It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from.

Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Möbius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

Let $\mathbb{K}$ be a field of characteristic zero, and $*\,=\,t$ the transpose involution for the matrix algebra ${{M}_{2}}\left( \mathbb{K} \right)$. Let $\mathfrak{U}$ be a proper subvariety of the variety of algebras with involution generated by $\left( {{M}_{2}}\left( \mathbb{K} \right),\,* \right)$. We define two sequences of algebras with involution ${{R}_{p}},\,{{S}_{q}}$, where $p,\,q\,\in \,\mathbb{N}$. Then we show that ${{T}_{*}}\left( \mathfrak{U} \right)$ and ${{T}_{*}}\left( {{R}_{p}}\oplus \,{{S}_{q}} \right)$ are $*$-asymptotically equivalent for suitable $p,\,q$.

Let $V$ be an analytic variety in some open set in ${{\mathbb{C}}^{n}}$ which contains the origin and which is purely $k$-dimensional. For a curve $\gamma $ in ${{\mathbb{C}}^{n}}$, defined by a convergent Puiseux series and satisfying $\gamma (0)\,=\,0$, and $d\,\ge \,1$, define ${{V}_{t}}\,:=\,{{t}^{-d}}\,\left( V\,-\,\gamma \left( t \right) \right)$. Then the currents defined by ${{V}_{t}}$ converge to a limit current ${{T}_{\gamma ,d}}\left[ V \right]$ as $t$ tends to zero. ${{T}_{\gamma ,d}}\left[ V \right]$ is either zero or its support is an algebraic variety of pure dimension $k$ in ${{\mathbb{C}}^{n}}$. Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragmén-Lindelöf condition that was used by Hörmander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on ${{\mathbb{R}}^{n}}$.

Let ${{\mathcal{H}}_{n}}$ be the real linear space of $n\,\times \,n$ complex Hermitian matrices. The unitary (similarity) orbit $\mathcal{U}\left( C \right)$ of $C\,\in \,{{\mathcal{H}}_{n}}$ is the collection of all matrices unitarily similar to $C$. We characterize those $C\,\in \,{{\mathcal{H}}_{n}}$ such that every matrix in the convex hull of $\mathcal{U}\left( C \right)$ can be written as the average of two matrices in $\mathcal{U}\left( C \right)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}\left( C \right)$, the convexity of images of $\mathcal{U}\left( C \right)$ under linear transformations, and some related questions concerning the joint $C$-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.

Let $f:\,E\,\to \,B$ be a dominant morphism, where $E$ and $B$ are smooth irreducible complex quasi-projective varieties, and let ${{F}_{b}}$ be the general fiber of $f$. We present conditions under which the homomorphism ${{\text{ }\!\!\pi\!\!\text{ }}_{1}}\left( {{F}_{b}} \right)\,\to \,{{\text{ }\!\!\pi\!\!\text{ }}_{1}}\,\left( E \right)$ induced by the inclusion is injective.

Let $\phi :\,X\,\to \,M$ be a morphism from a smooth irreducible complex quasi-projective variety $X$ to a Grassmannian variety $M$ such that the image is of dimension ≥ 2. Let $D$ be a reduced hypersurface in $M$, and $\gamma $ a general linear automorphism of $M$. We show that, under a certain differential-geometric condition on $\phi (X)$ and $D$, the fundamental group ${{\text{ }\!\!\pi\!\!\text{ }}_{1}}\left( {{\left( \gamma \,o\,\phi \right)}^{-1}}\,\left( M\,\backslash \,D \right) \right)$ is isomorphic to a central extension of ${{\pi }_{1}}\left( M\,\backslash \,D \right)\,\,\times \,{{\pi }_{1}}\left( X \right)$ by the cokernel of ${{\pi }_{2}}\left( \phi \right)\,:\,{{\pi }_{2}}\left( X \right)\,\to {{\pi }_{2}}\left( M \right)$.

James gave an integral homotopy decomposition of $\sum \Omega \sum X$, Hilton-Milnor one for $\Omega (\sum X\,\vee \,\sum Y)$, and Cohen-Wu gave $p$-local decompositions of $\Omega \sum X$ if $X$ is a suspension. All are natural. Using idempotents and telescopes we show that the James and Hilton-Milnor decompositions have analogues when the suspensions are replaced by coassociative $\text{co-}H$ spaces, and the Cohen-Wu decomposition has an analogue when the (double) suspension is replaced by a coassociative, cocommutative $\text{co-}H$ space.

Let ${{l}^{\Phi }}$ and ${{L}^{\Phi }}\left( \Omega \right)$ be the Orlicz sequence space and function space generated by $N$-function $\Phi (u)$ with Orlicz norm. We give equivalent expressions for the nonsquare constants ${{C}_{J}}\left( {{l}^{\Phi }} \right),\,{{C}_{J}}\left( {{L}^{\Phi }}\left( \Omega \right) \right)$ in sense of James and ${{C}_{S}}\left( {{l}^{\Phi }} \right),\,{{C}_{S}}\left( {{L}^{\Phi }}\left( \Omega \right) \right)$ in sense of Schäffer. We are devoted to get practical computational formulas giving estimates of these constants and to obtain their exact value in a class of spaces ${{l}^{\Phi }}$ and ${{L}^{\Phi}}\left(\Omega \right)$.