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Register a new userCharacterizing semigroups $X$ with commutative extensions $varphi(X)$, $lambda(X)$, $N_2(X)$, $upsilon(X)$
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We characterize semigroups $X$ whose semigroups of filters $varphi(X)$, maximal linked systems $lambda(X)$, linked upfamilies $N_2(X)$, and upfamilies $upsilon(X)$ are commutative.
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Surveys in the next decade will deliver large samples of galaxy clusters that transform our understanding of their formation. Cluster astrophysics and cosmology studies will become systematics limited with samples of this magnitude. With known properties, hydrodynamical simulations of clusters provide a vital resource for investigating potential systematics. However, this is only realized if we compare simulations to observations in the correct way. Here we introduce the textsc{Mock-X} analysis framework, a multiwavelength tool that generates synthetic images from cosmological simulations and derives halo properties via observational methods. We detail our methods for generating optical, Compton-$y$ and X-ray images. Outlining our synthetic X-ray image analysis method, we demonstrate the capabilities of the framework by exploring hydrostatic mass bias for the IllustrisTNG, BAHAMAS and MACSIS simulations. Using simulation derived profiles we find an approximately constant bias $bapprox0.13$ with cluster mass, independent of hydrodynamical method or subgrid physics. However, the hydrostatic bias derived from synthetic observations is mass-dependent, increasing to $b=0.3$ for the most massive clusters. This result is driven by a single temperature fit to a spectrum produced by gas with a wide temperature distribution in quasi-pressure equilibrium. The spectroscopic temperature and mass estimate are biased low by cooler gas dominating the emission, due to its quadratic density dependence. The bias and the scatter in estimated mass remain independent of the numerical method and subgrid physics. Our results are consistent with current observations and future surveys will contain sufficient samples of massive clusters to confirm the mass dependence of the hydrostatic bias.
We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in Z[x]; we include a new bound which was latent in a paper by Mignotte, and a few minor improvements to some existing bounds. We compare these bounds and show that none is universally better than the others. In the second part of the paper we give several concrete examples of factorizations where the factors have unexpectedly large coefficients. These examples help us understand why the bounds must be larger than you might expect, and greatly extend the collection published by Collins.
It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $langle B rangle$ denote the subring of the full matrix ring generated by $B$. Then $langle B rangle$ is a free $mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $langle B rangle$ with given finite index. Thus, the formal Dirichlet series $zeta_{langle B rangle}(s)=sum_{ngeq 1}a_n n^{-s}$ is well-defined where $a_n$ is the number of ideals of $langle B rangle$ with index $n$. In this article we aim to find an explicit form of $zeta_{langle B rangle}(s)$ when $B$ has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, $langle B rangle$ is isomorphic to $mathbb{Z}[x]/m(x)mathbb{Z}[x]$ where $m(x)$ is the minimal polynomial of $B$ over $mathbb{Q}$, and $mathbb{Z}[x]/m(x)mathbb{Z}[x]$ is isomorphic to $mathbb{Z}[x]/m(x+gamma)mathbb{Z}[x]$ for each $gammain mathbb{Z}$. Thus, the problem is reduced to counting the number of ideals of $mathbb{Z}[x]/x(x-alpha)(x-beta)mathbb{Z}[x]$ with given finite index where $0,alpha$ and $beta$ are distinct integers.
Let $mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Yin mathcal C$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $mathcal C$-$closed$ if for each congruence $approx$ on $X$ the quotient semigroup $X/_approx$ is $mathcal C$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $Isubseteq X$ there are elements $x,yin I$ such that $xy otin{x,y}$. We prove that a semigroup $X$ is $mathcal C$-closed if it admits a homomorphism $h:Xto E$ to a chain-finite semilattice $E$ such that for every $ein E$ the semigroup $h^{-1}(e)$ is $mathcal C$-closed. Applying this theorem, we prove that a commutative semigroup $X$ is $mathcal C$-closed if and only if $X$ is periodic, chain-finite, all subgroups of $X$ are bounded, and for any infinite set $Asubseteq X$ the product $AA$ is not a singleton. A commutative semigroup $X$ is projectively $mathcal C$-closed if and only if $X$ is chain-finite, all subgroups of $X$ are bounded and the union $H(X)$ of all subgroups in $X$ has finite complement $Xsetminus H(X)$.
We investigate the possible rescattering effects which may contribute to the process $B^+to J/psiphi K^+$. It is shown that the $D_{s}^{*+}D_{s}^-$ rescattering via the open-charmed meson loops, and $psi^prime phi$ rescattering via the $psi^prime K_1$ loops may simulate the structures of $X(4140)$ and $X(4700)$, respectively. However, if the quantum numbers of $X(4274)$ ($X(4500)$) are $1^{++}$ ($0^{++}$), it is hard to to ascribe the observation of $X(4274)$ and $X(4500)$ to the $P$-wave threshold rescattering effects, which implies that $X(4274)$ and $X(4500)$ could be genuine resonances. We also suggest that $X(4274)$ may be the conventional orbitally excited state $chi_{c1}(3P)$.
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