Narrow Results

Space-time diagrams of a one-dimensional cellular automaton can be visualized as half-plane arrays of symbols. The set of rectangular blocks extracted from such arrays forms a two-dimensional (picture) language. We initiate a study of cellular automata through the associated two dimensional languages by investigating cellular automata whose two-dimensional languages are factorial-local. We show that these cellular automata have the same characterization as one-sided cellular automata with SFT traces.

Let be the Jiang–Su algebra and let τ be its unique tracial state. We prove that for all , the following statements are equivalent:

(1) a is a finite sum of commutators.

(2) a is a sum of five commutators.

(3) τ(a) = 0.

The off-line (or post-mortem) analysis of execution event traces is a popular approach to understand the performance of HPC applications that use the message passing paradigm. Combining this analysis with simulation makes it possible to “replay” the application execution to explore “what if?” scenarios, e.g., assessing application performance in a range of (hypothetical) execution environments. However, such off-line analysis faces scalability issues for acquiring, storing, or replaying large event traces.

We first present two previously proposed and complementary frameworks for off-line replaying of MPI application event traces, each with its own objectives and limitations. We then describe how these frameworks can be combined so as to capitalize on their respective strengths while alleviating several of their limitations. We claim that the combined framework affords levels of scalability that are beyond that achievable by either one of the two individual frameworks. We evaluate this framework to illustrate the benefits of the proposed combination for a more scalable off-line analysis of MPI applications.

Let $X$ be a separable Banach space endowed with a nondegenerate centered Gaussian measure $\mu$ and let $w$ be a positive function on $X$ such that $w\in W^{1,s}(X,\mu )$ and $logw\in W^{1,t}(X,\mu )$ for some $s>1$ and $t>s^{\prime }$. In this paper, we introduce and study Sobolev spaces with respect to the weighted Gaussian measure $\nu :=w\mu$. We obtain results regarding the divergence operator (i.e. the adjoint in $L^2$ of the gradient operator along the Cameron–Martin space) and the trace of Sobolev functions on hypersurfaces $\left\{x\in X|G(x)=0\right\}$, where $G$ is a suitable version of a Sobolev function.

The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to Korevaar and Schoen on the one hand, and Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to Ambrosio (for BV maps into metric spaces), Reshetnyak and finally to the notion of Newtonian–Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of Bourgain, Brezis and Mironescu in terms of "limit" of the space W^s,p as s → 1, 0 < s < 1, and finally following the approach proposed by Nguyen. We also establish the regularity of traces of maps in W^s,p (0 < s ≤ 1 < sp).

For any three $n\times n$ matrices $A,B,X$ over a commutative ring $S$, we prove that $\mathrm{\text{det}}(A+B-AXB)=\mathrm{\text{det}}(A+B-BXA)\in S$. This apparently new formula may be regarded as a “ternary generalization” of Sylvester’s classical determinantal formula $\mathrm{\text{det}}(I_n-AB)=\mathrm{\text{det}}(I_n-BA)$ for any pair of $n\times n$ matrices $A,B$ over $S$. By suitably specializing our ternary formula above, we have also obtained several other binary determinantal and tracial identities that are hitherto unknown.

Trace Base Management System (TBMS) offers processing and querying functionalities for traces that may be of interest to users of tracked systems. Our goal is to ensure the importing of various external traces into kernel for Trace-Based System (kTBS), which is a TBMS developed in the LIRIS laboratory. To overcome the problem of traces heterogeneity, we propose to define a generic collector. To this end, a user with enough knowledge of the tracked system is prompted to define its kTBS trace model and correspondences between the elements of this model and the elements of the trace to import. The system generalises the mappings previously elicited by the user through interaction to create mapping rules. After this phase, the collector will generate modelled traces from the existing ones and the already defined mapping rules.

In this paper, we first present further improvement of squared Young-type inequalities with double weights and their reverses. As a consequence of our results, we show the reverses of main results obtained by Nasiri and Askari [Some refined Young-type inequalities using different weights, Asian-Eur. J. Math. 15 (2022) 7]. As a further application, we provide some related inequalities for unitarily invariant norms, traces, and determinants.

We study optimality of function spaces that appear in Sobolev embeddings. We focus on rearrangement-invariant Banach function spaces.