Conharmonically Flat Vaisman-Gray Manifold
|
|
- Juliana Ray
- 5 years ago
- Views:
Transcription
1 American Journal of Mathematics and Statistics 207, 7(): DOI: /j.ajms Conharmonically Flat Vaisman-Gray Manifold Habeeb M. Abood *, Yasir A. Abdulameer Department of Mathematics, College of Education for Pure Sciences, Basra University, Basra, Iraq Abstract This paper is devoted to study some geometrical properties of conharmonic curvature tensor of Vaisman-Gray manifold. In particular, we have found the necessary and sufficient condition that flat conharmonic Vaisman-Gray manifold is an Einstein manifold. Keywords Almost Hermitian Manifold, Vaisman-Gray manifold, Conharmonic tensor. Introduction One of the representative work of differential geometry is an almost Hermitian structure. Gray and Hervalla [] found that the action of the unitary group UU(nn) on the space of all tensors of type (3,0) decomposed this space into sixteen classes. The conditions that determined each one of these classes belongs to the type of almost Hermitian structure have been identified. These conditions were formulated by using the method of Kozel's operator [2]. The Russian researcher Kirichenko found an interesting method to study the different classes of almost Hermitian manifold. This method depending on the space of the principal fiber bundle of all complex frames of manifold MM with structure group is the unitary group UU(nn). This space is called an adjoined GG-structure space, more details about this space can be found in [3-6]. One of the most important classes of almost Hermitian structures is denoted by WW WW 4, where WW and WW 4 respectively denoted to the nearly Ka hler manifold and local conformal Ka hler manifold. A harmonic function is a function whose Laplacian vanishes. Related to this fact, Y. Ishi [7] has studied conharmonic transformation which is a conformal transformation that preserves the harmonicity of a certain function. Agaoka, et al. [8] studied the twisted product manifold with vanishing conharmonic curvature tensor. Agaoka, et al. [9] studied the fibred Riemannian space with flat conharmonic curvature tensor, in particular, they proved that a conharmonically flat manifold is locally the product manifold of two spaces of constant curvature tensor with constant scalar curvatures. Siddiqui and Ahsan [0] gave an interesting application when they studied the conharmonic curvature tensor on the four dimensional space-time that satisfy the Einstein field equations. Abood and Lafta [] * Corresponding author: iraqsafwan2006@gmail.com (Habeeb M. Abood) Published online at Copyright 207 Scientific & Academic Publishing. All Rights Reserved studied the conharmonic curvature tensor of nearly Ka hler and almost Ka hler manifolds. The present work devoted to study the flatness of conharmonic curvature tensor of Vaisman-Gray manifold by using the methodology of an adjoined GG-structure space. 2. Preliminaries Suppose that MM is 2nn -dimensional smooth manifold, CC (MM) is a set of all smooth functions on MM, XX(MM) is the module of smooth vector fields on MM. An almost Hermitian manifold (AAAA-manifold) is the set {MM, JJ, gg =<.,. >}, where MM is a smooth manifold, and JJ is an almost complex structure, and gg =<.,. > is a Riemannian metric, such that < JJJJ, JJJJ >=< XX, YY > ; XX, YY XX(MM). Suppose that TT cc pp (MM) is the complexification of tangent space TT pp (MM) at the point pp MM and {ee,, ee nn, JJee,, JJJJ nn } is a real adapted basis of AAAA-manifold. Then in the module TT cc pp (MM) there exists a basis given by {εε,, εε nn, εε,, εε nn} which is called adapted basis, where, εε = σσ(ee ) and εε = σσ (ee ) and σσ, σσ are two endomorphisms in the module XX cc (MM) which are defined by σσ = iiii 2 JJcc and σσ = iiii + 2 JJcc, such that, XX cc (MM) and JJ cc are the complexifications of XX(MM) and JJ respectively. The corresponding frame of this basis is {pp; εε,, εε nn, εε,, εε nn}. Suppose that the indexes ii, jj, kk and ll are in the range,2,,2nn and the indexes, bb, cc, dd and ff are in the range,2,, nn. And = + nn. The GG-structure space is the principal fiber bundle of all complex frames of manifold MM with structure group is the unitary group UU(nn). This space is called an adjoined GG-structure space. In the adjoined GG -structure space, the components matrices of complex structure JJ and Riemannian metric gg are given by the following: JJ ii jj = II nn 0, gg iiii = 0 II nn (2.) 0 II nn II nn 0 where II nn is the identity matrix of order nn.
