Narrow Results

Let $f:({ℝ}^n,0)\to (ℝ,0)$ be a definable function germ of class ${{𝒞}}^2$ and let $(X,0)\subset ({ℝ}^n,0)$ be a germ of a closed definable set. We investigate topological invariants associated with $f_{|X}$. In particular, we give several topological formulae for the Euler characteristics of related sets. We also relate the topology of $f_{|X}$ to the topology of a definable function with isolated critical point in the stratified case.

We define a new one-form H^A based on the second fundamental tensor , the Gauss–Bonnet–Chern form can be novelly expressed with this one-form. Using the ϕ-mapping theory we find that the Gauss–Bonnet–Chern density can be expressed in terms of the δ-function δ(ϕ) and the relationship between the Gauss–Bonnet–Chern theorem and Hopf–Poincaré theorem is given straightforwardly. The topological current of the Gauss–Bonnet–Chern theorem and its topological structure are discussed in details. At last, the Morse theory formula of the Euler characteristic is generalized.

In this work, we study the RVB method for calculating the Hawking temperature of different black holes and find that there is an undetermined integral constant in the temperature expression. We use this method to calculate the Hawking temperature of the black hole in Einstein gravity, and massive gravity, Einstein–Gauss–Bonnet gravity, Scalar–Tensor–Vector modified gravity and $f(R)$ gravity, respectively. By comparing with the temperature obtained by the Hawking temperature formula, we find that regardless of the gravitational theory from which the black hole solution is obtained, after the black hole metric is reduced to two dimensions, if there is no first-order term of $r$ in $g_{00}$, the integral constant is 0. If there is a first-order term of $r$ in $g_{00}$, the integral constant is determined by the coefficient in front of the first-order term of $r$.

This paper presents a new approach for calculating the Euler characteristic in 2D binary images. The problem is addressed using the Three OrThogonal symbol chain code (3OT code), using only one symbol for the calculation of the Euler characteristic. Using this code, it is possible to introduce new geometric concepts represented by the same symbol of the 3OT alphabet and to simplify the overall equation of the Euler characteristic. This process is supported by the proof of a set of theorems and their numerical validation, using a set of binary images with a variable number of holes. Thus, this research proves that the 3OT code can be used not only for image compression as reported in the literature, but also to simplify the expression of the Euler characteristic as well as for the analysis and simplification of the shape of contours.

An exact analytical approach to n-dimensional discrete images, objects, and surfaces is given. Objects are defined as finite grid point sets of Zⁿ. Using the 2n-neighborhood of Zⁿ, components are maximal connected sets of object points. Formulas are given for count measures of object (points, edges, faces, cells, marginal elements, etc.), and the Euler number ψ⁽ⁿ⁾. Such measures are defined in Z³ with respect to volume, surface content, and the mean curvature integral in E³. The generalization for Zⁿ allows the definition of similarities between n-dimensional objects. The Euler number and other object characteristics can be expressed only by numbers of marginal elements of single object surfaces. Finally, a general formula for ψ⁽³⁾ is given.

Let $G$ be the simple group $\mathrm{\text{PSL}}(3,2^p)$, where $p$ is a prime number. For any subgroup $H$ of $G$, we compute the Möbius function $\mu (H)$ of $H$ in the subgroup lattice of $G$. To this aim, we describe the intersections of maximal subgroups of $G$. We point out some connections of the Möbius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of $r$-subgroups of $\mathrm{\text{PGL}}(3,q)$, for any prime $r$ and any prime power $q$.

We compute the slice euler characteristic of certain links by using the so-called generalized adjunction formula, which is proved by Kronheimer, Mrowka, Morgan, Szabó and Taubes. Furthermore, for links obtained from such links by band surgery, we estimate the unknotting numbers, the 4-dimensional clasp numbers, and the slice euler characteristic.

Let $X$ and $Y$ be the complementary regions of a closed hypersurface $M$ in $S^4=X{\cup }_{M}Y$. We use the Massey product structure in $H^{\ast }(M;\mathbb{Z})$ to limit the possibilities for $\chi (X)$ and $\chi (Y)$. We show also that if ${\pi }_1(X)\ne 1$ then it may be modified by a 2-knot satellite construction, while if $\chi (X)\le 1$ and ${\pi }_1(X)$ is abelian then ${\beta }_1(M)\le 4$ or ${\beta }_1(M)=6$. Finally we use TOP surgery to propose a characterization of the simplest embeddings of $F\times S^1$.

