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In this paper, motivated by recent work of Schnider and Axelrod-Freed and Soberón, we study an extension of the classical Grünbaum-Hadwiger-Ramos mass partition problem to mass assignments. Using the Fadell-Husseini index theory we prove that for a given family of j mass assignments µ1, . . ., µj on the Grassmann manifold Gl(Rd) and a given integer k ≥ 1 there exist a linear subspace L ∈Gl(Rd) and k affine hyperplanes in L that equipart the masses µ L 1, . . ., µL j assigned to the subspace L, provided that d ≥ j + (2k−1 − 1)2[log2 j] [ABSTRACT FROM AUTHOR]

The Conley index theory is a powerful topological tool for describing the basic structure of dynamical systems. One important feature of this theory is the attractor-repeller decomposition of isolated invariant sets. In this decomposition, all points in the invariant set belong to the attractor, its associated dual repeller, or a connecting region. In this connecting region, points tend towards the attractor in forwards time and the repeller in backwards time. This decomposition is also, in a certain topological sense, stable under perturbation. Conley theory is well-developed for flows and homeomorphisms, and has also been extended to some more abstract settings such as semiflows and relations. In this paper we aim to extend the attractor-repeller decomposition, including its stability under perturbation, to continuous time set-valued dynamical systems. The most common of these systems are differential inclusions such as Filippov systems. Of particular importance for this generalization is the perturbation of Filippov systems to nearby smooth systems. [ABSTRACT FROM AUTHOR]

Conley index theory is a powerful topological tool for obtaining information about invariant sets in dynamical systems. A key feature of Conley theory is that the index is robust under perturbation; given a continuous family of flows {φλ}, the index remains constant over a range of parameter values, avoiding many of the complications associated with bifurcations. This theory is well-developed for flows and homeomorphisms, and has even been extended to certain classes of semiflows. However, in recent years mathematicians and scientists have become interested in differential inclusions. Here the theory has also been studied for inclusions which satisfy certain bounding properties. In this paper we extend some of these results–in particular, the stability of isolating neighbourhoods under perturbation–to inclusions which do not satisfy these bounding properties. We do so by utilizing a novel approach to the solution set of differential inclusions which results in an object called a multiflow. This perspective allows us to relax the assumptions of the earlier work and also to develop tools needed to extend the continuation of Conley’s attractor-repeller decomposition to differential inclusions, a result which is addressed in subsequent work. Our interest in these results is in the study of piecewise-continuous differential equations–which are typically reframed as a certain type of differential inclusion called Filippov systems–and how these discontinuous equations relate to families of smooth systems which limit to them. Therefore this paper also discusses in some detail how the generalization of Conley index theory applies to Filippov systems. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices then the c-dominating matrix of G is the square matrix of order n whose (i, j)-entry is equal to 1 if the i-th and j-th vertex of G are adjacent, is also equal to 1 if the i = j, vi 2 Dc and zero otherwise. The c-dominating energy of a graph G, is defined as the sum of the absolute values of all eigenvalues the c-dominating matrix. The main purposes of this paper are to introduce the c-dominating Estrada index of a graph. Moreover, obtain upper and lower bounds for thec-dominating Estrada index and investigate the relations between the c-dominating Estrada index and the c-dominating energy. [ABSTRACT FROM AUTHOR]

It seems that the index theory for non-compact spaces has found its ultimate formulation in the realm of coarse spaces and K -theory of related operator algebras. Relative and partitioned index theorems may be mentioned as two important and interesting examples of this program. In this paper we formulate a combination of these two theorems and establish a partitioned-relative index theorem. [ABSTRACT FROM AUTHOR]

This paper concerns the following nonlinear Choquard equation:−ε2Δw + V(x)w = ε−θW(x) (Iθ ∗ (W|w|p))|w|p−2w, x ∈ RN, (∗) where ε > 0, N > 2, Iθ is the Riesz potential with order θ ∈ (0,N), p ∈ [2, N+θ/N−2), min V > 0 and infW > 0. Under proper assumptions, we explore the existence, concentration, convergence and decay estimate of semiclassical solutions for (∗). The multiplicity of solutions is established via pseudo-index theory. The existence of sign-changing solutions is achieved by minimizing the energy on Nehari nodal set. [ABSTRACT FROM AUTHOR]

