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We examine the family of generalized Laguerre polynomials L n (n) (x) . In 1989, Gow discovered that if n is even, then the discriminant of L n (n) (x) is a nonzero square of a rational number. Additionally, in the case where the polynomial L n (n) (x) is irreducible over the rationals, the associated Galois group is the alternating group A n . Filaseta et al. (2012) established the irreducibility of L n (n) (x) for every n > 2 satisfying 2 (mod 4) . They also demonstrated that if n is 0 (mod 4) , then L n (n) (x) has a linear factor if it is not irreducible. The question of whether L n (n) (x) has a linear factor when n is 0 (mod 4) remained unanswered. We resolve this question by proving that L n (n) (x) does not have a linear factor for sufficiently large n. This conclusion completes the classification of generalized Laguerre polynomials having Galois group the alternating group, excluding a finite set of exceptions. [ABSTRACT FROM AUTHOR]

Let d and n be positive integers, and E/F be a separable field extension of degree m = n + d n . We show that if | F | > 2 , then there exists a point P ∈ P n (E) which does not lie on any degree d hypersurface defined over F. In other words, the m Galois conjugates of P impose independent conditions on the m-dimensional F-vector space of degree d forms in x 0 , x 1 , ... , x n . As an application, we determine the maximal dimensions of linear systems L 1 and L 2 of hypersurfaces in P n over a finite field F, where every F-member of L 1 is reducible and every F-member of L 2 is irreducible. [ABSTRACT FROM AUTHOR]

In this paper, random environment and random time delay are introduced into general nonlinear autoregressive models to improve the applicability of these models. The introduction of random environment makes nonlinear autoregressive models more flexible and more widely used. Random time delay is a new way for models to be applied in new fields. In order to analyze the properties of the proposed new model, we construct a new stochastic process and prove that it is an irreducible and aperiodic time-homogeneous Markov chain. The nonrecurrence of the proposed new type of models is considered, and some sufficient conditions for nonrecurrence are investigated. [ABSTRACT FROM AUTHOR]

The eccentricity matrix, ɛ (G) , of a graph G is derived from the distance matrix by letting the u v -th element to be equal to the distance between two vertices u and v , if the distance is the minimum of their eccentricities and zero otherwise. In this article, we study the spectrum of ɛ (G) and establish an upper bound for its ɛ -spectral radius when G is a self-centered graph. Further, we explore the structure of ɛ (G ∘ H) , where G ∘ H is the corona product of a self-centered graph G and a graph H. We characterize the irreducibility of ɛ (G ∘ H) and, in this process, find that it is independent of ɛ (H) , which allows us to construct infinitely many graphs with irreducible eccentricity matrix. Moreover, we compute the complete spectrum of ɛ (G ∘ H) including its ɛ -eigenvectors, ɛ -energy, and ɛ -inertia. Finally, we conclude that several non-isomorphic ɛ -co-spectral graphs can be generated using the corona product of two graphs. [ABSTRACT FROM AUTHOR]

For positive integers n ≥ m , let P n , m (x) : = ∑ j = 0 m n j x j = n 0 + n 1 x + ... + n m x m be the truncated binomial expansion of (1 + x) n consisting of all terms of degree ≤ m. It is conjectured that for n > m + 1 , the polynomial P n , m (x) is irreducible. We confirm this conjecture when 2 m ≤ n < (m + 1) 1 0. Also we show for any r ≥ 1 0 and 2 m ≤ n < (m + 1) r + 1 , the polynomial P n , m (x) is irreducible when m ≥ max { 1 0 6 , 2 r 3 }. Under the explicit abc-conjecture, for a fixed m , we give an explicit n 0 , n 1 depending only on m such that ∀ n ≥ n 0 , the polynomial P n , m (x) is irreducible. Further ∀ n ≥ n 1 , the Galois group associated to P n , m (x) is the symmetric group S m. [ABSTRACT FROM AUTHOR]

We explicitly provide numbers d, e such that each irreducible factor of a polynomial f(x) with integer coefficients has a degree greater than or equal to d and f(x) can have at most e irreducible factors over the field of rational numbers. Moreover, we prove our result in a more general setup for polynomials with coefficients from the valuation ring of an arbitrary valued field. [ABSTRACT FROM AUTHOR]

