Exponential prefixed polynomial equations

A prefixed polynomial equation is an equation of the form $P(t_1,\ldots,t_n) = 0$, where $P$ is a polynomial whose variables $t_1,\ldots,t_n$ range over the natural numbers, preceded by quantifiers over some, or all, of its variables. Here, we consider exponential prefixed polynomial equations (EPPEs), where variables can also occur as exponents. We obtain a relatively concise EPPE equivalent to the combinatorial principle of the Paris-Harrington theorem for pairs (which is independent of primitive recursive arithmetic), as well as an EPPE equivalent to Goodstein's theorem (which is independent of Peano arithmetic). Some new devices are used in addition to known methods for the elimination of bounded universal quantifiers for Diophantine predicates

Fast matrix multiplication techniques based on the Adleman-Lipton model

On distributed memory electronic computers, the implementation and association of fast parallel matrix multiplication algorithms has yielded astounding results and insights. In this discourse, we use the tools of molecular biology to demonstrate the theoretical encoding of Strassen's fast matrix multiplication algorithm with DNA based on an $n$-moduli set in the residue number system, thereby demonstrating the viability of computational mathematics with DNA. As a result, a general scalable implementation of this model in the DNA computing paradigm is presented and can be generalized to the application of \emph{all} fast matrix multiplication algorithms on a DNA computer. We also discuss the practical capabilities and issues of this scalable implementation. Fast methods of matrix computations with DNA are important because they also allow for the efficient implementation of other algorithms (i.e. inversion, computing determinants, and graph theory) with DNA.Comment: To appear in the International Journal of Computer Engineering Research. Minor changes made to make the preprint as similar as possible to the published versio

A New MCMC Sampling Based Segment Model for Radar Target Recognition

One of the main tools in radar target recognition is high resolution range profile (HRRP)‎. ‎However‎, ‎it is very sensitive to the aspect angle‎. ‎One solution to this problem is to assume the consecutive samples of HRRP identically independently distributed (IID) in small frames of aspect angles‎, ‎an assumption which is not true in reality‎. ‎However, b‎‎ased on this assumption‎, ‎some models have been developed to characterize the sequential information contained in the multi-aspect radar echoes‎. ‎Therefore‎, ‎they only consider the short dependency between consecutive samples‎. ‎Here‎, ‎we propose an alternative model‎, ‎the segment model‎, ‎to address the shortcomings of these assumptions‎. ‎In addition‎, ‎using a Markov chain Monte-Carlo (MCMC) based Gibbs sampler as an iterative approach to estimate the parameters of the segment model‎, ‎we will show that the proposed method is able to estimate the parameters with quite satisfying accuracy and computational load‎

Quantum lower bound for inverting a permutation with advice

Given a random permutation $f: [N] \to [N]$ as a black box and $y \in [N]$, we want to output $x = f^{-1}(y)$. Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but \emph{not} on the input $y$. Classically, there is a data structure of size $\tilde{O}(S)$ and an algorithm that with the help of the data structure, given $f(x)$, can invert $f$ in time $\tilde{O}(T)$, for every choice of parameters $S$, $T$, such that $S\cdot T \ge N$. We prove a quantum lower bound of $T^2\cdot S \ge \tilde{\Omega}(\epsilon N)$ for quantum algorithms that invert a random permutation $f$ on an $\epsilon$ fraction of inputs, where $T$ is the number of queries to $f$ and $S$ is the amount of advice. This answers an open question of De et al. We also give a $\Omega(\sqrt{N/m})$ quantum lower bound for the simpler but related Yao's box problem, which is the problem of recovering a bit $x_j$, given the ability to query an $N$-bit string $x$ at any index except the $j$-th, and also given $m$ bits of advice that depend on $x$ but not on $j$.Comment: To appear in Quantum Information & Computation. Revised version based on referee comment

On The Positive Definiteness of Polarity Coincidence Correlation Coefficient Matrix

Polarity coincidence correlator (PCC), when used to estimate the covariance matrix on an element-by-element basis, may not yield a positive semi-definite (PSD) estimate. Devlin et al. [1], claimed that element-wise PCC is not guaranteed to be PSD in dimensions p>3 for real signals. However, no justification or proof was available on this issue. In this letter, it is proved that for real signals with p<=3 and for complex signals with p<=2, a PSD estimate is guaranteed. Counterexamples are presented for higher dimensions which yield invalid covariance estimates.Comment: IEEE Signal Processing Letters, Volume 15, pp. 73-76, 200

Eye of the Mind: Image Processing for Social Coding

Developers are increasingly sharing images in social coding environments alongside the growth in visual interactions within social networks. The analysis of the ratio between the textual and visual content of Mozilla's change requests and in Q/As of StackOverflow programming revealed a steady increase in sharing images over the past five years. Developers' shared images are meaningful and are providing complementary information compared to their associated text. Often, the shared images are essential in understanding the change requests, questions, or the responses submitted. Relying on these observations, we delve into the potential of automatic completion of textual software artifacts with visual content.Comment: This is the author's version of ICSE 2020 pape

Radar HRRP Modeling using Dynamic System for Radar Target Recognition

High resolution range profile (HRRP) is being known as one of the most powerful tools for radar target recognition. The main problem with range profile for radar target recognition is its sensitivity to aspect angle. To overcome this problem, consecutive samples of HRRP were assumed to be identically independently distributed (IID) in small frames of aspect angles in most of the related works. Here, considering the physical circumstances of maneuver of an aerial target, we have proposed dynamic system which models the short dependency between consecutive samples of HRRP in segments of the whole HRRP sequence. Dynamic system (DS) is used to model the sequence of PCA (principal component analysis) coefficients extracted from the sequence of HRRPs. Considering this we have proposed a model called PCA+DS. We have also proposed a segmentation algorithm which segments the HRRP sequence reliably. Akaike information criterion (AIC) used to evaluate the quality of data modeling showed that our PCA+DS model outperforms factor analysis (FA) model. In addition, target recognition results using simulated data showed that our method based on PCA+DS achieves better recognition rates compared to the method based on FA