1,720,974 research outputs found
Divergence zero quaternionic vector fields and Hamming graphs
Full text linkWe give a possible extension of the definition of quaternionic power
series, partial derivatives and vector fields in the case of two (and then several)
non commutative (quaternionic) variables. In this setting we also investigate
the problem of describing zero functions which are not null functions in the for-
mal sense. A connection between an analytic condition and a graph theoretic
property of a subgraph of a Hamming graph is shown, namely the condition
that polynomial vector field has formal divergence 0 is equivalent to connect-
edness of subgraphs of Hamming graphs H(d, 2). We prove that monomials in
variables z and w are always linearly independent as functions only in bidegrees
(p, 0), (p, 1), (0, q), (1, q) and (2, 2)
Quaternionic Cartan coverings and applications
Full text linkWe present the topological foundation for solvability of Multiplicative
Cousin problems formulated on an axially symmetric domain In particular, we provide a geometric construction of quaternionic
Cartan coverings, which are generalizations of (complex) Cartan coverings as
presented in Section 4 of [FP]. Because of the requirements of symmetry
inherent to the domains of definition of quaternionic regular functions, the
existence of quaternionic Cartan coverings of is not a consequence of
existence of complex Cartan coverings, because for the latter there are no
requirements for the symmetries with respect to the real axis. Due to the
special role of the real axis, also the covering restricted to has to have additional properties. All these required properties
were achieved by starting from a particular symmetric tiling of the symmetric
set . Finally we provide an application
of these results to prove the vanishing of 'antisymmetric' cohomology groups of
planar symmetric domains for
Slice conformality and Riemann manifolds on quaternions and octonions
Full text linkIn this paper we establish quaternionic and octonionic analogs of the
classical Riemann surfaces. The construction of these manifolds has nice
peculiarities and the scrutiny of Bernhard Riemann approach to Riemann
surfaces, mainly based on conformality, leads to the definition of slice
conformal or slice isothermal parameterization of quaternionic or octonionic
Riemann manifolds. These new classes of manifolds include slice regular
quaternionic and octonionic curves, graphs of slice regular functions, the
and dimensional spheres, the helicoidal and catenoidal and
dimensional manifolds. Using appropriate Riemann manifolds, we also give a
unified definition of the quaternionic and octonionic logarithm and -th root
function.Comment: Published online in: Math. Z. (2022) - (open access
On a definition of logarithm of quaternionic functions
Full text linkFor a slice-regular quaternionic function , the classical exponential function is not slice-regular in general. An alternative definition of an exponential function, the -exponential , was given in the work by Altavilla and de Fabritiis (2019): if is a slice-regular function, then is a slice-regular function as well. The study of a -logarithm of a slice-regular function becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a depends only on the structure of the zero set of the vectorial part of the slice-regular function , besides the topology of its domain of definition. We also show that, locally, every slice-regular nonvanishing function has a -logarithm and, at the end, we present an example of a nonvanishing slice-regular function on a ball which does not admit a -logarithm on that ball
On a continuation of quaternionic and octonionic logarithm along curves and the winding number
Full text linkThis paper focuses on the problem of finding a continuous extension of the
hypercomplex logarithm along a path. While a branch of the complex logarithm
can be defined in a small open neighbourhood of a strictly negative real point,
no continuous branch of the hypercomplex logarithm can be defined in any open
set which contains a strictly negative real
point (here represents the algebra of quaternions or
octonions).
To overcome these difficulties, we introduced the logarithmic manifold
and then showed that if then
is an
immersion and a diffeomorphism between and .
In this paper, we consider lifts of paths in to the
logarithmic manifold ; even though is simply connected, in general, given a path in , the existence of a lift of this path to
is not guaranteed. There is an obvious equivalence
between the problem of lifting a path in and the
one of finding a continuation of the hypercomplex logarithm
along this path.Comment: 30 pages, 4 figure
Earthquake model simulation
Full text linkV delu predstavimo model izračuna pričakovane škode nastale po potresnem sunku. Obravnavamo potrese stopnje VI, VII, VIII in IX po EMS ter predpostavimo, da so njihove pojavitve med seboj neodvisne in porazdeljene z enakim porazdelitvenim zakonom, ki je odvisnen od stopnje in lokacije potresa. Pričakovano škodo računamo po regijah, zato je v vsaki regiji potrebno oceniti povprečen čas pojavitve med dvema potresoma in izračunati verjetnosti dogodka, da se potres v določeni regiji sploh zgodi, ali da se jih morebiti zgodi več hkrati. Število pojavitev potresov je odvisno tudi od dolžine obravnavenega obdobja.
Poleg jakosti potresa na nastalo škodo vpliva struktura zgradb in leto gradnje. Vse stavbe na območju Slovenije smo razdelili v 6 ranljivostnih razredov, ki se med seboj razlikujejo po materialu nosilne konstrukcijo, ta pa določa njihovo potresno odpornost. Določiti je bilo potrebno delež stavb v vsakem ranljivostnem razredu, ki se poruši ob potresu določene intenzitete. Rezultate smo pustili izračunane v m^2, tako lahko v vsaki regiji določimo primerno denarno oceno pričakovane škode.We present a model for calculating the expected damage caused by an earthquake. We consider earthquakes of levels VI, VII, VIII and IX according to the EMS and assume that their occurrences are mutually independent and follow the same distribution law, which depends on the level and location of the earthquake. Estimated damage is calculated by region, therefore in each region we have to estimate the average time between two earthquakes and to calculate the probabilities that one or more earthquakes would occur in a particular region. The number of earthquakes also depends on the length of the observation period.
In addition to the magnitude of the earthquake, the damage caused is also affected by the structure of the buildings and the the year of construction. All buildings in the territory of Slovenia were divided into 6 vulnerability classes, according the material of the load-bearing structure, which determines their earthquake resistance. We determine the proportion of buildings in each vulnerability class which collapses upon an earthquake of given intensity intensity. We have left the results calculated in m^2, so we can determine an appropriate monetary estimate of the expected damage in each region
Quaternionic Iteration
No full textGlavni cilj diplomskega dela je opis in vizualizacija napolnjenih Juliajevih množic za kvaternionske kvadratne polinome kvaternionske spremenljivke, ki so posplošitev znanih napolnjenih Juliajevih množic za kompleksne kvadratne polinome kompleksne spremenljivke. Izpeljan je kriterij za neomejenost orbit ter opisane so napolnjene Juliajeve množice za določen razred kvadratnih polinomov. Njihov presek z dvema določenima ravninama sovpada z že znanima kompleksnima napolnjenima Juliajevima množicama, presek z ostalimi ravninami pa je s tem natanko določen. Priložena in opisana sta tudi koda za računanje z regularnimi funkcijami v Mathematici in iterativen algoritem za računanje orbit regularnih funkcij.The main goal of this work is the description and the visualization of filled Julia sets for quaternionic quadratic polynomials of a quaternionic variable which are a generalization of the known filled Julia sets for complex quadratic polynomials of a complex variable. A criterion is described for unboundedness of orbits and filled Julia sets of a certain class of polynomials are described. Their intersection with two specific planes coincides with the already known complex filled Julia sets and the intersection with the other planes is thereby precisely determined. Code for computing with regular functions in Mathematica is also attached and described, as well as an iterative algorithm for computing orbits of regular functions
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