theory of semrngs. However, n general, they do not concde wth the usual rng deals. So many results n rng theory have no analogues n semrngs usng only deals. In order to overcome ths dffculty n [6] Henrksen defned a more restrcted class of deals n semrngs, called k-deals, wth the property that f a semrng R s a rng, then a comple n R s a k-deal f and only f t s a rng deal. Another more restrcted class of deals n hemrngs, called now h-deals, has been gven and nvestgated by Izuka [7]. In 1965, Zadeh [8] ntroduced the concept of fuzzy set. Snce then many efforts have been made to use ths concept n algebrac structures. In ths connecton, Rosenfeld [9] ntroduced the noton of fuzzy subgroups. In [10] J. Ahsan et. al ntated the study of fuzzy semrngs. The notons of "belongngness ( )" and "quasconcdence (q)" of fuzzy ponts and fuzzy sets proposed and dscussed n [11, 12]. In [13], Zhan and Dudek ntroduced the concept of prme fuzzy h-deals of a hemrng and establshed some nterestng results. In [14], Kumbhojkar addressed the questons rased by Zhan and Dudek, redefned prme fuzzy h-deals, defned semprme fuzzy h-deals of a hemrng and constructed spectrum of prme fuzzy h-deals. Snce prme spectrum on the set of prme deals of a commutatve hemrng wth unty plays an mportant role n commutatve algebra, algebrac geometry, lattce theory and semprme deals of hemrngs determne the topology. So keepng n vew all these facts n ths paper we etend the deas of Kumbhojkar and defne (, q)-fuzzy prme h-deal, (, q)-fuzzy semprme h-deal. We also construct spectrum of (, q)-fuzzy prme h-deals n two dfferent ways. In the last secton we also dscuss mplcaton-based (, q)-fuzzy prme h-deals. PRELIMINARIES For basc defntons we refer to [4, 14]. A fuzzy subset ƒ of a unverse X s a functon ƒ: X [0,1]. A fuzzy subset ƒ of X of the form t (0,1] f y= f ( y) = 0 f y s called the fuzzy pont wth support and value t and s denoted by t. Furthermore for t (0,1], level subset of ƒ s denoted and defned by U(f,t) = { X:f() t}. Correspondng Author: T. Mahmood, Department of Mathematcs and Statstcs, Internatonal Islamc Unversty, Islamabad, Pakstan 1815
2 World Appl. Sc. J., 17 (12): , 2012 In [12] Pu and Lu defned and dscussed t αƒ, α,q, q, q. A fuzzy pont t s sad where { } to belong to (resp. quas-concdent wth) ƒ, wrtten t resp. f + t > 1. as t ƒ ( resp. qf t ), f f qf ( resp. qf) t ( resp. f and qf ) t means that t ƒ or t qƒ t t. To say that t α f means that t αƒ does not hold. For any two fuzzy subsets ƒ and g of X, ƒ g means that, for all X, ƒ() g(). For two fuzzy subsets ƒ and g of X, ƒ g and ƒ g wll mean the followng fuzzy subsets of X 2 (1 a) t f t qf Theorem: For a fuzzy subset ƒ of R the followng are equvallent: 2 (1 a) t f t qf (1 b) 2 f() f 0.5 for all R and t (0,1]. Corollary: A fuzzy h-deal ƒ of R s an (, q)-fuzzy semprme h-deal of R f and only f t satsfes (1 b). for all X. (f g)() = f() g() (f g)() = f() g() Lemma: A fuzzy h-deal ƒ of R s an (, q)-fuzzy semprme h-deal of R f and only f for all t (0,0.5], the non-empty level subset U(ƒ,t) s ether semprme h- deal of R or s R tself. (, q)-fuzzy PRIME AND (, q)-fuzzy SEMIPRIME H-IDEALS From now to onward throughout R wll denote a commutatve hemrng wth 1. Defnton: A fuzzy h-deal ƒ of R s sad to be an (, q)-fuzzy prme h-deal of R f t s non constant and for all,y R and t (0,1] (1a) (y) t f t qf or yt qf