hahn banach theorem extension of linear functionals

... Multilinear PCA Multilinear Principal Component Analysis (MPCA) [1, 2] is a multilinear extension of PCA MPCA computes a linear subspace representing the variance of data due to the variation of each factor ... calculated as the average of parts of the Gram matrix G Each entry of these three Gram-like matrices is the average of same-color entries of G (a) Gsubj consists of averages of dot products which ... which represent the averages of the pairwise relationships between a group of subjects (b) Gview consists of averages of dot products which represent the averages of the pairwise relationships...

Ngày tải lên: 21/06/2014, 16:20

Ngày tải lên: 05/08/2014, 15:20

Ngày tải lên: 05/08/2014, 09:46

... convex set of X, for example as K11 = K1 , then we can use a well-known result (Theorem 2.2) From here the statement of the theorem follows For the proof of case 3, one may use the proof of Theorem ... that the validity of results of this type also follows from results obtained by Phelps in [25] concerning the uniqueness of the extension of a linear functional to the whole of a Banach space ∗ ... all the conditions of Theorem 2.6 ∗ (or of Theorem 2.2) are fulfilled relative to the pair (Y0 ,Y0 ) Consequently, we obtain the correctness of Theorem 3.1 with the aid of Theorem 2.7 The next...

Ngày tải lên: 22/06/2014, 19:20

... “A generalization of Greguˇ fixed point theorem, ” Journal of Applied Sciences, vol 6, s no 15, pp 3160–3163, 2006 [5] C E Chidume, “Geometric properties of Banach spaces and nonlinear iterations,” ... iteration sequence [16] of a class of quasicontractive operators, called Zamfirescu operators, defined on a closed convex subset C of a Banach space X converges to the fixed point of T The first author ... contradiction with the definition of i Hence inf {F(Tx − x) : x ∈ C } = To prove that the infimum is attained is the easy part of the proof Take the following system of sets: Kn = {x : F(x − Tx) ≤...

Ngày tải lên: 22/06/2014, 22:20

... “A generalization of Greguˇ fixed point theorem, ” Journal of Applied Sciences, vol 6, s no 15, pp 3160–3163, 2006 [5] C E Chidume, “Geometric properties of Banach spaces and nonlinear iterations,” ... iteration sequence [16] of a class of quasicontractive operators, called Zamfirescu operators, defined on a closed convex subset C of a Banach space X converges to the fixed point of T The first author ... contradiction with the definition of i Hence inf {F(Tx − x) : x ∈ C } = To prove that the infimum is attained is the easy part of the proof Take the following system of sets: Kn = {x : F(x − Tx) ≤...

Ngày tải lên: 22/06/2014, 22:20

Ngày tải lên: 05/08/2014, 09:46

Ngày tải lên: 05/08/2014, 09:46

... analysis can be expressed as functionals of ESD of some random matrices Indeed, a parameter θ of the population can often be expressed as: θ= f (x)dF(x) Let Fn (x) be the ESD of the random matrix which ... Product of Two Random Matrices 18 2.2 Limits of Extreme Eigenvalues 19 2.3 Convergence Rate of ESD 21 2.4 Literature Review CLT of Linear ... asymptotic properties of the ESD of the corresponding LDRM ix List of Notations List of Notations Chapter Wn (xi j )1≤i, j≤n , Wigner matrix Fn (x) F F(x) semicircular law U the open set of the real line...

Ngày tải lên: 12/09/2015, 09:39

... 0.874 mg of the tetrazine derivatized twofold with dansyl, was dissolved in a few drops of DMSO and then added to the reaction product of A The Discussion Here we demonstrate the synthesis of exemplary ... synthesis of molecules suitable for functionalization of PNA: Fmoc-C2-glycine-tert-butyl ester reacts with the carbonic acid chloride of the DAR component X 14 to the corresponding tert-butyl ester of ... of these ligation methods Joining molecular pieces is as easy as ‘‘clicking’’ together the two pieces of a buckle.[8] Some attributes of this philosophy are applicable to the broad spectrum of...

Ngày tải lên: 26/10/2012, 08:57

... Introduction Much of the sophistication of complicated linear equation-solving packages” is devoted to the detection and/or correction of these two pathologies As you work with large linear sets of equations, ... consists of a particular solution xp added to any linear combination of (typically) N − M vectors (which are said to be in the nullspace of the matrix A) The task of finding the solution space of A ... the sum of the squares of the differences between the left- and right-hand sides of equation (2.0.1) be minimized, then the overdetermined linear problem reduces to a (usually) solvable linear...

Ngày tải lên: 15/12/2013, 04:15

... row in A by a linear combination of itself and any other row, as long as we the same linear combination of the rows of the b’s and (which then is no longer the identity matrix, of course) • Interchanging ... two rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y Rather, it just corresponds to writing the same set of linear equations ... any two columns of A gives the same solution set only if we simultaneously interchange corresponding rows of the x’s and of Y In other words, this interchange scrambles the order of the rows in...

