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We introduce the class of network right $*$-abundant semigroups. These are based on networks that extend the notion of a directed graph. This class properly contains the class of graph inverse semigroups. We investigate the structure of network right $*$-abundant semigroups. We show that two network right $*$-abundant semigroups are isomorphic if and only if the underlying networks are isomorphic.

We let $\mathcal{F}$ be a finite family of sets closed under taking unions and $\emptyset ot \in \mathcal{F}$, and call an element abundant if it belongs to more than half of the sets of $\mathcal{F}$. In this notation, the classical Frankl's conjecture (1979) asserts that $\mathcal{F}$ has an abundant element. As possible strengthenings, Poonen (1992) conjectured that if $\mathcal{F}$ has precisely one abundant element, then this element belongs to each set of $\mathcal{F}$, and Cui and Hu (2019) investigated whether $\mathcal{F}$ has at least $k$ abundant elements if a smallest set of $\mathcal{F}$ is of size at least $k$. Cui and Hu conjectured that this holds for $k = 2$ and asked whether this also holds for the cases $k = 3$ and $k > \frac{n}{2}$ where $n$ is the size of the largest set of $\mathcal{F}$. We show that $\mathcal{F}$ has at least $k$ abundant elements if $k \geq n - 3$, and that $\mathcal{F}$ has at least $k - 1$ abundant elements if $k = n - 4$, and we construct a union-closed family with precisely $k - 1$ abundant elements for every $k$ and $n$ satisfying $n - 4 \geq k \geq 3$ and $n \geq 9$ (and for $k = 3$ and $n = 8$). We also note that $\mathcal{F}$ alw

In this paper the dependence of the spectra of cosmic ray nuclei on the charges of nuclei was studied, according to the data of the NUCLEON space experiment. First, we studied the dependence of the spectral index of magnetic rigidity spectra on the charge for abundant nuclei. Secondly, for the charge range $Z=9\div20$, the differences in the total spectra of rare odd and abundant even nuclei were studied. Using the GALPROP package, the inverse problem of CR propagation from a source (near supernova) to an observer was solved, a component-by-component spectrum in the source was reconstructed, and it was shown that a systematic change in the spectral index in the source exist. It is supposed that this change may be interpreted as incomplete ionization of cosmic rays at the stage of acceleration in the supernova remnant shock. The ratio of the total spectra of magnetic rigidity for low-abundance odd and abundant even nuclei from the charge range $Z=9\div20$ is obtained, and it was shown that the spectra of odd rare nuclei are harder than the stpectra of abundat even nuclei in the rigidity range 300--10000~GV.

We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant superintegrable systems on Riemannian manifolds of constant sectional curvature fall into this class. We explicitly compute the natural Hessian coordinates for examples of non-degenerate second-order superintegrable systems in dimensions two and three.

Biochemical processes typically involve many chemical species, some in abundance and some in low molecule numbers. Here we first identify the rate constant limits under which the concentrations of a given set of species will tend to infinity (the abundant species) while the concentrations of all other species remains constant (the non-abundant species). Subsequently we prove that in this limit, the fluctuations in the molecule numbers of non-abundant species are accurately described by a hybrid stochastic description consisting of a chemical master equation coupled to deterministic rate equations. This is a reduced description when compared to the conventional chemical master equation which describes the fluctuations in both abundant and non-abundant species. We show that the reduced master equation can be solved exactly for a number of biochemical networks involving gene expression and enzyme catalysis, whose conventional chemical master equation description is analytically impenetrable. We use the linear noise approximation to obtain approximate expressions for the difference between the variance of fluctuations in the non-abundant species as predicted by the hybrid approach and

The divisor function $σ(n)$ sums the divisors of $n$. We call $n$ abundant when $σ(n) - n > n$ and perfect when $σ(n) - n = n$. I recently introduced the recursive divisor function $a(n)$, the recursive analog of the divisor function. It measures the extent to which a number is highly divisible into parts, such that the parts are highly divisible into subparts, so on. Just as the divisor function motivates the abundant and perfect numbers, the recursive divisor function motivates their recursive analogs, which I introduce here. A number is recursively abundant, or ample, if $a(n) > n$ and recursively perfect, or pristine, if $a(n) = n$. There are striking parallels between abundant and perfect numbers and their recursive counterparts. The product of two ample numbers is ample, and ample numbers are either abundant or odd perfect numbers. Odd ample numbers exist but are rare, and I conjecture that there are such numbers not divisible by the first $k$ primes -- which is known to be true for the abundant numbers. There are infinitely many pristine numbers, but that they cannot be odd, apart from 1. Pristine numbers are the product of a power of two and odd prime solutions to cer

