Abstract:
In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $\varphi(q,p(x))$ be the Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$ is a finite field with $q$ elements. We prove that $\varphi(q,p(x))|(q^{{\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducibles one each of degree $1, 2$ and $3$.

Abstract:
Let $X$ be a smooth projective curve over a finite field $\mathbb{F}$ with $q$ elements. For $m\geq 1,$ let $X_m$ be the curve $X$ over the finite field $\mathbb{F}_m$, the $m$-th extension of $\mathbb{F}.$ Let $K_n(m)$ be the $K$-group $K_n(X_m)$ of the smooth projective curve $X_m.$ In this paper, we study the structure of the groups $K_n(m).$ If $l$ is a prime, we establish an analogue of Iwasawa theorem in algebraic number theory for the orders of the $l$-primary part $K_n(l^m)\{l\}$ of $K_n(l^m)$. In particular, when $X$ is an elliptic curve $E$ defined over $\mathbb{F},$ our method determines the structure of $K_n(E).$ Our results can be applied to construct an efficient {\bf DL} system in elliptic cryptography.

Abstract:
Lehmer's conjecture for Ramanujan's tau function says that $\tau(n) \neq 0$ for all $n$. In this paper, we generalize D. H. Lehmer's result to give a sufficient condition for level one cusp forms $f$ such that the smallest $n$ for which the Fourier coefficients $a_n(f)=0$ must be a prime. For the unique cusp form $\Delta_{k}$ of level one and weight k with $k=16, 20, 22$, we achieve a large bound $B_k$ of $n$ such that $a_n(\Delta_k)\ne0$ for all $n

Abstract:
In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $\Lambda$, we show that the right self-injective dimension of $\Lambda$ is at most $k$ if and only if each $M \in$mod $\Lambda$ satisfying the condition that Ext$_{\Lambda}^i(M, \Lambda)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $\Lambda$ are identical. In addition, we show that if $\Lambda$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $\Lambda$ with grade$M$ finite, then Ext$_{\Lambda}^i($Ext$_{\Lambda ^{op}}^i($Ext$_{\Lambda}^{{\rm grade}M}(M, \Lambda), \Lambda), \Lambda)=0$ if and only if $i\neq$grade$M$. For a 2-Gorenstein ring $\Lambda$, we show that a non-zero proper left ideal $I$ of $\Lambda$ is reflexive if and only if $\Lambda /I$ has no non-zero pseudo-null submodule.

Abstract:
A distributive lattice $L$ with minimum element $0$ is called decomposable if $a$ and $b$ are not comparable elements in $L$ then there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b), b=\overline{b}\vee(a\wedge b)$ and $\overline{a}\wedge \overline{b}=0$. The main purpose of this paper is to study the structure of decomposable lattices determined by their prime ideals. The properties for five special decomposable lattices are derived.

Abstract:
A distributive lattice $L$ with minimum element $0$ is called decomposable lattice if $a$ and $b$ are not comparable elements in $L$ there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b), b=\overline{b}\vee(a\wedge b)$ and $\overline{a}\wedge \overline{b}=0$. The main purpose of this paper is to investigate prime ideals, minimal prime ideals and special ideals of a decomposable lattice. These are keys to understand the algebraic structure of decomposable lattices.

Abstract:
In the framework of a chiral constituent quark model, considering the contributions of $\pi$ annihilation and one-gluon annihilation, the proton-antiproton $S$-wave elastic scattering cross section experimental data can be reproduced by adjusting properly one-gluon annihilation coupling constant. Meanwhile, using the fixed model parameter, we do a dynamical calculation for all possible $S$-wave nucleon-antinucleon states, the results show that, there is no $S$-wave bound state as indicated by a strong enhancement at threshold of $p\bar{p}$ in $J/\psi$ and $B$ decays.

Abstract:
The fucoidanase from Fusarium sp. (LD8) was obtained by solid-state fermentation. The fermented solid medium was extracted by citric acid buffer, and the extracts were precipitated by acetone and purified by Sephadex G-100 successively. The results showed that the specific fucoidanase activity of purified enzyme was 22.7-fold than that of the crude enzyme. The recovery of the enzyme was 23.9%. The purified enzyme gave a single band on SDS-PAGE gel, and the molecular weight of fucoidanase was about 64？kDa. The isoelectric point of the enzyme was 4.5. The enzyme properties were also studied. The results showed that the optimum temperature and pH were 60°C and 6.0, respectively; the temperature of half inactivation was 50°C, and the most stable pH for the enzyme was 6.0. , and the of the enzyme was 8.9？mg·L？1 and 2.02？mg·min？1·mL？1 by using fucoidan from Fucus vesiculosus as substrate. The compositions of the secondary structure of fucoidanase were estimated by FTIR, the second derivative spectra, and the curve-fitting analysis of the amide I bands in their spectra. The results showed that -sheet was the dominant component (58.6%) and -helix was the least (12%); the content of -turn and random coil were 15.39% and 14.5%, respectively. 1. Introduction Sulfated polysaccharides, named fucoidan, could be extracted from many marine brown algae. The main differences of structural characterization in algal fucoidan originate from their species; the backbone of fucoidan isolated from Laminaria cichorioides [1] was consisting of (1–3) linked fucopyranose residues, and that from L. japonica was primarily (1–3) linked fucopyranose residues (75%) and a few (1–4) fucopyranose linkages (25%) [2]; The fucoidan from Fucus serratus [3], F. evanescens [4], and F. distichus [5] was shown to contain a backbone consisting of alternating (1–3) and (1–4) linked a-L-fucopyranose residues. Investigators were interested in screening fucoidan which was isolated from different brown algal species; fucoidan was attributed several different bioactivities including anticoagulant [6–8], antithrombotic/antithrombin activity [9], antiviral [10], and other activities [11]. The biological effects of fucoidan depended on the molecular mass, sulfate content, and sugar constituents [12, 13]. The low molecular weight fucoidan (LMWF) could be obtained by fucoidanase (E.C.3.2.1.44) enzymolysis, and it could hydrolyze fucoidan to sulfated LMWF without removal of its side substitute groups. Fucoidanase could be extracted from hepatopancreas of invertebrates [14, 15], marine bacteria [16–25], and

Abstract:
The Omega-Omega (SIJ=-6,0,0) dibaryon state is studied with the extended quark-delocalization color-screening model including a pi meson exchange tail, which reproduces the properties of the deuteron quantitatively. We find the mass of the di-Omega to be about 45 MeV lower than the Omega-Omega threshold. The effect of channel coupling due to the tensor force has been calculated and found to be small in this case. We have also studied the effect of other pseudoscalar meson exchanges and sensitivity to the short-range cutoff radius, r_0, for the meson exchanges.

Abstract:
The influence of gluon and Goldstone boson induced tensor interactions on the dibaryon masses and D-wave decay widths has been studied in the quark delocalization, color screening model. The effective S-D wave transition interactions induced by gluon and Goldstone boson exchanges decrease rapidly with increasing strangeness of the channel. The tensor contribution of K and $\eta$ mesons is negligible in this model. There is no six-quark state in the light flavor world studied so far that can become bound by means of these tensor interactions besides the deuteron. The partial D-wave decay widths of the $IJ^p={1/2}2^+$ N$\Omega$ state to spin 0 and 1 $\Lambda\Xi$ final states are 12.0 keV and 21.9 keV respectively. This is a very narrow dibaryon resonance that might be detectable in relativistic heavy ion reactions by existing RHIC detectors through the reconstruction of the vertex mass of the decay product $\Lambda\Xi$ and by the COMPAS detector at CERN or at JHF in Japan and the FAIR project in Germany in the future.