In the age of globalisation, implementation and commercialisation of new technologies are perceived as key elements determining competitiveness of particular countries, therefore, the growth of innovativeness is seen as the predominant direction of European Union society’s transformation into information society. The aim of the paper is to propose a procedure of measurement of innovativeness growth over time, with the Summary Innovation Index (SII) methodology as a starting point. The considered issue can be expressed by the following main question: how to measure the innovation performance dynamics for a selected group of countries (such as the EU- 28, EU-15 or EU-13 countries) and for time intervals (not only for two moments of observations). This is an important inquiry since well-known innovativeness indices (SII, GII, or IOI) concentrate mainly on the provision of information about countries’ innovation performance for a specific year of observations. Due to this fact, changes occurring over longer time periods are rather neglected. The main result of the paper is a proposition of average innovativeness growth index. The index uses weights describing the employment share of a selected group of specialists (e.g.: scientists and engineers, research and development personnel) in relation to the economically active population. [ABSTRACT FROM AUTHOR]

This article presents the definition of Markov chain indexed by homogeneous trees in Markovian environment. Then we mainly study the strong limit theorems for a Markov chain indexed by homogeneous trees in Markovian environment. We also establish the strong law of large numbers and the Shannon-McMillan theorems for finite Markov chains indexed by a homogeneous tree in a Markovian environment with finite state space. We only prove the results on a Bethe tree and then just state the analogous results on a rooted Cayley tree. There are abundant achievements in research of Markov chains in determined environments, but few results about this topic of Markov chains indexed by trees in random environment. The results of this manuscript are very meaningful to give a good start to establish the strong limit properties for Markov chains indexed by trees in different random environments. [ABSTRACT FROM AUTHOR]

In this paper, we study the ℤ / 2 {\mathbb{Z}/2} action on the real Grassmann manifolds G n ⁢ (ℝ 2 ⁢ n) {G_{n}(\mathbb{R}^{2n})} and G ~ n ⁢ (ℝ 2 ⁢ n) {\widetilde{G}_{n}(\mathbb{R}^{2n})} given by taking the (appropriately oriented) orthogonal complement. We completely evaluate the related ℤ / 2 {\mathbb{Z}/2} Fadell–Husseini index utilizing a novel computation of the Stiefel–Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For n = 2 a ⁢ (2 ⁢ b + 1) {n=2^{a}(2b+1)} , k = 2 a + 1 - 1 {k=2^{a+1}-1} , a convex body C in ℝ 2 ⁢ n {\mathbb{R}^{2n}} , and k real-valued functions α 1 , ... , α k {\alpha_{1},\ldots,\alpha_{k}} continuous on convex bodies in ℝ 2 ⁢ n {\mathbb{R}^{2n}} with respect to the Hausdorff metric, there exists a subspace V ⊆ ℝ 2 ⁢ n {V\subseteq\mathbb{R}^{2n}} such that projections of C to V and its orthogonal complement V ⊥ {V^{\perp}} have the same value with respect to each function α i {\alpha_{i}} , that is, α i ⁢ (p V ⁢ (C)) = α i ⁢ (p V ⊥ ⁢ (C)) {\alpha_{i}(p_{V}(C))=\alpha_{i}(p_{V^{\perp}}(C))} for all 1 ≤ i ≤ k {1\leq i\leq k}. [ABSTRACT FROM AUTHOR]

The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the ℤ \mathbb{Z} /3 ℤ \mathbb{Z} symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda–Haagerup subfactor which emerged from our study of its Brauer–Picard groupoid. More specifically, we construct a new subfactor 𝒮 {\mathcal{S}} which is a ℤ \mathbb{Z} /4 ℤ \mathbb{Z} × \times ℤ \mathbb{Z} /2 ℤ \mathbb{Z} analogue of the Haagerup subfactor and we show that the even parts of the Asaeda–Haagerup subfactor are higher Morita equivalent to an orbifold quotient of 𝒮 {\mathcal{S}}. This gives a new construction of the Asaeda–Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda–Haagerup subfactor: calculating its Drinfeld center, classifying all extensions of the Asaeda–Haagerup fusion categories, finding the full higher Morita equivalence class of the Asaeda–Haagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda–Haagerup categories. The details of the applications will be given in subsequent papers. [ABSTRACT FROM AUTHOR]

