Brouwer's Theorem – 90^{th} Next

Mathematica Moravica, Vol. **3** (1999), 1–4.

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Strictly Convex Metric Spaces and Fixed Points

Mathematica Moravica, Vol. **3** (1999), 5–16.

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Abstract and keywords

**Abstract.** In this article we examine strictly convex metric space and strictly convex metric space
with convex round balls. These objects generalize well known concept of strictly convex Banach space. We prove fixed point theorems for nonexpansive,
quasi-nonexpansive and asymptotically nonexpansive mappings in strictly convex metric space with convex round balls.
These results extend previous result of R. de Marr, F.E. Browder, W.A. Kirk, K. Goebel, W.G. Dotson, T.C. Lim and some others.

**Keywords.** Strictly convex Banach space, nonexpansive mappings, quasi-nonexpansive mappings, asymptotically nonexpansive mappings,
strictly convex metric space, strictly convex metric space with convex round balls, normal structure, fixed point theorem.

Normalizations of Fuzzy BCC-ideals in BCC-algebras

Mathematica Moravica, Vol. **3** (1999), 17–24.

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Abstract and keywords

**Abstract.** We introduce the notion of normal fuzzy BCC-ideals, maximal
fuzzy BCC-ideal and completely normal fuzzy BCC-ideal in BCC-algebras. We investigate some properties of normal
(resp. maximal, completely normal) BCC-ideals. We show that for every non-constant normal fuzzy BCC-ideal which is
a maximal element of $(N(X), \subseteq)$ takes only the values 0 and 1, and every maximal fuzzy BCC-ideal is completely normal.

**Keywords.** BCC-algebra, fuzzy BCC-ideal, normal fuzzy BCC-ideal, maximal fuzzy BCC-ideal, completely normal fuzzy BCC-ideal.

On Embedding Hilbert Algebras in BCK-algebrass

Mathematica Moravica, Vol. **3** (1999), 25–28.

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Abstract and keywords

**Abstract.** We prove that the class of Hilbert algebras maybe embedded into the class of all BCK-algebras.

**Keywords.** Hilbert algebra, BCK-algebra.

A Note on the Range of Compact Multipliers of Mixed-norm Sequence Space

Mathematica Moravica, Vol. **3** (1999), 29–32.

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Abstract and keywords

**Abstract.** In this note we consider the range as a range space of compact multipliers
of mixed norm sequence spaces $l^{p,q}$, $0 \leq p,q \leq \infty$. In contrast to the general case we show that a compact multiplier
always remain compact under reduction of the final space to the range space of multipliers.

**Keywords.** Range of compact multipliers, Mixed-norm sequence space, Measures of noncompactness,
Range space for compact operators, Hardy's inequality, Mean operators.

Some Characterizations of Lorentzian Spherical Space-like Curves

Mathematica Moravica, Vol. **3** (1999), 33–37.

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Abstract and keywords

**Abstract.** In this paper, some characterizations of a unit speed space-like curve
whose image lies on a Lorentzian sphere in $R_{1}^{3}$ Minkowski 3-space are given.

**Keywords.** Lorentzian spherical curves, Minkowski's space, Minkowski 3-space, Lorentzian sphere,
Frenet trihedron, space-like curve, time-like curve.

Difference Equations and New Equivalents of the Kurepa Hypothesis

Mathematica Moravica, Vol. **3** (1999), 39–42.

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Abstract and keywords

**Abstract.** By making use of the difference equations we deduce, for the first time,
two new equivalent statements of the Guy's unsolved problem B44, also known as the Kurepa hypothesis for the left factorial,

**Keywords.** Kurepa's hypothesis, left factorial, difference equations.

A Fixed Point Theorem for Mappings in $d$-complete Topological Spaces

Mathematica Moravica, Vol. **3** (1999), 43–48.

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Abstract and keywords

**Abstract.** A general fixed point for four mappings satisfying an implicit
relation in $d$-complete topological spaces which generalize Theorem 3.7 of [1] is proved.

**Keywords.** $d$-complete topological spaces, semi-compatible mappings, fixed points.

A Note on Generalized Inverse Functions

Mathematica Moravica, Vol. **3** (1999), 49–52.

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Abstract and keywords

**Abstract.** Robin Harte generalized an observation of Hochwald and Morell.
In this note we offer a generalization of Harte's result.

**Keywords.** Generalized inverses, Drazin inverse (invertible), Harte's observation,
Observation of Hochwald and Morell, Generalized Drazin inverse.

Über eine Zahlentheoretische Funktion

Mathematica Moravica, Vol. **3** (1999), 53–62.

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Abstract and keywords

**Abstract.** We prove elementary and asymptotic properties of an arithmetical function.

**Keywords.** Arithmetical functions, multiplicative theory, asymptotic theory of arithmetical functions.

A Remark on the Location of the Zeros of Polynomials

Mathematica Moravica, Vol. **3** (1999), 63–66.

