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## Prolongement d'un morphisme d'espaces analytiques complexes

### Fonctions de Plusieurs Variables Complexes II (1975-01-01) 482: 274-281 , January 01, 1975

## A selection-migration model in population genetics

### Journal of Mathematical Biology (1975-09-01) 2: 219-233 , September 01, 1975

### Summary

We consider a model with two types of genes (alleles) A_{1} A_{2}. The population lives in a bounded habitat *R*, contained in *r*-dimensional space (*r*= 1, 2, 3). Let *u* (*t, x*) denote the frequency of *A*_{1} at time t and place *x* ɛ *R*. Then *u* (*t, x*) is assumed to obey a nonlinear parabolic partial differential equation, describing the effects of population dispersal within *R* and selective advantages among the three possible genotypes *A*_{1}*A*_{1}, *A*_{1}*A*_{2}, *A*_{2}*A*_{2}. It is assumed that the selection coefficients vary over *R*, so that a selective advantage at some points *x* becomes a disadvantage at others. The results concern the existence, stability properties, and bifurcation phenomena for equilibrium solutions.

## Newton’s Method

### Calculus I (1975-01-01): 67-71 , January 01, 1975

Many of the equations arising in practical problems are of a type difficult or impossible to solve by the standard algebraic methods. For example, the equations:
$$ 2\sin {\kern 1pt} x - x = 0{\kern 1pt} {\kern 1pt} {e^x} - 2x - 1 = 0,{\kern 1pt} {\kern 1pt} {x^6} - 3x + 1 = 0 $$
have roots which we may estimate, by graphing the functions and finding where the graphs cut the *x*-axis, but which we cannot find exactly. In these cases a numerical procedure known as Newton’s method allows us to use a value *x*_{0} which is an approximate root of the equation:
$$ f(x) = 0 $$
in order to obtain a *better* approximation *x*_{1}.

## The Schrödinger equation

### Besov Spaces and Applications to Difference Methods for Initial Value Problems (1975-01-01) 434: 132-151 , January 01, 1975

## Scharfn-fach transitive permutationsmengen

### Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (1975-08-01) 43: 144-145 , August 01, 1975

## A bound of the exterior arcs for a univalent mapping

### Mathematical notes of the Academy of Sciences of the USSR (1975-09-01) 18: 807-813 , September 01, 1975

In this paper we consider the intersection of the circle ¦w|=x with the image of the disc ¦z|≤r, 0<r<1, under the mapping of a function of the form*f*(z)=z+c_{2}z^{2}+... which is univalent analytic in ¦z|<1. Earlier I. E. Bazilevich proved that for x≥c^{π/e}r the measure of the above intersection does not exceed the measure of the intersection produced by the function*f*^{*}(z)=z/(1−*ηz*)^{2}, η¦=1. In this paper I. E. Bazilevich's ideas are used to strengthen some of his results.

## Partially ordered sets and the rogers-ramanujan identities

### aequationes mathematicae (1975-02-01) 12: 94-107 , February 01, 1975

## Associative products of graphs

### Monatshefte für Mathematik (1975-12-01) 80: 277-281 , December 01, 1975

It is shown that there exist exactly twenty products of graphs defined on the Cartesian product of the vertex sets of the factors where the adjacencies of the vertices only depend on the adjacencies in the factors.

## Iteration Rules with Weak Memory

### Stochastic Processes with Learning Properties (1975-01-01) 84: 62-77 , January 01, 1975

In this chapter we are concerned with constraints under which a broad class of machine learning procedures, governed by iteration rules with memory, converge. More distinctly, we assume iteration rules with weak (or just finite) memory with respect to previous data and previously proposed estimates. Procedures of this sort offer rich possibilities to find convergent learning algorithms with powerful processing steps, and reduce in this way the time necessary for learning. We prove, in what follows, relations of interest in devising such sort of iteration rules. Use of these results, e.g., for modifying convergent learning algorithms heuristically, while retaining their convergence, and other related topics are subjects we consider in Chapter 5.