On singular Lüroth quartics

Plane quartics containing the ten vertices of a complete pentalateral and limits of them are called Lüroth quartics.The locus of singular Lüroth quartics has two irreducible components,both of codimension two in P14.We compute the degree of them and discuss the consequences of this computation on the explicit form of the Lüroth invariant.One important tool is the Cremona hexahedral equations of the cubic surface.We also compute the class in M 3 of the closure of the locus of nonsingular Lüroth quartics.     

A Class of singular pseudo-homogeneous pseudodi fferential operator

In this paper, we considered a class of weighted Sobolev space and in an uniform way researched Fourier-Bessel singular pseudodifferential operator (below denoted by SP_sDO) associated with pseudo-homogeneous (include homogeneous) symbol. Let     

Constructive factorizations of unitons via singular Darboux transformations

Some new factorizations of unitons into Lie groups are established via singular Darboux transformations. The factorization processes for Grassmannian unitons are also considered. Furthermore, a purely algebraic method for constructing Grassmannian unitons is presented.     

High order finite volume methods for singular perturbation problems

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.     

On a class of measure-valued processes:singular cases

A measure-valued diffusion process describing how the measures evolve under flows or "imaginary" flows on Rd is constructed in this paper. The interest of the process is that on the one hand, it can be viewed as a measure-valued flow; on the other hand, the general stochastic flows of measurable maps or kernels do not cover it.     

Padé polynomial asymptotics from a singular integral equation

We derive a singular integral equation satisfied by the remainder function associated with the polynomials forming a diagonal Padé approximant. From this equation, the asymptotic behavior of the high-order polynomials is deduced for certain classes of functions being approximated.Communicated by Edward B. Saff.     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

Fourth-order Schrödinger type operator with singular potentials

We prove that a deformation of a hypersurface in an (n + 1)-dimensional real space form \({{\mathbb S}^{n+1}_{p,1}}\) induces a Hamiltonian variation of the normal congruence in the space \({{\mathbb L}({\mathbb S}^{n+1}_{p,1})}\) of oriented geodesics. As an application, we show that every Hamiltonian minimal submanifold in \({{\mathbb L}({\mathbb S}^{n+1})}\) (resp. \({{\mathbb L}({\mathbb H}^{n+1})}\)) with respect to the (para-)Kähler Einstein structure is locally the normal congruence of a hypersurface \({\Sigma}\) in \({{\mathbb S}^{n+1}}\) (resp. \({{\mathbb H}^{n+1}}\)) that is a critical point of the functional \({{\mathcal W}(\Sigma) = \int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}}\), where k_i denote the principal curvatures of \({\Sigma}\) and \({\epsilon \in \{-1, 1\}}\). In addition, for \({n = 2}\), we prove that every Hamiltonian minimal surface in \({{\mathbb L}({\mathbb S}^{3})}\) (resp. \({{\mathbb L}({\mathbb H}^{3})}\)), with respect to the (para-)Kähler conformally flat structure, is the normal congruence of a surface in \({{\mathbb S}^{3}}\) (resp. \({{\mathbb H}^{3}}\)) that is a critical point of the functional \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K+1}}\) (resp. \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K-1}}\)), where H and K denote, respectively, the mean and Gaussian curvature of \({\Sigma}\).     

Invariant theory for singular α-determinants

From the irreducible decompositions' point of view, the structure of the cyclic GLn(C)-module generated by the α-determinant degenerates when (1?k?n−1) (see [S. Matsumoto, M. Wakayama, Alpha-determinant cyclic modules of gln(C), J. Lie Theory 16 (2006) 393-405]). In this paper, we show that -determinant shares similar properties which the ordinary determinant possesses. From this fact, one can define a new (relative) invariant called a wreath determinant. Using (GLm,GLn)-duality in the sense of Howe, we obtain an expression of a wreath determinant by a certain linear combination of the corresponding ordinary minor determinants labeled by suitable rectangular shape tableaux. Also we study a wreath determinant analogue of the Vandermonde determinant, and then, investigate symmetric functions such as Schur functions in the framework of wreath determinants. Moreover, we examine coefficients which we call (n,k)-sign appeared at the linear expression of the wreath determinant in relation with a zonal spherical function of a Young subgroup of the symmetric group Snk.     

