Ultra Exponent Matrices
The paper studies ultra exponent matrices, that is, reduced exponent matrices for which the usual triangle inequality is strengthened to an ultrametric-type inequality. It is proved that every ultra exponent matrix is an exponent matrix, and that every exponent matrix whose entries belong to {0, 1}...
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| author | Зеленський, Олексій |
| author_facet | Зеленський, Олексій |
| author_institution_txt_mv | [
{
"author": "Олексій Зеленський",
"institution": "Kamianets-Podilskyi Ivan Ohiienko National University"
}
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| description | The paper studies ultra exponent matrices, that is, reduced exponent matrices for which the usual triangle inequality is strengthened to an ultrametric-type inequality. It is proved that every ultra exponent matrix is an exponent matrix, and that every exponent matrix whose entries belong to {0, 1} is an ultra exponent matrix. The behaviour of ultra matrices under elementary transformations is analysed: transformations of the first type do not preserve the ultra property in general, while simultaneous permutations of rows and columns do preserve it. It is shown that a quiver obtained from a reduced ultra exponent matrix with at least one entry greater than one is not rigid. A counterexample demonstrates that not every admissible quiver with a loop at every vertex can be represented by an ultra exponent matrix. Several structural characterizations are also established, including monotone deformations preserving the associated quiver, a filtration description in terms of transitive threshold relations, a connection between 0-1 ultra matrices and partial orders, a minimax interpretation, and closure under componentwise maximum. From the viewpoint of mathematical modelling, ultra exponent matrices may be interpreted as discrete directed distance models with bottleneck-type constraints, where the value assigned to a transition is determined by the strongest restriction along admissible paths rather than by an additive cost. This makes them useful for modelling hierarchical systems, priority relations, minimax optimization, constrained network flows, clustering-like structures, and algebraic or combinatorial systems whose essential information is encoded by admissible weighted quivers. The obtained results provide tools for comparing such models and for reducing redundant representations without losing the underlying directed structure. |
| doi_str_mv | 10.32626/2308-5878.2026-30.72-90 |
| first_indexed | 2026-06-09T01:00:12Z |
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ISSN 2308-5878. Mathematical and computer modelling.
Series: Physical and mathematical sciences. 2026. Issue 30. P. 72-90.
72
UDC 512.5525
DOІ: 10.32626/2308-5878.2026-30.72-90
Zelenskiy O. V.
ORCІD: 0000-0002-4969-0132,
Cand. of Phys. and Math. Sciences, Kamianets-Podilskyi
Ivan Ohiienko National University, Kamianets-Podilskyi, Ukraine,
E-maіl: esteticcode@gmail.com
ULTRA EXPONENT MATRICES
The paper studies ultra exponent matrices, that is, reduced ex-
ponent matrices for which the usual triangle inequality is strength-
ened to an ultrametric-type inequality. It is proved that every ultra
exponent matrix is an exponent matrix, and that every exponent
matrix whose entries belong to {0, 1} is an ultra exponent matrix.
The behaviour of ultra matrices under elementary transformations
is analysed: transformations of the first type do not preserve the ul-
tra property in general, while simultaneous permutations of rows
and columns do preserve it. It is shown that a quiver obtained from
a reduced ultra exponent matrix with at least one entry greater than
one is not rigid. A counterexample demonstrates that not every
admissible quiver with a loop at every vertex can be represented by
an ultra exponent matrix. Several structural characterizations are
also established, including monotone deformations preserving the
associated quiver, a filtration description in terms of transitive
threshold relations, a connection between 0-1 ultra matrices and
partial orders, a minimax interpretation, and closure under compo-
nentwise maximum. From the viewpoint of mathematical model-
ling, ultra exponent matrices may be interpreted as discrete di-
rected distance models with bottleneck-type constraints, where the
value assigned to a transition is determined by the strongest re-
striction along admissible paths rather than by an additive cost.
This makes them useful for modelling hierarchical systems, priori-
ty relations, minimax optimization, constrained network flows,
clustering-like structures, and algebraic or combinatorial systems
whose essential information is encoded by admissible weighted
quivers. The obtained results provide tools for comparing such
models and for reducing redundant representations without losing
the underlying directed structure.
Key words: exponent matrix, reduced exponent matrix, admis-
sible quiver, ultra exponent matrix, rigid quiver, weight function,
mathematical modelling.
5 Стаття надійшла до редакції: 14.05.2026
Рекомендовано до друку: 24.05.2026
Оприлюднено (online): 29.05.2026
Ця стаття розповсюджується на умовах ліцензії CC Attribution-NonCommercial-NoDerivatives 4.0
© Zelenskiy O. V., 2026
ISSN 2308-5878. Mathematical and computer modelling.
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73
Introduction. One important aspect of ring theory is the study of
properties of rings by means of graph-theoretic methods. Every tiled order
over a discrete valuation ring is completely determined by its exponent
matrix and by the discrete valuation ring itself [1]. Many structural proper-
ties of such orders are determined by their exponent matrices, in particular
by the corresponding quivers [1-3].
Exponent matrices have relatively recently become an independent ob-
ject of study. In [4-8], admissible quivers, rigid quivers, unit cycles, unit quiv-
ers, and weight functions defining admissible quivers were investigated. The
introduction of weight functions made it possible to formulate properties of
exponent matrices in the language of weighted directed graphs.
In this paper we study ultra exponent matrices. Their main feature is
that they satisfy not only the usual triangle inequality
,ij jk ik
but also the stronger ultrametric-type inequality
max , .ij jk ik
Throughout the paper, as is customary for exponent matrices of tiled or-
ders, the entries of exponent matrices are assumed to be nonnegative integers.
From the viewpoint of mathematical modelling, an ultra exponent
matrix can be regarded as an integer-valued directed distance table in
which the distance from one vertex to another is controlled by the largest
restriction on intermediate paths. Such a model is useful when a dominant
transition cost, bottleneck constraint, hierarchy level, or priority relation is
more significant than the sum of local costs. Therefore, ultra matrices may
be applied to discrete network models, minimax optimization, clustering-
type structures, and algebraic models in which admissible directed graphs
encode essential structural constraints.
Basic definitions.