Extending existing work in small dimensions, Dessai computed the Euler characteristic, signature, and elliptic genus for $8$-manifolds of positive sectional curvature in the presence of torus symmetry. He also computes the diffeomorphism type by restricting his results to classes of manifolds known to admit non-negative curvature, such as biquotients. The first part of this paper extends Dessai’s calculations to even dimensions up to $16$. In particular, we obtain a first characterization of the Cayley plane in such a setting. The second part studies a closely related family of manifolds called positively elliptic manifolds, and we prove a conjecture of Halperin in this context for dimensions up to $16$ or Euler characteristics up to $16$.

The authors generalize the works in [5] and [6] to prove a Hopf index theorem associated to a smooth section of a real vector bundle with non-isolated zero points.

In this paper we compute the Euler characteristic of the Selmer groups associated to modular forms over certain Kummer extensions of the field of rational numbers. We also discuss the Euler characteristic of Λ-adic deformations of Galois representations associated to modular forms.

In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.

By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all nondegenerate. For such Morse functions, Saeki and Yamamoto have previously defined a certain notion of cusp cobordism, and computed the unoriented cusp cobordism group of Morse functions on surfaces. In this paper, we compute unoriented and oriented cusp cobordism groups of Morse functions on manifolds of any dimension by employing Levine’s cusp elimination technique as well as the complementary process of creating pairs of cusps along fold lines. We show that both groups are cyclic of order two in even dimensions, and cyclic of infinite order in odd dimensions. For Morse functions on surfaces our result yields an explicit description of Saeki–Yamamoto’s cobordism invariant which they constructed by means of the cohomology of the universal complex of singular fibers.

A 2-rainbow dominating function (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2\}$ such that for any vertex $v\in V(G)$ with $f(v)=\varnothing$, the condition ${\bigcup }_{u\in N(v)}f(u)=\{1,2\}$ is fulfilled. The weight of a 2RDF $f$ is the value $\omega (f)={\sum }_{v\in V(G)}|f(v)|$. The $2$-rainbow domination number of a graph $G$, denoted by ${\gamma }_{r2}(G)$, is the minimum weight of a 2RDF of $G$. The rainbow bondage number $b_{r2}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $E^{\prime }\subseteq E(G)$ for which ${\gamma }_{r2}(G-E^{\prime })>{\gamma }_{r2}(G)$. Dehgardi, Sheikholeslami and Volkmann, [The $k$-rainbow bondage number of a graph, Discrete Appl. Math.174 (2014) 133–139] proved that the rainbow bondage number of a planar graph does not exceed 15. In this paper, we generalize their result for graphs which admit a $2$-cell embedding on a surface with non-negative Euler characteristic.

The vertex arboricity $\mathrm{\text{va}}(G)$ of a graph $G$ is the minimum number of colors the vertices of the graph $G$ can be colored so that every color class induces an acyclic subgraph of $G$. There are many results on the vertex arboricity of planar graphs. In this paper, we replace planar graphs with graphs which can be embedded in a surface $\mathrm{\Sigma }$ of Euler characteristic $\chi (\mathrm{\Sigma })\ge 0$. We prove that for the graph $G$ which can be embedded in a surface $\mathrm{\Sigma }$ of Euler characteristic $\chi (\mathrm{\Sigma })\ge 0$ if no $3$-cycle intersects a $5$-cycle, or no $5$-cycle intersects a $6$-cycle, then $\mathrm{\text{va}}(G)\le 2$ in addition to the $4$-regular quadrilateral mesh.

For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with fixed boundary conditions and prescribed singular pattern, by which we mean the combinatorial information of how the 1-dimensional singular locus meets the boundary. Each step of our algorithm modifies the given map only locally by either creating or eliminating a pair of cusps. We show that the number of cusps is an invariant modulo 2 and can be reduced to at most one, and we compute the minimal number of components of the singular locus in terms of the prescribed data. Applications include a discussion of pseudo-immersions as well as the computation of state sums in Banagl’s positive topological field theory.

An analysis of a number of modern experiments with nuclei having a deformation in the framework of the Riemannian geometry with positive and negative geodesic curvature is given.

We present an analytic proof of the Poincaré-Hopf index theorem. Our proof makes use of an old idea of Atiyah and works for the case where the isolated zeros of the vector field can be degenerate.