We study factorizations of HOMFLY polynomials of certain prime knots and oriented links. We begin with a computer analysis of prime knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs. [ABSTRACT FROM AUTHOR]

A polynomial f(x) over a field K is said to be stable if all its iterates are irreducible over K. L. Danielson and B. Fein have shown that over a large class of fields K, if f(x) is an irreducible monic binomial, then it is stable over K. In this paper it is proved that this result no longer holds over finite fields. Necessary and sufficient conditions are provided under which a given binomial is stable over Fq. These conditions are used to construct a table listing the stable binomials over Fq of the form f(x) = xd - a, a ∈ Fq \ {0, 1}, for q ≤ 27 and d ≤ 10. The paper ends with a brief link to Mersenne primes. [ABSTRACT FROM AUTHOR]

Let S be a smooth projective surface over \mathbb {C}. Let S^{[n_1,\dots,n_k]} denote the nested Hilbert scheme which parametrizes zero-dimensional subschemes \xi _{n_1} \subset \ldots \subset \xi _{n_k} where \xi _i is a closed subscheme of S of length i. We show that S^{[n,m]}, S^{[n,m,m+1]}, S^{[n,n+1,m]}, S^{[n,n+1,m,m+1]}, S^{[n,n+2,m]} and S^{[n,n+2,m,m+1]} are irreducible. [ABSTRACT FROM AUTHOR]

We show that for a norm continuous quantum Markov semigroup irreducibility and positivity improvement for any time t > 0 are equivalent under some general conditions that hold, in particular, in the finite-dimensional case. This result is applied to quantum Markov semigroups of weak coupling limit type. We also discuss the discrete-time case. [ABSTRACT FROM AUTHOR]

Elementary irreducibility criteria are established for f (x p) where f (x) ∈ ℤ [ x ] is irreducible over ℚ and p is a prime. For instance, our main criterion implies that if f (x p) is reducible over ℚ , then f (x) divides f (x p) modulo p 2 . Among several applications, it is shown that if f (x) has coefficients in { − 1 , 1 } , then f (x 2) is irreducible over ℚ excluding a couple of obvious exceptions. As another application, it is proved that if n > 4 and a 1 , a 2 , ... , a n are distinct integers, then for ∈ { − 1 , 1 } , the polynomial (x 2 − a 1) (x 2 − a 2) ⋯ (x 2 − a n) + is irreducible over ℚ unless n is odd and = − 1. Some emphasis is given to the non-cyclotomic monic polynomials f (x) with f (0) ∈ { − 1 , 1 }. In these cases, among other things, it is shown that if p ≫ (deg f) log max { 2 , H (f) } , where H (f) denotes the height of f (x) , then f (x p) is irreducible over ℚ. Proofs of the irreducibility criteria rest upon a general result of Capelli concerning the factorization of f (x m). [ABSTRACT FROM AUTHOR]

Given a continuum X, Cn(X) denotes the hyperspace of nonempty closed subsets of X with at most n components. A strong size level is a subset of the form σ−1(t), where σ is a strong size map for Cn(X) and t ∈ (0, 1]. In this paper, answering a question by Capulín-Pérez, Fuentes-Montes de Oca, Lara-Mejía and Orozco-Zitli, we prove that for each n ≥ 2, no strong size level for Cn(X) is irreducible. [ABSTRACT FROM AUTHOR]

In this note, we provide a new criterion of polynomials’s irreducibility over Fq[X], where Fq is a finite field [ABSTRACT FROM AUTHOR]

Considering irreducibility is fundamental for studying the ergodicity of stochastic dynamical systems. In this paper, we establish the irreducibility of stochastic complex Ginzburg-Laudau equations driven by pure jump noise. Our results are dimension free and the conditions placed on the driving noises are very mild. A crucial role is played by criteria developed by the authors of this paper and T. Zhang for the irreducibility of stochastic equations driven by pure jump noise. As an application, we obtain the ergodicity of stochastic complex Ginzburg-Laudau equations. We remark that our ergodicity result covers the weakly dissipative case with pure jump degenerate noise. [ABSTRACT FROM AUTHOR]