Definition 1.1 [1, p. 353]. A matrix
0ij nM
is called an exponent matrix if the following conditions hold:
for all , , 1, , ,ij jk ik i j k n
0 for all 1, , .ii i n
An exponent matrix is called reduced if, in addition, the following
condition holds:
1 for all , 1, , , .ij ji i j n i j
Let ij be a reduced exponent matrix. Introduce the matrix
1
,ij n nE M
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where nE is the identity matrix. Also introduce the matrix
2
1
, min .ij n ij ik kj
k n
M
Definition 1.2 [1, p. 357]. The quiver of a reduced exponent matrix is
the quiver Q Q whose adjacency matrix is defined by the formula
2 1
.Q
Definition 1.3 [1]. Reduced exponent matrices 1 and 2 are called
equivalent if one of them can be obtained from the other by a finite se-
quence of elementary transformations of the following two types:
1) subtract an integer t from all entries of the i -th row and add the same
integer t to all entries of the i -th column;
2) interchange two rows and simultaneously interchange the two columns
with the same numbers.
Definition 1.4 [1, p. 357]. A quiver Q is called admissible if there
exists a reduced exponent matrix such that
.Q Q
Definition 1.5. A quiver ,Q QQ V A is called weighted if a function
: QA
is defined on the set of its arrows. The function is called a weight func-
tion, and its value on an arrow is called the weight of this arrow. The sum
of the weights of all arrows along a path is called the weight of this path.
Theorem 1 [4]. A strongly connected quiver ,Q QQ V A is admis-
sible if and only if there exists a weight function satisfying the follow-
ing conditions:
1) the weight of an arrow from vertex i to vertex j is smaller than the
weight of any path from i to j of length 2l ;
2) the weight of a loop at vertex i is smaller than the weight of any cycle
of length 2l passing through vertex i ;
3) the weight of every cycle is greater than or equal to 1 ;
4) the weight of every loop is equal to 1 ;
5) through every vertex without a loop there passes a cycle of length
2l whose weight is equal to 1 .
Remark 1.1. Conditions (4) and (5) imply that through every vertex
of an admissible quiver there passes a cycle of weight 1 .
Definition 1.6. A weight function satisfying all conditions of Theo-
rem 1 is called an admissible weight function.
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Construction. Suppose that a quiver Q and an admissible weight
function are given. Then one can construct an exponent matrix
ij as follows: if the quiver Q contains an arrow ij , then
;ij ij
if such an arrow does not exist, then ij is defined as the weight of a lightest,
that is, a minimum-weight, path from the vertex iv to the vertex jv .
Definition 1.7. A simple cycle in an admissible quiver whose weight
is equal to 1 is called a unit cycle.
Definition 1.8 [6]. An admissible quiver Q is called rigid if there ex-
ists, up to equivalence, a unique reduced exponent matrix such that
.Q Q
Main results.
Definition 2.1. A matrix 0ij nM is called an ultra ex-
ponent matrix if the following conditions hold:
max , for all , , 1, , ,ij jk ik i j k n
0 for all 1, , .ii i n
If, in addition, the condition
1 for all , 1, , ,ij ji i j n i j
holds, then is called a reduced ultra exponent matrix. In what fol-
lows, unless otherwise stated, an ultra exponent matrix means a reduced
ultra exponent matrix.
Proposition 2.1. Every ultra exponent matrix is an exponent matrix.
Proof. Let ij be an ultra exponent matrix. By definition,
max ,ij jk ik
for all , ,i j k . Since all entries of the matrix are nonnegative, we have
max , .ij jk ij jk
Therefore,
.ij jk ik
Moreover, by the definition of an ultra matrix, 0ii for all i . Thus
satisfies all conditions in the definition of an exponent matrix. The
proposition is proved.
Proposition 2.2. An exponent matrix with entries 0 and 1 is an ul-
tra exponent matrix.
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Proof. Let ij be an exponent matrix, and suppose that all its en-
tries belong to the set 0,1 . We must prove that, for all , ,i j k , the inequality
max ,ij jk ik
holds. If 0ik , the inequality is obvious. Let 1ik . Since is an
exponent matrix, we have
1.ij jk ik
Because , 0,1ij jk , this inequality implies that at least one of
the two entries ij and jk is equal to 1 . Hence
max , 1 .ij jk ik
Thus the ultrametric inequality holds for all , ,i j k . The proposition
is proved.
Proposition 2.3. An elementary transformation of the first type does
not, in general, preserve the property of being an ultra matrix, whereas an
elementary transformation of the second type preserves this property.
Proof. First we show that a transformation of the second type pre-
serves the property of being an ultra matrix. Such a transformation only
simultaneously permutes rows and columns with the same numbers; that
is, it simply renumbers the vertices. If the inequality
max ,ij jk ik
held for the original matrix, then after a simultaneous permutation of rows
and columns the same inequality holds for the correspondingly renum-
bered indices. Hence a transformation of the second type preserves the
property of being an ultra matrix.
Now we show that a transformation of the first type does not pre-
serve this property in general. Consider the ultra matrix
0 0 0
1 0 0 .
2 2 0
It is a reduced ultra exponent matrix. Perform an elementary trans-
formation of the first type for the first row and the first column with pa-
rameter 1t , that is, add 1 to all entries of the first row and subtract 1
from all entries of the first column. We obtain the matrix
0 1 1
' 0 0 0 .
1 2 0
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The matrix ' is no longer an ultra matrix. Indeed, for the indices
3i , 1j , 2k we have
31 12 32max ' , ' max 1,1 1 2 ' .
Thus the ultrametric inequality is violated. The proposition is
proved.
Proposition 2.4. If all off-diagonal entries of a reduced ultra expo-
nent matrix are positive, then the quiver obtained from it is a complete
quiver with a loop at every vertex.
Proof. Let ij be a reduced ultra exponent matrix and suppose that
0 for all .ij i j
We show that, for every pair of distinct vertices i and k , the quiver
Q contains an arrow i k .
For i k , the arrow i k is absent if and only if there exists an in-
dex ,j i k such that
.ik ij jk
However, all off-diagonal entries are positive, so
max , .ij jk ij jk
On the other hand, the ultra inequality gives
max , .ik ij jk
Therefore, the equality ik ij jk is impossible. Hence the ar-
row i k exists for all i k .
Moreover, for each i and each j i we have 0ij and 0ji ,
and hence
2.ij ji
This means that the quiver has a loop at every vertex. Therefore the
quiver is complete and has a loop at every vertex. The proposition is proved.
