Stability of the equation of homomorphism and completeness of the underlying space

We prove that all assumptions of a Theorem of Forti and Schwaiger (cf. [G. L. Forti, J. Schwaiger, Stability of homomorphisms and completeness, C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 215–220]) on the coherence of stability of the equation of homomorphism with the completeness of the space of values of all these homomorphisms, are essential. We give some generalizations of this theorem and certain examples of applications

Multi-variable translation equation which arises from homothety

In many regular cases, there exists a (properly defined) limit of iterations of a function in several real variables, and this limit satisfies the functional equation (1-z)f(x)=f(f(xz)(1-z)/z); here z is a scalar and x is a vector. This is a special case of a well-known translation equation. In this paper we present a complete solution to this functional equation in case f is a continuous function on a single point compactification of a 2-dimensional real vector space. It appears that, up to conjugation by a homogeneous continuous function, there are exactly four solutions. Further, in a 1-dimensional case we present a solution with no regularity assumptions on f.Comment: 15 page

The projective translation equation and unramified 2-dimensional flows with rational vector fields

Let X=(x,y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1-z)f(X)=f(f(Xz)(1-z)/z); here f(X)=(u(x,y),v(x,y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in C^2\{union of curves}) projective flows whose vector field is still rational. We prove that, up to conjugation with 1-homogenic birational plane transformation, these are of 6 types: 1) the identity flow; 2) one flow for each non-negative integer N - these flows are rational of level N; 3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; 4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; 5) the level 4 flow expressable in terms of lemniscatic elliptic functions; 6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Polya-Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus.Comment: 34 pages, 2 figure

Visibly transparent & radiopaque inorganic organic composites from flame-made mixed-oxide fillers

Radiopaque composites have been produced from flame-made ytterbium/silica mixed oxide within a crosslinked methacrylate resin matrix. The refractive index of the filler powder increased with ytterbium oxide loading. A high transparency was achieved for a matching refractive index of the filler powder and the polymer in comparison to commercial materials with 52wt% ceramic filling. It was demonstrated that powder homogeneity with regard to particle morphology and distribution of the individual metal atoms is essential to obtain a highly transparent composite. In contrast, segregation of crystalline single-oxide phases drastically decreased the composite transparency despite similar specific surface areas, refractive indices and overall composition. The superior physical strength, transparency and radiopacity compared to composites made from conventional silica based-fillers makes the flame-made mixed-oxide fillers especially attractive for dental restoration material

Les équations de pré-Schröder

On donne la revue des résultats au sujet des équations de pré-Schröder introduites par G. Targonski. On formule aussi des problèmes ouverts

Sur un théorème de S. Gołąb

On examine en quel degré les suppositions dans un théorème de S. Gołąb, au sujet de la détermination au juste de 1’isomorphisme de 1’addition et de la multiplication simples, sont essentielles. On démontre aussi une modification de ce théorème et on pose un problème ouvert

Sur des fonctions des niveaux invariantement ordonnés

On considère des fonctions remplissantes la condition (3) et dans le cas particulate l\u27implication (1). On donne des conditions sous lesquelles ces fonctions sont généralament homogènes

Synthesis and evaluation of novel dental monomer with branched carboxyl acid group

To enhance the water miscibility and increase the mechanical properties of dentin adhesives, a new glycerol-based monomer with vinyl and carboxylic acid, 4-((1,3-bis(-methacryloyloxy)propan-2-yl)oxy)-2-methylene-4-oxobutanoic acid (BMPMOB), was synthesized and characterized. Dentin adhesive formulations containing 2-hydroxyethyl methacrylate (HEMA), 2,2-bis[4-(2-hydroxy-3-methacryloxypropoxy) phenyl]propane (BisGMA), and BMPMOB were characterized with regard to real-time photopolymerization behavior, water sorption, dynamic mechanical analysis, and microscale three-dimensional internal morphologies and compared with HEMA/BisGMA controls. The experimental adhesive copolymers showed higher glass transition temperature and rubbery moduli, as well as improved water miscibility compared to the controls. The enhanced properties of the adhesive copolymers indicated that BMPMOB is a promising comonomer for dental restorative materials

The projective translation equation and rational plane flows. I

Let X=(x,y). A plane flow is a function F(X,t): R^2*R->R^2 such that F(F(X,s),t)=F(X,s+t) for (almost) all real numbers x,y,s,t (the function F might not be well-defined for certain x,y,t). In this paper we investigate rational plane flows which are of the form F(X,t)=f(Xt)/t; here f is a pair of rational functions in 2 real variables. These may be called projective flows, and for a description of such flows only the knowledge of Cremona group in dimension 1 is needed. Thus, the aim of this work is to completely describe over R all rational solutions of the two dimensional translation equation (1-z)f(X)=f(f(Xz)(1-z)/z). We show that, up to conjugation with a 1-homogenic birational plane transformation (1-BIR), all solutions are as follows: a zero flow, two singular flows, an identity flow, and one non-singular flow for each non-negative integer N, called the level of the flow. The case N=0 stands apart, while the case N=1 has special features as well. Conjugation of these canonical solutions with 1-BIR produce a variety of flows with different properties and invariants, depending on the level and on the conjugation itself. We explore many more features of these flows; for example, there are 1, 4, and 2 essentially different symmetric flows in cases N=0, N=1, and N>=2, respectively. Many more questions will be treated in the second part of this work.Comment: 54 pages, 6 figures. Final version before proof