Characterizing local rings via homological dimensions and regular sequences

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If the G_C-dimension of M/aM is finite for all ideals a generated by an R-regular sequence of length at most d−t then either the G_C-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.     

Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

Notes on Jordan (σ, τ)*-derivations and Jordan triple (σ, τ)*-derivations

Let R be a 2-torsion free semiprime *-ring, σ, τ two epimorphisms of R and f, d : R → R two additive mappings. In this paper we prove the following results: (i) d is a Jordan (σ, τ)*-derivation if and only if d is a Jordan triple (σ, τ)*-derivation. (ii) f is a generalized Jordan (σ, τ)*-derivation if and only if f is a generalized Jordan triple (σ, τ)*-derivation.     

Gradient estimate for solutions to Poisson equations in metric measure spaces

Let (X,d) be a complete, pathwise connected metric measure space with a locally Ahlfors Q-regular measure μ, where Q>1. Suppose that (X,d,μ) supports a (local) (1,2)-Poincaré inequality and a suitable curvature lower bound. For the Poisson equation Δu=f on (X,d,μ), Moser-Trudinger and Sobolev inequalities are established for the gradient of u. The local Hölder continuity with optimal exponent of solutions is obtained.     

Lifting measures to Markov extensions

Generalizing a theorem ofHofbauer (1979), we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. Using these results we prove:

1. IfT is anS-unimodal map with an attracting invariant Cantor set, then ∫log|T′|dμ=0 for the unique invariant measure μ on the Cantor set.
2. IfT is piecewise invertible, iff is the Radon-Nikodym derivative ofT with respect to a σ-finite measurem, if logf has bounded distortion underT, and if μ is an ergodicT-invariant measure satisfying a certain lower estimate for its entropy, then μ?m iffh _μ (T)=Σlogf dμ.

     

Smooth metric measure spaces with weighted Poincaré inequality

In this paper, we study smooth metric measure space (M, g, e ^?f dv) satisfying a weighted Poincaré inequality and establish a rigidity theorem for such a space under a suitable Bakry–Émery curvature lower bound. We also consider the space of f-harmonic functions with finite energy and prove a structure theorem.     

Intersection patterns of convex sets

LetK ₁,…K_n be convex sets inR ^d. For 0≦i denote byf _ithe number of subsetsS of {1,2,…,n} of cardinalityi+1 that satisfy ∩{K _i∶i∈S}≠Ø. We prove:Theorem.If f _d+r=0 for somer r>=0, then {fx161-1} This inequality was conjectured by Katchalski and Perles. Equality holds, e.g., ifK ₁=…=K_r=R^d andK _r+1,…,K_n aren?r hyperplanes in general position inR ^d. The proof uses multilinear techniques (exterior algebra). Applications to convexity and to extremal set theory are given.     

On a global upper bound for Jensen's inequality

In this paper we shall give a global upper bound for Jensen's inequality without restrictions on the target convex function f. We also introduce a characteristic c(f) i.e. an absolute constant depending only on f, by which the global bound is improved.     

A Liouville-Type Theorem for Smooth Metric Measure Spaces

For smooth metric measure spaces (M,g,e ^?f d _vol ) we prove a Liouville-type theorem when the Bakry–Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case f is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spaces with Bakry–Emery Ricci tensor bounded from below.     

On the solutions to certain laplace inequalities with applications to geometry of submanifolds

LetM be a complete Riemannian manifold with Ricci curvature bounded from below. We give an explicit estimate for the size of the negative sets of solutions to the differential inequality Δu ≥λu where Δ is the Laplacian and λ is a negative constant. This inequality arises naturally when we study the lengthH of the mean curvature of an isometric immersionf of M into another Riemannian manifoldN with curvature bounded above by some constantκ. Suppose that the image f(M) does not meet the cut locus of some pointo ∈ N. As a consequence of our estimate, we prove that, givenρ > 0, if supH is less than a certain explicit expression μ(κ, ρ) inρ andκ on any domainU that contains an inscribed ball of radius greater than an explicitly computable numberR, then the diameter of the setf(U) inN must exceed 2ρ. Moreover, if supH = μ(κ, ρ) onM and the diameter off(M) inN equals 2ρ, thenf is a minimal immersion into a distance sphere of radiusρ inN.     

Generalized Lebesgue points for Sobolev functions

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space (X, d, μ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B(x, r) converge to f(x) when r converges to 0. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function f ∈ M ^s,p (X), 0 < s ≤ 1, 0 < p < 1, where X is a doubling metric measure space, has generalized Lebesgue points outside a set of \(\mathcal{H}^h\)-Hausdorff measure zero for a suitable gauge function h.     

Stabilizers and orbits of smooth functions

Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?○f is right equivalent to f, i.e. there exists a diffeomorphism such that ?○f=f○h at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C^∞(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings M→S¹.     