We shall use the following auxiliary lemma in the proof of the main
theorem.
Lemma 2.5. Let ij be a reduced ultra exponent matrix. De-
fine the function
0, 0,
1, 1,
1, 2,
x
x x
x x
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and define the matrix ij by
.ij ij
Then is a reduced ultra exponent matrix and
.Q Q
Proof. Since 0 0 , we have 0ii for all i . The function is
strictly increasing on the set of nonnegative integers. Therefore, from the
inequality
max ,ik ij jk
we obtain
max , max , max , .ik ik ij jk ij jk ij jk
Hence is an ultra matrix. Reducedness is also preserved: if i j ,
then the inequality 1ij ji implies that at least one of the entries ij
and ji is positive; therefore at least one of the entries ij and ji is also
positive, and hence
1.ij ji
It remains to prove that the quivers Q and Q coincide.
For i k , the arrow i k is absent in Q if and only if there ex-
ists an index ,j i k such that
.ik ij jk
We prove that this equality is preserved after applying the function
. Suppose that
.ik ij jk
Since the matrix is ultra, we have
max , .ik ij jk
For nonnegative numbers, however, one always has
max , .ij jk ij jk
Therefore the equality ik ij jk can hold only when one of the
two summands ij and jk is zero and the other is equal to ik . Hence,
after applying , we again have
.ik ij jk
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Conversely, if
,ik ij jk
then the ultra inequality for the matrix implies that
max , .ik ij jk
Since the left-hand side is the sum of two nonnegative numbers, this is
possible only when one of the summands ij and jk is zero. By injectivity
of on 0 , we obtain the corresponding equality for the original entries:
.ik ij jk
Thus, for all i k , the absence or presence of the arrow i k is the
same in Q and in Q .
For loops we argue as follows: a loop at the vertex i is absent if and
only if there exists j i such that
1.ij ji
The function does not change the values 0 and 1 , so the equality
1ij ji holds if and only if
1.ij ji
Therefore loops are also preserved. Consequently,
.Q Q
The lemma is proved.
Theorem 2. A quiver obtained from an ultra matrix having at least
one entry greater than one is not rigid.
Proof. Let ij be a reduced ultra exponent matrix, and sup-
pose that there exists at least one entry 1pq . Construct the matrix
by the rule of Lemma 2.5:
,ij ij
where
0, 0,
1, 1,
1, 2.
x
x x
x x
By Lemma 2.5, the matrix is a reduced ultra exponent matrix and
gives the same quiver:
.Q Q
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We show that the matrices and are not equivalent. Consider the
sum of all entries of a matrix
1 1
.
n n
ij
i j
S
An elementary transformation of the first type does not change this
sum: if a number t is subtracted from the i -th row, the total sum decreas-
es by nt ; if the same number t is added to the i -th column, the total sum
increases by nt . Thus the total change is zero. An elementary transfor-
mation of the second type merely permutes the entries of the matrix and
therefore also preserves their total sum.
Hence S is an invariant of equivalence of reduced exponent ma-
trices.
Since the matrix contains an entry 1pq , when passing to
this entry increases by 1 . All other entries either remain unchanged or also
increase by 1 if they are greater than 1 . Therefore
.S S
It follows that and cannot be equivalent. Thus the same quiver
Q is obtained from at least two nonequivalent reduced exponent matrices,
namely and . Therefore this quiver is not rigid. The theorem is proved.
The following result shows that the assertion that every admissible
quiver with a loop at every vertex can be obtained from an ultra matrix is
false in general.
Theorem 3. There exists an admissible quiver with a loop at every
vertex that cannot be obtained from any ultra exponent matrix.
Proof. Consider the quiver Q with the set of vertices
1,2,3,4QV
and the adjacency matrix
1 1 0 0
1 1 1 0
.
0 1 1 1
0 0 1 1
Q
This quiver has a loop at every vertex. Moreover, it contains the ar-
rows
1 2, 2 3, 3 4,
and contains no other non-loop arrows.
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First we show that Q is admissible. Define a weight function by
assigning weight 1 to every arrow, including the loops:
1 for every arrow .QA
Then the weight of any path of length l is equal to l . In particular, the
weight of every arrow is equal to 1 and is smaller than the weight of any path
of length 2l . The weight of every loop is also equal to 1 and is smaller
than the weight of any cycle of length 2l . The weight of every cycle is
greater than or equal to 1 , and the weight of every loop is equal to 1 . Since
every vertex has a loop, the condition concerning vertices without loops is
satisfied automatically. By Theorem 1, the quiver Q is admissible.
Now we prove that Q cannot be obtained from an ultra matrix. Sup-
pose, to the contrary, that there exists a reduced ultra exponent matrix
ij
such that
.Q Q
Denote the weights of the arrows along the path 1 2 3 4 by
12 23 34, , ,a b c
and the weights of the arrows in the opposite direction by
21 32 43' , ' , ' .a b c
Since the quiver Q has no arrow 1 4 , while the only simple di-
rected path from vertex 1 to vertex 4 is
1 2 3 4,
the corresponding matrix entry must be equal to the weight of the lightest
path from 1 to 4 :
14 .a b c
On the other hand, since is an ultra matrix, we successively obtain
13 max , ,a b
and also
14 13max , max , , .c a b c
Hence
14 max , , .a b c a b c
But , ,a b c are nonnegative integers, so always
max , , .a b c a b c
Thus
max , , .a b c a b c
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This is possible only if at least two of the numbers , ,a b c are equal
to zero.
Analogously, since Q has no arrow 4 1 , while the only simple di-
rected path from 4 to 1 is
4 3 2 1,
we obtain
41 ' ' '.c b a
The ultra inequality implies
41 max ', ', ' .c b a
Therefore at least two of the numbers ', ', 'a b c are equal to zero.
Consequently, among the three pairs
, ' , , ' , , 'a a b b c c
there is at least one pair in which both entries are equal to zero. Indeed,
among , ,a b c there are at least two zeros, and among ', ', 'a b c there are
also at least two zeros; therefore, for some adjacent pair of vertices, we
simultaneously have
0 and 0.ij ji
This contradicts the reducedness of the matrix, since for i j one
must have
1.ij ji
Thus the assumption that there exists an ultra matrix with
Q Q is false. Therefore there exists an admissible quiver with a loop
at every vertex that cannot be obtained from an ultra exponent matrix. The
theorem is proved.
