Epihypocurves and epihypocyclic surfaces with arbitrary base curve

If a circle rolls around another motionless circle then a point bind with the rolling circle forms a curve. It is called epicycloid, if a circle is rolling outside the motionless circle; it is called hypocycloid if the circle is rolling inside the motionless circle. The point bind to the rolling circle forms a space curve if the rolling circle has the constant incline to the plane of the motionless circle. The cycloid curve is formed when the circle is rolling along a straight line. The geometry of the curves formed by the point bind to the circle rolling along some base curve is investigated at this study. The geometry of the surfaces formed when the circle there is rolling along some curve and rotates around the tangent to the curve is considered as well. Since when the circle rotates in the normal plane of the base curve, a point rigidly connected to the rotating circle arises the circle, then an epihypocycloidal cyclic surface is formed. The vector equations of the epihypocycloid curve and epihypocycloid cycle surfaces with any base curve are established. The figures of the epihypocycloids with base curves of ellipse and sinus are got on the base of the equations obtained. These figures demonstrate the opportunities of form finding of the surfaces arised by the cycle rolling along different base curves. Unlike epihypocycloidal curves and surfaces with a base circle, the shape of epihypocycloidal curves and surfaces with a base curve other than a circle depends on the initial rolling point of the circle on the base curve.

About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function

A simple algorithm for calculating Christoffel symbols, a covariant projection of the result of the Laplace operator's action on the vector, vector curl and other similar operations in an arbitrary oblique base are proposed. For an arbitrary base with ortho ei is found the expressions of vector projections (ΔA) i and (rot A) i , where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA) i and (rot A) i for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (ΔA) i and (rot A) i as an example are made in cases of spherical and cylindrical coordinate systems.

Bootstrapping ADE M-strings

We study elliptic genera of ADE-type M-strings in 6d (2,0) SCFTs from their modularity and explore the relation to topological string partition functions. We find a novel kinematical constraint that elliptic genera should follow, which determines elliptic genera at low base degrees and helps us to conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of related geometries. Using this, we can bootstrap the elliptic genera to arbitrary base degree, including D/E-type theories for which explicit formulas are only partially known. We utilize our results to obtain the 6d Cardy formulas and the superconformal indices for (2,0) theories.

Regular Hom-algebras admitting a multiplicative basis

Let {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field {{\mathbb{F}}}. A basis {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of {{\mathfrak{H}}} is called multiplicative if for any {i,j\in I}, we have that {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some {k,p\in I}. We show that if {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case {{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

Multifunctional Reflective Digital Metasurfaces of Arbitrary Base Based on Liquid Crystal Technology

This paper presents a multifunctional reflective digital metasurface of arbitrary base based on voltage tunable liquid crystal (LC). The reflective digital metamaterial can be multiplexed for different desirable functions by properly biasing the LC for different code patterns. Simulation results of three significant functions, beam steering with a steering elevation angle 27° at 75 GHz, RCS reduction of at least 15dB along the incident direction, and beam shaping with different beam shapes have been presented to prove the concept.

Local Shtukas and Divisible Local Anderson Modules

We develop the analog of crystalline Dieudonné theory for$p$-divisible groups in the arithmetic of function fields. In our theory$p$-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian$t$-modules and$t$-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.