Tau functions and the limit of block Toeplitz determinants

A classical way to introduce tau functions for integrable hierarchies of solitonic equations is by means of the Sato-Segal-Wilson infinite-dimensional Grassmannian. Every point in the Grassmannian is naturally related to a Riemann-Hilbert problem on the unit circle, for which Bertola proposed a tau function that generalizes the Jimbo-Miwa-Ueno tau function for isomonodromic deformation problems. In this paper, we prove that the Sato-Segal-Wilson tau function and the (generalized) Jimbo-Miwa-Ueno isomonodromy tau function coincide under a very general setting, by identifying each of them to the large-size limit of a block Toeplitz determinant. As an application, we give a new definition of tau function for Drinfeld-Sokolov hierarchies (and their generalizations) by means of infinite-dimensional Grassmannians, and clarify their relation with other tau functions given in the literature.Comment: 22 page

From the Pearcey to the Airy process

Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian motions on the real line for the eigenvalues, as was discovered by Dyson. Applying scaling limits to the random matrix models, combined with Dyson's dynamics, then leads to interesting, infinite-dimensional diffusions for the eigenvalues. This paper studies the relationship between two of the models, namely the Airy and Pearcey processes and more precisely shows how to approximate the multi-time statistics for the Pearcey process by the one of the Airy process with the help of a PDE governing the gap probabilities for the Pearcey process.Comment: 21 pages, 2 figure

Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials

We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painleve' IV differential equation for each family.Comment: Final version, accepted for publication on CMP: Communications in Mathematical Physics. 24 pages, 1 figur

Penerapan Markerless Augmented Reality Pada Brosur Perumahan Berbasis Android(Studi Kasus Pada CV. Anugerah Barat Jaya Palembang)

Every company, especially housing developers, have their own way to market their home. In support of marketing activities, there are several ways that are often carried out by the company, such as creating mockups promotional media, brochures, banners, and more. However, not all companies developer home using interactive promotional media such as mockups for needed capital cost is quite expensive. While, the media campaign brochures are not able to provide clear and complete information due to lack of interactive, augmented realiy technology built on Android-based housing brochure. Augmented reality technology is able to add a virtual objects into the real world at a time. There are two methods commonly used Markerless and markerbased tracking. Markerless different from tracking markerbased which has a black background and white marker and need space as a place marker, methods Markerless make computers able to track all the real object which is connected to a 3D object to be displayed as an image, the text as a marker without the need of space as a place marker. The software is built using the Waterfall Model of software development and applied based on android, for android is a medium that is practical and can always be taken anywhere. The results of this research is application-based electronic brochure augmented reality object type houses with 45 and 100

From Hide to Handbag: A Holistic Review of the Leather Industry and Its Environmental Impact

Leather is a universally versatile material used in various products in the global market. Today, leather, specifically from the commercial cow, is still desired by manufacturers for its durability, accessibility, and quality. However, there are adverse environmental and ethical consequences of leather products using chrome tanning, in particular, that consumers do not recognize. This paper analyzes, evaluates, and contextualizes the current methods of leather production and the environmental issues associated with the trade. Furthermore, it discusses some sustainable efforts made regarding material choice and alternative production methods. With a holistic review of leather production and its implications, this paper examines the EU Strategy for Sustainable Circular Economy and Textiles, Chain of Custody - Transparency in Leather Manufacturing Pilot Program, and the Sustainable Development Goals in the Leather Industry EU. These policies aim to foster a more environmentally aware and sustainable leather trade. This paper will determine their effectiveness, feasibility, and potential drawbacks, then recommend reform options reflecting the current policies.https://digitalcommons.salve.edu/glo100/1002/thumbnail.jp