Additional structural properties of ultra matrices. In this section
we present several additional results that clarify the internal structure of
ultra exponent matrices and show that they can be described not only by
an ultrametric-type inequality, but also by transitive relations, minimax
operations, and monotone deformations of values.
Theorem 4.1. Let 0ij nM be a reduced ultra exponent
matrix. Let
0 0:f
be a strictly increasing function such that
0 0, 1 1.f f
Construct the matrix
.ijf f
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Then f is a reduced ultra exponent matrix and
.Q f Q
Proof. Since is an ultra matrix, for all , ,i j k we have
max , .ik ij jk
The function f is strictly increasing, hence
max , .ik ij jkf f
Since f is increasing, we have
max , max , .ij jk ij jkf f f
Therefore,
max , .ik ij jkf f f
This means that the matrix f satisfies the ultrametric inequality.
Moreover, since 0ii and 0 0f , we have
0iif
for all i . Hence f is an ultra matrix.
Let us verify reducedness. For i j , we have
1.ij ji
Since the entries are nonnegative, at least one of the entries ij and ji
is not less than 1 . By strict monotonicity of f and the equality 1 1f , the
corresponding entry after applying f is also not less than 1 . Therefore
1.ij jif f
Thus f is a reduced ultra exponent matrix.
It remains to prove that the quiver does not change. For distinct i
and k , the arrow i k is absent in Q if and only if there exists an
index ,j i k such that
.ik ij jk
In the case of an ultra matrix, such an equality is possible only when
one of the summands ij and jk is zero. Indeed, if both summands are
positive, then
max , ,ij jk ij jk
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whereas the ultra inequality gives
max , ,ik ij jk
which contradicts the equality ik ij jk . Hence, if such an equality
holds, one summand is zero and the other is equal to ik . Since 0 0f ,
after applying f we obtain
.ik ij jkf f f
Conversely, if
,ik ij jkf f f
then, applying the already proved ultra inequality to the matrix f , we
again obtain that one summand on the right-hand side is zero. Because
0 0f and f is strictly increasing, the corresponding original sum-
mand is also zero. Thus the decomposition equality for and for f
holds simultaneously.
Hence all non-loop arrows in the quivers Q and Q f coincide.
For loops the preservation is analogous. A loop at vertex i is absent
if and only if there exists j i such that
1.ij ji
Since 0 0f , 1 1f , and strict monotonicity implies 1f x
for 2x , an equality with sum 1 is preserved if and only if it held before
applying f . Thus loops are also preserved. Therefore
.Q f Q
The theorem is proved.
Remark 4.2. Lemma 2.5 is a special case of Theorem 4.1 for the
function
0, 0,
1, 1,
1, 2.
x
f x x
x x
This special case is used in the proof of non-rigidity of quivers ob-
tained from ultra matrices having at least one entry greater than one.
Theorem 4.3. Let 0ij nM be a matrix with zero main
diagonal. For each integer 0r , define a relation rR on the set
1,2, ,n by
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.r ijiR j r
Then is an ultra exponent matrix if and only if, for every 0r ,
the relation rR is transitive.
Proof. Suppose first that is an ultra matrix. Assume that riR j and
rjR k . This means that
, .ij jkr r
Then
max , .ij jk r
By the ultra inequality,
max , .ik ij jk r
Hence riR k . Thus rR is transitive for every 0r .
Conversely, suppose that for every 0r the relation rR is transi-
tive. Take arbitrary indices , ,i j k and put
max , .ij jkr
Then
, ,ij jkr r
that is, riR j and rjR k . By transitivity of rR , we have riR k , that is,
max , .ik ij jkr
This is precisely the ultra inequality. Therefore is an ultra matrix.
The theorem is proved.
Corollary 4.4. If ij is a reduced ultra exponent matrix, then
the relation on the set 1,2, ,n defined by
0iji j
is a partial order.
Proof. Reflexivity follows from the equality 0ii . Transitivity fol-
lows from Theorem 4.3 for 0r : if 0ij and 0jk , then 0ik .
Let us verify antisymmetry. If for i j the equalities
0, 0ij ji
held simultaneously, then
0,ij ji
which contradicts the reducedness of the matrix. Hence is a partial
order. The corollary is proved.
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Corollary 4.5. Reduced ultra exponent matrices with entries 0 and 1
are in one-to-one correspondence with partial orders on the set 1,2, ,n .
More precisely, if a partial order is given on the set 1,2, ,n ,
then the corresponding matrix is defined by
0, ,
1, .
ij
i j
i j
Proof. Let be a reduced 0,1-ultra matrix. By Corollary 4.4, the re-
lation
0iji j
is a partial order.
Conversely, suppose that a partial order is given on the set
1,2, ,n and that the matrix ij is defined by the formula above.
Clearly, 0ii for all i . If 0ik , then the ultra inequality
max ,ik ij jk
holds automatically. Let 1ik . This means that i k . If both 0ij
and 0jk , then we would have i j and j k , whence, by transitivi-
ty, i k . This contradicts the condition 1ik . Therefore at least one of
the entries ij and jk is equal to 1 , and hence
max , 1,ij jk
so the ultra inequality holds.
Reducedness also holds. If i j , then by antisymmetry of the partial
order the relations i j and j i cannot hold simultaneously. Therefore
the two entries ij and ji cannot both be equal to zero. Hence
1.ij ji
The corollary is proved.
Theorem 4.6. For matrices ijA a and ijB b with nonnega-
tive entries, define the minimax operation
1
min max , .ik kjij k n
A B a b
Let 0ij nM be a matrix with zero main diagonal. Then
is an ultra exponent matrix if and only if
.
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87
Proof. Let be an ultra matrix. Then for all , ,i j k we have
max , .ij ik kj
Therefore,
1
min max , .ij ik kj
k n
On the other hand, if we take k i , then
max , max 0, .ii ij ij ij
Thus
1
min max , .ik kj ij
k n
Combining the two inequalities, we obtain
1
min max , .ik kj ij
k n
Hence .
Conversely, suppose that . Then for all ,i j we have
1
min max , .ij ik kj
k n
It follows that, for every k ,
max , .ij ik kj
This is exactly the ultra inequality. Therefore is an ultra matrix.
The theorem is proved.
