Volume 30, Issue 6

It is known that the free locally convex space L⁢(X){L(X)} on a space X is metrizable only if X is finite and that L⁢(X){L(X)} is barrelled if and only if X is discrete. We significantly generalize these results by proving that L⁢(X){L(X)} is a Mackey space if and only if X is discrete. Noting that real locally convex spaces which are Mackey groups are always Mackey spaces, but that the converse is false, it is also proved here that L⁢(X){L(X)} is a Mackey group if and only if it is a Mackey space.

We investigate the sharp Lp→Lr{L^{p}\to L^{r}} estimates for the restricted averaging operator AC{A_{C}} over the cone C of the d -dimensional vector space 𝔽qd{\mathbb{F}_{q}^{d}} over the finite field 𝔽q{\mathbb{F}_{q}} with q elements. The restricted averaging operator AC{A_{C}} for the cone C is defined by the relation AC⁢f=f∗σ|C{A_{C}f=f\ast\sigma|_{C}}, where σ denotes the normalized surface measure on the cone C , and f is a complex-valued function on the space 𝔽qd{\mathbb{F}_{q}^{d}} with the normalized counting measure dx . In the previous work [D. Koh, C.-Y. Shen and I. Shparlinski, Averaging operators over homogeneous varieties over finite fields, J. Geom. Anal. 26 2016, 2, 1415–1441], the sharp boundedness of AC{A_{C}} was obtained in odd dimensions d≥3{d\geq 3}, but only partial results were given in even dimensions d≥4{d\geq 4}. In this paper we prove the optimal estimates in even dimensions d≥6{d\geq 6} in the case when the cone C⊂𝔽qd{C\subset\mathbb{F}_{q}^{d}} contains a d/2{{d/2}}-dimensional subspace.

Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge. We also show how our results from sub-Riemannian geometry can give improved approximations for the case of studying expansions of holonomy of principal bundles over the Euclidean space.

Let f:ℝn→ℝ{f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}} be a polynomial and 𝒵⁢(f){\mathcal{Z}(f)} its zero set. In this paper, in terms of the so-called Newton polyhedron of f , we present a necessary criterion and a sufficient condition for the compactness of 𝒵⁢(f){\mathcal{Z}(f)}. From this we derive necessary and sufficient criteria for the stable compactness of 𝒵⁢(f){\mathcal{Z}(f)}.

We use the unit-graphs and the special unit-digraphs on matrix rings to show that every n×n{n\times n} nonzero matrix over 𝔽q{{\mathbb{F}}_{q}} can be written as a sum of two SLn{\operatorname{SL}_{n}}-matrices when n>1{n>1}. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and we prove that if X is a subset of Mat2⁡(𝔽q){\operatorname{Mat}_{2}({\mathbb{F}}_{q})} with size |X|>2⁢q3⁢q/(q-1){\lvert X\rvert>2q^{3}\sqrt{q}/(q-1)}, then X contains at least two distinct matrices whose difference has determinant α for any α∈𝔽q∗{\alpha\in{\mathbb{F}}_{q}^{\ast}}. Using this result, we also prove a sum-product type result: if A,B,C,D⊆𝔽q{A,B,C,D\subseteq{\mathbb{F}}_{q}} satisfy |A|⁢|B|⁢|C|⁢|D|4=Ω⁢(q0.75){\sqrt[4]{\lvert A\rvert\lvert B\rvert\lvert C\rvert\lvert D\rvert}=\Omega(q^{% 0.75})} as q→∞{q\rightarrow\infty}, then (A-B)⁢(C-D){(A-B)(C-D)} equals all of 𝔽q∗{{\mathbb{F}}_{q}^{\ast}}. In particular, if A is a subset of 𝔽q{{\mathbb{F}}_{q}} with cardinality |A|>32⁢q3/4{\lvert A\rvert>\frac{3}{2}q^{3/4}}, then the subset (A-A)⁢(A-A){(A-A)(A-A)} equals all of 𝔽q{{\mathbb{F}}_{q}}. We derive some identities involving character sums of the entries of 2×2{2\times 2} matrices over finite fields. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.

Right φ-amenability and right character amenability have been introduced for Banach algebras. Here, these concepts will be generalized for Fréchet algebras. Then some of the previous available results about right φ-amenability and right character amenability for the case of Banach algebras will be verified for Fréchet algebras. Related results about Segal Fréchet algebras are provided.

We study a real analytic Jacobi–Eisenstein series of matrix index and deduce several arithmetically interesting properties. In particular, we prove the followings: (a) Its Fourier coefficients are proportional to the average values of the Eisenstein series on higher-dimensional hyperbolic space. (b) The associated Dirichlet series of two variables coincides with those of Siegel, Shintani, Peter and Ueno. This makes it possible to investigate the Dirichlet series by means of techniques from modular form.

We prove that a commutative parasemifield S is additively idempotent, provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.

In this work, we are concerned with the study of the N-Lusin property in metric measure spaces. A map satisfies that property if sets of measure zero are mapped to sets of measure zero. We prove a new sufficient condition for the N-Lusin property using a weak and pointwise Lipschitz-type estimate. Relations with approximate differentiability in metric measure spaces and other sufficient conditions for the N-Lusin property will be provided.

This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions E/F{E/F}, where F is a finite unramified field extension of ℚ2{\mathbb{Q}_{2}}, fall into two cases that we call Case 1 and Case 2 . In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1 . In this paper, we obtain the local density formula for the remaining Case 2 , by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice (L,H){(L,H)}, when a base field is unramified over ℚ{\mathbb{Q}} at a prime (2){(2)}.

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) condition), which means that in each tangent plane, there exists a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank 2 non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space S3×ℝ{S^{3}\times\mathbb{R}} which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on S2×S3{S^{2}\times S^{3}} and S6×S7{S^{6}\times S^{7}} which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on S3×S3{S^{3}\times S^{3}}, shedding some light on the long standing general Hopf conjecture.

In this paper, we study the ℤ/2{\mathbb{Z}/2} action on the real Grassmann manifolds Gn⁢(ℝ2⁢n){G_{n}(\mathbb{R}^{2n})} and G~n⁢(ℝ2⁢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=2a⁢(2⁢b+1){n=2^{a}(2b+1)}, k=2a+1-1{k=2^{a+1}-1}, a convex body C in ℝ2⁢n{\mathbb{R}^{2n}}, and k real-valued functions α1,…,αk{\alpha_{1},\ldots,\alpha_{k}} continuous on convex bodies in ℝ2⁢n{\mathbb{R}^{2n}} with respect to the Hausdorff metric, there exists a subspace V⊆ℝ2⁢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 αi{\alpha_{i}}, that is, αi⁢(pV⁢(C))=αi⁢(pV⊥⁢(C)){\alpha_{i}(p_{V}(C))=\alpha_{i}(p_{V^{\perp}}(C))} for all 1≤i≤k{1\leq i\leq k}.

In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in graded restricted module categories of basic classical Lie superalgebras over an algebraically closed field of odd characteristic, and obtain a sum formula in the corresponding Grothendieck groups. Consequently, we formulate a strong linkage principle for such categories.

We verify the maximum conjecture on the rigidity of totally nondegenerate model CR manifolds in the following two cases: (i) for all models of CR dimension one, (ii) for the so-called full-models, namely those in which their associated symbol algebras are free CR. In particular, we discover that in each arbitrary CR dimension and length ≥3{\geq 3}, there exists at least one totally nondegenerate model, enjoying this conjecture. Our proofs rely upon some recent results in the Tanaka theory of transitive prolongation of fundamental algebras.