Ammonium hydrosulphide has long since been postulated to exist at least in certain layers of the giant planets. Its radiation products may be the reason for the red colour seen on Jupiter. Several ammonium salts, the products of NH3 and an acid, have previously been detected at comet 67P/Churyumov-Gerasimenko. The acid H2S is the fifth most abundant molecule in the coma of 67P followed by NH3. In order to look for the salt NH4+SH-, we analysed in situ measurements from the Rosetta/ROSINA Double Focusing Mass Spectrometer during the Rosetta mission. NH3 and H2S appear to be independent of each other when sublimating directly from the nucleus. However, we observe a strong correlation between the two species during dust impacts, clearly pointing to the salt. We find that NH4+SH- is by far the most abundant salt, more abundant in the dust impacts than even water. We also find all previously detected ammonium salts and for the first time ammonium fluoride. The amount of ammonia and acids balance each other, confirming that ammonia is mostly in the form of salt embedded into dust grains. Allotropes S2 and S3 are strongly enhanced in the impacts, while H2S2 and its fragment HS2 are not de

The aim of this paper is to study the Okounkov bodies associated to abundant divisors. As a main result, we prove that the valuative Okounkov bodies of an abundant divisor encode all the numerical properties. We apply this result to recover the asymptotic base loci of an abundant divisor from the valuative Okounkov bodies. We also give a criterion of when the valuative and limiting Okounkov bodies of an abundant divisor coincide by comparing their Euclidean volumes. To obtain these results, we prove some variants of Fujita's approximations for Okounkov bodies using Iitaka fibrations.

A weakly U abundant is a class of semigroups characterized using some generalized Green' relations. In this paper we discuss the variants of weakly U - abundant semigroups and it is shown that the idempotent variants of these semigroups are again weakly U abundant. Further we also discuss the natural partial order on variants of a weakly U abundant semigroup.

The previous detection of two species related to the non polar molecule cyanogen (NCCN), its protonated form (NCCNH+) and one metastable isomer (CNCN), in cold dense clouds supported the hypothesis that dicyanopolyynes are abundant in space. Here we report the first identification in space of NC4NH+, which is the protonated form of NC4N, the second member of the series of dicyanopolyynes after NCCN. The detection was based on the observation of six harmonically related lines within the Yebes 40m line survey of TMC-1 QUIJOTE. The six lines can be fitted to a rotational constant B = 1293.90840 +/- 0.00060 MHz and a centrifugal distortion constant D = 28.59 +/- 1.21 Hz. We confidently assign this series of lines to NC4NH+ based on high-level ab initio calculations, which supports the previous identification of HC5NH+ by Marcelino et al. (2020) from the observation of a series of lines with a rotational constant 2 MHz lower than that derived here. The column density of NC4NH+ in TMC-1 is (1.1 +1.4 -0.6)e10 cm-2, which implies that NC4NH+ is eight times less abundant than NCCNH+. The species CNCN, previously reported toward L483 and tentatively in TMC-1, is confirmed in this latter sour

A semigroup is \emph{regular} if it contains at least one idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class. A regular semigroup is \emph{inverse} if satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class, or (ii) the idempotents commute. Analogously, a semigroup is \emph{abundant} if it contains at least one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. An abundant semigroup is \emph{adequate} if its idempotents commute. In adequate semigroups, there is a unique idempotent in each $\mathcal{R}^*$ and $\mathcal{L}^*$-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each $\mathcal{R}^*$ and $\mathcal{L}^*$-class, must the idempotents commute? In this note we use ideal extensions to provide a negative answer to this question.

