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The Central Sets Theorem, a fundamental result in Ramsey theory, is a joint extension of both Hindman's theorem and van der Waerden's theorem. It was originally introduced by H. Furstenberg using methods from topological dynamics. Later, using the algebraic structure of the Stone-$Č$ech compactification $β$ S of a semigroup S, N. Hindman and V. Bergelson extended the theorem in 1990. H. Shi and H. Yang established a topological dynamical characterization of central sets in an arbitrary semigroup (S,+), and showed it to be equivalent to the usual algebraic characterization. D. De, N. Hindman, and D. Strauss later proved a stronger version of the Central Sets Theorem for semigroups in 2008. D. Phulara further genaralized the result for commutative semigroups in 2015. Recently in his work, Zhang generalized it further and proved the central sets theorem for uncountably many central sets. We extend the theorem to arbitrary adequate partial semigroups and VIP systems.

We analyze generalized Bopp shifts and Darboux normalization in two-dimensional noncommutative quantum mechanics (NCQM) from the viewpoint of the unitary representation theory of the kinematical symmetry group \(G_{\mathrm{NC}}\). This group is a step-two nilpotent Lie group with three-dimensional center, and the regular part of its unitary dual \(\widehat{G_{\mathrm{NC}}}\) is labelled by central characters \((\hbar,\vartheta,B_{\mathrm{in}})\). Ordinary two-dimensional quantum mechanics (QM) appears inside \(\widehat{G_{\mathrm{NC}}}\) as the family of Weyl-Heisenberg representations inflated along the quotient \(G_{\mathrm{NC}}\rightarrow G_{\mathrm{WH}}\), with central character \((\hbar,0,0)\). We prove that a generic nondegenerate NCQM sector \((\hbar_0,\vartheta_0,B_0)\), with \(\hbar_0,\vartheta_0,B_0 eq 0\) and \(\hbar_0-B_0\vartheta_0 eq 0\), is not unitarily equivalent to the ordinary QM sector \((\hbar_0,0,0)\) as a \(G_{\mathrm{NC}}\)-representation. Consequently, generalized Bopp shifts and Darboux normalizations, although they can produce auxiliary operator quadruples satisfying canonical commutation relations, do not establish kinematical equivalence of the correspo

Let G be a simple, finite, connected, and undirected graph. The middle graph M(G) of G is obtained from the subdivision graph S(G) after joining pairs of subdivided vertices that lie on adjacent edges of G and the central graph C(G) of G is obtained from S(G) after joining all non-adjacent vertices of G. We show that if the order of G is at least 4, then Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic (as abstract groups) and apply this result to obtain new upper bounds of the distinguishing number and the distinguishing index of C(G) and M(G) and provide examples showing that these bounds cannot be improved in general. Moreover, we use idempotent commutative Latin squares and a theorem of Galvin on list edge colorings of bipartite graphs to study the total distinguishing chromatic number of central graphs.

A new 2-parameter family of central structures in trees, called central forests, is introduced. Minieka's $m$-center problem and McMorris's and Reid's central-$k$-tree can be seen as special cases of central forests in trees. A central forest is defined as a forest $F$ of $m$ subtrees of a tree $T$, where each subtree has $k$ nodes, which minimizes the maximum distance between nodes not in $F$ and those in $F$. An $O(n(m+k))$ algorithm to construct such a central forest in trees is presented, where $n$ is the number of nodes in the tree. The algorithm either returns with a central forest, or with the largest $k$ for which a central forest of $m$ subtrees is possible. Some of the elementary properties of central forests are also studied.

We define the notion of central orderings for a general commutative ring $A$ which generalizes the notion of central points of irreducible real algebraic varieties. We study a central and a precentral loci which both live in the real spectrum of the ring $A$ and allow to state central Positivestellensätze in the spirit of Hilbert 17th problem.

We consider the $R$-matrix presentations of the quantum queer superalgebra $U_q(q_n)$ and its affine counterpart $U_q(\widehat q_n)$. We derive crossing symmetry relations for the $R$-matrices and use them to construct central elements in both superalgebras. We also produce an epimorphism $ev:U_q(\widehat q_n)\to U_q(q_n)$ identical on the subalgebra isomorphic to $U_q(q_n)$.

