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Mathematics
Interesse 1 My interests as far as Science is concerned is almost only mathematics in the "pure" sense ! .
* Interesse 2 In addition to mathematics a little physics and other science.
Description of Hobbies , projects and scientific interests mainly in mathematics.
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LIST of Mathematical publications
1) Christensen, J.P.R. (1969), Uniform Measures,Proc. of the functional analysis week, March 3--7, Various publications series , Mathematics Institut,Århus University.
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The paper contains various results on uniformly distributed measures which are stronger than the results published below in 3) . The measures considered in 1) and in 3) have basically a stronger property than being uniformly distributed , a property which permits a theory of Spherical functions. On a metric space ( X ,d ) a Radon measure u is uniformly distributed ( by definition ) if the u measure of a ball depends only of the radius and not of the center. There need NOT be any Group structure , i.e. the group of isometrics might not be transitive even if a uniformly distributed measure exists . But the uniformly distributed measure are analogous to Haar measure t.ex. such a measure is unique up to a positive scalar factor !
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2)Christensen, J.P.R. (1970a),On some measures analogous to Haar measure, Math. Scand.103--196.
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This paper gives uniqueness theorems for uniformly distributed measures and some geometric properties of metric spaces on which such a measure exists . On a metric space ( X , d ) a Radon measure u is uniformly distributed ( by definition ) if the u measure of a ball depends only of the radius and not of the center. There need not be a transitive group of isometrics on X even if there is a uniformly distributed measure – if there are its unique up to multiplication with a positive scalar.
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3)Christensen, J.P.R. (1970b),Uniform measures and spherical harmonics, Math. Scand., 293--302.
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Here a Special class of uniformly distributed measures u on a metric space is studied, who has the property that the measure of the intersection of ball B1 with ball B2 is a function of the distance between the centers of B1 and B2 and the radius of B1 and B2 respectively. This property is MUCH stronger than being uniformly distributed and makes it possible to develop a theory of Spherical functions as functions which in their definition need no Group structure but only the measure and the metric . Still most classical theorems of Spherical functions carry over to this case!
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4) Christensen, J.P.R. (1971a),Borel structures in groups and semi groups,
Math. Scand.vol 28.pages 124--128.
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Some relations between the Borel structure and the Topology of Groups and semi groups of a particular type is studied with application to continuity of measurable homomorphisms . Theorems of locally compact groups may have analogues even for semi groups even if no invariant measure exists because the (semi) Group might NOT be locally compact !
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5)Christensen, J.P.R. (1971b),Borel structures and a topological zero -- one law, Math. Scand.
vol 29.pages 245--255.
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A similar result to 4) but much deeper is considered . Countably additive follows from finitely additive defined on a sigma-field AND "reasonable" measurability assumptions!oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
6)Christensen, J.P.R. (1971c),On Some Properties of Effros Borel Structure on Spaces of
Closed Subsets, Math.Ann.vol 195,pages 17--23.
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A typical result is that the space of compact subsets of a metric space X is an analytic space precisely when X is Polish . Similar but later results are due to the French mathematician J. Saint-Raymond which did obtain them independently ( different method ) and almost simultaneous! The theory gives one of the ( Today ) known ”simple” sets , which are analytic but NOT co-analytic , thus not Borel.
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7)Christensen, J.P.R. (1972a),Measurability Problems in a Metrical Convex Compact Set,
Israel Journal of Mathematics, Vol. 12, No 2.
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The set of primary points of a compact metrizable convex set in a locally convex Vector space is the complement of an analytic set hence universally measurable! Each point in the compact metrizable convex set has thus a desintegration after the primary points in the ordinary measure-theoretic sense – the concept of ”primary point ” is of course analogous to ”extreme point”.Although this ”should be” applied t.ex. in he classical theory of moments (??) good applications of this theory is only known in the theory of Operator algebras . What are the primary points in the set of solutions to a classical indeterminate moment problem (??) – that's a very good question!
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8)Christensen, J.P.R. (1972b),On Sets of Haar Measure Zero in Abelian Polish Groups, Israel
Journal of Mathematics, Vol. 13, Nos 3--4, 255--260.
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This gives the first and still the simplest concept of measure-theoretic zero set in a non locally compact Polish group. It is also the concept for which most results from the locally compact case carries over. Since this work those measure-theoretic null sets in arbitrary complete separable metric groups has found many striking applications , but open questions still abound!
