The p-Frobenius and p-Sylvester numbers for Fibonacci and Lucas triplets.

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $ a_1, a_2, \dots, a_l $ be positive integers such that their greatest common divisor is one. For a nonnegative integer $ p $, denote the $ p $-Frobenius number by $ g_p (a_1, a_2, \dots, a_l) $, which is the largest integer that can be represented at most $ p $ ways by a linear combination with nonnegative integer coefficients of $ a_1, a_2, \dots, a_l $. When $ p = 0 $, the $ 0 $-Frobenius number is the classical Frobenius number. When $ l = 2 $, the $ p $-Frobenius number is explicitly given. However, when $ l = 3 $ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly. It is even more difficult when $ p > 0 $, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers [1] or of repunits [2] for the case where $ l = 3 $. In this paper, we show the explicit formula for the Fibonacci triple when $ p > 0 $. In addition, we give an explicit formula for the $ p $-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $ p $ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.

A novel IKKα inhibitor, noraristeromycin, blocks the chronic inflammation associated with collagen-induced arthritis in mice.

OBJECTIVES: To evaluate the therapeutic efficacy of a novel inhibitor for IκB kinase alpha (IKKα), noraristeromycin (NAM), for murine experimental model of rheumatoid arthritis, collagen- induced arthritis (CIA). METHODS: NAM has been chemically synthesized as reported earlier. CIA was induced in DBA/1JNCrlj mice by intradermal inoculation of bovine type II collagen (col II) together with Freund Complete Adjuvant. Following the Day 21 booster injection of col II with Freund Incomplete Adjuvant, the animals were monitored for the development of arthritis and clinically evaluated. NAM was administered orally at different doses prior to induction (prophylactic protocol) or after the emergence of definitive arthritis (therapeutic protocol). RESULTS: Here we demonstrate the experimental evidence that oral administration of NAM could completely prevent the occurrence of experimental arthritis in CIA mouse model at 0.3 mg/kg with ED50 value of approximately 0.1 mg/kg twice daily. Moreover, twice daily oral therapeutic dosage of 1 mg/kg of NAM significantly inhibited the paw swelling and disease progression even after the occurrence of experimental CIA. In addition, NAM exhibited an excellent pharmacokinetics in mice and oral administration of NAM could suppress the production of TNFα elicited by lipopolysaccharide (LPS) in a dose-dependent manner. CONCLUSIONS: These results indicated that IKKα inhibition is an effective novel therapy for the treatment of chronic inflammatory processes such as those associated with RA and other related conditions.

Induction of antioxidant and phase 2 drug-metabolizing enzymes by falcarindiol isolated from Notopterygium incisum extract, which activates the Nrf2/ARE pathway, leads to cytoprotection against oxidative and electrophilic stress.

In the present study, we isolated falcarindiol from Notopterygium incisum and investigated the effect of falcarindiol on the expression of antioxidant enzymes (AOEs), such as catalase, and phase 2 drug-metabolizing enzymes (DMEs), such as glutathione S-transferase and NAD(P)H:quinone oxidoreductase 1, in a cultured cell line from normal rat liver, Clone 9 cells. Exposure of Clone 9 cells to falcarindiol resulted in the significant induction of AOEs and phase 2 DMEs. Western blot analysis and transfection studies using a luciferase reporter construct demonstrated that the induction of AOEs and phase 2 DMEs by falcarindiol was caused through the Nrf2/ARE (nuclear factor-E2-related factor 2/antioxidant response element) pathway. Pretreatment of cells with falcarindiol accelerated the detoxification of a potentially toxic quinone (menadione) and mitigated menadione-induced cytotoxicity. We found that falcarindiol was a novel inducer of AOEs and phase 2 DMEs and falcarindiol might exhibit chemopreventive activity.