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Abstract and keywords

**Abstract.** In this paper we determine, in the complex plane, regions containing
the zeros of the polynomial
\[P(z)=z^n+a_1z^{n-1}+a_2z^{n-2}+ \cdots +a_{n-1}z+a_n,\qquad n \geq 3.\]
We also obtain two expressions which represent upper bounds for the moduli of the zeros of $P(z)$
with greater precision than those obtained by Cauchy and P. Montel.

**Keywords.** Zeros of polynomials, upper bounds for zeros moduli.

A Procedure for Obtaining Iterative Formulas of Higher Order

Mathematica Moravica, Vol. **3** (1999), 67–75.

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Abstract and keywords

**Abstract.** In this paper a procedure for obtaining iterative formulas of higher order is obtained.
In particular, a family of iterative formulas of higher order is given. The family includes several already known results.

**Keywords.** Iteration formulae, approximate solutions of equations.

Fixed Points on Transversal Probabilistic Spaces

Mathematica Moravica, Vol. **3** (1999), 77–82.

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Abstract and keywords

**Abstract.** In this paper we introduce a notion of the probabilistic contraction
on a lower transversal probabilistic space and prove two fixed point statements. The lower transversal probabilistic spaces are
a natural extension of Menger's and probabilistic spaces.

**Keywords.** Fixed points, upper or lower transversal probabilistic spaces, Menger's space, probabilistic contraction,
lower (probabilistic) transverse, bisection functions.

Generalization of Hardy-Littlewood-Pólya Majorization Principle

Mathematica Moravica, Vol. **3** (1999), 83–92.

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Abstract and keywords

**Abstract.** This paper continues the study of the general convex functions.
In this paper we extension our the former objects of the function in contact and of the function in circled contact.

The following main result is proved: Let $J \subset \mathbf{R}$ be an open interval and let $x_{i},y_{i}\in J$, $(i=1,\dots,n)$,
be real numbers such that fulfilling
\[x_{i}\geq \cdots \geq x_{n},\qquad y_{i}\geq \cdots \geq y_{n}.\]
Then, a necessary and sufficient condition in order that
\[\sum_{i=1}^{n} f(x_{i}) \geq 2\sum_{i=1}^{n} f(y_{i})-n \max\Bigl\{f(a), f(b), g\bigl(f(a), f(b)\bigr)\Bigr\}\]
holds for every general convex function $f: J\to \mathbf{R}$ which is in contact with function $g: f^2(J) \to \mathbf{R}$ and for arbitrary
$a,b \in J$, ($a \leq x_{i}\leq b$ for $i=1,\dots,n$), is that
\[\sum_{i=1}^{k} y_{i}\leq \sum_{i=1}^{k} x_{i},\quad (k=1, \dots, n-1),\quad \sum_{i=1}^{n} y_{i} = \sum_{i=1}^{n} x_{i}.\]
**Keywords.** General convex functions, Convex functions, Inequalities, Majorization principle,
Hardy-Littlewood-Pólya majorization principle, Karamata theorem, Characterization of majorization.

New Measures of Noncompactness

Mathematica Moravica, Vol. **3** (1999), 93–96.

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Abstract and keywords

**Abstract.** In this paper are consider some new measures of noncompactness.
They notations have proved useful in several areas of nonlinear functional analysis. Also, they measures of noncompactness
suggested by some simple characterizations of relative compactness.

**Keywords.** Measures of noncompactness, Cantor intersection theorem, Kuratowski's function, Darbo's theorem,
Sadovskij's theorem, Hausdorff's function, Istrăţescu's function, Fixed point theory.

Continuity of General J-convex Functions

Mathematica Moravica, Vol. **3** (1999), 97–104.

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Abstract and keywords

**Abstract.** In this paper we continue the study of the general J-convex functions,
which are introduced in our former paper (Tasković, Math. Japonica, 37 (1992), 367-372).
We prove that if $D \subset \mathbb{R}^n$ a convex and open set, and if $f: D \to \mathbb{R}$ is a general J-inner function
with the property of local oscillation in $D$, then it is continuous in $D$.

Since every J-convex function (also an additive function) is general J-inner function, we obtain as a particular case of
the preceding statement the result of F. Bernstein and G. Doetsch.

**Keywords.** Convex functions, Quasiconvex functions, General convex functions, Extremal problems,
Measurability, Continuity, General inner functions.

Characterization of General Convex Functions and its Applications

Mathematica Moravica, Vol. **3** (1999), 105–110.

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Abstract and keywords

**Abstract.** In this paper we continue the study of the general convex functions,
which we introduced in our former paper (Tasković, Math. Japonica, 37 (1992), 367-372).
This paper presents a new characterization of general convex functions in term of general level sets.
Applications in convex analysis are considered.

**Keywords.** Convex functions, Quasi-convex functions, General convex functions, Extremal problems,
Level sets, General level sets, Characterization of general convexity.

A Note on Topological $n$-groups

Mathematica Moravica, Vol. **3** (1999), 111–115.

