We study the existence and uniqueness of non-negative smooth bounded solutions to the Cauchy problem for a system of non-linear parabolic equations with monotone and concave non-linearities. Under certain conditions on the non-linear terms and the coefficients of the system, a constructive theorem ensuring the existence of such a solution is established. In addition, an equivalent system of non-linear Hammerstein-Volterra type integral equations is considered. For this system, a special iterative scheme is proposed, which, under an additional condition on the non-linearities, is shown to uniformly converge as geometric progression to a smooth solution of the original Cauchy problem. A solution is shown to be unique in the class of bounded smooth vector functions with non-negative components. Such non-linear parabolic systems are interesting not only in mathematics, but also in biochemistry and chemical kinetics. The paper concludes with applied examples satisfying all the assumptions of our theoretical results.
We solve the problem of unboundedness of the lengths of periods and quasiperiods of functional continued fractions constructed in the field of formal power series K((1/x)) for elements of a given hyper-elliptic field L defined over a field K of characteristic different from 2. It is proven that if in a hyperelliptic field L there exist elements with a periodic expansion into a functional continued fraction, then, for each integer N, there exists an element in the field L, whose continued fraction period length is greater than N. Moreover, it is shown that, under the assumption that L contains elements with periodic continued fraction expansions, the supremum over all elements of L of the period lengths in the sequences of degrees of partial quotients is unbounded.
The structure of periodic E-functions is described. It is proved that each such function f(z) can be represented as f(z) = e-N alpha zP(e alpha z), where N is a non-negative integer, alpha is an algebraic function, and P is a polynomial with algebraic coefficients.
We show that if an upper semi-continuous function $\varphi$ on $\Omega \subset \mathbb{C}^n$ satisfies some natural conditions, and for any non-negative psh function $\psi$ on $\mathbb{D}\times\Omega$, the fiberwise Bergman kernel associated with $p_2^*\varphi+\psi$ is log-psh, then $\varphi$ must be psh. This offers a version of converse to Berndtsson's theorem concerning log-plurisubharmonic variation of Bergman kernels and generalizes a previous result of Li-Zhou. Additionally, we obtain a parallel result for the positivity of Hermitian holomorphic line bundles on Kähler manifolds.
The article is concerned with Halpern's first-order logics of probability, which we denote by '1 and '2 - the first of these deals with probability distributions on the domain, while the second employs distributions on external sets of possible worlds. The proofs of [1] of the complexity lower bound results for '1 and '2 rely heavily on using polynomials. We shall obtain the same lower bounds for small fragments of '1 and '2 in which neither addition nor multiplication is allowed. Further, it will be studied what happens if we exclude field variables, and hence quantifiers over reals; the upper bound proofs here will utilize suitable analogues of the (downward) Lo & uml;wenheim-Skolem theorem.