We give another alternative proof to the Kawamata semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant. After the first version of this article was posted to the e-print Arxiv, Prof. Fujino notified the author that the quick and essential proof ([Fujino. On Kawamata's theorem.(EMS 2011), Rem 2.7]) is already known. The author would like to thank him. More precisely, Prof. Fujino already gave the quick and essential proof ([Fujino. On Kawamata's thm.(EMS 2011), Rem 2.7], [Fujino. Finite generation of the lc ring in dim 4. (Kyoto J. Math. 50 (2010)), Rem 3.15]) from the finite generation thm (Birkar-Cascini-Hacon-McKernan [BCHM]) of the lc rings for klt pairs and from the fact (cf. Mourougane-Russo [MoRu, C.R.A.S. Math. 325 (1997)]) that a nef and abundant $\mathbf{Q}$-divisor $D$ is semiample if its graded ring is finitely generated:"For a nef and abundant lc divisor which is klt, the lc ring is finitely generated, thus it is semiample." [BCHM] first proved that the minimal model program runs for big klt lc divisors and next implied the finite generation of the lc rings for klt lc divisors which are not necessarily big from the Fujino-Mori

We show the invariance of plurigenera for generalized polarized pairs with abundant nef parts and generalized canonical singularities. This is obtained by investigating a type of newly introduced multiplier ideal sheaf which is of bimeromorphic nature.

We study the behavior of generalized lc pairs with $\mathrm{\textbf b}$-log abundant nef part, a meticulously designed structure on algebraic varieties. We show that this structure is preserved under the canonical bundle formula and sub-adjunction formulas, and is also compatible with the non-vanishing conjecture and the abundance conjecture in the classical minimal model program.

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $κ$ is supercompact if and only if every $Π^1_1$ sentence true in a structure $M$ (of any size) containing $κ$ in a language of size less than $κ$ is also true in a substructure $m\prec M$ of size less than $κ$ with $m\capκ\inκ$.

A positive integer $n$ is called an abundant number if $σ(n)\ge 2n$, where $σ(n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H os proved that there are two positive constants $c_1$ and $c_2$ such that $c_1\log\log\log x\le E(x)\le c_2\log\log\log x$. In this paper, we resolve this old problem by proving that, $E(x)/\log \log\log x$ tends to a limit as $x\to +\infty$, and the limit value has an explicit form which is between $3$ and $4$.

A $(v,k,t)$ packing of size $b$ is a system of $b$ subsets (blocks) of a $v$-element underlying set such that each block has $k$ elements and every $t$-set is contained in at most one block. $P(v,k,t)$ stands for the maximum possible $b$. A packing is called abundant if $b> v$. We give new estimates for $P(v,k,t)$ around the critical range, slightly improving the Johnson bound and asymptotically determine the minimum $v=v_0(k,t)$ when abundant packings exist. For a graph $G$ and a positive integer $c$, let $χ_\ell(G,c)$ be the minimum value of $k$ such that one can properly color the vertices of $G$ from any assignment of lists $L(v)$ such that $|L(v)|=k$ for all $v\in V(G)$ and $|L(u)\cap L(v)|\leq c$ for all $uv\in E(G)$. Kratochvíl, Tuza and Voigt in 1998 asked to determine $\lim_{n\rightarrow \infty} χ_\ell(K_n,c)/\sqrt{cn}$ (if exists). Using our bound on $v_0(k,t)$, we prove that the limit exists and equals $1$. Given $c$, we find the exact value of $χ_\ell(K_n,c)$ for infinitely many $n$.

In this paper, we use canonical bundle formulas to prove some generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs $(X,B)$ with $K_X+B$ effective, $L \in Pic (X)$ nef, nefness and abundance of $K_X+B+L$ implies semiampleness. This essentially generalizes Kawamata's theorem to the setting of generalized abundance.

The isotope masses and relative abundances for each element are fundamental chemical knowledge. Computing the isotope masses of a compound and their relative abundances is an important and difficult analytical chemistry problem. We demonstrate that this problem is equivalent to sorting $Y=X_1+X_2+\cdots+X_m$. We introduce a novel, practically efficient method for computing the top values in $Y$. then demonstrate the applicability of this method by computing the most abundant isotope masses (and their abundances) from compounds of nontrivial size.