Aug 2026· Periodica Mathematica Hungarica· 0 citations· 22 references
Abstract
<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>
<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>
J. Vicente-Pérez, Margarita M. L. Rodríguez· TOP - An Official Journal of...· 0 citations
<jats:p>
The thermal partition function,
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>Z</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>Z</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, of a
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>CFT_d</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mi>F</mml:mi>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mi>d</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
on
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>S^{d-1}</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msup>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mo>−</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
is parameterized by the inverse temperature
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\beta</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>β</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
along with
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\lfloor d/2\rfloor</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mo stretchy="false" form="prefix">⌊</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>/</mml:mi>
<mml:mn>2</mml:mn>
<mml:mo stretchy="false" form="postfix">⌋</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
angular velocities
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\omega_i</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msub>
<mml:mi>ω</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
. In this paper, we investigate the behaviour of this partition function when
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>n</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>n</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of the
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\omega_i</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msub>
<mml:mi>ω</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
are scaled to unity (the largest allowed value) at fixed values of the other
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>(\lfloor d/2\rfloor-n)</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mo stretchy="false" form="prefix">(</mml:mo>
<mml:mo stretchy="false" form="prefix">⌊</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>/</mml:mi>
<mml:mn>2</mml:mn>
<mml:mo stretchy="false" form="postfix">⌋</mml:mo>
<mml:mo>−</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo stretchy="false" form="postfix">)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
angular velocities. We argue that
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\ln Z</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mrow>
<mml:mi mathvariant="normal">ln</mml:mi>
<mml:mo></mml:mo>
</mml:mrow>
<mml:mi>Z</mml:mi>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
develops a simple pole in
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>(1-\omega_i)</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mo stretchy="false" form="prefix">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>−</mml:mo>
<mml:msub>
<mml:mi>ω</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo stretchy="false" form="postfix">)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
for each
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\omega_i</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msub>
<mml:mi>ω</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\beta</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>β</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>n</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>n</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
special angular momenta
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>J_1\ldots J_n</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:msub>
<mml:mi>J</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mi>…</mml:mi>
H. Anand, N. Benjamin, Vipul Kumar et al.· SciPost Physics· 8 citations· ⚡3
<jats:p>
In this paper, we introduce the notion of generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules respect to some subclass
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, extending the classical notion of Gorenstein projective modules. By exploiting the correspondence between projective modules over the endomorphism ring
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{End}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>End</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of a module
<jats:italic>C</jats:italic>
and elements of its additive closure
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}=\textrm{Add}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>W</mml:mi>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>Add</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, we establish a fundamental correspondence between Gorenstein projective
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{End}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>End</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-modules and generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{Add}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>Add</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules. This result refines existing relative homological settings and provides a natural extension of well-known results in Gorenstein homological algebra. We explore key properties, such as closure under direct summands and sums, and identify conditions under which the class of generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules coincides with other classes of modules, like Gorenstein projective modules.
</jats:p>
Driss Bennis, C. Lomp, Abderrazak Nassir· Algebras and Representation...· 0 citations
<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
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