Theorem: For a fuzzy subset ƒ of R the followng are equvallent: (1a) (y) t f t qf or yt qf (1b) f() f(y) f(y) 0.5 for all,y R and t (0,1]. Corollary: A fuzzy h-deal ƒ of R s an (, q)-fuzzy prme h-deal of R f and only f t satsfes (1b). Lemma: A non-empty subset P of R s semprme h- deal of R f and only f C P s an (, q)-fuzzy semprme h-deal of R. Remark: Every (, q)-fuzzy prme h-deal of R s (, q)-fuzzy semprme h-deal of R, however converse s not true n general. Eample: Let N 0 = {0} N and p 1, p 2, p 3, be the dstnct prme numbers n N 0. If J 0 = N 0 and J l = p 1 p 2 p 3 p l N 0, l = 1,2,3,, then J 0 J 1 J 2 J n J n+1 As every non-zero element of N 0 has unque prme factorzaton, J l s a semprme h-deal for l = 2,3, but not a prme h-deal. Then for such values of l, by Lemma 3.10, C s an (, q)-fuzzy semprme h- l J deal of R, but by Lemma 3.5, C l s not an (, q)- J fuzzy prme h-deal of R. In our net dscusson Ω denotes the set of all (, q)-fuzzy prme h-deals of R, also we assume for every f Ω, ƒ(0) = 1. Lemma: A fuzzy h-deal ƒ of R s an (, q)-fuzzy prme h-deal of R f and only f for all t (0,0.5], the non-empty level subset U(f,t) = {:f() t} s ether prme h-deal of R or s R tself. Lemma: A non-empty subset P of R s prme h-deal of R f and only f C P s an (, q)-fuzzy prme h-deal of R. Defnton: A fuzzy h-deal ƒ of R s sad to be an (, q)-fuzzy semprme h-deal of R f t s non constant and for all R and t (0,1]. SPECTRUM OF (, q)-fuzzy PRIME H-IDEALS We denote and defne Ω = {ƒ:ƒ s (, q)- fuzzy prme h-deal of R}. If λ Ω, then V( λ ) = {f Ω: λ f}. If λ= C {a}, where a R, then V( λ ) = V(a) = {f Ω :C{a} f} = {f Ω :f(a) = 1} Further r() h λ = {f Ω: λ f} s called (, q)- fuzzy h-prme radcal of λ. Further 1816
3 World Appl. Sc. J., 17 (12): , 2012 V( λ ) =Ω V( λ ) = {f Ω : λ> f} Lemma: r h (λ) s (, q)-fuzzy semprme h-deal contanng λ. Proof: Proof s straghtforward, hence omtted. Lemma: If λ,µ are fuzzy subsets of R. Then Proof () As V(o) =φ and V( R ) =Ω, where R Ω, defned by R() = 1, for all R. So φω IΩ,. () Let {V( λ): I} be an arbtrary collecton of elements of I(Ω). To prove IV( λ ) s n I(Ω). () λ µ mples V( µ ) V( λ ) () V( λ) V( µ ) V( λ µ ) () If A and B are h-deals of R, then V(C A) V(C B) = V(C A B) (v) If η s (, q)-fuzzy h-deal of R generated by λ and r h (η) s (, q)-fuzzy h-prme radcal of η. Then V( λ ) = V( η ) = V(r h ( η )). (v) If { λ : Λ } s a famly of fuzzy subsets of R, then V( λ { : Λ }) = λ { : Λ } (v) If A R, then V(C A) = {V(a): a A} (v) V(a) V(b) = V(ab), for all a,b R. Proof () Let f V( µ ). Then µ f λ µ f f V( λ ). () As λ µ λ and λ µ µ, by () V( λ) V( λ µ ) and V( µ ) V( λ µ ) V( λ) V( µ ) V( λ µ ). () By usng Lemma 2.5 and () V(C A) V(C B) V(C A B). Now let f V(C A B). Whch mples CA B f f() = 1 for all A B. Then ƒ() = 1 for all A or ƒ() = 1 for all B CA f or CB f f V(C A) or f V(C B) f V(C A) V(C B). Hence V(C A) V(C B) = V(C A B). (v) Frst we prove V( λ ) = V( η ). Let Γ be the class of all (, q)-fuzzy h-deals of R contanng λ. Then η= Γ. Then by the fact λ f η f, result follows mmedately. Now we prove V( η= ) V(r ( η )). Let Γ be the class of all (, q)- h fuzzy prme h-deals of R contanng η. Then r() η = Γ. Then by the fact r() η f η f, h result follows mmedately. (v) Let f λ { : Λ }. Then f λ for all Λ λ f for all Λ λ { : Λ} f f V( λ { : Λ }) λ { : Λ} V( λ { : Λ }). Workng backward we get the desred