Smooth Spaces Research Articles

Whitney stratifications are conically smooth

The notion of conically smooth structure on a stratified space was introduced by Ayala, Francis and Tanaka. This is a very well behaved analogue of a differential structure in the context of stratified topological spaces, satisfying good properties such as the existence of resolutions of singularities and handlebody decompositions. In this paper we prove Ayala, Francis and Tanaka’s conjecture that any Whitney stratified space admits a canonical conically smooth structure. We thus establish a connection between the theory of conically smooth spaces and the classical examples of stratified spaces from differential topology.

Using educational technologies (Padlet) for student engagement – reflection from the Australian classroom

PurposeEducational technologies such as Padlet have begun to transform the way teachers teach, students learn, and teachers and students interact. However, relatively little emphasis has been given to explore collaborative learning (CL) using educational technologies such as Padlet as a means of student engagement. Thus, the purpose of this reflective article is to gain insight into the use of educational technologies Padlet in the tutorial workshop in accounting program at an Australian university.Design/methodology/approachThis study adopts a qualitative podcast research approach with a specific focus on the author's personal reflection over the course of a semester in an Australian university classroom.FindingsThe findings of this article (based on the author's own reflection from the classroom) show that a digitally innovative useful tool Padlet works on a variety of different devices and does not require student's technical know-how to use it. The findings further suggest that Padlet technology as a great equaliser has helped students' engagement in classroom activities.Research limitations/implicationsDespite some noticeable limitations, this paper suggests that using Padlet in classroom engagement activities has provided a smoothing space for CL. The findings of this reflective article would be an interest to students, academics, curriculum designers and educational technology experts.Originality/valueThis study provides interesting insights into how Padlet contributes to the student's engagement.

Efficient generator of mathematical expressions for symbolic regression

We propose an approach to symbolic regression based on a novel variational autoencoder for generating hierarchical structures, HVAE. It combines simple atomic units with shared weights to recursively encode and decode the individual nodes in the hierarchy. Encoding is performed bottom-up and decoding top-down. We empirically show that HVAE can be trained efficiently with small corpora of mathematical expressions and can accurately encode expressions into a smooth low-dimensional latent space. The latter can be efficiently explored with various optimization methods to address the task of symbolic regression. Indeed, random search through the latent space of HVAE performs better than random search through expressions generated by manually crafted probabilistic grammars for mathematical expressions. Finally, EDHiE system for symbolic regression, which applies an evolutionary algorithm to the latent space of HVAE, reconstructs equations from a standard symbolic regression benchmark better than a state-of-the-art system based on a similar combination of deep learning and evolutionary algorithms.

A toolkit for ortho-symplectic dualities

We propose new confining dualities in 3dN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauge theories with orthogonal gauge groups, with and without monopole superpotentials. Deriving some of those dualities requires a sequence of gauging and Higgsing for a ℤ2 symmetry. This prevents the gauge theory from developing a smooth quantum moduli space and affects the global structure of the gauge group, muting it from SO to O+.The confining dualities provide tools to deconfine fields transforming in the symmetric rank-2 representation of classical gauge groups. As an application, we propose and derive S-confining dualities for SO(N) (Sp(N)) gauge theories with an adjoint and 1 (2) fundamentals, respectively. From these S-confining dualities, we readily obtain various duality appetizers and the 3d mirror of A2N Argyres-Douglas theories.

Solvibility of the Kahn-Hillard model in Quasi-Sobolev space

In this paper, a theoretical investigation of the Kahn-Hillard Model In Quasi-Sobolev Space is fulfilled by constructing a smooth space of a Kahn-Hillard Model in a bounded domain with a smooth boundary.

G1 spline functions for point cloud fitting

In this work we present a new construction of basis functions that generate the space of geometrically smooth splines on an unstructured quadrilateral mesh. The basis is represented in terms of biquintic Bézier polynomials on each quadrilateral face. The gluing along the face boundaries is achieved using quadratic gluing data functions, leading to globally G1–smooth spaces. We analyze the latter space and provide a combinatorial formula for its dimension as well as an explicit basis construction. Moreover, we assess the use of this basis in point cloud fitting problems. To apply G1 least squares fitting, a quadrilateral structure as well as parameters in each quadrilateral is required. Even though the general problem of segmenting and parametrizing point clouds is beyond the focus of the present work, we describe a procedure that produces such a structure as well as patch-local parameters. Our experiments demonstrate the accuracy and smoothness of the obtained reconstructed models in several challenging instances.

