We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions," and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected," and to avoid local extrema over one-dimensional submanifolds of the range, in which case the Morse 2-function is "indefinite." This is founda...

Supersymmetric solution of PT-/non-PT-symmetric and non-Hermitian Morse potential is studied to get real and complex-valued energy eigenvalues and corresponding wave functions. Hamiltonian Hierarchy method is used in the calculations. PACS numbers: 03.65.-w; 03.65.Ge

The existence of multiple solutions for a class of fourth-order elliptic equation with respect to the generalized asymptotically linear conditions is established by using the minimax method and Morse theory.

We prove that a morphism h over a two-letter alphabet {a, b} is overlap-free, i.e., maps overlap-free words into overlap-free words, iff the word h(abbabaab) is overlap-free. As a consequence, we obtain a simple proof of the fact that the only infinite overlap-free words that can be obtained by iterating a morphism are the Thue-Morse sequence and its opposite. RLsumP Nous prouvons qu’un morphis...

Background: Tinea capitis (TC) is the most prevalent pediatric superficial dermatophyte infection. Scalp dermoscopy or “trichoscopy” represents a valuable, noninvasive technique for evaluation of patients with hair loss due to TC. Purpose: To characterize trichoscopic findings in children clinical suggestive Case: A 13-year-old boy was presented scaled plaque on his scalp that had appeared 1 mo...

In this work, we design a nearly linear time discrete Morse theory based algorithm for computing homology groups of 2-manifolds, thereby establishing the fact that computing homology groups of 2-manifolds is remarkably easy. Unlike previous algorithms of similar flavor, our method works with coefficients from arbitrary abelian groups. Another advantage of our method lies in the fact that our al...

In 1974, Mark Goresky and Robert MacPherson began their development of intersection homology theory (see [24] in these volumes), and their first paper on this topic appeared in 1980; see [12]. At that time, they were missing a fundamental tool which was available for the study of smooth manifolds; they had no Morse Theory for stratified spaces. Goresky and MacPherson wished to have a Stratified...

In this paper we consider the φF category associated to the family of circle valued Morse functions defined on a closed manifold M . This number will be called the Morse-Smale characteristic of manifold M for circle-valued Morse functions and and it will be denoted by γs1(M). We present some basic notions and results concerning circle-valued Morse functions and we prove some properties for γs1(...

Two implementations of a homology algorithm based on the Forman’s discrete Morse theory combined with the coreduction method are presented. Their efficiency is compared with other implementations of homology algorithms.