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9)Andersen, N.J.M. and J.P.R. Christensen (1973a),Some Results on Borel Structures with
Applications to Sub series Convergence in Abelian Topological Groups, Israel Journal
of Mathematics, Vol. 15, No 4, 414--420.
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Strong results of the type that measurable finitely additive mappings are countably additive ! And this holds (almost ) no matter what groups the mappings takes values !
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10)Christensen, J.P.R. (1973b),Compact Convex Sets and Compact Choquet Simplexes,
Inventiones Math.vol 19.pages 1--4.
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A weak form of Nuclearity of ALL locally convex Frechet spaces is shown. Any compact
convex set is contained in the vector sum of 2 choquet simplexes!
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11)Christensen, J.P.R. (1973c),Measure Theoretic Zero Sets in Infinite Dimensional Spaces
and Applications to Differentiability of Lipschitz Mappings,
II -- Coll. Anal. Fonct. 29--39, Publications du Department de Math`ematiques, Lyon.
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A deeper study of measure theoretic thinness in non locally compact Polish groups. The concept is applied to automatic continuity AND to a non finite dimensional version of the classical Rademacher theorem on differentiability of Lipschitz mappings almost everywhere! Although the concept is analogous to first category , the topological concept of thinness , it is entirely different but a few properties are very similar!
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12)Christensen, J.P.R. (1973d),Necessary and Sufficient Conditions for the Measurability
of Certain Sets of Closed Subsets, Math. Ann.vol 200, pages 189--193.
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The results in 6) are considerably improved .
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13)Berg, Christian and J.P.R. Christensen (1974a),On the Relation
of Locally Compact Groups and the Norms of Convolution Operators,
Math. Ann.vol 208,pages149--153.
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Deliberately simplified : Amenability of a Locally compact group implies that the norm of the convolution with a non negative measure is the mass of that measure ! The converse ( suitably formulated ) is also true !
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14)Berg, Christian and J.P.R. Christensen (1974b),Sur la norme des op`erateurs de convolution,
Inventiones Math.vol 23,pages 173--178.
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Roughly speaking the converse of 13) is shown . Since semi simple Lie groups are NOT amenable , this gives a new result on convolution operators on semi simple Lie Groups and on the structure of convolution semi groups on such groups!
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15)Christensen, J.P.R. (1974d),Topology and Borel structure, MONOGRAPH - North--Holland
Mathematics Studies Nr. 10.
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This Doctors thesis collects and considerably improves upon results on the properties of Topology and Borel structure and their relations with many applications to automatic continuity of measurable additive measures. The concept of measure theoretic thinness in non locally compact groups is also discussed in great detail.
Many deep results are not published elsewhere!
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16)Christensen, J.P.R. (1975a),Submeasures and the Problem of the Existence of Control measure,
Proceedings from the conference on Measure theory in Oberwolfach.
Springer Lecture Notes in Mathematics Nr. 541.
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This is used to give considerable progress in the solution of the Maharam control measure problem . Most literature in the field uses this article at some point their work ! But ALAS the Maharam control measure problem is still unsolved and might be as difficult as the Riemann-conjecture!? Recently it has been solved in the negative by Michel Talagrand !!
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17)Christensen, J.P.R. and Wojchiech Herer (1975b),On the Existence of Pathological Submeasures and the Construction of Exotic Topological Groups, Math. Ann.vol 213,pages 203--210.
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On the existence of an abelian Polish groups G with the property , that all positive definite continuous functions on G are constants , are constructed using a sub measure who is a type of counterexample to the Maharam conjecture (but without sequential point continuity - thus not really a counterexample!). The basic idea of the proof is some variant of the Cantor Diagonal argument !
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18)Christensen, J.P.R. (1976a),Codimension of some Subspaces in a Fr`echet Algebra,
Proceedings of the American Mathematical Society Vol. 57, Number 2.
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A problem due to Curtis from automatic continuity is solved essentially using results from 15) .
If the co dimension of some algebraically defined subspaces is at most countable it is finite
(under some descriptive properties of the spaces involved). This solves at once a seemingly
difficult problem from automatic continuity ( automatic continuity of some point derivations
under assumption that they form a finite-dimensional linear space! ).
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19)Christensen, J.P.R. (1976b),Positive Definite Functions on Abelian Semi groups,
Math. Ann.vol 233,pages 253--272.