On Non-linear Mappings Preserving the Semi-inner Product

We say that a smooth normed space X has a property (SL), if every mapping f:X rightarrow X preserving the semi-inner product on X is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space ell _p, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach spaces in terms of subspaces of quotient spaces. As a consequence, we prove that the space ell _p have the property (SL) for every 1< p < infty . Finally, using a variant of the Gowers–Maurey space, we construct an infinite-dimensional uniformly smooth Banach space X such that every smooth Banach space isomorphic to X has the property (SL).

Co-creation of educational commons spaces to reverse inequalities: project SMOOTH and the Children's Club.

This article presents an action-research project from the EU-funded SMOOTH project, which focuses on the potential of Educational Commons to address educational inequalities. The project adopts an emergent paradigm that views spaces for collaboration, content co-creation, socialization, governance, and play as catalysts for reversing inequalities. The action-research, conducted in a disadvantaged non-formal education setting in northern Portugal, involved children aged 8-10 years old. Over a span of 10 months, the innovative action-research program aimed to achieve several objectives: (1) reversing inequalities faced by vulnerable social groups, (2) strengthening inter-cultural and inter-generational dialogue and social integration, (3) developing essential social and personal skills, and (4) creating smooth spaces of democratic citizenship based on equality, collaboration, sharing, and caring. By understanding the tensions and conflicts that emerge in children's everyday situations, the project sought to build and foster community through embracing differences. This article analyzes the characteristics, behaviors, challenges, and strengths observed during the 30 sessions. The results provide insights into the dimensions of Children as commoners, in terms of sharing and care, cooperation and collective creativity and active citizenship. This research contributes to the exploration of Educational Commons as a means to promote equity and transform educational contexts.

Sampling numbers of smoothness classes via ℓ1-minimization

Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in L2. In particular, we show that in relevant cases such as mixed and isotropic weighted Wiener classes or Sobolev spaces with mixed smoothness, sampling numbers in L2 can be upper bounded by best n-term trigonometric widths in L∞. We describe a recovery procedure from m function values based on ℓ1-minimization (basis pursuit denoising). With this method, a significant gain in the rate of convergence compared to recently developed linear recovery methods is achieved. In this deterministic worst-case setting we see an additional speed-up of m−1/2 (up to log factors) compared to linear methods in case of weighted Wiener spaces. For their quasi-Banach counterparts even arbitrary polynomial speed-up is possible. Surprisingly, our approach allows to recover mixed smoothness Sobolev functions belonging to SprW(Td) on the d-torus with a logarithmically better rate of convergence than any linear method can achieve when 1<p<2 and d is large. This effect is not present for isotropic Sobolev spaces.

Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

Given a pair of normally hyperbolic operators over (possibnly different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called Møller operator, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called paracausal relation. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a *-isomorphism between (generally off-shell) CCR-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this *-isomorphism.

Global Critical Chart for Local Calabi–Yau Threefolds

Abstract In this paper, we investigate Keller’s deformed Calabi–Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total space of an $\omega _{Y}$-torsor as a certain deformed $(n+1)$-Calabi–Yau completion of the derived category of $Y$. As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, that is, a Calabi–Yau threefold of the form $\textrm{Tot}_{C}(N)$, where $C$ is a smooth projective curve and $N$ is a rank two vector bundle on $C$. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar–Vafa invariants.

A Surface Family with a Mutual Geodesic Curve in Galilean 3-Space G3

This article gives an approach for establishing a surface family with a mutual geodesic curve in Galilean 3-space G3. Given a smooth space curve, we derive the sufficient and necessary condition for the given curve to be geodesic on it. Furthermore, we resolve the conditions when the surface is a ruled surface. Consequently, its ability to develop with the mutual geodesic of the elements of the surface family is researched. Meanwhile, we explain this approach by submitting considerable examples.

