Existence of strong minimizers for the Griffith static fracture model in dimension two

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate $SB{D}^p$ functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSB{D}^p$ functions by $SB{V}^p$ ones.     

Degree-equipartite graphs

A graph $G$ of order $2n$ is called degree-equipartite if for every $n$-element set $A?V\left(G\right)$, the degree sequences of the induced subgraphs $G\left[A\right]$ and $G[V\left(G\right)?A]$ are the same. In this paper, we characterize all degree-equipartite graphs. This answers Problem 1 in the paper by Grünbaum et al. [B. Grünbaum, T. Kaiser, D. Král, and M. Rosenfeld, Equipartite graphs, Israel J. Math. 168 (2008) 431–444].     

Markov processes conditioned on their location at large exponential times

Suppose that ${\left(X_t\right)}_{t\ge 0}$ is a one-dimensional Brownian motion with negative drift $?\mu$. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like $X$ conditioned to hit 0, after which time it behaves as $X$ killed at the last time $X$ visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift $+\mu$ when it is negative, like Brownian motion with negative drift $?\mu$ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state $a$ at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.     

On several problems about automorphisms of the free group of rank two

Let $F_n$ be a free group of rank n generated by $x_1,\dots ,x_n$. In this paper we discuss three algorithmic problems related to automorphisms of $F_2$.A word $u=u(x_1,\dots ,x_n)$ of $F_n$ is called positive if no negative exponents of $x_i$ occur in u. A word u in $F_n$ is called potentially positive if $?(u)$ is positive for some automorphism ? of $F_n$. We prove that there is an algorithm to decide whether or not a given word in $F_2$ is potentially positive, which gives an affirmative solution to problem F34a in [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] for the case of $F_2$.Two elements u and v in $F_n$ are said to be boundedly translation equivalent if the ratio of the cyclic lengths of $?(u)$ and $?(v)$ is bounded away from 0 and from ∞ for every automorphism ? of $F_n$. We provide an algorithm to determine whether or not two given elements of $F_2$ are boundedly translation equivalent, thus answering question F38c in the online version of [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] for the case of $F_2$.We also provide an algorithm to decide whether or not a given finitely generated subgroup of $F_2$ is the fixed point group of some automorphism of $F_2$, which settles problem F1b in [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] in the affirmative for the case of $F_2$.     

Tighter approximation bounds for LPT scheduling in two special cases

$Q||C_{\max }$ denotes the problem of scheduling n jobs on m machines of different speeds such that the makespan is minimized. In the paper two special cases of $Q||C_{\max }$ are considered: case I, when $m?1$ machine speeds are equal, and there is only one faster machine; and case II, when machine speeds are all powers of 2 (2-divisible machines). Case I has been widely studied in the literature, while case II is significant in an approach to design so called monotone algorithms for the scheduling problem.We deal with the worst case approximation ratio of the classic list scheduling algorithm ‘Largest Processing Time (LPT)’. We provide an analysis of this ratio $\mathit{Lpt}/\mathit{Opt}$ for both special cases: For ‘one fast machine’, a tight bound of $(\sqrt{3}+1)/2\approx 1.3660$ is given. For 2-divisible machines, we show that in the worst case $1.3673<\mathit{Lpt}/\mathit{Opt}<1.4$. Besides, we prove another lower bound of $955/699>(\sqrt{3}+1)/2$ when LPT breaks ties arbitrarily.To our knowledge, the best previous lower and upper bounds were $(4/3,3/2?1/2m]$ in case I [T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM Journal on Computing 6 (1) (1977) 155–166], respectively $[4/3?1/3m,3/2]$ in case II [R.L. Graham, Bounds on multiprocessing timing anomalies, SIAM Journal on Applied Mathematics 17 (1969) 416–429; A. Kovács, Fast monotone 3-approximation algorithm for scheduling related machines, in: Proc. 13th Europ. Symp. on Algs. (ESA), in: LNCS, vol. 3669, Springer, 2005, pp. 616–627]. Moreover, Gonzalez et al. conjectured the lower bound 4/3 to be tight in the ‘one fast machine’ case [T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM Journal on Computing 6 (1) (1977) 155–166].     

Collision and escape orbits in a generalized Hénon–Heiles model

The motion of a material point of unit mass in a field determined by a generalized Hénon–Heiles potential $U=A{q}_1^2+B{q}_2^2+C{q}_1^2{q}_2+D{q}_2^3$, with $(q_1,q_2)=$ standard Cartesian coordinates and $(A,B,C,D)\in {(0,\infty )}^2\times R^2$, is addressed for two limit situations: collision and escape. Using McGehee-type transformations, the corresponding collision and infinity boundary manifolds pasted on the phase space are determined. These are fictitious manifolds, but, due to the continuity with respect to initial data, their flow determines the near by orbit behaviour.The dynamics on the collision and infinity manifolds is fully described. The topology of the flow on the collision manifold is independent of the parameters. In the full phase space, while spiraling collision orbits are present, most of the orbits avoid collision. The topology of the flow on the infinity manifold changes as the ratio between $C$ and $D$ varies. More precisely, there are two symmetric pitchfork bifurcations along the line $2C?3D=0$, due to the reshaping of the potential along the bifurcation line. Besides rectilinear and spiraling orbits, the near-escape dynamics includes oscillatory orbits, for which angular momentum alternates sign.     

