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We introduce the notion of PC cancellative additive monoids with infinity and use it to characterize cancellative additive principal ideal domains with infinity. Our characterization improves various known characterizations from the literature, both, in the context of the commutative cancellative monoids, as well as in the context of the analogues of the statements from the commutative ring theory.

We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie (2004), but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived algebraic geometry, and the algebraic geometry of commutative ring spectra as in Lurie (2004), but also to differential topology, complex geometry, the theory of supermanifolds, derived manifolds etc., where it produces a theory of higher generalized orbifolds appropriate for these settings. This universal framework yields new insights into the general theory of Deligne-Mumford stacks and orbifolds, including a representability criterion which gives a categorical characterization of such generalized Deligne-Mumford stacks. This specializes to a new categorical description of classical Deligne-Mumford stacks, a result sketched in Carchedi (2019), which extends to derived and spectral Deligne-Mumford stacks as well.

The paper investigates the asymptotic behavior at infinity of ring Q -homeomorphisms with respect to the p -modulus for p > 2 on the complex plane. A sufficient condition for the mean integral value of the function Q under which the mapping has an exponential growth at infinity has been found. An example has been constructed that shows the accuracy of the obtained results.

We develop a 4 × 4-matrix model based on temporal coupled mode theory (TCMT) to elucidate the intricate energy exchange within a non-Hermitian, resonant photonic structure, based on the recently described infinity-loop micro-resonator (ILMR). We consider the structure to consist of four coupled resonant modes, with clockwise and counterclockwise propagating optical fields, the interplay between which gives rise to a rich spectral form with both overlapping and non-overlapping resonances within a single free spectral range (FSR). Our model clarifies the precise conditions for exceptional points (EPs) in this system by examining neighboring resonances over the device free spectral range (FSR). We find that the system is robust to the conditions for observing an EP, despite the presence of non-zero coupling of signals, or crosstalk, between the resonant modes.

The purpose of this article is to prove the Iwasawa main conjecture for CM fields in certain cases through a detailed study on the divisibility relation between pp-adic LL-functions for CM fields and Eisenstein ideals of unitary groups of degree three.

The commutative differential graded algebra $A_{\\mathrm {PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\\mathcal {I}}(X)$ of $A_{\\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\\mathcal {I}$ to model $E_{\\infty }$ differential graded algebras (dga) by strictly commutative objects, called commutative $\\mathcal {I}$-dgas. We define a functor $A^{\\mathcal {I}}$ from simplicial sets to commutative $\\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\\infty }$ dga of cochains. The functor $A^{\\mathcal {I}}$ shares many properties of $A_{\\mathrm {PL}}$, and can be viewed as a generalization of $A_{\\mathrm {PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\\mathcal {I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.

This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

We study the empirical measure L An of the eigenvalues of nonnormal square matrices of the form A n = U n T n V n with U n , V n independent Haar distributed on the unitary group and T n real diagonal. We show that when the empirical measure of the eigenvalues of T n converges, and T n satisfies some technical conditions, L An converges towards a rotationally invariant measure μ on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single ring theorem [6]. We also consider the case where U n , V n are independently Haar distributed on the orthogonal group.

We study Drinfeld modular forms for the modular group Γ = GL(r, 𝔽 q [T]) on the Drinfeld symmetric space Ω r , wherer≥ 2. Results include the existence of a (q− 1)-th root (up to constants)hof the discriminant function Δ, the description of the growth/decay of the standard formsg₁,g₂, . . .gr−1 , Δ on the fundamental domain 𝓕 of Γ, and the reduction of these forms on the central part 𝓕ₒof 𝓕. The results are exemplified in detail forr= 3.