We construct the complex simple Lie algebras using elementary algebraic geometry. We use our construction to obtain a new proof of the classification of complex simple Lie algebras that does not appeal to the classification of root systems.

It is shown that, under some algebraic conditions on fixed reals α1, α2, . . . , αn+1 and vectors a1,a2, . . . ,an+1 ∈ R, every continuous at a point function f : R → R satisfying the simultaneous system of inequalities f(x + ai) ≤ αi + f(x), x ∈ R, i = 1, 2, . . . , n + 1, has to be of the form f(x) = p · x + f(0), x ∈ R, with uniquely determined p ∈ R. For mappings with values in a Banach spa...

How can di¤erent individuals’probability assignments to some events be aggregated into a collective probability assignment? Although there are several classic results on this problem, they all assume that the ‘agenda’of relevant events forms a -algebra, an overly demanding assumption for many practical applications. We drop this assumption and explore probabilistic opinion pooling on general ag...

In [2] W. Rautenberg proved that (the content of) any 2–element matrix is finitely axiomatizable and asked if the same is true for any finite matrix. This conjecture was disproved by P. Wojtylak, who in [5] (see also [4]) costructed a 5–element matrix with 2 designated elements, which is not finitely axiomatizable. Recently, W. Dziobiak presented in [1] a 4–element algebraic matrix (i.e. a matr...

We produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group PLo(I) of piecewise-linear orientation-preserving homeomorphisms of the unit interval under the operation of composition, and also of the generalized R. Thompson groups Fn. The first description is as a set of isomorphism classes of groups which is closed under three algebraic operat...

This paper provides a unifying mathematical framework for orders of magnitude models used in Qualitative Physics. An axiomatic of the qualitative equality is provided and a general algebraic structure called qualita­ tive algebra is defined. It is shown that the usual model (+,-,0,?) and the extended model recently introduced by Dubois and Prade are particular cases in the class of mo­ dels tha...

Strictly competitive games are a class of 2-player games often quoted in the literature to be a proper generalization of zero-sum games. Other times it is claimed, e.g. by Aumann, that strictly competitive games are only payoff transformations of zero-sum games. But to the best of our knowledge there is no proof of such claim. We shed light to this point of confusion in the literature, showing ...

We study entire holomorphic curves in the algebraic torus, and show that they can be characterized by the “growth rate” of their derivatives.

We develop a new and uniform approach to the three families of simple groups of Lie type discovered by Suzuki and Ree, without using Lie algebras. A novel type of algebraic structure is defined, whose automorphism groups are the groups in question. This leads to elementary proofs of the group orders and simplicity, as well as much information on subgroup structure and geometry.

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact we establish a slightly stronger version which applies to a wider class of boundary 1-cycles. Arguments here are based on the Hahn-Banach Theorem and some geometric m...