Jan Haluska | Slovak Academy of Sciences (original) (raw)

Papers by Jan Haluska

Mathematica Slovaca, 1997

Acta Universitatis Carolinae. Mathematica et Physica, 1991

If Condition (GB), introduced in [7] and [8] is fulfilled, then the everywhere convergence of the... more If Condition (GB), introduced in [7] and [8] is fulfilled, then the everywhere convergence of the net of measurable functions implies the convergence of these functions with respect to the semivariation on a set of the finite variation of the measure m : Σ → L(X, Y), where Σ is a σ-algebra of subsets of the set T = ∅, X, Y are both locally convex spaces. The generalized strong continuity of the semivariation of the measure, introduced in this paper, implies Condition (GB).

arXiv (Cornell University), Jul 31, 2020

An operation of associative, commutative and distributive multiplication on Euclidean vector spac... more An operation of associative, commutative and distributive multiplication on Euclidean vector space E 4 is introduced by a skew circulant matrix. The resulting algebra W over R is isomorphic to C×C. The related algebraic, geometrical, and topological properties are given.There are subplanes of W isomorphic to the Gauss and Clifford complex number planes. A topology on W is given by a norm which is a sum of two norms. A hint how to apply this 4 dimensional algebra over R to the 12-tone Equally Tempered Tuning algebra is given.

Real Analysis Exchange, 1999

The Egoroff theorem for measurable X-valued functions and operator-valued measures m: Σ→ L( X, Y)... more The Egoroff theorem for measurable X-valued functions and operator-valued measures m: Σ→ L( X, Y), where Σ is a σ-algebra of subsets of T ≠∅ and X, Y are both locally convex spaces, is proved. The measure is supposed to be atomic and the convergence of functions is net.

It is known that the perfect fifth and octave can be expressed as XY 3 and XY 5 respectively, whe... more It is known that the perfect fifth and octave can be expressed as XY 3 and XY 5 respectively, where X = 256/243 = 2/3 is the minor and Y = 9/8 : X = 3/21 the major Pythagorean semitone. In this paper, Pythagorean system is studied when couples of semitones need not be considered in rational numbers; we deal with semitone couples (X,Y ) in algebraic numbers expressed with dth roots, d = 1, 2, 3, . . ..

Mathematica Slovaca, 2004

A Fubini theorem in vector spaces for the Kurzweil integral with respect to operator measures is ... more A Fubini theorem in vector spaces for the Kurzweil integral with respect to operator measures is proven.

We divide the set of all diatonic scales into three classes P,G,R, the intersection of which cont... more We divide the set of all diatonic scales into three classes P,G,R, the intersection of which contains the major diatonic scale. The class G contains Gypsy scales, the class P – Pythagorean heptatonic, and the class R – Redfield Scale. The paper could be of interest for the music theorists, mathematicians, as well as producers of modern key music instruments, interpreters, composers, and electro-acoustical studios.

In the paper we explain the notion of geometrical net from the view of coding of music informatio... more In the paper we explain the notion of geometrical net from the view of coding of music information. A direct, elementary, and very short alternative proof of the assertion that there are no transcendental semitones generating Pythagorean system, is given. This is a conclusion of the negation of the psychological Weber – Fechner’s law. Further, we discuss about a kind of uncertainty bounded with the melodic and harmonic structures in music.

We introduce a generalized Kolmogoroff integral of the first type with respect to the operator va... more We introduce a generalized Kolmogoroff integral of the first type with respect to the operator valued measure in complete bornological locally convex topological vector spaces and show that, in the equal setting, the class of integrable functions coincide with the class of integrable functions in the generalized Dobrakov integral sense, [8].

We can observe an effort of linguists, musicologists, and also natural scientists (M. Boroda, G. ... more We can observe an effort of linguists, musicologists, and also natural scientists (M. Boroda, G. Altmann, G. Wimmer, Z. Martináková, R. Köhler) to find the basic units of the human speech and, specially, music. The dominated power of the present western music can be denoted shortly with the word ”clavier”. Grubby spoken, music scores are still pressed into the frame of 12 different music degrees within the octave, the discrete choice of pitch frequencies, and the octave equivalence. The similar general picture holds also for rhythm. The exception proves the rule. In fact, tunings are doubtless classical kinds of segmentation of the western music. The idea is not new: to consider as segments the all relative frequency intervals within a tone system as segments. Every interval should be derived from a few basic intervals a the minimal set of all basic intervals must carry the whole information about the tone system. Note that in general that basic intervals need not be the smallest in...

A construction of product measures in complete bornological locally convex topological vector spa... more A construction of product measures in complete bornological locally convex topological vector spaces is given. Two theorems on the existence of the bornological product measure are proved. A Fubini-type theorem is given. Mathematics Subject Classification 2000: Primary 46G10, Secondary 28B05

Modern Real Analysis, 2015