Related papers: Origami: real structure, enumeration and quantum m…

We propose a novel computational framework for modeling and simulating origami structures. In this framework, bilinear solid-shell elements are employed to model the origami panels while crease folding is considered through the angle…

We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show…

Given an n-tuple of positive real numbers, Konno defines an algebraic variety called a hyperpolygon space, a hyperkahler analogue of the Kahler variety parametrizing spacial polygons with fixed edge lengths. The ordinary polygon space can…

In this paper we study certain real functions defined in a very simple way by Zagier as sums of infinite powers of quadratic polynomials with integer coefficients. These functions give the even parts of the period polynomials of the modular…

Origami is an ancient art that continues to yield both artistic and scientific insights to this day. In 2012, Buhler, Butler, de Launey, and Graham extended these ideas even further by developing a mathematical construction inspired by…

We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…

The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…

The quantum modular invariant of a real number is defined as a discontinuous, PGL(2,Z)-invariant multi-valued map using the distance-to-the-nearest-integer function. On the rationals, the quantum modular invariant is shown to be infinity…

In this century, a square-tiled translation surface (an origami) is intensively studied as an object with special properties of its translation structure and its $SL(2,\mathbb{R})$-orbit embedded in the moduli space. We generalize this…

We study the real components of modular curves. Our main result is an abstract group-theoretic description of the real components of a modular curve defined by a congruence subgroup of level N in terms of the corresponding subgroup of…

Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…

Physical systems, characterized by an ensemble of interacting elementary constituents, can be represented and studied by different algebras of observables or operators. For example, a fully polarized electronic system can be investigated by…

Kirigami, art of paper cutting, enables two-dimensional sheets transforming into unique shapes which are also hard to reshape once with prescribed cutting patterns. Rare kirigami designs manipulate cuts on three-dimensional objects to…

We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as…

We introduce an additive approach for the design of a class of transformable structures based on two-bar linkages ("scissor mechanisms") joined at vertices to form a two dimensional lattice. Our discussion traces an underlying mathematical…

We present a study of real Hurwitz numbers enumerating a special kind of real meromorphic functions, which we call simple framed purely real functions. We deduce partial differential equations of cut-and-join type for generating functions…

We present the realization of Hurwitz algebras in terms of 2x2 vector matrices, which maintain the correspondence between the geometry of the vector spaces used in the classical physics and the underlined algebraic foundation of the quantum…

We present a formalization of geometric instruments that considers separately geometric and arithmetic aspects of them. We introduce the concept of tool, which formalizes a physical instrument as a set of axioms representing its geometric…

The subject of Polynomiography deals with algorithmic visualization of polynomial equations, having many applications in STEM and art, see [Kal04]. Here we consider the polynomiography of the partial sums of the exponential series. While…

We continue the study of generalized gauge theory called gauge origami, based on the quantum algebraic approach initiated in [arXiv:2310.08545]. In this article, we in particular explore the D2 brane system realized by the screened vertex…