result. (v) Follows drectly from (v). (v) Straghtforward. Theorem: The set I( Ω ) = {V( λ): λ Ω } forms a topology on the set Ω. h Note that equal to ƒ(), then 1817 V( ) { f : f} { f : k I such that f} λ = Ω λ > = Ω λ > λ1( a1) λ2( a 2)... λ = λ1( b1) λ2( b 2)... I + a1+ a = b1 + b where a,b R for all and also only a fnte number of the as and bs are not zero. Snce λ (0) = 1, therefore we are consderng the nfmum of a fnte number of terms because 1 s are effectvely not beng consdered. Now, f for some k I, λ k >ƒ, then there ests R such that λ k ()>ƒ(). Consder the partcular epresson for whch a k =, b k = 0 and a = b = 0 for all k. We see that λ k () s an element of the set whose supremum s defned to be ( λ ). k. Ths mples I Thus ( λ ) λ > f ( λ ) > f that s λ > f. Hence λ k >ƒ for some k I mples λ > f. Conversely, suppose that ests R such that ( λ ) > k then there λ > f f. λ1 a1 λ2 a 2... > f λ a a... z b b... z 1 b1 λ2 b = λ1 a1 λ2 a 2... λ1 b1 λ2 b 2... > f ( ) ( a1 a 2... b1 b = + + beng the correspondng breakup of, where only a fnte number of as and bs are not zero.) Now on the contrary f we suppose that (*) does not hold, that s f all the elements of the set (whose supremum we are takng) are ndvdually less than are
4 World Appl. Sc. J., 17 (12): , 2012 λ1( a1) λ2( a 2)... λ = λ a 1( b1) λ2( b 2)... + I 1+ a z= b1+ b z f whch s a contradcton. Thus (*) holds. Thus, Let and λ1( 1) λ2( 2) λ1( 1) λ2( 2) ( 1) ( 2) ( 1) ( 2) a a... b b... > f f a f a... f b f b... ( a ) ( a )... ( b ) ( b )... ( ) λ1 1 λ2 2 λ1 1 λ2 2 =λ l l where 1 l ( 1) ( 2) ( 1) ( 2) = ( l) f a f a...f b f b... f So, l( l) f( l) λ > t follows that λ l >ƒ for some l I. Hence λ > f mples that λ l >ƒ for some l I. Hence the two statements some l I are equvalent. Hence As { } V( λ ) = f Ω : λ > f λ Ω and λ l >ƒ for λ > f = f Ω : λ > f = V( λ) I I thus, IV( ) () Let V( ),V( ) λ I Ω. λ1 λ2 IΩ. Proof: As for a R,V(a) = {f Ω :f(a) = 1}, so V(a) = {f Ω :f(a) 1}. Now as V (1) = Ω, so {V(a):a R} =Ω. Also by usng (v) of Lemma 4.2, V(ab) = V(a) V(b) for all a,b R Thus B s base for some topology T on Ω. Now let U T,-then for some A R, U = {V(a):a A} = Ω { V(a): a A} =Ω {V(a): a A} =Ω V(C ) (by (v) of Lemma 4.2) = V(C ) A Further as V(C A) = V(C A ) = V(r h(c A )). Hence T s completely determned by (, q)-fuzzy semprme h-deals of R. IMPLICATION-BASED (, PRIME H-IDEALS A q)-fuzzy Fuzzy propostonal calculus s an etenson of the Arstotelan propostonal calculus. In fuzzy propostonal calculus the truth set s taken [0, 1] nstead of {0, 1}, whch s the truth set n Arstotelan propostonal calculus. In fuzzy logc some of the operators lke,,, can be defned by usng truth tables. One can also use the etenson prncple to obtan the defntons of these operators. In fuzzy logc the truth value of a fuzzy proposton λ s denoted by [λ]. In the followng we gve fuzzy logc and ts correspondng set theoretcal notatons, whch we wll use n the paper hereafter. [ λ=λ ] (), [ λ ] = 1 λ () [P Q] = mn{[p],[q]}, [P Q] = ma{[p],[q]} [P Q] = mn{1,1 [P] + [Q]} [ P()] = nf[p()] P f and only f [P] = 1. Then { } V( λ1) V( λ 2 ) = f Ω : λ 1 > f and λ 2 > f = V( λ1 λ2) Hence t follows that I(Ω) forms a topology on the set Ω. Operator name Early Zadeh Lukasewcz Standard star (Godel) Contraposton of Godel Theorem: If a R and V(a) =Ω V(a). Then Gans-Rescher B = {V(a):a R} s a base for a topology T on Ω. Then Kleen-Denes open sets of ths topology are V(C A ),A R. Further Goguen ths topology s completely determned by (, q)- fuzzy semprme h-deals of R Defnton of the operator I m (,y) = ma{1,mn{,y}} I a (,y) = mn{1,1 + y} 1 f y I g(,y) = y f y< 1 f y I cg(,y) = 1 f y < 1 f y I gr(,y) = 0 f y< I b(,y) = ma{1,y} 1 f y I gg(,y) = y f y<
