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We consider a Mikusiński-type convolution algebra C α , including functions with power-type singularities at the origin as well as all functions continuous on [ 0 , ∞) . Algebraic properties of this space are derived, including its ideal structure, filtered and graded structure, and Jacobson radical. Applications to operators of fractional calculus and the associated integro-differential equations are discussed. [ABSTRACT FROM AUTHOR]

Let $f\colon M^{2n}\to \mathbb {R}^{2n+p}$ , $2\leq p\leq n-1$ , be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng (2013, Michigan Mathematical Journal 62, 421–441) conjectured that if the codimension is $p\leq 11$ , then, along any connected component of an open dense subset of $M^{2n}$ , the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces in the kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\mathbb {R}^{2n+p}$ of larger dimension than $2n$. This bold conjecture was proved by Dajczer and Gromoll just for codimension 3 and then by Yan and Zheng for codimension 4. In this paper, we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step for a possible classification of the nonholomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss algebraic rough set and its relationships with convex space, rough set and generalized neighborhood space. Specifically, the notion of algebraic relations is introduced and a pair of lower approximation operator and upper approximation operator are presented. Then, several conditions of algebraic relations such as seriality, reflexivity, (resp., weak, primitive) symmetry and (resp., strong) transitivity are characterized by algebraic approximation operators. Based on this, relationships among algebraic rough sets, convex structures and generalized neighborhood systems are investigated. It is proved that the category of reflexive and transitive algebraic rough spaces is isomorphic to the category of convex spaces. In particular, the category of reflexive, weakly symmetric and transitive algebraic rough spaces is isomorphic to the category of convex matroids and the category of reflexive, weakly symmetric and transitive algebraic generalized neighborhood spaces. [ABSTRACT FROM AUTHOR]

By making use of Halperin's local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau–de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout. [ABSTRACT FROM AUTHOR]

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras. [ABSTRACT FROM AUTHOR]

Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions of a three-dimensional Calabi–Yau category and its associated Donaldson–Thomas invariants. In subsequent work, T. Bridgeland and I. Strachan showed that Joyce structures satisfying a certain non-degeneracy condition encode a complex hyperkähler structure on the tangent bundle of the base of the Joyce structure. In this work we give a definition of an analogous structure over an affine special Kähler (ASK) manifold, which we call a special Joyce structure. Furthermore, we show that it encodes a real hyperkähler (HK) structure on the tangent bundle of the ASK manifold, possibly of indefinite signature. Particular examples include the semi-flat HK metric associated to an ASK manifold (also known as the rigid c-map metric) and the HK metrics associated to certain uncoupled variations of BPS structures over the ASK manifold. Finally, we relate the HK metrics coming from special Joyce structures to HK metrics on the total space of algebraic integrable systems. [ABSTRACT FROM AUTHOR]

The moduli space Y=Y(E6)$Y = Y(E_6)$ of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth‐century work of Cayley and Salmon. Modern interest in Y$Y$ was restored in the 1980s by Naruki's explicit construction of a W(E6)$W(E_6)$‐equivariant smooth projective compactification Y¯${\overline{Y}}$ of Y$Y$, and in the 2000s by Hacking, Keel, and Tevelev's construction of the Kollár–Shepherd‐Barron–Alexeev (KSBA) stable pair compactification Y∼${\widetilde{Y}}$ of Y$Y$ as a natural sequence of blowups of Y¯${\overline{Y}}$. We describe generators for the cones of W(E6)$W(E_6)$‐invariant effective divisors and curves of both Y¯${\overline{Y}}$ and Y∼${\widetilde{Y}}$. For Naruki's compactification Y¯${\overline{Y}}$, we further obtain a complete stable base locus decomposition of the W(E6)$W(E_6)$‐invariant effective cone, and as a consequence find several new W(E6)$W(E_6)$‐equivariant birational models of Y¯${\overline{Y}}$. Furthermore, we fully describe the log minimal model program for the KSBA compactification Y∼${\widetilde{Y}}$, with respect to the divisor KY∼+cB+dE$K_{{\widetilde{Y}}} + cB + dE$, where B$B$ is the boundary and E$E$ is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points. [ABSTRACT FROM AUTHOR]

The original article can be found online at https://doi.org/10.1007/s00222-024-01257-1The publication of this article unfortunately contained mistakes. The style of the references was not correct. The corrected reference list is given below.The original article has been corrected.Publisher's NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.By Yuan Liu; Melanie Matchett Wood and David Zureick-BrownReported by Author; Author; Author [Extracted from the article]

