Israel Journal of Mathematics最新文献

We study the Poisson bracket invariant, which measures the level of Poisson noncommutativity of a smooth partition of unity, on closed symplectic surfaces. Motivated by a general conjecture of Polterovich [P3] and building on preliminary work of Buhovsky–Tanny [BT], we prove that for any smooth partition of unity subordinate to an open cover by discs of area at most c, and under some localization condition on the cover when the surface is a sphere, then the product of the Poisson bracket invariant with c is bounded from below by a universal constant. Similar results were obtained recently by Buhovsky–Logunov–Tanny [BLT] for open covers consisting of displaceable sets on all closed surfaces, and their approach was extended by Shi–Lu [SL] to open covers by nondisplaceable discs. We investigate the sharpness of all these results.

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.

We prove the centrality of K₂(F₄, R) for an arbitrary commutative ring R. This completes the proof of the centrality of K₂(Φ, R) for any root system Φ of rank ≥ 3. Our proof uses only elementary localization techniques reformulated in terms of pro-groups. Another new result of the paper is the construction of a crossed module on the canonical homomorphism St(Φ, R) → G_sc(Φ, R), which has not been known previously for exceptional Φ.

Let P be a bounded polyhedron defined as the intersection of the non-negative orthant ℝⁿ₊ and an affine subspace of codimension m in ℝⁿ. We show that a simple and computationally efficient formula approximates the volume of P within a factor of γ^m, where γ > 0 is an absolute constant. The formula provides the best known estimate for the volume of transportation polytopes from a wide family.

Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr–Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If H is an acyclic digraph, the oriented Ramsey number of H, denoted (overrightarrow {{r_1}} (H)), is the least N such that every tournament on N vertices contains a copy of H. We show that for any Δ ≥ 2 and any sufficiently large n, there exists an acyclic digraph H with n vertices and maximum degree Δ such that

This proves that (overrightarrow {{r_1}} (H)) is not always linear in the number of vertices for bounded-degree H. On the other hand, we show that (overrightarrow {{r_1}} (H)) is nearly linear in the number of vertices for typical bounded-degree acyclic digraphs H, and obtain linear or nearly linear bounds for several natural families of bounded-degree acyclic digraphs.

For multiple colors, we prove a quasi-polynomial upper bound (overrightarrow {{r_k}} (H) = {2^{{{(log ,n)}^{{O_k}(1)}}}}) for all bounded-de gree acyclic digraphs H on n vertices, where (overrightarrow {{r_k}} (H)) is the least N such that every k-edge-colored tournament on N vertices contains a monochromatic copy of H. For k ≥ 2 and n ≥ 4, we exhibit an acyclic digraph H with n vertices and maximum degree 3 such that (overrightarrow {{r_k}} (H) ge {n^{Omega (log ,n/log log ,n)}}), showing that these Ramsey numbers can grow faster than any polynomial in the number of vertices.

We establish Hanner’s inequality for arbitrarily many functions in the setting where the Rademacher distribution is replaced with higher dimensional random vectors uniform on Euclidean spheres.

For a sequence of Boolean functions ({f_n}:{{ - 1,1} ^{{V_n}}} to { - 1,1} ), defined on increasing configuration spaces of random inputs, we say that there is sparse reconstruction if there is a sequence of subsets U_n ⊆ V_n of the coordinates satisfying ∣U_n∣ = o(∣V_n∣) such that knowing the coordinates in U_n gives us a non-vanishing amount of information about the value of f_n.

We first show that, if the underlying measure is a product measure, then no sparse reconstruction is possible for any sequence of transitive functions. We discuss the question in different frameworks, measuring information content in L² and with entropy. We also highlight some interesting connections with cooperative game theory. Beyond transitive functions, we show that the left-right crossing event for critical planar percolation on the square lattice does not admit sparse reconstruction either. Some of these results answer questions posed by Itai Benjamini.

Radially symmetric global unbounded solutions of the chemotaxis system

are considered in a ball Ω = B_R(0) ⊂ ℝⁿ, where n ≥ 3 and R > 0.

Under the assumption that D and S suitably generalize the prototypes given by D(ξ) = (ξ + ι)^m−1 and S(ξ) = (ξ + 1)^−λ−1 for all ξ > 0 and some m ∈ ℝ, λ >0 and ι ≥ 0 fulfilling (m + lambda < 1 - {2 over n}), a considerably large set of initial data u₀ is found to enforce a complete mass aggregation in infinite time in the sense that for any such u₀, an associated Neumann type initial-boundary value problem admits a global classical solution (u, v) satisfying

as well as

with some C > 0.

Random integers, sampled uniformly from [1, x], share similarities with random permutations, sampled uniformly from S_n. These similarities include the Erdős–Kac theorem on the distribution of the number of prime factors of a random integer, and Billingsley’s theorem on the largest prime factors of a random integer. In this paper we extend this analogy to non-uniform distributions.

Given a multiplicative function α: ℕ → ℝ_≥0, one may associate with it a measure on the integers in [1, x], where n is sampled with probability proportional to the value α(n). Analogously, given a sequence {θ_i}_i≥1 of non-negative reals, one may associate with it a measure on S_n that assigns to a permutation a probability proportional to a product of weights over the cycles of the permutation. This measure is known as the generalized Ewens measure.

We study the case where the mean value of α over primes tends to some positive θ, as well as the weights α(p) ≈ (log p)^γ. In both cases, we obtain results in the integer setting which are in agreement with those in the permutation setting.

We study the coset covering function ℭ(r) of an infinite, finitely generated group: the number of cosets of infinite index subgroups needed to cover the ball of radius r. We show that ℭ(r) is of order at least (sqrt{r}) for all groups. Moreover, we show that ℭ(r) is linear for a class of amenable groups including virtually nilpotent and polycyclic groups, and that it is exponential for property (T) groups.