The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence u1, . . . , un of norm 1 vectors in a real Hilbert space H , there exists a unit vector v ∈ H , such that X 1 〈ui, v〉2 6 n . The 2-dimensional case is proved b...

Twisted geometry is a piecewise-flat geometry less rigid than Regge geometry. In Loop Gravity, it provides the classical limit for each step of the truncation utilized in the definition of the quantum theory. We define the torsionless spin-connection of a twisted geometry. The difficulty given by the discontinuity of the triad is addressed by interpolating between triads. The curvature of the r...

Rigid E -unification is the problem of unifying two expressions modulo a set of equations, with the assumption that every variable denotes exactly one term (rigid semantics). This form of unification was originally developed as an approach to integrate equational reasoning in tableau-like proof procedures, and studied extensively in the late 80s and 90s. However, the fact that simultaneous rigi...

Hot torsion test (HTT) has extensively been used to analysis and physically model flow behavior and microstructure evolution of materials and alloys during hot deformation processes. In this test, the specimen geometry has a great influence in obtaining reliable test results. In this paper, the interaction of thermal-mechanical conditions and geometry of the HTT specimen was studied. The commer...

We describe the Lipschitz geometry of complex curves. For the most part this is well known material, but we give a stronger version even of known results. In particular, we give a quick proof, without any analytic restrictions, that the outer Lipschitz geometry of a germ of a complex plane curve determines and is determined by its embedded topology. This was first proved by Pham and Teissier, b...

In this paper, we study how to improve the innovative ability of students based on the teaching practice of the Analytic Geometry and Functional Analysis, how to permeate its important ideas, cultivate and improve students' self-learning ability, the ability of independent thinking and problem-solving ability through the teaching of the Analytic Geometry and Functional Analysis, and put forward...

1 Kasner, "Conformal Classification of Analytic Arcs or Elements, Poincare's Local Problem of Conformal Geometry," Trans. Am. Math. Soc., 16, 333-349 (1915). The theory of a pair of regular arcs, including the horn angle, is given in "Conformal Geometry," Proceedings Cambridge International Congress Mathematicians, 1912, and a paper appearing in Scripta Mathematica, 1945. 2 Kasner and De Cicco,...

Let A be a commutative ring, B a commutative A-algebra and M a complex of B-modules. We begin by constructing the square SqB/A M , which is also a complex of B-modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism ρ : M ≃ −→ SqB/A M then the pair (M,ρ) is called a rigid complex over B relative to A...