The development of finiteness in the Transeurasian languages

Nonfinite verb forms can gradually acquire morphological and syntactic properties of finiteness. Across the languages of the world, such developments can follow various pathways with various results. In this article, I first discuss the four major pathways mechanisms for developing finite function on formerly nonfinite forms. Next, I argue that the Transeurasian languages (i. e., Japanese, Korean, Tungusic, Mongolic, and Turkic languages) share a common mechanism. Historical reconstruction indicates that these languages all show the tendency to reanalyze directly a nonfinite verb form as a finite one, without the omission of a specific matrix verb. I refer to this tendency as “indirect insubordination”. I argue that the recurrence of indirect insubordination on formally related suffixes can be taken as an indication of common ancestorship.

On the Computational Complexity of Finite Operations

The essential variables in a finite function f are defined as variables which occur in f and weigh with the values of that function. The number of essential variables is an important measure of complexity for discrete functions. When replacing some variables in a function with constants the resulting functions are called subfunctions, and when replacing all essential variables in a function with constants we obtain an implementation of this function. Such an implementation corresponds with a path in an ordered decision diagram (ODD) of the function which connects the root with a leaf of the diagram. The sets of essential variables in subfunctions of f are called separable in f. In this paper we study several properties of separable sets of variables in functions which directly affect the number of implementations and subfunctions in these functions. We define equivalence relations which classify the functions of k-valued logic into classes with the same number of: (i) implementations; (ii) subfunctions; and (iii) separable sets. These relations induce three transformation groups which are compared with the lattice of all subgroups of restricted affine group (RAG). This allows us to solve several important computational and combinatorial problems.

Dynamics of $(e^{z} - 1)/z$: the Julia set and bifurcation

We describe the dynamical behaviour of the entire transcendental non-critically finite function $f_\lambda (z) = \lambda(e^z - 1)/z$, $\lambda > 0$. Our main result is to obtain a computationally useful characterization of the Julia set of $f_\lambda (z)$ as the closure of the set of points with orbits escaping to infinity under iteration, which in turn is applied to the generation of the pictures of the Julia set of $f_\lambda (z)$. Such a characterization was hitherto known only for critically finite entire transcendental functions [11]. We find that bifurcation in the dynamics of $f_\lambda (z)$ occurs at $\lambda = \lambda^{*}$ ($\approx 0.64761$) where $\lambda^\ast = {(x^{*})}^{2} /({e}^{x^{*}} -1)$ and $x^{*}$ is the unique positive real root of the equation $e^{x}(2 -x ) -2 = 0$.

Asymptotic Expansion of Solutions of Parabolic Equations with a Small Parameter

The heat equation with a small parameter, is considered, where ε ∈ (0, 1), 𝑚 < 1 and χ is a finite function. A complete asymptotic expansion of the solution in powers ε is constructed.