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John Nash initiated the study of arcs in algebraic geometry in a quite unusual paper written in 1968, at the beginning of his long illness. This paper raised many interesting and still-open questions that can be investigated using elementary methods. The tone of the Nash article is discursive, and through a series of examples and comments it reports on "some interesting possible truths which were encountered. "We also follow this approach and present the intricacies and beauties of this subject using concrete examples. In addition we discuss some recent results and unsolved problems that are accessible to undergraduates. [ABSTRACT FROM AUTHOR]

We discuss rotation number associated with the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon η, we consider the outer billiard table T constructed by cutting out a fixed area A from the interior of η. T is piecewise hyperbolic and the polygon η is an invariant curve of T under the billiard map φ. We show that, if β η is a periodic point under φ with rational rotation number τ = p q , then φq is not the local identity at β. This proves that the rotation number τ as a function of the parameter A is a devil's staircase function. [ABSTRACT FROM AUTHOR]

We study the differential properties of generalized arc schemes and geometric versions of Kolchin's Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves. [ABSTRACT FROM AUTHOR]

We present a decomposition of the space of twisted arcs of a toric stack. As a consequence, we give a combinatorial description of the motivic integral associated to a torus-invariant divisor of a toric stack. [ABSTRACT FROM AUTHOR]

Dynamic geometry software (DGS) such as Cabri and Geometers' Sketchpad has been regularly used worldwide for teaching and learning Euclidean geometry for a long time. The DGS with its inductive nature allows students to learn Euclidean geometry via explorations. However, with respect to non-Euclidean geometries, do we need to introduce them to students in a deductive manner? Do students have quite different experiences in non-Euclidean environment? This study addresses these questions by illustrating the student mathematics teachers' actions in dynamic spherical geometry environment. We describe how student mathematics teachers explore new conjectures in spherical geometry and how their conjectures lead them to find proofs in DGS. [ABSTRACT FROM AUTHOR]

The article reports on enumerative algebraic geometry of conics. It discusses solutions to problems on conics which include one provided by Jakob Steiner who was a professor of geometry at the University of Berlin. Steiner's problem asks how many conics are tangent to all five given conics in a plane. The problems presented involve algebraic geometry and depend in the conics, lines, and points. Definitions for conic are presented as well as a moduli space for conics. The article also argues that Steiner's solution the problem was incorrect and proposes a different method. An overview of the report is provided.

A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C × F)/G. In this article, we classify the surfaces of general type with pg = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out that they belong to four families, that we call surfaces of type I, II, III, IV. The moduli spaces I, II, IV are irreducible, whereas III is the disjoint union of two irreducible components. In the last section we start the analysis of the case where G is not abelian, by constructing several examples. [ABSTRACT FROM AUTHOR]

The article offers information on dynamical systems and irrational angle construction by paper-folding. Construction of rational angle to any desired accuracy by folding a strip of paper requires approximating method. The construction uses a periodic sequence composed of two types of folding moves, guided by number-theoretic properties of the angle's fractional representation. The number-theoretic scheme must be replaced by something new, the Folds and Switchfold prescription, in the construction of irrational angles.

The article focuses on commensurable triangles. It discusses the Pythagorean theorem in geometry wherein three positive integers called a Pythagorean triple satisfy the equation. The article mentions that commensurable triangles also involve triple integers. It discusses that in commensurable triangles, one angle is a rational multiple of another angle. The article presents an analysis of triples in commensurable triangles and examines formulas in solving polynomial equations involving triples and their generation.

Let W → 2 be the universal Weierstrass family of cubic curves over . For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 2. Since W → 2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S3 with monodromy in SL2 (/N). [ABSTRACT FROM AUTHOR]

The article discusses the mathematical principles and applications concerning the families of plane algebraic curves that contain a special set of points. The case of curves in cages is associated with the elementary methods that presume the familiarity with algebraic geometry and geometrical applications. These applications can be considered as natural generalizations of classical theorems in projective geometry and are associated with familiarity with Pascal's theorem and its comparison with the classical Pascal diagram.

