Neighborhoods and Manifolds of Immersed Curves

We present some fine properties of immersions ℐ : M ⟶ N between manifolds, with particular attention to the case of immersed curves c : S 1 ⟶ ℝ n . We present new results, as well as known results but with quantitative statements (that may be useful in numerical applications) regarding tubular coordinates, neighborhoods of immersed and freely immersed curve, and local unique representations of nearby such curves, possibly “up to reparameterization.” We present examples and counterexamples to support the significance of these results. Eventually, we provide a complete and detailed proof of a result first stated in a 1991-paper by Cervera, Mascaró, and Michor: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold.

Periodic ultradiscrete plane transformation with a period of 12

V.A. Bykovskii constructed a new periodic ultradiscrete plane transformation with a period of 12. In his work only the idea of proving this periodicity was proposed. We provide a complete and detailed proof of this statement.

Periodic ultradiscrete transformations of the plane with periods of 5, 7, 8, 9

V.A. Bykovskii constructed three new periodic ultradiscrete transformations of the plane In addition to the two well-known. In his work, only the idea of proving these statements was proposed. We give a complete and detailed proof of them for sequences with periods 5, 7, 8, 9.

A polyfold proof of the Arnold conjecture

We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $${{\mathbb {C}}{\mathbb {P}}}^1\times M$$ C P 1 × M to $${{\mathbb {C}}}^+ \times M$$ C + × M and $${{\mathbb {C}}}^-\times M$$ C - × M , as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

The Inverse of Ackermann Function is Computable in Linear Time

We propose a detailed proof of the fact that the inverse of Ackermann function is computable in linear time.

Schrödinger’s Ballot: Quantum Information and the Violation of Arrow’s Impossibility Theorem

We study Arrow’s Impossibility Theorem in the quantum setting. Our work is based on the work of Bao and Halpern, in which it is proved that the quantum analogue of Arrow’s Impossibility Theorem is not valid. However, we feel unsatisfied about the proof presented in Bao and Halpern’s work. Moreover, the definition of Quantum Independence of Irrelevant Alternatives (QIIA) in Bao and Halpern’s work seems not appropriate to us. We give a better definition of QIIA, which properly captures the idea of the independence of irrelevant alternatives, and a detailed proof of the violation of Arrow’s Impossibility Theorem in the quantum setting with the modified definition.

Outline of the Upcoming Proof

‘Outline of the Upcoming Proof’ provides a comprehensive outline of the proof of the disc embedding theorem. The disc embedding theorem for topological 4-manifolds, due to Michael Freedman, underpins virtually all our understanding of topological 4-manifolds. The famously intricate proof utilizes techniques from both decomposition space theory and smooth manifold topology. The latter is used to construct an infinite iterated object, called a skyscraper, and the former to construct homeomorphisms from a given topological space to a quotient space. The detailed proof of the disc embedding theorem is the core aim of this book. In this chapter, a comprehensive outline of the proof is provided, indicating the chapters in which each aspect is discussed in detail.

Skyscrapers Are Standard: The Details

‘Skyscrapers Are Standard: The Details’ provides a thorough and detailed proof that every skyscraper is homeomorphic to the standard 2-handle, relative to the attaching region. Results from decomposition space theory established in Part I and the constructive results from Part II are combined. The idea is to construct a subset of a skyscraper called the design, define an embedding of this subset into the standard 2-handle, and then consider the decomposition spaces obtained by quotienting out the connected components of the complement of this common subset. It is shown that the decomposition spaces are homeomorphic, and that both quotient maps are approximable by homeomorphisms. This chapter also shows that everything can be done fixing a neighbourhood of the attaching region. It is then deduced that skyscrapers are standard, as desired.

On the Distribution of the Nontrivial Zeros for the Dirichlet L-Functions

This paper addresses a variant of the product for the Dirichlet $L$--functions. We propose a completely detailed proof for the truth of the generalized Riemann conjecture for the Dirichlet $L$--functions, which states that the real part of the nontrivial zeros is $1/2$. The Wang and Hardy--Littlewood theorems are also discussed with removing the need for it. The results are applicable to show the truth of the Goldbach's conjecture.

TRIPLE SEQUENCES AND THEIR LIMITATIONS

In that paper, we have to established the belief of a sequence and about limits of sequences and the idea of convergence and divergence for a sequence and also it includes the double sequences and triple sequences. We will be developing the principles for triple sequences that is equivalent to the principles of double sequences and single sequence. Also we have an investigate a basic principles for triple sequences and also gives detailed proof of them.