Open associahedra and scattering forms

We continue the study of open associahedra associated with bi-color scattering amplitudes initiated in ref. [1]. We focus on the facet geometries of the open associahedra, uncovering many new phenomena such as fiber-product geometries. We then provide novel recursion procedures for calculating the canonical form of open associahedra, generalizing recursion relations for bounded polytopes to unbounded polytopes.

The existence of supersingular curves of genus 4 in arbitrary characteristic

We prove that there exists a supersingular nonsingular curve of genus 4 in arbitrary characteristic $$p>0$$ p > 0 . For $$p>3$$ p > 3 we shall prove that the desingularization of a certain fiber product over $$\mathbf{P }^1$$ P 1 of two supersingular elliptic curves is supersingular.

On finitary properties for fiber products of free semigroups and free monoids

We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and $$\mathcal {J}$$ J -trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.

Almost Gorenstein rings arising from fiber products

The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times _T S$ of Cohen–Macaulay local rings R, S of the same dimension $d>0$ over a regular local ring T with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are R and S. In addition, the other generalizations of Gorenstein properties are also explored.

Mixed-characteristic shtukas

This chapter looks at mixed-characteristic shtukas. Much of the theory of mixed-characteristic shtukas is motivated by the structures appearing in (integral) p-adic Hodge theory. The chapter assesses Drinfeld's shtukas and local shtukas. In the mixed characteristic setting, X will be replaced with Spa Zp. The test objects S will be drawn from Perf, the category of perfectoid spaces in characteristic p. For an object, a shtuka over S should be a vector bundle over an adic space, together with a Frobenius structure. The product is not meant to be taken literally (if so, one would just recover S), but rather it is to be interpreted as a fiber product over a deeper base. Motivated by this, the chapter then defines an analytic adic space and shows that its associated diamond is the appropriate product of sheaves on Perf.

The Comparison on Mechanical Bonding Properties of Untreated Coconut Fiber towards Synthetic Fiber for Fiberglass Boat Building

The development of high performance materials made from natural resources is increasing worldwide. The interest in natural fiber reinforced polymer composite materials is rapidly growing both in terms of their industrial applications and fundamental research. They are renewable, cheap, completely or partially recyclable, and biodegradable. The coconut fiber can be a potential candidate to replace the industrial core and foam which its application are worldwide and it is used to increase the thickness of the fiberglass boat. In this research, three types of testing panel are constructed by using 10 mm of Coconut Fiber, 3D Core Foam and Infusion Grooved PVC Foam as the sandwich core. The resin infusion method which is produce quality final products have been used. The findings be obtained by conducting two testing methods, for the Flatwise Tensile Strength testing, the specimens taken from Coconut Fiber product yielded a higher value of strength which is 3.005 MPa compared to the specimens taken from Infusion Grooved PVC Foam and 3D Core Foam which is 2.963 MPa and 1.264 MPa respectively. For the Flatwise Compressive Strength testing, the specimens taken from the Coconut Fiber product had higher value of compressive stress compared to the value of specimens taken from Infusion Grooved PVC Foam and 3D Core Foam which is 29.66 MPa, 2.58 MPa and 4.68 MPa respectively. This research has proved that the Coconut Fiber is quite suitable to become as one of the laminating schedule for the construction of the fiberglass boat hull.