English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let      GF ( q ) {\text {GF}}(q)       denote a finite field of characteristic two. Let      V n {V_n}       denote an n-dimensional vector space over      GF ( q ) {\text {GF}}(q)      . An      n ×  n n \times n       symmetric matrix  A  over      GF ( q ) {\text {GF}}(q)       is said to be an alternate matrix if  A  has zero diagonal. Let  A  be an      n ×  n n \times n       alternate matrix over      GF ( q ) {\text {GF}}(q)       and let  C  be an      s ×  s s \times s       symmetric matrix over      GF ( q ) {\text {GF}}(q)      . By using Albert’s canonical forms for symmetric matrices over fields of characteristic two, the number      N ( A , C , n , s , r ) N(A,C,n,s,r)       of      s ×  n s \times n       matrices  X  of rank  r  over      GF ( q ) {\text {GF}}(q)       such that      X A X T = C XA{X^T} = C       is determined. A symmetric bilinear form on      V n ×  V n {V_n} \times {V_n}       is said to be alternating if      f ( x , x ) = 0 f(x,x) = 0      , for each  x  in      V n {V_n}      . Let  f  be such a bilinear form. A basis      ( x 1 , …  , x ρ  , y 1 , …  , y ρ  ) , n = 2 ρ  ({x_1}, \ldots ,{x_\rho },{y_1}, \ldots ,{y_\rho }),n = 2\rho      , for      V n {V_n}       is said to be a symplectic basis for      V n {V_n}       if      f ( x i , x j ) = f ( y i , y j ) = 0 f({x_i},{x_j}) = f({y_i},{y_j}) = 0       and      f ( x i , y j ) = δ  i j f({x_i},{y_j}) = {\delta _{ij}}      , for each  i ,      j = 1 , 2 , …  , ρ  j = 1,2, \ldots ,\rho      . In determining the number      N ( A , C , n , s , r ) N(A,C,n,s,r)      , it is shown that a symplectic basis for any subspace of      V n {V_n}      , can be extended to a symplectic basis for      V n {V_n}      . Furthermore, the number of ways to make such an extension is determined.

Rank 𝑟 solutions to the matrix equation 𝑋𝐴𝑋^{𝑇}=𝐶,𝐴 alternate, over 𝐺𝐹(2^{𝑦})