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The paper analyzes a spectral approach to reconstructing a scalar field on the sphere, given only information about a masked version of the field, together with precise information about the (smooth) mask. The theory is developed for a general mask and later specialized to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realization of a Gaussian random field with an artificial angular power spectrum of moderate degree (l≤100). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise. [ABSTRACT FROM AUTHOR]

Let C (S m) denote the set of continuous maps from the unit sphere S m in the Euclidean space R m + 1 into itself endowed with the supremum norm. We prove that the set { f n : f ∈ C (S m) and n ≥ 2 } of iterated maps is not dense in C (S m) . This, in particular, proves that the periodic points of the iteration operator of order n are not dense in C (S m) for all n ≥ 2 , providing an alternative proof of the result that these operators are not Devaney chaotic on C (S m) proved in Veerapazham et al. (Proc Am Math Soc 149(1):217–229, 2021). [ABSTRACT FROM AUTHOR]

In this paper, we study the rigidity phenomena on the (n + 1) -dimensional anti-invariant submanifolds of the unit sphere of dimension (2 n + 1) from the intrinsic and extrinsic aspects, respectively. First of all, we establish a basic inequality for such submanifolds relative to the norm of the covariant differentiation of both the second fundamental form h and mean curvature vector field H. Secondly, the lower bound of the norm of H is further derived by means of a general inequality. Finally, in dealing with those minimal anti-invariant submanifolds with η -Einstein induced metrics, we obtain an inequality in terms of the Weyl curvature tensor, squared norm S of h, and scalar curvature. In particular, these inequalities above are optimal in the sense that all the submanifolds attaining the equalities are completely determined. [ABSTRACT FROM AUTHOR]

Let $X$ , $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$ -regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$ , we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$ -regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$ -regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$ -dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$ -regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$. [ABSTRACT FROM AUTHOR]

We discuss strong local and global well–posedness for the three–dimensional NLS equation with nonlinearity concentrated on S 2 . Precisely, local well–posedness is proved for any C 2 power–nonlinearity, while global well–posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point–concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas. [ABSTRACT FROM AUTHOR]

It is known that many statistical and machine learning approaches heavily rely on pairwise distance between data points. The choice of distance function on the underlying manifold has a fundamental impact on performance of these processes. This is closely related to questions of how to appropriately calculate distances, and hence, fundamental solutions (heat kernels) for heat operators can be obtained. In general, it is not so easy to obtain a closed form for heat kernels. We first survey results of heat kernels on radially symmetric Riemannian manifolds, e.g., Euclidean spaces and unit spheres in Rn. For the cases n = 1;2; 3, we may construct the heat kernel explicitly. But, the computation is much more complicated when n > 3. However, by results of Nagase, we may construct parametrices for the heat kernel by using elementary functions so that the error terms can be under controlled. In the second part of the paper, we discuss some results on subRiemannian manifolds, especially 3-dimensional sphere in C² as a CR-manifold. We study geodesics connecting two given points on S³ respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of S³ and some partial results are obtained in the general case. The Carnot-Carath'eodory distance is calculated. We also present some motivations related to quantum mechanics. Then we give a brief discussion of Greiner's methods on the heat kernel for the Cauchy-Riemann subLaplacian on S2n+1. We provide a brief discussion on applications of these heat kernels to graph kernels in the last part of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the quasi-H o ¨ lder mappings between unit spheres of p-normed spaces. The quasi-H o ¨ lder map is a natural generalization of the H o ¨ lder map, Lipschitz map, quasi-isometry and ε -isometry, etc. Concretely, we study the extension of a quasi-H o ¨ lder embedding between unit spheres of p-normed spaces ( 0 < p ≤ 1 ) under the quasi-H o ¨ lder or anti-H o ¨ lder type assumption. We generalize the main results in Xiao and Lu (Ann Funct Anal 13(2):Paper No. 25, 2022) from the r-isometric case to the wider case of quasi-H o ¨ lder mappings. Moreover, the method we adopted here is different; and the Hahn–Banach property, required as a condition in the main theorems of [30] has been removed in our results. [ABSTRACT FROM AUTHOR]

Abstract We obtain sharp inequalities of Hardy type for functions in the Sobolev space W 1 , p $W^{1,p}$ on the unit sphere S n − 1 $\mathbb{S}^{n-1}$ in R n $\mathbb{R}^{n}$ . We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities. We apply our results to obtain lower bounds for the the first eigenvalue of the p-Laplacian on the sphere.

We obtain sharp inequalities of Hardy type for functions in the Sobolev space W 1 , p on the unit sphere S n − 1 in R n . We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities. We apply our results to obtain lower bounds for the the first eigenvalue of the p-Laplacian on the sphere. [ABSTRACT FROM AUTHOR]

