Evaluating testing methods by delivered reliability

There are two main goals in testing software: (1) to achieve adequate quality (debug testing), where the objective is to probe the software for defects so that these can be removed, and (2) to assess existing quality (operational testing), where the objective is to gain confidence that the software is reliable. Debug methods tend to ignore random selection of test data from an operational profile, while for operational methods this selection is all-important. Debug methods are thought to be good at uncovering defects so that these can be repaired, but having done so they do not provide a technically defensible assessment of the reliability that results. On the other hand, operational methods provide accurate assessment, but may not be as useful for achieving reliability. This paper examines the relationship between the two testing goals, using a probabilistic analysis. We define simple models of programs and their testing, and try to answer the question of how to attain program reliability: is it better to test by probing for defects as in debug testing, or to assess reliability directly as in operational testing? Testing methods are compared in a model where program failures are detected and the software changed to eliminate them. The “better” method delivers higher reliability after all test failures have been eliminated. Special cases are exhibited in which each kind of testing is superior. An analysis of the distribution of the delivered reliability indicates that even simple models have unusual statistical properties, suggesting caution in interpreting theoretical comparisons

Non locality, closing the detection loophole and communication complexity

It is shown that the detection loophole which arises when trying to rule out local realistic theories as alternatives for quantum mechanics can be closed if the detection efficiency $\eta$ is larger than $\eta \geq d^{1/2} 2^{-0.0035d}$ where $d$ is the dimension of the entangled system. Furthermore it is argued that this exponential decrease of the detector efficiency required to close the detection loophole is almost optimal. This argument is based on a close connection that exists between closing the detection loophole and the amount of classical communication required to simulate quantum correlation when the detectors are perfect.Comment: 4 pages Latex, minor typos correcte

Urinary Exosomes Contain MicroRNAs Capable of Paracrine Modulation of Tubular Transporters in Kidney

Exosomes derived from all nephron segments are present in human urine, where their functionality is incompletely understood. Most studies have focused on biomarker discovery rather than exosome function. Through sequencing we identified the miRNA repertoire of urinary exosomes from healthy volunteers; 276 mature miRNAs and 345 pre-miRNAs were identified (43%/7% of reads). Among the most abundant were members of the miR-10, miR-30 and let-7 families. Targets for the identified miRNAs were predicted using five different databases; genes encoding membrane transporters and their regulators were enriched, highlighting the possibility that these miRNAs could modulate key renal tubular functions in a paracrine manner. As proof of concept, cultured renal epithelial cells were exposed to urinary exosomes and cellular exosomal uptake was confirmed; thereafter, reduced levels of the potassium channel ROMK and kinases SGK1 and WNK1 were observed in a human collecting duct cell line, while SPAK was unaltered. In proximal tubular cells, mRNA levels of the amino acid transporter gene SLC38A2 were diminished and reflected in a significant decrement of its encoded protein SNAT2. Protein levels of the kinase SGK1 did not change. Thus we demonstrated a novel potential function for miRNA in urinary exosomes.This work was supported by the Wellcome Trust (grant 088489/Z/09/Z to FEKF and Strategic award 100140/Z/12/Z to the Cambridge Institute for Medical Research). The sequencing facility is supported by the Cambridge Biomedical Research Centre

Chromatic number, clique subdivisions, and the conjectures of Haj\'os and Erd\H{o}s-Fajtlowicz

For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for every graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant $C$ such that $\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In this paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on $n$ vertices with independence number $\alpha$.Comment: 14 page

A note on linear Sperner families

In an earlier work we described Gr\"obner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors $\mathbf{v} \in \{0,1\}^n$ of the complete $d$ unifom set family over the ground set $[n]$. In particular, it turns out that the standard monomials of the above ideal are {\em ballot monomials}. We give here a partial extension of the latter fact. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation $a_1v_1+\cdots +a_nv_n=k$, where $0<a_q\leq a_2\leq \cdots \leq a_n$ and $k$ are integers. As an application, we confirm a conjecture of Frankl for linear Sperner systems.Comment: 12 page

Quantum fingerprinting

Classical fingerprinting associates with each string a shorter string (its fingerprint), such that, with high probability, any two distinct strings can be distinguished by comparing their fingerprints alone. The fingerprints can be exponentially smaller than the original strings if the parties preparing the fingerprints share a random key, but not if they only have access to uncorrelated random sources. In this paper we show that fingerprints consisting of quantum information can be made exponentially smaller than the original strings without any correlations or entanglement between the parties: we give a scheme where the quantum fingerprints are exponentially shorter than the original strings and we give a test that distinguishes any two unknown quantum fingerprints with high probability. Our scheme implies an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity. We optimize several aspects of our scheme.Comment: 8 pages, LaTeX, one figur

Small ball probability, Inverse theorems, and applications

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page