Abstract:
In this paper, we use Clifford algebra and the spinor calculus to study the complex structures on Euclidean space $R^8$ and the spheres $S^4,S^6$. By the spin representation of $G(2,8)\subset Spin(8)$ we show that the Grassmann manifold G(2,8) can be looked as the set of orthogonal complex structures on $R^8$. In this way, we show that G(2,8) and $CP^{3}$ can be looked as twistor spaces of $S^6$ and $S^4$ respectively. Then we show that there is no almost complex structure on sphere $S^4$ and there is no orthogonal complex structure on the sphere $S^6$.

Abstract:
In this paper, we show that the twistor space ${\cal J}(R^{2n+2})$ on Euclidean space $R^{2n+2}$ is a Kaehler manifold and the orthogonal twistor space $\widetilde{\cal J}(S^{2n})$ of the sphere $S^{2n}$ is a Kaehler submanifold of ${\cal J}(R^{2n+2})$. Then we show that an orthogonal almost complex structure $J_f$ on $S^{2n}$ is integrable if and only if the corresponding section $f\colon S^{2n}\to \widetilde{\cal J}(S^{2n}) $ is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere $S^{2n}$ for $n>1$.

Abstract:
This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.

Abstract:
This paper shows that the Grassmann Manifolds $G_{\bf F}(n,N)$ can all be imbedded in an Euclidean space $M_{\bf F}(N)$ naturally and the imbedding can be realized by the eigenfunctions of Laplacian $\triangle$ on $G_{\bf F}(n,N)$. They are all minimal submanifolds in some spheres of $M_{\bf F}(N)$ respectively. Using these imbeddings, we construct some degenerate Morse functions on Grassmann Manifolds, show that the homology of the complex and quaternion Grassmann Manifolds can be computed easily.

Abstract:
We show that there is no complex structure in a neighborhood of the space of orthogonal almost complex structures on the sphere $S^{2n}, \ n>1$. The method is to study the first Chern class of vetcor bundle $T^{(1,0)}S^{2n}$.

Abstract:
T follicular helper (Tfh) cell is a new subpopulation of CD4+ T cell family, whose differentiation is affected by Bcl-6, Blimp-1, STAT3, STAT5 and so on, and it could affect or decide the development of other subsets of CD4+ T cells. The important function of Tfh cell is to help B cell mediate humoral immunity, many researches have proved that Tfh cells participate in the development of autoimmune disease, immunodeficient disease, tumor and infectious diseases.

Abstract:
In this paper, we use characteristic classes of the canonical vector bundles and the Poincar\' {e} dualality to study the structure of the real homology and cohomology groups of oriented Grassmann manifold $G(k, n)$. Show that for $k=2$ or $n\leq 8$, the cohomology groups $H^*(G(k,n),{\bf R})$ are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poincar\' {e} dualality: $H^q(G(k,n),{\bf R}) \to H_{k(n-k)-q}(G(k,n),{\bf R})$ can be given explicitly.

Abstract:
In this paper we investigate the spectral norm for circulant matrices, whose entries are modified Fibonacci numbers and Lucas numbers. We obtain the identity estimations for the spectral norms. Some numerical test results are listed to verify the results using those approaches.

Abstract:
The explicit formulae of spectral norms for circulant-type matrices are investigated; the matrices are circulant matrix, skew-circulant matrix, and -circulant matrix, respectively. The entries are products of binomial coefficients with harmonic numbers. Explicit identities for these spectral norms are obtained. Employing these approaches, some numerical tests are listed to verify the results. 1. Introduction The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers [1]. In numerical analysis, circulant matrices (named “premultipliers” in numerical methods) are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. Furthermore, circulant, skew-circulant, and -circulant matrices play important roles in various applications, such as image processing, coding, and engineering model. For more details, please refer to [2–13] and the references therein. The skew-circulant matrices were collected to construct preconditioners for LMF-based ODE codes; Hermitian and skew-Hermitian Toeplitz systems were considered in [14–17]; Lyness employed a skew-circulant matrix to construct -dimensional lattice rules in [18]. Recently, there are lots of research on the spectral distribution and norms of circulant-type matrices. In [19], the authors pointed out the processes based on the eigenvalue of circulant-type matrices and the convergence to a Poisson random measure in vague topology. There were discussions about the convergence in probability and distribution of the spectral norm of circulant-type matrices in [20]. The authors in [21] listed the limiting spectral distribution for a class of circulant-type matrices with heavy tailed input sequence. Ngondiep et al. showed that the singular values of -circulants in [22]. Solak established the lower and upper bounds for the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries in [23]. ？pek investigated an improved estimation for spectral norms in [24]. In this paper, we derive some explicit identities of spectral norms for some circulant-type matrices with product of binomial coefficients with harmonic numbers. The outline of the paper is as follows. In Section 2, the definitions and preliminary results are listed. In Section 3, the spectral norms of some circulant matrices are studied. In Section 4, the formulae of spectral norms for skew-circulant matrices are established. Section 5 is devoted to investigate the

Abstract:
The aim of the present study was to investigate the
effect of Lycium barbarum polysaccharides (LBP) on the treatment of ITP mice and to explore its
mechanism. Forty idiopathic thrombocytopenic purpura (ITP) mice were divided
randomly into a model control group and LBP groups I, II, III and IV. ITP mice
in LBP groups I, II, III, and IV were administered LBP at four different doses
(50, 100, 200 and 400 mg·kg^{-}^{1}·d^{-1}, respectively)
for 7 days by gavage. Blood samples were collected from the tail veins of the
mice after treatment. Platelet counts were determined, and the total antioxidant
status (TAS), total oxidant status (TOS) were measured with ELISA kits. The
platelet count was (30.28 ± 13.42) × 10^{9}/L in the model control group, and
the number of platelets in all LBP groups was higher than that in the model
control group. The platelet count increased, and it reached (67.09 ± 10.81) × 10^{9}/L in LBP
group I; the platelet counts in the other three groups increased significantly
compared to LBP group I, and they did not differ significantly. TAS
concentrations in the LBP groups were significantly increased compared to the model
control group, whereas TOS concentrations were significantly decreased. Taken
together, these results indicate that LBP is effective at increasing the number
of platelet (PLT), and LBP may treat ITP mice via suppressing oxidative stress.