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Using classical Choquet theory disintegrations of various classes of functions on abelian
semi groups are studied. This paper were a predecessor to the classical monograph 39)
although of course with MUCH fewer results !
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20)Christensen, J.P.R. (1977),Some results with relation to the control measure problem,
Proceedings from the conference on Vector Space Measures and applications. Springer
Lecture Notes in Mathematics 645.
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Still some progress ( if modest ) in the Maharam control measure problem . Today Michel Talagrand has solved this problem in the negative !
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21)Christensen, J.P.R. (1978a),The Small Ball Theorem for Hilbert Spaces, Math. Ann.vol 237,
pages 273--276.
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The first result of its type - that a measure is determined by its value on small balls for any Hilbert space - and some related results . The Geometric measure theory on INFINITE dimensional Hilbert spaces is indeed VERY different from finite dimensional spaces and most finite dimensional results are NOT valid – but still a measure is uniquely determined by its values on ”small” balls !
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22)Christensen, J.P.R. and Paul Ressel (1978b),Functions Operating
on Positive Definite Matrices and a Theorem of Schoenberg,Transactions of the American Mathematical Society. Volume 243,p. 89--95.
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This extends substantially earlier results of the same type obtained by the authors together with Berg . Those results are t.ex. discussed in Chapter 5 in the monograph 39) .
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23)Christensen, J.P.R. (1979a),A Survey of Small Ball Theorems and Problems, Proceedings of the conference on Measure theory in Oberwolfach. Springer Lecture Notes in Mathematics Nr. 794.
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Improvements ( if slight ) of results in 21) . The subject can now be called Geometric measure theory in INFINITE dimensional Banach spaces . Most classical results are wrong but a few is still valid but with entirely different proofs....
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24)Christensen, J.P.R. and J. Westerstrøm (1979b),A Note on Extreme Positive Definite Matrices,
Math. Ann.vol 24
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In many closed convex sets of matrices the structure of extreme points is a mystery. The paper gives some positive results with bounds of rank and other bounds for extremal matrices of a special type. The set of Primary points is indeed an even greater mystery. In particular what is a primary point in the set of solutions to an indeterminate case of the classical moment problem ??
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25)Berg, C., J.P.R. Christensen and C.U. Jensen (1979c), A Remark on the Multidimensional Moment Problem, Math. Ann.vol 243,pages 163-169.
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The main result is perhaps the simplest possible positive polynomial of 2 variables (with integer coefficients) who is NOT a finite sum of squares of polynomials . This is applied to disintegration of positive definite multisequences.
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26)Berg, Christian and J.P.R. Christensen (1981a),Density Questions in the Classical Theory of Moments , Extrait des Annales Tome XXXI, Fascicule 3.
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Density for various spaces of polynomials with respect to a p-norm (wrt. a measure) is studied and some preliminary results obtained. Recently other authors has made considerable progress in the subject with different methods.
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27)Christensen, J.P.R. (1981b), Joint Continuity of Separately Continuous Functions , Proceedings of the American Mathematical Society, Volume 82, Number 3, p.455--461.
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The classical topological game of Banach-Mazur as modified by Gustave Choquet is still further modified in various ways to be applied to give very simple and short proof of much improved versions of theorem of the type that separate continuity implies joint continuity in a topologically FAT set . The class of spaces for which such theorems holds is greatly extended . The ( female ) player beta STARTS the game by choosing an open set in a Hausdorff topological space (X, P) and the ( male ) player Alfa chooses an open subset of the set previously choosen by beta ( his choice is restricted in various ways in various versions of the game adapted for specific applications ) . The male wins if the intersection of the countably many sets created this way is non empty . It turns out that the topological spaces suitable for various proofs are those spaces
which has some advantage for the male in various modifications of the above game. The strongest assumption is that of being a Male-chauvinist space , this is more then really needed but stable under products! To be Anti-feminist is weaker but sufficient for most proofs – on the other hand it is not stable under products . Let us define Male-chauvinist space ( or strongly Alfa-favorable ) : The modification of the original Banach-Mazur game described above is that the Male Alfa chooses a point in his set . The Female beta might try to "escape" the points determined by the male in the course of the game . The Male wins ( by definition ) if his sequence formed in the course of the game accumulates to at least one point of the intersections of all open sets formed by the Male ( or Female which is here the same ) during the course of the game. If the Male has a winning strategy depending ONLY on the previous move by the Female , the space X is called Male-chauvinist . If for any strategy of the Female depending on the first finitely many moves THERE EXISTS a game consistent with the rules in which the Male wins the space X is called Anti-feminist (or weakly beta-defavorable ) ! The last weaker requirement suffices for most proofs but is not stable under topological products as are the first !