A polynomial f (x) is referred to as reciprocal if it satisfies f (x) = ± x deg ⁡ f f (1 / x). It is established that if f (x) ∈ Q [ x ] is reciprocal and irreducible over Q , and r > 0 is an odd integer, then f (x r) has an irreducible reciprocal factor over Q. Furthermore, if f (x) has a root on the unit circle, and r > 0 is an arbitrary integer, then every irreducible factor of f (x r) is reciprocal. This generalizes a related result of Filaseta and Meade for reciprocal 0 , 1 -polynomials. As an outcome of our methods, we obtain some information on the irreducible factors over Q of f (x r) for an arbitrary f (x) ∈ Q [ x ]. [ABSTRACT FROM AUTHOR]

In this paper, the authors give some sufficient conditions for an amalgamated 3-manifold along a compact connected surface F with boundary to be ∂-irreducible in terms of distances between some kinds of vertex subsets of the curve complex and the arc complex of F. [ABSTRACT FROM AUTHOR]

A commutative ring $R$ with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length $e$ is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length $2$. Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length $e$.

We provide an elementary way to obtain an irreducibility criterion for generalized principal series of p-adic reductive groups and the finite central covering groups, extending known and frequently employed results for principal series. Basically our approach is a counting argument which rests on a semi-direct product decomposition of the relative Weyl group and its action on generalized principal series, in conjunction with the theory of Jacquet module. We thus circumvent the obstacle that the relative Weyl group of a general Levi subgroup is not a Coxeter group. Along the way, we observe a structural irreducibility criterion with respect to Levi subgroups. Indeed, following Speh–Vogan's argument for real groups which is essentially a corollary of intertwining operator theory, the former result can be generalized to parabolic inductions inducing from essentially tempered representations, but the latter one can't be obtained in this way. In view of the lack of intertwining operator theory for mod ℓ representations, our counting argument may play a role there, as opposed to the failure of Speh–Vogan's argument. [ABSTRACT FROM AUTHOR]

Let C(A)=A∘A−T{\mathcal{C}}\left(A)=A\circ {A}^{-T} be the combined matrix of an invertible matrix AA, where ∘\circ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an HH-matrix whose comparison matrix is singular. In particular, we focus on C(A){\mathcal{C}}\left(A) when AA is diagonally equipotent, and we study whether C(A){\mathcal{C}}\left(A) is an HH-matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices.

Introduction. The tendency to strengthen the priority of the technical side of life and the activity of a modern person, characterized by increasing complexity and ambiguity in the definition of a human and his world, aggravates the problem of preserving and reproduction of a living principle in a human. A problematic question arises: is a person – a living person – possible in a technical and technologized world in general? Theoretical analysis. The article notes that when solving the indicated problem, the rejection of the technique, the process of technicalization, technical in general are not provided. However, the absolutization of the meaning of the technical in the life and activity of a human is not supposed, when the existence of a person is reduced to the technical. A modern person needs a kind of reorientation – not to be included in the world of technology and techniques, but, on the contrary, to include them in his life in order to remain modernly alive. All the diversity of life at the expense of new techniques is ensured. Conclusion. In relations between the human and the technical, it is proposed to be based on the principles of the eco-humanistic approach (integrity, coordination, consistency and optimality), when favorable development opportunities, the realization of the internal potential of each side are provided.

Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly irreducible, irreducible, completely irreducible, proper, minimal, primary, nil, nilpotent, regular, radical, principal, finitely generated ideals. We characterise ideal spaces that are sober. We introduce the notion of a strongly disconnected spaces and show that for a ring with zero Jacobson radical, strongly disconnected ideal spaces containing all maximal ideals of the ring imply existence of nontrivial idempotent elements in the ring. We also give a sufficient condition for a spectrum to be connected. [ABSTRACT FROM AUTHOR]

A commutative ring R with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length e is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length 2. Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length e. [ABSTRACT FROM AUTHOR]

