Application of the $$\bar\partial$$ -dressing method to a $$(2+1)$$ -dimensional equation

Theoretical and Mathematical Physics - A remarkable method for investigating solutions of nonlinear soliton equation is the $$\bar\partial$$ -dressing method. Although there are other methods that...     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

The $$ \ell _1 $$-Analysis in Phase Retrieval with Redundant Dictionary

This article presents new results concerning the recovery of a signal from the magnitude only measurements where the signal is not sparse in an orthonormal basis but in a redundant dictionary, which we call it phase retrieval with redundant dictionary for short. To solve this phaseless problem, we analyze the \( \ell _1 \)-analysis model. Firstly we investigate the noiseless case with presenting a null space property of the measurement matrix under which the \( \ell _1 \)-analysis model provides an exact recovery. Secondly we introduce a new property (S-DRIP) of the measurement matrix. By solving the \( \ell _1 \)-analysis model, we prove that this property can guarantee a stable recovery of real signals that are nearly sparse in overcomplete dictionaries.     

Towards characterising polynomiality of $$\frac{1-q^b}{1-q^a}{n\brack m}$$ and applications

In this note, we shall give conditions which guarantee that \(\frac{1-q^b}{1-q^a}\Big [\begin{array}{l} n\\ m\\ \end{array}\Big ]\in \mathbb {Z}[q]\) holds. We shall provide a full characterisation for \(\frac{1-q^b}{1-q^a}\Big [\begin{array}{l} ka\\ m\\ \end{array}\Big ]\in \mathbb {Z}[q]\). This unifies a variety of results already known in literature. We shall prove new divisibility properties for the binomial coefficients and a new divisibility result for a certain finite sum involving the roots of the unity.     

1-Dimensional solutions of the $$\lambda $$-self shrinkers

We examine the solutions of 1-dimensional \(\lambda \)-self shrinkers and show that for certain \(\lambda <0\), there are some closed, embedded solutions other than circles. For negative \(\lambda \) near zero, there are embedded solutions with 2-symmetry. For negative \(\lambda \) with large absolute value, there are embedded solutions with m-symmetry, where m is greater than 2.     

Resplendent models and $${\Sigma_1^1}$$ -definability with an oracle

In this article we find some sufficient and some necessary -conditions with oracles for a model to be resplendent or chronically resplendent. The main tool of our proofs is internal arguments, that is analogues of classical theorems and model-theoretic constructions conducted inside a model of first-order Peano Arithmetic: arithmetised back-and-forth constructions and versions of the arithmetised completeness theorem, namely constructions of recursively saturated and resplendent models from the point of view of a model of arithmetic. These internal arguments are used in conjunction with Pabion’s theorem that ensures that certain oracles are coded in a sufficiently saturated model of arithmetic. Examples of applications are provided for the theories of dense linear orders and of discrete linear orders. These results are then generalised to other ω-categorical theories and theories with a unique countable recursively saturated model.      

Mathematical Analysis of a Cauchy Problem for the Time-Fractional Diffusion-Wave Equation with $$ \alpha \in \left( 0,2\right) $$

This paper deals with a theoretical mathematical analysis of a Cauchy problem for the time-fractional diffusion-wave equation in the upper half-plane, \(x\in \mathbb {R}\), \(t\in \mathbb {R}^+\), where the Caputo fractional derivative of order \(\alpha \in \left( 0,2\right) \) is considered. An explicit solution to this Cauchy problem is obtained via separation of variables. A first proof of the validity of the obtained results is provided for a certain kind of initial conditions. Throughout this work a new expression of the solution to this problem and its utility for carrying out rigurous proofs are presented. Finally, several new properties of the solution are obtained.     

Conservative Means of Orthogonal Series and the Spaces $$L^p [0;1] $$ , $$p \in (1;\infty ) $$

Necessary and sufficient conditions for an orthogonal series to be the Fourier series of a function in the space , , are obtained. In the special case of regular summation methods we recover the classical results of Orlicz and Lomnicki.     

A model of $$\mathsf {ZFA}+ \mathsf {PAC}$$ with no outer model of $$\mathsf {ZFAC}$$ZFAC with the same pure part

We produce a model of \(\mathsf {ZFA}+ \mathsf {PAC}\) such that no outer model of \(\mathsf {ZFAC}\) has the same pure sets, answering a question asked privately by Eric Hall.     

The log-behavior of $$\root n \of {p(n)}$$ and $$\root n \of {p(n)/n}$$p(n)/nn

Let p(n) denote the partition function and let \(\Delta \) be the difference operator with respect to n. In this paper, we obtain a lower bound for \(\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}\), leading to a proof of a conjecture of Sun on the log-convexity of \(\{\root n \of {p(n)/n}\}_{n\ge 60}\). Using the same argument, it can be shown that for any real number \(\alpha \), there exists an integer \(n(\alpha )\) such that the sequence \(\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}\) is log-convex. Moreover, we show that \(\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}\). Finally, by finding an upper bound for \(\Delta ^2 \log \root n-1 \of {p(n-1)}\), we establish an inequality on the ratio \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\).     