A food-chain model with Crowley-Martin functional response in the unstirred chemostat is considered. First, the global framework of coexistence solutions is discussed by the maximum principle and bifurcation theory. We obtain the sufficient and necessary conditions for coexistence of steadystate. Second, the stability and uniqueness of coexistence solutions are investigated by means of the combination of the perturbation theory and fixed point index theory. Our results indicate that if the magnitude of interference among predator is sufficiently large, the model has only one unique linearly stable coexistence solution when the maximal growth rate of predator belongs to certain range. Finally, some numerical simulations are carried out to verify and complement the theoretical results. [ABSTRACT FROM AUTHOR]

In this letter, generalized frequency division multiplexing (GFDM) is combined with index modulation (IM) exploiting the block-based structure of GFDM in an innovative way. The proposed GFDM-IM scheme uses flexible IM numerology to reduce out-of-band (OOB) emission and provides a novel multilayer system model to further benefit from the enhanced distance spectrum of IM. The proposed scheme adapts maximum likelihood successive interference cancellation based near-optimum detector to improve the error performance and reduce the computational complexity. It has been shown that the proposed GFDM-IM scheme achieves better error performance than classical and IM-based orthogonal frequency division multiplexing and GFDM schemes at the same spectral efficiency, and provides an interesting tradeoff among OOB emission, spectral efficiency, and latency. [ABSTRACT FROM AUTHOR]

We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit construction of a geometric cobordism bicategory which incorporates general background fields in a stack, and together with the theory of symmetric monoidal bicategories we use it to provide the concrete forms of invertible extended quantum field theories which capture anomalies in both the path integral and Hamiltonian frameworks. Specialising this situation by using the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose, we obtain a new Hamiltonian perspective on the parity anomaly. We compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space, which is defined in a parity-symmetric way by suitably augmenting the standard chiral fermionic Fock spaces with Lagrangian subspaces of zero modes of the Dirac Hamiltonian that naturally appear in the index theorem. We describe the significance of our constructions for the bulk-boundary correspondence in a large class of time-reversal invariant gauge-gravity symmetry-protected topological phases of quantum matter with gapless charged boundary fermions, including the standard topological insulator in 3 + 1 dimensions. [ABSTRACT FROM AUTHOR]

We consider the following parabolic system whose nonlinearity has no gradient structure: { ∂ t u = Δ u + | v | p − 1 v , ∂ t v = μ Δ v + | u | q − 1 u , u ( ⋅ , 0 ) = u 0 , v ( ⋅ , 0 ) = v 0 , in the whole space R N , where p , q > 1 and μ > 0 . We show the existence of initial data such that the corresponding solution to this system blows up in finite time T ( u 0 , v 0 ) simultaneously in u and v only at one blowup point a , according to the following asymptotic dynamics: { u ( x , t ) ∼ Γ [ ( T − t ) ( 1 + b | x − a | 2 ( T − t ) | log ⁡ ( T − t ) | ) ] − ( p + 1 ) p q − 1 , v ( x , t ) ∼ γ [ ( T − t ) ( 1 + b | x − a | 2 ( T − t ) | log ⁡ ( T − t ) | ) ] − ( q + 1 ) p q − 1 , with b = b ( p , q , μ ) > 0 and ( Γ , γ ) = ( Γ ( p , q ) , γ ( p , q ) ) . The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case μ = 1 ; and the fact that the case μ ≠ 1 breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data. [ABSTRACT FROM AUTHOR]

In the variational study of singular Lagrange systems, the zero energy solutions play an important role. The anisotropic Kepler problem is such a singular system introduced by physicist M. Gutzwiller to reveal connections between classic and quantum chaos. In this paper we find a simple way of computing the Morse indices of the zero solutions in this problem. In particular, we show the Morse indices are connected with the oscillating behaviors of these solutions. Our results can also be applied to some problems in celestial mechanics. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we use the Limit Index Theory due to Li and the fractional version of concentration compactness principle to study the multiplicity of solutions for a class of noncooperative fractional p‐Laplacian elliptic system with homogeneous Dirichlet boundary conditions involving the critical exponents. [ABSTRACT FROM AUTHOR]