5 World Appl. Sc. J., 17 (12): , 2012 A functon I:[0,1] [0,1] [0,1] s called fuzzy mplcaton f ts monotonc wth respect to both varables separately and fulflls the bnary mplcaton truth table By monotoncty of I I(1,0) = 0,I(0,0) = I(0,1) = I(1,1) = 1 I(0,) = I(,1) = 1 for all [0,1]. There have been defned many mplcaton operators n [15]. We consder n the followng some mportant Implcaton Operators: In the followng defnton, we consder the mplcaton operators n Lukasewcz system of contnuous-valued logc. Defnton: A fuzzy set λ of R s called a fuzzfed left (rght) prme h-deal f t satsfes: () For any,y R, [ λ] [y λ] [+ y λ ], () For any,y R, [y λ] [y λ]( resp. [ λ] [y λ ]), () For any,y R, [y λ] [ λ] [y λ ], (v) For any a,b,,z R wth + a+ z = b + z, [a λ ] [b λ ] [ λ ]. A fuzzy set λ of a hemrng R s called a fuzzfed prme h- deal f and only f t s both fuzzfed left and rght prme h-deal of R. Corollary: Let I be an mplcaton operator, t (0,1] s a fed number and λ be a fuzzy set of R. Then λ s a t-mplcaton based fuzzy prme h-deal of R f and only f the followng condtons hold: () For any,y R,I( λ() λ(y), λ (+ y)) t, () For any,y R,I( λ(), λ(y)) t( resp. I( λ(y), λ(y)) t), () For any,y R,I( λ(y), λ() λ(y)) t, (v) For any a,b,,z R wth + a + z = b+ z,i( λ(a) λ(b), λ()) t. Theorem (1) Let I = I gr. Then λ s a 0.5-mplcaton-based fuzzy prme h-deal of R f and only f λ s a fuzzy prme h-deal of R. (2) Let I = I g. Then λ s a 0.5-mplcaton-based fuzzy prme h-deal of R f and only f λ s an (, q)- fuzzy prme h-deal of R. Proof: (1) Let us assume λ be a 0.5-mplcaton-based fuzzy prme h-deal of R. Then () For any,y R,I gr ( λ() λ(y), λ (+ y)) 0.5, () For any,y R,I gr( λ(), λ(y)) 0.5( resp. I( λ(y), λ(y)) 0.5), () For any,y R,I gr( λ(y), λ() λ(y)) 0.5, (v) For any a,b,,z R wth + a + z = b+ z,i ( λ(a) λ(b), λ()) 0.5. gr In [23] the concept of t-tautology s gven, that s tp f and only f [P] t. Now we defne the mplcaton based fuzzy h-deal: Defnton: A fuzzy set λ of R s called a t-mplcatonbased, t (0,1] s a fed number, left (rght) prme h- deal f t satsfes: () For any,y R, t[ λ] [y λ] [+ y λ ], () For any,y R, t[y λ] [y λ ]( resp. [ λ] [y λ ]), () For any,y R, t[y λ] [ λ] [y λ ], (v) For any a,b,,z R wth + a + z = b + z, [a λ] [b λ ] [ λ ]. A fuzzy t set λ of a hemrng R s called a t-mplcatonbased, t (0,1] s a fed number, prme h-deal f and only f t s both t-mplcaton-based left and t-mplcaton-based rght prme h-deal of R Now from () the only case whch s possble s that λ ( + y) λ() λ (y). Smlarly other condtons can be proved for λ to be prme fuzzy h-deal of R. Converse of the Theorem s straghtforward. (2) Let us assume λ be a 0.5-mplcaton-based fuzzy prme h-deal of R. Then () For any,y R,I g( λ() λ(y), λ ( + y)) 0.5, () For any,y R,I g( λ(), λ(y)) 0.5( resp. I( λ(y), λ(y)) 0.5), () For any,y R,I g( λ(y), λ() λ(y)) 0.5, (v) For any a,b,,z R wth + a+ z= b+ z,i ( λ(a) λ(b), λ()) 0.5. g Now () mples λ ( + y) λ() λ (y) or λ() λ (y) >λ ( + y) 0.5. Then ( y) mn{ () (y) 0.5}. λ + λ λ () and (v) can be proved smlarly.