Solving bivariate systems using Rational Univariate Representations

Given two coprime polynomials $P$ and $Q$ in $\mathbb{Z}[x,y]$ of degree bounded by $d$ and bitsize bounded by $\tau$, we address the problem of solving the system $\{P,Q\}$. We are interested in certified numerical approximations or, more precisely, isolating boxes of the solutions. We are also interested in computing, as intermediate symbolic objects, rational parameterizations of the solutions, and in particular Rational Univariate Representations (RURs), which can easily turn many queries on the system into queries on univariate polynomials. Such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system.We present new algorithms for computing linear separating forms, RUR decompositions and isolating boxes of the solutions. We show that these three algorithms have worst-case bit complexity ${\tilde{O}}_B(d^6+d^5\tau )$, where $\tilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity. We also present probabilistic Las Vegas variants of our two first algorithms, which have expected bit complexity ${\tilde{O}}_B(d^5+d^4\tau )$. A key ingredient of our proofs of complexity is an amortized analysis of the triangular decomposition algorithm via subresultants, which is of independent interest.     

Multiplicity results for two kinds of equivariant systems

Two equivariant problems of the form $\epsilon \mathrm{\Delta }u=\nabla F\left(u\right)$ are considered, where $F$ is a real function which is invariant under the action of a group $G$, and, using Morse theory, for each problem an arbitrarily great number of orbits of solutions is founded, choosing $\epsilon$ suitably small.The first problem is a $O\left(2\right)$-equivariant system of two equations, which can be seen as a complex Ginzburg-Landau equation, while the second one is a system of $m$ equations which is equivariant for the action of a finite group of real orthogonal matrices $m\times m$.     

Lattice BGK simulations of double diffusive natural convection in a rectangular enclosure in the presences of magnetic field and heat source

In this paper, we develop a temperature-concentration lattice Bhatnagar-Gross-Krook (TCLBGK) model, with a robust boundary scheme for simulating the two-dimensional, hydromagnetic, double-diffusive convective flow of a binary gas mixture in a rectangular enclosure, in which the upper and lower walls are insulated, while the left and right walls are at a constant temperature and concentration and a uniform magnetic field is applied in the $x$-direction. In the model, the velocity, temperature and concentration fields are solved by three independent LBGK equations which are combined into a coupled equation for the whole system. In our simulations, we take the Prandtl number $Pr=1$, the Lewis number $Le=2$, the thermal Rayleigh number $R{a}_T={10}^5,{10}^6$, the Hartmann number $Ha=0,10,25,50$, the dimensionless heat generation or absorption $?=0.0,?1.0$, the buoyancy ratio $N=0.8,1.3$, and the aspect ratio $A=2$ for the enclosure. The numerical results are found to be in good agreement with those of previous studies [A.J. Chamkha, H. Al-Naser, Hydromagnetic double-diffusive convection in a rectangular enclosure with opposing temperature and concentration gradients, Int. J. Heat Mass Transfer 45 (2002) 2465C2483].     

Interval conjectures for level Hilbert functions

We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the Interval Conjecture (IC): If, for some positive integer α, $(1,h_1,\dots ,h_i,\dots ,h_e)$ and $(1,h_1,\dots ,h_i+\alpha ,\dots ,h_e)$ are both level h-vectors, then $(1,h_1,\dots ,h_i+\beta ,\dots ,h_e)$ is also level for each integer $\beta =0,1,\dots ,\alpha$. In the Gorenstein case, i.e. when $h_e=1$, we also supply the Gorenstein Interval Conjecture (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetric entries, say $h_i$ and $h_{e?i}$, of a Gorenstein h-vector.These conjectures are inspired by the research performed in this area over the last few years. A series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of all level Hilbert functions. Therefore, our conjectures would at least provide the existence of a very strong — and natural — form of order in the structure of such important and complicated sets.We are still far from proving the conjectures at this point. However, we will already solve a few interesting cases, especially when it comes to the IC, in this paper. Among them, that of Gorenstein h-vectors of socle degree 4, that of level h-vectors of socle degree 2, and that of non-unimodal level h-vectors of socle degree 3 and any given codimension.     

On the existence of retransmission permutation arrays

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of $\{1,\dots ,n\}$, and for $1?i?n$, all $n$ symbols occur in each $i\times ?\frac{n}{i}?$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-1, 2, 3, 4 $\mathrm{RPA}\left(n\right)$ and a type-1, 2 latin $\mathrm{RPA}\left(n\right)$.     

The Cauchy problem for a generalized Camassa–Holm equation

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces $H^s$, $s>3/2$ for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in $H^r$-topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.     

Some statistics on Stirling permutations and Stirling derangements

A permutation of the multiset $\{1,1,2,2,\dots ,n,n\}$ is called a Stirling permutation of order $n$ if every entry between the two occurrences of $i$ is greater than $i$ for each $i\in \{1,2,\dots ,n\}$. In this paper, we introduce the definitions of block, even indexed entry, odd indexed entry, Stirling derangement, marked permutation and bicolored increasing binary tree. We first study the joint distribution of ascent plateaux, even indexed entries and left-to-right minima over the set of Stirling permutations of order $n$. We then present an involution on Stirling derangements.