Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.

Calculation of the Collapsing Critical Load of a Translating Cam Mechanism and Roller Translating Follower Actuating an Elevator

The above-ground parking lot above Dâmbovița river has stemmed from the need to cover the lack of parking lots in Bucharest. The parking lot may be serviced, in case of emergency, by auxiliary elevators that are operated by mechanisms provided with translating cams and roller translating followers. In this paper, a calculation of the collapsing critical load shall be performed. One shall also analyze here the rod model and the model with several rods of the mechanism provided with a translating cam and roller translating follower. In order for the rods to resist collapsing, one must be found in the elastic field, thus verifying Euler’s formula.

Arithmetic proof of the multiplicity-weighted Euler characteristic for symmetrically arranged space-filling polyhedra

The puzzling observation that the famous Euler's formula for three-dimensional polyhedra V − E + F = 2 or Euler characteristic χ = V − E + F − I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χm = V m − E m + F m − I m = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.

Unification Theories: Means and Generalized Euler Formulas

The main concepts in this paper are the means and Euler type formulas; the generalized mean which incorporates the harmonic mean, the geometric mean, the arithmetic mean, and the quadratic mean can be further generalized. Results on the Euler’s formula, the (modified) Yang–Baxter equation, coalgebra structures, and non-associative structures are also included in the current paper.

Mathematics and Poetry • Unification, Unity, Union

We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.

Mathematics and Poetry • Unification, Unity, Union

We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.