2 American Journal of Mathematics and Statistics 207, 7(): Definition 2. [2] The Riemannian curvature tensor RR of a smooth manifold MM is an 4-covariant tensor RR: TT pp (MM) TT pp (MM) TT pp (MM) TT pp (MM) R which is defined by: RR(XX, YY, ZZ, WW) = gg(rr(zz, WW)YY, XX), where RR (XX, YY)ZZ = [ XX, YY ] [XX,YY] ZZ; XX, YY, ZZ, WW TT pp (MM) and satisfies the following properties: i) RR(XX, YY, ZZ, WW) = RR(YY, XX, ZZ, WW); ii) RR(XX, YY, ZZ, WW) = RR(XX, YY, WW, ZZ); iii) RR(XX, YY, ZZ, WW) + RR(XX, ZZ, WW, YY) + RR(XX, WW, YY, ZZ) = 0; iv) RR(XX, YY, ZZ, WW) = RR(ZZ, WW, XX, YY). Definition 2.2 [3] The Ricci tensor is a tensor of type (2,0) which is defined as follows: kk rr iiii = RR iiiiii = gg kkkk RR kkkkkkkk Definition 2.3 [7] The conharmonic tensor of an AAAA-manifold is a tensor TTof type (4,0) which is defined as the form: TT iiiiiiii = RR iiiiiiii 2(nn ) [rr iiiigg jjjj rr jjjj gg iiii + rr jjjj gg iiii rr iiii gg jjjj ] where rr, RR and gg are respectively Ricci tensor, Riemannian curvature tensor and Riemannian metric. Similar to the properties of Riemannian curvature tensor, the conharmonic tensor has the following properties: TT iiiiiiii = TT jjjjjjjj = TT iiiiiiii = TT kkkkkkkk Definition 2.4. An AAAA -manifold is called a conharmonically flat if the conharmonic tensor vanishes. Definition 2.5 [4] In the adjoined GG-structure space, an AAAA-manifold {MM, JJ, gg =<.,. >} is called a Vaisman-Gray manifold ( VVVV -manifold) if BB = BB bbbbbb, BB cc = αα [ δδ bb] cc ; is called a locally conformal K a hler manifold (LLLLLL-manifold) if BB = 0 and BB cc = αα [ δδ bb] cc ; and is called a nearly Ka hler manifold (NNNN-manifold) if BB = BB bbbbbb and BB cc = 0, where BB =, BB = cc JJ 2 bb,cc 2 JJ [bb,cc ] and αα = δδ FF JJ is a Lie form; FF is a nn Ka hler form which is defined by F(XX, YY) =< JJJJ, YY >, δδ is a coderivative and XX, YY XX(MM) and the bracket [ ] denote to the antisymmetric operation. Theorem 2.6 [5] In the adjoined GG-structure space, the components of Riemannian curvature tensor of VVVV-manifold are given by the following forms: i) RR = 2(ΒΒ [cccc ] + αα [ ΒΒ bb]cccc ); ii) RR bbbbbb = 2ΑΑ bbbbbb ; iii) RR bb cccc = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ); iv) RR bbbb dd = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb, where, {ΑΑ bbbbbb } are some functions on adjoined GG- structure space and AA bbbb are system of fuctions in the adjoined GG -structure space which are symmetric by the lower and upper indices which are called components of holomorphic sectional curvature tensor. and {αα bb bb, αα } are the components of the covariant differential structure tensor of first and second type and {αα, αα } are the components of the Lee form on adjoint GG-structure space such that: dddd a + αα bb ωω bb = αα bb ωω bb + αα ωω bb, and dddd αα bb ωω bb = αα bb ωω bb + αα ωω bb, where, {ωω, ωω } are the components of mixture form, {ωω bb } are the components of Riema-nnian connection of metric gg. The other components of Riemannian curvature tensor RR can be obtained by the property of symmetry for RR. There are three special classes of almost Hermitian manifold depending on the components of the Riemannian curvature tensor. Their conditions are embodied in the following definition: Definition 2.7. [6] In the adjoined GG-structure space, an AAAA-manifold is a manifold of class: ) RR if and only if, RR = RR bbbbbb = RR bb cccc = 0; 2) RR 2 if and only if, RR = RR bbbbbb = 0; 3) RR 3 (RRRR-manifold) if and only if, RR bbbbbb = 0. It easy to see that RR RR 2 RR 3. Theorem 2.8 [5] In the adjoined GG-structure space, the components of Ricci tensor of VVVV-manifold are given by the following forms: ) rr = nn (αα 2 + αα bbbb + αα αα bb ); 2) rr bb = 3ΒΒ cccch ΒΒ cccch ΑΑ cccc bbbb + nn 2 (αα αα bb αα h αα h ) αα h 2 hδδ bb + (nn 2)αα bb, and the others are conjugate to the above components. Definition 2.9 [7] A Riemannian manifold is called an Einstein manifold, if the Ricci tensor satisfies the equation rr iiii = CCgg iiii, where, CC is an Einstein constant. Definition 2.0 [8] An AAAA -manifold has JJ-invariant Ricci tensor, if JJ rr = rr JJ. The following Lemma gives a fact about Ricci tensor in the adjoined GG-structure space. Lemma 2. [6] An AAAA-manifold has JJ-invariant Ricci tensor if and only if, we have rr bb = rr = 0. Definition 2.2 [3] Define two endomorphisms on ττ 0 rr (VV) as follows: i) Symmetric mapping Sym: ττ 0 rr (VV) ττ 0 rr (VV) by: ssssss(tt)(vv,, vv rr ) = tt vv rr! σσ(),, vv σσ(rr) σσ SS rr. ii) Antisymmetric mapping Alt: ττ 0 rr (VV) ττ 0 rr (VV) by: AAAAAA(tt)(vv,, vv rr ) = εε(σσ)tt vv rr! σσ(),, vv σσ(rr) σσ SS rr. The symbols ( ) and [ ] are usually used to denote the symmetric and antisymmetric respectively. 3. Main Results Theorem 3.. In the adjoined GG -structure space, the components of conharmonic tensor of VVVV -manifold are given by the following forms:
