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.     

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.     

Cell decomposition of some unitary group Rapoport–Zink spaces

In this paper we study the \(p\) -adic analytic geometry of the basic unitary group Rapoport–Zink spaces \(\mathcal {M}_K\) with signature \((1,n-1)\) . Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die reine und angewandte Mathematik 645:1–39, 2010), Fargues (Théorie de la réduction pour les groupes p-divisibles, prépublications. http://www.math.jussieu.fr/~fargues/Prepublications.html, 2010), and the Bruhat–Tits stratification of the reduced special fiber \(\mathcal {M}_{red}\) defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain \(\mathcal {D}_K\) in \(\mathcal {M}_K\) for the action of \(G(\mathbb {Q}_p)\times J_b(\mathbb {Q}_p)\) , the product of the associated \(p\) -adic reductive groups, and prove that \(\mathcal {M}_K\) admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic spaces (Preprint). arXiv:1011.1720, 2013).     

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).     

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).     

Extremal geometry of a Brownian porous medium

The path \(W[0,t]\) of a Brownian motion on a \(d\) -dimensional torus \(\mathbb T ^d\) run for time \(t\) is a random compact subset of \(\mathbb T ^d\) . We study the geometric properties of the complement \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) for \(d\ge 3\) . In particular, we show that the largest regions in \(\mathbb T ^d{{\setminus }} W[0,t]\) have a linear scale \(\varphi _d(t)=[(d\log t)/(d-2)\kappa _d t]^{1/(d-2)}\) , where \(\kappa _d\) is the capacity of the unit ball. More specifically, we identify the sets \(E\) for which \(\mathbb T ^d{{\setminus }} W[0,t]\) contains a translate of \(\varphi _d(t)E\) , and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) and the \(\varepsilon \) -cover time of \(\mathbb T ^d\) as \(\varepsilon \downarrow 0\) . Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1–14, 2003), are based on a large deviation estimate for the shape of the component with largest capacity in \(\mathbb T ^d{{\setminus }} W_{\rho (t)}[0,t]\) , where \(W_{\rho (t)}[0,t]\) is the Wiener sausage of radius \(\rho (t)\) , with \(\rho (t)\) chosen much smaller than \(\varphi _d(t)\) but not too small. The idea behind this choice is that \(\mathbb T ^d {{\setminus }} W[0,t]\) consists of “lakes”, whose linear size is of order \(\varphi _d(t)\) , connected by narrow “channels”. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of \(\mathbb T ^d {{\setminus }} W_{\rho (t)}[0,t]\) as \(t\rightarrow \infty \) . Our results give a complete picture of the extremal geometry of \(\mathbb T ^d{{\setminus }} W[0,t]\) and of the optimal strategy for \(W[0,t]\) to realise extreme events.     

The \bar{\partial }-Equation on an Annulus Between Two Strictly q-Convex Domains with Smooth Boundaries

In this paper, we study the global boundary regularity of the \(\bar{\partial }\) - equation on an annulus domain \(\Omega \) between two strictly \(q\) -convex domains with smooth boundaries in \(\mathbb{C }^n\) for some bidegree. To this finish, we first show that the \(\bar{\partial }\) -operator has closed range on \(L^{2}_{r, s}(\Omega )\) and the \(\bar{\partial }\) -Neumann operator exists and is compact on \(L^{2}_{r,s}(\Omega )\) for all \(r\ge 0\) , \(q\le s\le n-q- 1\) . We also prove that the \(\bar{\partial }\) -Neumann operator and the Bergman projection operator are continuous on the Sobolev space \(W^{k}_{r,s}(\Omega )\) , \(k\ge 0\) , \(r\ge 0\) , and \(q\le s\le n-q-1\) . Consequently, the \(L^{2}\) -existence theorem for the \(\bar{\partial }\) -equation on such domain is established. As an application, we obtain a global solution for the \(\bar{\partial }\) equation with Hölder and \(L^p\) -estimates on strictly \(q\) -concave domain with smooth \(\mathcal C ^2\) boundary in \(\mathbb{C }^n\) , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).     

Classifying codimension two multigerms

We generalise the operations of augmentation and concatenations defined in Cooper et al. (Compos Math 131(2):121–160, 2002) in order to obtain multigerms of analytic (or smooth) maps \((\mathbb {K}^n,S)\rightarrow (\mathbb {K}^p,0)\) with \(\mathbb {K}=\mathbb {C}\) or \(\mathbb {R}\) from monogerms and some special multigerms. We then prove that any corank 1 codimension 2 multigerm in Mather’s nice dimensions \((n,p)\) with \(n\ge p-1\) can be constructed using augmentations and these operations.     

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).     

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 \(l^{m}(\beta )>0\) .     

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).     

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).     

