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This paper is concerned with existence, non-existence and uniqueness of positive (coexistence) steady states to a predator-prey system with prey-dependent dispersal. To overcome the analytical obstacle caused by the cross-diffusion structure embedded in the prey-dependent dispersal, we use a variable transformation to convert the problem into an elliptic system without cross-diffusion structure. The transformed system and pre-transformed system are equivalent in terms of the existence or non-existence of positive solutions. Then we employ the index theory alongside the method of the principle eigenvalue to give a nearly complete classification for the existence and non-existence of positive solutions. Furthermore we show the uniqueness of positive solutions and characterize the asymptotic profile of solutions for small or large diffusion rates of species. Our results pinpoint the positive role of prey-dependent dispersal on the population dynamics for the first time by showing that the prey-dependent dispersal in the predator-prey system is a beneficial strategy increasing the chance of predator's survival and hence promoting the coexistence of species. [ABSTRACT FROM AUTHOR]

We introduce the topologically twisted index for four-dimensional N = 1 gauge theories quantized on AdS 2 × S 1 . We compute the index by applying supersymmetric localization to partition functions of vector and chiral multiplets on AdS 2 × T 2 , with and without a boundary: in both instances we classify normalizability and boundary conditions for gauge, matter and ghost fields. The index is twisted as the dynamical fields are coupled to a R-symmetry background 1-form with non-trivial exterior derivative and proportional to the spin connection. After regularization, the index is written in terms of elliptic gamma functions, reminiscent of four-dimensional holomorphic blocks, and crucially depends on the R-charge. [ABSTRACT FROM AUTHOR]

The aim of this study was to compare selection indices for important traits in intensive Spanish goat breeds in four economic scenarios, using the Florida as most representative breed of this production system in Spain. For this analysis, we considered the following traits: milk yield (MY), fat plus protein yields (FPY), casein yield (CY), somatic cell score (SCS), reproductive efficiency (RE), litter size (LS), mammary system (MS), final score (FS), body capacity index (BCI), and length of productive life (LPL). We estimated the genetic parameters and EBVs of most of these traits with REML methodology, while LPL was modeled through survival analysis. Four scenarios were proposed, depending on the overall objective for improvement: (1) milk production, (2) milk production and cheese extract, (3) cheese extract, and (4) milk production, cheese extract and sale of animals. Then, within each scenario, three different types of indices were designed using the different primary and secondary objectives/criteria considered suitable to improve the overall objective. The results indicated that selecting only for primary traits yielded the highest genetic response for all the scenarios. Including secondary traits led to positive correlated responses in those traits, but a decrease in the responses in the primary criteria. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the existence of radial solutions for a class of biharmonic equation with a convection term, involving two real parameters A and p. We mainly use a combination of the fixed point index theory and the Banach contraction theorem to prove that there are A0 > 0 and p0 > 0 such the equation admits at least one radial solution for all (A, p) e [-Ao, x [0, p0]. [ABSTRACT FROM AUTHOR]

The aim of this study was to compare selection indices for important traits in intensive Spanish goat breeds in four economic scenarios, using the Florida as most representative breed of this production system in Spain. For this analysis, we considered the following traits: milk yield (MY), fat plus protein yields (FPY), casein yield (CY), somatic cell score (SCS), reproductive efficiency (RE), litter size (LS), mammary system (MS), final score (FS), body capacity index (BCI), and length of productive life (LPL). We estimated the genetic parameters and EBVs of most of these traits with REML methodology, while LPL was modeled through survival analysis. Four scenarios were proposed, depending on the overall objective for improvement: (1) milk production, (2) milk production and cheese extract, (3) cheese extract, and (4) milk production, cheese extract and sale of animals. Then, within each scenario, three different types of indices were designed using the different primary and secondary objectives/criteria considered suitable to improve the overall objective. The results indicated that selecting only for primary traits yielded the highest genetic response for all the scenarios. Including secondary traits led to positive correlated responses in those traits, but a decrease in the responses in the primary criteria.

We describe the structure of diffeological bundle of non formal classical pseudo-differential operators over formal ones, and its structure group. For this, we give results on diffeological principal bundles with (a priori) no local trivialization including an Ambrose-Singer theorem, use the smoothing connections alrealy exhibited by the author in previous works, and finish with open questions.