3 40 Habeeb M. Abood et al.: Conharmonically Flat Vaisman-Gray Manifold ) TT = 2 ΒΒ [cccc ] + αα [ ΒΒ bb]cccc ; 2) TT bbbbbb = 2ΑΑ bbbbbb + (rr 2(nn ) bbbb δδ cc rr bbbb δδ dd ); 3) TT bb cccc = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ) 4) TT bbbb dd = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb + [ δδ cc bb] + rr cc [bb δδ dd ] ); (rr (nn ) dd (rr ( (nn ) (cc δδ dd) bb) ), and the others are conjugate to the above components. ) Put ii =, jj = bb, kk = cc and ll = dd, we have TT = RR 2(nn ) (rr gg bbbb rr bbbb gg + rr bbbb gg rr gg bbbb ) According to the equation (2.) we deduce that TT = RR = 2 ΒΒ [cccc ] + αα [ ΒΒ bb]cccc. 2) Put ii =, jj = bb, kk = cc and ll = dd, we get TT bbbbbb = RR bbbbbb 2(nn ) (rr ddgg bbbb rr bbbb gg cc + rr bbbb gg dd rr cc gg bbbb ) = RR bbbbbb + 2(nn ) (rr bbbb gg cc rr bbbb gg dd ) = 2ΑΑ bbbbbb + (rr 2(nn ) bbbb δδ cc rr bbbb δδ dd ). 3) Put ii =, jj = bb, kk = cc and ll = dd, it follows that TT bb cccc = RR bb cccc 2(nn ) (rr ddgg bb cc rr bb ddgg cc + rr bb ccgg dd rr cc gg bb dd) = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ) (rr 2(nn ) ddδδ bb cc rr bb ddδδ cc + rr bb ccδδ dd rr cc δδ bb dd ) = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ) (rr 2(nn ) dd δδ bb cc rr bb dd δδ cc + rr bb cc δδ dd rr cc δδ bb dd ) = 2( ΒΒ h ΒΒ hcccc + αα [ [cc ) (rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd ). 4) Put ii =, jj = bb, kk = cc and ll = dd, we obtain TT bbbb dd = RR bbbb dd 2(nn ) (rr dd gg bbbb rr bbdd gg cc + rr bbbb gg dd rr cc gg bbdd ) = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb + (rr 2(nn ) bbdd δδ cc + rr cc δδ dd bb ) = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb + (rr 2(nn ) bb dd δδ cc + rr cc δδ dd bb ) = ΑΑ bbbb + ΒΒ h ΒΒ hbbbb ΒΒ h ccββ dd hbb + rr ( (nn ) (cc δδ dd) bb). Definition 3.2. In the adjoined GG-structure space, an almost Hermitian manifold is a manifold of class: TTRR if and only if, TT = TT bbbbbb = TT bb cccc = 0; TTRR 2 if and only if, TT = TT bbbbbb = 0; TTRR 3 (TTTTTT-manifold) if and only if, TT bbbbbb = 0. We call TTRR a conharmonic paraka hler manifold. Theorem 3.3. Let MM be a VVVV-manifold of class TTRR with JJ-invariant Ricci tensor, then MM is a manifold of class RR if and only if, MM is a manifold of flat Ricci tensor. To prove MM is a manifold of class RR, we must prove that RR = RR bbbbbb = RR bb cccc = 0. Let MM be a manifold of class TTRR, according to definition 3.2, we have According to theorem 3., we have TT = TT bbbbbb = TT bb cccc = 0 TT = 0 (3.)
4 American Journal of Mathematics and Statistics 207, 7(): By using the equation (3.), we get 2 ΒΒ [cccc ] + αα [ ΒΒ bb]cccc = 0 RR = 0. (3.2) TT bbbbbb = 0 According to theorems 2.6 and 3., we deduce RR bbbbbb + 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) = 0 Since, MM has JJ-invariant Ricci tensor, then Also, by the equation (3.), we deduce TT bb cccc = 0 By using theorems 2.6 and 3., we have RR bb cccc (nn ) rr [ dd δδ bb] cc + rr [bb cc δδ ] dd = 0 Suppose that MM is a manifold of flat Ricci tensor, then Hence, according to the equations, (3.2), (3.3) and (3.4), we get Conversely, by using the equation (3.), we have By using theorem 3., it follows that RR bbbbbb = 0. (3.3) RR bb cccc = 0. (3.4) RR = RR bbbbbb = RR bb cccc = 0. TT = TT bbbbbb = TT bb cccc = 0 2 ΒΒ [cccc ] + αα [ ΒΒ bb]cccc = 2ΑΑ bbbbbb + (rr 2(nn ) bbbb δδ cc rr bbbb δδ dd ) = 2 ΒΒ h ΒΒ hcccc + αα [ [cc rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd Let MM be a manifold of class RR, then, according to theorem 2.6 and definition 2.7, we obtain 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) + (nn ) rr [ dd δδ bb] cc + rr [bb cc δδ ] dd = 0 Symmetrizing by the indexes (bb, cc), we get rr 2(nn ) bbbb δδ cc (rr 2 bbbb δδ dd + rr cccc δδ dd ) + rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd Antisymmetrizing by the indexes (bb, cc), we have rr 2(nn ) bbbb δδ cc 2 (rr 2 bbbb δδ dd rr cccc δδ dd ) + (rr 2 cccc δδ dd rr bbbb δδ dd ) + rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd rr 2(nn ) bbbb δδ cc + rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd rr 2(nn ) bbbb δδ cc + rr 2(nn ) dd δδ bb cc rr bb dd δδ cc + rr bb cc δδ dd rr cc δδ bb dd Symmetrizing by the indexes (, bb), we deduce rr 2(nn ) bbbb δδ cc + 2(nn ) rr 2 dd δδ bb cc + rr bb dd δδ cc rr 2 dd bb δδ cc + rr dd δδ bb cc + rr 2 cc bb δδ dd + rr cc δδ bb dd (rr 2 cc δδ bb dd + rr bb cc δδ dd ) 2(nn ) rr bbbb δδ cc = 0 rr bbbb δδ cc = 0 Contracting by the indexes (, dd), we have rr bbbb δδ cc = 0 rr bbbb = 0 Since MM has JJ-invariant Ricci tensor, then rr iiii = 0. Theorem 3.4. Suppose that MM is VVVV-manifold of class TTRR 3 with JJ-invariant Ricci tensor, then MM is a manifold of class RR 3 if and only if, MM is a manifold of flat Ricci tensor. Suppose that MM is a manifold of class TTRR 3. According to definition 3.2., we have TT bbbbbb = 0. By using the Theorem 3., we deduce