Purpose: This study aims to determine which intertrochanteric (IT) hip fracture and patient characteristics predict the necessity for adjunct reduction aides prior to prep and drape aiming for a more efficient surgery. Materials and Methods: Institutional fracture registries from two academic medical centers from 2017-2022 were analyzed. Data on patient demographics, comorbidities, fracture patterns identified on radiographs including displacement of the lesser trochanter (LT), thin lateral wall (LW), reverse obliquity (RO), subtrochanteric extension (STE), and number of fracture parts were collected, and the need for additional aides following traction on fracture table were collected. Fractures were classified using the AO/OTA classification. Regression analyses identified significant risk factors for needing extra reduction aides. Results: Of the 166 patients included, the average age was 80.84±12.7 years and BMI was 24.37±5.3 kg/m2. Univariate regression revealed increased irreducibility risk associated with RO (odds ratio [OR] 27.917, P= 0.001), LW (OR 24.882, P<0.001), and STE (OR 5.255, P=0.005). Multivariate analysis significantly correlated RO (OR 120.74, P<0.001) and thin LW (OR 131.14, P<0.001) with increased risk. However, STE (P=0.36) and LT displacement (P=0.77) weren't significant. Fracture types 2.2, 3.2, and 3.3 displayed elevated risk (P<0.001), while no other factors increased risk. Conclusion: Elderly patients with IT fractures with RO and/or thin LW are at higher risk of irreducibility, necessitating adjunct reduction aides. Other parameters showed no significant association, suggesting most fracture patterns can be achieved with traction manipulation alone. [ABSTRACT FROM AUTHOR]

By defining consciousness as nothingness or simply as “nothing,” Sartre plays with several meanings of these terms: negativity and negation, distance, indetermination, irreducibility. The nothingness of consciousness takes on an ontological meaning: it is a “tearing away” from being-in-itself, a transcendence understood as the capacity to transcend what is, while retaining an epistemological meaning: it is what cannot be positively determined as “something” or as a property of being. Still, on the epistemological level as well as on the ontological level, it is indeed from “something,” from physical and social being, that the nothingness of consciousness draws its existence and its capacities. In my article, I examine different meanings that can be given to the “nothing” of consciousness in the light of the thought of Sartre himself, emphasizing the difference between two major meanings of negation: as opposition and as indetermination. Then I confront Sartre’s concept of consciousness with more recent considerations of different inspiration, notably from researchers like Chalmers, Damasio, Gallagher, and Zahavi. My thesis is that the Sartrean concept, semi-transcendental and semi-naturalist, does admit the search for a naturalist explanation of consciousness, but assigns its limit precisely through the concept of nothingness. [ABSTRACT FROM AUTHOR]

Since Hochster's work, spectral spaces have attracted increasing interest. Through this note we give a new self-contained and constructible topology-independent proof of the fact that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space.

We study the dynamics of the unicritical polynomial family f d , c (z) = z d + c ∈ ℂ [ z ]. The c -values for which f d , c has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when d is a prime power and c is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of f d , c . We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the family of binomials xaι ∙ ∙ ∙ xmm -- y11 ∙ ∙ ∙ ynn with gcd(aι,..., am, 6χ,..., bn) = 1 is irreducible by identifying the connection between the irreducibility of a binomial in C[x1, . . ., xm, y1, . . ., yn] and C(x2, . . ., xm, y1, . . ., yn)[x1]. Then we show that the necessary and sufficient conditions for the irreducibility of this family of binomials is equivalent to the existence of a unimodular matrix Ui with integer entries such that (a1, . . ., am, b1, . . ., bn)T = Uiei for i ∈ {1, . . ., m + n}, where ei is the standard basis vector. [ABSTRACT FROM AUTHOR]

We use the variational approach to investigate periodic measures for a class of stochastic partial differential equations (SPDEs) with regime-switching. The hybrid system is driven by degenerate Lévy noise. We use the Lyapunov function method to study the existence of periodic measures and show the uniqueness of periodic measures by establishing the strong Feller property and irreducibility of the associated time-inhomogeneous semigroup. The main results are applied to stochastic fractional porous medium equations with regime-switching. [ABSTRACT FROM AUTHOR]

Let X be a metric continuum and n a positive integer. Let Fn (X) be the hyperspace of nonempty subsets of X with at most n points. If 0 < m < n, we consider the quotient space Fnm (X) = Fn (X)/Fm (X). Given a mapping f from X into X, we consider the induced mappings fn from Fn (X) into Fn (X) and fnm from Fnm (X) into Fnm (X). In this paper we study the relations among the dynamics of the mappings f, fn, and fnm and we answer some questions, by F. Barragán, A. Santiago-Santos and J. Tenorio, related to the properties: minimality, irreducibility, strong transitive and turbulence.

