Finitary approximations of coarse structures

A coarse structure $ \mathcal{E}$ on a set $X$ is called finitary if, for each entourage $E\in \mathcal{E}$, there exists a natural number $n$ such that $ E[x]< n $ for each $x\in X$. By a finitary approximation of a coarse structure $ \mathcal{E}^\prime$, we mean any finitary coarse structure $ \mathcal{E}$ such that $ \mathcal{E}\subseteq \mathcal{E}^\prime$.If $\mathcal{E}^\prime$ has a countable base and $E[x]$ is finite for each $x\in X$ then $ \mathcal{E}^\prime$has a cellular finitary approximation $ \mathcal{E}$ such that the relations of linkness on subsets of $( X,\mathcal{E}^\prime)$ and $( X, \mathcal{E})$ coincide.This answers Question 6 from [8]: the class of cellular coarse spaces is not stable under linkness. We define and apply the strongest finitary approximation of a coarse structure.

Deep Residual Transform for Multi-scale Image Decomposition

Multi-scale image decomposition (MID) is a fundamental task in computer vision and image processing that involves the transformation of an image into a hierarchical representation comprising of different levels of visual granularity from coarse structures to fine details. A well-engineered MID disentangles the image signal into meaningful components which can be used in a variety of applications such as image denoising, image compression, and object classification. Traditional MID approaches such as wavelet transforms tackle the problem through carefully designed basis functions under rigid decomposition structure assumptions. However, as the information distribution varies from one type of image content to another, rigid decomposition assumptions lead to inefficiently representation, i.e., some scales can contain little to no information. To address this issue, we present Deep Residual Transform (DRT), a data-driven MID strategy where the input signal is transformed into a hierarchy of non-linear representations at different scales, with each representation being independently learned as the representational residual of previous scales at a user-controlled detail level. As such, the proposed DRT progressively disentangles scale information from the original signal by sequentially learning residual representations. The decomposition flexibility of this approach allows for highly tailored representations cater to specific types of image content, and results in greater representational efficiency and compactness. In this study, we realize the proposed transform by leveraging a hierarchy of sequentially trained autoencoders. To explore the efficacy of the proposed DRT, we leverage two datasets comprising of very different types of image content: 1) CelebFaces and 2) Cityscapes. Experimental results show that the proposed DRT achieved highly efficient information decomposition on both datasets amid their very different visual granularity characteristics.

Coarse structures on groups defined by conjugations

For a group G, we denote by G↔ the coarse space on G endowed with the coarse structure with the base {{(x,y)∈G×G:y∈xF}:F∈[G]<ω}, xF={z−1xz:z∈F}. Our goal is to explore interplays between algebraic properties of G and asymptotic properties of G↔. In particular, we show that asdim G↔=0 if and only if G/ZG is locally finite, ZG is the center of G. For an infinite group G, the coarse space of subgroups of G is discrete if and only if G is a Dedekind group.

Minmax bornologies

A bornology $\mathcal{B}$ on a set $X$ is called minmax, if the smallest and largest coarse structures on $X$ compatible with $\mathcal{B}$ coincide. We prove that $\mathcal{B}$ is minmax, if and only if the family $\mathcal B^\sharp=\{p\in\beta X:\{X\setminus B:B\in\mathcal B\}\subset p\}$ consists of ultrafilters which are pairwise non-isomorphic via $\mathcal B$-preserving bijections of $X$. In addition, we construct a minmax bornology $\mathcal B$ on $\omega$ such that the set $\mathcal B^\sharp$ is infinite. We deduce this result from the existence of a closed infinite subset in $\beta\omega$ that consists of pairwise non-isomorphic ultrafilters.

Influence of Pouring Temperature and Alloy Additions on the Quality of a Semisolid Material Dragged during the Continuous-Casting Strip Processing of Al–Si A413

Al–Si A413 treated and untreated alloys were cast and poured at approximately 720 oC, 700 oC, and 680 oC in a cooling slope to obtain the semisolid material feeding the ceramic nozzle (150 cm3) at the lower roll (single-roll melt-dragged processing)—this drags the metallic slurry via the chill/columnar layers at a rate of 0.2 m/s, forming a molten-metal strip with a thickness of 2 mm and width 45 mm, approximately. The untreated alloy poured at 720 oC formed coarse structures of α-Al dendrites, as well as a coarse eutectic of Al–Si and microshrinking on the surface of the casting strip facing the atmosphere. The Al–Si 413 alloy poured at 680 oC and treated with Al5Ti1B (0.1%) led to microstructural refinement, resulting in α-Al globular structures, the absence of microporosities on the surface facing the atmosphere, and a finer and more homogeneous distribution of the eutectic grains with smaller Si particles. The AlTiB master alloys are not used as a grain refiner in Al–Si alloys because of Si poisoning. This subject is discussed in this paper. The addition of the inoculant and 0.2% of the Al–Si eutectic morphology modifying agent (Al–10%Sr) refined both the α-Al and eutectic phases more efficiently in the cast strip poured at 700 oC and 680 oC. This suggests that the inoculant did not interfere with the action of the modifying agent. As a result, molten metal strips of higher mechanical strengths and ductilities were obtained.