γ-sets and other singular sets of real numbers

A family of $\text{J}$ of open subsets of the real line is called an ω-cover of a set X iff every finite subset of X is contained in an element of $\text{J}$. A set of reals X is a γ-set iff for every ω-cover $\text{J}$ of X there exists $\text{〈D}{}_{\mathrm{n}}\text{: n < ω〉?}\text{J}{}^{\mathrm{\omega }}$ such that

In this paper we show that assuming Martin's axiom there is a γ-set X of cardinality the continuum.     

Schrödinger operators with highly singular oscillating potentials

We investigate the Feynman-Kac semigroupP _t ^V and its densityp ^V(t,.,.),t>0, associated with the Schrödinger operator ?1/2Δ+V on ?^d\{0}.V will be a highly singular, oscillating potential like $V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$ with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ? such that for allt, x, y p ^V(t, x, y)≤γ·p(βt, x, y)·e^at. On the other hand, also conditions are derived which imply thatp ^V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies.     

Numerical solutions to singular ϕ-Laplacianwith Dirichlet boundary conditions

In this note we are concerned with numerical solutions to Dirichlet problem $$[\phi(u')]' =f(x) \quad \mbox{in} [\alpha, \beta]; \quad u(\alpha)=A, \; u(\beta)=B, $$ where \(\phi :(-\eta , \eta ) \to \mathbb {R}\) \((\eta <+ \infty )\) is an increasing diffeomorphism with \(\phi '(y)\geq d >0\) for all \(y\in (-\eta , \eta )\) . The obtained algorithm combines the shooting method with Euler’s method and it is convergent whenever the problem is solvable. We provide numerical experiments confirming the theoretical aspects.     

Sixth-order Cahn–Hilliard equations with singular nonlinear terms

Our aim in this paper is to study the well-posedness for a class of sixth-order Cahn–Hilliard equations with singular nonlinear terms. More precisely, we prove the existence and uniqueness of variational solutions, based on a variational inequality.     

Homogenization of unbounded singular integrals in W 1,∞

We study homogenization by ??-convergence, with respect to the L ¹-strong convergence, of periodic multiple integrals in W ^1,?? when the integrand can take infinite values outside of a convex bounded open set of matrices.     

Narrow and ℓ2-strictly singular operators from L p

In the first part of the paper we prove that for 2 < p, r < ∞ every operator T: L _p → ? _r is narrow. This completes the list of sequence and function Lebesgue spaces X with the property that every operator T : L _p → X is narrow. Next, using similar methods we prove that every ?₂-strictly singular operator from L _p , 1 < p < ∞, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H. P. Rosenthal asserts that if an operator T from L ₁[0, 1] to itself satisfies the assumption that for each measurable set A ? [0, 1] the restriction \(T{|_{{L_1}(A)}}\) is not an isomorphic embedding, then T is narrow. (Here L ₁(A) = {x ∈ L ₁ : supp x ? A}.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being ?₂-strictly singular, for operators from L _p [0, 1] to itself, 1 < p < 2, to be narrow. We define a notion of a “gentle” growth of a function and we prove that for 1 < p < 2 every operator T from L _p to itself which, for every A ? [0, 1], sends a function of “gentle” growth supported on A to a function of arbitrarily small norm is narrow.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and Lipschitz function b∈Λβ0(Rn) is discussed from Lp(Rn) to Lq(Rn), 1/q=1/p-β0/n, and from Lp(Rn) to Triebel-Lizorkin space Fβ0,∞p. We also obtain the boundedness of generalized Toeplitz operatorθbα0 from LP(Rn) to Lq(Rn), 1/q =1/p-α0 β0/n. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and BMO function b is discussed on LP(Rn), 1 < p <∞.