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In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulae for calculating the poly-Bernoulli numbers of the second kind and the Stirling numbers of the second kind.
Consider a quantum computer in combination with a binary oracle of domain size N. It is shown how N/2+sqrt(N) calls to the oracle are sufficient to guess the whole content of the oracle (being an N bit string) with probability greater than…
We consider the problem of decomposing a higher-order tensor with binary entries. Such data problems arise frequently in applications such as neuroimaging, recommendation system, topic modeling, and sensor network localization. We propose a…
Processors may find some elementary operations to be faster than the others. Although an operation may be conceptually as simple as some other operation, the processing speeds of the two can vary. A clever programmer will always try to…
Given a set $P$ of $n$ points and a set $S$ of $n$ segments in the plane, we consider the problem of computing for each segment of $S$ its closest point in $P$. The previously best algorithm solves the problem in $n^{4/3}2^{O(\log^*n)}$…
We present reversible classical circuits for performing various arithmetic operations aided by dirty ancillae (i.e. extra qubits in an unknown state that must be restored before the circuit ends). We improve the number of clean qubits…
In this paper, we present a novel algorithm and its three variations for solving the Rubik's cube more efficiently. This algorithm can be used to solve the complete $n \times n \times n$ cube in $O(\frac{n^2}{\log n})$ moves. This algorithm…
Solving random subset sum instances plays an important role in constructing cryptographic systems. For the random subset sum problem, in 2013 Bernstein et al. proposed a quantum algorithm with heuristic time complexity…
This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order…
In this paper, we consider the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.
A Bernoulli factory is a model for randomness manipulation that transforms an initial Bernoulli random variable into another Bernoulli variable by applying a predetermined function relating the output bias to the input one. In literature,…
In the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.
The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(N\epsilon^{-2/3} + \epsilon^{-8/3})$…
In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum…
We offer multiplication method for factoring big natural numbers which extends the group of the Fermat's and Lehman's factorization algorithms and has run-time complexity $O(n^{1/3})$. This paper is argued the finiteness of proposed…
Two algorithms for computing $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, are described, and using a combination of these two algorithms, the resulting algorithm is $O(n^{3/2})$. The second algorithm uses a list…
Recently, Farhi, Goldstone, and Gutmann gave a quantum algorithm for evaluating NAND trees that runs in time O(sqrt(N log N)) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm…
Let {\alpha} be the maximal value such that the product of an n x n^{\alpha} matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic operations. In this paper we show that \alpha>0.30298, which improves the previous…
We present an O(\sqrt{N}) discrete query quantum algorithm for evaluating balanced binary NAND formulas and an O(N^{{1/2}+O(\frac{1}{\sqrt{\log N}})}) discrete query quantum algorithm for evaluating arbitrary binary NAND formulas.
There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity $O(n^3)$ to advanced tensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possible bound achieved), a lot of…