Complete congruence lattices of two related modular lattices

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable \({\omega}\)-categorical structures. Finally, we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable \({\omega}\)-categorical structures.     

Non-Abelian tensor square and related constructions of p-groups

Archiv der Mathematik - Let G be a group. We denote by $$nu (G)$$ a certain extension of the non-Abelian tensor square $$[G,G^{varphi }]$$ by $$G times G$$. We prove that if G is a finite potent...     

CCZ-equivalence of bent vectorial functions and related constructions

We observe that the CCZ-equivalence of bent vectorial functions over ${{\bf F}_2^n}$ (n even) reduces to their EA-equivalence. Then we show that in spite of this fact, CCZ-equivalence can be used for constructing bent functions which are new up to EA-equivalence and therefore to CCZ-equivalence: applying CCZ-equivalence to a non-bent vectorial function F which has some bent components, we get a function F?? which also has some bent components and whose bent components are CCZ-inequivalent to the components of the original function F. Using this approach we construct classes of nonquadratic bent Boolean and bent vectorial functions.     

Connections between congruence-lattices and polynomial properties

This paper gives characterisations of properties of varieties such as congruence-distributivity (CD), filtrality (FI), CD and having complemented principal congruences — these last two properties are shown to be equivalent- and having restricted equationally definable principal congruences (REDPC), in terms of the existence of some kind of polynomials. These are generalisations both of Jónsson's famous theorem characterizing CD as well as results concerning the (dual) discriminator. The methods are applied to show that REDPC implies CD, which was a problem asked in [2]. A generalisation of the concept of the Mal'cev condition — the so called Pixley-condition — is defined, and it is shown that filtrality and REDPC are Pixley-conditions. The relations between several concepts connected with the above ones are also investigated.The definitions can be found in section 2, and the results are contained in section 4, section 5, and section 6. We call attention to the figure at the end of the paper, which contains most of our results and gives a survey of the concepts examined.Presented by G. Grätzer.Thanks are also due to Peter Krauss, who called our attention to some gaps in our proofs in a previous version of the present paper.     

Connections between Romanovski and other polynomials

A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schrödinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.     

A survey of tensor products and related constructions in two lectures

We survey tensor products of lattices with zero and related constructions focused on two topics: amenable lattices and box products. Received August 21, 1998; accepted in final form September 9, 1998.     

Harmonic Hopf constructions between spheres II

Calculus of Variations and Partial Differential Equations -     

A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950s.     

Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock

We study the ground state solutions of the Dirac-Fock model in the case of weak electronic repulsion, using bifurcation theory. They are solutions of a minmax problem. Then we investigate a max-min problem coming from the electronpositron field theory of Bach-Barbaroux-Helffer-Siedentop. We show that given a radially symmetric nuclear charge, the ground state of Dirac-Fock solves this maxmin problem for certain numbers of electrons. But we also exhibit a situation in which the max-min level does not correspond to a solution of the Dirac-Fock equations together with its associated self-consistent projector.submitted 19/03/04, accepted 30/07/04     

Some Connections between Topological and Modal Logic

We study modal logics based on neighbourhood semantics using methods and theorems having their origin in topological model theory. We thus obtain general results concerning completeness of modal logics based on neighbourhood semantics as well as the relationship between neighbourhood and Kripke semantics. We also give a new proof for a known interpolation result of modal logic using an interpolation theorem of topological model theory.     

Connections between Size Functions and Critical Points

A mathematical approach to the concept of shape of a submanifold ℳ︁ of a Euclidean space had previously been given by means of ‘measuring functions’ (e.g. diameter or volume) and of the derived ‘size functions’. This paper relates the study and the computation of any such size function to the structure of critical points of the associated measuring function.     

Connections between projective geometry and superassociative algebra

This paper establishes a connection between projective geometry and the superassociative algebra of multiplace functions. In Part I a quaternary operation is defined for the points on a line j in a projective plane π relative to a fixed quadrangle and the result of operating on A,B,C,D is denoted by A(B,C,D). It is shown that π is a Pappus plane if and only if this operation is superassociative: (A(B,C,D))(E,F,G)=A(B(E,F,G),C(E,F,G),D(E,F,G)) for any points A,B,C,D,E,F,G on j. Desargues and certain special Desargues planes are also characterized by restrictions of the superassociative law. In Part II projective geometry is developed over superassociative systems.     

Connections between Deddens algebras and extended eigenvectors

A complex number λ is called an extended eigenvalue of the shift operator S, Sf = zf, on the disc algebra $C_A (\mathbb{D})$ if there exists a nonzero operator $A:{\text{ }}C_A (\mathbb{D}) \to C_A (\mathbb{D})$ satisfying the equation AS = λSA. We describe the set of all extended eigenvectors of S in terms of multiplication operators and composition operators. It is shown that there are connections between the Deddens algebra associated with S and the extended eigenvectors of S.