5 42 Habeeb M. Abood et al.: Conharmonically Flat Vaisman-Gray Manifold RR bbbbbb + 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) = 0 (3.5) Let MM be a manifold of class RR 3, then 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) = 0. Symmetrizing by the indexes (bb, cc), we get (rr 2(nn ) bbbb δδ cc (rr 2 bbbb δδ dd +rr cccc δδ dd )) = 0. Antisymmetrizing by the indexes (bb, cc), we have (rr 2(nn ) bbbb δδ cc ((rr 2 bbbb δδ dd rr cccc δδ dd ) + (rr cccc δδ dd rr bbbb δδ dd ))) = 0. rr 2(nn ) bbbb δδ cc = 0. rr bbbb δδ cc = 0. Contracting by the indexes (, dd), it follows that rr bbbb δδ cc = 0. rr bbbb = 0. Since MM has JJ-invariant Ricci tensor, then rr iiii = 0. Conversely, by using the equation (3.5), we have RR bbbbbb + 2(nn ) (rr bbbb δδ cc rr bbbb δδ dd ) = 0 Suppose that MM is a manifold of flat Ricci tensor, then RR bbbbbb = 0. Therefore, MM is a manifold of class RR 3. The following theorem gives the necessary and sufficient condition in which an VVVV-manifold is an Einstein manifold. Theorem 3.5. Suppose that MM is conharmonically flat VVVV -manifold with JJ -invariant Ricci tenor. Then the necessary and sufficient condition that MM an Einstein manifold, is αα cc = kkδδ cc, where kk is a constant. Let MM be a conharmonically flat VVVV -manifold. According to the definition 2.4 and theorem 3., we have 2 ΒΒ h ΒΒ hcccc + αα [ [cc rr [ (nn ) dd δδ bb] cc + rr [bb cc δδ ] dd Contracting by the indexes (bb, dd), it follows that 2 ΒΒ h ΒΒ hcccc + αα [ [cc δδ bb] bb] rr [ (nn ) bb δδ bb] cc + rr [bb cc δδ ] bb 2ΒΒ h ΒΒ hcccc + 2nnαα cc 2 rr (nn ) cc = 0. ΒΒ h ΒΒ hcccc + nnαα cc rr (nn ) cc = 0. Symmetrizing by the indexes (, bb), we deduce 2 (ΒΒ h ΒΒ hcccc + ΒΒ bbbb h ΒΒ hcccc ) + nnαα cc rr (nn ) cc = 0. Antisymmetrizing by the indexes (, bb), it follows that 2 ( 2 (ΒΒ h ΒΒ hcccc ΒΒ bbbb h ΒΒ hcccc ) + 2 (ΒΒbbbb h ΒΒ hcccc ΒΒ h ΒΒ hcccc )) + nnαα cc rr (nn ) cc = 0. nnαα cc rr (nn ) cc = 0. (3.6) nnαα cc = rr (nn ) cc. Let MM be an Einstein manifold, then nnαα cc = CC δδ (nn ) cc. αα cc = kkδδ cc. Conversely, by using the equation (3.6), we have nnαα cc rr (nn ) cc = 0. Since, αα cc = kkδδ cc, we deduce nnkkδδ cc = rr (nn ) cc. rr cc = kkkk(nn )δδ cc. rr cc = CCδδ cc, where, CC represent an Einstein constant. Since MM has JJ-invariant Ricci tensor. Therefore, MM is an Einstein manifold. 4. Conclusions The present work is devoted to study the flatness of conharmonic curvature tensor of Vaisman-Gray manifold. We found out an interesting application in theoretical physics. In particular, we found the necessary and sufficient condition that a conharmonically flat Vaisman-Gray manifold is an Einstein manifold. REFERENCES [] Gray A., Hervella L. M., Sixteen classes of almost Hermitian manifold and their linear invariants, Ann. Math. Pure and Appl., Vol. 23, No. 3, 35-58, 980. [2] Kozal J. L., Varicies Kahlerian-Notes, Sao Paolo, 957. [3] Kirichenko V.F., Some properties of tensors on K-spaces, Journal of Moscow state University, Math-Mach. department, Vol. 6, 78-85, 975. [4] Kirichenko V. F., K spaces of constant type, Seper. Math. J., Vol.7, No. 2, , 975. [5] Kirichenko V. F., K-spaces of constant holomorphic sectional curvature, Mathematical Notes, V.9, No.5, , 976. [6] Kirichenko V. F. Differential geometry of K spaces, problems f Geometry, V.8, 39-60, 977. [7] Ishi Y. On conharmonic transformation Tensor, N. S. 7, 73-80, 957. [8] Agaoka Y., Kim B. H., Lee H. J., Conharmonic Transformation of Twisted Produced Manifolds. Mem. Fac. Integrated Arts and Sci., Hiroshima Univ., ser. IV, Vol. 25, -20, Dec.999. [9] Agaoka Y., Kim B. H., Lee S. D., Conharmonic flat fibred Riemannian space, Mem. Fac. Integrated Arts and Sci., Hiroshima Univ., ser. IV, Vol. 26, 09-5, Dec [0] Siddiqui S. A., Ahsan Z., Conharmonic curvature tensor and the Space-time of General Relativity, Differential
6 American Journal of Mathematics and Statistics 207, 7(): Geometry-Dynamical Systems, Vol.2, p , 200. [] Abood H. M., Lafta G. R., Conharmonic Tensor of Certain Classes of Almost Hermitian manifolds, American Institute of Physics, CP 309, International Conference of Mathe-matical Sciences, 200. [2] Kobayashi S., Nomizu K., Foundations of differential geometry, V., John Wily and Sons, 963. [3] Raševskiĭ PK., Riemannian geometry and tensor analysis, M. Nauka, 964. [4] Banaru M., A new characterization of the Gray-Hervella classes of almost Hermi-tian manifolds, 8th International Conference on Differential Geometry and its Applications, Opava- Czech Republic, 27-3 August; 200. [5] Lia A. Ignatochkina, Vaisman Gray manifolds with J invariant conformal curvature tensor, Sb. Math., 94:2, , [6] Tretiakova E. V., Curvature identities for almost K hler manifold, VINITE, Moscow, No. 208-B99, 999. [7] Petrov A. Z., Einstein space, Phys-Math. Letr. Moscow, p. 463, 96. [8] Kirichenko V. F., Arseneva O. E., Self-dual geometry of generelized Hermitian surfaces Mat. Sbornik, 89, No., 2-44, 998.
Worksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 1-9. Algebra
Worksheets for GCSE Mathematics Algebraic Expressions Mr Black 's Maths Resources for Teachers GCSE 1-9 Algebra Algebraic Expressions Worksheets Contents Differentiated Independent Learning Worksheets
More informationHolomorphic Geodesic Transformations. of Almost Hermitian Manifold
International Mathematical Forum, 4, 2009, no. 46, 2293-2299 Holomorphic Geodesic Transformations of Almost Hermitian Manifold Habeeb M. Abood University of Basrha, Department of mathematics, Basrah-Iraq
More informationA linear operator of a new class of multivalent harmonic functions
International Journal of Scientific Research Publications, Volume 7, Issue 8, August 2017 278 A linear operator of a new class of multivalent harmonic functions * WaggasGalibAtshan, ** Ali Hussein Battor,
More informationWorksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra
Worksheets for GCSE Mathematics Quadratics mr-mathematics.com Maths Resources for Teachers Algebra Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation
More information(1) Correspondence of the density matrix to traditional method
(1) Correspondence of the density matrix to traditional method New method (with the density matrix) Traditional method (from thermal physics courses) ZZ = TTTT ρρ = EE ρρ EE = dddd xx ρρ xx ii FF = UU
More informationIntegrating Rational functions by the Method of Partial fraction Decomposition. Antony L. Foster
Integrating Rational functions by the Method of Partial fraction Decomposition By Antony L. Foster At times, especially in calculus, it is necessary, it is necessary to express a fraction as the sum of
More informationVariations. ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra
Variations ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra Last Time Probability Density Functions Normal Distribution Expectation / Expectation of a function Independence Uncorrelated
More information10.4 The Cross Product
Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb
More informationSECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems
SECTION 7: FAULT ANALYSIS ESE 470 Energy Distribution Systems 2 Introduction Power System Faults 3 Faults in three-phase power systems are short circuits Line-to-ground Line-to-line Result in the flow
More information7.3 The Jacobi and Gauss-Seidel Iterative Methods
7.3 The Jacobi and Gauss-Seidel Iterative Methods 1 The Jacobi Method Two assumptions made on Jacobi Method: 1.The system given by aa 11 xx 1 + aa 12 xx 2 + aa 1nn xx nn = bb 1 aa 21 xx 1 + aa 22 xx 2
More informationInternational Journal of Mathematical Archive-5(3), 2014, Available online through ISSN
International Journal of Mathematical Archive-5(3), 214, 189-195 Available online through www.ijma.info ISSN 2229 546 COMMON FIXED POINT THEOREMS FOR OCCASIONALLY WEAKLY COMPATIBLE MAPPINGS IN INTUITIONISTIC
More information(1) Introduction: a new basis set
() Introduction: a new basis set In scattering, we are solving the S eq. for arbitrary VV in integral form We look for solutions to unbound states: certain boundary conditions (EE > 0, plane and spherical
More informationReview for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa
Review for Exam3 12. 9. 2015 Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Assistant Research Scientist IIHR-Hydroscience & Engineering, University
More informationNon-Parametric Weighted Tests for Change in Distribution Function
American Journal of Mathematics Statistics 03, 3(3: 57-65 DOI: 0.593/j.ajms.030303.09 Non-Parametric Weighted Tests for Change in Distribution Function Abd-Elnaser S. Abd-Rabou *, Ahmed M. Gad Statistics
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 1 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles 5.6 Uncertainty Principle Topics 5.7
More informationReview for Exam Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa
57:020 Fluids Mechanics Fall2013 1 Review for Exam3 12. 11. 2013 Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa 57:020 Fluids Mechanics Fall2013 2 Chapter
More informationSecondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet
Secondary H Unit Lesson Worksheet Simplify: mm + 2 mm 2 4 mm+6 mm + 2 mm 2 mm 20 mm+4 5 2 9+20 2 0+25 4 +2 2 + 2 8 2 6 5. 2 yy 2 + yy 6. +2 + 5 2 2 2 0 Lesson 6 Worksheet List all asymptotes, holes and
More informationAs always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).
An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted
More informationWork, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition
Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.
More information(2) Orbital angular momentum
(2) Orbital angular momentum Consider SS = 0 and LL = rr pp, where pp is the canonical momentum Note: SS and LL are generators for different parts of the wave function. Note: from AA BB ii = εε iiiiii
More informationMathematics Ext 2. HSC 2014 Solutions. Suite 403, 410 Elizabeth St, Surry Hills NSW 2010 keystoneeducation.com.
Mathematics Ext HSC 4 Solutions Suite 43, 4 Elizabeth St, Surry Hills NSW info@keystoneeducation.com.au keystoneeducation.com.au Mathematics Extension : HSC 4 Solutions Contents Multiple Choice... 3 Question...
More informationEstimate by the L 2 Norm of a Parameter Poisson Intensity Discontinuous
Research Journal of Mathematics and Statistics 6: -5, 24 ISSN: 242-224, e-issn: 24-755 Maxwell Scientific Organization, 24 Submied: September 8, 23 Accepted: November 23, 23 Published: February 25, 24
More information10.1 Three Dimensional Space
Math 172 Chapter 10A notes Page 1 of 12 10.1 Three Dimensional Space 2D space 0 xx.. xx-, 0 yy yy-, PP(xx, yy) [Fig. 1] Point PP represented by (xx, yy), an ordered pair of real nos. Set of all ordered
More informationRotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition
Rotational Motion Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 We ll look for a way to describe the combined (rotational) motion 2 Angle Measurements θθ ss rr rrrrrrrrrrrrrr
More informationDressing up for length gauge: Aspects of a debate in quantum optics
Dressing up for length gauge: Aspects of a debate in quantum optics Rainer Dick Department of Physics & Engineering Physics University of Saskatchewan rainer.dick@usask.ca 1 Agenda: Attosecond spectroscopy