Long-term behaviors of biochemical systems are described by steady states in deterministic models and stationary distributions in stochastic models. Obtaining their analytic solutions can be done for limited cases, such as linear or finite-state systems, as it generally requires solving many coupled equations. Interestingly, analytic solutions can be easily obtained when underlying networks have special topologies, called weak reversibility (WR) and zero deficiency (ZD), and the kinetic law follows a generalized form of mass-action kinetics. However, such desired topological conditions do not hold for the majority of cases. Thus, translating networks to have WR and ZD while preserving the original dynamics was proposed. Yet, this approach is limited because manually obtaining the desired network translation among the large number of candidates is challenging. Here, we prove necessary conditions for having WR and ZD after translation, and based on these conditions, we develop a user-friendly computational package, TOWARDZ, that automatically and efficiently identifies translated networks with WR and ZD. This allows us to quantitatively examine how likely it is to obtain WR and ZD after translation depending on the number of species and reactions. Importantly, we also describe how our package can be used to analytically derive steady states of deterministic models and stationary distributions of stochastic models. TOWARDZ provides an effective tool to analyze biochemical systems. [ABSTRACT FROM AUTHOR]

Let M be a torsion-free module over an integral domain R. The main results of this article provide analogue of Dumas irreducibility criterion and an extension of Eisenstein's irreducibility criterion for factorial modules. Furthermore, we provide some applications of these results in R [ x ] -module M [ x ] . [ABSTRACT FROM AUTHOR]

Copyright of Journal Of Sichuan University (Natural Sciences Division) / Sichuan Daxue Xuebao-Ziran Kexueban is the property of Editorial Department of Journal of Sichuan University Natural Science Edition and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The anti-adjacency (aka eccentricity) matrix A (G) of a graph G is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. The motivation for this paper is to find answers to two problems: "For which connected graphs the eccentricity (anti-adjacency) matrix is either reducible or irreducible?" and "Characterize the graphs with a small number of distinct anti-adjacency eigenvalues". In this context, we get the sufficient and necessary conditions of the graphs which have an irreducible matrix. For the second problem, we characterize the graphs with exactly two distinct eigenvalues different from 0. We also obtain some spectral properties of the anti-adjacency matrix. [ABSTRACT FROM AUTHOR]

The present work is concerned with two-dimensional stochastic subcritical and critical convective Brinkman-Forchheimer (2 D SCBF) equations perturbed by a white noise (non-degenerate) in smooth bounded domains in R 2. We establish two important properties of the Markov semigroup associated with the solutions of 2 D SCBF equations (for the absorption exponent r = 1, 2, 3), that is, irreducibility and strong Feller property. These two properties implies the uniqueness of invariant measures and ergodicity also. Then, we discuss the ergodic behavior of 2 D SCBF equations by providing a Large Deviation Principle (LDP) for the occupation measure for large time (Donsker-Varadhan), which describes the exact rate of exponential convergence. [ABSTRACT FROM AUTHOR]

Abstract Background Hernia of the umbilical cord (HUC) is an uncommon type of abdominal wall defect characterized by a hernia of the midgut into the proximal section of the umbilical cord. This is occasionally coupled with other congenital abnormalities. This is frequently misdiagnosed and referred to as “omphalocele minor.” In certain cases, unintentional cord clamping causes iatrogenic intestinal harm. The purpose of this retrospective study is to highlight various aspects of the HUC therapy in 30 patients with typical and unusual presentations in a tertiary care facility as well as intraoperative findings and associated abnormalities. Methods Thirty neonates (21 males and 9 females) with usual and unusual presentations of HUC diagnosed and treated at the King Abdelaziz University Hospital, Jeddah, Saudi Arabia, over 15 years period from 2005 to 2020. Results Out of 30 cases included, 17 patients were reducible with simple classical HUC repair. While 13 patients had different presentations, six cases presented with irreducible content due to variable reasons, four cases presented with patent vitellointestinal duct (PVID), 2 cases presented with intestinal obstruction, and one case presented late with epithelialized HUC sac. Conclusions Attention to HUC should be paid by obstetric professionals in order not to miss it on antenatal ultrasound and careful umbilical cord examination at delivery to avoid clamping of visceral content if present in HUC.

