Narrow Results

We show that, for a closed orientable $n$-manifold, with $n$ not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex $(n-1)$-space ensures the existence of a totally real embedding into complex $n$-space. This implies that a closed orientable $(4k+1)$-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex $4k$-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.

We briefly discuss the concepts of immersion and embedding of space-times in higher-dimensional spaces. We revisit the classical work by Kasner in which he constructs a model of immersion of the Schwarzschild exterior solution into a six-dimensional pseudo-Euclidean manifold. We show that, from a physical point of view, this model is not entirely satisfactory, since the causal structure of the immersed space-time is not preserved by the immersion.

An immersion $f:{𝒟}\to {𝒞}$ between $\mathrm{\Delta }$-complexes is a $\mathrm{\Delta }$-map that induces injections from star sets of ${𝒟}$ to star sets of ${𝒞}$. We study immersions between finite-dimensional connected $\mathrm{\Delta }$-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szakács on immersions into $2$-dimensional $CW$-complexes.

This paper is the first one in the sequence of papers about a simple class of framed 4-graphs; the goal of this paper is to collect some well-known results on planarity and to reformulate them in the language of minors. The goal of the whole sequence is to prove analogs of the Robertson–Seymour–Thomas theorems for framed 4-graphs: namely, we shall prove that many minor-closed properties are classified by finitely many excluded graphs. From many points of view, framed 4-graphs are easier to consider than general graphs; on the other hand, framed 4-graphs are closely related to many problems in graph theory.

In the present paper, we proceed with the study of framed 4-graph minor theory initiated in [V. O. Manturov, Framed 4-valent graph minor theory I: Intoduction planarity criterion, arxiv: 1402.1564v1 [Math.Co]] and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in $ℝP^2$.

It is an open problem whether Kirk’s $\sigma$-invariant is the complete obstruction to a link map $f:S_{+}^2\cup S_{-}^2\to S^4$ being link homotopic to the trivial link. The link homotopy invariant associates to such a link map $f$ a pair $\sigma (f)=({\sigma }_{+}(f),{\sigma }_{-}(f))$, and we write $\sigma =({\sigma }_{+},{\sigma }_{-})$. With the objective of constructing counterexamples, Li proposed a link homotopy invariant $\omega =({\omega }_{+},{\omega }_{-})$ such that ${\omega }_{\pm }$ is defined on the kernel of ${\sigma }_{\pm }$ and which also obstructs link null-homotopy. We show that, when defined, the invariant ${\omega }_{\pm }$ is determined by ${\sigma }_{\mp }$, and is strictly weaker. In particular, this implies that if a link map $f$ has $\sigma (f)=(0,0)$, then after a link homotopy the self-intersections of $f(S_{+}^2)$ may be equipped with framed, immersed Whitney disks in $S^4\setminus f(S_{-}^2)$ whose interiors are disjoint from $f(S_{+}^2)$.

Recently V. Arnold introduced Strangeness and J^± invariants of generic immersions of an oriented circle to ℝ². Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface F. We explicitly describe all the invariants satisfying axioms, which naturally generalize the axioms used by V. Arnold.

We discuss in the present paper the following natural question: is the space of all Morse functions with fixed number of minima and maxima on a closed surface linearly connected? We give an algorithm for reduction of any Morse function on a closed orientable surface to some canonical form. We apply this result to the new representation for the inversion of 2-sphere in Euclidean 3-space, in terms of Reeb graph of the height function.

In general it is not clear which kind of information is supposed to be used for calculating the fair value of a contingent claim. Even if the information is specified, it is not guaranteed that the fair value is uniquely determined by the given information. A further problem is that asset prices are typically expressed in terms of a risk-neutral measure. This makes it difficult to transfer the fundamental results of financial mathematics to econometrics. I show that the aforementioned problems evaporate if the financial market is complete and sensitive. In this case, after an appropriate choice of the numéraire, the discounted price processes turn out to be uniformly integrable martingales under the real-world measure. This leads to a Law of One Price and a simple real-world valuation formula in a model-independent framework where the number of assets as well as the lifetime of the market can be finite or infinite.

The exploratory study presents the role of individual cognition and immersion in individual knowledge development. Mediation effect of knowledge essence on the role is also examined. Data from 296 valid subjects from elementary and junior high school teachers are analysed and reveal several findings. Knowledge essence alternates the effect of individual cognition on knowledge development. Consistent cognition is significantly associated with knowledge development in richness and metacognition in usefulness. Similarity in knowledge context mediates the effect of metacognition on knowledge development in usefulness, but alternates the effect of cognition consistency on richness. Knowledge explicitness mediates the effect of situational immersion on rich and useful knowledge development, but alternates the mediation effect of context similarity on useful knowledge development. Implications and suggestions are also addressed.

Let be the n-dimensional pseudo-Euclidean space of index p and M(n, p) the group of all transformations of generated by pseudo-orthogonal transformations and parallel translations. We describe the system of generators of the differential field of all M(n, p)-invariant differential rational functions of a map of an open subset . Using this result, we prove analogues of the Bonnet theorem for immersions of an n-dimensional C^∞-manifold J in . These analogues are given in terms of the pseudo-Riemannian metric, the volume form, and the connection on J induced by the immersion of J in .

The statement of the Hanna Neumann Conjecture (HNC) is purely algebraic: for a free group Γ and any nontrivial finitely generated subgroups A and B of Γ,

The goal of this paper is to systematically develop machinery that would allow for generalizations of HNC and to exhibit their relations with topology and analysis. On the topological side we define immersions of complexes, leafages, systems of complexes, flowers, gardens, and atomic decompositions of graphs and surfaces. The analytic part involves working with the classical Murray–von Neumann (!) dimension of Hilbert modules.

This also gives an approach to the Strengthened Hanna Neumann Conjecture (SHNC) and to its generalizations. We present three faces of it named, respectively, the square approach, the diagonal approach, and the arrangement approach. Each of the three comes from the notion of a system, and each leads to questions beyond graphs and free groups. Partial results, sufficient conditions, and generalizations of the statement of SHNC are presented.

In this work we propose a new procedure on how to extract global information of a space-time. We consider a space-time immersed in a higher dimensional space and formulate the equations of Einstein through the Frobenius conditions of immersion. Through an algorithm and implementation into algebraic computing system we calculate normal vectors from the immersion to find the second fundamental form. We make an application for a static space-time with spherical symmetry. We solve Einstein's equations in the vacuum and obtain space-times with different topologies.

We investigate the topology of Schwarzschild's black holes through the immersion of this space-time in space of higher dimension. Through the immersions of Kasner and Fronsdal we calculate the extension of the Schwarzschilds black hole.

Titanium dioxide is added into calcium phosphate bio-glass (CPG) to have crystalline phases of titanium phosphoric (TiP₂O₇) and calcium phosphoric (CaP₂O₇) on its surfaces. The bio-glass synthesis with the addition of titanium dioxide herein is denoted as TCPG. To elucidate their surface morphologies, both specimens of CPG and TCPG were immersed in Hanks' solution for two days before soaking in the mixed solutions of (NH₄)₂HPO₄ and Ca(NO₃)₂ at 70°. Crystalline layers of titanium phosphoric were observed on the surfaces of TCPG from immersing in Hanks' solution. After which calcium pyrophosphate appeared on the second step of soaking process from the calcium ion contained solutions. Due to the absence of crystalline phases on the surfaces of CPG specimen, it can be deduced that the addition of titania (TiO₂) causes the hydroxyapatite formation on the surface of bio-glass.