Mendelian Dynamics and Sturtevant ’ s Paradigm
This a brief introduction to the formal genetics centered around two mathematical ideas going back to Gregor Mendel and Alfred Sturtevant. Preambule. The road from biology (or from any branch of science on the fundamental level) to mathematics goes in several ( often Brownian rather than straight) paths in parallel. • Identifying a class of phenomena–”particular trees in a forest”–that appear with a regularity suggesting an underlying (mathematical) structure. (Are there non-mathematical structures?) •• Designing and performing experiments/observations purifying and amplifying what is seen by the naked eye (e.g. by planting our ”trees” to an ”artificial soil”). •••Making (often implicitly) ad hoc hypotheses, (e.g. continuity, symmetry, functoriality)– that provide a logical framework for the experimental data. For example, the ”theory of coin tossing” derives its mathematical beauty and the (probabilistic) predictive power not from such ”definitions” as ”the probability is a measure of uncertainty” but from the assumption that the the probability distribution on the space Z2 of the imaginary outcomes (binary nsequences) equals the (normalized) Haar measure that is, moreover, invariant under the permutation group Sn. Such hypotheses (assumptions), fragments of the grammer of the language in which Nature delievers her messages, are what a mathematician is primerly interested in, while a scientist is concerned with the ”meaning” of a message–the structure that is harder to formilize. Desiphering the grammer of Nature, or, biologically speaking, guessing the design of a seed by looking at (sample branches of) the grown tree, is rarely (if ever) done by mathematicians (Newtons do not count), even when the experimental data are abundant (as in the present day molecular biology). The past mathematical experience channals your imagination toward the old rather than new mathematical concepts. Even rigorously reformulating such hypotheses is not a straighforward task. For example, the Dirac δ-function needed the theory of distributions to be accepted by mathematicians and the functoriality of the (derivation of the) Boltzmann equation was recognized (albeit not much exploited till now) only with the advent of the ”functoriality paradigm” within pure mathematics. When a ”seed” is cultivated in a ”mathematical soil”, what grows out of it– mathematician’s tree–might look not quite as the real one. But a math-
MATHEMATICAL CONCEPTS OF EVOLUTION ALGEBRAS IN NON-MENDELIAN GENETICS
Evolution algebras are not necessarily associative algebras satisfying eiej = 0 whenever ei, ej are two distinct basis elements. They mimic the self-reproduction of alleles in non-Mendelian genetics. We present elementary mathematical properties of evolution algebras that are of importance from the biological point of view. Several models of Mendelian [2, 4, 12, 6, 8, 11] and non-Mendelian genetics [1, 5] exist. Based on the self-reproduction rule of non-Mendelian genetics [1, 7], the first author introduced a new type of algebra [10], called evolution algebra. In this paper we discuss some basic properties of evolution algebras. 1. Evolution algebras and subalgebras Let K be a field. A vector space E over K equipped with multiplication is an algebra (not necessarily associative) if u(v +w) = uv +uw, (u+ v)w = uw + vw, (αu)v = α(uv) = u(αv) for every u, v, w ∈ E and α ∈ K. Let {ei; i ∈ I} be a basis of an algebra E. Then eiej = ∑ k∈I aijkek for some aijk ∈ K, where only finitely many structure constants aijk are nonzero for a fixed i, j ∈ I. The multiplication in E is fully determined by the structure constants aijk, thanks to the distributive laws. Let E be an algebra. Then F ⊆ E is a subalgebra of E if F is a subspace of E closed under multiplication. It is not difficult to show that the intersection of subalgebras is a subalgebra. Thus, given a subset S of E, there is the smallest subalgebra of E containing S. We call it the subalgebra generated by S, and denote it by 〈S〉. As usual: Lemma 1.1. Let S be a subset of an algebra E. Then 〈S〉 consists of all elements of the form α1(s1,1 · · · s1,m1)+ · · ·+αk(sk,1 · · · sk,mk), where k ≥ 1, 1991 Mathematics Subject Classification. Primary: 17A20, 92C15; Secondary: 20N05, 37N25.
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...