Over the moduli space of pointed smooth algebraic curves, the projectivized k th Hodge bundle is the space of k -canonical divisors. The incidence loci are defined by requiring the k -canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized k th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails. [ABSTRACT FROM AUTHOR]

Let (X,Δ)$(X,\varDelta)$ be a projective klt pair, and f:X→Y$f\colon X\rightarrow Y$ a fibration to a smooth projective variety Y$Y$ with strictly nef relative anti‐log canonical divisor −(KX/Y+Δ)$-(K_{X/Y}+\varDelta)$. We prove that f$f$ is a locally trivial fibration with rationally connected fibres, and the base Y$Y$ is a canonically polarized hyperbolic manifold. In particular, when Y$Y$ is a single point, we establish that X$X$ is rationally connected. Moreover, when dimX=3$\dim X=3$ and −(KX+Δ)$-(K_X+\varDelta)$ is strictly nef, we prove that −(KX+Δ)$-(K_X+\varDelta)$ is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds. [ABSTRACT FROM AUTHOR]

We prove an additivity result for the log Kodaira dimension of algebraic fiber spaces over abelian varieties, a superadditivity result for fiber spaces over varieties of maximal Albanese dimension, as well as a subadditivity result for log pairs. [ABSTRACT FROM AUTHOR]

It is known that every effect algebra can be represented as the effect algebra of perspectivity classes of some E-test space. We describe when there exists join and meet of two perspectivity classes of events of some algebraic E-test space. Moreover we give the formula for join and meet of perspectivity classes mentioned above, using only tests. We obtain an example of finite, non-homogeneous effect algebra E such that sharp elements of E form a lattice, whereas E is not a lattice. [ABSTRACT FROM AUTHOR]

Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2, which is used in the Walsh analysis. The role of harmonics is played by some analogues of the classical Walsh functions. The concept of Fourier transform is introduced, and the Poisson summation formula, Plancherel theorem, Vilenkin-Chrestenson formulas and so on are proved. A kind of analogue of the Schwartz class is studied. This class consists of functions such that both the function itself and its Fourier transform have compact support. [ABSTRACT FROM AUTHOR]

We show that for each algebraic space that is separated and of finite type over a field, and whose affinization is connected and reduced, there is a universal morphism to a para-abelian variety. The latter are schemes that acquire the structure of an abelian variety after some ground field extension. This generalizes classical results of Serre on universal morphisms from algebraic varieties to abelian varieties. Our proof relies on corresponding facts for the proper case, together with the structural properties of group schemes, removal of singularities by alterations, and ind-objects. It turns out that the formation of the Albanese variety commutes with base-change up to universal homeomorphisms. We also give a detailed analysis of Albanese maps for algebraic curves and algebraic groups, with special emphasis on imperfect ground fields. [ABSTRACT FROM AUTHOR]

In this article, we propose to study a general notion of a down‐up Markov chain for multifurcating trees with n$$ n $$ labeled leaves. We study in detail down‐up chains associated with the (α,γ)$$ \left(\alpha, \gamma \right) $$‐model of Chen et al. (Electron. J. Probab. 14 (2009), 400–430.), generalizing and further developing previous work by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018; Random Struct. Algoritm. 54 (2020), 745–769; Electron. J. Probab. 25 (2020), 1–46.) in the binary special cases. The technique we deploy utilizes the construction of a growth process and a down‐up Markov chain on trees with planar structure. Our construction ensures that natural projections of the down‐up chain are Markov chains in their own right. We establish label dynamics that at the same time preserve the labeled alpha‐gamma distribution and keep the branch points between the k$$ k $$ smallest labels for order n2$$ {n}^2 $$ time steps for all k≥2$$ k\ge 2 $$. We conjecture the existence of diffusive scaling limits generalizing the "Aldous diffusion" by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018.) as a continuum‐tree‐valued process and the "algebraic α$$ \alpha $$‐Ford tree evolution" by Löhr et al. (Ann. Probab. 48 (2020), 2565–2590.) and by Nussbaumer and Winter (arXiv:2006.09316, 2020.) as a process in a space of algebraic trees. [ABSTRACT FROM AUTHOR]

Let A be an associative ring and M a finitely generated projective A-module. We introduce a category RBS (M) and prove several theorems which show that its geometric realisation functions as a well-behaved unstable algebraic K-theory space. These categories RBS (M) naturally arise as generalisations of the exit path ∞ -category of the reductive Borel–Serre compactification of a locally symmetric space, and one of our main techniques is to find purely categorical analogues of some familiar structures in these compactifications. [ABSTRACT FROM AUTHOR]

