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Succinct Partial Sums and Fenwick Trees [PDF]
Lecture Notes in Computer Science, 2017 We consider the well-studied partial sums problem in succint space where one is to maintain an array of n k-bit integers subject to updates such that partial sums queries can be efficiently answered. We present two succint versions of the Fenwick Tree - which is known for its simplicity and practicality. Our results hold in the encoding model where one
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On the boundedness of general partial sums
Periodica Mathematica Hungarica, 2023The author studies problems of convergence of general Fourier series with respect to ONS \( (\varphi_n) \) for the class \(C_V\) of functions with derivative of bounded variation on \( [0,1]. \) The function \[ T_n(x)=\max_{1\leq i\leq n}\left\vert \int_0^{i/n}Q_n(u,x)du\right\vert ,\] where \( Q_n(u,x)=\sum_{k=1}^ng_k(u)\varphi_k(x), \) \( g_n(x ...
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On the Partial Sums of a Fourier Series
Constructive Approximation, 2007We give sharp lower estimates for the partial sums of the Fourier series $\sin x+\frac{\cos2x}{2}+\frac{\sin 3x}{3}+\frac{\cos 4x}{4}+\cdots,$ with both an even and odd number of terms.
Alzer, H. +3 more
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Bulletin of the London Mathematical Society, 1988
The present paper is concerned with the estimation of sums of the type \[ S_ a(N,M)=\sum^{M}_{\ell =N+1}\chi (\ell)e^{2\pi ia\ell /p}, \] where \(\chi\) denotes a non-principal character modulo the prime p. In the special case of \(a=0\), the author has already obtained in [Proc. Lond. Math. Soc., III. Ser.
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The present paper is concerned with the estimation of sums of the type \[ S_ a(N,M)=\sum^{M}_{\ell =N+1}\chi (\ell)e^{2\pi ia\ell /p}, \] where \(\chi\) denotes a non-principal character modulo the prime p. In the special case of \(a=0\), the author has already obtained in [Proc. Lond. Math. Soc., III. Ser.
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Partial Sum Process for Records
Extremes, 2005Let \(X_{L_1},X_{L_2},\dots,X_{L_n}\) be a sequence of records for i.i.d. \(X_i\) with CDF \(F\). The authors derive a functional CLT for the process \[ Z_n(t)={1\over \psi(n+\sqrt{n})-\psi(n)} \left( \sum_{j=1}^{nt} X_{L_j}-\psi([nt]) \right), \] where \(\psi(x)=F^{-1}(1-\exp(-x))\) in the case where the normalized distribution of \(X_{L_n ...
Bose, Arup +2 more
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On the Complexity of Maintaining Partial Sums
SIAM Journal on Computing, 1985Let $F = \{ ({\bf r}_i ,s_i )|0 \leqq i < n\} $ be a file of n records, where ${\bf r}_i $ are d-dimensional vectors and $s_i $ are elements of a commutative semigroup S. We are interested in the partial sum problem, in which queries of the form “$\sum_{r_i \leqq {\bf a}} {s_i = ?} $” are to be answered.
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2014
In this paper we introduce and study a class of partially ordered sets that can be interpreted as partial sums of indeterminate real numbers. An important example of these partially ordered sets, is the classical Young lattice Y of the integer partitions.
Chiaselotti, Giampiero +2 more
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In this paper we introduce and study a class of partially ordered sets that can be interpreted as partial sums of indeterminate real numbers. An important example of these partially ordered sets, is the classical Young lattice Y of the integer partitions.
Chiaselotti, Giampiero +2 more
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On the fluctuations of partial sums
Series Statistics, 1979For non-negative integral valued interchangeable random variables Takacs [6, 7] has derived the probability distributions of certain statistics viz. concerning the partial sums . This paper deals with the probability distributions of some other statistics viz. concerning the partial sums , of geometric random variables .
Saran, Jagdish, Sen, Kanwar
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PRIMUS, 2008
Abstract In this article we consider an example suitable for investigation in many mid and upper level undergraduate mathematics courses. Fourier series provide an excellent example of the differences between uniform and non-uniform convergence. We use Dirichlet's test to investigate the convergence of the Fourier series for a simple periodic saw tooth
Michael A. Brilleslyper +1 more
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Abstract In this article we consider an example suitable for investigation in many mid and upper level undergraduate mathematics courses. Fourier series provide an excellent example of the differences between uniform and non-uniform convergence. We use Dirichlet's test to investigate the convergence of the Fourier series for a simple periodic saw tooth
Michael A. Brilleslyper +1 more
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1996
The name “Donsker class of functions” was chosen in honor of Donsker’s theorem on weak convergence of the empirical distribution function. A second famous theorem by Donsker concerns the partial-sum process \( {\mathbb{Z}_n}(s) = \frac{1}{{\sqrt n }}\sum\limits_{i = 1}^{\left[ {ns} \right]} {{Y_i}} = \frac{1}{{\sqrt n }}\sum\limits_{i = 1}^k {{Y_i ...
Aad W. van der Vaart, Jon A. Wellner
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The name “Donsker class of functions” was chosen in honor of Donsker’s theorem on weak convergence of the empirical distribution function. A second famous theorem by Donsker concerns the partial-sum process \( {\mathbb{Z}_n}(s) = \frac{1}{{\sqrt n }}\sum\limits_{i = 1}^{\left[ {ns} \right]} {{Y_i}} = \frac{1}{{\sqrt n }}\sum\limits_{i = 1}^k {{Y_i ...
Aad W. van der Vaart, Jon A. Wellner
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