Mathematics

We introduce generalized local homology which is in some sense dual to generalized local cohomology, and study some properties of generalized local homology modules for artinian modules, such as the artinianness, noetherianness and the characterization of Width I(M) by generalized local homology. By using duality, we get back some properties of generalized local cohomology.

We prove some results about the finiteness of co-associated primes of generalized local homology modules inspired by a conjecture of Grothendieck and a question of Huneke. We also show some equivalent properties of minimax local homology modules. By duality, we get some properties of Herzog’s generalized local cohomology modules.

Human immunodeficiency virus-1 (HIV-1) infection is associated with numerous effects on the nervous system, including pain and peripheral neuropathies. We now demonstrate that cultured rat dorsal root ganglion (DRG) neurons express a wide variety of chemokine receptors, including those that are thought to act as receptors for the HIV-1 coat protein glyco-protein120 (gp120). Chemokines that activate all of the known chemokine receptors increased [Ca 2ϩ ] i in subsets of cultured DRG cells. Many neurons responded to multiple chemokines and also to bradykinin, ATP, and capsaicin. Immunohistochemical studies demonstrated the expression of the CXCR4 and CCR4 chemokine receptors on populations of DRG neurons that also expressed substance P and the VR1 vanilloid receptor.

We previously demonstrated that chemokine receptors are expressed by neural progenitors grown as cultured neurospheres. To examine the significance of these findings for neural progenitor function in vivo, we investigated whether chemokine receptors were expressed by cells having the characteristics of neural progenitors in neurogenic regions of the postnatal brain. Using in situ hybridization we demonstrated the expression of CCR1, CCR2, CCR5, CXCR3, and CXCR4 chemokine receptors by cells in the dentate gyrus (DG), subventricular zone of the lateral ventricle, and olfactory bulb. The pattern of expression for all of these receptors was similar, including regions where neural progenitors normally reside. In addition, we attempted to colocalize chemokine receptors with markers for neural progenitors. In order to do this we used nestin-EGFP and TLX-LacZ transgenic mice, as well as labeling for Ki67, a marker for dividing cells. In all three areas of the brain we demonstrated colocalization of chemokine receptors with these three markers in populations of cells. Expression of chemokine receptors by neural progenitors was further confirmed using CXCR4-EGFP BAC transgenic mice. Expression of CXCR4 in the DG included cells that expressed nestin and GFAP as well as cells that appeared to be immature granule neurons expressing PSA-NCAM, calretinin, and Prox-1. CXCR4-expressing cells in the DG were found in close proximity to immature granule neurons that expressed the chemokine SDF-1/CXCL12. Cells expressing CXCR4 frequently coexpressed CCR2 receptors. These data support the hypothesis that chemokine receptors are important in regulating the migration of progenitor cells in postnatal brain. J. Comp. Neurol. 500:1007–1033, 2007. © 2006 Wiley-Liss, Inc.

The present work is an original evaluation of the microenvironmental pH (pHM) and crystallinity of an ionizable drug in order to enhance its dissolution using alkalizers in polyethylene glycol 6000 (PEG 6000) based solid dispersions (SDs). Telmisartan (TEL) was chosen as a model drug due to its poor and pH-dependent water solubility. The nine alkalizers used to modify the pH of TEL were MgO, NaOH, KOH, Na2CO3, NaHCO3, bentonite, Na2HPO4, K2HPO4 and arginine. MgO, NaOH, KOH and Na2CO3 in the SD system significantly increased the drug dissolution rate in intestinal fluid (pH 6.8) and water. Modulation of pHM was clearly observed as a function of time at different fractional dimensions of tablet. Structural change in drug crystallinity to an amorphous form was also a contributing factor based on differential scanning calorimetry (DSC) thermograms and powder X-ray diffraction (PXRD) patterns. The drug frequency of the C

We define a new invariant of quadratic Lie algebras and give a complete study and classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is related to O(n)-adjoint orbits in o(n). 0. INTRODUCTION Let g be a non-Abelian quadratic Lie algebra equipped with a bilinear form B. We can associate to (g, B) a canonical non-zero 3-form I ∈ 3 (g) g defined by I(X ,Y, Z) := B([X ,Y ], Z), ∀ X ,Y, Z ∈ g. Let {•, •} be the super-Poisson bracket on (g). The 3-form I satisfies (see [PU07]): {I, I} = 0. Conversely, given a quadratic vector space (g, B) and a non-zero 3-form I ∈ 3 (g) such that {I, I} = 0, there is a non-Abelian quadratic Lie algebra structure on g such that I is the canonical 3-form associated to g ([PU07]). Let Q(n) be the set of non-Abelian quadratic Lie algebra structures on the quadratic vector space C n. We identify Q(n) ↔ I ∈ 3 (C n) | {I, I} = 0 and Q(n) is an affine variety in 3 (C n) (Proposition 2.8). The dup-number of a non-Abelian quadratic Lie algebra g is defined by dup(g) := dim ({α ∈ g * | α ∧ I = 0}) , where I is the 3-form associated to g. It measures the decomposability of the 3form I and its range is {0, 1, 3} (Proposition 1.1). For instance, I is decomposable if, and only if, dup(g) = 3 and the corresponding quadratic Lie algebras are classified in [PU07], up to i-isomorphism (i.e. isometric isomorphism). It is easy to check that the dup-number of g is invariant by i-isomorphism, that is, two iisomorphic quadratic Lie algebras have the same dup-number (Lemma 2.1). We shall prove in this paper, a much stronger result: the dup-number of g is invariant by isomorphism.

In this paper, we generalize some results on quadratic Lie algebras to quadratic Lie superalgebras, by applying graded Lie algebras tools. We establish a one-toone correspondence between non-Abelian quadratic Lie superalgebra structures and nonzero even super-antisymmetric 3-forms satisfying a structure equation. An invariant number of quadratic Lie superalgebras is then defined, called the dup-number. Singular quadratic Lie superalgebras (i.e. those with nonzero dupnumber) are studied. We show that their classification follows the classifications of O(m)-adjoint orbits of o(m) and Sp(2n)-adjoint orbits of sp(2n). An explicit formula for the quadratic dimension of singular quadratic Lie superalgebras is also provided. Finally, we discuss a class of 2-nilpotent quadratic Lie superalgebras associated to a particular super-antisymmetric 3-form.

First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra. We give an isomorphic characterization of 2-step nilpotent pseudo-Euclidean Jordan algebras. Next, we find a Jordan-admissible condition for a Novikov algebra $\N$. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7. Comment: 33 pages

In this paper, we classify solvable Lie algebras of dimensions $\leq 8$ endowed with a nondegenerate invariant symmetric bilinear form over an algebraically closed field. This classification (up to isomorphisms) is mainly based on the double extension method.

We give a characterization of symplectic quadratic Lie algebras that their Lie algebra of inner derivations has an invertible derivation. A family of symplectic quadratic Lie algebras is introduced to illustrate this situation. Finally, we calculate explicitly the Betti numbers of a family of solvable Lie algebras in two ways: using the cohomology of quadratic Lie algebras and applying a Pouseele's result on extensions of the onedimensional Lie algebra by Heisenberg Lie algebras.

In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions smaller than 6, we use the Witt decomposition given in [Bou59] and a result in [PU07] to obtain two non-Abelian indecomposable solvable quadratic Lie algebras. In the case of dimension 6, by applying the method of double extension given in [Kac85] and [MR85] and the classification result of singular quadratic Lie algebras in [DPU], we have three families of solvable quadratic Lie algebras which are indecomposable and not isomorphic.

In this paper, we give an expansion of two notions of double extension and $T^*$-extension for quadratic and odd quadratic Lie superalgebras. Also, we provide a classification of quadratic and odd quadratic Lie superalgebras up to dimension 6. This classification is considered up to isometric isomorphism, mainly in the solvable case and the obtained Lie superalgebras are indecomposable.

Khi trả bài kiểm tra một tiết cho học sinh xong, giáo viên quay lên bục giảng để bắt đầu bài mới thì bỗng "roạc", "xoạt, xoạt", hình như là tiếng xé và vò giấy. Giáo viên quay lại thì thấy Nam đã xé tan bài làm được một điểm của mình trước sự ngơ ngác của các bạn trong lớp. Khi được hỏi tại sao em xé bài, thì Nam trả lời tỉnh queo: "Bài của em thì em xé". Trước sự việc đó,người giáo viên phải giải quyết ra sao?