Flag codes are a recent network coding strategy based on linear algebra. Fuzzy vector subspaces extend the notions of classical linear algebra. They can be seen as abstractions of flags to the point that several fuzzy vector subspaces can be identified to the same flag, which naturally induces an equivalence relation on the set of fuzzy vector subspaces. The main contributions of this work are the methodological abstraction of flags and flag codes in terms of fuzzy vector subspaces, as well as the generalisation of three distinct equivalence relations that originated from the fuzzy subgroup theory and study of their connection with flag codes, computing the number of equivalence classes in the discrete case, which represent the number of essentially distinct flags, and a comprehensive analysis of such relations and the properties of the corresponding quotient sets. [ABSTRACT FROM AUTHOR]

This article focuses on the implications of a noncommutative formulation of branch‐cut quantum gravity. Based on a mini‐superspace structure that obeys the noncommutative Poisson algebra, combined with the Wheeler–DeWitt equation and Hořava–Lifshitz quantum gravity, we explore the impact of a scalar field of the inflaton‐type in the evolution of the Universe's wave function. Taking as a starting point the Hořava–Lifshitz action, which depends on the scalar curvature of the branched Universe and its derivatives, the corresponding wave equations are derived and solved. The noncommutative quantum gravity approach adopted preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner Formalism. In this work we delve deeper into a mini‐superspace of noncommutative variables, incorporating scalar inflaton fields and exploring inflationary models, particularly chaotic and nonchaotic scenarios. We obtained solutions to the wave equations without resorting to numerical approximations. The results indicate that the noncommutative algebraic space captures low and high spacetime scales, driving the exponential acceleration of the Universe. [ABSTRACT FROM AUTHOR]

Prediction of renewable energy consumption structure (RECS) can provide important guidance for energy development planning and energy structure transformation. The RECS refer to the proportion of various renewable energy consumptions and belong to compositional data, which could reflect the structural shapes of a complete system better. The multivariate compositional data's vector autoregressive model (CDVAR) on the basis of the Simplex space and its algebraic system is proposed in this study aiming at the multi-dimensional small sample size. Firstly, the algebraic system of the Simplex space is introduced and the statistics of the compositional data are defined. Secondly, the novel model with the form of the compositional data is obtained and the least square parameter estimation of the model is derived according to Aitchison geometry. Third, the validation of the novel model is verified by the data on RECS in countries (China, USA, and Canada). The validation presents that the proposed model performs better in fitting, prediction, stability, and applicability compared with other five models under transformation. Last, the proposed model is applied to analyze and forecast the RECS of the above countries in 2021–2025 to provide an important basis for the optimization of the RECS. [ABSTRACT FROM AUTHOR]

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients. [ABSTRACT FROM AUTHOR]

We prove a topological decomposition of the space of meromorphic germs at zero in several variables with prescribed linear poles as a sum of spaces of holomorphic and polar germs. Evaluating the resulting holomorphic projection at zero gives rise to a continuous evaluator (at zero) on the space of meromorphic germs in several variables. Our constructions are carried out in the framework of Silva spaces and use an inner product on the underlying space of variables. They generalise to several variables, the topological direct decomposition of meromorphic germs at zero as sums of holomorphic and polar germs previously derived by the first and third author and provide a topological refinement of a known algebraic decomposition of such spaces previously derived by the second author and collaborators. [ABSTRACT FROM AUTHOR]

For a smooth projective curve, we derive a closed formula for the generating series of its Gromov–Witten invariants in genus g$g$ and degree zero. It is known that the calculation of these invariants can be reduced to that of the λg$\lambda _g$ and λg−1$\lambda _{g-1}$ integrals on the moduli space of stable algebraic curves. The closed formula for the λg$\lambda _g$ integrals is given by the λg$\lambda _g$ conjecture, proved by Faber and Pandharipande. We compute in this paper the λg−1$\lambda _{g-1}$ integrals via solving the degree zero limit of the loop equation associated to the complex projective line. [ABSTRACT FROM AUTHOR]

We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space and prove a semistable reduction theorem for points of algebraic stacks equipped with a Θ -stratification. These results provide a generalization of the Keel–Mori theorem to moduli problems whose objects have positive dimensional automorphism groups and give criteria on the moduli problem to have a separated or proper good moduli space. To illustrate our method, we apply these results to construct proper moduli spaces parameterizing semistable G -bundles on curves and moduli spaces for objects in abelian categories. [ABSTRACT FROM AUTHOR]