Kubernetes, an open-source platform for automating the deployment, scaling, and management of containerized applications, is widely used for its efficiency and scalability. However, its complexity and extensive configuration options often lead to security vulnerabilities if not managed properly. This paper presents a detailed analysis of misconfigurations in Kubernetes environments and their significant impact on system reliability and security. A centralized logging solution was developed to detect such misconfigurations, detailing the integration process with a Kubernetes cluster and the implementation of role-based access control. Utilizing a combination of open-source tools, the solution systematically identifies misconfigurations and aggregates diagnostic data into a central repository. The effectiveness of the solution was evaluated using specific metrics, such as the total cycle time for running the central logging solution against the individual open source tools.

We use hyperelliptic Shimura curves to find triple product $L$-functions of Hilbert newforms with central vanishing orders proved to be at least 3.

Central Bank Digital Currency (CBDC) can be defined as a virtual currency based on node network and digital encryption algorithm issued by a country which has a legal credit protection. CBDCs are supported by Distributed Ledger Technologies (DLTs), and they may allow a universal means of payments for the digital era. There are many ways to proceed, they all require central banks to develop technological expertise. Considering these points, it is important to understand the new IT governance in the financial markets due to CBDC and digital economy. Information Technology is an essential driver that will allow the new financial industry design. This paper has the objective to answer two questions through an updated Systematic Literature Review (SLR). The first question is What IT resources and tools have been considered or applied to set the governance of CBDC adoption? The second; Identify IT governance models in the financial market due to CBDC adoption. Bank for International Settlements (BIS) publications, Scopus and Web of Science were considered as sources of studies. After the strings and including criteria were applied, fourteen papers were analyzed. This paper finds many IT

We establish a central limit theorem for the central values of Dirichlet $L$-functions with respect to a weighted measure on the set of primitive characters modulo $q$ as $q \rightarrow \infty$. Under the Generalized Riemann Hypothesis (GRH), we also prove a weighted central limit theorem for the joint distribution of the central $L$-values corresponding to twists of two distinct primitive Hecke eigenforms. As applications, we obtain (under GRH) positive proportions of twists for which the central $L$-values simultaneously grow or shrink with $q$ as well as a positive proportion of twists for which linear combinations of the central $L$-values are nonzero.

It is a long-standing open problem raised by Starostin to describe all finite groups with soluble centralizers of involutions. One can observe that if the centralizer fusion system of an involution is nilpotent, then the centralizer of that involution is soluble. In this paper, we classify the cases when the centralizer fusion system of a central involution in a finite group whose all involutions have soluble centralizers is a nilpotent fusion system. Indeed, we analyse the case when the solvable radical has odd order and the corresponding factor group is simple.

In a pQCD-based model, we have analyzed the STAR data on the high $p_T$ suppression of charged hadrons, in Au+Au collisions at $\sqrt{s}$=200 GeV. In the jet quenching or the energy loss picture, $p_T$ spectra of charged hadrons as well as the $p_T$ dependence of nuclear modification factor, in all the centrality ranges, are well explained, with nearly a constant relative energy loss, $ΔE/E=0.56\pm 0.03$. Centrality independence of relative energy loss indicate that the matter produced in central and in peripheral collisions are different, otherwise relative energy loss would have shown strong centrality dependence. Qualitatively, centrality independence of relative energy loss can be understood, if in central Au+Au collisions deconfined matter is produced and the matter remain confined in peripheral collisions.

While the radiation mechanism of fast radio bursts (FRBs) is unknown, coherent curvature radiation and synchrotron maser are promising candidates. We find that both radiation mechanisms work for a neutron star (NS) central engine with $B\gtrsim 10^{12}$ G, while for the synchrotron maser, the central engine can also be an accreting black hole (BH) with $B\gtrsim 10^{12}$ G and a white dwarf (WD) with $B\sim 10^8-10^9$ G. We study the electromagnetic counterparts associated with such central engines, i.e., nebulae for repeating FRBs and afterglows for non-repeating FRBs. In general, the energy spectrum and flux density of the counterpart depend strongly on its size and total injected energy. We apply the calculation to the nebula of FRB 121102 and find that the persistent radio counterpart requires the average energy injection rate into the nebula to be between $2.7\times10^{39}~{\rm erg/s}$ and $1.5\times10^{44}~{\rm erg/s}$, and the minimum injected energy be $6.0\times10^{47}~{\rm erg}$ in around $7$ yr. Consequently, we find that for FRB 121102 and its nebula: (1) WD and accretion BH central engines are disfavored; (2) a rotation-powered NS central engine works when $1.2\times10