The Ising magnetization exponent on \mathbb{Z }^2 is 1/15

We prove that for the Ising model defined on the plane \(\mathbb Z ^2\) at \(\beta \,{=}\,\beta _c,\) the average magnetization under an external magnetic field \(h>0\) behaves exactly like $$\begin{aligned} \langle \sigma _0\rangle _{\beta _c, h} \asymp h^{\frac{1}{15}}. \end{aligned}$$ The proof, which is surprisingly simple compared to an analogous result for percolation [i.e. that \(\theta (p)=(p-p_c)^{5/36+o(1)}\) on the triangular lattice (Kesten in Commun Math Phys 109(1):109–156, 1987; Smirnov and Werner in Math Res Lett 8(5–6):729–744, 2001)] relies on the GHS inequality as well as the RSW theorem for FK percolation from Duminil-Copin et al. (Commun Pure Appl Math 64:1165–1198, 2011). The use of GHS to obtain inequalities involving critical exponents is not new; in this paper we show how it can be combined with RSW to obtain matching upper and lower bounds for the average magnetization.     

Algebraic quotient modules and subgroup depth

In Kadison J Pure Appl Alg 218:367–380, (2014) it was shown that subgroup depth may be computed from the permutation module of the left or right cosets: this holds more generally for a Hopf subalgebra, from which we note in this paper that finite depth of a Hopf subalgebra \(R \subseteq H\) is equivalent to the \(H\) -module coalgebra \(Q = H/R^+H\) representing an algebraic element in the Green ring of \(H\) or \(R\) . This approach shows that subgroup depth and the subgroup depth of the corefree quotient lie in the same closed interval of length one. We also establish a previous claim that the problem of determining if \(R\) has finite depth in \(H\) is equivalent to determining if \(H\) has finite depth in its smash product \(Q^* \# H\) . A necessary condition is obtained for finite depth from stabilization of a descending chain of annihilator ideals of tensor powers of \(Q\) . As an application of these topics to a centerless finite group \(G\) , we prove that the minimum depth of its group \(\mathbb {C}\,\) -algebra in the Drinfeld double \(D(G)\) is an odd integer, which determines the least tensor power of the adjoint representation \(Q\) that is a faithful \(\mathbb {C}\,G\) -module.     

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.     

Right-convergence of sparse random graphs

The paper is devoted to the problem of establishing right-convergence of sparse random graphs. This concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper-graphs \(\mathbb{G }_N, N\ge 1\) to some target graph \(W\) . The theory of dense graph convergence, including random dense graphs, is now well understood (Borgs et al. in Ann Math 176:151–219, 2012; Borgs et al. in Adv Math 219:1801–1851, 2008; Chatterjee and Varadhan in Eur J Comb 32:1000–1017, 2011; Lovász and Szegedy in J Comb Theory Ser B 96:933–957, 2006), but its counterpart for sparse random graphs presents some fundamental difficulties. Phrased in the statistical physics terminology, the issue is the existence of the limits of appropriately normalized log-partition functions, also known as free energy limits, for the Gibbs distribution associated with \(W\) . In this paper we prove that the sequence of sparse Erdös-Rényi graphs is right-converging when the tensor product associated with the target graph \(W\) satisfies a certain convexity property. We treat the case of discrete and continuous target graphs \(W\) . The latter case allows us to prove a special case of Talagrand’s recent conjecture [more accurately stated as level III Research Problem 6.7.2 in his recent book (Talagrand in Mean Field Models for Spin Glasses: Volume I: Basic examples. Springer, Berlin, 2010)], concerning the existence of the limit of the measure of a set obtained from \(\mathbb{R }^N\) by intersecting it with linearly in \(N\) many subsets, generated according to some common probability law. Our proof is based on the interpolation technique, introduced first by Guerra and Toninelli (Commun Math Phys 230:71–79, 2002) and developed further in (Abbe and Montanari in On the concentration of the number of solutions of random satisfiability formulas, 2013; Bayati et al. in Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010; Contucci et al. in Antiferromagnetic Potts model on the Erdös-Rényi random graph, 2011; Franz and Leone in J Stat Phys 111(3/4):535–564, 2003; Franz et al. in J Phys A Math Gen 36:10967–10985, 2003; Montanari in IEEE Trans Inf Theory 51(9):3221–3246, 2005; Panchenko and Talagrand in Probab Theory Relat Fields 130:312–336, 2004). Specifically, Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) establishes the right-convergence property for Erdös-Rényi graphs for some special cases of \(W\) . In this paper most of the results in Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) follow as a special case of our main theorem.     

A Note on Recent Papers by Grafakos and Teschl,and Estrada

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in \(\mathbb R^n\) and the Fourier transform of the same function in \(\mathbb R^{n+2}\) and \(\mathbb R^{n+1}\) , respectively, are located within known results on transplantation for Hankel transforms.     

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.