Quasi-stationary stefan problem with values on the front depending on its geometry

The problem presented below is a singular-limit problem of the extension of the Cahn-Hilliard model obtained via introducing the asymmetry of the surface tension tensor under one of the truncations (approximations) of the inner energy [2, 5–8, 10, 12, 13].     

Mutually-adjoint boundary-value problems with deviation from the characteristic for a multidimensional Gellerstedt equation

In this paper, we study mutually-adjoint boundary-value problems with a deviation from the characteristic for multidimensional Gellerstedt equation. In [3, 4], for the equation of the vibration of a string, the boundary-value problem with a deviation from the characteristic was studied, where the main attention was paid to the study of such problems for hyperbolic equations. For hyperbolic equations on the plane, this problem was studied in [5, 9].     

Further results on factorization of meromorphic solutions of partial differential equations

This paper is a continuation of Hu-Yang [2]. Here we extend Malmquist type theorem ofalgebraic differential equations of Steinmetz [3] and Tu [4] to higher order partial differential equations. The results also generalize Theorems 4.2 and 4.3 in [2].     

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence C_N . The mixed middle curvature and mixed curvature on C_N are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].     

Generalizations Of A “folk-Theorem” On Simple Functional Equations In A Single Variable

It is known (“mathematical folklore”) that, to every function defined on [1,2], there exists a solution of f(2x) = 2f(x) on ]0,∞[ of which the given function is a restriction to [1,2]. With a little care in the definition on [1,2], with still a lot of arbitrariness left, the resulting solution will be continuous, even C^∞ on ]0,∞[ (a behaviour markedly different from that of the Cauchy equation f(x + y) = f(x) + f(y), which has f(x) = cx as only continuous solution on ]0,∞[, even though, with y = x, it degenerates into the above equation). If 0 is added to the domain and we choose the “arbitrary function” bounded on [1,2[, then the solution will even be continuous (from the right) at 0. However, if f is supposed to be differentiable at 0 (from the right), then f(x) = cx is the only solution on [0,∞[. p In this paper we present similar and further results concerning general, Cⁿ (n ≤ ∞), analytic, locally monotonie or γ-th order convex solutions of the somewhat more general equation f(kx) = k^γf(x) (k ≠ 1 a positive, γ a real constant), which seems to be of importance in meterology. Some of the results are not quite what one expects.     

Exact Lagrangian immersions with one double point revisited

We study exact Lagrangian immersions with one double point of a closed orientable manifold $K$ into $\mathbb{C }^{n}$ . We prove that if the Maslov grading of the double point does not equal $1$ then $K$ is homotopy equivalent to the sphere, and if, in addition, the Lagrangian Gauss map of the immersion is stably homotopic to that of the Whitney immersion, then $K$ bounds a parallelizable $(n+1)$ -manifold. The hypothesis on the Gauss map always holds when $n=2k$ or when $n=8k-1$ . The argument studies a filling of $K$ obtained from solutions to perturbed Cauchy–Riemann equations with boundary on the image $f(K)$ of the immersion. This leads to a new and simplified proof of some of the main results of Ekholm and Smith (Exact Lagrangian immersions with a single double point 2011)). which treated Lagrangian immersions in the case $n=2k$ by applying similar techniques to a Lagrange surgery of the immersion, as well as to an extension of these results to the odd-dimensional case.     

Exponential polynomials on commutative hypergroups

Polynomials and exponential polynomials play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative groups. Recently several new results have been published in this field [2–4,6]. Spectral analysis and spectral synthesis has been studied on some types of commutative hypergroups, as well. However, a satisfactory definition of exponential monomials on general commutative hypergroups has not been available so far. In [5,7,8] and [9], the authors use a special concept on polynomial and Sturm–Liouville-hypergroups. Here we give a general definition which covers the known special cases.     

Tensor product varieties,perverse sheaves,and stability conditions

The space spanned by the class of simple perverse sheaves in Zheng (2008) without localization is isomorphic to the tensor product of a Verma module with a tensor product of irreducible integrable highest weight modules of the quantum enveloping algebra associated with a graph. Under the isomorphism, the simple perverse sheaves get identified with the canonical basis elements of the tensor product module. The two stability conditions coincide with the localization process in Zheng (2008), by using supports and singular supports of complexes of sheaves, respectively.     

Classification of Sol lattices

Sol geometry is one of the eight homogeneous Thurston 3-geometries $${\bf E}^{3}, {\bf S}^{3}, {\bf H}^{3}, {\bf S}^{2}\times{\bf R}, {\bf H}^{2}\times{\bf R}, \widetilde{{\bf SL}_{2}{\bf R}}, {\bf Nil}, {\bf Sol}.$$ In [13] the densest lattice-like translation ball packings to a type (type I/1 in this paper) of Sol lattices has been determined. Some basic concept of Sol were defined by Scott in [10], in general. In our present work we shall classify Sol lattices in an algorithmic way into 17 (seventeen) types, in analogy of the 14 Bravais types of the Euclidean 3-lattices, but infinitely many Sol affine equivalence classes, in each type. Then the discrete isometry groups of compact fundamental domain (crystallographic groups) can also be classified into infinitely many classes but finitely many types, left to other publication. To this we shall study relations between Sol lattices and lattices of the pseudoeuclidean (or here rather called Minkowskian) plane [1]. Moreover, we introduce the notion of Sol parallelepiped to every lattice type. From our new results we emphasize Theorems 3?C6. In this paper we shall use the affine model of Sol space through affine-projective homogeneous coordinates [6] which gives a unified way of investigating and visualizing homogeneous spaces, in general.     