Given a manifold with corners X, we associate to it the corner structure simplicial complex Σx. Its reduced K-homology is isomorphic to the K-theory of the C *-algebra Kb (X) of b-compact operators on X. Moreover, the homology of Σx is isomorphic to the conormal homology of X. In this note, we construct for an arbitrary abstract finite simplicial complex E a manifold with corners X such that Σx ≅ Σ. As a consequence, the homology and K-homology which occur for finite simplicial complexes also occur as conormal homology of manifolds with corners and as K-theory of their b-compact operators. In particular, these groups can contain torsion. [ABSTRACT FROM AUTHOR]

We develop index and degree theories for extensive form games allowing the identification of equilibria that are robust to payoff perturbations directly from the extensive form. Our approach is based on index and degree theories for games where the strategy sets are polytopes (and not necessarily simplices) and payoff functions are multiaffine. Polytope strategy sets arise naturally from topologically identifying equivalent mixed strategies of a normal form game. [ABSTRACT FROM AUTHOR]

Let B be a nonunital separable simple Jiang-Su-stable C*-algebra with stable rank one. We show that M(B) is the closed linear span of its projections, which implies Property I for B. We also show that the corona algebra C (B) is properly infinite if and only if T(B) is weak* compact. We also provide a number of other equivalent characterizations. [ABSTRACT FROM AUTHOR]

We apply the Atiyah-Singer index theorem and tensor products of elliptic complexes to the cohomology of transitive Lie algebroids. We prove that the Euler characteristic of a representation of a transitive Lie algebroid A over a compact manifold M vanishes unless A D TM, and prove a general Künneth formula. As applications, we give a short proof of a vanishing result for the Euler characteristic of a principal bundle calculated using invariant differential forms, and show that the cohomology of certain Lie algebroids are exterior algebras. The latter result can be seen as a generalization of Hopf's theorem regarding the cohomology of compact Lie groups. [ABSTRACT FROM AUTHOR]

The main purpose of this manuscript is to investigate the Sturm-Liouville BVP for non-autonomous Lagrangian systems. Under the suitable assumptions, we establish an existence theorem for three nonnegative solutions via Bonanno-Candito's three critical point theory. As an application in the complete Sturm-Liouville equations with Sturm-Liouville BVC, we get an existence theorem of three nonnegative solutions. Meanwhile, we give three examples to show the correctness of our results.

Abstract This paper is concerned with the existence of periodic solutions for asymptotically linear second-order delay differential equations. We will establish an index theory for the linear system directly in the sense that we do not need to change the problem of the original linear system into the problem of an associated Hamiltonian system. By using the critical point theory and the index theory, some new existence results are obtained.

We describe the structure of diffeological bundle of non formal classical pseudo-differential operators over formal ones, and its structure group. For this, we give results on diffeological principal bundles with (a priori) no local trivialization including an Ambrose-Singer theorem, use the smoothing connections alrealy exhibited by the author in previous works, and finish with open questions. [ABSTRACT FROM AUTHOR]

The weighted arithmetic mean is used in a wide variety of applications. An infinite number of possible decompositions of the change in the weighted mean are available, and it is therefore an open question which of the possible decompositions should be applied. In this article, we derive a decomposition of the change in the weighted mean based on a two-stage Bennet decomposition. Our proposed decomposition is easy to employ and interpret, and we show that it satisfies the difference counterpart to the index number time reversal test. We illustrate the framework by decomposing aggregate earnings growth from 2020Q4 to 2021Q4 in Norway and compare it with some of the main decompositions proposed in the literature. We find that the wedge between the identified compositional effects from the proposed two-stage Bennet decomposition and the one-stage Bennet decomposition is substantial, and for some industries, the compositional effects have opposite signs. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of periodic solutions for asymptotically linear second-order delay differential equations. We will establish an index theory for the linear system directly in the sense that we do not need to change the problem of the original linear system into the problem of an associated Hamiltonian system. By using the critical point theory and the index theory, some new existence results are obtained. [ABSTRACT FROM AUTHOR]

On filtered manifolds one can define a different notion of order for the differential operators. In this paper, we use generalized fixed point algebras to construct a pseudodifferential extension that reflects this behaviour. In the corresponding calculus, the principal symbol of an operator is a family of operators acting on certain nilpotent Lie groups. The role of ellipticity as a Fredholm condition is replaced by the Rockland condition on these groups. Our approach allows to understand this in terms of the representations of the corresponding algebra of principal symbols. Moreover, we compute the K-theory of this algebra. [ABSTRACT FROM AUTHOR]

