Volume 27, Issue 3

Let X be an aspherical polyhedron of the homotopy type of the figure-eight and let f : X → X be a self-map. The Wagner algorithm [Trans. Amer. Math. Soc. 351 (1999), 41–62] provides computations for the Nielsen number of self-maps of X satisfying the remnant condition. If f is without remnant, then using the concept of mutant by Jiang [Math. Ann. 311 (1998), 467–479] we may assume that f # ( b ) is an initial segment of f # ( a ), where f # is the induced endomorphism of π 1 ( X ) and a , b are generators of π 1 ( X ). Let f # ( b ) = U and f # ( a ) = U n R , where n is the maximal such positive integer. If R is not an initial segment of U , we say that f is of Type Y . In this paper, we prove that if f is of Type Y , then f can be mutated either to a map that has remnant or to an exceptional form for which we can calculate the Nielsen number directly. Not all self-maps of X are of Type Y . However, making use of the results in this paper, an algorithm is presented by Kim [J. Pure Appl. Algebra 216 (2012), 1652–1666] that does compute the Nielsen number for all self-maps of X .

By a unital ℓ-group we mean a lattice-ordered abelian group with a distinguished order unit. This paper is concerned with the category 𝖴fp$\mathsf {U}_{\textup {fp}}$ of finitely presented unital ℓ-groups. Using the duality between 𝖴fp$\mathsf {U}_{\textup {fp}}$ and a category of rational polyhedra, we will provide (i) a construction of finite limits and co-limits in 𝖴fp$\mathsf {U}_{\textup {fp}}$; (ii) a Cantor–Bernstein–Schröder theorem for finitely presented unital ℓ-groups; (iii) a proof that the fibered product of finitely generated projective unital ℓ-groups is projective; (iv) a geometrical characterization of exact unital ℓ-groups.

We prove analogues for odd primes of results of Snaith and Barker–Snaith. Let ℓ denote the p -complete connective Adams summand and consider the group of left ℓ-module automorphisms of ℓ∧ℓ in the stable homotopy category which induce the identity on mod p homology. We prove a group isomorphism between this group and a certain group of infinite invertible upper triangular matrices with entries in the p -adic integers. We determine information about the matrix corresponding to the automorphism 1∧Ψ q of ℓ∧ℓ, where Ψ q is the Adams operation and q is an integer which generates the p -adic units.

It is known that a general quintic equation cannot be solved by an ordinary origami construction. Alperin and Lang showed that three simultaneous folds leads to a solution of the general quintic equation. In this paper we improve their result and show that two simultaneous folds are sufficient to solve the general quintic equation.

Orthogonality of homogeneous orthogonally additive polynomials and orthosymmetry of multilinear maps are in general independent properties. In this paper, we show, for some important vector lattices, that the vector space of all orthogonally additive scalar-valued polynomials and the vector space of orthosymmetric scalar-valued multilinear maps are isomorphic.

In this paper, we study the moduli space of 4-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. Because the space of 4-dimensional algebras is large, we only classify the non-nilpotent algebras in this paper.

Let ( M , g ) be a compact, boundaryless manifold of dimension n with the property that either (i) n = 2 and ( M , g ) has no conjugate points, or (ii) the sectional curvatures of ( M , g ) are nonpositive. Let Δ be the positive Laplacian on M determined by g . We study the L 2 → L p mapping properties of a spectral cluster of (Δ) 1/2 of width 1/log λ. Under the geometric assumptions above, Bérard [Math. Z. 155 (1977), 249–276] obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a (log λ) 1/2 improvement for Hörmander's estimate on the L ∞ norms of eigenfunctions. In this paper we extend this improvement to the L p estimates for all p > 2( n +1)/( n -1).

In this paper we are interested in investigating the principal series representations of a certain rank 2 Kac–Moody group. This group (denoted by SL ˜2(𝔽q((z)))${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$) is introduced as a central extension by 𝔽q*${\mathbb {F}_q^*}$ of the group of two-by-two unimodular matrices over the non-archimedean local field, 𝔽q((z))${\mathbb {F}_q((z))}$. We begin by determining a formula for a cocycle that defines this central extension. This is used to classify all irreducible representations of the centrally extended torus inside SL ˜2(𝔽q((z)))${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$ and, as a consequence of this classification, we are able to determine the irreducible unramified principal series representations of SL ˜2(𝔽q((z)))${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$.

In a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.