Embeddings of Decomposition Spaces

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two? A decomposition space D ( Q , L p , Y ) \mathcal {D}(\mathcal {Q}, L^p, Y) is determined by a covering Q = ( Q i ) i ∈ I \mathcal {Q} = (Q_i)_{i \in I} of the frequency domain, an integrability exponent p p , and a sequence space Y ⊂ C I {Y \subset \mathbb {C}^I} . Given these ingredients, the decomposition space norm of a distribution g g is defined as ‖ g ‖ D ( Q , L p , Y ) = ‖ ( ‖ F − 1 ( φ i ⋅ g ^ ) ‖ L p ) i ∈ I ‖ Y , { \left \Vert g \right \Vert _{\mathcal {D}(\mathcal {Q}, L^p, Y)} = \left \Vert \left ( \left \Vert \mathcal {F}^{-1} \left ( \varphi _{i} \cdot \widehat {g} \right ) \right \Vert _{L^{p}} \right )_{i \in I} \right \Vert _{Y} } , where ( φ i ) i ∈ I (\varphi _i)_{i \in I} is a suitable partition of unity for Q \mathcal {Q} . We establish readily verifiable criteria which ensure the existence of a continuous inclusion (“an embedding”) D ( Q , L p 1 , Y ) ↪ D ( P , L p 2 , Z ) \mathcal {D}(\mathcal {Q},L^{p_1},Y) \hookrightarrow \mathcal {D}(\mathcal {P},L^{p_2},Z) , mostly concentrating on the case where Y = ℓ w q 1 ( I ) Y = \ell _{w}^{q_{1}} ( I ) and Z = ℓ v q 2 ( J ) Z = \ell _{v}^{q_{2}} (J) . Under suitable assumptions on Q , P \mathcal {Q}, \mathcal {P} , we will see that the relevant sufficient conditions are p 1 ≤ p 2 p_{1} \leq p_{2} and finiteness of a nested norm of the form \[ ‖ ( ‖ ( α i β j ⋅ v j / w i ) i ∈ I j ‖ ℓ t ) j ∈ J ‖ ℓ s , with I j = { i ∈ I : Q i ∩ P j ≠ ∅ } for j ∈ J . \left \Vert \left ( \left \Vert \left ( \alpha _{i} \beta _{j} \cdot v_{j} / w_{i} \right )_{ i \in I_{j}} \right \Vert _{\ell ^{t}} \right )_{j\in J} \right \Vert _{\ell ^{s}} \, , \quad \text {with} \quad I_{j} = \left \{ i \in I \,:\, Q_{i} \cap P_{j} \neq \varnothing \right \} \quad \text {for } j \in J \,. \] Like the sets I j I_j , the exponents t , s t, s and the weights α , β \alpha , \beta only depend on the quantities used to define the decomposition spaces. In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings. These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of p 1 , p 2 p_1, p_2 , our criteria yield a complete characterization for the existence of the embedding. The same holds for arbitrary values of p 1 , p 2 p_1, p_2 under more strict assumptions on the coverings. We also prove a rigidity result, namely that—for ( p 1 , q 1 ) ≠ ( 2 , 2 ) (p_{1}, q_{1}) \neq (2, 2) —two decomposition spaces D ( Q , L p 1 , ℓ w q 1 ) \mathcal {D}(\mathcal {Q}, L^{p_1}, \ell _w^{q_1}) and D ( P , L p 2 , ℓ v q 2 ) \mathcal {D}(\mathcal {P}, L^{p_2}, \ell _v^{q_2}) can only coincide if their “ingredients” are equivalent, that is, if p 1 = p 2 p_{1} = p_{2} and q 1 = q 2 q_{1} = q_{2} and if the coverings Q , P \mathcal {Q}, \mathcal {P} and the weights w , v w, v are equivalent in a suitable sense. The resulting embedding theory is illustrated by applications to α \alpha -modulation and Besov spaces. All known embedding results for these spaces are special cases of our approach; often, we improve considerably upon the state of the art.

Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier–Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. Here the scaling of the average degree, as a function of the graph size, can range from nearly logarithmic to nearly linear.

