| 12. 5. 2022 |
Ján Haluška – An ensemble of 6 ordered vector algebras, epimorphic to 6 generalized complex numbers
is a generalization of 12-TET |
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Abstract:
A $12$-dimensional linearly ordered vector algebra $\mathscr{W}_{12}$ over $\mathbb{R}$ is inspired with 12-Tone Equal Temperament Tuning system (12-TET) which is generally known in mathematical acoustics.
In the paper introduced definition of multiplication $\otimes$ is based on a chain of 85 vectors of the Pythagorean approximation cycle and the notion of octave equivalence. From the operation $\otimes$, it is derived an operation $\oplus$ (operation of transposition, shift of vectors). The operation $\otimes$ is associative, commutative and distributive with respect to $\oplus$. In the paper, there are considered first 6 sub-algebras of $\mathscr{W}_{12}$; $\mathscr{W}_{1} \subset \mathscr{W}_{2} \subset \mathscr{W}_{3} \subset \mathscr{W}_{6} \subset \mathscr{W}_{12}$, $\mathscr{W}_{2} \subset \mathscr{W}_{4} \subset \mathscr{W}_{12}$. We can compute all possible invertible vectors (if they exist) in all ordered sub-algebras of $\mathscr{W}_{12}$.
We describe only some interconnections among various substructures of $\mathscr{W}_{12}$, neither its numerous non-mathematical applications.
According to results of the paper, we can claim that each tonal European musical composition is algebraically isomorphic with a sequence of lineals (=chords) of vectors (=tones) in $\mathscr{W}_{12}$ in time.
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| 26. 5. 2022 |
Peter Eliaš – On tensor product |
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Abstract:
We compare construction of tensor product in the category of vector space, where morphisms are linear maps, and the category
of complete lattices, where morphisms are join-preserving maps.
Tensor product of vector spaces $V$, $W$ is a vector space $T=V\otimes W$, together with a map $\otimes\colon V\times W\to T$,
possessing a universal property with respect to all bilinear maps from the cartesian product $V\times W$ into arbitrary vector space $U$.
Tensor product of complete lattices $K$, $L$ (in the category where morphisms are join-preserving maps)
is the lattice $G$ of all Galois connections between $K$, $L$, together with an appropriate map $\varphi\colon K\times L\to G$.
This map possess a universal property with respect to all maps defined of $K\times L$ that preserve joins in each variable separately.
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| 9. 6. 2022 |
Michal Hospodár – Operations on subregular languages and nondeterministic state complexity |
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Abstract:
We study the nondeterministic state complexity of basic regular
operations on subregular language families. In particular, we focus
on the classes of combinational, finitely generated left ideal, group,
star, comet, two-sided comet, ordered, and power-separating languages,
and consider the operations of intersection, union, concatenation, power,
Kleene star, reversal, and complementation. We get the exact complexity
in all cases, except for complementation of group languages where we
only have an exponential lower bound. The complexity of all operations
on combinational languages is given by a constant function, except for
the $k$-th power where it is $k+1$. For all considered operations, the known
upper bounds for left ideals are met by finitely generated left ideal languages.
The nondeterministic state complexity of the $k$-th power, star, and
reversal on star languages is $n$. In all the remaining cases, the nondeterministic
state complexity of all considered operations is the same as in
the regular case, although sometimes we need to use a larger alphabet
to describe the corresponding witnesses.
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| 23. 6. 2022 |
Emília Halušková – On discrete properties of several real functions |
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Abstract:
Monounary algebras are the most simple algebraic structures. They are defined as a pair $(A, f)$, where $A$ is a nonempty set and $f$ is a mapping from $A$ to $A$.
They are represented by oriented graphs in which exactly one edge emerges from each vertex. They are interdisciplinary in nature. We will illustrate this
in Sharkovsky's theorem (1964) from the theory of dynamical systems
and Łojasiewicz's theorem (1951)
from the theory of iterative roots.
Next, we present the classification of monounary algebras defined on an interval of real numbers with a continuous strictly monotone function
according to decompositions into connected components, more precisely according to the types of non-isomorphic connected components that occur in these algebras.
Monounary algebras defined intervals of real number with a continuous strictly monotone operation contain at most 4 mutually nonisomorphic components.
If the interval is closed and bounded, then 8 combinations of these components are possible. For other intervals, 10 combinations are possible.
The bijectivity of the function, the set of fixed points of the function or its second iteration and the bounding of the function on the maximal subset of the domain
that does not contain the fixed points of the function play a role in the criteria.
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