We obtain new results on the existence and multiplicity of fixed points of Hammerstein equations in very general cones. In order to achieve this, we combine a new formulation of cones in terms of continuous functionals with fixed point index theory. Many examples and an application to boundary value problems are also included. [ABSTRACT FROM AUTHOR]

We consider the following parabolic system whose nonlinearity has no gradient structure: { ∂ t u = Δ u + e p v , ∂ t v = μ Δ v + e q u , u ( ⋅ , 0 ) = u 0 , v ( ⋅ , 0 ) = v 0 , p , q , μ > 0 , in the whole space R N . We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem. [ABSTRACT FROM AUTHOR]

We construct a solution for the Complex Ginzburg-Landau equation in a critical case which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows us to prove the stability of the constructed solution. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n vertices and let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 be the Laplacian eigenvalues of G . The k-th Laplacian spectral moment of G is defined to be L M k ( G ) = ∑ i = 1 n μ i k ( G ) , where k is a non-negative integer; and the Laplacian Estrada index of G is defined as L E E ( G ) = ∑ i = 1 n e μ i . In this paper, we first estimate these two indices for almost all graphs, and then we give lower and upper bounds to these two indices for almost all multipartite graphs. [ABSTRACT FROM AUTHOR]

In this paper we provide a detailed description of the K-theory torsor constructed by S. Saito for a Tate R-module, and its analogue for general idempotent complete exact categories. We study the classifying map of this torsor in detail, construct an explicit simplicial model, and link it to the index theory of Fredholm operators. The torsor is also related to canonical central extensions of loop groups. More precisely, we compare the K-theory torsor to previously studied dimension and determinant torsors. [ABSTRACT FROM AUTHOR]

We investigate the existence and multiplicity of positive solutions for a system of Riemann-Liouville fractional differential equations, subject to multipoint boundary conditions that contain fractional derivatives, by using some theorems from the fixed point index theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following fourth-order three-point boundary value problem ... where n ∊ [1/3, 1). In spite of sign-changing Green's function, for arbitrary positive integer n (≥ 2), we still obtain the existence of at least n - 1 decreasing positive solutions to the above problem by imposing some suitable conditions on f . The main tool used is the fixed point index theory. [ABSTRACT FROM AUTHOR]

In this paper, we present a new index called min-max distance degree index, for molecular graphs. It appears that this index correlates well with properties of 33 and 28 nonanes like, energy of formation and density, and its correlation coefficient is 0.812 and 0.754, respectively. The basic mathematical features and graph operations are presented. In addition, the linear regression analysis of predicted and experimental boiling point of acyclic hydrocarbons is shown, where predicted boiling point is determined by implementing this index with Wiener's model. [ABSTRACT FROM AUTHOR]

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors' knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrized by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrized by bounded intervals. [ABSTRACT FROM AUTHOR]

We show that the bulk-edge correspondence for two-dimensional type A and type topological insulators follows directly from the cobordism invariance of the index. [ABSTRACT FROM AUTHOR]

We give a heat kernel proof of the algebraic index theorem for deformation quantization with separation of variables on a pseudo-Kähler manifold. We use normalizations of the canonical trace density of a star product and of the characteristic classes involved in the index formula for which this formula contains no extra constant factors. [ABSTRACT FROM AUTHOR]

In this article we present a unification of the theory of algebraic, singular, topological and directional transition matrices by introducing the (generalized) transition matrix which encompasses each of the previous four. Some transition matrix existence results are presented as well as verification that each of the previous transition matrices are cases of the (generalized) transition matrix. Furthermore we address how applications of the previous transition matrices to the Conley index theory carry over to the (generalized) transition matrix. [ABSTRACT FROM AUTHOR]

Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the Batalin–Vilkovisky (BV) quantization of a one-dimensional sigma model with target X . This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello's homotopic theory of effective renormalization. We show that Fedosov's Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the author discusses articles in the issue on topics including mathematics education, index theory and mathematics research.

We investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions of a predator-prey model with Dinosaur functional response are established. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied when b and d are small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when k>0 is sufficiently large. [ABSTRACT FROM AUTHOR]

Motivation to revisit the Conley index theory for discrete multivalued dynamical systems stems from the needs of broader real applications, in particular in sampled dynamics or in combinatorial dynamics. The new construction of the index in [B. Batko and M. Mrozek, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1143-1162] based on weak index pairs, under the circumstances of the absence of index pairs caused by relaxing the isolation property, seems to be a promising step toward this direction. The present paper is a direct continuation of [B. Batko and M. Mrozek, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1143-1162] and concerns properties of the index defined therin, namely, the Ważewski property, the additivity property, the homotopy (continuation) property, and the commutativity property. We also present the construction of weak index pairs in an isolating block. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for the noncooperative Schrödinger-Kirchhoff-type system involving the fractional p-Laplacian and critical nonlinearities in $${\mathbb {R}}^{N}$$ . By applying the Limit Index Theory due to Li (Nonlinear Anal 25:1371-1389, 1995) and the fractional version of concentration-compactness principle, we obtain the existence and multiplicity of solutions for the above systems under some suitable assumptions. To our best knowledge, it seems that this is the first time to exploit the existence of solutions for the noncooperative Schrödinger-Kirchhoff-type system involving the fractional p-Laplacian and critical nonlinearity in $${\mathbb {R}}^{N}$$ . [ABSTRACT FROM AUTHOR]

The Narumi-Katayama index of a graph was introduced in 1984 for representing the carbon skeleton of a saturated hydrocarbons and is defined as the product of degrees of all the vertices of the graph. In this paper, we examine the Narumi-Katayama index of different total transformation graphs. [ABSTRACT FROM AUTHOR]

We consider what some authors call “parabolic Möbius subgroups” of matrices over Z , Q , and R and focus on the membership problem in these subgroups and complexity of relevant algorithms. [ABSTRACT FROM AUTHOR]

In this article, we establish a new index theory defined for the general non-degenerate matrix paths in GL + ( 2 ) . This is done by the complete homotopy classification for such paths. The parity relation theorem is established for relating this index to the Morse index of the corresponding differential operator. As an application, we establish the existence result of a non-trivial periodic solution of the asymptotically linear differential equation systems under an index twist condition. [ABSTRACT FROM AUTHOR]

We investigate solutions to nonlinear operator equations which are difficult to investigate with variational methods and obtain some abstract existence results by topology degree methods. These results apply to ordinary differential systems with impulsive effects satisfying nonlinear boundary value conditions, and we obtain some new results. [ABSTRACT FROM AUTHOR]

Spatial Air Index (SAI) has been proposed for improving query performance of k-nearest neighbour queries in road networks. SAI has been effectively utilized the usage of Adaptive Cooperative Caching (ACC) and reduced search space. Experiments have been conducted for evaluated query result, the experimental result show that SAI outperform compared to state-of-the-art Network Partition Index (NPI). [ABSTRACT FROM AUTHOR]

Objective of the present work is to improve range query performance using Hybrid Spatial Air Index (HSAI). HSAI has been designed with combination of both cache management and network coding for processing range queries in road networks. HSAI has been utilized the advantage of both cache management and network coding and reduce client search space. The experiments have been conducted for evaluating performance, the experimental results show that HSAI outperform. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for every irreversible Finsler n -dimensional real projective space ( R P n , F ) with reversibility λ and flag curvature K satisfying 16 9 ( λ 1 + λ ) 2 < K ≤ 1 with λ < 3 , there exist at least n − 1 non-contractible closed geodesics. In addition, if the metric F is bumpy with 64 25 ( λ 1 + λ ) 2 < K ≤ 1 and λ < 5 3 , then there exist at least 2 [ n + 1 2 ] non-contractible closed geodesics, which is the optimal lower bound due to Katok's example. The main ingredients of the proofs are the Fadell–Rabinowitz index theory of non-contractible closed geodesics on ( R P n , F ) and the S 1 -equivariant Poincaré series of the non-contractible component of the free loop space on R P n . [ABSTRACT FROM AUTHOR]