arXiv subjects

Paul Watts

Publications and source records attributed to Paul Watts.

At least 19 recordsLinked to original sources

arXiv2026

Stickel-type key exchange with hidden subspaces

We give a witness-finding cryptanalysis of Stickel-type key exchange schemes, which involve two-sided multiplication of $n \times n$ matrices over $\mathbb{F}_p$, where these matrices are drawn from public subspaces with a particular commuting structure. This analysis covers Stickel's original proposal , Shpilrain's polynomial extension of that scheme, Nager's algebraic extension of that scheme, and more generally all Stickel-type approaches using public subspaces over matrix algebra in finite fields: all such schemes can be broken in polynomial time. We also describe a new key establishment scheme using two-sided matrix multiplication in which the commuting subspaces used to form the key are hidden via conjugation by private terms, blocking this specific public-subspace analysis; the witness-finding problem in this new scheme has a direct reduction from a standard NP-hard problem (Edmonds' problem).

cs.CR↗

arXiv2023

Underestimating the uncertainty of aggregated results: the case of W-Boson mass

Estimates of uncertainty or variance in experimental means are central to physics. This is especially the case for `world averages' of fundamental physical parameters in particle physics, which aggregate results from a number of experiments to express current knowledge about these parameters and where variance in these world averages reflect uncertainty in that knowledge. The standard aggregation method used to estimate such parameters is a form of hierarchical fixed-effect meta-analysis. This method is inherently unsuitable for estimating these parameters because it ignores sources of random variation between experiments (that is, variation in experimental accuracy, which becomes increasingly important as experimental precision rises). This problem is well-known in the statistical literature, where the typical recommendation is to use random -- rather than fixed-effect techniques. We illustrate this problem by applying random-effect meta-analysis to estimates of the W-Boson mass.

hep-ex↗

arXiv2022

How to Tell When a Result Will Replicate: Significance and Replication in Distributional Null Hypothesis Tests

There is a well-known problem in Null Hypothesis Significance Testing: many statistically significant results fail to replicate in subsequent experiments. We show that this problem arises because standard `point-form null' significance tests consider only within-experiment but ignore between-experiment variation, and so systematically underestimate the degree of random variation in results. We give an extension to standard significance testing that addresses this problem by analysing both within- and between-experiment variation. This `distributional null' approach does not underestimate experimental variability and so is not overconfident in identifying significance; because this approach addresses between-experiment variation, it gives mathematically coherent estimates for the probability of replication of significant results. Using a large-scale replication dataset (the first `Many Labs' project), we show that many experimental results that appear statistically significant in standard tests are in fact consistent with random variation when both within- and between-experiment variation are taken into account in this approach. Further, grouping experiments in this dataset into `predictor-target' pairs we show that the predicted replication probabilities for target experiments produced in this approach (given predictor experiment results and the sample sizes of the two experiments) are strongly correlated with observed replication rates. Distributional null hypothesis testing thus gives researchers a statistical tool for identifying statistically significant and reliably replicable results.

stat.ME↗

arXiv2022

Why bother with Bayesian t-tests?

Given the well-known and fundamental problems with hypothesis testing via classical (point-form) significance tests, there has been a general move to alternative approaches, often focused on the Bayesian t-test. We show that the Bayesian t-test approach does not address the observed problems with classical significance testing, that Bayesian and classical t-tests are mathematically equivalent and linearly related in order of magnitude (so that the Bayesian t-test providing no further information beyond that given by point-form significance tests), and that Bayesian t-tests are subject to serious risks of misinterpretation, in some cases more problematic than seen for classical tests (with, for example, a negative sample mean in an experiment giving strong Bayesian t-test evidence in favour of a positive population mean). We do not suggest a return to the classical, point-form significance approach to hypothesis testing. Instead we argue for an alternative distributional approach to significance testing, which addresses the observed problems with classical hypothesis testing and provides a natural link between the Bayesian and frequentist approaches.

math.ST↗

arXiv2022

Homeostatic behavioural response to COVID-19 infections returns R to a set-point of 1

