A multi-dimensional SRBM: geometric views of its product form stationary distribution

We present a geometric interpretation of a product form stationary distribution for a \(d\) -dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative orthant. The \(d\) -dimensional SRBM data can be equivalently specified by \(d+1\) geometric objects: an ellipse and \(d\) rays. Using these geometric objects, we establish necessary and sufficient conditions for characterizing product form stationary distribution. The key idea in the characterization is that we decompose the \(d\) -dimensional problem to \(\frac{1}{2}d(d-1)\) two-dimensional SRBMs, each of which is determined by an ellipse and two rays. This characterization contrasts with the algebraic condition of Harrison and Williams (Ann Probab 15:115–137, 1987b). A \(d\) -station tandem queue example is presented to illustrate how the product form can be obtained using our characterization. Drawing the two-dimensional results in Avram et al. (Queueing Syst 37:259–289, 2001), Dai and Miyazawa (Queueing Syst 74:181–217, 2013), we discuss potential optimal paths for a variational problem associated with the three-station tandem queue.     

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let \(K\subset \mathbb R ^N\) be a convex body containing the origin. A measurable set \(G\subset \mathbb R ^N\) with positive Lebesgue measure is said to be uniformly \(K\) -dense if, for any fixed \(r>0\) , the measure of \(G\cap (x+r K)\) is constant when \(x\) varies on the boundary of \(G\) (here, \(x+r K\) denotes a translation of a dilation of \(K\) ). We first prove that \(G\) must always be strictly convex and at least \(C^{1,1}\) -regular; also, if \(K\) is centrally symmetric, \(K\) must be strictly convex, \(C^{1,1}\) -regular and such that \(K=G-G\) up to homotheties; this implies in turn that \(G\) must be \(C^{2,1}\) -regular. Then for \(N=2\) , we prove that \(G\) is uniformly \(K\) -dense if and only if \(K\) and \(G\) are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on \(K\) and \(G\) , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near \(r=0\) for the measure of \(G\cap (x+r\,K)\) (needed in 2008).     

The Gasca–Maeztu conjecture for n=5

An \(n\) -poised set in two dimensions is a set of nodes admitting unique bivariate interpolation with polynomials of total degree at most \(n\) . We are interested in poised sets with the property that all fundamental polynomials are products of linear factors. Gasca and Maeztu (Numer Math 39:1–14, 1982) conjectured that every such set necessarily contains \(n+1\) collinear nodes. Up to now, this had been confirmed only for \(n\le 4\) , the case \(n=4\) having been proved for the first time by Busch (Rev Un Mat Argent 36:33–38, 1990). In the present paper, we prove the case \(n=5\) with new methods that might also be useful in deciding the still open cases for \(n\ge 6\) .     

A vanishing result for strictly p-convex domains

In view of Andreotti and Grauert (Bull Soc Math France 90:193–259, 1962) vanishing theorem for \(q\) -complete domains in \(\mathbb C ^{n}\) , we reprove a vanishing result by Sha (Invent Math 83(3):437–447, 1986), and Wu (Indiana Univ Math J 36(3):525–548, 1987), for the de Rham cohomology of strictly \(p\) -convex domains in \(\mathbb R ^n\) in the sense of Harvey and Lawson (The foundations of \(p\) -convexity and \(p\) -plurisubharmonicity in riemannian geometry. arXiv:1111.3895v1 [math.DG]). Our proof uses the \({L}^2\) -techniques developed by Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Publishing Co, Amsterdam 1990), and Andreotti and Vesentini (Inst Hautes Études Sci Publ Math 25:81–130, 1965).     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