6 World Appl. Sc. J., 17 (12): , 2012 Net () mples λ() λ(y) λ (y) or λ (y) >λ() λ(y) 0.5. Then λ() λ(y) λ(y) 0.5. Hence λ s an (, q)-fuzzy prme h-deal of R. Converse s straghtforward. REFERENCES 1. Vandver, H.S., Note on a smple type of algebra n whch cancellaton law of addton does not hold. Bull. Amer. Math. Soc., 40: Glazek, K., A gude to lterature on semrngs and ther applcatons n mathematcs and nformaton scences: Wth complete bblography, Kluwer Acad. Publ. Nederland. 3. Wechler, W., The concept of fuzzness n automata and language theory. Academc verlog, Berln. 4. Golan, J.S., Semrngs and ther applcatons, Kluwer Acad. Publ. 5. Mordeson, J.N. and D.S. Malk, Fuzzy Automata and Languages. Theory and Applcatons, Computatonal Mathematcs Seres, Chapman and Hall/CRC, Boca Raton. 6. Henrksen, M., Ideals n semrngs wth commutatve addton. Amer. Math. Soc. Notces, Izuka, K., On Jacobson radcal of a semrng. Tohoku Math. J., 11: Zadeh, L.A., Fuzzy Sets. Informaton and Control, 8: Rosenfeld, A., Fuzzy groups. J. Math. Anal. Appl., 35: Ahsan, J., K. Safullah and M.F. Khan, Fuzzy Semrngs. Fuzzy Sets Syst., 60: Mural, V., Fuzzy ponts of equvalent fuzzy subsets. Inform. Sc., 158: Pu, P.M. and Y.M. Lu, Fuzzy topology I, neghborhood structure of a fuzzy pont and Moore-Smth convergence. J. Math. Anal. Appl., 76: Zhan, J. and W.A. Dudek, Fuzzy h-deals of hemrngs. Inform. Sc., 177: Kumbhojkar, H.V., Spectrum of prme L- fuzzy h-deals of a hemrng, Fuzzy Sets and Systems. do: /j.fss Nguyen, H.T. and E.A. Walker, A frst course n fuzzy logc. Chapman and Hall/CRC, Boca Raton. 1820
APPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationSoft Neutrosophic Bi-LA-semigroup and Soft Neutrosophic N-LA-seigroup
Neutrosophc Sets and Systems, Vol. 5, 04 45 Soft Neutrosophc B-LA-semgroup and Soft Mumtaz Al, Florentn Smarandache, Muhammad Shabr 3,3 Department of Mathematcs, Quad--Azam Unversty, Islamabad, 44000,Pakstan.
More informationMAT 578 Functional Analysis
MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these
More informationNeutrosophic Bi-LA-Semigroup and Neutrosophic N-LA- Semigroup
Neutrosophc Sets Systems, Vol. 4, 04 9 Neutrosophc B-LA-Semgroup Neutrosophc N-LA- Semgroup Mumtaz Al *, Florentn Smarache, Muhammad Shabr 3 Munazza Naz 4,3 Department of Mathematcs, Quad--Azam Unversty,
More informationA CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS
Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationCHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION
CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationSUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION
talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationSmarandache-Zero Divisors in Group Rings
Smarandache-Zero Dvsors n Group Rngs W.B. Vasantha and Moon K. Chetry Department of Mathematcs I.I.T Madras, Chenna The study of zero-dvsors n group rngs had become nterestng problem snce 1940 wth the
More informationThe Degrees of Nilpotency of Nilpotent Derivations on the Ring of Matrices
Internatonal Mathematcal Forum, Vol. 6, 2011, no. 15, 713-721 The Degrees of Nlpotency of Nlpotent Dervatons on the Rng of Matrces Homera Pajoohesh Department of of Mathematcs Medgar Evers College of CUNY
More informationSemilattices of Rectangular Bands and Groups of Order Two.
