The Proof that the Standard Transformations of E and B are not the Lorentz Transformations. Clifford Algebra Formalism

In this paper it is exactly proved by using the Clifford algebra formalism that the standard transformations of the three-dimensional (3D) vectors of the electric and magnetic fields E and B are not the Lorentz transformations of well-defined quantities from the 4D spacetime but the 'apparent' transformations of the 3D quantities. Thence the usual Maxwell equations with the 3D E and B are not in agreement with special relativity. The 1-vectors E and B, as well-defined 4D quantities, are introduced instead of ill-defined 3D E and B.Comment: 16 pages, section 3.3. is added and it contains the same proof for Hestenes' and Jancewicz's E and

Controlled surgery and L\mathbb{L}-homology

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $\sigma^c (f, b) \in H_n (B, \mathbb{L})$, where $M^n, X^n$ are topological manifolds of dimension $n \geq 5$. Our proof uses essentially the geometrically defined $\mathbb{L}$-spectrum as described by Nicas (going back to Quinn) and some well known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describe the assembly map $H_n (B, \mathbb{L}) \rightarrow L_n (\pi_1 (B))$ in terms of forms in the case $n \equiv 0 (4)$. Finally, we explicitly determine the canonical map $H_n (B, \mathbb{L}) \rightarrow H_n (B, L_0)$

Global reachability of 2D structured systems

In this paper the new concept of 2D structured system is defined and a characterization of global reachability is obtained. This extends a well known result for 1D structured systems, according to which (A ,B ) is (generically) reachable if and only if the matrix [A B] is full generically row rank and irreducible