The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov–Lawson for generalised Dirac operators as well as the result of Bär–Ballmann for Dirac-type operators. The theorem is seen through the point of view of boundary value problems, using the graphical decomposition of elliptically regular boundary conditions for general first-order elliptic operators due to Bär–Bandara. Splitting, decomposition and the Phi-relative index theorem are proved on route to the relative index theorem. [ABSTRACT FROM AUTHOR]

We study the Maslov-type index theory under a special Lagrangian boundary condition, namely (P , L 0) boundary condition, which comes naturally in the study of brake orbits of P -invariant Hamiltonian systems with certain orthogonal symplectic matrix P satisfying P p = I d. In this paper, we give some new iteration inequalities of Maslov-type index under this boundary condition. As applications, we consider the minimal cyclic period problems for brake orbits of first order nonlinear autonomous reversible P -symmetric Hamiltonian systems. We prove that if P = R (θ 1) ⋄ ... ⋄ R (θ n) and θ i ∈ [ 0 , π ] for each 1 ≤ i ≤ n , and the Hamiltonian is strictly convex and superquadratic at zero and infinity, then for each T > 0 , the Hamiltonian system possesses a nonconstant P -cyclic brake orbit with minimal P -cyclic period belonging to { p T , p T p + 1 }. For P = R (− 2 π p) ⋄ n and p large enough, the system possesses a nonconstant P -cyclic brake orbit with prescribed minimal P -cyclic period. [ABSTRACT FROM AUTHOR]

The paper presents a first step towards a family index theorem for classical self-adjoint boundary value problems.We address here the simplest non-trivial case of manifolds with boundary, namely the case of two-dimensional manifolds. The first result of the paper is an index theorem for families of first order self-adjoint elliptic differential operators with local boundary conditions, parametrized by points of a compact topological space X. We compute the K(X)-valued index in terms of the topological data over the boundary. The second result is the universality of the index: we show that the index is a universal additive homotopy invariant for such families if the vanishing on families of invertible operators is assumed. [ABSTRACT FROM AUTHOR]

Given an isometric action of a discrete group on a compact manifold with boundary and a -invariant elliptic boundary value problem on , we consider its twisting by projections over the crossed product algebra . The twisted problem is Fredholm and we compute its index in terms of the equivariant Chern character of the principal symbol of and a noncommutative Chern character of . In the special case, when is the Dirichlet problem for the Euler operator, the index is expressed as a linear combination of the Euler characteristics of the fixed point submanifolds of the group action. [ABSTRACT FROM AUTHOR]

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of S 7 -bundles over S 8 which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional curvature on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an infinite family of manifolds, the moduli space of positive Ricci curvature metrics has infinitely many path components. [ABSTRACT FROM AUTHOR]

In this paper, we establish a noncommutative refinement for the Z / k Z -index formula of Atiyah-Patodi-Singer (Math. Proc. Cambridge Philos. Soc. 79, 71-99 (1976)) in the case of Spin c Dirac operators. [ABSTRACT FROM AUTHOR]

The present research investigates the following delayed problem (0.1) ν′′+a(t)f(t,νt)=0,0

This article concerns topologically non-trivial interface Hamiltonians that find many applications in materials science and geophysical fluid flows. The non-trivial topology manifests itself in the existence of topologically protected, asymmetric edge states at the interface between two two-dimensional half spaces. The asymmetric transport is characterized by a quantized interface conductivity. The objective of this article is to compute such a conductivity and show its stability under perturbations. We present two methods. The first one computes the conductivity using the winding number of branches of absolutely continuous spectrum of the interface Hamiltonian. This calculation is independent of any bulk properties but requires a sufficient understanding of the spectral decomposition of the Hamiltonian. In the fluid flow setting, it also applies in cases where the so-called bulk-interface correspondence fails. The second method establishes a bulk-interface correspondence between the interface conductivity and a so-called bulk-difference invariant. We introduce the bulk-difference invariants characterizing pairs of half spaces. We then relate the interface conductivity to the bulk-difference invariant by means of a Fedosov–Hörmander formula, which computes the index of a related Fredholm operator and is obtained using semiclassical calculus. The two methods are used to compute invariants for representative 2 × 2 and 3 × 3 systems of equations that appear in the applications. [ABSTRACT FROM AUTHOR]

