Israel Journal of Mathematics最新文献

We derive the equations of chains for path geometries on surfaces by solving the equivalence problem of a related structure: sub-Riemannian geometry of signature (1, 1) on a contact 3-manifold. This approach is significantly simpler than the standard method of solving the full equivalence problem for path geometry. We then use these equations to give a characterization of projective path geometries in terms of their chains (the chains projected to the surface coincide with the paths) and study the chains of four examples of homogeneous path geometries. In one of these examples (horocycles in the hyperbolic planes) the projected chains are bicircular quartics.

Let T: X → X be a bounded operator on Banach space, whose spectrum σ(T) is included in the closed unit disc (overline{mathbb{D}}). Assume that the peripheral spectrum (sigma(T)capmathbb{T}) is finite and that T satisfies a resolvent estimate

We prove that T admits a bounded polygonal functional calculus, that is, an estimate ∥ϕ(T)∥ ≲ sup{∣ϕ(z)∣: z ∈ Δ} for some polygon Δ ⊂ ⅅ and all polynomials ϕ, in each of the following two cases: (i) either X = L^p for some 1 < p < ∞, and T: L^p → L^p is a positive contraction; or (ii) T is polynomially bounded and for all (xiinsigma(T)capmathbb{T}), there exists a neighborhood (cal{V}) of ξ such that the set ({(xi-z)(z-T)^{-1}:zincal{V}capoverline{mathbb{D}}^{c}}) is R-bounded (here X is arbitrary). Each of these two results extends a theorem of de Laubenfels concerning polygonal functional calculus on Hilbert space. Our investigations require the introduction, for any finite set (Esubsetmathbb{T}), of a notion of Ritt_E operator which generalizes the classical notion of Ritt operator. We study these Ritt_E operators and their natural functional calculus.

Let G/Γ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups H of G that are large enough, the orbits of H on G/Γ intersect nontrivially with a fixed compact set. As a consequence, we deduce finiteness results for totally geodesic submanifolds of arithmetic quotients of symmetric spaces that do not admit nontrivial deformation and with bounded volume. Our work generalizes previous work of Tomanov–Weiss and Oh on this topic.

This paper is concerned with the model-theoretic study of pairs (K, F) where K is an algebraically closed field and F is a distinguished subfield of K allowing extra structure. We study the basic model-theoretic properties of those pairs, such as quantifier elimination, model-completeness and saturated models. We also prove some preservation results of classification-theoretic notions such as stability, simplicity, NSOP₁, and NIP. As an application, we conclude that a PAC field is NSOP₁ iff its absolute Galois group is (as a profinite group).

Let k(B₀) and l(B₀) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B₀ of a finite group G. We prove that, if k(B₀)−l(B₀) = 1, then l(B₀) ≥ p − 1 or else p = 11 and l(B₀) = 9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B₀) ≥ p − 1 or else p = 11 and (G/{{bf{O}}_{{p^prime }}}(G) cong C_{11}^2, rtimes,{rm{SL}}(2,5)). These results are useful in the study of principal blocks with few characters.

We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least (2sqrt {p - 1} + 1 - {k_p}(G)), where k_p(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

We prove a version of Temkin’s local altered uniformization theorem. We show that for any rig-smooth, quasi-compact and quasi-separated admissible formal ({{cal O}_K})-model (mathfrak{X}), there is a finite extension K′/K such that ({mathfrak{X}_{{{cal O}_{{K^prime }}}}}) locally admits a rig-étale morphism (g:{mathfrak{X}^prime } to {mathfrak{X}_{{{cal O}_{{K^prime }}}}}) and a rig-isomorphism (h:{mathfrak{X}^{prime prime }} to {mathfrak{X}^prime}) with ({mathfrak{X}^prime }) being a successive semi-stable curve fibration over ({{cal O}_{{K^prime }}}) and ({mathfrak{X}^{prime prime }}) being a polystable formal ({{cal O}_{{K^prime }}})-scheme. Moreover, ({mathfrak{X}^prime }) admits an action of a finite group G such that (g:{mathfrak{X}^prime } to {mathfrak{X}_{{{cal O}_{{K^prime }}}}}) is G-invariant, and the adic generic fiber (mathfrak{X}_{{K^prime }}^prime ) becomes a G-torsor over its quasi-compact open image (U = {g_{{K^prime }}}(mathfrak{X}_{{K^prime }}^prime )). Also, we study properties of the quotient map ({mathfrak{X}^prime }/G to {mathfrak{X}_{{{cal O}_{{K^prime }}}}}) and show that it can be obtained as a composition of open immersions and rig-isomorphisms.

A well-known conjecture of Babai states that if G is any finite simple group and X is a generating set for G, then the diameter of the Cayley graph Cay(G, X) is bounded by log ∣G∣^c for some universal constant c. In this paper, we prove such a bound for Cay(G, X) for G = PSL(n, q), PSp(n, q) or PSU(n, q) where q is odd, under the assumptions that X contains a transvection and q ≠ 9 or 81.

We study the multiplicity of concentrating solutions for the following class of (p, q)-Laplacian problems

where ε > 0 is a small parameter, (gamma in {0,1},,1 < p < q < N,,,{q^*} = {{Nq} over {N - q}}) is the critical Sobolev exponent, ({Delta _s}u = {rm{div}}(|nabla u{|^{s - 2}}nabla u)), with s ∈ {p, q}, is the s-Laplacian operator, V: ℝ^N → ℝ is a positive continuous potential such that inf_∂Λ V > inf_Λ V for some bounded open set Λ ⊂ ℝ^N, and f: ℝ → ℝ is a continuous nonlinearity with subcritical growth. The main results are obtained by combining minimax theorems, penalization technique and Ljusternik–Schnirelmann category theory. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. As far as we know, all these results are new.

Denote by ({cal K}_0^n) the family of all closed convex sets A ⊂ ℝⁿ containing the origin 0 ∈ ℝⁿ. For (A in {cal K}_0^n), its polar set is denoted by A°. In this paper, we investigate the topological nature of the polar mapping A → A° on (({cal K}_0^n,{d_{AW}})), where d_AW denotes the Attouch–Wets metric. We prove that (({cal K}_0^n,{d_{AW}})) is homeomorphic to the Hilbert cube (Q = prodnolimits_{i = 1}^infty {[ - 1,1]} ) and the polar mapping is topologically conjugate with the standard based-free involution σ: Q → Q, defined by σ(x) = −x for all x ∈ Q. We also prove that among the inclusion-reversing involutions on ({cal K}_0^n) (also called dualities), those and only those with a unique fixed point are topologically conjugate with the polar mapping, and they can be characterized as all the maps (f:{cal K}_0^n to {cal K}_0^n) of the form f(A) = T(A°), with T a positive-definite linear isomorphism of ℝⁿ.

An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper continues the investigation on integrals of groups started in the work [1]. We study:

• A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2).

• The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1).

• Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups (Section 5).

• The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class (Sections 6 and 7).

• Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1).

• Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2).

We end the paper with a number of open problems.