Takao Komatsu

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Given a set of positive integers A = {a1, . . . , an}, we study the number pA(t) of nonnegative integer solutions (m1, . . . ,mn) to ∑n j=1mjaj = t. We derive an explicit formula for the polynomial part of pA. Let A = {a1, . . . , an} be a set of positive integers with gcd(a1, . . . , an) = 1. The classical Frobenius problem asks for the largest integer t(More)
We discuss the partial infinite sum ∑∞ k=n u −s k for some positive integer n, where uk satisfies a recurrence relation of order s, un = aun−1 + un−2 + · · ·+ un−s (n ≥ s), with initial values u0 ≥ 0, uk ∈ N (0 ≤ k ≤ s− 1), where a and s(≥ 2) are positive integers. If a = 1, s = 2, and u0 = 0, u1 = 1, then uk = Fk is the k-th Fibonacci number. Our results(More)
with θ irrational and taken to satisfy 0 < θ < 1; plainly this may be assumed without loss of generality. Evidently (fn) is a sequence of zeros and ones. Denote by w0 and w1 words on the alphabet {0, 1} ; that is, finite strings in the letters 0 and 1. Then the sequence (fn) is said to be invariant under the substitution W given by W : 0 −→ w0, 1 −→ w1, if(More)
— Let α > 1 be irrational. Several authors studied the numbers (α) = inf{ |y| : y ∈ Λm, y = 0}, where m is a positive integer and Λm denotes the set of all real numbers of the form y = 0α + 1αn−1 + · · · + n−1α + n with restricted integer coefficients | i| ≤ m. The value of 1(α) was determined for many particular Pisot numbers and m(α) for the golden(More)
In this paper we consider infinite sums derived from the reciprocals of the Fibonacci numbers, and infinite sums derived from the reciprocals of the square of the Fibonacci numbers. Applying the floor function to the reciprocals of these sums, we obtain equalities that involve the Fibonacci numbers.
and M−(θ, φ) = lim inf q→+∞ q‖qθ + φ‖ = lim inf q→−∞ |q|‖qθ − φ‖. ThenM(θ, φ) = min(M+(θ, φ),M−(θ, φ)). These notations are introduced by Cusick, Rockett and Szüsz [2].M(θ, φ) orM+(θ, φ) has been treated by Cassels [1], Descombes [3], Sós [9], Cusick et al. [2] and the author [5] by using several algorithms for inhomogeneous Diophantine approximation in(More)
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(More)
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(More)