SIMETRISASI BENTUK KANONIK JORDAN

If the characteristic polynomial of a linear operator is completely factored in scalar field of then Jordan canonical form of can be converted to its rational canonical form of , and vice versa. If the characteristic polynomial of linear operator is not completely factored in the scalar field of ,then the rational canonical form of can still be obtained but not its Jordan canonical form matrix . In this case, the rational canonical form of can be converted to its Jordan canonical form by extending the scalar field of to Splitting Field of minimal polynomial of , thus forming the Jordan canonical form of over Splitting Field of . Conversely, converting the Jordan canonical form of over Splitting Field of to its rational canonical form uses symmetrization on the Jordan decomposition basis of so as to form a cyclic decomposition basis of which is then used to form the rational canonical matrix of

THE INDUCTIVE BLOCKWISE ALPERIN WEIGHT CONDITION FOR TYPE AND THE PRIME

We establish the inductive blockwise Alperin weight condition for simple groups of Lie type $\mathsf C$ and the bad prime $2$ . As a main step, we derive a labelling set for the irreducible $2$ -Brauer characters of the finite symplectic groups $\operatorname {Sp}_{2n}(q)$ (with odd q), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.

On approximately monotone and approximately Hölder functions

A real valued function f defined on a real open interval I is called $$\Phi $$ Φ -monotone if, for all $$x,y\in I$$ x , y ∈ I with $$x\le y$$ x ≤ y it satisfies $$\begin{aligned} f(x)\le f(y)+\Phi (y-x), \end{aligned}$$ f ( x ) ≤ f ( y ) + Φ ( y - x ) , where $$ \Phi :[0,\ell (I) [ \rightarrow \mathbb {R}_+$$ Φ : [ 0 , ℓ ( I ) [ → R + is a given nonnegative error function, where $$\ell (I)$$ ℓ ( I ) denotes the length of the interval I. If f and $$-f$$ - f are simultaneously $$\Phi $$ Φ -monotone, then f is said to be a $$\Phi $$ Φ -Hölder function. In the main results of the paper, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for $$\Phi $$ Φ -monotonicity and $$\Phi $$ Φ -Hölder property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper $$\Phi $$ Φ -monotone and $$\Phi $$ Φ -Hölder envelopes. We also introduce a generalization of the classical notion of total variation and we prove an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.