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection (generalized SIS model). Specifically, after each infection, the infectivity is a random function of the time elapsed since the infection, and each recovered individual loses immunity gradually (equivalently, becomes gradually susceptible) after some time according to a random susceptibility function. The epidemic dynamics is described by the average infectivity and susceptibility processes in the population together with the numbers of infected and susceptible/uninfected individuals. In \cite{forien-Zotsa2022stochastic}, a functional law of large numbers (FLLN) is proved as the population size goes to infinity, and asymptotic endemic behaviors are also studied. In this paper, we prove a functional central limit theorem (FCLT) for the stochastic fluctuations of the epidemic dynamics around the FLLN limit. The FCLT limit for the aggregate infectivity and susceptibility processes is given by a system of stochastic non-linear integral equation driven by a two-dimensional Gaussian process.

Network analysis has emerged as a key technique in communication studies, economics, geography, history and sociology, among others. A fundamental issue is how to identify key nodes, for which purpose a number of centrality measures have been developed. This paper proposes a new parametric family of centrality measures called generalized degree. It is based on the idea that a relationship to a more interconnected node contributes to centrality in a greater extent than a connection to a less central one. Generalized degree improves on degree by redistributing its sum over the network with the consideration of the global structure. Application of the measure is supported by a set of basic properties. A sufficient condition is given for generalized degree to be rank monotonic, excluding counter-intuitive changes in the centrality ranking after certain modifications of the network. The measure has a graph interpretation and can be calculated iteratively. Generalized degree is recommended to apply besides degree since it preserves most favourable attributes of degree, but better reflects the role of the nodes in the network and has an increased ability to distinguish among their importa

Variation of empirical Fréchet means on a metric space with curvature bounded above is encoded via random fields indexed by unit tangent vectors. A central limit theorem shows these random tangent fields converge to a Gaussian such field and lays the foundation for more traditionally formulated central limit theorems in subsequent work.

The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI-modules, where FI is the category of finite sets and injective maps. We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion of $d$-step central stability, and prove that if the ideal of relations of a category is generated in degrees at most $d$, then every module presented in finite degrees is $d$-step centrally stable.

We study the algebraic geometry and combinatorics of the central degeneration (the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves) in type A. More specifically, we elucidate the central degeneration of semi-infinite orbits and explain its relations with Levi restriction. Also, we discuss the central degeneration of Mirkovi$\acute{\text{c}}$-Vilonen cycles in the affine Grassmannian, and the corresponding transformations of Mirkovi$\acute{\text{c}}$-Vilonen polytopes. In addition, we shed some light on the geometry of Iwahori MV cycles in the affine Grassmannian and generalized MV cycles in the affine flag variety, which are closely related to Demazure modules and affine Deligne-Lusztig varieties respectively.

We explore the redshift evolution of a curious correlation between the star-formation properties of central galaxies and their satellites (`galactic conformity') at intermediate to high redshift ($0.4<z<1.9$). Using an extremely deep near-infrared survey, we study the distribution and properties of satellite galaxies with stellar masses, ${\rm log} ({\rm M}_*/{\rm M}_{\odot})>9.7$, around central galaxies at the characteristic Schechter function mass, ${\rm M} \sim {\rm M}^{\ast}$. We fit the radial profiles of satellite number densities with simple power laws, finding slopes in the range -1.1 to -1.4 for mass-selected satellites, and -1.3 to -1.6 for passive satellites. We confirm the tendency for passive satellites to be preferentially located around passive central galaxies at $3σ$ significance and show that it exists to at least $z\sim2$. Meanwhile, the quenched fraction of satellites around star-forming galaxies is consistent with field galaxies of equal stellar masses. We find no convincing evidence for a redshift-dependent evolution of these trends. One simple interpretation of these results is that only passive central galaxies occupy an environment that is capable