More informationManipulator Dynamics (1) Read Chapter 6
Manipulator Dynamics (1) Read Capter 6 Wat is dynamics? Study te force (torque) required to cause te motion of robots just like engine power required to drive a automobile Most familiar formula: f = ma
More informationP.3 Division of Polynomials
00 section P3 P.3 Division of Polynomials In this section we will discuss dividing polynomials. The result of division of polynomials is not always a polynomial. For example, xx + 1 divided by xx becomes
More informationCharge carrier density in metals and semiconductors
Charge carrier density in metals and semiconductors 1. Introduction The Hall Effect Particles must overlap for the permutation symmetry to be relevant. We saw examples of this in the exchange energy in
More informationOn the 5-dimensional Sasaki-Einstein manifold
Proceedings of The Fourteenth International Workshop on Diff. Geom. 14(2010) 171-175 On the 5-dimensional Sasaki-Einstein manifold Byung Hak Kim Department of Applied Mathematics, Kyung Hee University,
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationM.5 Modeling the Effect of Functional Responses
M.5 Modeling the Effect of Functional Responses The functional response is referred to the predation rate as a function of the number of prey per predator. It is recognized that as the number of prey increases,
More informationClassical RSA algorithm
Classical RSA algorithm We need to discuss some mathematics (number theory) first Modulo-NN arithmetic (modular arithmetic, clock arithmetic) 9 (mod 7) 4 3 5 (mod 7) congruent (I will also use = instead
More informationProperty Testing and Affine Invariance Part I Madhu Sudan Harvard University
Property Testing and Affine Invariance Part I Madhu Sudan Harvard University December 29-30, 2015 IITB: Property Testing & Affine Invariance 1 of 31 Goals of these talks Part I Introduce Property Testing
More informationRevision : Thermodynamics
Revision : Thermodynamics Formula sheet Formula sheet Formula sheet Thermodynamics key facts (1/9) Heat is an energy [measured in JJ] which flows from high to low temperature When two bodies are in thermal
More informationDiffusive DE & DM Diffusve DE and DM. Eduardo Guendelman With my student David Benisty, Joseph Katz Memorial Conference, May 23, 2017
Diffusive DE & DM Diffusve DE and DM Eduardo Guendelman With my student David Benisty, Joseph Katz Memorial Conference, May 23, 2017 Main problems in cosmology The vacuum energy behaves as the Λ term in
More informationElastic light scattering
Elastic light scattering 1. Introduction Elastic light scattering in quantum mechanics Elastic scattering is described in quantum mechanics by the Kramers Heisenberg formula for the differential cross
More informationModule 7 (Lecture 25) RETAINING WALLS
Module 7 (Lecture 25) RETAINING WALLS Topics Check for Bearing Capacity Failure Example Factor of Safety Against Overturning Factor of Safety Against Sliding Factor of Safety Against Bearing Capacity Failure
More informationLesson 1: Successive Differences in Polynomials
Lesson 1 Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below. 2.4 3.4 4.4 5.4 6.4 5.76 11.56 19.36 29.16 40.96
More informationThe Elasticity of Quantum Spacetime Fabric. Viorel Laurentiu Cartas
The Elasticity of Quantum Spacetime Fabric Viorel Laurentiu Cartas The notion of gravitational force is replaced by the curvature of the spacetime fabric. The mass tells to spacetime how to curve and the
More informationSolving Fuzzy Nonlinear Equations by a General Iterative Method
2062062062062060 Journal of Uncertain Systems Vol.4, No.3, pp.206-25, 200 Online at: www.jus.org.uk Solving Fuzzy Nonlinear Equations by a General Iterative Method Anjeli Garg, S.R. Singh * Department
More informationOn The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays
Journal of Mathematics and System Science 6 (216) 194-199 doi: 1.17265/2159-5291/216.5.3 D DAVID PUBLISHING On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time
More informationTerms of Use. Copyright Embark on the Journey
Terms of Use All rights reserved. No part of this packet may be reproduced, stored in a retrieval system, or transmitted in any form by any means - electronic, mechanical, photo-copies, recording, or otherwise
More informationWave Motion. Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition
Wave Motion Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 Waves: propagation of energy, not particles 2 Longitudinal Waves: disturbance is along the direction of wave propagation
More informationChapter 5: Spectral Domain From: The Handbook of Spatial Statistics. Dr. Montserrat Fuentes and Dr. Brian Reich Prepared by: Amanda Bell
Chapter 5: Spectral Domain From: The Handbook of Spatial Statistics Dr. Montserrat Fuentes and Dr. Brian Reich Prepared by: Amanda Bell Background Benefits of Spectral Analysis Type of data Basic Idea
More informationAngular Momentum, Electromagnetic Waves
Angular Momentum, Electromagnetic Waves Lecture33: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay As before, we keep in view the four Maxwell s equations for all our discussions.
More informationF.4 Solving Polynomial Equations and Applications of Factoring
section F4 243 F.4 Zero-Product Property Many application problems involve solving polynomial equations. In Chapter L, we studied methods for solving linear, or first-degree, equations. Solving higher
More informationGradient expansion formalism for generic spin torques
Gradient expansion formalism for generic spin torques Atsuo Shitade RIKEN Center for Emergent Matter Science Atsuo Shitade, arxiv:1708.03424. Outline 1. Spintronics a. Magnetoresistance and spin torques
More informationReview for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa
Review for Exam2 11. 13. 2015 Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Assistant Research Scientist IIHR-Hydroscience & Engineering, University
More informationTECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES
COMPUTERS AND STRUCTURES, INC., FEBRUARY 2016 TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES Introduction This technical note
More informationQuantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc.
