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In two previous papers [Katsanikas & Wiggins, 2021a, 2021b], we developed two methods for the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom. We applied the first method (see [Katsanikas & Wiggins, 2021a]) in the case of a quadratic Hamiltonian system in normal form with three degrees of freedom, constructing a geometrical object that is the section of a 4D toroidal structure in the 5D energy surface with the space x = 0. We provide a more detailed geometrical description of this object within the family of 4D toratopes. We proved that this object is a dividing surface and it has the no-recrossing property. In this paper, we extend the results for the case of the full 4D toroidal object in the 5D energy surface. Then we compute this toroidal object in the 5D energy surface of a coupled quadratic normal form Hamiltonian system with three degrees of freedom. [ABSTRACT FROM AUTHOR]

When we wanted to further investigate the results of Bourgain–Pavlović, Koch–Tataru and Li–Xiao–Yang, we encountered the problem of stability of the parameter flow. For initial data in some space X , norm inflation implies that the solution does not belong to C (X). For BMO − 1 , Auscher–Dubois–Tchamitchian proved that Koch–Tataru's solution is stable in their defined space C 0 . In this paper, we consider Koch–Tataru's iteration space and construct a non-Gauss flow function to show that C 0 -stability may have meaning different to L ∞ (( BMO − 1) n) , which supports Chemin and Gallagher's point: well-posedness and norm inflation may have no conflict. [ABSTRACT FROM AUTHOR]

Let G = 핊d, d = 0, 1 or 3, act freely on a finitistic connected space X. This paper gives the cohomology classification of X if the mod 2 or rational cohomology of the orbit space X/G is isomorphic to the product of a projective space and sphere 픽ℙn קm, where 픽 = ℝ, ℂ or ℍ, respectively. For a free involution on X, a lower bound of covering dimension of the coincidence set of a continuous map f : X → ℝk is also determined. [ABSTRACT FROM AUTHOR]

Let ℰ (X) denote the group of self-homotopy equivalences of an elliptic space X of formal dimension n. In this paper, we show that if ℰ (X) is finite, then ℰ (X) is isomorphic to a certain subgroup of ℰ (X n − 1). Moreover, we define the subgroup ℬ n (X) of aut (π n − 1 (X)) and we show that ℬ n (X) ≅ ℤ 2 provided that n is odd. Here X n − 1 denotes the (n − 1) -skeleton of X. [ABSTRACT FROM AUTHOR]

In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on C (X) to characterize spaces X when every irreducible ideal of C (X) is pseudoprime. We also characterize the rings of continuous functions C (X) in which every pseudoprime ideal is strongly irreducible. We give a negative answer to a question raised by Gilmer and McAdam. [ABSTRACT FROM AUTHOR]

Gálvez-Carrillo et al. [Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals, Adv. Math. 334 (2018) 544–584] constructed a decomposition space U of all Möbius intervals, as a recipient of Lawvere's interval construction for Möbius categories, and conjectured that U enjoys a certain universal property: for every Möbius decomposition space X , the space of culf functors from X to U is contractible. In this paper, we work at the level of homotopy 1-types to prove the first case of the conjecture, namely for locally discrete decomposition spaces. This provides also the first substantial evidence for the general conjecture. This case is general enough to cover all locally finite posets, Cartier–Foata monoids, Möbius categories and strict (directed) restriction species. The proof is 2-categorical. First, we construct a local strict model of U , which is then used to show by hand that the Lawvere interval construction, considered as a natural transformation, does not admit other self-modifications than the identity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stochastic Cahn–Hilliard–Navier–Stokes equations in two-dimensional spaces. Applying the Maslowski–Seidler method, we establish the existence of invariant measure in state space L x 2 × H 1 with the weak topology. We also prove the existence of global pathwise solutions using the stochastic compactness argument. [ABSTRACT FROM AUTHOR]

In this paper, we consider the spectral theory. More precisely, we study the spectral families and their corresponding operators on reflexive Banach spaces. Assume A is a well-bounded operator on reflexive Lebesgue spaces then the operator A is a scalar-type spectral operator. We establish that if a weak spectral family E (λ) is concentrated on [ a , b ] then there exists a linear well-bounded operator A ∈ L (X) on the reflexive Banach space X , such that 〈 A (x) , y * 〉 = b 〈 x , y * 〉 − ∫ [ a , b ] 〈 x , E (λ) y * 〉 d λ holds for all x ∈ X and y * ∈ X * . We prove that by assuming (Φ , X) is a functional calculus on the measurable space (Z , Σ) , then there are a semi-finite measure space (Ω , F , μ) and solitary operator U : X → L p (Ω , F , μ) , and an injective pointwise continuous *-homomorphism F : M (Z , Σ) → M (Ω , F) , such that Φ (f) = U M F f U − 1 holds for any operator of the multiplication M F f that multiplies by function f. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation phenomena in a diffusion system with the Belousov–Zhabotinskii chemical reaction in ℝ N under the bistability assumption. We prove that there is a new type of entire solution originated from three moving planar traveling fronts, and evolved to a V-shaped traveling front as time changes, which means that the profile of this solution is not invariant at all. Here an entire solution is referred to a solution that is defined for all time t  ∈  ℝ and in the whole space x  ∈  ℝ N . Furthermore, we show that not only the entire solution but also all transition fronts share the same global mean speed by constructing suitable radially symmetric expanding and retracting super- and subsolutions. [ABSTRACT FROM AUTHOR]

