In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere $\Bbb S^d$ in $\Bbb R^{d+1}$. First we obtain strong equivalences of the approximation numbers for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>0$, which gives a clue to Open problem 3 as posed by Krieg and Vyb\'iral in \cite{KV}. Second, for the optimal quadrature errors for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $\alpha>d/2$, and apply Hilbert space structure and Vyb\'iral's theorem about Schur product theory to obtain lower bounds in the case $\alpha=d/2,\,\beta>1/2$ of small smoothness, which confirms the conjecture as posed by Grabner and Stepanyukin in \cite{GS} and solves Open problem 2 in \cite{KV}. Finally, we employ the weighted least squares operators and the least squares quadrature rules to obtain approximation theorems and quadrature errors for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$, which are order optimal.
相關內容
We continue the study of balanceable graphs, defined by Caro, Hansberg, and Montejano in 2021 as graphs $G$ such that any $2$-coloring of the edges of a sufficiently large complete graph containing sufficiently many edges of each color contains a balanced copy of $G$. While the problem of recognizing balanceable graphs was conjectured to be NP-complete by Dailly, Hansberg, and Ventura in 2021, balanceable graphs admit an elegant combinatorial characterization: a graph is balanceable if and only there exist two vertex subsets, one containing half of all the graph's edges and another one such that the corresponding cut contains half of all the graph's edges. We consider a special case of this property, namely when one of the two sets is a vertex cover, and call the corresponding graphs simply balanceable. We prove a number of results on balanceable and simply balanceable regular graphs. First, we characterize simply balanceable regular graphs via a condition involving the independence number of the graph. Second, we address a question of Dailly, Hansberg, and Ventura from 2021 and show that every cubic graph is balanceable. Third, using Brooks' theorem, we show that every $4$-regular graph with order divisible by $4$ is balanceable. Finally, we show that it is NP-complete to determine if a $9$-regular graph is simply balanceable.
Soft set theory serves as a mathematical framework for handling uncertain information, and hesitant fuzzy sets find extensive application in scenarios involving uncertainty and hesitation. Hesitant fuzzy sets exhibit diverse membership degrees, giving rise to various forms of inclusion relationships among them. This article introduces the notions of hesitant fuzzy soft $\beta$-coverings and hesitant fuzzy soft $\beta$-neighborhoods, which are formulated based on distinct forms of inclusion relationships among hesitancy fuzzy sets. Subsequently, several associated properties are investigated. Additionally, specific variations of hesitant fuzzy soft $\beta$-coverings are introduced by incorporating hesitant fuzzy rough sets, followed by an exploration of properties pertaining to hesitant fuzzy soft $\beta$-covering approximation spaces.
It is known that singular values of idempotent matrices are either zero or larger or equal to one \cite{HouC63}. We state exactly how many singular values greater than one, equal to one, and equal to zero there are. Moreover, we derive a singular value decomposition of idempotent matrices which reveals a tight relationship between its left and right singular vectors. The same idea is used to augment a discovery regarding the singular values of involutory matrices as presented in \cite{FasH20}.
Given a square pencil $A+ \lambda B$, where $A$ and $B$ are $n\times n$ complex (resp. real) matrices, we consider the problem of finding the singular complex (resp. real) pencil nearest to it in the Frobenius distance. This problem is known to be very difficult, and the few algorithms available in the literature can only deal efficiently with pencils of very small size. We show that the problem is equivalent to minimizing a certain objective function $f$ over the Riemannian manifold $SU(n) \times SU(n)$ (resp. $SO(n) \times SO(n)$ if the nearest real singular pencil is sought), where $SU(n)$ denotes the special unitary group (resp. $SO(n)$ denotes the special orthogonal group). This novel perspective is based on the generalized Schur form of pencils, and yields competitive numerical methods, by pairing it with { algorithms} capable of doing optimization on { Riemannian manifolds. We propose one algorithm that directly minimizes the (almost everywhere, but not everywhere, differentiable) function $f$, as well as a smoothed alternative and a third algorithm that is smooth and can also solve the problem} of finding a nearest singular pencil with a specified minimal index. We provide numerical experiments that show that the resulting methods allow us to deal with pencils of much larger size than alternative techniques, yielding candidate minimizers of comparable or better quality. In the course of our analysis, we also obtain a number of new theoretical results related to the generalized Schur form of a (regular or singular) square pencil and to the minimal index of a singular square pencil whose nullity is $1$.