Exponents of polar codes using algebraic geometric code kernels

Reed–Solomon and BCH codes were considered as kernels of polar codes by Mori and Tanaka (IEEE Information Theory Workshop, 2010, pp 1–5) and Korada et al. (IEEE Trans Inform Theory 56(12):6253–6264, 2010) to create polar codes with large exponents. Mori and Tanaka showed that Reed–Solomon codes over the finite field \(\mathbb {F}_q\) with \(q\) elements give the best possible exponent among all codes of length \(l \le q\) . They also stated that a Hermitian code over \(\mathbb {F}_{2^r}\) with \(r \ge 4\) , a simple algebraic geometric code, gives a larger exponent than the Reed–Solomon matrix over the same field. In this paper, we expand on these ideas by employing more general algebraic geometric (AG) codes to produce kernels of polar codes. Lower bounds on the exponents are given for kernels from general AG codes, Hermitian codes, and Suzuki codes. We demonstrate that both Hermitian and Suzuki kernels have larger exponents than Reed–Solomon codes over the same field, for \(q \ge 3\) ; however, the larger exponents are at the expense of larger kernel matrices. Comparing kernels of the same size, though over different fields, we see that Reed–Solomon kernels have larger exponents than both Hermitian and Suzuki kernels. These results indicate a tradeoff between the exponent, kernel matrix size, and field size.     

Smooth Fano Polytopes with Many Vertices

The \(d\) -dimensional simplicial, terminal, and reflexive polytopes with at least \(3d-2\) vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves on previous results of Casagrande (Ann Inst Fourier (Grenoble) 56(1):121–130, 2006) and Øbro (Manuscr Math 125(1): 69–79, 2008). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics.     

Non-Lipschitz points and the {\textit{SBV}} regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function \(T\) under controllability conditions which do not imply the Lipschitz continuity of \(T\) . We consider first the case of normal linear control systems with constant coefficients in \({\mathbb {R}}^N\) . We characterize points around which \(T\) is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call \(\mathcal {S}\) . Furthermore, we show that \(\mathcal {S}\) is countably \(\mathcal {H}^{N-1}\) -rectifiable with positive \(\mathcal {H}^{N-1}\) -measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set \(\mathcal {S}\) in a neighborhood of the origin in a similar way and prove the \(\mathcal {H}^1\) -rectifiability of \(\mathcal {S}\) and that \(\mathcal {H}^1(\mathcal {S})>0\) . In both cases, \(T\) is known to have epigraph with positive reach, hence to be a locally \(BV\) function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of \(DT\) must be concentrated in \(\mathcal {S}\) , our analysis yields that \(T\) is locally \(SBV\) , i.e., the Cantor part of \(DT\) vanishes. Our results imply also that \(T\) is differentiable outside a \(\mathcal {H}^{N-1}\) -rectifiable set. With small changes, our results are valid also in the case of multiple control input.     

Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve

A projective nonsingular plane algebraic curve of degree \(d\ge 4\) is called maximally symmetric if it attains the maximum order of the automorphism groups for complex nonsingular plane algebraic curves of degree \(d\) . For \(d\le 7\) , all such curves are known. Up to projectivities, they are the Fermat curve for \(d=5,7\) ; see Kaneta et al. (RIMS Kokyuroku 1109:182–191, 1999) and Kaneta et al. (Geom. Dedic. 85:317–334, 2001), the Klein quartic for \(d=4\) , see Hartshorne (Algebraic Geometry. Springer, New York, 1977), and the Wiman sextic for \(d=6\) ; see Doi et al. (Osaka J. Math. 37:667–687, 2000). In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every \(d\ge 8\) showing that the Fermat curve is the unique maximally symmetric nonsingular curve of degree \(d\) with \(d\ge 8\) , up to projectivity. For \(d=11,13,17,19\) , this characterization of the Fermat curve has already been obtained; see Kaneta et al. (Geom. Dedic. 85:317–334, 2001).     

On Sub-determinants and the Diameter of Polyhedra

We derive a new upper bound on the diameter of a polyhedron \(P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}\) , where \(A \in {\mathbb {Z}}^{m\times n}\) . The bound is polynomial in \(n\) and the largest absolute value of a sub-determinant of \(A\) , denoted by \(\Delta \) . More precisely, we show that the diameter of \(P\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\) . If \(P\) is bounded, then we show that the diameter of \(P\) is at most \(O(\Delta ^2 n^{3.5}\log n\Delta )\) . For the special case in which \(A\) is a totally unimodular matrix, the bounds are \(O(n^4\log n)\) and \(O(n^{3.5}\log n)\) respectively. This improves over the previous best bound of \(O(m^{16}n^3(\log mn)^3)\) due to Dyer and Frieze (Math Program 64:1–16, 1994).     

