Hilbert problem

With the following words, David Hilbert opened the Second International Congress of Mathematicians in Paris in the year 1900  

We will, but only briefly, comment on five of the 23 Hilbert problems, just to throw some light on a part of mathanatics of which we in chanistry rarely have the opportunity to hear about, yet nevertheless, we can appreciate their content to a degree. These are problems 3,6,7, 8, and 18. Let start with 3  

Hilbert Problem 3 Given any two tetrahedra and Tj with equal base area and equal height (and therefore equal volume), is it always possible to find a finite number of tetrahedra, so that when these tetrahedra are glued in some way to Tj and also glued to T2, the resulting polyhedra are scissors congruent  

Hilbert Problem 6 Mathematical treatment of the axioms of physics. 

Hilbert s problem 6 is really more about theoretical physics than mathematics. Fully stated, it is as follows  

---

Rigorous foundation of Schubert s enumerative calculus, Math. Development from Hilbert problems, Proceedings of Symposia in pure Mathematics of the AMS, Volume 28 (1983), 445-482. 

The problem of higher dimensional crystallographic groups has been formulated as problem 18 of the famous Hilbert problems. Hilbert in 1900, proposed 23 fairly general problems to stimulate mathematical research. In 1910, this problem was solved by Bieberbach He proved that in any dimension that there were only finitely many groups [20]. Around 60 years later it was found that there are 4783 groups in the four-dimensional space. 

Hilbert Problem 7 Irrationality and transcendence of certain numbers. 

Hilbert Problem 8 Problems (with the distribution) of prime numbers. 

Hilbert Problem 18 Building up space from congruent polyhedra. 

Our method of attacking plane, non-inertial problems will be, in the first instance, to reduce (2.8.9) to a Hilbert problem, in precisely the manner developed by Muskhelishvili (1963), and then to handle the specifically viscoelastic aspects, essentially by the methods outlined in Sects. 2.4-6. We remark that an alternative way of approaching the first stage is the dual integral equation method originally used in this context by Sneddon (1951) but with a long history of mathematical development summarized by Gladwell (1980). 

Our object is to apply (2.8.9) to problems involving loads on viscoelastic halfspaces. For such problems, it is desirable to re-express these equations in an alternative form [Muskhelishvili (1963)] which facilitates reduction to a Hilbert problem. Let the material occupy the upper half-plane > >0 so that (piz, t) is analytic in this region. It is convenient to extend the region of analyticity of (p(z, t) to the lower half-plane also. Then, as we shall see, it is possible to explore the discontinuities in this function across the real axis, which gives a Hilbert problem. Another approach, possibly more direct, is that of Galin, mentioned previously, which leads to problems of the Riemann-Hilbert type. These however are somewhat more difficult to deal with, from a mathematical point of view. 

Relation (3.3.2) constitutes a Hilbert problem if v(x,t) is known. This however is not generally the case for problems with varying boundary regions, which is the source of the added difficulty of non-inertial viscoelastic problems over elastic problems. Nevertheless, in this section, we will proceed as if v(x,t) were known. 

We now write down a general solution to (3.3.2), based on the solution of the Hilbert problem, given in Sect. A2.3. For the moment, we allow the possibility of singularities at the end points of the contact region. 

Consider the case where the contact area C(t) is contracting, or at least nonexpanding, for all t. This is the region on which displacement, or rather its derivative, is specified. It follows that v x, t) is known in C(t), so that the general solutions of the Hilbert problem (3.3.2) discussed in Sect. 3.3 are in fact final solutions of the problem. These are identical in form to the corresponding elastic solutions but where v x,t) takes the place of the displacement derivative. This is a special case of the Extended Correspondence Principle discussed in Sect. 2.6. 

The problem of the linear relationship may be stated as follows given the equation connecting the boundary values of an analytic function on a line of discontinuity together with some condition on the function at infinity, determine the analytic function, either uniquely or very nearly uniquely. This has been termed the Hilbert problem by Muskhelishvili (1953, 1963) and the Riemann problem by Gakhov (1966). We adopt the former terminology here. 

A closely related problem is given a linear relationship between the real and imaginary parts of the function on the boundaries of a domain in which it is analytic, find the function throughout the domain. This is termed the Riemann-Hilbert problem by Galin (1961) and also here. 

So, in the case of the Riemann-Hilbert problem, a function is analytic in a given domain bounded by a particular contour, and information is supplied about the limiting values of the function on this contour. In the case of the Hilbert problem, the region of analyticity contains open or closed contours on which the function is not defined. These are cuts in the region of analyticity across which the function is discontinuous. The supplied information is a relationship between the limiting values on either side of these cuts or internal boundaries. 

Plane elastic, and in a partial sense, viscoelastic boundary value problems can often be phrased in terms of either a Hilbert or a Riemann-Hilbert problem. Galin (1961) for example, bases his whole approach on the latter method. On the other hand, Muskhelishvili (1963) manages to cast various boundary value problems in the form of Hilbert problems, thereby solving them. Gakhov (1966) discusses the connection between the two problems. The Hilbert approach is adopted here, on the grounds that the theory is a little easier, though this may be a matter of taste. 

The Riemann-Hilbert problem is therefore not discussed. The treatment of the Hilbert problem is not very general, being confined in particular to the case of constant coefficients. The reader is referred to the standard texts mentioned above for more complete treatments. 

The Hilbert problem with constant coefficient may now be posed. The discussion is based on that of Muskhelishvili (1%3). Given L, we seek a sectionally analytic function F(z) such that its limits at a point u on L, from, S , namely (m), F (m) obey the relation... 

Equation (A2.4.1) is a singular integral equation for 0(x) in terms of g(x). The principle value of the integral is understood. It will emerge that the solution of this integral equation is closely related to the solution of the Hilbert problem discussed in the previous section. 

The theory is developed without any serious attempt at mathematical rigour. However, we also avoid the use of merely heuristic arguments which are particularly common in the literature on fracture. The orientation of the book is applied mathematical, though with the ultimate aim of extracting physically interesting results. Certain required techniques and results, notably the statement and solution of the Hilbert problem and the use of Hilbert transforms, are discussed in several mathematical appendices. Short tables of integrals and other relations are included. 

---