Quantum Mechanics An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Fall 2018 Prof. Sergio B. Mendes 1 CHAPTER 3 Experimental Basis of
More informationECE 6540, Lecture 06 Sufficient Statistics & Complete Statistics Variations
ECE 6540, Lecture 06 Sufficient Statistics & Complete Statistics Variations Last Time Minimum Variance Unbiased Estimators Sufficient Statistics Proving t = T(x) is sufficient Neyman-Fischer Factorization
More informationThe Bose Einstein quantum statistics
Page 1 The Bose Einstein quantum statistics 1. Introduction Quantized lattice vibrations Thermal lattice vibrations in a solid are sorted in classical mechanics in normal modes, special oscillation patterns
More informationInternational Journal of Mathematical Archive-5(12), 2014, Available online through ISSN
International Journal of Mathematical Archive-5(12), 2014, 75-79 Available online through www.ijma.info ISSN 2229 5046 ABOUT WEAK ALMOST LIMITED OPERATORS A. Retbi* and B. El Wahbi Université Ibn Tofail,
More informationJasmin Smajic1, Christian Hafner2, Jürg Leuthold2, March 23, 2015
Jasmin Smajic, Christian Hafner 2, Jürg Leuthold 2, March 23, 205 Time Domain Finite Element Method (TD FEM): Continuous and Discontinuous Galerkin (DG-FEM) HSR - University of Applied Sciences of Eastern
More informationCHAPTER 2 Special Theory of Relativity
CHAPTER 2 Special Theory of Relativity Fall 2018 Prof. Sergio B. Mendes 1 Topics 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 Inertial Frames of Reference Conceptual and Experimental
More informationLast Name _Piatoles_ Given Name Americo ID Number
Last Name _Piatoles_ Given Name Americo ID Number 20170908 Question n. 1 The "C-V curve" method can be used to test a MEMS in the electromechanical characterization phase. Describe how this procedure is
More informationSECTION 7: STEADY-STATE ERROR. ESE 499 Feedback Control Systems
SECTION 7: STEADY-STATE ERROR ESE 499 Feedback Control Systems 2 Introduction Steady-State Error Introduction 3 Consider a simple unity-feedback system The error is the difference between the reference
More informationPhysics 371 Spring 2017 Prof. Anlage Review
Physics 71 Spring 2017 Prof. Anlage Review Special Relativity Inertial vs. non-inertial reference frames Galilean relativity: Galilean transformation for relative motion along the xx xx direction: xx =
More informationSECTION 5: POWER FLOW. ESE 470 Energy Distribution Systems
SECTION 5: POWER FLOW ESE 470 Energy Distribution Systems 2 Introduction Nodal Analysis 3 Consider the following circuit Three voltage sources VV sss, VV sss, VV sss Generic branch impedances Could be
More informationF.1 Greatest Common Factor and Factoring by Grouping
1 Factoring Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers.
More informationME5286 Robotics Spring 2017 Quiz 2
Page 1 of 5 ME5286 Robotics Spring 2017 Quiz 2 Total Points: 30 You are responsible for following these instructions. Please take a minute and read them completely. 1. Put your name on this page, any other
More informationGeneral Strong Polarization
General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) May 1, 018 G.Tech:
More informationInformation Booklet of Formulae and Constants
Pearson BTEC Level 3 Nationals Engineering Information Booklet of Formulae and Constants Unit 1: Engineering Principles Extended Certificate, Foundation Diploma, Diploma, Extended Diploma in Engineering
More informationA Numerical Integration for Solving First Order Differential Equations Using Gompertz Function Approach
American Journal of Computational and Applied Mathematics 2017, 7(6): 143-148 DOI: 10.5923/j.ajcam.20170706.01 A Numerical Integration for Solving First Order Differential Equations Using Gompertz Function
More informationIntroduction to Density Estimation and Anomaly Detection. Tom Dietterich
Introduction to Density Estimation and Anomaly Detection Tom Dietterich Outline Definition and Motivations Density Estimation Parametric Density Estimation Mixture Models Kernel Density Estimation Neural
More informationNational 5 Mathematics. Practice Paper E. Worked Solutions
National 5 Mathematics Practice Paper E Worked Solutions Paper One: Non-Calculator Copyright www.national5maths.co.uk 2015. All rights reserved. SQA Past Papers & Specimen Papers Working through SQA Past
More informationAtomic fluorescence. The intensity of a transition line can be described with a transition probability inversely
Atomic fluorescence 1. Introduction Transitions in multi-electron atoms Energy levels of the single-electron hydrogen atom are well-described by EE nn = RR nn2, where RR = 13.6 eeee is the Rydberg constant.
More informationA THEOREM ON COMPACT LOCALLY CONFORMAL KAHLER MANIFOLDS
proceedings of the american mathematical society Volume 75, Number 2, July 1979 A THEOREM ON COMPACT LOCALLY CONFORMAL KAHLER MANIFOLDS IZU VAISMAN Abstract. We prove that a compact locally conformai Kahler
More informationModule 7 (Lecture 27) RETAINING WALLS
Module 7 (Lecture 27) RETAINING WALLS Topics 1.1 RETAINING WALLS WITH METALLIC STRIP REINFORCEMENT Calculation of Active Horizontal and vertical Pressure Tie Force Factor of Safety Against Tie Failure
More informationLesson 9: Law of Cosines
Student Outcomes Students prove the law of cosines and use it to solve problems (G-SRT.D.10). Lesson Notes In this lesson, students continue the study of oblique triangles. In the previous lesson, students
More informationA Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions
Lin Lin A Posteriori DG using Non-Polynomial Basis 1 A Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions Lin Lin Department of Mathematics, UC Berkeley;
More information9. Switched Capacitor Filters. Electronic Circuits. Prof. Dr. Qiuting Huang Integrated Systems Laboratory
9. Switched Capacitor Filters Electronic Circuits Prof. Dr. Qiuting Huang Integrated Systems Laboratory Motivation Transmission of voice signals requires an active RC low-pass filter with very low ff cutoff
More informationGeneral Strong Polarization
General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) December 4, 2017 IAS:
More informationLocality in Coding Theory
Locality in Coding Theory Madhu Sudan Harvard April 9, 2016 Skoltech: Locality in Coding Theory 1 Error-Correcting Codes (Linear) Code CC FF qq nn. FF qq : Finite field with qq elements. nn block length
More informationOn the Twisted KK-Theory and Positive Scalar Curvature Problem
