Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator


 Jodie Wilkerson
 3 years ago
 Views:
Transcription
1 Wan Boundary Value Problems (2015) 2015:239 DOI /s R E S E A R C H Open Access Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator Liquan Wan * * Correspondence: School of Accounting, Henan University of Economics and Law, Zhengzhou, , China Abstract In this paper, we not only give the asymptotic behavior at the origin for the maximum modulus of weak solutions of the stationary Schrödinger equation in a cone but also obtain the property of the negative parts of them, which generalize the PhragménLindelöf type theorems for subfunctions. Keywords: stationary Schrödinger equation; weak solution; cone 1 Introduction and main results Let R n be the ndimensional Euclidean space, where n 2. Let E be an open set in R n,the boundary and the closure of it are denoted by E and E,respectively.ApointP is denoted by (X, x n ), where X =(x 1, x 2,...,x n 1 ). For P R n and r >0,letB(P, r) denote the open ball with center at P and radius r in R n. We introduce a system of spherical coordinates (r, ), =(θ 1, θ 2,...,θ n 1 ), in R n which are related to the cartesian coordinates (X, x n )=(x 1, x 2,...,x n 1, x n )byx n = r cos θ 1. Let S n 1 and S n 1 + denote the unit sphere and the upper half unit sphere, respectively. For S n 1,apoint(1, ) ons n 1,andtheset{ ;(1, ) } are simply denoted by and respectively. The set {(r, ) R n ; r,(1, ) } in R n is simply denoted by, where R + and S n 1. Especially, the set R + by C n ( ), where R + is the set of positive real number and S n 1. Let C n ( ; I) ands n ( ; I) denotethesetsi and I,respectively,whereI is an interval on R and R is the set of real numbers. Especially, the set S n ( ) denotes S n ( ;(0,+ )), which is C n ( ) {O}. Let be the spherical part of the Laplace operator (see [1]), = n 1 r r + 2 r + 2 r, 2 and be a domain on S n 1 with smooth boundary. We consider the Dirichlet problem (see [2], p.41) ( + λ ) ϕ( )=0 on, ϕ( )=0 on Wan. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
2 Wan Boundary Value Problems (2015) 2015:239 Page 2 of 10 The least positive eigenvalue of the above boundary value problem is denoted by λ and the normalized positive eigenfunction corresponding to λ by ϕ( ), ϕ2 ( ) d =1,where d denotes the (n 1)dimensional volume element. We put a rather strong assumption on : ifn 3, then is a C 2,α domain (0 < α <1) on S n 1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [3], pp.8889, for the definition of the C 2,α domain). Let A a denote the class of nonnegative radial potentials a(p), i.e. 0 a(p) =a(r), P = (r, ) C n ( ), such that a L b loc (C n( )) with some b > n/2 if n 4andwithb =2ifn =2 or n =3. Let I be the identical operator. If a A a, then the stationary Schrödinger operator SSE a = + a(p)i can be extended in the usual way from the space C 0 (C n( )) to an essentially selfadjoint operator on L 2 (C n ( )) (see [4],Chapter13).WewilldenoteitSSE a as well. This last one has a GreenSch function G a (P, Q)whichispositiveonC n( ) and its inner normal derivative G a (P, Q)/ n Q 0, where / n Q denotes the differentiation at Q along the inward normal into C n ( ). In this paper, we are concerned with the weak solutions of the inequality SSE a u(p) 0, (1.1) where P =(r, ) C n ( ). We will also consider the class B a, consisting of the potentials a A a such that there exists the finite limit lim r r 2 a(r)=k [0, ), moreover, r 1 r 2 a(r) k L(1, ). We denote by SbH a ( ) the class of all weak solutions of the inequality (1.1) for any P =(r, ) C n ( ), which are continuous when a B a (see [5]). We denote by SpH a ( )the class of u(p)satisfying u(p) SbH a ( ). If u(p) SbH a ( )andu(p) SpH a ( ), then u(p) is the solution of SSE a u(p) = 0 for any P =(r, ) C n ( ). In our terminology we follow Nirenberg [6]. Other authors have under similar circumstances used various terms such as subfunctions, subsolutions, submetaharmonic function, subelliptic functions, panharmonic functions, etc.; see, for example, Duffin, Littman, Qiao et al., Topolyansky, Vekua (see [7 11]). Solutions of the ordinary differential equation (r) n 1 ( ) λ (r)+ r r + a(r) (r)=0, 0<r <, (1.2) 2 play an essential role in this paper. It is well known (see, for example, [12]) that if the potential a A a,thenequation(1.2) has a fundamental system of positive solutions {V, W} such that V is nondecreasing with 0 V(0+) V(r) as r +, and W is monotonically decreasing with + = W(0+) > W(r) 0 asr +.