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**Abstract.** In the present paper is proved the following proposition.
Let $(Q,A)$ be an $n$-group, $^{-1}$ its inversing operation, $n \geq 2$ and $Q$ is equipped with a topology $\mathcal{O}$. Also let
\[^{-1}A(x,a_{1}^{n-2},y)=z\quad \overset{def}{\iff}\quad A(z,a_{1}^{n-2},y)=x,\]
\[^{-1}A(x,a_{1}^{n-2},y)=z\quad \overset{def}{\iff}\quad A(x,a_{1}^{n-2},z)=y,\]
for all $x,y,z \in Q$ and for every sequence $a_{1}^{n-2}$ over $Q$. Then the following statements are equivalent:

(i) the $n$-ary operation $A$ is continuous in $\mathcal{O}$ and the $(n-1)$-ary operation $^{-1}$ is continuous in $\mathcal{O}$;

(ii) the $n$-ary operation $^{-1}A$ is continuous in $\mathcal{O}$; and

(iii) the $n$-ary operatin $A^{-1}$ is continuous in $\mathcal{O}$. [See, also Remark 2.2.]

**Keywords.** $n$-semigroups, $n$-quasigroup, $n$-group, $\{1,n\}$-neutral operations on $n$-groupoids,
inversing operation on $n$-group, topological $n$-groups.

A Comment of Power in $n$-group

Mathematica Moravica, Vol. **3** (1999), 117–126.

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**Abstract.** Let $n \geq 2$, Let $(Q,A)$ be an $n$-group, $e$ its $\{1,n\}$-neutral operation
[:[6], 1.3], and $^{-1}$ its inversing operation [:[7], 1.3]. Let also $Z$ be an set of all integers.
Then, in this paper, we say that $a^{m}$, $(m \in Z)$, is an $m$-th power of the element $a$ in $(Q,A)$ iff: (1) $a^{1}\overset{def}{=}a$;
(2) $a^{k+1}\overset{def}{=}A(a^{k},\overset{n-2}{a},a)$, $k\geq 1$; (3) $a^{0}\overset{def}{=}e(\overset{n-2}{a})$, and
(4) $a^{-k}\overset{def}{=}(\overset{n-2}{a},a^{k})^{-1}$, $k\geq 1$ [:2.3]. Furthermore, for all $a \in Q$ and for all
$s \in Z$ the following equality holds: $a^{\langle s\rangle}=a^{s+1}$, where $a^{\langle s\rangle}$
well-known is the $s$-th $n$-adic power of the element $a$ in $(Q,A)$ [:2.5]; $\langle s\rangle =s(n-1)+1$.
Among others, in the paper is proved the following proposition. For every $\alpha, \alpha_{1}, \dots, \alpha_{n}\in Z$, $(n \geq 3)$,
the following equalities hold:
\[e(a^{\alpha_1}, \dots, a^{\alpha_{n-2}})=a^{-\sum_{i=1}^{n-2}\alpha_{i}+n-2},\]
\[(a^{\alpha_1}, \dots, a^{\alpha_{n-2}}, a^{\alpha})^{-1}=a^{-\alpha-2(\sum_{i=1}^{n-2}\alpha_i-n+2)},\]
\[A(a^{\alpha_1}, \dots, a^{\alpha_n})=a^{\sum_{i=1}^{n}\alpha_i-n+2}\text{ [:2.7,2.8,footnote 4)]}.\]
**Keywords.** $n$-semigroups, $n$-quasigroups, $n$-groups, $\{1,n\}$-neutral operations on $n$-groupoids,
inversing operation on $n$-group, $nHG$-algebras.

Note on $(n,m)$-groups

Mathematica Moravica, Vol. **3** (1999), 127–139.

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**Abstract.** Among the results of the paper is the following proposition.
Let $2m \leq n < 3m$ and let $(Q,A)$ be an $(n,m)$-groupoid $(n,m \in N)$. Then, $(Q,A)$ is an $(n,m)$-group iff there
are mappings $^{-1}$ and $\mathbf{e}$ respectively of the sets $Q^{n-m}$ and $Q^{n-2m}$ into the set $Q^{m}$ such that the
following laws hold in the algebra $(Q,A,^{-1},\mathbf{e})$:
\[A(A(x_{1}^{n}),x_{n+1}^{2n-m}) = A(x_{1},A(x_{2}^{n+1},x_{n+2}^{2n-m}),\]
\[A(A(x_{1}^{n}),x_{n+1}^{2n-m}) = A(x_{1}^{n-m},A(x_{n-m+1}^{2n-m})),\]
\[A(x_{1}^{m}, a_{1}^{n-2m}, e(a_{1}^{n-2m})) = x_{1}^{m},\]
\[A(x_{1}^{m}, a_{1}^{n-2m}, (a_{1}^{n-2m},x_{1}^{m})^{-1}) = \mathbf{e}(a_{1}^{n-2m}).\]

**Keywords.** $(n,m)$-groupoids, $(n,m)$-groups, $\{i,j\}$-neutral operations on $(n,m)$-groupoids,
inversing operation on $(n,m)$-groups.