Since Hochster’s work, spectral spaces have attracted increasing interest. Through this note we give a new self-contained and constructible topology-independent proof of the fact that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space. [ABSTRACT FROM AUTHOR]

In this note, we give criteria for the irreducibility of functions in $ C^{m}\left[ 0,1\right] $, where $ m \in \{1,2,3,...\}\cup\{\infty\} \cup \{\omega\} $. We also discuss irreducibility in multivariable differentiability classes. Moreover, we characterize irreducible functions and maximal ideals in $ C^{\infty}\left[ 0,1\right] $. In fact, irreducible and prime smooth functions are the same, and every maximal ideal of $ C^{\infty}\left[ 0,1\right] $ is principal. [ABSTRACT FROM AUTHOR]

A path system P ${\mathscr{P}}$ in a graph G=(V,E) $G=(V,E)$ is a collection of paths with a unique uv $uv$ path for every two vertices u,v∈V $u,v\in V$. We say that P ${\mathscr{P}}$ is consistent if for any path P∈P $P\in {\mathscr{P}}$, every subpath of P $P$ is also in P ${\mathscr{P}}$. It is metrizable if there exists a positive weight function w:E→R>0 $w:E\to {{\mathbb{R}}}_{\gt 0}$ such that P ${\mathscr{P}}$ is comprised of w $w$‐shortest paths. We call P ${\mathscr{P}}$irreducible if there does not exist a partition V=A⊔B $V=A\bigsqcup B$ such that P ${\mathscr{P}}$ restricts to a path system on both G[A] $G[A]$ and G[B] $G[B]$. In this paper, we construct an infinite family of nonmetrizable irreducible consistent path systems on certain Paley graphs. [ABSTRACT FROM AUTHOR]

Copyright of Endoxa is the property of Editorial UNED and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The eccentricity matrix of a digraph is obtained from the distance matrix by keeping the largest distances in each row and each column, and leaving 0 in the remaining ones. Randić (2013) first defined D MAX -matrix of a graph and renamed it as eccentricity matrix by Wang et al. (2018). In this paper, we extend the concept of the eccentricity matrix of a graph to a digraph. We consider the irreducibility of the eccentricity matrix of a digraph with diameter 2 and obtain the lower bounds of ɛ -energy of a digraph with diameter 2. We characterize the lower bound of the spectral radius of the eccentricity matrix of a digraph and the corresponding extremal digraphs. Moreover, we give the ɛ -spectra and ɛ -energies of some digraphs. [ABSTRACT FROM AUTHOR]

For an arbitrary real number α and a positive integer n , the Generalized Laguerre Polynomials (GLP) is a family of polynomials defined by L n (α) (x) = (− 1) n ∑ j = 0 n (n + α) (n − 1 + α) ⋯ (j + 1 + α) (n − j) ! j ! (− x) j. Following the work of Banerjee, Filaseta, Finch and Leidy [2] which described the set A 0 ∪ A ∞ of integer pairs (n , α) for which the discriminant of L n (α) (x) is a nonzero square, where A 0 is finite and A ∞ is explicitly given infinite set, it was conjectured by Banerjee in [1] that for α ≠ − 1 , the only pair (n , α) ∈ A ∞ for which the associated Galois group of L n (α) (x) is not A n is (4 , 23). In this paper, we verify this conjecture for (α , n) ∈ A ∞ with α ∈ { − 2 n , − 2 n − 2 , − 2 n − 4 }. In fact, we prove more general results concerning the irreducibility and Galois groups of the Generalized Laguerre polynomials L n (α) (x) for n ∈ N and integers α such that α ∈ [ − 2 n − 4 , − 2 n ]. The case α = − 2 n − 1 corresponds to the Bessel polynomials which have been studied earlier by Filaseta and Trifonov [7] and Grosswald [9]. Our ideas give a simpler proof of their results. [ABSTRACT FROM AUTHOR]

We study quasi-stationary distribution of the continuous-state branching process with competition introduced by Berestycki et al. (2018). This process is defined as the unique strong solution to a stochastic integral equation with jumps. An important example is the logistic branching process proposed by Lambert (2005). We establish the strong Feller property, trajectory Feller property, Lyapunov condition, weak Feller property and irreducibility, respectively. These properties together allow us to prove that if the competition is strong enough near + ∞ , then there is a unique quasi-stationary distribution, which attracts all initial distributions with exponential rates. [ABSTRACT FROM AUTHOR]