Reiterman’s Theorem on Finite Algebras for a Monad

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes of finite algebras closed under finite products, subalgebras and quotients. In this article, Reiterman’s theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category D: we prove that a class of finite T -algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad T ^ associated to the monad T , which gives a categorical view of the construction of the space of profinite terms.

Certain Inequalities Pertaining to Some New Generalized Fractional Integral Operators

In this paper, we introduce the generalized left-side and right-side fractional integral operators with a certain modified ML kernel. We investigate the Chebyshev inequality via this general family of fractional integral operators. Moreover, we derive new results of this type of inequalities for finite products of functions. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. From our above-mentioned results, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view. Furthermore, two important results and some interesting consequences for convex functions in the framework of the defined class of generalized fractional integral operators are established. Finally, two basic examples demonstrated the significance of our results.

Angle Sums of Schläfli Orthoschemes

We consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and $$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$ K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 0 } , which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of $$K_n^A$$ K n A and $$K_n^B$$ K n B . This setting contains sums of external and internal angles of $$K_n^A$$ K n A and $$K_n^B$$ K n B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.

Performance of Small-Scale Sawmilling Operations: A Case Study on Time Consumption, Productivity and Main Ergonomics for a Manually Driven Bandsaw

Sawmilling operations represent one of the most important phases of the wood supply chain, because they connect the conversion flow of raw materials into finite products. In order to maintain a high volume of processed wood, sawmills usually adopt different processing strategies in terms of equipment and methods, which can increase the value or volume of the lumber produced from logs. In this study, the performance of small-scale sawmilling operations was monitored, whilst also evaluating the exposure of workers to harmful factors. An assessment of time consumption, productivity, and main ergonomics was conducted during the use of a manually driven bandsaw. In addition, the exposure to noise was investigated to complement the knowledge in this regard. The results indicated a rather high time utilization in productive tasks, which may come at the expense of exposure to noise and to poor working postures. The modelling approach resulted in statistically significant time consumption models for different phases (blade adjustment, effective sawing, returning, unloading lumber, and loading and fixing lumber). The exposure to noise was close to 92 dB (A) (8 h) and, therefore, the level of emitted noise is likely to depend on the condition of the used blades, species sawn and on the dimensional characteristics of the logs. In terms of ergonomic risks, the poorest postures were those related to tasks such as moving the logs, loading the logs, fixing the logs, rotating and removing the logs, as well as unloading the lumber.

Essentials of General Topology

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of topology. Among the topics treated are normality, regularity, and second countability. The proof of Tychonoff’s theorem for finite products appears in section 8. The section on locally compact spaces is the transition between the core of the chapter and the more advanced sections on metrization, compactification, and the product of infinitely many spaces. The highlights include the one-point compactification, the Urysohn metrization theorem, and Tychonoff’s theorem. Little subsequent material is based on the last three sections. At various points in the book, it is explained how results stated for the metric case can be extended to topological spaces, especially locally compact Hausdorff spaces. Some such results are developed in the exercises.

Allowing a Real Collaboration Between Humans and Robots

Robotics researchers are spending many efforts in developing methodologies and techniques that allow robots to work side by side with humans, with the aim of improving the manufacturing processes. Such a level of interaction does not require just the safe coexistence in a common space, which is something completely achieved by the current state of the art. In scenarios like co-assemblies, humans and robots have to execute alternating tasks, with the aim of realizing a set of possible finite products. This requires the robots to adapt, synchronize and actively cooperate with the humans. This work will show that this goal can be reached by providing the cobots with three main abilities: recognizing the human behaviour, predicting the human actions and optimally planning the robotic ones.