In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs. [ABSTRACT FROM AUTHOR]

Let BG be the classifying space of an algebraic group G over the field ${\mathbb C}$ of complex numbers. There are smooth projective approximations X of $BG\times {\mathbb P}^{\infty}$ , by Ekedahl. We compute a new stable birational invariant of X defined by the difference of two coniveau filtrations of X , by Benoist and Ottem. Hence we give many examples such that two coniveau filtrations are different. [ABSTRACT FROM AUTHOR]

The article is focuses on the study of the theory of rings of continuous functions with countable images on a topological space X; the algebraic properties of Cc(X). The connection between the properties of X and the algebraic properties of Cc(X) is investigated. [ABSTRACT FROM AUTHOR]

Heredity of open subspaces and closed subspaces of meet-continuous spaces are studied, some equivalent characterizations of the meet-continuous spaces are given. Particularly, relationship between algebraic continuous spaces and algebraic quasi-continuous spaces is illuminated by taking the meet-continuous spaces as an example. [ABSTRACT FROM AUTHOR]

In this paper, the solution to the Dirichlet problem for the wave equation on the star graph is constructed. To begin, we solve the boundary value problem on the interval (on one edge of the graph). We use the generalized functions method to obtain the wave equation with a singular right-hand side. The solution to the Dirichlet problem is determined through the convolution of the fundamental solution with the singular right-hand side of the wave equation. Thus, the solution found on the interval is determined by the initial functions, boundary functions, and their derivatives (the unknown boundary functions). A resolving system of two linear algebraic equations in the space of the Fourier transform in time is constructed to determine the unknown boundary functions. Following inverse Fourier transforms, the solution to the Dirichlet problem of the wave equation on the interval is constructed. After determining all the solutions on all edges and taking the continuity condition and Kirchhoff joint condition into account, we obtain the solution to the wave equation on the star graph. [ABSTRACT FROM AUTHOR]

The analysis and prediction of the energy consumption structure provide an important basis for promoting the optimization of the energy consumption structure, energy saving, and emission reduction. The structure data of energy consumption can reflect the proportion of various energy consumptions and belong to component data. In this study, an autoregressive moving average model of component data (CDARMA) based on simplex space and its algebraic system is established to study the energy consumption structure. First, various mathematical concepts in simplex space are presented to lay a theoretical modeling foundation. Second, the autocorrelation and partial autocorrelation coefficients of the component data are defined, and the CDARMA model in the simplex space is established. The parameters are estimated using least squares. Third, the energy consumption structure data of Europe, Japan, and China are used to verify the effectiveness of the proposed model. And the results are compared with other four models under eight kinds of transformation. Overall, the new model has a better fitting effect, prediction effect, and stability than the compared models. Lastly, the CDARMA model is used to predict the energy consumption structure of the three regions in 2019–2025 to provide a reference for the adjustment of the energy structure. [ABSTRACT FROM AUTHOR]

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well-known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate a natural question raised by Osin: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer Osin's question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study two new classes of spaces, called binary g*-pre regular and binary g*-pre normal spaces. Some basic properties of these separation axioms are studied by utilizing binary g*-pre closed and binary g*-pre open sets. [ABSTRACT FROM AUTHOR]

Let |$Z\to X$| be a closed immersion of smooth affine schemes over a field |$k$|⁠ , and let |$X^h_Z$| denote the henselisation of |$X$| along |$Z$|⁠. We prove that |$E(X^h_Z)\simeq E(Z)$| for every additive presheaf |$E\colon \textbf {SH}(k)^{\textrm {op}}\to \textrm {Ab}$| on the stable motivic homotopy category over |$k$| that is |$l_\varepsilon $| -torsion or |$l$| -torsion, where |$l\in \mathbb Z$| is coprime to |$\operatorname {char} k$|⁠ , and |$l_\varepsilon =\sum _{i=1}^n \langle (-1)^i \rangle $|⁠. More generally, the isomorphism holds for any homotopy invariant |$l_\varepsilon $| -torsion or |$l$| -torsion linear |$\sigma $| -quasi-stable framed additive presheaf on |$\textrm {Sm}_{k}$|⁠. This generalises the result known earlier for local schemes. We prove the above isomorphism by constructing (stable) |${\mathbb {A}}^1$| -homotopies of motivic spaces via algebraic geometry. To achieve this, we replace Quillen's trick with an alternative and more general construction that provides relative curves required in our setting. [ABSTRACT FROM AUTHOR]

