On type-cases, union elimination, and occurrence typing

We extend classic union and intersection type systems with a type-case construction and show that the combination of the union elimination rule of the former and the typing rules for type-cases of our extension encompasses occurrence typing . To apply this system in practice, we define a canonical form for the expressions of our extension, called MSC-form. We show that an expression of the extension is typable if and only if its MSC-form is, and reduce the problem of typing the latter to the one of reconstructing annotations for that term. We provide a sound algorithm that performs this reconstruction and a proof-of-concept implementation.

Spectra inhabiting the left half-plane that are universally realizable

Let Λ = {λ1, λ2, . . ., λ n } be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λ i ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.

The algebra generated by nilpotent elements in a matrix centralizer

For an arbitrary square matrix $S$, denote by $C(S)$ the centralizer of $S$, and by $C(S)_N$ the set of all nilpotent elements in $C(S)$. In this paper, we use the Weyr canonical form to study the subalgebra of $C(S)$ generated by $C(S)_N$. We determine conditions on $S$ such that $C(S)_N$ is a subalgebra of $C(S)$. We also determine conditions on $S$ such that the subalgebra generated by $C(S)_N$ is $C(S).$

THEORY OF SURFACES IN FOUR-DIMENSIONAL GALILEAN SPACE

By means of a special choice of coordinate lines of the surface in four-dimensional Galilean space, the first and second quadratic shape of the surface is defined. It has been proved that the second-order surface equation in three-dimensional space can be converted to a canonical form by means of a special transformation, which is the rotation of the coordinate axes of three-dimensional Galilean space. Furthermore, the transformation matrix is an element of the Heisenberg group that is neither symmetric nor orthogonal. In four-dimensional space R41 - the concept of a surface indicator is introduced and the main curvature of the surface is defined.

Fault Identification in Non-Stationary Systems Based on Sliding Mode Observers

The paper is devoted to the problem of fault identification in technical systems described by non-stationary nonlinear dynamic equations under unmatched disturbances. To solve the problem, sliding mode observers are used. The suggested ap- proach is based on the model of the original system of minimal dimension having different sensitivity to the faults and distur- bances in contrast to the traditional approaches to sliding observer design which are based on the original system. Additionally it is assumed that matrices describing such a model have the canonical form and are constant. The main purpose of using such a model is possibility to take into account the non-stationary feature of the systems. As a result, the model has stationary dynamic and non-stationary additional term that allows to promote sliding mode design. Besides, the new approach to design sliding mode observers is suggested. The peculiarity of this approach is that it does not require that original systems should be minimum phase and detectable. According to the traditional approaches stability of the observer is provided by minimum phase and detectability properties. In our approach, stability of the observer is achieved due to the canonical form of the matrices describing the model. In addition, the matching condition is not necessary. This allows to extend a class of systems for which sliding mode observers can be designed. Theoretical results are illustrated by practical example of electric servoactuator.

Stworzenie z niczego, czyli słów kilka o monografii Jacka Zielińskiego „Koncepcja creatio ex nihilo w myśli Apologetów greckich II wieku”

The monograph of Jacek Zieliński, The Concept of Creatio ex Nihilo in the Thought of the Greek Apologists of the 2nd century, published by Wroclaw’s Atut in 2013, discusses an important problem of the theory of creation from nothing. It also asks an important question, how far the elements of the concept, articulated in its final form only by Augustine of Hippo can be found in the writings of the Christian apologists of the 2nd century. It is an important question, especially that the Bible in its canonical form, the early extra-biblical Jewish literature or – even more – Greek tradition does not unambiguously advocate the concept creatio ex nihilo. Hence the question how, when and why the concept was articulated, since it played such an important role in the Christian thought of later centuries. In addition, the book of Jacek Zielinski is important in Polish market, because there are only a few serious publications dealing with the issue. The article gives a description and short analysis of the book, pinpointing its strong sights and showing areas, which could be strengthened in this and — hopefully — following publications on creatio ex nihilo by Jacek Zieliński.

Danse devant le miroir de François de Curel, ou d’une “pièce bien faite” à une “pièce bien défaite”

François de Curel ha pasado a la posteridad como autor de la “obra bien hecha”. Sin embargo, al estudiar su obra, en este caso La Danse devant le miroir (La danza frente al espejo), nos damos cuenta de que el escritor no respeta todas las reglas del arte dramático. De hecho, el escritor parece, en primer lugar, cuestionar la fábula de la forma canónica, que se construye sobre un mecanismo lógico de acción que conduce, a través de una tensión cada vez mayor, al desenlace final. La composición clásica se desvanece en la obra detrás de la evocación de los estados anímicos de los protagonistas, presos de la angustia existencial y, por tanto, notamos el paso de un drama agonista a un drama ontológico. El trabajo de socavar el "drama absoluto" también se manifiesta en un nuevo enfoque del personaje. Desprovisto de sus atributos activos, prefiere cuestionar su vida, repetir sus contratiempos antes que actuar, el personaje en acción deja paso al personaje en reflexión François de Curel has gone down to posterity as an author of the “well-made play”. However, while studying his work, in this case, La Danse devant le miroir (The Dance in Front of the Mirror), we realize that the writer does not respect all the rules of dramatic art. In fact, the writer seems, in the first place, to call into question the fable of the canonical form which is built on a logical mechanism of action leading through ever-increasing tension to the final denouement. The classical composition fades into the play behind the evocation of the protagonists’ emotional states in the grip of existential anxiety and, therefore, we note the passage from an agonistic drama to an ontological drama. The work of undermining the “absolute drama” is also manifested by a new approach to the character. Deprived of his active attributes, he prefers to call into question his life, rehash his setbacks rather than act. The character in action is giving way to the character in reflection.

Centrosymmetric universal realizability

A list $\Lambda =\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable, if it is the spectrum of an entrywise nonnegative matrix $A$. In this case, $A$ is said to be a realizing matrix. $\Lambda$ is said to be universally realizable, if it is realizable for each possible Jordan canonical form (JCF) allowed by $\Lambda$. The problem of the universal realizability of spectra is called the universal realizability problem (URP). Here, we study the centrosymmetric URP, that is, the problem of finding a nonnegative centrosymmetric matrix for each JCF allowed by a given list $\Lambda $. In particular, sufficient conditions for the centrosymmetric URP to have a solution are generated.

Amplituhedron-like geometries

We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical form of these geometries corresponds to the product of parity conjugate amplitudes at tree as well as loop level. The product of amplitudes in superspace lifts to a star product in bosonised superspace which we give a precise definition of. We give an alternative definition of amplituhedron-like geometries, analogous to the original amplituhedron definition, and also a characterisation as a sum over pairs of on-shell diagrams that we use to prove the conjecture at tree level. The union of all amplituhedron-like geometries has a very simple definition given by only physical inequalities. Although such a union does not give a positive geometry, a natural extension of the standard definition of canonical form, the globally oriented canonical form, acts on this union and gives the square of the amplitude.