The value VF,n(a0, . . . , an, b) = a0 + a1b+ a2b 2 + · · ·+ anb of a polynomial of degree n ≥ 1 over a field F can be computed by Horner’s rule using no more than nmultiplications in F , and it is optimal (for many fields, including the reals R and the complexes C) for the number of multiplications and/or divisions required, cf. Pan [1966], Winograd [1967, 1970]. We show that Horner’s rule is ...

The index of a pseudo B-Fredholm operator will be defined and generalize the usual operator. This concept used to extend some known results in Fredholm’s theory. Among other results, nullity, deficiency, ascent descent extended for pseudo-Fredholm

In this article, we formalize in Mizar [7] the definition of “torsion part” of Z-module and its properties. We show Z-module generated by the field of rational numbers as an example of torsion-free non free Z-modules. We also formalize the rank-nullity theorem over finite-rank free Z-modules (previously formalized in [1]). Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lo...

The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implie...

Let A be a bounded operator on a separable, infinite dimensional Hilbert space H. Moreover, assume that A is not in the set C + K(H) of operators expressible as a sum of a scalar multiple of identity and a compact operator. What is the smallest similarity invariant semigroup containing operator A? Equivalently, which operators can be expressed as products of operators, similar to A? A partial a...

Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgra...