ISSN 1439-8516 CN 11-2039/O1
Fourier integral operators play an important role in Fourier analysis and partial differential equations. In this paper, we deal with the boundedness of the bilinear and bi-parameter Fourier integral operators, which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators. We consider such FIOs when they have compact support in spatial variables. If they contain a real-valued phase φ(x, ξ, η) which is jointly homogeneous in the frequency variables ξ, η, and amplitudes of order zero supported away from the axes and the antidiagonal, we can show that the boundedness holds in the local-L2 case. Some stronger boundedness results are also obtained under more restricted conditions on the phase functions. Thus our results extend the boundedness results for bilinear and one-parameter FIOs and bilinear and bi-parameter pseudo-differential operators to the case of bilinear and bi-parameter FIOs.
Let R and S be associative rings and SVR a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR(IV (R),-) and HomR(-, IV (R)) exact exact complex …→I1.jpg){kind=small}
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I1→… of V-injective modules Ii and Ii, i∈N0, such that N≌Im(I0→I0). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class AV (R) which leads to the fact that V-Gorenstein injective modules admit exact right IV (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N⊕E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if ExtIV(R)≥n+1(I, N)=0 for all modules I with finite IV (R)-injective dimension.
The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmüller space AT(D) are studied in this paper. It is proved that if μ is asymptotically extremal in[[μ]] with hζ*(μ)<h*(μ) for some point ζ∈∂D, then there exist infinitely many geodesic segments joining[[0]] and [[μ]], and infinitely many holomorphic geodesic disks containing[[0]] and [[μ]] in AT(D).
By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the objective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces (Z, d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain X⊂Z is countably compact in any Hausdorff topology weaker than that induced by d. When (Z, d) is a Féchet space (i.e., a complete metrizable locally convex space), our existence result only requires that the domain X⊂Z is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems, which extend and improve the related known results.
We compute the derivations of the positive part of the two-parameter quantum group Ur,s(B3) and show that the Hochschild cohomology group of degree 1 of this algebra is a threedimensional vector space over the base field C. We also compute the groups of (Hopf) algebra automorphisms of the augmented two-parameter quantized enveloping algebra ?r,s≥0(B3).
A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv∈E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let χ'Σ(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges (we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G)<5/2, then χ'Σ(G)≤5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.
The aim of this paper is to identify the volatility function in Dupire's equation from given option prices. This inverse problem is formulated as an infinite-dimensional minimization problem with PDE constraints. The computational cost of solving the discretized problem on a fine discretization level is expensive. A multi-grid method is proposed to explore the hierarchical structures of discretized problems on different levels. Computational examples are presented to demonstrate the efficiency of our method.
Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G=HK and H∩K=1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.
In this paper, we obtain the estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation on the unit ball of complex Banach spaces. Meanwhile, we also establish the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cn. Especially, the above estimates are only sharp for biholomorphic starlike mappings and starlike mappings of order α under restricted conditions. Our derived results generalize many known results.
Broersma and Veldman proved that every 2-connected claw-free and P6-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P6-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P6-o-heavy but not hamiltonian. In this paper, we further give some Ore-type degree conditions restricting to induced copies of P6 of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.