論文
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Belnap-Dunn logic, also knows as the logic of First-Degree Entailment, is a logic that can serve as the underlying logic of theories that are inconsistent or incomplete. For various reasons, different expansions of Belnap-Dunn logic with non-classical connectives have been studied. This paper investigates the question whether those expansions are interdefinable with an expansion whose connectives include only classical connectives. This is worth knowing because it is difficult to say how close a logic with non-classical connectives is related to classical logic. The notion of interdefinability of logics used is based on a general notion of definability of a connective in a logic that seems to have been forgotten.
The study of homomorphisms of $(n,m)$-graphs, that is, adjacency preserving vertex mappings of graphs with $n$ types of arcs and $m$ types of edges was initiated by Ne\v{s}et\v{r}il and Raspaud [Journal of Combinatorial Theory, Series B 2000]. Later, some attempts were made to generalize the switch operation that is popularly used in the study of signed graphs, and study its effect on the above mentioned homomorphism. In this article, we too provide a generalization of the switch operation on $(n,m)$-graphs, which to the best of our knowledge, encapsulates all the previously known generalizations as special cases. We approach to study the homomorphism with respect to the switch operation axiomatically. We prove some fundamental results that are essential tools in the further study of this topic. In the process of proving the fundamental results, we have provided yet another solution to an open problem posed by Klostermeyer and MacGillivray [Discrete Mathematics 2004]. We also prove the existence of a categorical product for $(n,m)$-graphs on with respect to a particular class of generalized switch which implicitly uses category theory. This is a counter intuitive solution as the number of vertices in the Categorical product of two $(n,m)$-graphs on $p$ and $q$ vertices has a multiple of $pq$ many vertices, where the multiple depends on the switch. This solves an open question asked by Brewster in the PEPS 2012 workshop as a corollary. We also provide a way to calculate the product explicitly, and prove general properties of the product. We define the analog of chromatic number for $(n,m)$-graphs with respect to generalized switch and explore the interrelations between chromatic numbers with respect to different switch operations. We find the value of this chromatic number for the family of forests using group theoretic notions.
In this work we introduce a memory-efficient method for computing the action of a Hermitian matrix function on a vector. Our method consists of a rational Lanczos algorithm combined with a basis compression procedure based on rational Krylov subspaces that only involve small matrices. The cost of the compression procedure is negligible with respect to the cost of the Lanczos algorithm. This enables us to avoid storing the whole Krylov basis, leading to substantial reductions in memory requirements. This method is particularly effective when the rational Lanczos algorithm needs a significant number of iterations to converge and each iteration involves a low computational effort. This scenario often occurs when polynomial Lanczos, as well as extended and shift-and-invert Lanczos are employed. Theoretical results prove that, for a wide variety of functions, the proposed algorithm differs from rational Lanczos by an error term that is usually negligible. The algorithm is compared with other low-memory Krylov methods from the literature on a variety of test problems, showing competitive performance.
We often rely on censuses of triangulations to guide our intuition in $3$-manifold topology. However, this can lead to misplaced faith in conjectures if the smallest counterexamples are too large to appear in our census. Since the number of triangulations increases super-exponentially with size, there is no way to expand a census beyond relatively small triangulations; the current census only goes up to $10$ tetrahedra. Here, we show that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations. We use this idea to find counterexamples to three conjectures which ask, for certain $3$-manifolds, whether one-vertex triangulations always have a "distinctive" edge that would allow us to recognise the $3$-manifold.
In this paper, we propose an approach for identifying linear and nonlinear discrete-time state-space models, possibly under $\ell_1$- and group-Lasso regularization, based on the L-BFGS-B algorithm. For the identification of linear models, we show that, compared to classical linear subspace methods, the approach often provides better results, is much more general in terms of the loss and regularization terms used, and is also more stable from a numerical point of view. The proposed method not only enriches the existing set of linear system identification tools but can be also applied to identifying a very broad class of parametric nonlinear state-space models, including recurrent neural networks. We illustrate the approach on synthetic and experimental datasets and apply it to solve the challenging industrial robot benchmark for nonlinear multi-input/multi-output system identification proposed by Weigand et al. (2022). A Python implementation of the proposed identification method is available in the package \texttt{jax-sysid}, available at \url{//github.com/bemporad/jax-sysid}.
We propose an efficient algorithm for matching two correlated Erd\H{o}s--R\'enyi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- \alpha+o(1)}$ for a constant $\alpha \in [0,1)$, we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of $q$) when the edge correlation is below the square root of the Otter's constant (which is $\approx 0.338$).
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