1 Semlattces of Rectangular Bs Groups of Order Two R A R Monzo Abstract We prove that a semgroup S s a semlattce of rectangular bs groups of order two f only f t satsfes the dentty y y, y y, y S 1 Introducton
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationA note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 7, Number 2, December 203 Avalable onlne at http://acutm.math.ut.ee A note on almost sure behavor of randomly weghted sums of φ-mxng
More informationGenericity of Critical Types
Genercty of Crtcal Types Y-Chun Chen Alfredo D Tllo Eduardo Fangold Syang Xong September 2008 Abstract Ely and Pesk 2008 offers an nsghtful characterzaton of crtcal types: a type s crtcal f and only f
More informationSelf-complementing permutations of k-uniform hypergraphs
Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty
More informationSTEINHAUS PROPERTY IN BANACH LATTICES
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT STEINHAUS PROPERTY IN BANACH LATTICES DAMIAN KUBIAK AND DAVID TIDWELL SPRING 2015 No. 2015-1 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookevlle, TN 38505 STEINHAUS
More informationSubset Topological Spaces and Kakutani s Theorem
MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered
More informationCHAPTER 4. Vector Spaces
man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear
More informationOn the smoothness and the totally strong properties for nearness frames
Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo
More informationA CLASS OF RECURSIVE SETS. Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, USA
A CLASS OF RECURSIVE SETS Florentn Smarandache Unversty of New Mexco 200 College Road Gallup, NM 87301, USA E-mal: smarand@unmedu In ths artcle one bulds a class of recursve sets, one establshes propertes
More informationLEVEL SET OF INTUITIONTISTIC FUZZY SUBHEMIRINGS OF A HEMIRING
LEVEL SET OF INTUITIONTISTIC FUZZY SUBHEMIRINGS OF HEMIRING N. NITH ssstant Proessor n Mathematcs, Peryar Unversty PG Extn Centre, Dharmapur 636705. Emal : anthaarenu@gmal.com BSTRCT: In ths paper, we
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationThe Pseudoblocks of Endomorphism Algebras
Internatonal Mathematcal Forum, 4, 009, no. 48, 363-368 The Pseudoblocks of Endomorphsm Algebras Ahmed A. Khammash Department of Mathematcal Scences, Umm Al-Qura Unversty P.O.Box 796, Makkah, Saud Araba
More informationwhere a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets
11. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationAppendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
More informationThe Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices
Internatonal Mathematcal Forum, Vol 11, 2016, no 11, 513-520 HIKARI Ltd, wwwm-hkarcom http://dxdoorg/1012988/mf20166442 The Jacobsthal and Jacobsthal-Lucas Numbers va Square Roots of Matrces Saadet Arslan
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More informationP.P. PROPERTIES OF GROUP RINGS. Libo Zan and Jianlong Chen
Internatonal Electronc Journal of Algebra Volume 3 2008 7-24 P.P. PROPERTIES OF GROUP RINGS Lbo Zan and Janlong Chen Receved: May 2007; Revsed: 24 October 2007 Communcated by John Clark Abstract. A rng
More informationn-strongly Ding Projective, Injective and Flat Modules
Internatonal Mathematcal Forum, Vol. 7, 2012, no. 42, 2093-2098 n-strongly Dng Projectve, Injectve and Flat Modules Janmn Xng College o Mathematc and Physcs Qngdao Unversty o Scence and Technology Qngdao
More informationLecture 17 : Stochastic Processes II
: Stochastc Processes II 1 Contnuous-tme stochastc process So far we have studed dscrete-tme stochastc processes. We studed the concept of Makov chans and martngales, tme seres analyss, and regresson analyss
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationA Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras"
Chn. Ann. Math. 27B(4), 2006, 419{424 DOI: 10.1007/s11401-005-0025-z Chnese Annals of Mathematcs, Seres B c The Edtoral Oce of CAM and Sprnger-Verlag Berln Hedelberg 2006 A Note on \Modules, Comodules,
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationBallot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,
More informationApplication of Fuzzy Algebra in Automata theory
Amercan Journal of Engneerng Research (AJER) e-issn: 2320-0847 p-issn : 2320-0936 Volume-5, Issue-2, pp-21-26 www.ajer.org Research Paper Applcaton of Fuzzy Algebra n Automata theory Kharatt Lal Dept.