We make a study of ℓℓ-extensions of model category structures. We prove an existence result of ℓℓ-extensions, present some specific and some rather formal results about them and give an application of the existence result to the homotopy theory of categories enriched over a monoidal model category.

We obtain a necessary and sufficient condition for embeddings of integral Lipschitz classes Lip(α; p ) into classes ΛBV of functions of bounded Λ-variation.

Let R be a ring (associative, with 1), and let R 〈〈 a , b 〉〉 denote the power-series R -ring in two non-commuting, R -centralizing variables, a and b . Let A be an R -subring of R 〈〈 a 〉〉 and B be an R -subring of R 〈〈 b 〉〉. Let α denote the natural map A ⨿ R B → R 〈〈 a , b 〉〉. This article describes some situations where α is injective and some where it is not. We prove that if A is a right Ore localization of R [ a ] and B is a right Ore localization of R [ b ], then α is injective. For example, the group ring over R of the free group on {1+ a ,1+ b } is R [(1+ a ) ± ] ⨿ R R [(1+ b ) ± ], which then embeds in R 〈〈 a , b 〉〉. We thus recover a celebrated result of R. H. Fox, via a proof simpler than those previously known. We show that α is injective if R is Π- semihereditary , that is, every finitely generated, torsionless, right R -module is projective. (This concept was first studied by M. F. Jones, who showed that it is left-right symmetric. It follows from a result of I. I. Sahaev that if w.gl.dim R ≤ 1 and R embeds in a skew field, then R is Π-semihereditary. Also, it follows from a result of V. C. Cateforis that if R is right semihereditary and right self-injective, then R is Π-semihereditary.) The arguments and results extend easily from two variables to any set of variables. The article concludes with some results contributed by G. M. Bergman that describe situations where α is not injective. He shows that if R is commutative and w.gl.dim R ≥ 2, then there exist examples where the map α': A ⨿ R B → R 〈〈 a 〉〉 ⨿ R R 〈〈 b 〉〉 is not injective, and hence neither is α. It follows from a result of K. R. Goodearl that when R is a commutative, countable, non-self-injective, von Neumann regular ring, then the map α'': R 〈〈 a 〉〉 ⨿ R R 〈〈 b 〉〉 → R 〈〈 a , b 〉〉 is not injective. Bergman gives procedures for constructing other examples where α'' is not injective.

We consider periods along closed geodesics and along geodesic circles for eigenfunctions of the Laplace–Beltrami operator on a compact hyperbolic Riemann surface. We obtain uniform bounds for such periods as the corresponding eigenvalue tends to infinity. We use methods from the theory of automorphic functions and, in particular, the uniqueness of the corresponding invariant functionals on irreducible unitary representations of PGL 2 (ℝ).

Using fixed-point-free group actions, we set up a scheme to define nested classes of groups indexed over ordinals. Restricting to cellular actions on CW-complexes, we find new classes of groups as well as new characterizations for some well-known classes. We extend some of the properties of the cohomological dimension of a group to groups with jump (co)homology and study their implications to cellular actions on finite dimensional CW-complexes that satisfy a quite general homological condition.

Let p be a prime with p ≥ 7 and q =2( p -1). In this paper we prove the existence of a nontrivial product of filtration p + 2 in the stable homotopy groups of spheres π(pn+p3-p-2)q-6S${\pi _{(p^{n}+p^{3}-p-2)q-6}S}$. This nontrivial product is shown to be represented up to a nonzero scalar by the product element γ˜p-1h0bn-1∈ Ext 𝒜p+2,(pn+p3-p-2)q+p-4(ℤ/p,ℤ/p)$ {\protect \widetilde{\gamma }_{p-1}h_{0}b_{n-1}\in {\rm Ext}_{\mathcal {A}}^{p+2,(p^{n}+p^{3}-p-2)q+p-4}(\mathbb {Z}/p,\mathbb {Z}/p)} $ in the Adams spectral sequence where n ≥ 1 and n ≥ 4.

In this paper, we study the product and sum-product graphs defined over the residue ring mod m . We show that all (or almost all) systems of dot-product equations are solvable in any sufficiently large subset ℰ ⊂ ℤ m d . We also prove some results on sum-product estimates in ℤ m .

In this paper, we introduce a definition of BV functions for (non-Gaussian) differentiable measure in a Gelfand triple which is an extension of the definition of BV functions in [Ann. Probab. 40 (2012), 1759–1794], using Dirichlet form theory. By this definition, we can analyze the reflected stochastic quantization problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set Γ in a Banach space E . We prove the existence of a martingale solution of this problem if Γ is a regular convex set.