Corollary 4.7. Let A = (αij) be an ultra exponent matrix. For any dis-
tinct vertices i and j, the entry αij is equal to the corresponding minimax
directed distance:
: ,
min max ,ij uv
P i j u v P
where the minimum is taken over all directed paths from i to j in the com-
plete directed graph on the vertex set, and the weight of an edge (u, v) is
the matrix entry αuv.
Proof. The inequality
: ,
min max uv ij
P i j u v P
follows from the fact that among all paths there is a path of length 1 going
directly from i to j .
We prove the opposite inequality. Let
0 1: mP i i i i j
be an arbitrary path. Applying the ultra inequality successively, we obtain
0 1 1 2 1
max , , , .
m m
ij i i i i i i
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Series: Physical and mathematical sciences. 2026. Issue 30. P. 72-90.
88
Thus ij does not exceed the maximum weight of an edge on any
path from i to j . Therefore ij does not exceed the minimum of such
maxima over all paths. Together with the previous inequality, this gives
the required equality. The corollary is proved.
Theorem 4.8. Let ij and ij be ultra exponent matri-
ces of the same size. Define the matrix ijh by
max , .ij ij ijh
Then is also an ultra exponent matrix.
Proof. Since and are ultra matrices, for all , ,i j k we have
max , ,ik ij jk max , .ik ij jk
Then
max , max , , , .ik ik ik ij jk ij jkh
But
max , , max , .ij ij ij jk jk jkh h
Therefore,
max , .ik ij jkh h h
Moreover,
max , 0.ii ii iih
Let us also verify reducedness. For i j we have
1,ij ji ij jih h
because is a reduced exponent matrix. Hence is a reduced ultra ex-
ponent matrix. The theorem is proved.
Conclusions. The paper investigates ultra exponent matrices, that is, ma-
trices for which the usual triangle inequality is strengthened to an ultrametric
inequality. It is proved that every ultra exponent matrix is an exponent matrix,
and that every exponent matrix with entries 0 and 1 is an ultra matrix.
It is shown that elementary transformations of the first type do not pre-
serve the property of being an ultra matrix in general, whereas elementary
transformations of the second type preserve it. It is proved that a quiver ob-
tained from an ultra matrix having at least one entry greater than 1 is not rigid.
It is also established that not every admissible quiver with a loop at
every vertex can be obtained from an ultra exponent matrix. This shows
that the class of quivers corresponding to ultra exponent matrices is a
proper subclass of the class of admissible quivers with loops at all vertices.
ISSN 2308-5878. Mathematical and computer modelling.
Series: Physical and mathematical sciences. 2026. Issue 30. P. 72-90.
89
In addition, several structural characterizations of ultra matrices are
obtained. In particular, it is proved that monotone deformations of values
preserving 0 and 1 do not change the quiver of an ultra matrix. A filtra-
tion characterization of ultra matrices in terms of transitivity of the rela-
tions rR is established; the partial order generated by zero entries of a
reduced ultra matrix is described; and a classification of reduced 0,1-ultra
matrices in terms of partial orders is obtained. Furthermore, minimax
idempotency of ultra matrices and the closure of their class under compo-
nentwise maximum are proved.
References:
1. Hazewinkel M., Gubareni N., Kirichenko V. V. Algebras, Rings and Modules.
Dordrecht: Springer, 2004. Vol. 1. 380 p.
2. Hazewinkel M., Gubareni N., Kirichenko V. V. Algebras, Rings and Modules.
Dordrecht: Springer, 2007. Vol. 2. 400 p.
3. Kirichenko V. V., Zelenskiy O. V., Zhuravlev V. N. Exponent matrices and
tiled orders over discrete valuation rings. International Journal of Algebra and
Computation. 2005. Vol. 15, № 5-6. P. 997-1012.
4. Zelenskiy O. V. Rigid quivers of reduced exponent matrices. Bulletin of Kyiv
University. Series: Physics & Mathematics. 2007. № 3. P. 27-31.
5. Zhuravlev V. N. Admissible quivers. Journal of Mathematical Sciences. 2010.
Vol. 166, № 1. P. 24-33.
6. Kirichenko V. V., Khibina M. A., Zhuravlev V. N., Zelenskiy O. V. Quivers
and Latin square. São Paulo Journal of Mathematical Sciences. P. 1-15.
7. Zhuravlev V. M., Zelenskiy O. V., Darmosiuk V. M. Unit quivers of exponent
matrices. Bulletin of Taras Shevchenko National University of Kyiv. Series:
Physics & Mathematics. 2012. № 4. P. 27-31.
8. Zelenskiy O. V. Cycles of admissible quivers. Mathematical Studies. Vol. 42,
Issue 1. P. 3-8.
УЛЬТРА МАТРИЦІ ПОКАЗНИКІВ
У статті досліджено ультра матриці показників, тобто зведені ма-
триці показників, для яких звичайна трикутникова нерівність посилю-
ється до нерівності ультраметричного типу. Доведено, що кожна уль-
тра матриця показників є матрицею показників, а кожна матриця по-
казників з елементами з множини {0, 1} є ультра матрицею показни-
ків. Проаналізовано поведінку таких матриць відносно елементарних
перетворень: перетворення першого типу загалом не зберігає ультра-
властивість, тоді як одночасна перестановка рядків і стовпців її збері-
гає. Показано, що сагайдак, одержаний зі зведеної ультра матриці по-
казників, яка має хоча б один елемент, більший за одиницю, не є жор-
стким. Наведено контрприклад, який показує, що не кожний допус-
тимий сагайдак із петлею в кожній вершині може бути одержаний з
ультра матриці показників. Також встановлено низку структурних ха-
ISSN 2308-5878. Mathematical and computer modelling.
Series: Physical and mathematical sciences. 2026. Issue 30. P. 72-90.
90
рактеристик: описано монотонні деформації, які не змінюють відпо-
відний сагайдак, фільтраційну характеристику через транзитивні по-
рогові відношення, зв'язок 0-1 ультра матриць із частковими поряд-
ками, мінімаксну інтерпретацію та замкненість класу ультра матриць
відносно покомпонентного максимуму. З погляду математичного мо-
делювання ультра матриці показників можна розглядати як дискретні
моделі напрямлених відстаней із обмеженнями вузького місця, де
значення переходу визначається найсильнішим обмеженням на допу-
стимих шляхах, а не сумарною вартістю. Це дає змогу застосовувати
їх до моделювання ієрархічних систем, пріоритетних відношень, мі-
німаксної оптимізації, обмежених мережевих потоків, кластеризацій-
них структур та алгебраїчних або комбінаторних моделей, інформація
яких кодується допустимими зваженими сагайдаками. Одержані ре-
зультати дають інструменти для порівняння таких моделей і усунення
надлишкових представлень без втрати напрямленої структури.