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationOn the set of natural numbers
On the set of natural numbers by Jalton C. Ferrera Copyrght 2001 Jalton da Costa Ferrera Introducton The natural numbers have been understood as fnte numbers, ths wor tres to show that the natural numbers
More informationarxiv: v1 [math.co] 12 Sep 2014
arxv:1409.3707v1 [math.co] 12 Sep 2014 On the bnomal sums of Horadam sequence Nazmye Ylmaz and Necat Taskara Department of Mathematcs, Scence Faculty, Selcuk Unversty, 42075, Campus, Konya, Turkey March
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationVolume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].
Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland
More informationResearch Article Relative Smooth Topological Spaces
Advances n Fuzzy Systems Volume 2009, Artcle ID 172917, 5 pages do:10.1155/2009/172917 Research Artcle Relatve Smooth Topologcal Spaces B. Ghazanfar Department of Mathematcs, Faculty of Scence, Lorestan
More informationThe binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence
Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 (ISSN: 2347-2529 Journal homepage: www.jaamm.com IJAAMM Internatonal Journal of Advances n Appled Mathematcs and Mechancs The bnomal transforms of the
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationIdeal Amenability of Second Duals of Banach Algebras
Internatonal Mathematcal Forum, 2, 2007, no. 16, 765-770 Ideal Amenablty of Second Duals of Banach Algebras M. Eshagh Gord (1), F. Habban (2) and B. Hayat (3) (1) Department of Mathematcs, Faculty of Scences,
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationAn application of generalized Tsalli s-havrda-charvat entropy in coding theory through a generalization of Kraft inequality
Internatonal Journal of Statstcs and Aled Mathematcs 206; (4): 0-05 ISS: 2456-452 Maths 206; (4): 0-05 206 Stats & Maths wwwmathsjournalcom Receved: 0-09-206 Acceted: 02-0-206 Maharsh Markendeshwar Unversty,
More informationLecture 7: Gluing prevarieties; products
Lecture 7: Glung prevaretes; products 1 The category of algebrac prevaretes Proposton 1. Let (f,ϕ) : (X,O X ) (Y,O Y ) be a morphsm of algebrac prevaretes. If U X and V Y are affne open subvaretes wth
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationSMARANDACHE-GALOIS FIELDS
SMARANDACHE-GALOIS FIELDS W. B. Vasantha Kandasamy Deartment of Mathematcs Indan Insttute of Technology, Madras Chenna - 600 036, Inda. E-mal: vasantak@md3.vsnl.net.n Abstract: In ths aer we study the
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationALGEBRA HW 7 CLAY SHONKWILER
ALGEBRA HW 7 CLAY SHONKWILER 1 Whch of the followng rngs R are dscrete valuaton rngs? For those that are, fnd the fracton feld K = frac R, the resdue feld k = R/m (where m) s the maxmal deal), and a unformzer
More informationDISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization
DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.