We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to Z d -actions on a compact space can be lifted to an isomorphism of their foliation C ⁎ -algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the C ⁎ -algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of normalized solutions to the following Schrödinger equations with potentials and non-autonomous nonlinearities: - Δ u + V (x) u + λ u = f (x , u) in R N , ∫ R N | u (x) | 2 d x = a , u ∈ H 1 (R N) , where V (x) ≤ lim | x | → ∞ V (x) : = V ∞ ∈ (- ∞ , + ∞ ] and f(x, s) satisfies Berestycki–Lions type conditions with mass sub-critical growth. In the case V ∞ = + ∞ , we prove that for all a > 0 , the equation has a ground state solution, and if additionally f is odd, the equation has infinitely many normalized solutions with increasing energy. While in the case V ∞ < + ∞ , we prove that there exists a 0 ≥ 0 such that the ground state energy can be attained when a > a 0 , but not when 0 < a < a 0 . To this end, We develop robust arguments to show the conditional strict subadditivity of the ground state energy with respect to a. We also investigate the multiplicity of normalized radial solutions by index theory in this case. These results can be extended to other types of Schrödinger equations. [ABSTRACT FROM AUTHOR]

We discuss an approach to the problem of calculating the local topological index of vector fields given on complex spaces or varieties with singularities developed by the author over the past few years. Our method is based on the study of the homology of a contravariant version of the classical Poincaré–de Rham complex. This idea allows not only simplifying the calculations, but also clarifying the meaning of the basic constructions underlying many papers on the subject. In particular, in the graded case, the index can be expressed explicitly in terms of the elementary symmetric polynomials. We also considered some useful applications in physics, mechanics, control theory, the theory of bifurcations, etc. [ABSTRACT FROM AUTHOR]

A product set of pure strategies is a Nash block if it contains all best replies to the Nash equilibria of the game in which the players are restricted to the strategies in the block. This defines an intermediate block property, between curb (Basu and Weibull, Econ Lett 36(2):141–146, https://doi.org/10.1016/0165-1765(91)90179-O, http://www.sciencedirect.com/science/article/pii/016517659190179O, 1991) and coarse tenability (Myerson and Weibull (2015) Econometrica 83(3):943–976, https://doi.org/10.3982/ECTA11048). While the new concept is defined without reference to the consideration-set framework that defines tenability, the framework can be used to characterize Nash blocks in terms of potential conventions when large populations of individuals recurrently interact. Although weaker than curb, Nash blocks nevertheless maintain several robustness properties of curb sets. For example, every Nash block contains an essential component and is robust against payoff perturbations. [ABSTRACT FROM AUTHOR]

In this paper we characterize perturbation ideals of sets that generate the familiar spectra in Fredholm theory.

It follows from Browder (Summa Bras Math 4:183–191, 1960) that for every continuous function F : (X × Y) → Y , where X is the unit interval and Y is a nonempty, convex, and compact subset of a locally convex linear vector space, the set of fixed points of F, defined by C F : = { (x , y) ∈ X × Y : F (x , y) = y } , has a connected component whose projection to the first coordinate is X. We extend Browder's result to the case that X is a connected and compact Hausdorff space. [ABSTRACT FROM AUTHOR]

In this paper we characterize perturbation ideals of sets that generate the familiar spectra in Fredholm theory. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates the existence of solutions to subquadratic operator equations with convex nonlinearities and resonance by means of the index theory for self-adjoint linear operators developed by Dong and dual least action principle developed by Clarke and Ekeland. Applying the results to subquadratic convex Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions, generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions yield some new theorems concerning the existence of solutions or nontrivial solutions. In particular, some famous results about solutions to subquadratic convex Hamiltonian systems by Mawhin and Willem and Ekeland are special cases of the theorems.

This paper investigates solutions for subquadratic convex or $B$-concave operator equations. First, some existence results are obtained by the index theory and the critical point theory. Then, some applications to second order Hamiltonian systems satisfying generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions are pointed out. In particular, some well known theorems about periodic solutions for second order Hamiltonian systems are special cases of these results.

