Funktionalkalküle in mehreren Veränderlichen für stetige lineare Operatoren auf Banachräumen

Very important results and tools in the theory of generalized scalar operators are the theorem of support of C. Foias [10] (see also [8], Th. 3.1.6) and the theorem that every generalized scalar operator is decomposable ([8], Th. 3.1.19). This note contains results of this type for functional calculi in several variables. Moreover, we give (as in [15] in the case of one variable) a characterization of spectral distributions with support contained in ?ⁿ resp. in Γⁿ (where Г={z∈?:|z|=1}).     

Der spektrale Abbildungssatz für nichtanalytische Funktionalkalküle in mehreren Veränderlichen

In this note we continue the study of -functional calculi in several variables (introduced in [1] and [2]). In the situation of an inverse closed (see Def. 1.2) basic algebra of type I or II we characterize the algebra generated by an arbitrary -functional calculus φ and prove several variants of the spectral mapping theorem, thus generalizing corresponding results of F.-H. Vasilescu [10] (see also the more general version in [4], Th. 3.2.1) to the case of several commuting operators. By means of the spectral mapping theorem we prove the decomposability (in the sense of St. Frunz? [6]) of m-tuples of the type (φ(f₁),..., φ(f_m)) with f₁,...,f_mε .     

Decomposition theorems

The countable-decomposition theorem for linear functionals has become a useful tool in the theory of representing measures (see [4–7]). The original proof of this theorem was based on a rather involved study of extreme points in the state space of a convex cone. Recently M. Neumann [9] gave an independent proof using a refined form of Simons convergence lemma and Choquet's theorem. In this paper a (relatively) short proof of an extension (to a more abstract situation) of the countable-decomposition theorem is given. Furthermore a decomposition criterion is obtained which even works in the case when not all states are decomposable. All the work is based on a complete characterization of those states which are partially decomposable with respect to a given sequence of sublinear functionals.     

Some remarks on conditional entropy

Summary Using a definition of conditional entropy given by Hanen and Neveu [5, 10, 11] we discuss in this paper some properties of conditional entropy and mean entropy, in particular an integral representation of conditional entropy (§ 2), and the decomposition theorem of the KolmogorovSina¯i invariant (§ 3) (see also [6–8] and [12]).There is an essential difference between Jacob's proof of the last called theorem [6] and the proof given below. The definition of a Lebesgue space, given by Rokhlin [7–9] and [12], is not used in this paper.     

Zum Nachweis arithmetischer Cohen-Macaulay Varietäten

There exist some useful methods for the calculation of Hilbert's function without using a free resolution of polynomial ideals (see for example [4], [10], [11] and the references in these papers). Using Bezout's theorem (in the sense ofW. Gröbner [3], 144.5) these methods are suited for a proof that special homogeneous polynomial ideals are imperfect, but not for the arithmetically Cohen-Macaulay property. It is the theorem of this paper that these gaps can be filled. This theorem therefore provides some proof that an arbitrary homogeneous polynomial ideal is perfect or imperfect. Our methods are demonstrated in three examples, taking the third example from the paper ofG. A. Reisner [7], p. 35 and, using our methods, we rather easily obtain the result of [7], that the Cohen-Macaulay property depends on the characteristic of the field. In the second example, we give some remarks on the usefulness of the definition for perfeet ideals ofF. S. Macaulay [5] (see also [6]). This also illustrates whyF. S. macaulay could only construct imperfect ideals-except such one obtainable by using ideals of the principal class.

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Unserem Lehrer, Herrn Professor Dr. W. Gröbner, zum 80. Geburtstag in Verehrung gewidmet     

Schmüdgen's theorem for preorderings of higher level

The aim of this note is to generalize the Schmüdgen's theorem (e.g. see [7], [8], [9]) in the case of quadratic preorderings to preorderings of higher level. For the proof of this result we use the same technique developed in [6].     

A 1Ψ1 Summation Theorem for Macdonald Polynomials

In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8].     

On Newton's interpolation polynomials

In this paper functional analytic methods for nuclear locally convex spaces are applied to problems of analytic functions. The question is discussed whether the so-called Newton interpolation polynomials constitute a Schauder-basis in the space of analytic functions on the open unit circle (see Marku evi [3]). There are several different approaches to this problem, see, for instance, Walsh [7] and Gelfond [1]. Here we give a necessary and sufficient condition in terms of the interpolation points only. We consider the above space of analytic functions as a nuclear Kö-the-sequence space and use some deep theorems about nuclear spaces, such as the theorem of Dynin and Mitjagin (see Rolewicz [6], Pietsch An interesting connection with the theory of uniformly distributed sequences is mentioned.     

Diophantine Integral Geometry

In his works [1], [2] and [3], the author succeeded in establishing several inversion formulas for Radon transform on Euclidean space, Damek-Ricci space and also on a finite set. The present paper deals with Radon transform R on discrete hyperplanes in the lattice defined by linear diophantine equations. More precisely, we study carefully various natural questions in this context: specific properties of the discrete Radon transform R and its dual R*, inversion formula for R (see Theorem 4.1) and also an appropriate support theorem in the discrete case (see Theorem 5.3).      

General theorem for the existence of iterative roots of homeomorphisms with periodic points

We apply Zdun’s factorization theorem (see Zdun (2008) [3]) to give the conditions for the existence and the form of continuous and orientation-preserving iterative roots of homeomorphisms of the circle with a rational rotation number. Our theorem generalizes the previous results given by Jarczyk (2003) in [2], Zdun (2008) in [3] and Solarz (2003, 2009) in [4] and [5].     

