Unconditional uniqueness in the charge class for the Dirac–Klein–Gordon equations in two space dimensions

Recently, Grünrock and Pecher proved global well-posedness of the 2d Dirac–Klein–Gordon equations given initial data for the spinor and scalar fields in H ^s and H ^s+1/2 × H ^s-1/2, respectively, where s ≥ 0, but uniqueness was only known in a contraction space of Bourgain type, strictly smaller than the natural solution space C([0,T]; H ^s × H ^s+1/2 × H ^s-1/2). Here we prove uniqueness in the latter space for s ≥ 0. This improves a recent result of Pecher, where the range s > 1/30 was covered.     

Unconditional global well-posedness for the 3D Gross-Pitaevskii equation for data without finite energy

The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H ^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof uses the I-method which is complicated by the fact that no L ²-conservation law holds. This shows that earlier results of Bethuel-Saut for data of the form 1 + H ¹ and Gérard for finite energy data remain true for this class of rough data.     

On the influence of the prior distribution in image reconstruction

Two measures of the influence of the prior distribution p(θ) in Bayes estimation are proposed. Both involve comparing with alternative prior distributions proportional to p(θ) ^s , for s  ≥  0. The first one, the influence curve for the prior distribution, is simply the curve of parameter values which are obtained as estimates when the estimation is made using p(θ) ^s instead of p(θ). It measures the overall influence of the prior. The second one, it the influence rate for the prior, is the derivative of this curve at s = 1, and quantifies the sensitivity to small changes or inaccuracies in the prior distribution. We give a simple formula for the influence rate in marginal posterior mean estimation, and discuss how the influence measures may be computed and used in image processing with Markov random field priors. The results are applied to an image reconstruction problem from visual field testing and to a stylized image analysis problem.     

On a conjecture of Butler and Graham

Motivated by a hat guessing problem proposed by Iwasawa, Butler and Graham made the following conjecture on the existence of a certain way of marking the coordinate lines in [k] ⁿ : there exists a way to mark one point on each coordinate line in [k] ⁿ , so that every point in [k] ⁿ is marked exactly a or b times as long as the parameters (a, b, n, k) satisfies that there are nonnegative integers s and t such that s + t = k ⁿ and as + bt = nk ^n?1. In this paper we prove this conjecture for any prime number k. Moreover, we prove the conjecture for the case when a = 0 for general k.     

Infinitely many Gerretsen-Blundon style quadratic inequalities, all strongest in Blundon’s sense

Blundon has proved that if R, r and s are respectively the circumradius, the inradius and the semiperimeter of a triangle, then the strongest possible inequalities of the form q(R, r) ≤ s ² ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms q _B (R, r) = 16Rr ? 5r ² and Q _B (R, r) = 4R ² + 4Rr + 3r ²; strongest in the sense that if Q is a quadratic form and s ² ≤ Q(R, r) ≤ Q _B (R, r) for all triangles then Q(R, r) = Q _B (R, r), and similarly for q _B (R, r). In this paper we prove that Q _B (resp. q _B ) is just one of infinitely many forms that appear as minimal (resp. maximal) elements in the partial order induced by the comparability relation in a certain set of forms, and we conclude that all these minimal forms are strongest in Blundon’s sense. We actually find all possible such strongest forms. Moreover we find all possible quadratic forms q, Q for which q(R, r) ≤ s ² ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.     

On the optimality of list scheduling for online uniform machines scheduling

This paper considers classical online scheduling problems on uniform machines. We show the tight competitive ratio of LS for any combinations of speeds of three machines. We prove that LS is optimal when s ₃ ≥ s ₂ ≥ s ₁ = 1 and ${s_3^2\geq s_2^2+s_2s_3+s_2}$ , where s ₁, s ₂, s ₃ are the speeds of three machines. On the other hand, LS can not be optimal for all combinations of machine speeds, even restricted to the case of 1 = s ₁ = s ₂ < s ₃. For m ≥ 4 machines, LS remains optimal when one machine has very large speed, and the remaining machines have the same speed.     

Hölder type inequalities in Lorentz spaces

We study optimal Hölder type inequalities for the Lorentz spaces L ^p,s (R, μ), in the range 1 < p < ∞, 1 ≤ s ≤ ∞, for both the maximal and the dual norms. These estimates also give sharp results for the corresponding associate norms.     

Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation

For s < 3/2, it is shown that the Cauchy problem for the Degasperis-Procesi equation (DP) is ill-posed in Sobolev spaces H ^s . If 1/2 ≤ s < 3/2, then ill-posedness is due to norm inflation. This means that there exist DP solutions who are initially arbitrarily small and eventually arbitrarily large with respect to the H ^s norm, in an arbitrarily short time. Since DP solutions conserve a quantity equivalent to the L ²-norm, there is no norm inflation in H ⁰ for these solutions. In this case, ill-posedness is caused by failure of uniqueness. For all other s < 1/2, the situation is similar to H ⁰. Considering that DP is locally well-posed in H ^s for s > 3/2, this work establishes 3/2 as the critical index of well-posedness in Sobolev spaces.     

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff _c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S ¹, the geodesic distance on Diff _c (S ¹) vanishes if and only if ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff _c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff _c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for ${s=\frac12}$ , and the Camassa–Holm equation for s = 1.     

Hamiltonian cycles in cubic Cayley graphs: the 〈2,4k,3〉 case

It was proved by Glover and Maru?i? (J. Eur. Math. Soc. 9:775–787, 2007), that cubic Cayley graphs arising from groups G=〈a,x∣a ²=x ^s =(ax)³=1,…〉 having a (2,s,3)-presentation, that is, from groups generated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4, and have a Hamiltonian path when |G| is congruent to 0 modulo 4. In this article the existence of a Hamiltonian cycle is proved when apart from |G| also s is congruent to 0 modulo 4, thus leaving |G| congruent to 0 modulo 4 with s either odd or congruent to 2 modulo 4 as the only remaining cases to be dealt with in order to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs.     

Uniqueness of non-linear ground states for fractional Laplacians in {\mathbb{R}}

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$ ( ? Δ ) s Q + Q ? Q α + 1 = 0 i n R , where 0 < s < 1 and 0 < α < 4s/(1?2s) for ${s<\frac{1}{2}}$ s < 1 2 and 0 < α < ∞ for ${s\geq \frac{1}{2}}$ s ≥ 1 2 . Here (?Δ) ^s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for ${s=\frac{1}{2}}$ s = 1 2 and α = 1 in [5] for the Benjamin–Ono equation. As a technical key result in this paper, we show that the associated linearized operator L ₊ = (?Δ) ^s +1?(α+1)Q ^α is non-degenerate; i.e., its kernel satisfies ker L ₊ = span{Q′}. This result about L ₊ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.     

On the convergence of the proximal algorithm for nonsmooth functions involving analytic features

We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the ?jasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following ?jasiewicz’s original idea, we prove that any bounded sequence generated by the proximal algorithm converges to some generalized critical point. We also obtain convergence rate results which are related to the flatness of the function by means of ?jasiewicz exponents. Apart from the sharp and elliptic cases which yield finite or geometric convergence, the decay estimates that are derived are of the type O(k ^?s ), where s ∈ (0, + ∞) depends on the flatness of the function.     

A note on quadratic convergence of a smoothing Newton algorithm for the LCP

The linear complementarity problem (LCP) is to find ${(x,s)\in\mathfrak{R}^n\times\mathfrak{R}^n}$ such that (x, s) ≥ 0, s = Mx + q, x ^T s = 0 with ${M\in\mathfrak{R}^{n\times n}}$ and ${q\in\mathfrak{R}^n}$ . The smoothing Newton algorithm is one of the most efficient methods for solving the LCP. To the best of our knowledge, the best local convergence results of the smoothing Newton algorithm for the LCP up to now were obtained by Huang et al. (Math Program 99:423–441, 2004). In this note, by using a revised Chen–Harker–Kanzow–Smale smoothing function, we propose a variation of Huang–Qi–Sun’s algorithm and show that the algorithm possesses better local convergence properties than those given in Huang et al. (Math Program 99:423–441, 2004).     

