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 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.     

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).     

Cyclic parallel CR-submanifolds of maximal CR-dimension in a complex space form

We first classify \((2n-1)\) -dimensional cyclic parallel CR-submanifold \(M\) with CR-dimension \(n-1\) in a non-flat complex space form of constant holomorphic sectional curvature \(4c\) . Then, we prove that \(||\nabla h||^2\ge 4(n-1)c^2\) , where \(h\) is the second fundamental form on \(M\) . We also completely classify \((2n-1)\) -dimensional CR-submanifolds with CR-dimension \(n-1\) in a non-flat complex space form which satisfy the equality case of this inequality. This generalizes an inequality for real hypersurfaces in a non-flat complex space form obtained by Maeda (J Math Soc Jpn 28:529–540; 1976) and Chen et al. (Algebras Groups Geom 1:176–212; 1984) for complex projective and hyperbolic spaces, respectively.     

Modularity and monotonicity of games

The purpose of this paper is twofold. First, we generalize Kajii et al. (J Math Econ 43:218–230, 2007) and provide a condition under which for a game \(v\) , its Möbius inverse is equal to zero within the framework of the \(k\) -modularity of \(v\) for \(k \ge 2\) . This condition is more general than that in Kajii et al. (J Math Econ 43:218–230, 2007). Second, we provide a condition under which for a game \(v\) , its Möbius inverse takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of \(k\) -monotone games. Furthermore, this paper shows that the modularity of a game is related to \(k\) -additive capacities proposed by Grabisch (Fuzzy Sets Syst 92:167–189, 1997). To illustrate its application in the field of economics, we use these results to characterize a Gini index representation of Ben-Porath and Gilboa (J Econ Theory 64:443–467, 1994). Our results can also be applied to potential functions proposed by Hart and Mas-Colell (Econometrica 57:589–614, 1989) and further analyzed by Ui et al. (Math Methods Oper Res 74:427–443, 2011).     

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 \(n\ge 6\) .     

L_p-theory of free boundary problems of magnetohydrodynamics in multi-connected domains

We present the \(L_p\) -theory of solvability of free boundary problems of magnetohydrodynamics of viscous incompressible fluids in multi-connected domains constructed in the paper Solonnikov (Interf Free Bound 14:569–603, 2012) for \(p=2\) . The case of simply connected domains is studied in Padula and Solonnikov (J Math Sci 178:313–344, 2011), Solonnikov ( \(L_p\) -theory of free boundary problems of magnetohydrodynamics in simply connected domains, submitted to AMS Translations).     

C^{1,\alpha }-regularity for surfaces with H\in L^p

In this paper, we prove several results on the geometry of surfaces immersed in \(\mathbb {R}^3\) with small or bounded \(L^2\) norm of \(|A|\) . For instance, we prove that if the \(L^2\) norm of \(|A|\) and the \(L^p\) norm of \(H\) , \(p>2\) , are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded \(L^2\) norm of \(|A|\) , not necessarily small, then such a disk is graphical away from its boundary, provided that the \(L^p\) norm of \(H\) is sufficiently small, \(p>2\) . These results are related to previous work of Schoen–Simon (Surfaces with quasiconformal Gauss map. Princeton University Press, Princeton, vol 103, pp 127–146, 1983) and Colding–Minicozzi (Ann Math 160:69–92, 2004).     

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).     

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.     

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.     

Pseudo-effective cone of Grassmann bundles over a curve

Let \(E\) be a vector bundle over a smooth projective curve \(X\) defined over an algebraically closed field \(k\) . For any integer \(1\,\le \, r\, <\, \mathrm{rank}(E)\) , let \(\mathrm{Gr}_r(E)\,\longrightarrow \, X\) be a Grassmann bundle parametrizing all \(r\) dimensional quotients of the fibers of \(E\) . We compute the pseudo-effective cone in the real Néron–Severi group \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) . We prove that this cone coincides with the nef cone in \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) if and only if the vector bundle \(E\) is semistable (respectively, strongly semistable) when the characteristic of \(k\) is zero (respectively, positive). Examples are given to show that this characterization of (strong) semistability is not true for vector bundles on higher dimensional projective varieties.     

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).     

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 \(l^{m}(\beta )>0\) .     

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.     

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.