Jul 2026· TOP - An Official Journal of the Spanish Society of Statistics and Operations Research· 0 citations· 12 references
Abstract
<jats:p>
Lexicographical extensions of well-known separation theorems for convex sets in
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathbb {R}^n$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>R</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
are provided in the literature. Particularly, recent theorems regarding open and closed separation of a convex set from any outside point by linear operators from
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathbb {R}^n$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>R</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
to
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathbb {R}^m$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>R</mml:mi>
</mml:mrow>
<mml:mi>m</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, in the sense of the lexicographical order of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathbb {R}^m$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>R</mml:mi>
</mml:mrow>
<mml:mi>m</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, for each
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$m\in \{1,\ldots ,n\}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>m</mml:mi>
<mml:mo>∈</mml:mo>
<mml:mo>{</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>…</mml:mo>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo>}</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, allow to define two new families of properties for convex sets. Based on these results, that we review and extend in this paper, we provide dual characterizations for the consistency of two kinds of systems defined by weak and/or strict lexicographical linear inequalities, and for those inequalities which are satisfied for every solution of a given system. Such results are formulated in terms of appropriate convex hulls of certain sets depending on the coefficients of the system.
</jats:p>
<jats:p>
Seymour’s second neighborhood conjecture states that every oriented graph
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\vec {G}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>G</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
has a Seymour vertex, namely,
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\vec {G}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>G</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
has a vertex whose second-order out-neighborhood is at least as large as its first-order out-neighborhood. In this paper, we approach the conjecture by considering an inhomogeneous random graph
<jats:italic>G</jats:italic>
, where each edge
<jats:italic>e</jats:italic>
in the complete graph
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$K_n$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>K</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
appears independently with probability
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$p_n(e)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mi>p</mml:mi>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>e</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
. Under suitable density and regularity conditions, we show that every orientation of
<jats:italic>G</jats:italic>
contains a Seymour vertex with high probability, confirming the conjecture asymptotically. Moreover, if we consider an inhomogeneous random oriented graph
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\vec {G}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>G</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
by assigning an orientation to each edge of
<jats:italic>G</jats:italic>
independently with equal probability, we prove that
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\vec {G}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>G</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
contains a Seymour vertex with high probability across a broader range of regimes.
</jats:p>
Y. Shang· Periodica Mathematica Hungar...· 0 citations
<jats:p>
We provide a geometric model for the free
<jats:italic>X</jats:italic>
-generated
<jats:italic>F</jats:italic>
-restriction semigroup in the extended signature
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${(\cdot , ^+,^{\mathfrak {m}},\lambda )}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mo>·</mml:mo>
<mml:msup>
<mml:mo>,</mml:mo>
<mml:mo>+</mml:mo>
</mml:msup>
<mml:msup>
<mml:mo>,</mml:mo>
<mml:mi>m</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:mi>λ</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, where the unary operation
<jats:sup>m</jats:sup>
maps an element
<jats:italic>a</jats:italic>
to the maximum element
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$a^{\mathfrak {m}}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>a</mml:mi>
<mml:mi>m</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of its
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\sigma $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>σ</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-class, and the constant
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\lambda $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>λ</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
is the unique left identity. This model is based on a certain quotient of the Cayley graph expansion of the free monoid
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$X^*$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>X</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
with respect to the extended set of generators
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$X\cup \overline{X^*}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>X</mml:mi>
<mml:mo>∪</mml:mo>
<mml:mover>
<mml:msup>
<mml:mi>X</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, where the generators from
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\overline{X^*}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:msup>
<mml:mi>X</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
are in a bijection with the free monoid
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$X^*$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>X</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
and serve to capture the maximum elements of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\sigma $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>σ</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-classes of the quotient. We also provide models for the free
<jats:italic>X</jats:italic>
-generated strong and perfect
<jats:italic>F</jats:italic>
-restriction semigroups in the same extended signature. The constructed models enable us to solve the word problems for all the free objects under consideration.
</jats:p>
<jats:p>
We study the tail behavior of measurable functions under operators satisfying suitable
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$L^p$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mi>p</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
bounds in the framework of Grand Lebesgue Spaces. By exploiting the connection between the growth of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$L^p$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mi>p</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
norms and the Young–Fenchel transform, we derive explicit tail estimates. We also present model examples and apply the abstract result to the classical Riesz transforms, showing how the
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$L^p$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mi>p</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
growth of the operator interacts with the intrinsic tail behavior of the input function.
</jats:p>
M. R. Formica, E. Ostrovsky, L. Sirota· Ricerche di Matematica· 0 citations