International Journal of Advances in Mathematics Volume 2017, Number 3, Pages 9-15, 2017 eissn 2456-6098 c adv-math.com On the Twisted KK-Theory and Positive Scalar Curvature Problem Do Ngoc Diep Institute
More informationMath 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
More informationProf. Dr.-Ing. Armin Dekorsy Department of Communications Engineering. Stochastic Processes and Linear Algebra Recap Slides
Prof. Dr.-Ing. Armin Dekorsy Department of Communications Engineering Stochastic Processes and Linear Algebra Recap Slides Stochastic processes and variables XX tt 0 = XX xx nn (tt) xx 2 (tt) XX tt XX
More informationConstitutive Relations of Stress and Strain in Stochastic Finite Element Method
American Journal of Computational and Applied Mathematics 15, 5(6): 164-173 DOI: 1.593/j.ajcam.1556. Constitutive Relations of Stress and Strain in Stochastic Finite Element Method Drakos Stefanos International
More informationOn Einstein Nearly Kenmotsu Manifolds
International Journal of Mathematics Research. ISSN 0976-5840 Volume 8, Number 1 (2016), pp. 19-24 International Research Publication House http://www.irphouse.com On Einstein Nearly Kenmotsu Manifolds
More informationModeling of a non-physical fish barrier
University of Massachusetts - Amherst ScholarWorks@UMass Amherst International Conference on Engineering and Ecohydrology for Fish Passage International Conference on Engineering and Ecohydrology for Fish
More informationMath 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
More informationMath, Stats, and Mathstats Review ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD
Math, Stats, and Mathstats Review ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD Outline These preliminaries serve to signal to students what tools they need to know to succeed in ECON 360 and refresh their
More informationInteraction with matter
Interaction with matter accelerated motion: ss = bb 2 tt2 tt = 2 ss bb vv = vv 0 bb tt = vv 0 2 ss bb EE = 1 2 mmvv2 dddd dddd = mm vv 0 2 ss bb 1 bb eeeeeeeeeeee llllllll bbbbbbbbbbbbbb dddddddddddddddd
More informationON A CONTINUED FRACTION IDENTITY FROM RAMANUJAN S NOTEBOOK
Asian Journal of Current Engineering and Maths 3: (04) 39-399. Contents lists available at www.innovativejournal.in ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Journal homepage: http://www.innovativejournal.in/index.php/ajcem
More informationarxiv: v1 [math.dg] 1 Jul 2014
Constrained matrix Li-Yau-Hamilton estimates on Kähler manifolds arxiv:1407.0099v1 [math.dg] 1 Jul 014 Xin-An Ren Sha Yao Li-Ju Shen Guang-Ying Zhang Department of Mathematics, China University of Mining
More informationAnswers to Practice Test Questions 12 Organic Acids and Bases
Answers to Practice Test Questions 12 rganic Acids and Bases 1. (a) (b) pppp aa = llllll(kk aa ) KK aa = 10 pppp aa KK aa = 10 4.20 KK aa = 6.3 10 5 (c) Benzoic acid is a weak acid. We know this because
More informationOBE solutions in the rotating frame
OBE solutions in the rotating frame The light interaction with the 2-level system is VV iiiiii = μμ EE, where μμ is the dipole moment μμ 11 = 0 and μμ 22 = 0 because of parity. Therefore, light does not
More informationSolution of the Inverse Eigenvalue Problem for Certain (Anti-) Hermitian Matrices Using Newton s Method
Journal of Mathematics Research; Vol 6, No ; 014 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education Solution of the Inverse Eigenvalue Problem for Certain (Anti-) Hermitian
More informationYang-Hwan Ahn Based on arxiv:
Yang-Hwan Ahn (CTPU@IBS) Based on arxiv: 1611.08359 1 Introduction Now that the Higgs boson has been discovered at 126 GeV, assuming that it is indeed exactly the one predicted by the SM, there are several
More informationLecture 11. Kernel Methods
Lecture 11. Kernel Methods COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Andrey Kan Copyright: University of Melbourne This lecture The kernel trick Efficient computation of a dot product
More informationCHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
More informationTime Domain Analysis of Linear Systems Ch2. University of Central Oklahoma Dr. Mohamed Bingabr
Time Domain Analysis of Linear Systems Ch2 University of Central Oklahoma Dr. Mohamed Bingabr Outline Zero-input Response Impulse Response h(t) Convolution Zero-State Response System Stability System Response
More informationDiffusion modeling for Dip-pen Nanolithography Apoorv Kulkarni Graduate student, Michigan Technological University
Diffusion modeling for Dip-pen Nanolithography Apoorv Kulkarni Graduate student, Michigan Technological University Abstract The diffusion model for the dip pen nanolithography is similar to spreading an
More informationLinear Transformation on Strongly Magic Squares
American Journal of Mathematics and Statistics 2017, 7(3): 108-112 DOI: 10.5923/j.ajms.20170703.03 Linear Transformation on Strongly Magic Squares Neeradha. C. K. 1,*, T. S. Sivakumar 2, V. Madhukar Mallayya
More informationModule 4 (Lecture 16) SHALLOW FOUNDATIONS: ALLOWABLE BEARING CAPACITY AND SETTLEMENT
Topics Module 4 (Lecture 16) SHALLOW FOUNDATIONS: ALLOWABLE BEARING CAPACITY AND SETTLEMENT 1.1 STRIP FOUNDATION ON GRANULAR SOIL REINFORCED BY METALLIC STRIPS Mode of Failure Location of Failure Surface
More informationMath 171 Spring 2017 Final Exam. Problem Worth
Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:
More informationCSC 578 Neural Networks and Deep Learning
CSC 578 Neural Networks and Deep Learning Fall 2018/19 3. Improving Neural Networks (Some figures adapted from NNDL book) 1 Various Approaches to Improve Neural Networks 1. Cost functions Quadratic Cross
More informationAPPLICATIONS. A Thesis. Presented to. the Faculty of California Polytechnic State University, San Luis Obispo. In Partial Fulfillment
ASSESSING THE v 2 -f TURBULENCE MODELS FOR CIRCULATION CONTROL APPLICATIONS A Thesis Presented to the Faculty of California Polytechnic State University, San Luis Obispo In Partial Fulfillment of the Requirements
More informationMath 3 Unit 4: Rational Functions
Math Unit : Rational Functions Unit Title Standards. Equivalent Rational Expressions A.APR.6. Multiplying and Dividing Rational Expressions A.APR.7. Adding and Subtracting Rational Expressions A.APR.7.
More informationSome examples of radical equations are. Unfortunately, the reverse implication does not hold for even numbers nn. We cannot
40 RD.5 Radical Equations In this section, we discuss techniques for solving radical equations. These are equations containing at least one radical expression with a variable, such as xx 2 = xx, or a variable
More information