Invariant Theory by M. Neusel

By M. Neusel

Show description

Read or Download Invariant Theory PDF

Best linear books

A first course in linear algebra

A primary direction in Linear Algebra is an advent to the elemental thoughts of linear algebra, besides an advent to the recommendations of formal arithmetic. It starts with structures of equations and matrix algebra prior to entering into the idea of summary vector areas, eigenvalues, linear ameliorations and matrix representations.

Measure theory/ 3, Measure algebras

Fremlin D. H. degree idea, vol. three (2002)(ISBN 0953812936)(672s)-o

Elliptic Partial Differential Equations

Elliptic partial differential equations is likely one of the major and such a lot lively components in arithmetic. In our publication we learn linear and nonlinear elliptic difficulties in divergence shape, with the purpose of supplying classical effects, in addition to more moderen advancements approximately distributional suggestions. consequently the ebook is addressed to master's scholars, PhD scholars and someone who desires to start learn during this mathematical box.

Additional info for Invariant Theory

Sample text

H2 (b) H:,:,2 =   . .  .. 2) .. .. HJ1 HJ1 +1 · · · HJ1 +J2 −2 .. .. .. . .  HJ3 −1 HJ3 ··· HJ2 +J3 −3  . . .  . .. HJ3 (b) H:,:,J3 =   .. .  .. .. HJ1 +J3 −2 HJ1 +J3 −1 · · · HJ1 +J2 +J3 −4  HJ2 +J3 −2  ..  . .  HJ1 +J2 +J3 −4  HJ1 +J2 +J3 −3 Then the BHHB tensor H has the level-2 Vandermonde decomposition ⊗ Z1,1 ) ×2 (Z2,2 ⊗ Z1,2 ) · · · ×m (Z2,m ⊗ Z1,m ) , H = C ×1 (Z2,1 KR KR KR 44 CHAPTER 3. FAST TENSOR-VECTOR PRODUCTS where C is a diagonal tensor with diagonal entries {ck }K k=1 , each a Vandermonde matrix    Ip −1 2 2 1 z1,1 z1,1 · · · z1,1 1 z2,1 z2,1   Ip −1  2 2 1 z1,2 z1,2 · · · z1,2  1 z2,2 z2,2  Z1,p =  ..

Let A be a complex tensor, E be a nonnegative tensor, b be a complex vector, and d be a nonnegative vector. Define Σ := x ∈ Cn : A + E xm−1 = b + d, E ≤ E, d ≤ d . Then Σ = x ∈ Cn : Axm−1 − b ≤ E|x|m−1 + d . Proof. On the one hand, if x ∈ Σ, then Axm−1 − b = −Exm−1 + d ≤ E|x|m−1 + d. On the other hand, if Axm−1 − b ≤ E|x|m−1 + d, then there exist two signature matrices S1 and S2 such that S1 Axm−1 − b = Axm−1 − b and S2 x = |x|, which indicates that S1 Axm−1 − b ≤ ES2m−1 xm−1 + d. Thus there is a diagonal matrix D with |D| ≤ I such that Axm−1 − b = S1∗ DES2m−1 xm−1 + S1∗ Dd.

We first construct • 3rd -order square Hankel tensors of size n × n × n (n = 10, 20, . . , 100), and • 3rd -order square BHHB tensors of level-1 size n1 × n1 × n1 and level-2 size n2 × n2 × n2 (n1 , n2 = 5, 6, . . , 12). Then we compute the tensor-vector products H ×2 x2 ×3 x3 using • our proposed fast algorithm based on FFT, and • the non-structured algorithm based on the definition. The average running times of 1000 products are shown in Fig. 2. From the results, we can see that the running time of our algorithm increases far more slowly than that of the non-structured algorithm just as predicted by the theoretical analysis.

Download PDF sample

Rated 4.08 of 5 – based on 13 votes