Elliptic operators on elementary ramified spaces

Diffusion problems on topological networks (one-dimensional networks) have been introduced by G. Lumer [Lu. 1–4] and are also considered by F. Ali Mehmeti [AM] and the author [N.1–3]. According to the ideas of G. Lumer [Lu. 5], we develop here a local approach to diffusion problems on higher dimensional ramified spaces. We consider the variational formulation of such problems (see [L-U, G-T, Li, Sh, Lu. 5]). The transmission operator is the sum of weak Ventcel'-Visik boundary operators [B-C-P] (it is either a first order operator or a second order operator). Finally, like Gilbarg-Trudinger [G-T], we establish a continuity result which will be used in [N. 5] to show that one of the assumptions of the Lumer-Phillips theorem [P] (density of the range) is fulfilled.     

Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature

We give a simple proof of an extension of the existence results of Ricci flow of Giesen and Topping (2010, 2011) [15], [20], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi’s existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of Perelman [7] on Ricci flow. We will also give a simple proof of a special case of the existence theorem of Topping (2010) [16] without using the existence theorem of Shi (1989) [9].     

Weak self-adjointness and conservation laws for a porous medium equation

The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006, 2007) [4], [7]. In Ibragimov (2007) [6] a general theorem on conservation laws was proved. In Gandarias (2011) [3] we generalized the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. In this paper we find the subclasses of weak self-adjoint porous medium equations. By using the property of weak self-adjointness we construct some conservation laws associated with symmetries of the differential equation.     

Zariski density in lie groups

In [7] Furstenberg gave a proof of Borel’s density theorem [1], which depended not on complete reducibility but rather on properties of the action of a minimally almost periodic group on projective space. In [9] and [10] the basic idea of this proof was extended in various ways to deal with other particular classes of Lie groupsG and closed subgroupsH of cofinite volume. In [5] Dani gives a more general form of the density theorem in whichH need only be non-wandering. In the present paper we define the condition ofk-minimal quasiboundedness, and prove that this condition is necessary and sufficient for the density theorem to hold ((2.4) and (2.6)). Here we replace the arguments of [9] and [10] simply by proofs that the groups considered there satisfy this condition (2.10). We extend the results of [9] and [10] by considering groups which are analytic rather than algebraic, and in the solvable case we completely characterize thek-minimally quasibounded groups (2.9). In the last section we give two applications of the density theorem.     

Structure of radicals in bands of semigroups

It is well-known that semigroups of many important classes are decomposable into bands of their subsemigroups with more “rigid” structure. For efficient applications of radicals to semigroups of that kind, an information on the shape of radicals in bands of semigroups may be useful. Here we will investigate the Jacobson, Baer, Brown-McCoy and the least special radicals. The results were announced in [3]. The interaction of bands and radicals was also considered in [4] for the class of semigroups with zero; analogous problems for bands and radicals of associative rings were investigated in [5], [6] and [7].     

Orthomodular lattices with state-separated noncompatible pairs

In the logico-algebraic foundation of quantum mechanics one often deals with the orthomodular lattices (OML) which enjoy state-separating properties of noncompatible pairs (see e.g. [18], [9] and [15]). These properties usually guarantee reasonable richness of the state space—an assumption needed in developing the theory of quantum logics. In this note we consider these classes of OMLs from the universal algebra standpoint, showing, as the main result, that these classes form quasivarieties. We also illustrate by examples that these classes may (and need not) be varieties. The results supplement the research carried on in [1], [3], [4], [5], [6], [11], [12], [13] and [16].     

A remark on entire solutions of quasilinear elliptic equations

By applying a main comparison theorem of Pucci and Serrin (2007) [2] we cover, for general equations of p-Laplace type, the open cases of Theorems B, D, E of Farina and Serrin (submitted for publication) [1] as described in Problems 2 and 3 of Section 12 of Farina and Serrin (submitted for publication) [1]. Moreover, we provide significant improvements of Theorem C and Theorem 5 of Farina and Serrin (submitted for publication) [1], the latter in the context of mean curvature type operators, see Theorem 1.3 and Theorems 5.2-5.4 below.Finally, Theorem 1.1 provides a new Liouville theorem outside the context of work in Farina and Serrin (submitted for publication) [1].     

Some results on the multiplier conjecture for the casen=5n 1

Using the method presented in [1], we obtain some new results which improve on the result of MacFarland's theorem (see [2]) in this case.     

A Priori Estimate for the Discontinuous Oblique Derivative Problem for Elliptic Systems

Recently [6] an existence as well as a uniqueness theorem for the discontinuous oblique derivative problem for nonlinear elliptic system of first order in the plane, see [12, 19, 23] was proved, based on some a priori estimate from [20]. This estimate, however, is deduced by reductio ad absurdum. Therefore the constants in this estimate are unknown so that the estimate cannot be used for numerical procedures, e.g. for approximating the solution of a nonlinear problem by solutions of related linear problems, see [24, 3, 4]. In this paper a direct proof of an a priori estimate is given using some variations of results from [14], see also [11], where the constants can explicitely be estimated. For related a priori estimates see [1 – 5, 8, 16, 17, 20, 21, 24 – 26]. A basic reference for the oblique derivative problem is [9].