On the behavior of formal neighborhoods in the Nash sets associated with toric valuations: a comparison theorem

Let $ {\mathcal C} $ be an algebraic curve and c be an analytically irreducible singular point of ${\mathcal C}$ . The set ${\mathscr {L}_{\infty }}({\mathcal C})^c$ of arcs with origin c is an irreducible closed subset of the space of arcs on ${\mathcal C}$ . We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along ${\mathscr {L}_{\infty }}({\mathcal C})^c$ the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space.

We study the formal neighborhoods at rational non-degenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.

Vibrant neighborhoods play a pivotal role in the human well-being and social cohesion. However, existing research amalgamates vitality discussions of varied functions, failing to offer nuanced insights for specific neighborhood planning. Another gap emerges from the lack of research focusing on cultivating neighborhood vibrancy based on bottom-up spatial perceptions at the eye-level. Therefore, our study delves into the spatial and temporal disparities in vitality between formal neighborhoods and urban villages. Both real and virtual indicators are employed to measure the physical and digital vitality of neighborhoods. We incorporate eye-level perceptions as a new explanatory variable based on the physical environment. Our findings suggest that real vitality exhibits a modest morning and evening peak on weekends, while virtual vitality continually increases and peaks at night. In addition to enhancing green and enclosed neighborhood environments, improving public transportation options that replace reliance on motorized travel could facilitate formal neighborhood vitality. Regarding urban villages, clear rights-of-way can boost commuting efficiency, spatial accessibility and road safety, while the provision of ample public facilities and public spaces fosters the creation of “15-minute living circles.” The differentiated planning recommendations could offer valuable insights to promote livable neighborhoods and ultimately help to improve human well-being.

Let [Formula: see text] be a field. We introduce a new geometric invariant, namely the minimal formal models, associated with every curve singularity (defined over [Formula: see text]). This is a noetherian affine adic formal [Formula: see text]-scheme, defined by using the formal neighborhood in the associated arc scheme of a primitive [Formula: see text]-parametrization. For the plane curve [Formula: see text]-singularity, we show that this invariant is [Formula: see text]. We also obtain information on the minimal formal model of the so-called generalized cusp. We introduce various questions in the direction of the study of these minimal formal models with respect to singularity theory. Our results provide the first positive elements of answer. As a direct application of the former results, we prove that, in general, the isomorphisms satisfying the Drinfeld–Grinberg–Kazhdan theorem on the structure of the formal neighborhoods of arc schemes at non-degenerate arcs do not come from the jet levels. In some sense, this shows that the Drinfeld–Grinberg–Kazhdan theorem is not a formal consequence of the Denef–Loeser fibration lemma.

On a fixed point index method for the analysis of the asymptotic behavior and boundary value problems of infinite dimensional dynamical systems and processes

The spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data. In this paper, we investigate the concept of spatial distribution for data in infinite dimensional Banach spaces. Many technical difficulties are encountered in such spaces that are primarily due to the noncompactness of the closed unit ball. In this work, we prove some Glivenko–Cantelli and Donsker-type results for the empirical spatial distribution process in infinite dimensional spaces. The spatial quantiles in such spaces can be obtained by inverting the spatial distribution function. A Bahadur-type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived. A study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces. The spatial distribution can be used to define the spatial depth in infinite dimensional Banach spaces, and we study the asymptotic properties of the empirical spatial depth in such spaces. We also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data.

This study examines the transformation in housing typology from low-rise to apartment buildings in the formal neighborhood of Kabul city. These formal neighborhoods were developed according to plans from 1978. The majority of these neighborhoods were designed with detached houses that had courtyards. Literature reviews, field visits, opinions of residents, and a planning organization provided data for this study. In this study, the transformation of housing in planned neighborhoods is analyzed in relation to their social and environmental impacts. Researchers determined how varying housing typologies affected residents’ health and quality of life in these planned neighborhoods. Initially, we assessed the physical characteristics of the study area and evaluated how much transformation volume is present in the study area. Second, we examined residents’ views of residential development and its impacts, as well as their daily lives. In order to identify the relationship between these two aspects, the study examined the characteristics of the area (variables) from the perspectives of privacy, natural light, shading, sound pollution, air pollution, and energy use. We used several criteria to evaluate the accuracy of the physical characteristics and the respondents’ opinions. Lastly, we provided some recommendations and solutions to improve the current situation.