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28)Christensen, J.P.R. and Paul Ressel (1981c), A Probabilistic characterization of Negative Definite and Completely Alternating Functions, Z. Wahrscheinlichkeitstheorie und verwandte Gebiete, vol 57,pages 407--417.
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An inequality of a type studied by Hoeffding characterizes special classes of functions on semi groups. The inequality forces the representing measure to be supported by a certain set ! This theory is further developed in Chapter 7 in the monograph 39) . Recent results by Torben Maack Bisgaard has shown that many of those results are optimal !
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29)Christensen, J.P.R. and J.K. Pachl (1981d),Measurable Functionals on Function Spaces, Extrait des Annales de L`institut Fourier de L`universit`e de Grenoble, Tome XXXI, Fascicule 2.
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Measurable linear functionals on function spaces with respect to a Borel structure induced by measures are induced by measures on the set on which the functions are defined ( in a very large class of cases ) !
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30)Christensen, J.P.R. (1982a),Theorems of Namioka and R.E. Johnson Type for Upper
Semi continuous and Compact Valued Set-Valued Mappings, Proceedings of the American Mathematical Society, Volume 86, Number 4,p. 649--655.
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Considerable strengthening of results in a very preliminary form in 27) . Under general conditions minimal upper semi continuous compact valued ( non empty compact sets as values ) are single point valued and continuous in a topologically FAT set. This gives some type of continuous selection theorem for VERY non linear multi-valued mappings . Can be applied t.ex. to metric projection and other multivalued mappings!
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31)Christensen, J.P.R. and Paul Ressel (1982b),Positive Definite Kernels on the Complex Hilbert Sphere, Math.Z.,vol 180,pages 193--201.
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The results in 28) are greatly improved upon. Complex versions of those results are given !
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32)Christensen, Jens Peter Reus and Paul Ressel (1982c), Norm-dependent positive definite functions on Banach-spaces,Probability in Banach spaces, IV, Oberwolfach.
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For any INFINITE DIMENSIONAL Banach space a simple disintegration of such functions are given. The norm is to be composed with a function f of the variable x which is an integral mixture ( after the parameter t ) of functions of the type exp(-|t|*x^2) !
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33)Berg, Christian and Jens Peter Reus Christensen (1983a),Exposant critiques dans
le probl`eme des moments. C.R. Acad. Sc. Paris,vol 296,pages 661--663.
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Measures with a prescribed critical exponent of a particular type is constructed. Other authors have since considerably improved this result!
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34)Berg, Christian and Jens Peter Reus Christensen (1983b),Suites complement definies
positives, majoration de Schur et le probl`eme des moments de Stieltjes en dimension
k, C.R. Acad. Sc. Paris,vol 297,pages 45--48.
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For many "desirable" theorems counterexamples are given in dimension k =2 already.
This depends on polynomials in the tribe of the one studied in 25) .
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35)Christensen, J.P.R. (1983c),Dense Fr`echet Differentiability of Mackey Continuous
Convex Functions , Comptes rendus de l`Acad`emie bulgare des Sciences, Tome 36, No 6, 737--738.
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The results of 27) and 30) is applied to study sub differentials of convex functions and
strong results obtained . The subject can now be called Non-linear automatic continuity!
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36)Christensen, J.P.R. (1983d),Surwey on Density Questions in the Classical Theory of Moments,
Proceedings of the Conference Topology and Measure IV, Trassenheide GDR,
Greifswald, Part 1, pp56-59.
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This elaborates on the results of 26) and 33) .
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37)Christensen, J.P.R. (1983e), Remarks on Namioka Spaces and R.E. Johnson's Theorem on the Norm Separability of
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Further improvements essential for applications are given of the results in 27) and 30 ) .
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38)Christensen, J.P.R. and Petar Stojanov Kenderov (1984a), Dense Strong Continuity of Mappings and the Radon--Nikodym Property, Math. Scand. vol 54,pages 70--78.
