While the functions in (1.1) do not have ordinary derivatives at t = 0, they have symmetric derivatives: f s (0) = g (0) = h (0) = 0. For a deeper understanding of the symmetric derivative and its properties, we refer the reader to the specialized monograph [8]. Here we note that the symmetric quotient (f (t+ h)− f (t− h)) /(2h) has, in general, better convergence properties than the ordinary d...

we investigate buchbaum and eisenbud's construction of the second symmetric power $s_r(x)$ of a chain complex $x$ of modules over a commutative ring $r$. we state and prove a number of results from the folklore of the subject for which we know of no good direct references. we also provide several explicit computations and examples. we use this construction to prove the following ...

Okhotin showed an exponential trade-off in the conversion from nondeterministic unary automata to unambiguous nondeterministic unary automata. We show that the trade-off in the case of unary symmetric difference automata to finitely (structurally) ambiguous unary symmetric difference automata is linear with constant 1 in the number of states. In particular, for every n-state unary nondeterminis...

Abstract This article addresses the problem of computing an upper bound degree d a polynomial solution P ( x ) algebraic difference equation form G − τ 1 , … s + 0 = when such with coefficients in field K characteristic zero exists and where is non-linear s-variable [ ] . It will be shown that if quadratic constant then one can construct countable family polynomials f l u there (minimal) index ...