Ключові слова: матриця показників, зведена матриця показни-
ків, допустимий сагайдак, ультра матриця показників, жорсткий са-
гайдак, вагова функція, математичне моделювання.
Математичне та комп'ютерне моделювання
Серія: Фізико-математичні науки
Редакційна колегія:
Zb_F-M_1.pdf
Гвоздєв М. І.
ORCІD: 0000-0003-4022-9777, Харківський національний університет радіоелектроніки, м. Харків, Україна, E-maіl: mykyta.hvozdev@nure.ua
Сидоров М. В.
ORCІD: 0000-0001-8022-866X, д-р фіз.-мат. наук, професор, Харківський національний університет радіоелектроніки, м. Харків, Україна, E-maіl: maxim.sidorov@nure.ua
метод двобічних наближень у ЧИСЕЛЬНОМУ аналізі задачі Нав’є, що є математичноЮ моделлю одновимірної мікроелектромеханічної системи
Ключові слова: мікробалка, задача Нав’є, ізотонний опертор, інваріантний конусний відрізок, крайова задача, математичне моделювання, мікроелектромеханічна система, метод двобічних наближень, напівлінійне звичайне диференціальне рівняння четвертого пор...
Список використаних джерел:
References:
The Method of Two-Sided Approximations in the Numerical Analysis of the Navier Problem as a Mathematical Model of a One-Dimensional Microelectromechanical System
Key words: boundary value problem, deflection, fourth-order semilinear ordinary differential equation, Green’s function, Hammerstein equation, invariant cone segment, isotone operator, mathematical modeling, method of two-sided approximations, microbe...
Громик А. П.
ORCID: 0000-0003-3071-9756, канд. техн. наук, Заклад вищої освіти «Подільський державний університет», м. Кам’янець-Подільський, Україна, E-mail: gapon74@gmail.com
Конет І. М.
ORCID: 0000-0002-4241-0548, д-р фіз.-мат. наук, професор, Волинський національний університет імені Лесі Українки, м. Луцьк, Україна, E-mail: konet51@ukr.net
Пилипюк Т. М.
ORCID: 0000-0002-4676-9830, канд. фіз.-мат. наук, Кам’янець-Подільський національний університет імені Івана Огієнка, м. Кам’янець-Подільський, Україна, E-mail: pylypyuk.tetiana@kpnu.edu.ua
ГІПЕРБОЛІЧНІ КРАЙОВІ ЗАДАЧІ МАТЕМАТИЧНОЇ ФІЗИКИ В КУСКОВО-ОДНОРІДНОМУ КЛИНОВИДНОМУ ЦИЛІНДРИЧНО-КРУГОВОМУ ШАРІ З ПОРОЖНИНОЮ
Ключові слова: гіперболічне рівняння, початкові та крайові умови, умови спряження, інтегральні перетворення, гібридні інтегральні перетворення, головні розв’язки.
Список використаних джерел:
References:
HYPERBOLIC BOUNDARY VALUE PROBLEMS OF MATHEMATICAL PHYSICS IN A PIECEWISE HOMOGENEOUS WEDGE-SHAPED CYLINDRICAL- CIRCULAR LAYER WITH А CAVITY
Key words: hyperbolic equation, initial and boundary conditions, conjugation conditions, integral transforms, hybrid integral transforms, main solutions.
Гук Н. А.
ORCID: 0000-0001-7937-1039, д-р фіз.-мат. наук, Дніпровський національний університет імені Олеся Гончара, м. Дніпро, Україна, E-mail: huk_n@365.dnu.edu.ua
Сіліч-Балгабаєва В. Б.
ORCID: 0000-0002-9490-3600, канд. техн. наук, Дніпровський національний університет імені Олеся Гончара, м. Дніпро, Україна, E-mail: v_silichbalgabaeva@365.dnu.edu.ua
Степанова Н. І.
ORCID: 0000-0002-9490-3600, канд. фіз.-мат. наук, Дніпровський національний університет імені Олеся Гончара, м. Дніпро, Україна, E-mail: stepanova_n@365.dnu.edu.ua
DATA-DRIVEN ПІДХІД ДО ОБЕРНЕНИХ ЗАДАЧ БІФУРКАЦІЇ ТОНКОСТІННИХ СИСТЕМ НА ОСНОВІ НЕЙРОННИХ МЕРЕЖ
Ключові слова: тонкостінні системи, біфуркація, втрата стійкості, обернені задачі, нейронні мережі, data-driven підхід, динамічні системи, прогнозування, нелінійна динаміка, ідентифікація параметрів, інформаційні технології.
Список використаних джерел:
References:
DATA-DRIVEN APPROACH TO INVERSE BIFURCATION PROBLEMS OF THIN-WALLED SYSTEMS BASED ON NEURAL NETWORKS
Key words: thin-walled systems, bifurcation, loss of stability, inverse problems, neural networks, data-driven approach, dynamic systems, prediction, nonlinear dynamics, parameter identification, information technologies.
Жолтовський О. О.
ORCID: 0009-0009-9245-1725, аспірант, Чернівецький національний університет імені Юрія Федьковича, м. Чернівці, Україна, E-mail: olekszholt@gmail.com
Черевко І. М.
ORCID: 0000-0002-2690-2091, д-р фіз-мат. наук, професор, Чернівецький національний університет імені Юрія Федьковича, м. Чернівці, Україна, E-mail: i.cherevko@chnu.edu.ua
ЗАСТОСУВАННЯ МЕТОДУ СПЛАЙН-ФУНКЦІЙ ДО МОДЕЛЮВАННЯ ЛІНІЙНИХ КРАЙОВИХ ЗАДАЧ ІЗ ЗАПІЗНЕННЯМ
Ключові слова: крайова задача, запізнення, метод сплайн-колокацій, кубічні сплайни, комп’ютерне моделювання.
