Let
X, Y
be sets and let
Φ
,
Ψ
be mappings with the domains
X
2
and
Y
2
respectively. We say that
Φ
is
combinatorially similar
to
Ψ
if there are bijections
f
:
Φ
(
X
2
)
→
Ψ
(
Y
2
)
and
g
:
Y
→
X
...such that
Ψ
(
x
,
y
)
=
f
(
Φ
(
g
(
x
)
,
g
(
y
)
)
)
for all
x
,
y
∈
Y
. It is shown that the semigroups of binary relations generated by sets
{
Φ
-
1
(
a
)
:
a
∈
Φ
(
X
2
)
}
and
{
Ψ
-
1
(
b
)
:
b
∈
Ψ
(
Y
2
)
}
are isomorphic for combinatorially similar
Φ
and
Ψ
. The necessary and sufficient conditions under which a given mapping is combinatorially similar to a pseudometric, or strongly rigid pseudometric, or discrete pseudometric are found. The algebraic structure of semigroups generated by
{
d
-
1
(
r
)
:
r
∈
d
(
X
2
)
}
is completely described for nondiscrete, strongly rigid pseudometrics and, also, for discrete pseudometrics
d
:
X
2
→
R
.
On quasinearly subharmonic functions Dovgoshey, O.; Riihentaus, J.
Lobachevskii journal of mathematics,
03/2017, Letnik:
38, Številka:
2
Journal Article
Recenzirano
We recall the definition of quasinearly subharmonic functions, point out that this function class includes, among others, subharmonic functions, quasisubharmonic functions, nearly subharmonic ...functions and essentially almost subharmonic functions. It is shown that the sum of two quasinearly subharmonic functions may not be quasinearly subharmonic. Moreover, we characterize the harmonicity via quasinearly subharmonicity.
We study the conditions for the density of a subsequence of a statistically convergent sequence under which this subsequence is also statistically convergent. Some sufficient conditions of this type ...and almost converse necessary conditions are obtained in the setting of general metric spaces.
We study the properties of real functions
f
for which the compositions
f
◦
d
is a metric for every metric space (
X
,
d
). The explicit form is found for the invertible elements of the semigroup
of ...all such functions. The increasing functions
are characterized by the subadditivity condition and a maximal inverse subsemigroup in the set of these functions is explicitly described. The upper envelope of the set of functions
with
f
(1) = 1 is found and it leads to the exact constant in Harnack’s inequality for functions from
.
Let (
X
,
d
X
) and (
Y
,
d
Y
) be semimetric spaces with distance sets
D
(
X
) and
D
(
Y
), respectively. A mapping
F
:
X
→
Y
is a weak similarity if it is surjective and there exists a strictly ...increasing
f
:
D
(
Y
)→
D
(
X
) such that
d
X
=
f
∘
d
Y
∘(
F
⊗
F
). It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity
F
:
X
→
Y
is an isometry if
X
and
Y
are ultrametric and compact with
D
(
X
)=
D
(
Y
). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.
On spaces extremal for the Gomory-Hu inequality Dovgoshey, O.; Petrov, E.; Teichert, H. -M.
P-adic numbers, ultrametric analysis, and applications,
04/2015, Letnik:
7, Številka:
2
Journal Article
Recenzirano
Let (
X, d
) be a finite ultrametric space. In 1961 E.C. Gomory and T.C. Hu proved the inequality |Sp(
X
)| ⩽ |
X
| where Sp(
X
) = {
d
(
x, y
):
x, y
∈
X
}. Using weighted Hamiltonian cycles and ...weighted Hamiltonian paths we give new necessary and sufficient conditions under which the Gomory-Hu inequality becomes an equality. We find the number of non-isometric (
X, d
) satisfying the equality |Sp(
X
)| = |
X
| for given Sp(
X
). Moreover it is shown that every finite semimetric space
Z
is an image under a composition of mappings
f
:
X
→
Y
and
g
:
Y
→
Z
such that
X
and
Y
are finite ultrametric spaces,
X
satisfies the above equality,
f
is an
ɛ
-isometry with an arbitrary
ɛ
> 0, and
g
is a ball-preserving map.
Let ℝ
+
= 0,∞) and let
A
⊆ ℝ
+
n
. We have found the necessary and sufficient conditions under which a function Φ:
A
→ ℝ
+
has an isotone subadditive continuation on ℝ
+
n
. It allows us to describe ...the metrics, defined on the Cartesian product
X
1
×...×
X
n
of given metric spaces
, generated by the isotone metric preserving functions on ℝ
+
n
. It is also shown that the isotone metric preserving functions Φ: ℝ
+
n
→ ℝ
+
coincide with the first moduli of continuity of the nonconstant bornologous functions
g
: ℝ
+
n
→ ℝ
+
.
Let \(X\), \(Y\) be sets and let \(\Phi\), \(\Psi\) be mappings with domains \(X^{2}\) and \(Y^{2}\) respectively. We say that \(\Phi\) and \(\Psi\) are combinatorially similar if there are ...bijections \(f \colon \Phi(X^2) \to \Psi(Y^{2})\) and \(g \colon Y \to X\) such that \(\Psi(x, y) = f(\Phi(g(x), g(y)))\) for all \(x\), \(y \in Y\). Conditions under which a given mapping is combinatorially similar to an ultrametric or a pseudoultrametric are found. Combinatorial characterizations are also obtained for poset-valued ultrametric distances recently defined by Priess-Crampe and Ribenboim.
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