In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic p > 3, we confirm this conjecture for the Lie algebras of Cartan type S ˜ n and S n . Moreover, we show that the variety of nilpotent elements in S n is a complete intersection. Motivated by the proof of the irreducibility, we describe explicitly the ring of invariant polynomial functions on S n .

Let X be an arbitrary topological space and let Y be a closed connected oriented n -dimensional manifold. In this work we consider p maps f 1 ,..., f p : X → Y , p ≥ 2, define a Lefschetz class L ( f 1 ,..., f p ) ∈ H n ( p -1) ( X ;ℚ) and prove that L ( f 1 ,..., f p ) ≠ 0 implies f 1 ( x ) = f 2 ( x ) = ⋯ = f p ( x ) for some x ∈ X . In the particular case where Y is a homology sphere there are presented some formulas to calculate L ( f 1 ,..., f p ).

The natural pseudo-distance is a similarity measure conceived for the purpose of comparing shapes. In this paper we revisit this pseudo-metric from the point of view of quotients. In particular, we show that the natural pseudo-distance coincides with the quotient pseudo-metric on the space of continuous functions on a compact manifold, endowed with the uniform convergence metric, modulo self-homeomorphisms of the manifold. As applications of this result, the natural pseudo-distance is shown to be actually a metric on a number of function subspaces such as the space of topological embeddings, of isometries, and of simple Morse functions on surfaces.

We consider a parametric Neumann problem with an indefinite and unbounded potential. Using a combination of critical point theory with truncation and comparison techniques, with Morse theory and with invariance arguments for a suitable negative gradient flow, we prove two multiplicity theorems for certain values of the parameter. In the first theorem we produce three solutions and in the second five. For all solutions we provide sign information. Our work improves significantly results existing in the literature.

In this paper we investigate the heat kernel of the Schrödinger operator L=-ΔG+W$L=-\Delta _G+W$ on the nilpotent Lie group G , where Δ G is the sub-Laplacian on G and the non-negative potential W belongs to the reverse Hölder class B q 1 . The main aim of the paper is to give a pointwise estimate for the heat kernel of Schrödinger operators with non-negative potentials on the nilpotent Lie group G . As its applications, we obtain the L p estimates for parabolic Schrödinger operators with certain non-negative potentials.

Keating and Snaith modeled the Riemann zeta-function ζ( s ) by characteristic polynomials of random N × N unitary matrices, and used this to conjecture the asymptotic main term for the 2 k -th moment of ζ(1/2+ i t ) when k > -1/2. However, an arithmetical factor, widely believed to be part of the leading term coefficient, had to be inserted in an ad hoc manner. Gonek, Hughes and Keating later developed a hybrid formula for ζ( s ) that combines a truncation of its Euler product with a product over its zeros. Using it, they recovered the moment conjecture of Keating and Snaith in a way that naturally includes the arithmetical factor. Here we use the hybrid formula to recover a conjecture of Hughes, Keating and O'Connell concerning the discrete moments of the derivative of the Riemann zeta-function averaged over the zeros of ζ( s ), incorporating the arithmetical factor in a natural way.

We show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W , any Lagrangian submanifold with trivial first de Rham cohomology group, embedded on a (symplectic) page of the (induced) convex open book on ∂W$\partial W$, can be assumed to be Legendrian in ∂W$\partial W$ with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. We also show that the convex stabilization of a compact convex Lefschetz fibration on W yields a compact convex Lefschetz fibration on a Liouville domain W ' which is exact symplectomorphic to a positive expansion of W . In particular, with the induced structures ∂W$\partial W$ and ∂W'$\partial W^{\prime }$ are contactomorphic.

In this paper, we consider the endomorphism algebra of an infinitely generated tilting module of the form R 𝒰 ⊕ R 𝒰 / R over a tame hereditary k -algebra R with k an arbitrary field, where R 𝒰 is the universal localization of R at an arbitrary set 𝒰 of simple regular R -modules. We show that the derived module category of this endomorphism algebra is a recollement of the derived module category 𝒟( R ) of R and the derived module category 𝒟(𝔸 𝒰 ) of the adèle ring 𝔸 𝒰 associated with 𝒰. When k is an algebraically closed field, the ring 𝔸 𝒰 can be precisely described in terms of Laurent power series ring k (( x )) over k . Moreover, if 𝒰 is a union of finitely many cliques, we give two different stratifications of the derived category of this endomorphism algebra by derived categories of rings such that the two stratifications are of different finite lengths.