We explicitly compute the moduli space pointed algebraic curves with a given numerical semigroup as Weierstrass semigroup for many cases of genus at most seven and when possible describe the structure as cone over an explicitly given variety. We determine the dimension of the moduli space for all semigroups of genus seven. [ABSTRACT FROM AUTHOR]

One of the primary objectives of this paper is to establish compatibility between two different jet space functors in the most general context. We first show an adjunction between the jet and the Witt functor on algebras. Following Borger's approach, we then construct the algebraic jet space functor in the general setting where the base is an arbitrary prolongation sequence. We then show that this functor is representable in the category of schemes and that Buium's jet space can be recovered by the π -adic completion of this representable scheme. As an application, this allows us to strengthen a result of Buium on the relation between Greenberg's transform and the special fiber of jet spaces, including the ramified case. [ABSTRACT FROM AUTHOR]

Differential equations of fractional order arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. In this article, a numerical method based on Laplace transform and numerical inverse Laplace transform for the numerical modeling of differential equations of fractional order is developed. The analytic inversion can be very difficult for complex forms of the transform function. Therefore, numerical methods are used for the inversion of the Laplace transform. In general, the numerical inverse Laplace transform is an ill-posed problem. This difficulty has led to various numerical methods for the inversion of the Laplace transform. In this work, the Weeks method is utilized for the numerical inversion of the Laplace transform. In our proposed numerical method, first, the fractional-order differential equation is converted to an algebraic equation using Laplace transform. Then, the transformed equation is solved in Laplace space using algebraic techniques. Finally, the Weeks method is utilized for the inversion of the Laplace transform. Weeks method is one of the most efficient numerical methods for the computation of the inverse Laplace transform. We have considered five test problems for validation of the proposed numerical method. Based on the comparison between analytical results and the Weeks method results, the reliability and effectiveness of the Weeks method for fractional-order differential equations was confirmed. [ABSTRACT FROM AUTHOR]

We study the question of whether the Brauer group is isomorphic to the cohomological one in spectral algebraic geometry. For this, we prove the compact generation of the derived category of twisted sheaves in the setting of spectral algebraic stacks. In particular, we obtain the compact generation of the ∞$\infty$‐category of quasi‐coherent sheaves and the existence of compact perfect complexes with prescribed support for such stacks. We extend these results to derived algebraic geometry by studying the relationship between derived and spectral algebraic stacks. [ABSTRACT FROM AUTHOR]

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a d-dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter H. This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximation space. This paper presents a novel localization technique that leads to a super-exponential decay of its basis relative to H. This suggests that basis functions with supports of width \mathcal O(H|\log H|^{(d-1)/d}) are sufficient to preserve the optimal algebraic rates of convergence in H without pre-asymptotic effects. A sequence of numerical experiments illustrates the significance of the new localization technique when compared to the so far best localization to supports of width \mathcal O(H|\log H|). [ABSTRACT FROM AUTHOR]

We characterize finite elements, an order-theoretic concept in Archimedean vector lattices, in the setting of ordered Banach spaces with positive unconditional basis as vectors having finite support with respect to their basis representations. Using algebraic vector space bases, we further describe a class of infinite dimensional vector lattices in which each element is finite and even self-majorizing. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period maps are quasi-projective and that the restriction of the Griffiths bundle is ample. [ABSTRACT FROM AUTHOR]

Let p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum {\mathbb {S}} admits an "eigensplitting" that generalizes known splittings on K-theory and TC. We identify the summands in the fiber as the covers of {\mathbb {Z}}_{p}-Anderson duals of summands in the K(1)-localized algebraic K-theory of {\mathbb {Z}}. Analogous results hold for the ring {\mathbb {Z}} where we prove that the K(1)-localized fiber sequence is self-dual for {\mathbb {Z}}_{p}-Anderson duality, with the duality permuting the summands by i\mapsto p-i (indexed mod p-1). We explain an intrinsic characterization of the summand we call Z in the splitting TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j'\vee Z in terms of units in the p-cyclotomic tower of {\mathbb {Q}}_{p}. [ABSTRACT FROM AUTHOR]

We consider Lie algebroids over algebraic spaces (in short we call it as a-spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss about properties of universal enveloping algebroid U (O X , L) of a Lie algebroid L over an a-space (X , O X) . This is done by sheafification of the presheaf of universal enveloping algebras for Lie-Rinehart algebras. We review the extent to which structure of the universal enveloping algebroid of Lie algebroids (over special a-spaces) resembles a sheaf of bialgebras. In the sequel we present a version of Poincare-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem for the Lie algebraoid. [ABSTRACT FROM AUTHOR]