In this paper, the extension of isometries is considered for some operators between the unit spheres of two real p-normed spaces ( 0 < p ≤ 1 ) under some nonexpansivity and expansivity assumptions. In the light of the characteristics of p-normed spaces, the notion of isometry is generalized to the case of r-isometries. By virtue of the technique of dual spaces, it is proved that such operators can be extended from the unit sphere to the whole space. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach spaces. We prove that if their unit spheres S(X) and S(Y) are (α, β)-Hölder homeomorphic for some α, β ∈ (0, 1]. Then the unit spheres of Lp(Ω, μ, X) and Lq(Ω, μ, Y) are (min { p q , α } , min { q p , β }) -Hölder homeomorphic for every measure space (Ω, μ) and for every 1 ≤ p, q < ∞. In particular, the result yields an interesting application involving extrapolation phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we obtain results of nonexistence of nonconstant positive solutions, and also existence of an unbounded sequence of sign-changing solutions for some critical problems involving conformally invariant operators on the unit sphere, in particular to the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together with blow up arguments, a variant of Pohozaev's type identity, a refinement of regularity results for this type operators, and finally, by exploiting the symmetries of the sphere. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of fourth degree polynomial problems, which are NP-hard. First, we are concerned with the bi-quadratic optimization problem (Bi-QOP) over compact sets, which is proven to be equivalent to a multi-linear optimization problem (MOP) when the objective function of Bi-QOP is concave. Then, we introduce an augmented Bi-QOP (which can also be regarded as a regularized Bi-QOP) for the purpose to guarantee the concavity of the underlying objective function. Theoretically, both the augmented Bi-QOP and the original problem share the same optimal solutions when the compact sets are specified as unit spheres. By exploiting the multi-block structure of the resulting MOP, we accordingly propose a proximal alternating minimization algorithm to get an approximate optimal value of the problem under consideration. Convergence of the proposed algorithm is established under mild conditions. Finally, some preliminary computational results on synthetic datasets are reported to show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Let (H , 〈 , 〉) be a complex Hilbert space and B (H) the space of bounded linear operators in H. Any other equivalent inner product in H is of the form 〈 f , g 〉 A = 〈 A f , g 〉 (f , g ∈ H) for some positive invertible operator A ∈ B (H). In this paper we study the bundle M which consist of the unit sphere { f ∈ H : 〈 f , f 〉 A = 1 } over each (equivalent) inner product 〈 , 〉 A , which due to the observation above can be defined M = { (A , f) ∈ B (H) × H : A is positive and invertible and 〈 A f , f 〉 = 1 }. We prove that M is a complemented submanifold of the Banach space B (H) × H and a homogeneous space of the Banach-Lie group G (H) ⊂ B (H) of invertible operators. We introduce a reductive structure in M , and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of M , for instance, the one obtained when the positive elements A describing the inner products lie in a prescribed C⁎-algebra A ⊂ B (H). [ABSTRACT FROM AUTHOR]

For a singular Riemannian foliation ℱ on a Riemannian manifold, a curve is called horizontal if it meets the leaves of ℱ perpendicularly. For a singular Riemannian foliation ℱ on a unit sphere 𝕊 n , we show that if ℱ is a polar foliation or if ℱ is given by the orbits of an infinitesimally polar action, then the horizontal diameter of 𝕊 n is π , i.e. any two points in 𝕊 n can be connected by a horizontal curve of length ≤ π. [ABSTRACT FROM AUTHOR]

We consider the general problem formulation as the minimization of the sum of the numbers of the feasible subset of cardinality n out of the set of N numbers. Asymmetric travelling salesman problem, assignement problem and the problem of minimization of the sum of n numbers out of the set of N numbers can be formulated in these terms. The input of all these problems can be represented as the point from the non-negative part of the unit sphere after some intuitive scaling. In this settings we assume that the retrieval of the input data is difficult. We further propose to skip the retrieval of some input numbers and substitute them with the zeros. We call this problem the problem with incomplete input. We estimate the quality of the optimal solution for the problem with incomplete input used as the solution for the problem with the complete data. [ABSTRACT FROM AUTHOR]

The reflection map introduced by D'Angelo is applied to deduce simpler descriptions of nondegeneracy conditions for sphere maps and to the study of infinitesimal deformations of sphere maps. It is shown that the dimension of the space of infinitesimal deformations of a nondegenerate sphere map is bounded from above by the explicitly computed dimension of the space of infinitesimal deformations of the homogeneous sphere map. Moreover a characterization of the homogeneous sphere map in terms of infinitesimal deformations is provided. [ABSTRACT FROM AUTHOR]

The object of the present work is to investigate Sasakian manifolds satisfying certain conditions on Q tensor whose trace is well-known Z tensor. [ABSTRACT FROM AUTHOR]

Directional data are related to vectors on the unit sphere. When these vectors are not signed, this kind of data is called axial data. The Watson distribution is one of the main models for axial data. This model is equipped by two parameters: dominant axis and concentration. Based on the Rényi divergence and the Bhattacharya and Hellinger distances, we propose three hypothesis tests to check if two samples come from populations having the same concentration parameter. Results from synthetic and real data indicate that the proposed tests can yield good performance on Watson data. [ABSTRACT FROM AUTHOR]

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space X, the unit sphere SX always contains an uncountable (1+)-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of c0(ω1). Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically (1+ε)-separated subset of any regular cardinality not exceeding the density of X; should the space X be super-reflexive, the unit sphere of X contains such a subset of cardinality equal to the density of X. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones. [ABSTRACT FROM AUTHOR]

In this paper, based on the smooth point of the unit ball and its support linear functional, we show two equivalent formulations of the isometric extension problem between the unit spheres of strictly convex two-dimensional normed spaces. We prove that these equivalent formulations have a positive answer in a special case. [ABSTRACT FROM AUTHOR]