Ngày tải lên: 15/12/2013, 04:15

... solve linear systems In many applications only the part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv Sample page from NUMERICAL RECIPES IN C: THE ART OF ... purposes, because of its greater diagnostic capability in pathological cases.) Updating a QR decomposition Some numerical algorithms involve solving a succession of linear systems each of which differs ... d[], float b[]) Solves the set of n linear equations R · x = b, where R is an upper triangular matrix stored in a and d a[1 n][1 n] and d[1 n] are input as the output of the routine qrdcmp and are...

Ngày tải lên: 15/12/2013, 04:15

... || Vậy g 3.3 Các ứng dụng định lí Hahn- Banach Các định lí trên, đặc biệt định lí Hahn - Banach có nhiều ứng dụng Sau đây, ta xét số ứng dụng định lí Hahn - Banach 3.3.1 Mệnh đề Với x, y thuộc ... chuẩn gọi không gian định chuẩn 1.10 Định nghĩa không gian Banach Không gian định chuẩn đầy đủ (với mêtric sinh chuẩn) gọi không gian Banach 1.11 Không gian dãy x = ( xn ) y = ( yn ), lp Với ... Mệnh đề Công thức (1) xác định chuẩn L( E , F ) 1.14.3 Định lí Nếu F không gian Banach không gian L( E , F ) Banach 1.15 Định nghĩa không gian vectơ tôpô Giả sử E không gian vectơ trờng K Một...

Ngày tải lên: 20/12/2013, 22:36

... (2.2.4) is called backsubstitution The combination of Gaussian elimination and backsubstitution yields a solution to the set of equations The advantage of Gaussian elimination and backsubstitution ... and the increasing numbers of predictable zeros reduce the count to one-third), and N M times, respectively Each backsubstitution of a right-hand side is N executions of a similar loop (one multiplication ... matrix as part of the Gauss-Jordan scheme.) For computing the inverse matrix (which we can view as the case of M = N right-hand sides, namely the N unit vectors which are the columns of the identity...

Ngày tải lên: 24/12/2013, 12:16

... modify the loop of the above fragment and (e.g.) divide by powers of ten, to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of the absolute values of the factors ... Computer Solution of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations ... reduction free_vector(vv,1,n); 48 Chapter Solution of Linear Algebraic Equations To summarize, this is the preferred way to solve the linear set of equations A · x = b: float **a,*b,d; int n,*indx;...

Ngày tải lên: 24/12/2013, 12:16

... 2.5 Iterative Improvement of a Solution to Linear Equations Obviously it is not easy to obtain greater precision for the solution of a linear set than the precision of your computer’s floating-point ... limitations of bandec, and the above routine does take advantage of the opportunity In general, when TINY is returned as a diagonal element of U , then the original matrix (perhaps as modified by roundoff ... Unfortunately, for large sets of linear equations, it is not always easy to obtain precision equal to, or even comparable to, the computer’s limit In direct methods of solution, roundoff errors accumulate,...

Ngày tải lên: 24/12/2013, 12:16

... “7/8”; it is that factor at each hierarchical level of the recursion In total it reduces the process of matrix multiplication to order N log2 instead of N What about all the extra additions in (2.11.3)–(2.11.4)? ... submatrices Imagine doing the inversion of a very large matrix, of order N = 2m , recursively by partitions in half At each step, halving the order doubles the number of inverse operations But this means ... complicated nature of the recursive Strassen algorithm, you will find that LU decomposition is in no immediate danger of becoming obsolete If, on the other hand, you like this kind of fun, then try...

Ngày tải lên: 24/12/2013, 12:16

... discussion of the use of SVD in this application to Chapter 15, whose subject is the parametric modeling of data SVD methods are based on the following theorem of linear algebra, whose proof is beyond ... matrix Define the residual matrix R of B0 as 58 Chapter Solution of Linear Algebraic Equations We can define the norm of a matrix as the largest amplification of length that it is able to induce ... x[1 n] of the linear set of equations A · X = B The matrix a[1 n][1 n], and the vectors b[1 n] and x[1 n] are input, as is the dimension n Also input is alud[1 n][1 n], the LU decomposition of a...

Ngày tải lên: 24/12/2013, 12:16

... same permutation of the columns of U, elements of W, and columns of V (or rows of VT ), or (ii) forming linear combinations of any columns of U and V whose corresponding elements of W happen to ... Value Decomposition A A⋅x = b (a) null space of A solutions of A⋅x = d solutions of A ⋅ x = c′ SVD “solution” of A ⋅ x = c range of A d c′ c SVD solution of A⋅x = d (b) Figure 2.6.1 (a) A nonsingular ... subspace of b that can be “reached” by A, in the sense that there exists some x which is mapped there This subspace of b is called the range of A The dimension of the range is called the rank of A...

Ngày tải lên: 24/12/2013, 12:16