One clear aspect of behaviour in the COVID-19 pandemic has been people's focus on, and response to, reported or observed infection numbers in their community. We describe a simple model of infectious disease spread in a pandemic situation where people's behaviour is influenced by the current risk of infection and where this behavioural response acts homeostatically to return infection risk to a certain preferred level. This model predicts that the reproduction rate $R$ will be centered around a median value of 1, and that a related measure of relative change in the number of new infections will follow the standard Cauchy distribution. Analysis of worldwide COVID-19 data shows that the estimated reproduction rate has a median of 1, and that this measure of relative change calculated from reported numbers of new infections closely follows the standard Cauchy distribution at both an overall and an individual country level.

q-bio.PE↗

arXiv2020

Significance and Replication in simple counting experiments: Distributional Null Hypothesis Testing

Null Hypothesis Significance Testing (NHST) has long been of central importance to psychology as a science, guiding theory development and underlying the application of evidence-based intervention and decision-making. Recent years, however, have seen growing awareness of serious problems with NHST as it is typically used; this awareness has led to proposals to limit the use of NHST techniques, to abandon these techniques and move to alternative statistical approaches, or even to ban the use of NHST entirely. These proposals are premature, because the observed problems with NHST arise as a consequence of an historically contingent, essentially unmotivated, and fundamentally incorrect, choice: that of NHST testing against point-form null hypotheses. Using simple counting experiments we give a detailed presentation of an alternative, more general approach: that of testing against distributional nulls. We show that this distributional approach is well-motivated mathematically, practically and experimentally, and that the use of distributional nulls addresses various problems with the standard point-form NHST approach, avoiding issues to do with sample size and allowing a coherent estimation of the probability of replication of a given experimental result. Rather than abandoning NHST, we should use the NHST approach in its most general form, with distributional rather than point-form null hypotheses.

stat.ME↗

arXiv2020

Distributional Null Hypothesis Testing with the T distribution

Null Hypothesis Significance Testing (NHST) has long been central to the scientific project, guiding theory development and supporting evidence-based intervention and decision-making. Recent years, however, have seen growing awareness of serious problems with NHST as it is typically used, and hence to proposals to limit the use of NHST techniques, to abandon these techniques and move to alternative statistical approaches, or even to ban the use of NHST entirely. These proposals are premature, because the observed problems with NHST all arise as a consequence of a contingent and in many cases incorrect choice: that of NHST testing against point-form nulls. We show that testing against distributional, rather than point-form, nulls is better motivated mathematically and experimentally, and that the use of distributional nulls addresses many problems with the standard point-form NHST approach. We also show that use of distributional nulls allows a form of null hypothesis testing that takes into account both the statistical significance of a given result and the probability of replication of that result in a new experiment. Rather than abandoning NHST, we should use the NHST approach in its more general form, with distributional rather than point-form nulls.

stat.ME↗

arXiv2014

Optimizing for an arbitrary perfect entangler: I. Functionals

Optimal control theory is a powerful tool for improving figures of merit in quantum information tasks. Finding the solution to any optimal control problem via numerical optimization depends crucially on the choice of the optimization functional. Here, we derive a functional that targets the full set of two-qubit perfect entanglers, gates capable of creating a maximally-entangled state out of some initial product state. The functional depends on easily-computable local invariants and uniquely determines when a gate evolves into a perfect entangler. Optimization with our functional is most useful if the two-qubit dynamics allows for the implementation of more than one perfect entangler. We discuss the reachable set of perfect entanglers for a generic Hamiltonian that corresponds to several quantum information platforms of current interest.

quant-ph↗

arXiv2014

Surprisingly Rational: Probability theory plus noise explains biases in judgment

The systematic biases seen in people's probability judgments are typically taken as evidence that people do not reason about probability using the rules of probability theory, but instead use heuristics which sometimes yield reasonable judgments and sometimes systematic biases. This view has had a major impact in economics, law, medicine, and other fields; indeed, the idea that people cannot reason with probabilities has become a widespread truism. We present a simple alternative to this view, where people reason about probability according to probability theory but are subject to random variation or noise in the reasoning process. In this account the effect of noise is cancelled for some probabilistic expressions: analysing data from two experiments we find that, for these expressions, people's probability judgments are strikingly close to those required by probability theory. For other expressions this account produces systematic deviations in probability estimates. These deviations explain four reliable biases in human probabilistic reasoning (conservatism, subadditivity, conjunction and disjunction fallacies). These results suggest that people's probability judgments embody the rules of probability theory, and that biases in those judgments are due to the effects of random noise.

