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We consider the following critical Hartree equation 0.1 - Δ u = (∫ Ω u 2 μ ∗ (y) | x - y | μ d y) u 2 μ ∗ - 1 + ε u , in Ω , u = 0 , on ∂ Ω , where N ≥ 4 , 0 < μ ≤ 4 , ε > 0 is a small parameter, Ω is a smooth bounded open domain in R N , and 2 μ ∗ = 2 N - μ N - 2 is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. We first establish a nonlocal version of global compactness lemma. Then by using reduction arguments, we prove that, as ε → 0 , the solution u ε of (0.1) blows up exactly at a critical point of the Robin function that cannot be on the boundary of Ω . [ABSTRACT FROM AUTHOR]

Let l be a line in a projective space ℙn. We consider the blowing up ℙn(l) of ℙn along l. Assume that X is a smooth closed subvariety of ℙn. If the strict transform of X in ℙn(l) has a splitting tangent sequence and dim X is at least 2, then X is a linear subspace of ℙn. [ABSTRACT FROM AUTHOR]

We prove that a meromorphic mapping, which sends a peace of a real analytic strictly pseudoconvex hypersurface in C 2 to a compact subset of C N which does not contain germs of non-constant complex curves is continuous from the concave side of the hypersurface. This implies the analytic continuability along CR-paths of germs of holomorphic mappings from real analytic hypersurfaces with non-vanishing Levi form to the locally spherical ones in all dimensions. [ABSTRACT FROM AUTHOR]

The article develops techniques for solving equations $$G(x,y)=0$$ , where $$G(x,y)=G(x_1,\ldots ,x_n,y)$$ is a function in a given quasianalytic class (for example, a quasianalytic Denjoy-Carleman class, or the class of $${\mathcal C}^\infty $$ functions definable in a polynomially-bounded o-minimal structure). We show that, if $$G(x,y)=0$$ has a formal power series solution $$y=H(x)$$ at some point a, then H is the Taylor expansion at a of a quasianalytic solution $$y=h(x)$$ , where h( x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed. [ABSTRACT FROM AUTHOR]

We present the algorithms for computing the normal form of unimodular complete intersection surface singularities classified by C. T. C. Wall. He indicated in the list only the μ-constant strata and not the complete classification in each case. We give a complete list of surface unimodular singularities. We also give the description of a classifier which is implemented in the computer algebra system Singular. [ABSTRACT FROM AUTHOR]

We work with quasianalytic classes of functions. Consider a real-valued function y = f(x)onanopensubset U of ℝn, which satisfies a quasi-analytic equation G(x, y) = 0. We prove that f is arc-quasianalytic (i.e., its restriction to every quasianalytic arc is quasianalytic) if and only if f becomes quasianalytic after (a locally finite covering of U by) finite sequences of local blowings-up. This generalizes a theorem of the first two authors on arc-analytic functions. [ABSTRACT FROM AUTHOR]

This paper deals with the large time behavior of nonnegative solutions to the equation where p > 2, q > 0, and the function a( x) ≥ 0 has a compact support. We obtain the critical exponent for global existence q and the Fujita exponent q. In one-dimensional case N = 1, we have $$q_0 = \frac{{2(p - 1)}} {p}$$ and q = 2( p − 1). Particularly, all solutions are global in time if 0 < q ≤ q, but blow up if q < q ≤ q; while if q > q both blowing up solutions and global solutions exist. However, for the case N ≥ p > 2, these two critical exponents are exactly the same. Namely, q = p − 1 = q. [ABSTRACT FROM AUTHOR]

This paper deals with blowing up of solutions to the Cauchy problem for a class of generalized Zakharov system with combined power-type nonlinearities in two and three space dimensions. On the one hand, for c = +∞ we obtain two finite time blow-up results of solutions to the aforementioned system. One is obtained under the condition α ≥ 0 and $$1 + \tfrac{4} {N} \leqslant p < \tfrac{{N + 2}} {{N - 2}}$$ or α < 0 and $$1 < p < 1 + \tfrac{4} {N}$$ ( N = 2, 3); the other is established under the condition N = 3, $$1 < p < \tfrac{{N + 2}} {{N - 2}}$$ and α( p − 3) ≥ 0. On the other hand, for c < +∞ and α( p − 3) ≥ 0, we prove a blow-up result for solutions with negative energy to the Zakharov system under study. [ABSTRACT FROM AUTHOR]

In this paper, new regular incidence structures are presented. They arise from sets of conics in the affine plane blown–up at its rational points. The LDPC codes given by these incidence matrices are studied. These sparse incidence matrices turn out to be redundant, which means that their number of rows exceeds their rank. Such a feature is absent from random LDPC codes and is in general interesting for the efficiency of iterative decoding. The performance of some codes under iterative decoding is tested. Some of them turn out to perform better than regular Gallager codes having similar rate and row weight. [ABSTRACT FROM AUTHOR]

We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimension at most m − 3. [ABSTRACT FROM AUTHOR]

In this note we give a criterion for a line bundle on a general blowup of a ruled surface to be k-very ample. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a topological criterion for a chain of spheres in a two-dimensional complex manifold to be a composition of so-called triangular chains. Explicit formulas are given for the relevant topological characteristics of chains. Certain corollaries of this criterion related to the Jacobian conjecture are discussed. [ABSTRACT FROM AUTHOR]