More informationTHE RING AND ALGEBRA OF INTUITIONISTIC SETS
Hacettepe Journal of Mathematcs and Statstcs Volume 401 2011, 21 26 THE RING AND ALGEBRA OF INTUITIONISTIC SETS Alattn Ural Receved 01:08 :2009 : Accepted 19 :03 :2010 Abstract The am of ths study s to
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationOn fuzzification of algebraic and topological structures
On fuzzfcaton of algebrac and topologcal structures Sergejs Solovjovs Department of Mathematcs, Unversty of Latva 19 Rana Blvd., Rga LV-1586, Latva sergejs@lu.lv Abstract Gven a concrete category (A, U)
More informationR n α. . The funny symbol indicates DISJOINT union. Define an equivalence relation on this disjoint union by declaring v α R n α, and v β R n β
Readng. Ch. 3 of Lee. Warner. M s an abstract manfold. We have defned the tangent space to M va curves. We are gong to gve two other defntons. All three are used n the subject and one freely swtches back
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More informationBinomial transforms of the modified k-fibonacci-like sequence
Internatonal Journal of Mathematcs and Computer Scence, 14(2019, no. 1, 47 59 M CS Bnomal transforms of the modfed k-fbonacc-lke sequence Youngwoo Kwon Department of mathematcs Korea Unversty Seoul, Republc
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationOn the Operation A in Analysis Situs. by Kazimierz Kuratowski
v1.3 10/17 On the Operaton A n Analyss Stus by Kazmerz Kuratowsk Author s note. Ths paper s the frst part slghtly modfed of my thess presented May 12, 1920 at the Unversty of Warsaw for the degree of Doctor
More informationInternational Journal of Mathematical Archive-3(3), 2012, Page: Available online through ISSN
Internatonal Journal of Mathematcal Archve-3(3), 2012, Page: 1136-1140 Avalable onlne through www.ma.nfo ISSN 2229 5046 ARITHMETIC OPERATIONS OF FOCAL ELEMENTS AND THEIR CORRESPONDING BASIC PROBABILITY
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationMatrix-Norm Aggregation Operators
IOSR Journal of Mathematcs (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. PP 8-34 www.osrournals.org Matrx-Norm Aggregaton Operators Shna Vad, Sunl Jacob John Department of Mathematcs, Natonal Insttute of
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationRestricted divisor sums
ACTA ARITHMETICA 02 2002) Restrcted dvsor sums by Kevn A Broughan Hamlton) Introducton There s a body of work n the lterature on varous restrcted sums of the number of dvsors of an nteger functon ncludng
More informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationDouble Layered Fuzzy Planar Graph
Global Journal of Pure and Appled Mathematcs. ISSN 0973-768 Volume 3, Number 0 07), pp. 7365-7376 Research Inda Publcatons http://www.rpublcaton.com Double Layered Fuzzy Planar Graph J. Jon Arockaraj Assstant
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationSome Concepts on Constant Interval Valued Intuitionistic Fuzzy Graphs
IOS Journal of Mathematcs (IOS-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 6 Ver. IV (Nov. - Dec. 05), PP 03-07 www.osrournals.org Some Concepts on Constant Interval Valued Intutonstc Fuzzy Graphs
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationQuantum and Classical Information Theory with Disentropy
Quantum and Classcal Informaton Theory wth Dsentropy R V Ramos rubensramos@ufcbr Lab of Quantum Informaton Technology, Department of Telenformatc Engneerng Federal Unversty of Ceara - DETI/UFC, CP 6007
More informationCALCULUS CLASSROOM CAPSULES
CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules
More informationDeterminants Containing Powers of Generalized Fibonacci Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal
More informationPAijpam.eu SOME NEW SUM PERFECT SQUARE GRAPHS S.G. Sonchhatra 1, G.V. Ghodasara 2
Internatonal Journal of Pure and Appled Mathematcs Volume 113 No. 3 2017, 489-499 ISSN: 1311-8080 (prnted verson); ISSN: 1314-3395 (on-lne verson) url: http://www.jpam.eu do: 10.12732/jpam.v1133.11 PAjpam.eu
More informationHomotopy Type Theory Lecture Notes
15-819 Homotopy Type Theory Lecture Notes Evan Cavallo and Stefan Muller November 18 and 20, 2013 1 Reconsder Nat n smple types s a warmup to dscussng nductve types, we frst revew several equvalent presentatons
More informationFUZZY TOPOLOGICAL DIGITAL SPACE OF FLAT ELECTROENCEPHALOGRAPHY DURING EPILEPTIC SEIZURES
Journal of Mathematcs and Statstcs 9 (3): 180-185, 013 ISSN: 1549-3644 013 Scence Publcatons do:10.3844/jmssp.013.180.185 Publshed Onlne 9 (3) 013 (http://www.thescpub.com/jmss.toc) FUY TOPOLOGICAL DIGITAL
More informationThe L(2, 1)-Labeling on -Product of Graphs
Annals of Pure and Appled Mathematcs Vol 0, No, 05, 9-39 ISSN: 79-087X (P, 79-0888(onlne Publshed on 7 Aprl 05 wwwresearchmathscorg Annals of The L(, -Labelng on -Product of Graphs P Pradhan and Kamesh
More information