Let c be any square-free integer, p any odd prime such that (c/p) = -1, and n any positive integer. The quantity ./IT, which would ordinarily be defined (mod p) as one of the two solutions of the congruence: x E c (mod p n ) , does not exist. Nevertheless, we may deal with objects of the form a + b/c~(mod p), where a and b are integers, in much the same way that we deal with complex numbers, th...

Abstract. Let p > 3 be a prime, and let m be an integer with p ∤ m. In the paper we solve some conjectures of Z.W. Sun concerning Pp−1 k=0 2k k 3 /mk (mod p2), Pp−1 k=0 2k k 4k 2k /mk (mod p) and Pp−1 k=0 2k k 2 4k 2k /mk (mod p2). In particular, we show that P p−1 2 k=0 2k k 3 ≡ 0 (mod p2) for p ≡ 3, 5, 6 (mod 7). Let {Pn(x)} be the Legendre polynomials. In the paper we also show that P[ p 4 ]...

It is important to reduce the seismic response of houses in order to prevent from collapse of houses. Some control methods for seismic response of houses are developed. The oil dampers using the flow resistance of the silicon oil have been developed. In this paper, a Magnet Oil Damper (MOD) in which permanent magnets are added to the oil damper is proposed in order to improve the performance of...

1.2 Definition (modular reduction). Suppose we divide a and b by m, obtaining integer quotients and remainders, where the remainders are between 0 and m − 1. That is, a = q1m + r1 and b = q2m+ r2, where 0 ≤ r1 ≤ m− 1 and 0 ≤ r2 ≤ m− 1. Then it is not difficult to see that a ≡ b (mod m) if and only if r1 = r2. We will use the notation a mod m (without parentheses) to denote the remainder when a ...

The aim of the present study was to evaluate the fracture resistance of teeth restored with resin composite. Forty-eight maxillary premolar teeth were chosen and randomly divided to six groups: G1 (control): sound teeth; G2: MOD preparation, unrestored; G3: MOD + Adper Single Bond 2/P60; G4: MOD + Adper Easy One/P60; G5: MOD + P90 restorative system; G6: MOD + Adper Easy One/P90 Bond/P90. Speci...

l k=0 (−1) m−k l k m − k n 2k k − 2l + m = l k=0 l k 2k n n − l m + n − 3k − l. On the basis of this identity, for d ∈ {0, 1, 2,. .. } and ε ∈ {0, ±1} we construct explicit f ε (d) and g ε (d) such that for any prime p > d we have p−1 k=1 k ε C k+d ≡ f ε (d) if p ≡ 1 (mod 3), g ε (d) if p ≡ 2 (mod 3), where C n denotes the Catalan number 1 n+1 2n n ; for example, if p 5 is a prime then 0<k<p−4 ...

In this paper, we associate canonically a precyclic mod- ule to a regular multiplier Hopf algebra endowed with a group-like projection and a modular pair in involution satisfying certain con- dition

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the n’th random variable Xn is by a particular member Rn of a given family of distributions, whose variance increases with n. The basic assumption is that the ratio ...

This study examined the effect of heavy-intensity warm-up exercise on O(2) uptake (VO(2)) kinetics at the onset of moderate-intensity (80% ventilation threshold), constant-work rate exercise in eight older (65 +/- 2 yr) and seven younger adults (26 +/- 1 yr). Step increases in work rate from loadless cycling to moderate exercise (Mod(1)), heavy exercise, and moderate exercise (Mod(2)) were perf...

As real-time systems continue to grow, performance evaluation plays a critical role in the design of these systems since the computation time, the service time, and the responsive actions must satisfy the time constraints. One of these systems is the real-time distributed multimedia-on-demand (MOD) service system. The MOD system usually fails when it misses a task deadline. The main units of th...