We present some constructions of limits and colimits in pro-categories. These are critical tools in several applications. In particular, certain technical arguments concerning strict pro-maps are essential for a theorem about étale homotopy types. Also, we show that cofiltered limits in pro-categories commute with finite colimits.

Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschet...

(1.1) H = {(x, y) ∈ (C) × C : y1y2 + p(x) = z}, Here, p : (C) → C is the superpotential mirror to Y (following [7] or [9]), and z is any regular value of p. H is an affine threefold with trivial canonical bundle. Hence, it has a Fukaya category Fuk(H), whose objects are compact exact Lagrangian submanifolds equipped with gradings and Spin structures. This is an A∞-category over C. Consider the ...

We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R3. This ‘homotopy cardinality’ is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.

In this article, we study the fractional SIR epidemic model with Atangana–Baleanu–Caputo operator. We explore properties and applicability of ZZ transformation on operator as transform derivative. This is an application two power methods. obtain a special solution homotopy perturbation method (HPM) combined scheme; then present problem existence solution, also apply new to solving ABC The solut...

The purpose of this paper is to study geometrically simply-connected homotopy $4$-spheres by analyzing $n$-component links in $S^3$ with a Dehn surgery realizing $\#^n (S^1 \times S^2)$. We call such $n$R-links. Our main result that $4$-sphere can be built without $1$-handles and only two $2$-handles diffeomorphic the standard special case one attached along knot form $Q_{p,q} = T_{p,q} \# T_{-...

We define an obstruction for a knot to be ?[?]-homology ribbon, and use this provide restrictions on the integers that can occur as triple linking numbers of derivative links knots are either homotopy ribbon or doubly slice. Our main application finds new non-doubly slice knots. In particular, gives information solvable filtration Taehee Kim: algebraically need not (1)-solvable, (0.5, 1)-solvab...

in this study, a new efficient semi-analytical method is introduced to give approximate solutions of strongly nonlinear oscillators. the proposed method is based on combination of two different methods, the multi-step homotopy analysis method and spectral method, called the multi-step spectral homotopy analysis method (msham). in this method, firstly, we propose a new spectral homotopy analysis...