In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and...
In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For instance, the $...
Boundary effects in the stepwise structure of the Lyapunov spectra and the corresponding wavelike structure of the Lyapunov vectors are discussed numerically in quasi-one-dimensional systems consisting of many hard-disks. Four kinds of boundary con...
We introduce a definition of a "localization width" whose logarithm is given by the entropy of the distribution of particle component amplitudes in the Lyapunov vector. Different types of localization widths are observed, for example, a minimum loc...
We introduce a novel characterization of all Walrasian price vectors in terms of forbidden over- and under demanded sets for monotone gross substitute combinatorial auctions. For ascending and descending auctions we suggest a universal framework fo...
Graph partitioning schedules parallel calculations like sparse matrix-vector multiply (SpMV). We consider contiguous partitions, where the $m$ rows (or columns) of a sparse matrix with $N$ nonzeros are split into $K$ parts without reordering. We pr...
We initiate the analytical functional bootstrap study of conformal field theories with large $N$ limits. In this first paper we particularly focus on the 1D $O(N)$ vector bootstrap. We obtain a remarkably simple bootstrap equation from the $O(N)$ vec...
We investigate solutions $(\mathcal{M}, g)$ to Einstein's vacuum field equations with positive cosmological constant $Λ$ which admit a smooth past null infinity $\mathcal{J}^-$ à la Penrose and a Killing vector field whose associated Mars-Simon ten...
We provide a classification of $Λ>0$-vacuum spacetimes which admit a Killing vector field with respect to which the associated "Mars-Simon tensor" (MST) vanishes and having a conformally flat $\mathcal{J}^-$ (or $\mathcal{J}^+$). To that end, we als...
We study the stationary and axisymmetric non-convective differentially rotating perfect-fluid solutions of Einstein's field equations admitting one conformal symmetry. We analyse the two inequivalent Lie algebras not exhaustively considered in Mars...
? PageIndex: Document Index for Vectorless, Reasoning-based RAG (⭐ 20740)
A reusable leave-behind for enterprise customers showing the differentiator of using the best Vector Store in the world: Astra DB (⭐ 21)
Consider a linear regression model with n-dimensional response vector, regression parameter β= (β_1, ..., β_p) and independent and identically N(0, σ^2) distributed errors. Suppose that the parameter of interest is θ= a^T βwhere a is a specifie...
Consider a linear regression model with regression parameter beta and normally distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where c and t are spe...
Consider a linear regression model with regression parameter beta=(beta_1,..., beta_p) and independent normal errors. Suppose the parameter of interest is theta = a^T beta, where a is specified. Define the s-dimensional parameter vector tau = C^T bet...
For the comparison of inequality and welfare in multiple attributes the use of generalized Gini indices is proposed. Individual endowment vectors are summarized by using attribute weights and aggregated in a spectral social evaluation function. Such...
We report a simple extension of a model for noise reduction in signal processing, already introduced by Mourik et al., in the presence of binary coupling vectors, which turns to be more useful for practical and engineering implementations. We also...
Raster to Vector Graphics Converter (⭐ 5571)
We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasi-linea...
The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $ξ$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^ξ$ for $v\in V$ satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$. We prove that many...