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Substantial applications of 37) are given .
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39)Berg, Christian, Jens Peter Reus Christensen, Paul Ressel (1984b), Harmonic Analysis on Semigroups,Theory of Positive Definite and Related Functions, - MONOGRAPH - Graduate texts in Mathematics Nr. 100.
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This is a classical monograph of Radon measures on Hausdorff spaces following Topsoes program of proving all main theorems using the Kisynski lemma (exposition in Chapter 2 ) , and of Harmonic analysis on commutative semi groups with involution.
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40)Christensen, Jens Peter Reus (1984c), A short survey on modern nonlinear automatic continuity theory . Israel seminar on geometrical aspects of functional analysis, Tel Aviv University.
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The point of view is given that the modified Choquet game is a powerful tool in what could be called Non-linear automatic continuity .
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41)Christensen, Jens Peter Reus (1984d),Geometric measure theory in infinite dimensional Banach spaces. Israel seminar on geometrical aspects of functional analysis, Tel Aviv University .
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Almost all results in classical geometric measure theory are FALSE in infinite dimension but still a few .....
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42)Christensen, J.P.R. and Pal Fisher (1986), Positive definite doubly stochastic matrices and extreme points, Journal of Linear Algebra and Applications.
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Extreme points of a new class of matrices are given.
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43)Christensen, J.P.R. and Pal Fisher (1987), Small sets and a class of general functional
equations, Aequationes Mathematicae,vol 33,pages 18--22, Birkhauser Verlag, Basel.
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Measure theoretic thin sets are applied to automatic continuity theorems of a new type !
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44)Berg, Christian, J.P.R. Christensen and P.H. Maserick (1988a), Sequences with Finitely Many Negative Squares , Journal of functional analysis, Vol. 79, No 2, August 1988, p.260--287.
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Moments sequences and closely related sequences which are integer valued turns out to have
spectacular properties . An optimal result of this type is discussed ! Under a suitable boundedness condition ANY such sequence is the coefficients of a rational function - a strange result on the border of analysis and algebraic number theory !! Raises questions !!
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45)Christensen, J.P.R. and Pal Fischer (1988b), Linear Independence of Iterates and entire Solutions of Functional Equations , Proceedings of the American Mathematical Society, Volume 101, Number 4, August 1988, 1120--1124.
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Most interesting Functional equations such as the Feigenbaum equation has NO non trivial entire functions as solutions because the 2 sides of the equation would have a growth with effectively different speed!
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46)Christensen, J.P.R. and Pal Fisher (1988c) , Joint continuity of measurable biadditive
mappings, Proceedings of the American Mathematical Society, Volume 103, Number 4,
August 1988,p.1125--1128.
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Strong results similar to 43) .
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47)Christensen, J.P.R. and B.J. Oommen (1988d),Epsilon-Optimal Discretized Linear
Reward-Penalty Learning Automata, IEEE May/June 1988, Volume 18, Number 3,
p451--458, (ISSN 0018-9472).
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A learning automaton of a special type is studied and its convergence properties are derived .
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48)Christensen, J.P.R. and Carsten Rasmussen (1988e),Axiomatic characterization of
Inhomogeneous Poisson Point Processes (IPPP) their mixtures and random translations,
Published in Volume in Contemporary Mathematics ``Measure and Measurable Dynamics´´ in honor of Dorothy Maharam Stone.
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There is a one to one correspondence between Radon measures u and Poisson point processes , when u is intensity of the process . This is used to simplify and improve the proof of some equations in Telephone-traffic theory.
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49)Christensen, J.P.R. and Carsten Rasmussen (1989),On point processes which are IPPP at each point but not globally IPPP,their infinite divisibility and mixing. Proceedings of NTS in Helsinki (1989).
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The results of 48) are further studied . A point process can be locally a Poisson process
but not globally !
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50)Christensen, Jens Peter Reus and B.J. Oommen (1990),Epsilon-optimal Stubborn Learning
Mechanisms, IEEE Transactions on Systems ,Man, and Cybernetics ,September/October
1990 Volume 20 Nr. 5 ISSN 0018-9472 .
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A deeper and more extensive convergence theorem than the one in 47) is proved ! Convergence is independent of with what probability the linear updating is started!
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51)Christensen , Jens Peter Reus and Pal Fischer(1993), MULTIDIMENSIONAL STOCHASTIC MATRICES AND p183: p255-276 (1993) .