physics.data-an↗

arXiv2013

Graphical Calculus for the Double Affine Q-Dependent Braid Group

We define a double affine $Q$-dependent braid group. This group is constructed by appending to the braid group a set of operators $Q_i$, before extending it to an affine $Q$-dependent braid group. We show specifically that the elliptic braid group and the double affine Hecke algebra (DAHA) can be obtained as quotient groups. Complementing this we present a pictorial representation of the double affine $Q$-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation we can fully describe any DAHA. Specifically, we graphically describe the parameter $q$ upon which this algebra is dependent and show that in this particular representation $q$ corresponds to a twist in the ribbon.

math-ph↗

arXiv2013

Metric Structure of the Space of Two-Qubit Gates, Perfect Entanglers and Quantum Control

We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the invariant volume of the space of two-qubit perfect entanglers. We find that this volume corresponds to more than 84% of the total invariant volume of the space of two-qubit gates. This same metric is also used to determine the effective target sizes that selected gates will present in any quantum-control procedure designed to implement them.

quant-ph↗

arXiv2013

A short note on Jacobsthal's function

The function g(n) represents the smallest number Q such that every sequence of Q consecutive integers contains an integer coprime to n. We give a new and explicit upper bound on this function.

math.NT↗

arXiv2012

Jacobsthal's function and a generalisation of Euler's totient

Jacobsthal's function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to P_k, the product of the first k primes. The best known bound on h(k) is h(k) < C (k ln k)^2 for some unknown constant C, due to Iwaniec. We use a generalisation of Euler's totient function to give a stronger bound on h(k).

math.NT↗

arXiv2012

A computational upper bound on Jacobsthal's function

The function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to the first k primes. We give a new computational method for calculating strong upper bounds on h(k).

math.NT↗

arXiv2005

Pre-Big Bang Scenario on Self-T-Dual Bouncing Branes

We consider a new class of 5-dimensional dilatonic actions which are invariant under T-duality transformations along three compact coordinates, provided that an appropriate potential is chosen. We show that the invariance remains when we add a boundary term corresponding to a moving 3-brane, and we study the effects of the T-duality symmetry on the brane cosmological equations. We find that T-duality transformations in the bulk induce scale factor duality on the brane, together with a change of sign of the pressure of the brane cosmological matter. However, in a remarkable analogy with the Pre-Big Bang scenario, the cosmological equations are unchanged. Finally, we propose a model where the dual phases are connected through a scattering of the brane induced by an effective potential. We show how this model can realise a smooth, non-singular transition between a pre-Big Bang superinflationary Universe and a post-Big Bang accelerating Universe.

hep-th↗

arXiv2000

Derivatives and the Role of the Drinfel'd Twist in Noncommutative String Theory

We consider the derivatives which appear in the context of noncommutative string theory. First, we identify the correct derivations to use when the underlying structure of the theory is a quasitriangular Hopf algebra. Then we show that this is a specific case of a more general structure utilising the Drinfel'd twist. We go on to present reasons as to why we feel that the low-energy effective action, when written in terms of the original commuting coordinates, should explicitly exhibit this twisting.

hep-th↗

arXiv1999

Noncommutative String Theory, the R-Matrix, and Hopf Algebras

Motivated by the form of the noncommutative *-product in a system of open strings and Dp-branes with constant nonzero Neveu-Schwarz 2-form, we define a deformed multiplication operation on a quasitriangular Hopf algebra in terms of its R-matrix, and comment on some of its properties. We show that the noncommutative string theory *-product is a particular example of this multiplication, and comment on other possible Hopf algebraic properties which may underlie the theory.

hep-th↗

arXiv1999

Ward Identities and Anomalies in Pure W_4 Gravity

W_4 gravity is treated algebraically, represented by a set of transformations on classical fields. The Ward identities of the theory are determined by requiring the algebra to close. The general forms for the anomalies are found by looking for solutions to the Wess-Zumino consistency conditions, and some specific cases are considered.

hep-th↗