The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine homsets, monomorphisms, epimorpshisms and isomorphisms [6] within a free-object category [15] and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property, and the Pullback L...

in this paper‎, ‎we give the notions of crossed polymodule and cat$^1$-polygroup‎ ‎as a generalization of loday's definition‎. ‎then‎, ‎we define the pullback cat$^1$-polygroup and we obtain some results in this respect‎. ‎specially‎, ‎we prove that by a pullback cat$^1$-polygroup we can obtain a cat$^1$-group‎.

We consider the pullback attractors for non-autonomous dynamical systems generated by stochastic lattice differential equations with non-autonomous deterministic terms. We first establish a sufficient condition for existence of pullback attractors of lattice dynamical systems with both non-autonomous deterministic and random forcing terms. As an application of the abstract theory, we prove the ...

We’ve covered a lot of ground so far. I want to remind you that we’ve essentially defined a very few things, and spent all our energy on showing that they behave well with respect to each other. In particular: proper pushforward, flat pullback, c·, s·, s·(X, Y). Gysin pullback for divisors; intersecting with pseudo-divisors. Gysin pullback for 0sections of vector bundles. We know how to calcula...

It is a long-standing conjecture in 3-manifold theory that every homotopy equivalence f :M!N between closed, irreducible 3-manifolds M and N , with j 1(M)j (=j 1(N)j) = 1, is homotopic to a homeomorphism. In [Wa], Waldhausen proved that this conjecture was true, if we assume that N contains a 2-sided incompressible surface S. The proof consists of rst homotoping the map f so that the pullback s...

Let X be a projective manifold, ρ : X̃ → X its universal covering and ρ∗ : V ect(X) → V ect(X̃) the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map ρ∗ and the properties of the function theory on X̃. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holo...

AIMS In coronary bifurcations assessment, evaluation of side-branch (SB) ostia by an optical coherence tomography (OCT) pullback performed in the main branch (MB) could speed up lesion evaluation and minimize contrast volume. Dedicated software that reconstructs the cross-sections perpendicular to the SB centreline could improve this assessment. We aimed to validate a new method for assessing t...

In this paper, the existence and uniqueness of pullback attractors for the modified Swift-Hohenberg equation defined on R driven by both deterministic non-autonomous forcing and additive white noise are established. We first define a continuous cocycle for the equation in L(R), and we prove the existence of pullback absorbing sets and the pullback asymptotic compactness of solutions when the eq...

In this paper we study the pullback of a Siegel Eisenstein series on Sp2m+2n to Sp2m×Sp2n. There is a well-established literature on such pullbacks. In the case that m = n Garrett showed that the pullback is actually a cusp form in each variable separately. Here we generalize this result showing the pullback is cuspidal in the smaller variable in the case m 6= n. Such results have applications ...