Nonsingular solutions to the Einstein equations on piecewise-Lorentzian manifolds

We consider 4-dimensional spacetime manifolds that are piecewise Lorentzian, where the Lorentzian components of the manifold are separated by codimension-one planes (spacelike or timelike) on which the metric is degenerate. Such manifolds are of interest because they enlarge the smooth and nonsingular solution space of the Einstein equations. Planes of degeneracy that are perpendicular to each other can exist simultaneously. We describe various solutions of this type to the vacuum equations $G_{\mu\nu}=0$ and $G_{\mu\nu}+\Lambda g_{\mu\nu}=0$, and to $G_{\mu\nu}= 8\pi G T_{\mu\nu}$ for a perfect fluid. Novel examples include static gravitational lumps of finite curvature and a spacetime that responds to a cosmological constant via oscillations in time and/or space. A spacelike degeneracy plane can be used to avoid the big bang singularity, as we have further described elsewhere.

An adaptive regularization method in Banach spaces

This paper considers optimization of nonconvex functionals in smooth infinite dimensional spaces. It is first proved that functionals in a class containing multivariate polynomials augmented with a sufficiently smooth regularization can be minimized by a simple linesearch-based algorithm. Sufficient smoothness depends on gradients satisfying a novel two-terms generalized Lipschitz condition. A first-order adaptive regularization method applicable to functionals with β-Hölder continuous derivatives is then proposed, that uses the linesearch approach to compute a suitable trial step. It is shown to find an ϵ-approximate first-order point in at most evaluations of the functional and its first p derivatives.

Souplet–Zhang and Hamilton‐type gradient estimates for non‐linear elliptic equations on smooth metric measure spaces

AbstractIn this article, we present new gradient estimates for positive solutions to a class of non‐linear elliptic equations involving the f‐Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville‐type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed.

On generalized K-functionals in L_p for 0

We show that the Peetre K-functional between the space L_p with 0<p<1 and the corresponding smooth function space W_p^psi generated by the Weyl-type differential operator psi (D), where psi is a homogeneous function of any positive order, is identically zero. The proof of the main results is based on the properties of the de la Vallée Poussin kernels and the quadrature formulas for trigonometric polynomials and entire functions of exponential type.

Position Vectors of the Natural Mate and Conjugate of a Space Curve

The concept of the natural mate and the conjugate curves associated to a smooth curve in Euclidian 3-space were introduced initially by Dashmukh and others. In this paper, we give some extra results that add more properties of the natural mate and the conjugate curves associated with a smooth space curve in E 3 . The position vectors of the natural mate and the conjugate of a given smooth curve are investigated. Also, using the position vector of the natural mate, the necessary and sufficient condition for a smooth given curve to be a Bertrand curve is introduced. Moreover, a new characterization of a general helix is introduced.

Local linear independence of bilinear (and higher degree) B-splines on hierarchical T-meshes

We generate hierarchical T-meshes by repeatedly inserting new line segments, in order to adapt both the size and the shape of the cells to the specific requirements of the underlying application. The associated spaces of bilinear spline functions are spanned by the locally refined (LR) B-splines of Dokken et al. (2013), which are products of univariate B-splines defined over local knot vectors. Our method guarantees that these functions are linearly independent for the obtained class of hierarchical T-meshes. Furthermore, they even possess the property of local linear independence, since exactly 4 functions take non-zero values on each cell of the mesh, and they form a partition of unity without the need to perform additional scaling or truncation.The correctness of our refinement algorithm is verified by enumerating the (newly introduced) local configurations, which represent the possible topologies of the mesh in the vicinity of a cell. It is then shown that the method works well in all the resulting situations. The fairly large number (385) of cases sheds some light on the challenges associated with linear independence in the context of LR B-spline refinement.Additionally we apply our method to meshes that possess the additional property local semi-regularity (first studied by Weller and Hagen, 1995), where only 49 topologically different situations may arise. However, we observe that the resulting class of hierarchical T-meshes is less flexible, which leads to a substantial amount of excess refinement in applications.Finally we note that the same construction can be applied to generate basis functions for Cs–smooth spline spaces of degree p=2s+1, again with the property of local linear independence.