Generalized convexity and inequalities

Let R₊=(0,∞) and let M be the family of all mean values of two numbers in R₊ (some examples are the arithmetic, geometric, and harmonic means). Given m₁,m₂∈M, we say that a function is (m₁,m₂)-convex if f(m₁(x,y))?m₂(f(x),f(y)) for all x,y∈R₊. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m₁,m₂)-convexity on m₁ and m₂ and give sufficient conditions for (m₁,m₂)-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.     

Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth in fluid-like tissue. The model equations include a diffusion equation for the nutrient concentration, and the Stokes equation with a source which represents the proliferation of tumor cells. The proliferation rate μ and the cell-to-cell adhesiveness γ which keeps the tumor intact are two parameters which characterize the “aggressiveness” of the tumor. For any positive radius R there exists a unique radially symmetric stationary solution with radius r=R. For a sequence μ/γ=Mn(R) there exist symmetry-breaking bifurcation branches of solutions with free boundary r=R+εYn,0(θ)+O(ε²) (n even ?2) for small |ε|, where Yn,0 is the spherical harmonic of mode (n,0). Furthermore, the smallest Mn(R), say Mn_∗(R), is such that n_∗=n_∗(R)→∞ as R→∞. In this paper we prove that the radially symmetric stationary solution with R=RS is linearly stable if μ/γ<N_∗(RS,γ) and linearly unstable if μ/γ>N_∗(RS,γ), where N_∗(RS,γ)?Mn_∗(RS), and we prove that strict inequality holds if γ is small or if γ is large. The biological implications of these results are discussed at the end of the paper.     

Non-spectral problem for a class of planar self-affine measures

The self-affine measure μM,D corresponding to an expanding matrix M∈Mn(R) and a finite subset D⊂Rn is supported on the attractor (or invariant set) of the iterated function system {?d(x)=M⁻¹(x+d)}_d∈D. The spectral and non-spectral problems on μM,D, including the spectrum-tiling problem implied in them, have received much attention in recent years. One of the non-spectral problem on μM,D is to estimate the number of orthogonal exponentials in L²(μM,D) and to find them. In the present paper we show that if a,b,c∈Z, |a|>1, |c|>1 and ac∈Z?(3Z),

     

Embeddings in the 3/4 range

We give a complete obstruction to turning an immersion f:M^m→Rⁿ into an embedding when 3n?4m+5. It is a secondary obstruction, and exists only when the primary obstruction, due to André Haefliger, vanishes. The obstruction lives in a twisted cobordism group, and its vanishing implies the existence of an embedding in the regular homotopy class of f in the range indicated. We use Tom Goodwillie's calculus of functors, following Michael Weiss, to help organize and prove the result.     

The finiteness dimension of local cohomology modules and its dual notion

Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_a(M), the finiteness dimension of M with respect to a, and, its dual notion q_a(M), the Artinianness dimension of M with respect to a. When (R,m) is local and r?f_a(M) is less than , the m-finiteness dimension of M relative to a, we prove that is not Artinian, and so the filter depth of a on M does not exceed f_a(M). Also, we show that if M has finite dimension and is Artinian for all i>t, where t is a given positive integer, then is Artinian. This immediately implies that if q?q_a(M)>0, then is not finitely generated, and so f_a(M)≤q_a(M).     

Topological stability through extremely tame retractions

Suppose that F:(Rn×Rd,0)→(Rp×Rd,0) is a smoothly stable, Rd-level preserving germ which unfolds f:(Rn,0)→(Rp,0); then f is smoothly stable if and only if we can find a pair of smooth retractions r:(Rn+d,0)→(Rn,0) and s:(Rp+d,0)→(Rp,0) such that f°r=s°F. Unfortunately, we do not know whether f will be topologically stable if we can find a pair of continuous retractions r and s.The class of extremely tame (E-tame) retractions, introduced by du Plessis and Wall, are defined by their nice geometric properties, which are sufficient to ensure that f is topologically stable.In this article, we present the E-tame retractions and their relation with topological stability, survey recent results by the author concerning their construction, and illustrate the use of our techniques by constructing E-tame retractions for certain germs belonging to the E- and Z-series of singularities.     

Topological structure of complete Riemannian manifolds with cyclic holonomy groups

Let M be a complete m-dimensional Riemannian manifold with cyclic holonomy group, let X be a closed flat manifold homotopy equivalent to M, and let L→X be a nontrivial line bundle over X whose total space is a flat manifold with cyclic holonomy group. We prove that either M is diffeomorphic to X×Rm-dimX or M is diffeomorphic to L×Rm-dimX−1.     

Subsequent to a theorem of Markoff

Let f(x, y) be an indefinite binary quadratic form, d(f) its discriminant, m(f) the infimum of |f(x, y)| over all integers x, y not both zero, and put $\text{μ(f) = m(f)d(f)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. In this paper we prove the existence of countably many disjoint open intervals I_j contained in $\text{0 ≤ x ≤}\text{1}\text{3}$ such that there is no f with μ(f) in I_j (j = 1, 2,…) and such that for any interval I containing two intervals I_j, I_k there is an f with μ(f) in I.