The δ Equation on a Hilbert Space and Some Applications to Complex Analysis on Infinite Dimensional Vector Spaces

A Spectral Order for Infinite Dimensional Quantum Spaces: A Preliminary Report

Let M⊂CN be a real-analytic CR submanifold, M′⊂CN′ a Nash set and EM′ the set of points in M′ of D'Angelo infinite type. We show that if M is minimal, then, for every point p∈M, and for every pair of germs of C∞-smooth CR maps f,g:(M,p)→M′, there exists an integer k=kp such that if f and g have the same k-jets at p, and do not send M into EM′, then necessarily f=g. Furthermore, the map p↦kp may be chosen to be bounded on compact subsets of M. As a consequence, we derive the finite jet determination property for pairs of germs of CR maps from minimal real-analytic CR submanifolds in CN into Nash subsets in CN′ of D'Angelo finite type, for arbitrary N,N′≥2.

In this paper, we develop a method to construct holomorphic functions that exist only on infinite dimensional spaces. The following types of holomorphic functions f:U→ℂ on some open subsets U of an infinite dimensional complex Banach space are constructed: (1) f is bounded holomorphic on U and is continuously, but not uniformly continuously extended to U¯; (2) f is continuous on U¯ and holomorphic of bounded type on U, but f is unbounded on U; (3) f is holomorphic of bounded type on U and f cannot be continuously extended to U¯. The technique we develop is powerful enough to provide, in the cases (2) and (3) above, large algebraic structures formed by such functions (up to the zero function, of course).

One way to think of functional analysis is as the branch of mathematics that studies the extent to which the properties possessed by finite dimensional spaces generalize to infinite dimensional spaces. In the finite dimensional case there is only one natural linear topology. In that topology every linear functional is continuous, convex functions are continuous (at least on the interior of their domains), the convex hull of a compact set is compact, and nonempty disjoint closed convex sets can always be separated by hyperplanes. On an infinite dimensional vector space, there is generally more than one interesting topology, and the topological dual, the set of continuous linear functionals, depends on the topology. In infinite dimensional spaces convex functions are not always continuous, the convex hull of a compact set need not be compact, and nonempty disjoint closed convex sets cannot generally be separated by a hyperplane. However, with the right topology and perhaps some additional assumptions, each of these results has an appropriate infinite dimensional version.

Chapter 34 Equilibrium theory in infinite dimensional spaces

We consider the extension problem of a self-consistent family of infinite measures to a completely additive measure.For probability measures, Kolmogorov's extension theorem assures that the extension is uniquely possible.Our results are as follows:(a) For CT-finite measures, we can reduce the problem to the case of probability measures, so that the extension is uniquely possible.As an application, on an infinite dimensional vector space we can construct such a measure that is invariant both under rotations and homotheties with respect to the origin.It is obtained as the limit of ndimensional measure: Also we shall discuss about the Lorentz invariant measure on an infinite dimensional space.(b) If measures are not a-finite, under the additional condition (EG) in §6, the extension is possible but not unique.We shall mention about the largest and the smallest extension.As an application, we can consider the symbolic representation of a flow {7f} defined on an infinite measure space X, namely constructing an appropriate product space VV R and an appropriate measure on IV R , T$ on X is represented by a shift St on jj/R.w (.)_ w (. + t) m §1 Kolmogorov's extension theorem §2 Reduction to finite measures §3 Rotationally invariant measure §4 (0, oo)-type measures §5 Lorentz invariant measure §6 Non a-finite case §7 er-finite plus essentially infinite case §8 Symbolic representation of flows § 1. Kolmogoro¥ 9 s Extension Theorem Let Q be a set, and S3 be a a-r'mg of subsets of Q.The pair {£?,

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this manuscript we solve the “faceless problem”, that is, the informally posed open problem of characterizing those convex sets free of proper faces. Indeed, we prove that a non-singleton convex subset of a real vector space is free of proper faces if and only if every point of the convex set is an inner point. As a consequence, we prove the following dichotomy theorem: a point of a convex set of a real vector space is either a extreme point or an inner point of some face. We also characterize the non-singleton closed convex subsets of a topological vector space free of proper faces, which turn out to be the linear manifolds. An application of this allows us to show that the only possible minimal faces of a linearly bounded closed convex subset of a topological vector space are the extreme points. As an application of the technique we develop to prove our results, we easily construct dense proper faces of convex sets and non-dense proper faces whose closure is not a face. This fact allows us to equivalently renorm any infinite dimensional Banach space so that its unit ball contains a non-closed face. Even more, we prove that every infinite dimensional Banach space containing an isomorphic copy of $$c_0$$ or $$\ell _p$$ , $$1<p<\infty $$ , can be equivalently renormed so that its unit ball contains a face whose closure is not a face.