3 Wan Boundary Value Problems (2015) 2015:239 Page 3 of 10 Denote ι ± k = 2 n ± (n 2) 2 +4(k + λ), 2 then the solutions to equation (1.2) have the following asymptotic (see [3]): V(r) r ι+ k, W(r) r ι k, asr. Let u(p) (P =(r, ) C n ( )) be a function. We introduce the following notations: u + = max{u,0}, u = min{u,0}, =sup u(p), l = max ϕ( ), u(p) S u (r)=sup ϕ( ), L u = lim sup S u (r) W(r), J u = sup P C n ( ) u(p) W(r)ϕ( ). For any two positive numbers δ an r,we put E u 0 (r; δ)={ : u(p) δw(r) } and ξ u (δ)=lim sup ϕ( ) d. E0 u(r;δ) The integral u(r, )ϕ( ) d, is denoted by N u (r), when it exists. The finite or infinite limits N u (r) lim r V(r) and N u (r) lim W(r) are denoted by μ u and η u, respectively, when they exist. We shall say that u(p) (P =(r, ) C n ( )) satisfies the PhragménLindelöf boundary condition on S n ( ), if lim sup u(p) 0 P Q,Q S n ( ) for every Q S n ( ). Throughout this paper, unless otherwise specified, we will always assume that u(p) SbH a ( ) and satisfy the PhragménLindelöf boundary condition on S n ( ). Recently, about the PhragménLindelöf theorems for subfunctions in a cone, Qiao and Deng (see [9], Theorem 3) proved the following result. Theorem A If μ u + = η u + =0,
4 Wan Boundary Value Problems (2015) 2015:239 Page 4 of 10 then u(p) 0 for any P =(r, ) C n ( ). A stronger version of a PhragménLindelöf type theorem is also due to Qiao and Deng (see [9], Theorem 3). Theorem B If and then lim inf r lim inf V(r) W(r) <+ (1.3) <+, (1.4) u(p) ( μ u V(r)+η u W(r) ) ϕ( ) (1.5) for any P =(r, ) C n ( ). However, they do not tell us in [9] whether or not the limit B u = lim W(r) exists. In this paper, we first of all answer this question positively and prove the following result. Theorem 1 If (1.3) is satisfied, then the limit B u (0 B u + ) exists and B u =(L u ) + l, (1.6) where (L u ) + = η u +. (1.7) Remark It is obvious that η u L u. On the other hand, we have η u L u from (1.5). Thus, if (1.3)and(1.4) are satisfied, then we have η u = L u. As an application of Theorem 1 we immediately have the following result by using Lemma 3 in Section 2. Corollary If lim inf r V(r) 0, (1.8)
5 Wan Boundary Value Problems (2015) 2015:239 Page 5 of 10 then B u =(J u ) + l. (1.9) In [9], the authors gave the properties of the positive part of weak solutions satisfying the PhragménLindelöf boundary condition on S n ( ). Finally, we shall show one of the properties of its negative part. From the remark, we have η u + = L u + =(L u ) + =(η u ) +. Since N u (r)=n u +(r) N u (r), Theorem 2 follows immediately. Theorem 2 Under the conditions of Theorem B, if η u 0, then N u (r) lim W(r) =0. 2 Some lemmas Lemma 1 (see [9], Lemma 8) (1) Both of the limits μ u and η u ( < μ u, η u + ) exist. (2) If η u 0, then V 1 (r)n u (r) is nondecreasing on (0, + ). (3) If μ u 0, then W 1 (r)n u (r) is nonincreasing on (0, + ). Lemma 2 If (1.3) is satisfied and there exists a positive number R such that u(p) 0 for any P =(r, ) C n ( ;(0,R)), then for any positive number δ, we have u(p) ( μ u V(r) δξ u (δ)w(r) ) ϕ( ) (2.1) for any P =(r, ) C n ( ). Proof Let δ be any given positive number and {r k } be a sequence such that lim r k =0 and lim ϕ( ) d = ξ u (δ). k k E0 u(r k;δ) Then we have N u (r k ) E u 0 (r k;δ) for any 0 < r k < R and hence u(r k, )ϕ( ) d δw(r k ) ϕ( ) d E0 u(r k;δ) η u δξ u (δ). Thus we obtain (2.1)fromTheoremB.