Orthogonality Principle for Bilinear Littlewood–Paley Decompositions

We explore Littlewood–Paley like decompositions of bilinear Fourier multipliers. Grafakos and Li (Am. J. Math. 128(1):91–119 2006) showed that a bilinear symbol supported in an angle in the positive quadrant is bounded from \(L^p\times L^q\) into \(L^r\) if its restrictions to dyadic annuli are bounded bilinear multipliers in the local \(L^2\) case \(p\ge 2\) , \(q\ge 2\) , \(r= 1/(p^{-1}+q^{-1})\le 2\) . We show that this range of indices is sharp and also discuss similar results for multipliers supported near axis and negative diagonal.     

A refined stable restriction theorem for vector bundles on quadric threefolds

Let \(E\) be a stable rank 2 vector bundle on a smooth quadric threefold \(Q\) in the projective 4-space \(P\) . We show that the hyperplanes \(H\) in \(P\) for which the restriction of \(E\) to the hyperplane section of \(Q\) by \(H\) is not stable form, in general, a closed subset of codimension at least 2 of the dual projective 4-space, and we explicitly describe the bundles \(E\) which do not enjoy this property. This refines a restriction theorem of Ein and Sols (Nagoya Math J 96:11–22, 1984) in the same way the main result of Coand? (J Reine Angew Math 428:97–110, 1992) refines the restriction theorem of Barth (Math Ann 226:125–150, 1977).     

A symmetry property for q-weighted Robinson–Schensted and other branching insertion algorithms

In [19], a \(q\) -weighted version of the Robinson–Schensted algorithm was introduced. In this paper, we show that this algorithm has a symmetry property analogous to the well-known symmetry property of the usual Robinson–Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin [5–8]. This approach, which uses ‘growth graphs’, can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other \(q\) -weighted versions of the Robinson–Schensted algorithm which have recently been introduced by Borodin–Petrov [2].     

Minimum degree of the difference of two polynomials over {\mathbb Q}, and weighted plane trees

A weighted bicolored plane tree (or just tree for short) is a bicolored plane tree whose edges are endowed with positive integral weights. The degree of a vertex is defined as the sum of the weights of the edges incident to this vertex. Using the theory of dessins d’enfants, which studies the action of the absolute Galois group on graphs embedded into Riemann surfaces, we show that a weighted plane tree is a graphical representation of a pair of coprime polynomials \(P,Q\in {\mathbb C}\,[x]\) such that: (a)  \(\deg P = \deg Q\) , and \(P\) and \(Q\) have the same leading coefficient; (b) the multiplicities of the roots of  \(P\) (respectively, of  \(Q\) ) are equal to the degrees of the black (respectively, white) vertices of the corresponding tree; (c) the degree of the difference \(P-Q\) attains the minimum which is possible for the given multiplicities of the roots of \(P\)  and  \(Q\) . Moreover, if a tree in question is uniquely determined by the set of its black and white vertex degrees (we call such trees unitrees), then the corresponding polynomials are defined over \({\mathbb Q}\) . The pairs of polynomials \(P,Q\) such that the degree of the difference \(P-Q\) attains the minimum, and especially those defined over \({\mathbb Q}\) , are related to some important questions of number theory. Dozens of papers, from 1965 (Birch et al. in Norske Vid Selsk Forh 38:65–69, 1965) to 2010 (Beukers and Stewart in J Number Theory 130:660–679, 2010), were dedicated to their study. The main result of this paper is a complete classification of the unitrees, which provides us with the most massive class of such pairs defined over  \({\mathbb Q}\) . We also study combinatorial invariants of the Galois action on trees, as well as on the corresponding polynomial pairs, which permit us to find yet more examples defined over  \({\mathbb Q}\) . In a subsequent paper, we compute the polynomials \(P,Q\) corresponding to all the unitrees.     