Список використаних джерел:
References:
THE APPLICATION OF SPLINE FUNCTION METHOD TO THE MODELING OF LINEAR BOUNDARY VALUE PROBLEMS WITH DELAY
Key words: boundary value problem, delayed argument, spline collocation method, cubic splines, computer modeling.
Zelenskiy O. V.
ORCІD: 0000-0002-4969-0132, Cand. of Phys. and Math. Sciences, Kamianets-Podilskyi Ivan Ohiienko National University, Kamianets-Podilskyi, Ukraine, E-maіl: esteticcode@gmail.com
ULTRA EXPONENT MATRICES
Key words: exponent matrix, reduced exponent matrix, admissible quiver, ultra exponent matrix, rigid quiver, weight function, mathematical modelling.
References:
УЛЬТРА МАТРИЦІ ПОКАЗНИКІВ
Ключові слова: матриця показників, зведена матриця показників, допустимий сагайдак, ультра матриця показників, жорсткий сагайдак, вагова функція, математичне моделювання.
Мусій Р. С.
ORCІD: 0000-0002-7169-2206, д-р фіз.-мат. наук, професор, Національний університет «Львівська політехніка», м. Львів, Україна, E-maіl: musiy@lp.edu.ua
Кунинець А. В.
ORCІD: 0000-0003-2481-3236, канд. фіз.-мат. наук, Національний університет «Львівська політехніка», м. Львів, Україна, E-maіl: andrii.v.kunynets@lpnu.ua
Свідрак І. Г.
ORCІD: 0000-0003-1811-2011, канд. техн. наук, Національний університет «Львівська політехніка», м. Львів, Україна, E-maіl: inha.h.svidrak@lpnu.ua
Тимошенко Н. М.
ORCІD: 0000-0002-5595-6531, канд. фіз.-мат. наук, Національний університет «Львівська політехніка», м. Львів, Україна, E-maіl: nadiia.m.tymoshenko@lpnu.ua
Шиндер В. К.
ORCІD: 0000-0002-9414-5619, канд. фіз.-мат. наук, Національний університет «Львівська політехніка», м. Львів, Україна, E-maіl: valentyn.k.shynder@lpnu.ua
ВИЗНАЧЕННЯ ТА АНАЛІЗ ТЕПЛА ДЖОУЛЯ І ПОНДЕРОМОТОРНОЇ СИЛИ У ПОРОЖНИСТОМУ МІДНОМУ ЦИЛІНДРІ ЗА ДІЇ ЕЛЕКТРОМАГНІТНОГО ІМПУЛЬСА
Ключові слова: порожнистий електропровідний циліндр, електромагнітний імпульс, осьова компонента вектора напруженості магнітного поля, тепло Джоуля, пондеромоторна сила.
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DETERMINATION AND ANALYSIS OF JOULE HEAT AND PONDEROMOTIVE FORCE IN A HOLLOW COPPER CYLINDER UNDER THE ACTION OF AN ELECTROMAGNETIC IMPULSE
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Нікітін А. В.
ORCІD: 0000-0001-5137-0114, д-р фіз.-мат. наук, професор, Національний університет «Острозька академія», м. Острог, Україна, E-maіl: anatolii.nikitin@oa.edu.ua
Шведюк В. В.
ORCІD: 0009-0002-2906-0958, аспірант, Національний університет «Острозька академія», м. Острог, Україна, E-maіl: volodymyr.v.shvediuk@oa.edu.ua
СТІЙКІСТЬ ДЕТЕРМІНОВАНОЇ ТА УСЕРЕДНЕНОЇ СИСТЕМ РАДІОФІЗИЧНИХ ПРОЦЕСІВ У МІКРОКОНТРОЛЕРНИХ СИСТЕМАХ БПЛА
Ключові слова: БПЛА, стійкість детермінованої системи, власні значення, метод Ляпунова, радіофізичні процеси, мікроконтролер, простір станів.
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STABILITY OF DETERMINISTIC AND AVERAGED SYSTEMS OF RADIOPHYSICAL PROCESSES IN UAV MICROCONTROLLER SYSTEMS
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Пархоменко В. Г.
ORCІD: 0009-0008-7309-0875, Харківський національний університет радіоелектроніки, м. Харків, Україна, E-maіl: vladyslav.parkhomenko1@nure.ua
аналіз методОМ двобічних наближень стаціонарної реактивно-дифузивної моделі у сферичній гранулі з кінетикою арреніуса
Ключові слова: гетеротонний оператор, інтегральне рівняння Гаммерштейна, метод двобічних наближень, напівлінійне еліптичне рівняння з оператором Лапласа, нелінійна крайова задача, радіально-симетричний додатний розв’язок, реакційно-дифузійна система, ...
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Analysis by the Method of Two-Sided Approximations of a Stationary Reaction–Diffusion Model in a Spherical Pellet with Arrhenius Kinetics
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ORCІD: 0009-0004-7547-8655, Харківський національний університет радіоелектроніки, м. Харків, Україна, E-maіl: anton.savchenko@nure.ua
Застосування методу двобічних наближень до аналізу впливу типів закріплення кінців на статичний прогин балки в моделях мікроелектромеханічних систем
Ключові слова: балка, жорстке закріплення, ізотонний оператор, інваріантний конусний відрізок, консольна балка, крайова задача, крайові умови для балки, математичне моделювання, метод двобічних наближень, мікроелектромеханічна система, прогин, рівнянн...
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APPLICATION OF THE METHOD OF TWO-SIDED APPROXIMATIONS TO THE ANALYSIS OF THE INFLUENCE OF BEAM END CONDITIONS ON THE STATIC DEFLECTION OF A BEAM IN MICROELECTROMECHANICAL SYSTEMS
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Сеньо П. С.
ORCІD: 0009-0005-7979-2905, д-р фіз.-мат. наук, професор, Львівський національний університет імені Івана Франка, м. Львів, Україна E-maіl: petro.seno@lnu.edu.ua
Заяць А. Р.
ORCІD: 0009-0004-7793-0238, аспірант, Львівський національний університет імені Івана Франка, м. Львів, Україна E-maіl: artur.zaiats@lnu.edu.ua
МОДИФІКАЦІЯ БАЗОВОГО ДВОСТОРОННЬОГО МЕТОДУ РОЗВ’ЯЗУВАННЯ ІНТЕГРАЛЬНИХ РІВНЯНЬ
Ключові слова: функціональні інтервали, інтегральні рівняння, двосторонні апроксимації, параболічний паралелограм, математичне моделювання, системний аналіз.
