In this paper, we have established some noiseless coding theorems for a generalized parametric ‘useful’ inaccuracy measure of order $\alpha $ and type $\beta $ and generalized mean codeword length. Further, lower bounds on exponentiated useful code length for the best 1:1 code have been obtained in terms of the useful inaccuracy of order $\alpha $ and type $\beta $ and the generalized average useful codeword length.

We present simultaneous reduction algorithms for two (nonsymmetric) matrices $A$ and $B$ to upper Hessenberg and lower Hessenberg forms, respectively. One is through the simultaneous similarity reduction and the other is through a Lanczos–Arnoldi-type iteration. The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorithm and the standard Arnoldi algorithm. We shall also apply our reduction to construct a model reduction for certain kind second-order single-input single-output system. It is proved that the model reduction has the desirable moment matching property.

We explain the Dirac–Segal approach to quantum field theory. We study local observables in this approach and the theory of deformations. We found out that this theory of deformation in the second-order coincides with the renormalization of the same theory, would it be considered in Polyakov approach. We conjecture that it is still true to all orders.

The octonions are distinguished in the $M$-theory in which Universe is the usual Minkowski space ${\mathbb {R}}^4$ times a $G_2$ manifold of very small diameter with $G_2$ being the automorphism group of the octonions. The multidimensional octonion analysis is initiated in this article, which extends the theory of several complex variables, such as the Bochner–Martinelli formula, the theory of non-homogeneous Cauchy–Riemann equations, and the Hartogs principle, to the non-commutative and non-associative realm.

A number of results about deriving further Sobolev inequalities from a given Sobolev inequality are presented. Various techniques are employed, including Bessel potentials and Riesz transforms. Combining these results with the $W^{1,2}$ Sobolev inequality along the Ricci flow established by the author in earlier papers then yields various new Sobolev inequalities along the Ricci flow.