6 Wan Boundary Value Problems (2015) 2015:239 Page 6 of 10 Lemma 3 Under the conditions of the corollary, L u > and J u = L u. Proof It is evident that J u L u. (2.2) Hence, we shall prove that J u = L u under the assumption that L u <+. Since(1.3) and (1.4) are satisfied and (1.8)givesμ u 0, we have u(p) η u W(r)ϕ( ) for any P =(r, ) C n ( )fromtheoremb,whichgives η u J u. (2.3) Since Lemma 1 and the remark give η u > and η u = L u,respectively,wehavethe conclusion from (2.2)and(2.3). Given a continuous function ψ defined on the truncated cone C n ( ;(R 1, R 2 )), where R 1 and R 2 are two positive real numbers satisfying R 1 < R 2, then the solution of the Dirichlet Sch problem on C n ( ;(R 1, R 2 )) with ψ is denoted by H ψ (P; C n ( ;(R 1, R 2 ))). Lemma 4 If μ u + <+ and η u + <+, (2.4) are satisfied, then B u η u +. Proof Take any P =(r, ) C n ( ) and any pair of numbers R 1, R 2 satisfying 0 < 2R 1 < r < 1 2 R 2 <.Ifψ(P) is a boundary function on C n ( ;(R 1, R 2 )) satisfying ψ(p)= then we have { u(r i, ) on{r i } (i =1,2), 0 ons n ( ;(R 1, R 2 )), ( ( u(p) H ψ P; Cn ;(R1, R 2 ) )) = u + (R 1, ) Ga C n ( ;(R 1,R 2 )) (P,(R 1, )) y R n 1 1 d u + (R 2, ) Ga C n ( ;(R 1,R 2 )) (P,(R 2, )) R n 1 2 d, y where G a C n ( ;(R 1,R 2 )) (, ) is the GreenSch function on C n( ;(R 1, R 2 )) with the pole at P. Here we use the following inequalities (see [1], p.124): (G a C n ( ;(R 1,R 2 )) (P,(R 1, ))) R c 1 W(r) W(R 1 ) ϕ( )ϕ( ) R n 1 1
7 Wan Boundary Value Problems (2015) 2015:239 Page 7 of 10 and (G a C n ( ;(R 1,R 2 )) (P,(R 2, ))) R c 2 V(r) V(R 2 ) where c 1 and c 2 are two positive constants. Then we have ϕ( )ϕ( ) R n 1, 2 u(p) c 3 W 1 (R 1 )N u +(R 1 )W(r)ϕ( )+c 4 V 1 (R 2 )N u +(R 2 )V(r)ϕ( ), (2.5) where c 3 and c 4 are two positive constants. As R 1 0andR 2 in (2.5), we obtain ( c 3 η u +W(r)+c 4 μ u +V(r) ) max ϕ( ) from Lemma 1, which gives the conclusion of Lemma 4 from (2.4). 3 Proof of Theorem 1 Put τ = lim inf W(r). Since N u (r) ϕ( ) d, and Lemma 1 gives η u >, (3.1) we immediately see that τ >. Now we distinguish two cases. Case 1 τ =+. In this case B u exists and is equal to +.Itisobviousthatforanypositivenumberr M u +(r) W(r) l S u +(r) W(r), (3.2) which gives L u =+. These results show that (1.7) holds in this case. Case 2 τ <+. From Theorem B,weseethatL u <+. On the other hand, we have L u > from (3.1). Subcase L u <+. There exists a positive number R ɛ such that u(p) (L u + ɛ)w(r)ϕ( ) for any ɛ >0,whereP =(r, ) C n ( ;(0,R ɛ )).
8 Wan Boundary Value Problems (2015) 2015:239 Page 8 of 10 This gives lim sup W(r) L ul. (3.3) Now, assume that τ < L u l.thereexistapositivenumberδ 1 and a set E u such that ϕ( ) d >0 E u and L u ϕ( ) τ 2δ 1 (3.4) for E u. We define v 1 (P)(P =(r, ) C n ( )) by v 1 (P)=u(P) (L u + ɛ)w(r)ϕ( ) (3.5) and apply Lemma 2 to v 1 (P). It gives u(p) [{ L u + ɛ δ 1 ξ v1 (δ 1 ) } W(r)+μ v1 V(r) ] ϕ( ) for any P =(r, ) C n ( ). So we have L u L u δ 1 ξ v1 (δ 1 ). If we can show that ξ v1 (δ 1 )>0, (3.6) then we have a contradiction. To prove (3.6), take a sequence {r k },withlim k r k =0,suchthat M u (r k ) W(r k ) τ + δ 1 (k =1,2,3,...). From (3.4)and(3.5)wehave v 1 (r k, ) W(r k ) for any E u,whichgives u(r k, ) W(r k ) L uϕ( ) δ 1 E u E v 1 0 (r k; δ 1 ) (k =1,2,3,...). Hence ξ v1 (δ 1 ) ϕ( ) d >0. E u
9 Wan Boundary Value Problems (2015) 2015:239 Page 9 of 10 Thus from (3.3) we can simultaneously prove the existence of B u and (1.6). Subcase 2.2 L u <0. Take any small number ɛ > 0 satisfying L u + ɛ < 0. There exists a positive number R ɛ such that u(p) (L u + ɛ)w(r)ϕ( ) for any P =(r, ) C n ( ;(0,R ɛ )). This gives lim sup W(r) 0. (3.7) Now suppose that τ <0.Thereareasequence{r k } tending to 0 and a positive number δ 2 such that M u (r k ) W(r k ) 2δ 2 (k =1,2,3,...). Define v 2 (P)(P =(r, ) C n ( )) by v 2 (P)=u(P) (L u + ɛ)w(r)ϕ( ) and apply Lemma 2 to v 2 (P). Then we obtain u(p) [{ L u + ɛ δ 2 ξ v2 (δ 2 ) } W(r)+μ v2 V(r) ] ϕ( ) for any P =(r, ) C n ( ), which gives L u L u δ 2 ξ v2 (δ 2 ). If we can show that ξ v2 (δ 2 )>0, (3.8) then we have a contradiction. To prove (3.8), write F u = { ; L u ϕ( ) δ 2 }. It is evident that ϕ( ) d >0. F u For every F u,wehave v 1 (r k, ) W(r k ) u(r k, ) W(r k ) L uϕ( ) δ 2,
10 Wan Boundary Value Problems (2015) 2015:239 Page 10 of 10 which shows that F u E v 2 0 (r k; δ 2 ) (k =1,2,3,...). Hence we have ξ v2 (δ 2 ) ϕ( ) d >0. F u Thus we can prove that τ 0. With (3.7), this also gives the existence of B u and B u =0=(L u ) + l. Lastly, we shall show that (1.7)holds. If η u + =+, thenitisevidentthatb u + =+. This together with (3.2) givesl u + =+. Since L u + =(L u ) +, (3.9) we know that (1.7)holds. Next suppose that η u + <+.WehaveB u <+ by Lemma 4 and hence L u + = η u + by the remark. With (3.9), this gives (1.7). Competing interests The author declares that there is no conflict of interests regarding the publication of this article. Acknowledgements The author would like to thank the referees for their comments. This work was supported by the National Social Science Foundation of China (No. 13BGL047), the Humanities and Social Science Fund of Ministry of Education (No ZD011) and the 2015 Universities Philosophy Social Sciences Innovation Team of Henan Province: the Institutional Arrangements for Mixed Ownership Reform of Stateowned Enterprises in Henan Province (No CXTD09). Received: 29 October 2015 Accepted: 8 December 2015 References 1. Azarin, V: Generalization of a theorem of Hayman on subharmonic functions in an mdimensional cone. Transl. Am. Math. Soc. 80, (1969) 2. Rosenblum, G, Solomyak, M, Shubin, M: Spectral Theory of Differential Operators. VINITI, Moscow (1989) 3. Gilbarg, D, Trudinger, N: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977) 4. Reed, M, Simon, B: Methods of Modern Mathematical Physics, vol. 3. Academic Press, New York (1970) 5. Simon, B: Schrödinger semigroups. Bull. Am. Math. Soc. 7, (1982) 6. Nirenberg, L: Existence Theorems in Partial Differential Equations. NYU Notes (1954) 7. Duffin, R: Yukawan potential theory. J. Math. Anal. Appl. 35, (1971) 8. Littman, W: A strong maximum principle for weakly Lsubharmonic functions. J. Math. Mech. 8, (1959) 9. Qiao, L, Deng, G: A theorem of PhragménLindelöf type for subfunctions in a cone. Glasg. Math. J. 53(3), (2011) 10. Topolyansky, D: A note on submetaharmonic functions. Dopov. Ukr. Akad. Nauk 4, (1960) (in Ukrainian) 11. Vekua, I: Metaharmonic functions. Proc. Tbilisi Math. Inst. 12, (1943) (inrussian) 12. Verzhbinskii, G, Maz ya, V: Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I. Sib. Mat. Zh. 12, (1971)
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More information1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1
Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between
More informationRelatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations
Wang and Agarwal Advances in Difference Equations (2015) 2015:312 DOI 10.1186/s1366201506500 R E S E A R C H Open Access Relatively dense sets, corrected uniformly almost periodic functions on time
More informationDIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER. 1. Introduction
DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER DMITRY KHAVINSON, ERIK LUNDBERG, HERMANN RENDER. Introduction A function u is said to be harmonic if u := n j= 2 u = 0. Given a domain
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationA SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIAN. Keywords: shape optimization, eigenvalues, Dirichlet Laplacian
A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA DORI BUCUR AD DARIO MAZZOLEI Abstract. In this paper we give a method to geometrically modify an open set such that the first k eigenvalues of
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationFALSE ALARMS IN FAULTTOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem
Opuscula Mathematica Vol. 32 No. 4 2012 http://dx.doi.org/10.7494/opmath.2012.32.4.751 FALSE ALARMS IN FAULTTOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of faulttolerant
More informationBOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam
Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationEXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHERORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH
EXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHERORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH Abstract. We consider the th order elliptic boundary value problem Lu = f(x,
More informationMonotone maps of R n are quasiconformal
Monotone maps of R n are quasiconformal K. Astala, T. Iwaniec and G. Martin For Neil Trudinger Abstract We give a new and completely elementary proof showing that a δ monotone mapping of R n, n is K quasiconformal
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationQuasi Contraction and Fixed Points
Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa00168, 6 Pages doi:10.5899/2012/jnaa00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady
More informationA Note on the Ruin Probability in the Delayed Renewal Risk Model
Southeast Asian Bulletin of Mathematics 2004 28: 1 5 Southeast Asian Bulletin of Mathematics c SEAMS. 2004 A Note on the Ruin Probability in the Delayed Renewal Risk Model Chun Su Department of Statistics
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More information1 Introduction, Notation and Statement of the main results
XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matemática Aplicada Sevilla, 2428 septiembre 2007 (pp. 1 6) Skewproduct maps with base having closed set of periodic points. Juan
More informationTHE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS
Scientiae Mathematicae Japonicae Online, Vol. 5,, 9 9 9 THE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS CHIKAKO HARADA AND EIICHI NAKAI Received May 4, ; revised
More informationOn the existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems
RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 97 (3), 2003, pp. 461 466 Matemática Aplicada / Applied Mathematics Comunicación Preliminar / Preliminary Communication On the existence of multiple principal
More informationON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS
ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute
More informationSCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
More informationON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. GacovskaBarandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. GacovskaBarandovska Smirnov (1949) derived four
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationAn improved online algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the PrePublished Version. An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean
More informationSOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction
italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University
More informationRESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS
RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS T. J. CHRISTIANSEN Abstract. We consider scattering by an obstacle in R d, d 3 odd. We show that for the Neumann Laplacian if
More informationON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath
International Electronic Journal of Algebra Volume 7 (2010) 140151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December
More informationEXISTENCE AND NONEXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7669. UL: http://ejde.math.txstate.edu
More informationContinuity of the Perron Root
Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North
More informationExtremal equilibria for reaction diffusion equations in bounded domains and applications.
Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal RodríguezBernal Alejandro VidalLópez Departamento de Matemática Aplicada Universidad Complutense de Madrid,
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationIn memory of Lars Hörmander
ON HÖRMANDER S SOLUTION OF THE EQUATION. I HAAKAN HEDENMALM ABSTRAT. We explain how Hörmander s classical solution of the equation in the plane with a weight which permits growth near infinity carries
More informationTenacity and rupture degree of permutation graphs of complete bipartite graphs
Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More informationTRIPLE FIXED POINTS IN ORDERED METRIC SPACES
Bulletin of Mathematical Analysis and Applications ISSN: 18211291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON
More informationMath212a1010 Lebesgue measure.
Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE
i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationWald s Identity. by Jeffery Hein. Dartmouth College, Math 100
Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationSign changes of Hecke eigenvalues of Siegel cusp forms of degree 2
Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree
More informationLabeling outerplanar graphs with maximum degree three
Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationShape Optimization Problems over Classes of Convex Domains
Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY email: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationStationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the nonnegative
More informationMATH 425, HOMEWORK 7, SOLUTIONS
MATH 425, HOMEWORK 7, SOLUTIONS Each problem is worth 10 points. Exercise 1. (An alternative derivation of the mean value property in 3D) Suppose that u is a harmonic function on a domain Ω R 3 and suppose
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More informationFINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS
FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationCHAPTER 5. Product Measures
54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue
More informationCacti with minimum, secondminimum, and thirdminimum Kirchhoff indices
MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 4758 (2010) Cacti with minimum, secondminimum, and thirdminimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang
More informationTail inequalities for order statistics of logconcave vectors and applications
Tail inequalities for order statistics of logconcave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.TomczakJaegermann Banff, May 2011 Basic
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More informationSome Problems of SecondOrder Rational Difference Equations with Quadratic Terms
International Journal of Difference Equations ISSN 09736069, Volume 9, Number 1, pp. 11 21 (2014) http://campus.mst.edu/ijde Some Problems of SecondOrder Rational Difference Equations with Quadratic
More informationMath 231b Lecture 35. G. Quick
Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the Jhomomorphism could be defined by
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationSEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov
Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s446715357 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationb i (x) u x i + F (x, u),
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 14, 1999, 275 293 TOPOLOGICAL DEGREE FOR A CLASS OF ELLIPTIC OPERATORS IN R n Cristelle Barillon Vitaly A. Volpert
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationQuasistatic evolution and congested transport
Quasistatic evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
More informationSOME RESULTS ON THE DRAZIN INVERSE OF A MODIFIED MATRIX WITH NEW CONDITIONS
International Journal of Analysis and Applications ISSN 22918639 Volume 5, Number 2 (2014, 191197 http://www.etamaths.com SOME RESULTS ON THE DRAZIN INVERSE OF A MODIFIED MATRIX WITH NEW CONDITIONS ABDUL
More informationNumerical Verification of Optimality Conditions in Optimal Control Problems
Numerical Verification of Optimality Conditions in Optimal Control Problems Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der JuliusMaximiliansUniversität Würzburg vorgelegt von
More informationTHE SETTLING TIME REDUCIBILITY ORDERING AND 0 2 SETS
THE SETTLING TIME REDUCIBILITY ORDERING AND 0 2 SETS BARBARA F. CSIMA Abstract. The Settling Time reducibility ordering gives an ordering on computably enumerable sets based on their enumerations. The
More informationTRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh
Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp . TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 114, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO
More informationSamuel Omoloye Ajala
49 Kragujevac J. Math. 26 (2004) 49 60. SMOOTH STRUCTURES ON π MANIFOLDS Samuel Omoloye Ajala Department of Mathematics, University of Lagos, Akoka  Yaba, Lagos NIGERIA (Received January 19, 2004) Abstract.
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationResearch Article The General Traveling Wave Solutions of the Fisher Equation with Degree Three
Advances in Mathematical Physics Volume 203, Article ID 65798, 5 pages http://dx.doi.org/0.55/203/65798 Research Article The General Traveling Wave Solutions of the Fisher Equation with Degree Three Wenjun
More informationMixedÀ¾ Ð½Optimization Problem via Lagrange Multiplier Theory
MixedÀ¾ Ð½Optimization Problem via Lagrange Multiplier Theory Jun WuÝ, Sheng ChenÞand Jian ChuÝ ÝNational Laboratory of Industrial Control Technology Institute of Advanced Process Control Zhejiang University,
More information