Israel Journal of Mathematics最新文献

In this paper, we are concerned with the tempered fractional operator (-(Delta+lambda)^{alphaover{2}}) with α ∈ (0, 2) and λ is a sufficiently small positive constant. We first establish various maximum principle principles and develop the direct moving planes and sliding methods for anti-symmetric functions involving tempered fractional operators. And then we consider tempered fractional problems. As applications, we extend the direct method of moving planes and sliding methods for the tempered fractional problem, and discuss how they can be used to establish symmetry, monotonicity, Liouville-type results and uniqueness results for solutions in various domains. We believe that our theory and methods can be conveniently applied to study other problems involving tempered fractional operators.

For the exactly solvable model of exponential last passage percolation on ℤ², consider the geodesic Γ_n joining (0, 0) and (n, n) for large n. It is well known that the transversal fluctuation of Γ_n around the line x = y is n^2/3+o(1) with high probability. We obtain the exponent governing the decay of the small ball probability for Γ_n and establish that for small δ, the probability that Γ_n is contained in a strip of width δn^2/3 around the diagonal is exp(−Θ(δ^−3/2)) uniformly in high n. We also obtain optimal small deviation estimates for the one point distribution of the geodesic showing that for ({t}over{2n}) bounded away from 0 and 1, we have ℙ(∣x(t) − y(t)∣ ≤ δn^2/3) = Θ(δ) uniformly in high n, where (x(t), y(t)) is the unique point where Γ_n intersects the line x + y = t. Our methods are expected to go through for other exactly solvable models of planar last passage percolation and also, upon taking the n → ∞ limit, expected to provide analogous estimates for geodesics in the directed landscape.

We study solutions of −Δu + Vu = λu on ℝⁿ. Such solutions localize in the ‘allowed’ region {x ∈ ℝⁿ: V(x) ≤ λ} and decay exponentially in the ‘forbidden’ region {x ∈ ℝⁿ: V(x) > λ}. One way of making this precise is Agmon’s inequality implying decay estimates in terms of the Agmon metric. We prove a complementary decay estimate in terms of harmonic measure which can improve on Agmon’s estimate, connect the Agmon metric to decay of harmonic measure and prove a sharp pointwise Agmon estimate.

For an untwisted affine Lie algebra we prove an embedding of any higher level Demazure module into a tensor product of lower level Demazure modules (e.g., level one in type A) which becomes in the limit (for anti-dominant weights) the well-known embedding of finite-dimensional irreducible modules of the underlying simple Lie algebra into the tensor product of fundamental modules. To achieve this goal, we first simplify the presentation of these modules extending the results of [13] in the ({mathfrak g})-stable case. As an application, we propose a crystal theoretic way to find classical decompositions with respect to a maximal semi-simple Lie subalgebra by identifying the Demazure crystal as a connected component in the corresponding tensor product of crystals.

We build a model of the P-ideal ichotomy (PID) and Martin’s axiom for ω₁ (({rm MA}_{{omega}_1})) in which there is a 2-entangled set of reals. In particular, it follows that the Open Graph Axiom or Baumgartner’s axiom for ω₁-dense sets are not consequences of ({rm PID} + {rm MA}_{{omega}_1}). We review Neeman’s iteration method using two type side conditions and provide an alternative proof for the preservation of properness.

A new series of central elements is found in the free alternative algebra. More exactly, let Alt[X] and SMalc[X] ⊂ Alt[X] be the free alternative algebra and the free special Malcev algebra over a field of characteristic 0 on a set of free generators X, and let f (x, y, x₁,…, x_n) ∈ SMalc[X] be a multilinear element which is trivial in the free associative algebra. Then the element u_n = u_n (x, x₁,…,x_n) = f (x², x, x₁,…,x_n) − f (x, x²,x₁,…,x_n) lies in the center of the algebra Alt[X]. The elements u_n(x, x₁,…, x_n) are uniquely defined up to a scalar for a given n (that is, they do not depend on f but only on deg f), and they are skew-symmetric on the variables x₁,…,x_n. Moreover, u_n = 0 for n = 4m + 2, 4m + 3 and u_n ≠ 0 for n = 4m, 4m + 1. The ideals generated by the elements u_4m, u_4m+1 lie in the associative center of the algebra Alt[X] and have trivial multiplication.

The folding entropy is a quantity originally proposed by Ruelle in 1996 during the study of entropy production in the non-equilibrium statistical mechanics [53]. As derived through a limiting process to the non-equilibrium steady state, the continuity of entropy production plays a key role in its physical interpretations. In this paper, the continuity of folding entropy is studied for a general (non-invertible) differentiable dynamical system with degeneracy. By introducing a notion called degenerate rate, it is proved that on any subset of measures with uniform degenerate rate, the folding entropy, and hence the entropy production, is upper semi-continuous. This extends the upper semi-continuity result in [53] from endomorphisms to all C^r (r > 1) maps.

We further apply our result in the one-dimensional setting. In achieving this, an equality between the folding entropy and (Kolmogorov–Sinai) metric entropy, as well as a general dimension formula are established. The upper semi-continuity of metric entropy and dimension are then valid when measures with uniform degenerate rate are considered. Moreover, the sharpness of the uniform degenerate rate condition is shown by examples of C^r interval maps with positive metric (and folding) entropy.

Sets of recurrence, which were introduced by Furstenberg, and van der Corput sets, which were introduced by Kamae and Mendés France, as well as variants thereof, are important classes of sets in Ergodic Theory. In this paper, we construct a set of strong recurrence which is not a van der Corput set. In particular, this shows that the class of enhanced van der Corput sets is a proper subclass of sets of strong recurrence. This answers some questions asked by Bergelson and Lesigne.

All components of complements of discriminant varieties of simple real function singularities are explicitly listed. A combinatorial algorithm enumerating the topological types of morsifications of real function singularities is promoted.

We study the problem of m-adic stability of F-singularities, that is, whether the property that a quotient of a local ring ((R,mathfrak{m})) by a non-zero divisor (xinmathfrak{m}) has good F-singularities is preserved in a sufficiently small (mathfrak{m})-adic neighborhood of x. We show that (mathfrak{m})-adic stability holds for F-rationality in full generality, and for F-injectivity, F-purity and strong F-regularity under certain assumptions. We show that strong F-regularity and F-purity are not stable in general. Moreover, we exhibit strong connections between stability and deformation phenomena, which hold in great generality.