The general theory of quaternion q-difference equations is completely different from traditional q-difference equations in the complex space for the special algebraic structure of the quaternion space, such as the non-commutativity of multiplication among elements. In this paper, some properties of basic quaternion q-discrete functions, such as the quaternion q-exponential function, are obtained. The existence, uniqueness, and extension theorems of the solution for higher-order linear quaternion q-difference equations (QQDCEs) are established through constructing quaternion q-stepwise approximation sequences, and the equivalent variable transform between higher-order linear QQDCEs and quaternion linear q-difference equations is given. Moreover, some basic results, such as the Wronskian formula, Liouville formula, and general solution structure theorems of higher-order linear QQDCEs with constant and variable coefficients, are established by applying the quaternion characteristic polynomial and the quaternion determinant algorithm. In addition, some particular and general solution formulas of non-homogeneous QQDCEs are obtained under the non-commutative condition of quaternion elements. Finally, several examples are provided to illustrate the feasibility of our obtained results. [ABSTRACT FROM AUTHOR]

Let k be a number field and X a smooth, geometrically integral quasi-projective variety over k. For any linear algebraic group G over k and any G-torsor g: Z \to X, we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for all twists of Z by elements in H^1_{\text {\'{e}t}}(k,G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for X. As an application, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent. [ABSTRACT FROM AUTHOR]

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In the present work, a spectral solver is developed for integration of certain differential equations of solid mechanics, namely static stress equilibrium in composite materials, described in cylindrical or spherical polar coordinates. The spectral approach is encompassed in approximating the displacement field using expansion into a series of Chebyshev polynomials in the radial coordinate and complex exponents in the angular direction. Consequently, differential operators in real space become algebraic operators in spectral space. The spatial heterogeneity and metric non-flatness pertinent to polar geometry are addressed by an iterative strategy, employing both second-order and first-order iterative solvers. The essence of the new contribution is in addressing the difficulty posed by the inherent nonsmoothness present in composite materials and the polar singularity. The interplay of the two produces instability, which is resolved in the proposed approach, specifically by using a new efficient linesearch algorithm, appropriate for the studied class of problems. The method is illustrated by analysis of 1D and 2D linear-elastic and linear-elastic–perfectly plastic response of composites to prescribed radial surface displacement. The developed method allows performing stress homogenization on polar representative volume elements, which has its conceptual advantages, while allowing similar runtime (for sufficient computing resources and an iterative strategy) to the one exhibited by spectral analysis in Cartesian coordinates. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to verify a conjecture of Gross under mild hypotheses: all reduced, separated, and excellent schemes have the resolution property away from a closed subset of codimension \geq 3. Our technique uses formal-local descent and the existence of affine flat neighborhoods to reduce the problem to constructing certain modules over commutative rings. Once in the category of modules we exhibit enough locally free sheaves directly, thereby establishing the resolution property for a specific class of algebraic spaces. A crucial step is showing it suffices to resolve a single coherent sheaf. [ABSTRACT FROM AUTHOR]

This paper considers the event‐triggered control design for the uniform sampled‐data set stabilization of switched delayed Boolean control networks (SDBCNs). First, using the algebraic state space representation method, SDBCNs are converted into the equivalent algebraic form. Second, using the algebraic form, the uniform sampled‐data reachable sets are constructed, based on which, a necessary and sufficient condition is obtained for the uniform sampled‐data set stabilization of SDBCNs. Finally, the event‐triggered mechanism is presented, and a sufficient condition is proposed to design the time‐variant state feedback event‐triggered controller for the uniform sampled‐data set stabilization of SDBCNs. [ABSTRACT FROM AUTHOR]

Inspired by the one-hundredth anniversary of the seminal works of Stern and Gerlach, our contribution is a proposal of how to use their famous experiment in a more contemporary perspective. Our main idea is to re-cast the experiment in the modern language of prepare-and-measure scenarios. By doing so, it is possible to connect geometric and algebraic aspects of the space of states with the physical space. We also discuss possible simulations of the SG experiment as well as some experimental properties of the experiment revealed at the statistical level. Merging a more modern perspective with a paradigmatic experiment, we hope this paper can serve as an entry door for quantum information theory and the. [ABSTRACT FROM AUTHOR]