We will prove that every pair of quasi-arithmetic means satisfying certain smoothness assumption has the supremum and the infimum in this set. More precisely, if f and g are C2 functions with nowhere ...vanishing first derivative then there exists a function h such that: (i) Af≤Ah, (ii) Ag≤Ah, and (iii) for every continuous strictly monotone function s:I→RAf≤AsandAg≤AsimpliesAh≤As (Af stands for a quasi-arithmetic mean). Moreover, h∈C2, h′≠0, and it is a solution of the differential equationh″h′=max(f″f′,g″g′). We also provide some extension to a finite family of means and dual (reflected) result.
We study the properties of the mean-type mappings
M
:
I
p
→
I
p
of the form
M
(
x
1
,
⋯
,
x
p
)
:
=
(
M
1
(
x
α
1
,
1
,
⋯
,
x
α
1
,
d
1
)
,
⋯
,
M
p
(
x
α
p
,
1
,
⋯
,
x
α
p
,
d
p
)
)
,
where
p
and
d
i
...-s are positive integers, each
M
i
is a
d
i
-variable mean on an interval
I
⊂
R
, and
α
i
,
j
-s are elements from
{
1
,
⋯
,
p
}
. We show that, under some natural assumption on
M
i
-s, the problem of existing the unique
M
-invariant mean can be reduced to the ergodicity of the directed graph with vertexes
{
1
,
⋯
,
p
}
and edges
{
(
α
i
,
j
,
i
)
:
i
,
j
admissible
}
.
We show that, for a (not necessarily continuous) weakly contractive mean-type mapping
M
:
I
p
→
I
p
(where
I
is an interval and
p
∈
N
), the functional equation
K
∘
M
=
K
has at most one solution in ...the family of continuous means
K
:
I
p
→
I
. Some general approach to this equation is also given.
We extend some approach to the family of symmetric means (i.e. symmetric functions
M
:
⋃
n
=
1
∞
I
n
→
I
with
min
≤
M
≤
max
;
I
is an interval). Namely, it is known that every symmetric mean can be ...written in a form
M
(
v
1
,
⋯
,
v
n
)
:
=
F
(
f
(
v
1
)
+
⋯
+
f
(
v
n
)
)
, where
f
:
I
→
G
and
F
:
G
→
I
(
G
is a commutative semigroup). For
G
=
R
k
or
G
=
R
k
×
Z
(
k
∈
N
) and continuous functions
f
and
F
we obtain two series of families (depending on
k
). It can be treated as a measure of complexity in a family of means (this idea is inspired by theory of regular languages and algorithmics). As a result we characterize the celebrated families of quasi-arithmetic means (
G
=
R
×
Z
) and Bajraktarević means (
G
=
R
2
under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families.
Error recognition in the Cantor cube Pasteczka, Paweł
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica,
01/2023, Volume:
22, Issue:
1
Journal Article
Peer reviewed
Open access
Based on the notion of thin sets introduced recently by T. Banakh, Sz. Głąb, E. Jabłońska and J. Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are ...associated with a set of admissible messages (i.e. subsets of the Cantor cube
).
Using Banach-Mazur games we prove that all protocols detecting errors are Baire spaces and generic (in particular maximal) ones are not neither Borel nor meager.
We also show that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit. This result is related to so-called xor-sets defined by D. Niwiński and E. Kopczyński in 2014.
The celebrated Hardy inequality can be written in the form
∫
0
∞
P
p
(
f
|
0
,
x
)
d
x
≤
(
1
-
p
)
-
1
/
p
∫
0
∞
f
(
x
)
d
x
for
p
∈
(
0
,
1
)
and
f
∈
L
1
with
f
≥
0
,
where
P
p
stands for the
p
...-th power mean. One can ask about possible generalizations of this property to another families (with sharp constant depending on the mean). Adapting the notion of Riemann integral, for every weighted mean we define the lower and the upper integral mean. We prove that every symmetric, monotone,
R
-weighted mean on
I
which is continuous in its entries and weights has at most one continuous extension to an integral one. Moreover this extension preserves the Hardy constant. This result allows to extend the latter inequality to the family of homogeneous, concave deviation means.
We define decision-making functions which arise from studying the multidimensional generalization of the weighted Bajraktarević means. It allows a nonlinear approach to optimization problems. These ...functions admit several interesting (from the point of view of decision-making) properties, for example, delegativity (which states that each subgroup of decision-makers can aggregate their decisions and efforts), casuativity (each decision affects the final outcome except two trivial cases) and convexity-type properties. Beyond establishing the most important properties of such means, we solve their equality problem, we introduce a notion of synergy and characterize the null-synergy decision-making functions of this type.
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