In 1978, Rabinowitz proved the existence of a non-constant T -periodic solution for nonlinear Hamiltonian systems on R 2 n with Hamiltonian function being super-quadratic at the infinity and zero for any given T > 0. Since the minimal period of this solution may be T / k for some positive integer k , he proposed the question whether there exists a solution with T as its minimal period for such a Hamiltonian system. This is the so-called Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, this conjecture has been deeply studied by many mathematicians. But under the original structural conditions of Rabinowitz, the conjecture is still open when n ≥ 2. In this paper, I give a brief survey on the studies of this conjecture and hope to lead to more interests on it. [ABSTRACT FROM AUTHOR]

In this paper we investigate the nuclear trace of vector‐valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo‐differential operators. First we characterise the nuclearity of pseudo‐differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector‐valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector‐valued Fourier multiplier in terms of its operator‐valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic Hörmander classes. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stationary solutions of the Dirac equation − i ∑ k = 1 3 α k ∂ k u + a β u + ω u + V (x) u = G u (x , u) , where G is asymptotically quadratic and is not assumed to be C 2. We present a new approach to construct an index theory for the associated linear Dirac equation and define the relative Morse index to measure the difference between the nonlinearity at the origin and at infinity. By building upon the idea of combining the index theory and a generalized linking theorem, we obtain existence and multiplicity of stationary solutions. [ABSTRACT FROM AUTHOR]

Abstract We investigate the existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance by means of the index theory, the critical point theory without compactness assumptions, the least action principle, the saddle point reduction theorem, and the minimax method. Applying the results to second-order HS satisfying periodic boundary value conditions, we obtain some new results.

This paper investigates multiple solutions for super-linear symmetric operator equations by optimization methods like index theory, Mountain pass theorem and mini-max methods. Applying the results to second order Hamiltonian systems with symmetry yields finite or infinitely many solutions. [ABSTRACT FROM AUTHOR]

Various fields of economic analysis (e.g. growth and productivity) and economic policy (e.g. monetary and social policy) rely on accurate measures of price change. Unfortunately, the price index formulae that most price statisticians consider as particularly accurate—the superlative indices of Fisher, Törnqvist, and Walsh—are believed to violate the property of consistency in aggregation. This property, however, is indispensable for economic studies that attempt to disaggregate the overall result into the contributions of individual entities such as sectors of the economy or groups of products. The present paper introduces a thoroughly motivated formal definition of consistency in aggregation and proves that, contrary to general perception, the three superlative price indices can be considered as consistent in aggregation. Furthermore, many other price indices are shown to be consistent in aggregation. The theoretical findings are applied to the Swedish consumer price index. [ABSTRACT FROM AUTHOR]

We discuss an approach to the problem of calculating the local topological index of vector fields given on complex spaces or varieties with singularities developed by the author over the past few years. Our method is based on the study of the homology of a contravariant version of the classical Poincaré–de Rham complex. This idea allows not only simplifying the calculations, but also clarifying the meaning of the basic constructions underlying many papers on the subject. In particular, in the graded case, the index can be expressed explicitly in terms of the elementary symmetric polynomials. We also considered some useful applications in physics, mechanics, control theory, the theory of bifurcations, etc.

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert C ∗ -module bundles and prove an index theorem for such operators. As an application, we derive an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when N is a closed spin manifold, we show that if the cylinder N × ℝ carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on N must vanish. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of solutions to subquadratic operator equations with convex nonlinearities and resonance by means of the index theory for self-adjoint linear operators developed by Dong and dual least action principle developed by Clarke and Ekeland. Applying the results to subquadratic convex Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions, generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions yield some new theorems concerning the existence of solutions or nontrivial solutions. In particular, some famous results about solutions to subquadratic convex Hamiltonian systems by Mawhin and Willem and Ekeland are special cases of the theorems. [ABSTRACT FROM AUTHOR]

In this paper, we will define the index pair (i A (B) , ν A (B)) by the dual variational method, and show the relationship between the indices defined by different methods. As applications, we apply the index (i A (B) , ν A (B)) to study the existence and multiplicity of homoclinic orbits of nonlinear Hamiltonian systems and solutions of nonlinear Dirac equations. [ABSTRACT FROM AUTHOR]

We use the index formula for fibered boundaries to compute the LL2-index of the Dirac operator twisted by an Anti-Self-Dual instanton defined on X S1, where X is a complete asymptotically conical three-manifold. As a particular case of this calculation, we get another derivation of the index formula for the Dirac operator twisted by a caloron on R3 x S1. [ABSTRACT FROM AUTHOR]

In 2014, J. Pipan and M. Schechter discussed periodic solutions for second-order Hamiltonian systems with resonance. In this paper, we will generalize their results. To this end, we will first establish an index theory for second-order linear Hamiltonian systems with coefficient matrix in L 1 . Then we propose generalized assumptions. We also investigate multiple solutions for symmetric second-order Hamiltonian systems. [ABSTRACT FROM AUTHOR]