Given an M×N array of reals, we want to find a rectangular contiguous subarray such that the sum of the entries in the subarray is maximized. Since Bentley posed this problem in his Programming Pearls column in 1984 with an O (NM ²) time solution, no progress on the sequential complexity has been reported to date. We give the first subcubic algorithm, by reducing the problem to “funny matrix multiplication”, where the scalar product and addition in usual matrix multiplication are replaced by addition and max operations, respectively. We also give a faster ε-approximation algorithm via the same reduction.

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Let p be a positive integer greater than 1 and denote the p-adic expansion of n ∈N by n =∑_i ≥0α_i (n )p ⁱ, where α_i (n ) ∈{0,1,…,p −1}. We set s (n,l ) =∑_i ≥01_{{αi (n)=l }} for l =1,2,…,p −1, and s (n ) =∑_l =1^p −1l s (n,l ). We define the power sum of sum of digits by S_k(N ) =∑_n =0^N −1s (n )^k, k ∈N, and the exponential sum of sum of digits by F (ξ ,N ) =∑_n =0^N −1e ^ξs(n), ξ ∈R for N ∈N. Explicit formulas of these sums have been investigated by many authors for a long time. In this paper, developing our previous results, we shall give explicit formulas of F (ξ ,N ) and S_k(N ).

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Voiculescu’s single variable free entropy is generalized to the free relative entropy in the microstate approach, and the integral representation of the free relative entropy is obtained by large deviation results for random matrices. It is also shown that the free relative entropy shares common properties of the classical relative entropy.

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Let f = (f_n ,F_n ) _n≥0 be a martingale on some filtered complete probability space (Ω,F,P ) with the usual conditions. We define the iterated martingale transforms I ^(m)(f ) = (I_n^(m), (F_n )) (m ≥1) with respect to f, the discrete analogues of the iterated stochastic integrals. We obtain the L^p (1≤p)-estimates of I ^(m)(f )

||I ^(m)*||_p ≤ (4mp)^m ||S ^m (f )||_p

and we also characterize a continuous martingale by the limit of the iterated martingale transforms.

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We determine explicitly the trace of the representation of SL(2,Z ⁄NZ ) in the space of modular forms of half integral weight and derive an explicit formula of the multiplicities of all irreducible components occuring in the representation of SL(2,Z ⁄pZ ) in the space of modular forms of half integral weight, which yields an analogue of Eichler’s results in the case of modular forms of half integral weight.

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This article attempts to explain why blackboard can be other colors than black while black board must be black. This difference in meaning between them is shown to be accounted for by assuming the independently motivated assumptions, including Heim's (1982) Tripartite Quantificational Structure. Building on the consideration of the two nominal expressions, blackboard and black board, I try to shed light on the question of what occurs at the interface where phonological, semantic, and syntactic components meet.

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This paper offers a solution to the following problems concerning frequency relatives like “When she was drunk, which was often, Mary smoked”: (a) Why do frequency relatives only co-occur with restrictive when/if -clauses in habitual/generic sentences?, (b) What is an antecedent of the relative pronoun which? and (c) What type of copular sentences do frequency relatives belong to? It is proposed that adverbs of quantification should explicitly specify sum situations in logical representations of habitual/generic sentences. Under the assumption that frequency relatives are specificational, it is argued that the pre-copular relative pronoun ought to have a logical representation containing a sum-situation variable, and the post-copular adverb of frequency functions as a value assigned to the variable.

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This paper attempts to built two analytical models of Japan and South Korea and perform a simulation of the household waste generation mechanism, and compares the effectiveness of Japan's household waste generation program with that of Korea's. The result of the simulation shows that there are common features on the mechanism of household waste generation of both countries, namely, the progress of service economy is the factor that increase the household waste generation etc.

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