To the question on the almost everywhere convergence of the Riesz means of double orthogonal series

Sufficient conditions of the classical type ensuring the almost everywhere (a.e.) convergence of the nonnegative-order Riesz means of double orthogonal series are indicated. Analogies of the onedimensional results of Kolmogoroff [7] and Kaczmarz?CZygmund [5, 12] have been obtained for the Cesaro means and those of Zygmund [13] for the Riesz means. These analogies establish the a.e. equiconvergence of the lacunary subsequences of rectangular partial sums and of the entire sequence of Riesz means, generalize the corresponding results of Moricz [9] for the Cesaro a.e. summability by (C, 1, 1), (C, 1, 0), and (C, 0, 1) methods of double orthogonal series, and were announced earlier without proofs in the author??s work [3].     

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009).     

Comparing Invariants of SK1

In this text, we compare an invariant of the reduced Whitehead group SK ₁ of a central simple algebra recently introduced by Kahn (2010) to other invariants of SK ₁. Doing so, we prove the non-triviality of Kahn’s invariant using the non-triviality of an invariant introduced by Suslin (1991) which is non-trivial for Platonov’s examples of non-trivial SK ₁ (Platonov, Math USSR Izv 10(2):211–243, 1976). We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras (Merkurjev 1995).     

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].     

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 remark on Feuerbach hyperbolas

Recently many authors have studied properties of triangles and the theory of perspective triangles in the Euclidean plane (see Kimberling et al. J Geom Graph 14:1–14, 2010; Kimberling et al. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html, 2012; Moses and Kimberling J Geom Graph 13:15–24, 2009; Moses and Kimberling Forum Geom 11:83–93, 2011; Odehnal Elem Math 61:74–80, 2006; Odehnal Forum Geom 10:35–40, 2010; Odehnal J Geom Graph 15: 45–67, 2011). The aim of this paper is to present a new approach to the construction of points on the Feuerbach hyperbola. Surprisingly, these points can be obtained as centers of perspectivity of a triangle ABC and a certain one-parametric set of triangles A′B′C′. The presented construction is based on partitions of the triangle’s sides and—in a way—dual to the construction of points on the Kiepert hyperbola. It can also be generalized to spherical triangles. The proofs are based on an affine property of triangles, which amazingly can also be used for the proof of the spherical theorem.     

Constructing exact Lagrangian immersions with few double points

We establish, as an application of the results from Eliashberg and Murphy (Lagrangian caps, 2013), an h-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space ${\mathbb{R}^6_{\rm st}}$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding ${S^1 \times S^2 \to \mathbb{R}^6_{\rm st}}$ with vanishing Maslov class.     

A Characterization of the Finite Veronesean by Intersection Properties

A combinatorial characterization of the Veronese variety of all quadrics in PG(n, q) by means of its intersection properties with respect to subspaces is obtained. The result relies on a similar combinatorial result on the Veronesean of all conics in the plane PG(2, q) by Ferri [Atti Accad. Naz. Lincei Rend. 61(6), 603?C610 (1976)], Hirschfeld and Thas [General Galois Geometries. Oxford University Press, New York (1991)], and Thas and Van Maldeghem [European J. Combin. 25(2), 275?C285 (2004)], and a structural characterization of the quadric Veronesean by Thas and Van Maldeghem [Q. J. Math. 55(1), 99?C113 (2004)].     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

Geometric rigidity for analytic estimates of Müller–?verák

In the paper Müller–?verák (J Differ Geom 42(2):229–258, 1995) conformally immersed surfaces with finite total curvature were studied. In particular it was shown that surfaces with total curvature ${\int_{\Sigma} |A|^2 < 8 \pi}$ in dimension three were embedded and conformal to the plane with one end. Here, using techniques from Kuwert–Li (W ^2,2-conformal immersions of a closed Riemann surface into R ⁿ . arXiv:1007.3967v2 [math.DG], 2010), we will show that if the total curvature ${ \int_{\Sigma}|A|^2\leq8\pi}$ , then we are either embedded and conformal to the plane, isometric to a catenoid or isometric to Enneper’s minimal surface. In fact the technique of our proof shows that if we are conformal to the plane, then if n?≥ 3 and ${ \int_{\Sigma} | A|^{2}\leq 16 \pi }$ then Σ is embedded or Σ is the image of a generalized catenoid inverted at a point on the catenoid. In order to prove these theorems, we prove a Gauss–Bonnet theorem for surfaces with complete ends and isolated finite area singularities which extends a theorem of Jorge-Meeks (Topology 22(2):203–221, 1983). Using this theorem, we then prove an inversion formula for the Willmore energy.     

Finite simple 3′-groups are cyclic or Suzuki groups

In this note we prove that all finite simple 3′-groups are cyclic of prime order or Suzuki groups. This is well known in the sense that it is mentioned frequently in the literature, often referring to unpublished work of Thompson. Recently an explicit proof was given by Aschbacher [3], as a corollary of the classification of ${\mathcal{S}_3}$ -free fusion systems. We argue differently, following Glauberman’s comment in the preface to the second printing of his booklet [8]. We use a result by Stellmacher (see [12]), and instead of quoting Goldschmidt’s result in its full strength, we give explicit arguments along his ideas in [10] for our special case of 3′-groups.