This note is of supplementary nature to our previous paper [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. Let S be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane ℙ 2 . Such S has appeared as the singular locus of certain hypergeometric system introduced in [J. Kaneko, K. Matsumoto and K. Ohara, A system of hypergeometric differential system in two variables of rank 9,Int. J. Math. 28 (2017) 1750015], and we have given generators and defining relations of the fundamental group of X = ℙ 2 ∖ S [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. X has a 9-fold Galois covering space X ̃ given by the complement of the Hesse configuration of 12 lines in ℙ 2 . Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of π 1 (X ̃) , which we carry out in this note. [ABSTRACT FROM AUTHOR]

This paper studies the bi-compact- G δ -open topology on the family C (X) of all continuous functions from a completely regular space X to the space ℝ. We compare this topology with open-point topology, bi-point-open topology and bi-compact-open topology and mention an example of a space asked in [P. Garg and S. Kundu, The compact- G δ -open topology on C (X) , Topology Appl. 159(8) (2012) 2082–2089] which satisfies C p (X) < C k z (X) < C k (X). We also investigate submertrizability, mertrizability, separability, Č ech-completeness and Baire property of bi-compact- G δ -open topology on C (X). [ABSTRACT FROM AUTHOR]

We characterize measure spaces such that the canonical map L ∞ → L 1 ∗ is surjective. In case of d -dimensional Hausdorff measure on a complete separable metric space X , we give two equivalent conditions. We give examples of X and d so that whether these conditions are met is undecidable in ZFC, including one with d equals the Hausdorff dimension of X. [ABSTRACT FROM AUTHOR]

Given N ≥ 3 , 1 < p < N , two measurable functions V (r) ≥ 0 and K (r) > 0 , and a continuous function A (r) > 0 (r > 0), we study the quasilinear elliptic equation − div (A (| x |) | ∇ u | p − 2 ∇ u) u + V (| x |) | u | p − 2 u = K (| x |) f (u) in ℝ N. We find existence of nonnegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space X into the sum of Lebesgue spaces L K q 1 + L K q 2 , and thus into L K q (= L K q + L K q ) as a particular case. Our results do not require any compatibility between how the potentials A , V and K behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of V and K , not of the potentials separately. The nonlinearity f has a double-power behavior, whose standard example is f (t) = min { t q 1 − 1 , t q 2 − 1 } , recovering the usual case of a single-power behavior when q 1 = q 2 . [ABSTRACT FROM AUTHOR]

A differential geometric statement of the noncommutative topological index theorem is worked out for covariant star products on noncommutative vector bundles. To start, a noncommutative manifold is considered as a product space X = Y × Z , wherein Y is a closed manifold, and Z is a flat Calabi–Yau m -fold. Also, a semi-conformally flat metric is considered for X which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of Z , the noncommutative star product is defined covariantly on vector bundles over X. This covariant star product leads to the celebrated Groenewold–Moyal product for trivial vector bundles and their flat connections, such as C ∞ (X). Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern–Weil theory is established by considering the covariant star product and the superconnection formalism. Finally, the index of the ⋆-noncommutative version of elliptic operators is studied and the noncommutative topological index theorem is stated accordingly. [ABSTRACT FROM AUTHOR]

We enhance the quandle counting invariants of oriented classical and virtual links using a construction similar to quandle modules but inspired by symplectic quandle operations rather than Alexander quandle operations. Given a finite quandle X and a vector space V over a field, sets of bilinear forms on V indexed by pairs of elements of X satisfying certain conditions yield new enhanced multiset- and polynomial-valued invariants of oriented classical and virtual links. We provide examples to illustrate the computation of the invariants and to show that the enhancement is proper. [ABSTRACT FROM AUTHOR]

Suppose a sequence M j of Alexandrov spaces collapses to a space X with only weak singularities. Yamaguchi constructed a map f j : M j → X called an almost Lipschitz submersion for large j. We prove that if M j has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of X , then f j is a locally trivial fibration. Moreover, we show some properties on the intrinsic metric and the volume of the fibers of f j . [ABSTRACT FROM AUTHOR]

Let G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris–Rips and Čech complexes of X/G. Whereas (co) homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes. [ABSTRACT FROM AUTHOR]

We introduce the space X of quaternion Hermitian forms of size n on a -adic field with odd residual characteristic, and define typical spherical functions ω (x ; s) on X and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to S n , and define a spherical Fourier transform on the Schwartz space (∖) which is Hecke algebra ℋ (,) -injective map into the symmetric Laurent polynomial ring of size n. Then, we determine the explicit formulas of ω (x ; s) by a method of the author's former result. In the last section, we give precise generators of (∖) and determine all the spherical functions for n ≤ 4 , and give the Plancherel formula for n = 2. [ABSTRACT FROM AUTHOR]

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location x , an individual must have a trait close to some optimal trait y opt (x). Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with y opt (x) = (x) , 0 < | | ≪ 1. We construct steady states solutions and, when is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable y we take advantage of a Hilbert basis of L 2 (ℝ) made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable x we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights. [ABSTRACT FROM AUTHOR]