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Important inequalities for linear AND non linear codes!
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52)Christensen, Jens Peter Reus and Pal Fischer (1994), LINEAR INDEPENDENCE OF
ITERATES AND MEROMORPHIC SOLUTIONS OF FUNCTIONAL EQUATIONS, PROCEEDINGS OF AMERICAN MATHEMATICAL SOCIETY Volume 120, Number 4 April (1994).
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The results of 45) are improved and similar new ones studied !
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53)Christensen, Jens Peter Reus and Pal Fischer (1996),Ergodic invariant probability measures and entire functions , ACTA MATHEMATICA HUNGARICA 73 (3) p 213 - 218 (1996) !
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Any non linear entire function f ( being a polynomial or not ) has in a sense "too many" invariant non discrete ergodic probabilities with compact invariant support K where K is an invariant subset of the Julia set J(f) for f . The idea of the proof uses a lemma due to Ahlfors and Andre Bloch to replace the property of being a polynomial – it is probably relatively easy to prove for polynomials !
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54) Christensen , Jens Peter Reus and Vladimir Kanovei , Michael Reeken , On Borel orderable groups ,TOPOLOGY AND ITS APPLICATIONS 109 (2001) 285-299 Elsevier
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A serie of results on Borel abelian ordered groups are presented and discussed
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55) Christensen , Jens Peter Reus , A Finitely Additive Measure Defined on a Sigma-field is Automatically Countably Additive , Atti Sem. Mat . Fis. Univ . Modena, IL, 509-511 (2001)
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The statement of the title is formulated and proved ! The result was presented at the
Ninth Meeting on Real Analysis and Measure Theory held in Grado during September 15-19 (2000).
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56)Christensen , Jens Peter Reus and Zoltan Sasvari , The Dimension of the Linear Space Spanned by ALL Partial Derivatives of a Symmetric Polynomial, Math. Nachr. 242 (2002), p 1-4 .
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A simple formula is proved for the dimension of the linear space spanned by all partial
derivatives of a symmetric polynomial .
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MISCELLANEOUS OPEN PROBLEMS IN RANDOM ORDER WHICH INTEREST ME OR
TO WHICH I HAVE CONTRIBUTED:
Problem 1) THE CONTROL MEASURE PROBLEM
This problem seem to be due to Dorothy Maharam and is at least 50 years old .
Perhaps it may be considered as the Riemann-problem of measure theory ?
Let ( X, Bo ) be a set X equipped with a Boolean algebra Bo of subsets of X
on which is defined a SUB MEASURE m i.e. a set-function satisfying the axioms
1) Increasing: A < B implies m(A) < m(B) ! 2)Subadditive : m(A union B) < m(A)+m(B) . It is part of the definition of Sub measure m that on the empty set Ø we have m( Ø )=0 ! The Sub measure m is called Normalized if m(X)=1 ! Continuous (which shall always mean sequential point continuous in the Dorothy Maharam sense unless specified otherwise ) means that if A1,A2,A3,….. is a sequence of sets in Bo whose indicator functions tends point wise to 0 then m(An) tends to 0 ! The only other continuity assumption of interest in this theory is Uniformly Exhaustive i.e. for all e > 0 there is a natural number N=N(e) such that for any set of N pair wise disjoint sets A1,A2,…,AN there is one of those sets on which m is less then or equal to e !
There is many equivalent formulations of the Control measure problem . It may be stated as the question whether a continuous Sub measure m did admit a positive measure u dominated by it (or with same nulsets in the sigma-field generated by Bo) ! Any who wants to study it can find the best and most easily readable account of the recent progresses in Alain Louveau`s basic paper referred below and in the literature to which he refers !Some researchers , such as this humble author , has tried to add the further assumption to the sub measure m of being defined on the sigma-field of Borel subsets of an abelian compact metric group G AND being translation invariant . No big progress ( it should dominate a multiple of the Haar measure if the world is reasonable and it is rather strange if not but …). The best positive result is the positive solution of the problem
for uniformly exhaustive sub measures ( due originally to Kalton and Roberts , see references in the work below ). It is not known whether a continuous Sub measure is automatically uniformly exhaustive and in fact this seems precisely to be the problem !
Michel Talagrand has recently shown that a continuous Sub measure does NOT NEED to be uniformly exhaustive !!!