The nonlocal Liouville-type equation in $$\mathbb {}$$ and conformal immersions of the disk with boundary singularities

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence \(u_k :\mathbb {R}\rightarrow \mathbb {R}\) of solutions to

$$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$

(1)

with \(K_k\) bounded in \(L^\infty \) and \(e^{u_k}\) bounded in \(L^1\) uniformly with respect to k, we show that up to extracting a subsequence \(u_k\) can blow-up at (at most) finitely many points \(B=\{a_1,\ldots , a_N\}\) and that either (i) \(u_k\rightarrow u_\infty \) in \(W^{1,p}_{{{\mathrm{loc}}}}(\mathbb {R}{\setminus } B)\) and \(K_ke^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}K_\infty e^{u_\infty }+ \sum _{j=1}^N \pi \delta _{a_j}\), or (ii) \(u_k\rightarrow -\infty \) uniformly locally in \(\mathbb {R}{\setminus } B\) and \(K_k e^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}\sum _{j=1}^N \alpha _j \delta _{a_j}\) with \(\alpha _j\ge \pi \) for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis–Merle and Li–Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (\(\alpha _j=\pi \) and \(\alpha _j\ge \pi \)) which are not known in dimension 2 under the weak assumption that \((K_k)\) be bounded in \(L^\infty \) and is allowed to change sign.

     

The $$MMAP\left[ K \right]/PH\left[ K \right]/1$$ queues with a last-come-first-served preemptive service discipline

This paper studies two queueing systems with a Markov arrival process with marked arrivals and PH-distribution service times for each type of customer. Customers (regardless of their types) are served on a last-come-first-served preemptive resume and repeat basis, respectively. The focus is on the stationary distribution of queue strings in the system and busy periods. Efficient algorithms are developed for computing the stationary distribution of queue strings, the mean numbers of customers served in a busy period, and the mean length of a busy period. Comparison is conducted numerically between performance measures of queueing systems with preemptive resume and preemptive repeat service disciplines. A counter-intuitive observation is that for a class of service time distributions, the repeat discipline performs better than the resume one. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multiple expansions of real numbers with digits set $$\left\{ 0,1,q\right\} $$ 0 , 1 , q

Mathematische Zeitschrift - For $$q&gt;1$$ we consider expansions in base q with digits set $$\left\{ 0,1,q\right\} $$ . Let $${{\mathcal {U}}}_q$$ be the set of points which have a unique...     

The sharp lower bound for the volume of threefolds of general type with $${\chi({\mathcal O}_X)=1}$$

Let V be a smooth projective threefold of general type. Denote by K ³, a rational number, the self-intersection of the canonical sheaf of any minimal model of V. One defines K ³ as a canonical volume of V. The paper is devoted to proving the sharp lower bound which can be reached by an example: . Lei Zhu was supported by Fudan Graduate Students’ Innovation Projects (EYH5928004).     

On the Convergence of Products γ^-sh1hn in the Adams Spectral Sequence

Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.     

Representation of an algebra associated with the Dynkin graph $$\tilde E_7$$

We describe the structure of pairs of self-adjoint operators A and B whose spectra belong to the set {±1/2, ±3/2} and for which (A+B)²=I. Such pairs of operators determine a representation of a *-algebra associated with the extended Dynkin graph .__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1193–1202, September, 2004.This revised version was published online in May 2005 with a corrected article title.     

Global well posedness of traffic flow models with phase transitions

This note is devoted to the study of traffic flow models that develop phase transitions. From the analytical point of view, this is a first example of a well posedness result for conservation laws developing phase transitions, which is independent from the number of phase boundaries in the initial data or in the solutions. We consider below the Cauchy problem as well as the problem with boundaries.     

On the structure of level sets of uniform and Lipschitz quotient mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{R} $$

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient mappings from . We show that if , is a uniform quotient mapping then for every has a bounded number of components, each component of separates and the upper bound of the number of components depends only on and the moduli of co-uniform and uniform continuity of .Next we prove that all level sets of any co-Lipschitz uniformly continuous mapping from to are locally connected, and we show that for every pair of a constant and a function with , there exists a natural number , so that for every co-Lipschitz uniformly continuous map with a co-Lipschitz constant and a modulus of uniform continuity , there exists a natural number and a finite set with card so that for all has exactly components, has exactly components and each component of is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of for are also described - they have a finite tree structure.