Список використаних джерел:
References:
MODIFICATION OF THE BASIC TWO-SIDED METHOD FOR SOLVING INTEGRAL EQUATIONS
Key words: functional intervals, integral equations, two-sided approximations, parabolic parallelogram, mathematical modelling, system analysis.
Янбеков Р. Я.
ORCІD: 0009-0004-6588-7121, Харківський національний університет радіоелектроніки, м. Харків, Україна, E-maіl: ravil.yanbekov@nure.ua
дослідження одновимірних стаціонарних задач термохімії методом двобічних наближень на основі використання функції Гріна
Ключові слова: математичне моделювання, термохімічні процеси, метод двобічних наближень, функція Гріна, перша крайова задача, напівлінійне звичайне диференціальне рівняння.
Список використаних джерел:
References:
Analysis of One-Dimensional Steady-State Thermochemical Problems Using the Method of Two-Sided Approximations with Green’s Function
Key words: mathematical modelling, thermochemical processes, two-sided approximation method, Green’s function, first boundary value problem, semilinear ordinary differential equation.
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Зміст
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Математичне та комп’ютерне моделювання
Серія: Фізико-математичні науки
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| id | mcm-mathkpnueduua-article-361006 |
| institution | Mathematical and computer modelling. Series: Physical and mathematical sciences |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-06-09T01:00:12Z |
| publishDate | 2026 |
| publisher | Кам'янець-Подільський національний університет імені Івана Огієнка |
| record_format | ojs |
| resource_txt_mv | mcm-mathkpnueduua/e2/2b70fcca6f01be1ed5acfa28adf163e2.pdf |
| spelling | mcm-mathkpnueduua-article-3610062026-06-08T08:10:39Z Ultra Exponent Matrices Ультра матриці показників Зеленський, Олексій The paper studies ultra exponent matrices, that is, reduced exponent matrices for which the usual triangle inequality is strengthened to an ultrametric-type inequality. It is proved that every ultra exponent matrix is an exponent matrix, and that every exponent matrix whose entries belong to {0, 1} is an ultra exponent matrix. The behaviour of ultra matrices under elementary transformations is analysed: transformations of the first type do not preserve the ultra property in general, while simultaneous permutations of rows and columns do preserve it. It is shown that a quiver obtained from a reduced ultra exponent matrix with at least one entry greater than one is not rigid. A counterexample demonstrates that not every admissible quiver with a loop at every vertex can be represented by an ultra exponent matrix. Several structural characterizations are also established, including monotone deformations preserving the associated quiver, a filtration description in terms of transitive threshold relations, a connection between 0-1 ultra matrices and partial orders, a minimax interpretation, and closure under componentwise maximum. From the viewpoint of mathematical modelling, ultra exponent matrices may be interpreted as discrete directed distance models with bottleneck-type constraints, where the value assigned to a transition is determined by the strongest restriction along admissible paths rather than by an additive cost. This makes them useful for modelling hierarchical systems, priority relations, minimax optimization, constrained network flows, clustering-like structures, and algebraic or combinatorial systems whose essential information is encoded by admissible weighted quivers. The obtained results provide tools for comparing such models and for reducing redundant representations without losing the underlying directed structure. У статті досліджено ультра матриці показників, тобто зведені матриці показників, для яких звичайна трикутникова нерівність посилюється до нерівності ультраметричного типу. Доведено, що кожна ультра матриця показників є матрицею показників, а кожна матриця показників з елементами з множини {0, 1} є ультра матрицею показників. Проаналізовано поведінку таких матриць відносно елементарних перетворень: перетворення першого типу загалом не зберігає ультравластивість, тоді як одночасна перестановка рядків і стовпців її зберігає. Показано, що сагайдак, одержаний зі зведеної ультра матриці показників, яка має хоча б один елемент, більший за одиницю, не є жорстким. Наведено контрприклад, який показує, що не кожний допустимий сагайдак із петлею в кожній вершині може бути одержаний з ультра матриці показників. Також встановлено низку структурних характеристик: описано монотонні деформації, які не змінюють відповідний сагайдак, фільтраційну характеристику через транзитивні порогові відношення, зв'язок 0-1 ультра матриць із частковими порядками, мінімаксну інтерпретацію та замкненість класу ультра матриць відносно покомпонентного максимуму. З погляду математичного моделювання ультра матриці показників можна розглядати як дискретні моделі напрямлених відстаней із обмеженнями вузького місця, де значення переходу визначається найсильнішим обмеженням на допустимих шляхах, а не сумарною вартістю. Це дає змогу застосовувати їх до моделювання ієрархічних систем, пріоритетних відношень, мінімаксної оптимізації, обмежених мережевих потоків, кластеризаційних структур та алгебраїчних або комбінаторних моделей, інформація яких кодується допустимими зваженими сагайдаками. Одержані результати дають інструменти для порівняння таких моделей і усунення надлишкових представлень без втрати напрямленої структури. Кам'янець-Подільський національний університет імені Івана Огієнка 2026-05-29 Article Article Рецензована Стаття application/pdf https://mcm-math.kpnu.edu.ua/article/view/361006 10.32626/2308-5878.2026-30.72-90 Mathematical and computer modelling. Series: Physical and mathematical sciences; 2026: Mathematical and computer modelling. Series: Physical and mathematical sciences. Issue 30; 72-90 Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки; 2026: Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки. Випуск 30; 72-90 2308-5878 10.32626/2308-5878.2026-30 en https://mcm-math.kpnu.edu.ua/article/view/361006/349671 Авторське право (c) 2026 Олексій Зеленський |
| spellingShingle | Зеленський, Олексій Ultra Exponent Matrices |
| title | Ultra Exponent Matrices |
| title_alt | Ультра матриці показників |
| title_full | Ultra Exponent Matrices |
| title_fullStr | Ultra Exponent Matrices |
| title_full_unstemmed | Ultra Exponent Matrices |
| title_short | Ultra Exponent Matrices |
| title_sort | ultra exponent matrices |
| url | https://mcm-math.kpnu.edu.ua/article/view/361006 |
| work_keys_str_mv | AT zelensʹkijoleksíj ultraexponentmatrices AT zelensʹkijoleksíj ulʹtramatricípokaznikív |