Une preuve de la conjecture d’Erdős,Joò et Komornik dans le cas des séries formelles

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdös, Joò and Komornik in (Bull Soc Math France 118:377–390, 1990), is the study of the set \(\Lambda _{m}(\beta )\) the spectrum of \(\beta \) and the determination of \(l^{m}(\beta )\) for Pisot number \(\beta \) , where \(\Lambda _{m}(\beta )\) denotes the set of numbers having at least one representation of the form \(\omega =\varepsilon _{n} \beta ^{n}+\varepsilon _{n-1}\beta ^{n-1}+\cdots +\varepsilon _{1}\beta +\varepsilon _{0},\) such that the \(\varepsilon _{i}\in \{-m,\ldots ,0,\ldots ,m\}\) , for all \(0\le i\le n\) , and \(l^{m}(\beta )=\inf \{|\omega |:\omega \in \Lambda _{m},\omega \ne 0\}.\) In this paper, we consider \(\Lambda _{m}(\beta )\) , where \(\beta \) is a formal power series over a finite field and the \(\varepsilon _{i}\) are polynomials of degree at most \(m\) for all \(0\le i\le n\) . Our main result is to give a full answer in the Laurent series case, to an old question of Erd?s and Komornik (Acta Math Hungar 79:57–83, 1998), as to whether \(l^{1}(\beta )=0\) for all non-Pisot numbers. More generally, we characterize the inequalities \(l^{m}(\beta )>0\) .     

Improved online algorithms for the batch scheduling of equal-length jobs with incompatible families to maximize the weighted number of early jobs

We consider the online scheduling of equal-length jobs with incompatible families on \(m\) identical batch machines. Each job has a release time, a deadline and a weight. Each batch machine can process up to \(b\) jobs (which come from the same family) simultaneously as a batch, where \(b\) is called the capacity of the machines. Our goal is to determine a preemption-restart schedule which maximizes the weighted number of early jobs. For this problem, Li et al. (Inf Process Lett 112:503–508, 2012) provided an online algorithm of competitive ratio \(3+2\sqrt{2}\) for both \(b=\infty \) and \(b<\infty \) . In this paper, we study two special cases of this problem. For the case that \(m=2\) , we first present a lower bound 2, and then provide an online algorithm with a competitive ratio of 3 for both \(b=\infty \) and \(b<\infty \) . For the case in which \(m=3\) , \(b=\infty \) and all jobs come from a common family, we present an online algorithm with a competitive ratio of \((8+2\sqrt{15})/3\approx 5.249\) .     

Adjunction and singular loci of hyperplane sections,II

Let \((X,L)\) be a smooth polarized variety of dimension \(n\) . Let \(A\in |L|\) be an irreducible hypersurface and let \(\Sigma \) be the singular locus of \(A\) . We assume that \(\Sigma \) is a smooth subvariety of dimension \(k\ge 2\) , and odd codimension \(\ge 3\) . Motivated from the results of Beltrametti et al. (J. Math. Soc. Jpn. 2014), we study the nefness and bigness of the adjoint bundle \(K_{\Sigma }+ (k-2)L_{\Sigma }\) in this framework. Several explicit examples show that the results are effective.     

Laplacian ideals,arrangements, and resolutions

The Laplacian matrix of a graph \(G\) describes the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann–Roch theory of \(G\) . The lattice ideal associated to the lattice generated by the columns of the Laplacian provides an algebraic perspective on this recently (re)emerging field. This binomial ideal \(I_G\) has a distinguished monomial initial ideal \(M_G\) , characterized by the property that the standard monomials are in bijection with the \(G\) -parking functions of the graph \(G\) . The ideal \(M_G\) was also considered by Postnikov and Shapiro (Trans Am Math Soc 356:3109–3142, 2004) in the context of monotone monomial ideals. We study resolutions of \(M_G\) and show that a minimal-free cellular resolution is supported on the bounded subcomplex of a section of the graphical arrangement of \(G\) . This generalizes constructions from Postnikov and Shapiro (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and of Perkinson et al. on the commutative algebra of Sandpiles. As a corollary, we verify a conjecture of Perkinson et al. regarding the Betti numbers of \(M_G\) and in the process provide a combinatorial characterization in terms of acyclic orientations.