Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the classical case. Unlike previous attempts at generalizing link homotopy to spatial graphs, our new relation allows analogues of some standard link homotopy res...

We propose a new sheaf-theoretical method for the calculation of the monodromy zeta functions of Milnor fibrations. As an application, classical formulas of Kushnirenko [11] and Varchenko [23] etc. concerning polynomials on C will be generalized to polynomial functions on any toric variety.

Introduction 1 1. Notation 8 2. Finite group action on products of curves 9 3. The fundamental group of (C1 × C2)/G. 12 4. The structure theorem for fundamental groups of quotients of products of curves 14 5. The classification of standard isotrivial fibrations with pg = q = 0 where X = (C1 × C2)/G has rational double points 20 5.1. The singularities of X 21 5.2. The signatures 23 5.3. The poss...

In the first part of this paper we present a formalization in Agda of the James construction in homotopy type theory. We include several fragments of code to show what the Agda code looks like, and we explain several techniques that we used in the formalization. In the second part, we use the James construction to give a constructive proof that π4(S ) is of the form Z/nZ (but we do not compute ...

We develop an obstruction theory for homotopy of homomorphisms f, g : M → N between minimal differential graded algebras. We assume that M = ΛV has an obstruction decomposition given by V = V0⊕V1 and that f and g are homotopic on ΛV0. An obstruction is then obtained as a vector space homomorphism V1 → H(N ). We investigate the relationship between the condition that f and g are homotopic and th...

We consider a category H ⊗ (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of H ⊗ to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct ...

in this paper, nonlinear klein-gordon equation with quadratic term is solved by means of an analytic technique, namely the homotopy analysis method (ham).comparisons are made between the adomian decomposition method (adm), the exact solution and homotopy analysis method. the results reveal that the proposed method is very effective and simple.

in this paper, the homotopy perturbation method (hpm) and elzaki transform is employed to obtain the approximate analytical solution of the sine gorden and the klein gorden equations. the nonlinear terms can be handled by the use of homotopy perturbation method. the proposed homotopy perturbation method is applied to reformulate the first and the second order initial value problems which leads ...

For logical thinking learners, it is important to learn problem solving formally and intuitively. Homotopy which is modern mathematics combines algebra and geometry together. Homotopy gives the most abstract level of explanation or the most basic invariant which is the most required characteristics for logical thinking. The homotopy lifting property gives good explanation of inductive reasoning...

A quasi-schemoid is a small category with a particular partition of the set of morphisms. We define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. The homotopy set of self-homotopy equivalences on a quasi-schemoid is used as a homotopy invariant in the study. The main theorem enables us to deduce that the homotopy invariant for the quasi-schemoid in...