On the second weight of generalized Reed-Muller codes

Not much is known about the weight distribution of the generalized Reed-Muller code RM _q (s,m) when q > 2, s > 2 and m ≥ 2. Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m – 1)(q – 1) we then find the first s + 1 – (m – 1)(q–1) weights. For the case m = 2 the second weight is now known for all values of s. The results are derived mainly by using Gröbner basis theoretical methods.     

Simplicity of Some Derivations of k[x,y]

If k is a field of characteristic zero, c ∈ k?0, and 1 ≤ s ≤ r are integers such that either r ? s + 1 divides s or s divides r ? s + 1, then it is shown that the derivation y ^r ? _x  + (xy ^s  + c)? _y of the polynomial ring k[x, y] is simple.     

Further study of entire radial solutions of a biharmonic equation with exponential nonlinearity

We study the structure of entire radial solutions of a biharmonic equation with exponential nonlinearity: $$\begin{array}{ll}\Delta^2 u = \lambda {\rm e}^u \;\; {\rm in}\; \mathbb{R}^N, N \geq 5 \quad\quad\quad (0.1)\end{array}$$ with λ = 8(N ? 2)(N ? 4). It is known from a recent interesting paper by Arioli et al. that (0.1) admits a singular solution U _s (r) = ln r ^?4. We show that for 5 ≤ N ≤ 12, any regular entire radial solution u with u(r) ? ln r ^?4 → 0 as r → ∞ of (0.1) intersects with U _s (r) infinitely many times. On the other hand, if N ≥ 13, then u(r) < U _s (r) for all r > 0, and the solutions are strictly ordered with respect to the initial value a = u(0). Moreover, the asymptotic expansions of the entire radial solutions near ∞ are also obtained. Our main results give a positive answer to a conjecture in Arioli et al. (J Differ Equ 230:743–770, 2006) [see lines ?11 to ?9, p. 747 of Arioli et al. (J Differ Equ 230:743–770, 2006)].     

Hemisystems of small flock generalized quadrangles

In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order (5², 5) and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order (s ², s) for s ≤ 11. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle H(3, s ²).     

Finite groups with given nearly s-embedded subgroups

Let G be a finite group and H a subgroup of G. We say that H is s-permutable in G if HP = PH for all Sylow subgroups P of G; H is s-semipermutable in G if HP = PH for all Sylow subgroups P of G with (|P|, |H|) = 1. Let H _s G be the subgroup of H generated by all those subgroups of G which are s-permutable in G and H ^sG the intersection of all such s-permutable subgroups of G contain H. We say that H is nearly s-embedded in G if G has an s-permutable subgroup T such that H ^sG = HT and \({H \cap T \leqq H_{ssG}}\) , where H _ssG is an s-semipermutable subgroup of G contained in H. In this paper, we study the structure of a finite group G under the assumption that some subgroups of prime power order are nearly s-embedded in G. A series of known results are improved and extended.     

The asymptotic formula in Waring’s problem for one square and seventeen fifth powers

Let R _k,s(n) denote the number of solutions of the equation \({n= x^2 + y_1^k + y_2^k + \cdots + y_s^k}\) in natural numbers x, y ₁, . . . , y _s . By a straightforward application of the circle method, an asymptotic formula for R _k,s(n) is obtained when k ≥ 3 and s ≥ 2^k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2^k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R _5,17(n) by this strategy.     

Suboptimal Polynomial Meshes on Planar Lipschitz Domains

We construct norming meshes with cardinality 𝒪(n ^s ), s = 3, for polynomials of total degree at most n on the closure of bounded planar Lipschitz domains. Such cardinality is intermediate between optimality (s = 2), recently obtained by Kroó on multidimensional C ² star-like domains, and that arising from a general construction on Markov compact sets due to Calvi and Levenberg (s = 4).