ALAIN LOUVEAU ; PROGRESS RECENTS SUR LE PROBLEME DE MAHARAM
D`APRES N.J.KALTON ET J.W.ROBERTS ;Seminaire Initiation à l 'Analyse .
G.CHOQUET , M.ROGALSKI, J. SAINT RAYMOND 23 année , 1983/84 , n 20,
8 p. 24 Maj 1984.
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Problem 2) The various problems on Ergodic invariant probabilities for Entire functions. Those problems takes their points of departure in 53) in the publication list above . The paper gives 2 families of Ergodic invariant probabilities for a non-linear entire function f mapping the complex plane into itself – the first family ALWAYS is non empty and compactly supported . The second family ( boundary measures ) might be empty (??) and might have unbounded support ! Other families may exists ?? Do there always exists an Ergodic invariant probability measure for f whose topological support is the whole Julia set J(f) for f ?? For the special entire function z à exp(z) it is well known that the Julia set is the
whole complex plane !!But it is NOT known ( seemingly ! ) whether t.ex. there is an Ergodic exp invariant probability measure on the complex plane with density w.r.t. Lebesgue measure ( or dense support) ??
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Problem 3) Various problems on measures and Balls.
The first of those problems is whether or not in a complete separable
( or compact ) metric space a measure (positive countably additive
finite Borel measure ) is uniquely determined by its values on balls !
After many erroneous proof of that false statement Roy O. Davies
gave the first known counterexample . See in the first papers how
t.ex. existence of a uniformly distributed measure forces enough
structure on the space for this problem to have a positive solution –
it also has a positive solution for a bounded subset of a Hilbert space
but only if we know the value of the measure for all sufficiently small
balls centered in the Hilbert space. Perhaps it is likely that if this is
only known for balls centered in the support of the measure the answer
is negative (???) see 21) in the list above !! The simplest entirely open
problem in this area is whether or not a ( positive ) measure u is
uniquely determined by its values on intersections of precisely
2 balls in a (compact ) metric space (X , d ) ? This is likely to be false
but very close to correct ?
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Problem 4 ) Various problems on the properties of extreme rays in certain convex cones and extreme points in convex sets !
A lot of interesting and mostly open problems of this type is inspired of 24) and 42) above ! Just to mention one case of a convex cone the structure of whose extreme rays are not known and where even a partial knowledge would be extremely interesting , let us consider the Cone P-plus of all everywhere non-negative polynomials in n variables ( only for n=1 something non trivial is known !) . A non-negative polynomial is a finite sum of squares of rational functions ( which for n=1 can be chosen to be polynomials ) . This ought to help a lot in identifying the extreme rays in P-plus , but nothing seems to be known ! Another cone whose extreme rays WOULD be highly interesting to identify is the cone of all multisequences in n variables ( with n > 1 ) which are positive definite in the Hamburger sense ?? One case of a ( compact ) convex set where the extreme points are not known ( but a little information is known ) is the case of all doubly stochastic positive definite matrices of order n !
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Problem 5) A problem on pairwise orthogonal Latin squares.
There is known some examples on pairs of orthogonal Latin squares of order 10 and even a case WHICH CANNOT be extended to a triple of pairwise orthogonal Latin squares of order 10 ! But it seems to be entirely open iff there exists a triple of pairwise orthogonal Latin squares of order 10 . This could be withinn reach of computation but even with a supercomputer perhaps is not unless some VERY substantial work reducing ideas are invented . The problem is equivalent with the question if a Code C of words of length 5 (!!) and alfabet {1,2,3,4,5,6,7,8,9,10} with the property that 2 different words is equal to each other in at most one place ( i.e. has Hamming distance at least 4!) can have precisely 100 codewords?! Empirically perhaps this is FALSE and it would be not impossible that the the maximal number of words is precisely 90 ( this is the maximum found so far and it seems rather likely that it is maximum !). The code with this maximum number ( so far empirically found) has some mathematical structure !! This structure is as follows : We start with a code C with 10 words who are multiples of each other ( entrywise ) as far as modulo 11 is concerned ! Then when a new word is found we take all multiples of it modulo 11 ( entrywise ) and add the 10 new words . In many cases ( not all ) we get 90 words as maximum , never more it seems !!
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Thursday 21.08.2008 August!!
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Mathematics