The article demonstrates that the two consequences of tetrahedrons circumscribing a sphere also holds true for more general solids called circumsolids whose faces can be both planar and curved. Any plane through the center of the inscribed sphere divides the circumsolid into two smaller solids whose surface areas are equal if and only if they have equal volumes. The centroid of the boundary surface of a circumsolid and the centroid of its volume are also collinear always with the center of the inscribed sphere at distances in the ratio of four is to three from the center.

We explicitly compute the equations and components of the jet schemes of a monomial subscheme of affine space from an algebraic perspective. [ABSTRACT FROM AUTHOR]

In this article, we deal with one new singular direction of meromorphic function of infinite order concerning shared-values. Moreover, the example shows the condition is necessary. [ABSTRACT FROM AUTHOR]

The article discusses theorems on triangle inequality and reverse inequality on normed spaces. Theorem 1 states that for nonzero vectors, given values for x and y, in a normed space the result would result into an inequality. If x = y = 1 then there is an equality. A generalization can be proven through a Massera-Schaffer inequality. A second theorem for triangle inequality and the corresponding proof is also provided which demonstrates the strengthening of the triangle inequality and its reverse inequality in normed spaces.

Type III unprojection plays a very important role in the birational geometry of Fano threefolds (cf. Corti et al., 2000; Reid, 2000). According to Reid (2000, p. 43), it was first introduced by A. Corti on his calculations of Fano threefolds of genus 6 and 7. It seems that at present a general definition of Type III unprojection is still missing. After proving some general facts about residual ideals in Section 2, we propose a definition for the generic Type III unprojection (Definition 3.3), and prove in Theorem 3.5 that it gives a Gorenstein ring. Communicated by W. Bruns [ABSTRACT FROM AUTHOR]

The Hilbert scheme of 3-folds in ℙ n , n ≥ 6 , that are scrolls over ℙ 2 or over a smooth quadric surface Q ⊂⃒ ℙ 3 or that are quadric or cubic fibrations over ℙ 1 is studied. All known such threefolds of degree 7 ≤ d ≤ 11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed. [ABSTRACT FROM AUTHOR]

In Keilen (2003) we gave sufficient conditions for the irreducibility of the family Virr([This symbol cannot be presented in ASCII format]l,...,[This symbol cannot be presented in ASCII format]r) of irreducible curves in the linear system \D\t with precisely r singular points of topological, respectively, analytical, types [This symbol cannot be presented in ASCII format]l,... ,[This symbol cannot be presented in ASCII format]r on several classes of smooth projective surfaces Σ. The conditions were of the form [This symbol cannot be presented in ASCII format] where τ* is some invariant of singularity types, KΣ is the canonical divisor of Σ, and γ is some constant. In the present paper we improve this condition, that is, the constant γ, by a factor of 9. [ABSTRACT FROM AUTHOR]

We prove that there exists an integer M such that if S ⊂ [Popf]4 is a smooth surface with deg(S) > M, then the canonical map of S is birational. Then we consider surfaces, S, satisfying hi([Iscr]S(3 − i)) = 0, 0 ≤ i ≤ 2 and show that they are regular and that their canonical system is base point free and very ample if deg(S) > M and S doesn't contain −2-curves. [ABSTRACT FROM AUTHOR]

Given 5 points in 3-space and two snapshots of these points, taken with a camera at two different positions. Then, in general, there are 10 possibilities for the second position of the camera relative to its first position. The result is well known. We prove it using the Thom-Porteous Formula. [ABSTRACT FROM AUTHOR]

The article offers an alternative elementary proof to the one published by J. P. May in the paper "Munshi's proof of the Nullstellensatz," in which some of the algebraic technicalities are avoided. The alternative proof requires a simple version of the Noether normalization lemma. The proof needs the resultant of two polynomials but in such a simple manner that only one property of it is required. The analysis is confined to the weak form of Nullstellensatz because the strong form is easily derived from the weak form using the Rabinowitsch trick.