1,722 research outputs found
Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators
Ground state energies and wave functions of quartic and pure quartic
oscillators are calculated by first casting the Schr\"{o}dinger equation into a
nonlinear Riccati form and then solving that nonlinear equation analytically in
the first iteration of the quasilinearization method (QLM). In the QLM the
nonlinear differential equation is solved by approximating the nonlinear terms
by a sequence of linear expressions. The QLM is iterative but not perturbative
and gives stable solutions to nonlinear problems without depending on the
existence of a smallness parameter. Our explicit analytic results are then
compared with exact numerical and also with WKB solutions and it is found that
our ground state wave functions, using a range of small to large coupling
constants, yield a precision of between 0.1 and 1 percent and are more accurate
than WKB solutions by two to three orders of magnitude. In addition, our QLM
wave functions are devoid of unphysical turning point singularities and thus
allow one to make analytical estimates of how variation of the oscillator
parameters affects physical systems that can be described by the quartic and
pure quartic oscillators.Comment: 8 pages, 12 figures, 1 tabl
How good are your fits? Unbinned multivariate goodness-of-fit tests in high energy physics
Multivariate analyses play an important role in high energy physics. Such
analyses often involve performing an unbinned maximum likelihood fit of a
probability density function (p.d.f.) to the data. This paper explores a
variety of unbinned methods for determining the goodness of fit of the p.d.f.
to the data. The application and performance of each method is discussed in the
context of a real-life high energy physics analysis (a Dalitz-plot analysis).
Several of the methods presented in this paper can also be used for the
non-parametric determination of whether two samples originate from the same
parent p.d.f. This can be used, e.g., to determine the quality of a detector
Monte Carlo simulation without the need for a parametric expression of the
efficiency.Comment: 32 pages, 12 figure
Enhanced Transmission Due to Disorder
The transmissivity of a one-dimensional random system that is periodic on
average is studied. It is shown that the transmission coefficient for
frequencies corresponding to a gap in the band structure of the average
periodic system increases with increasing disorder while the disorder is weak
enough. This property is shown to be universal, independent of the type of
fluctuations causing the randomness. In the case of strong disorder the
transmission coefficient for frequencies in allowed bands is found to be a non
monotonic function of the strength of the disorder. An explanation for the
latter behavior is provided.Comment: 9 pages, RevTeX 3.0, 4 Postscript figure
Ballistic deposition patterns beneath a growing KPZ interface
We consider a (1+1)-dimensional ballistic deposition process with
next-nearest neighbor interaction, which belongs to the KPZ universality class,
and introduce for this discrete model a variational formulation similar to that
for the randomly forced continuous Burgers equation. This allows to identify
the characteristic structures in the bulk of a growing aggregate ("clusters"
and "crevices") with minimizers and shocks in the Burgers turbulence, and to
introduce a new kind of equipped Airy process for ballistic growth. We dub it
the "hairy Airy process" and investigate its statistics numerically. We also
identify scaling laws that characterize the ballistic deposition patterns in
the bulk: the law of "thinning" of the forest of clusters with increasing
height, the law of transversal fluctuations of cluster boundaries, and the size
distribution of clusters. The corresponding critical exponents are determined
exactly based on the analogy with the Burgers turbulence and simple scaling
considerations.Comment: 10 pages, 5 figures. Minor edits: typo corrected, added explanation
of two acronyms. The text is essentially equivalent to version
Influence of dynamic content on visual attention during video advertisements
Purpose
Dynamic advertising, including television and online video ads, demands new theory and tools developed to understand attention to moving stimuli. The purpose of this study is to empirically test the predictions of a new dynamic attention theory, Dynamic Human-Centred Communication Systems Theory, versus the predictions of salience theory.
Design/methodology/approach
An eye-tracking study used a sample of consumers to measure visual attention to potential areas of interest (AOIs) in a random selection of unfamiliar video ads. An eye-tracking software feature called intelligent bounding boxes (IBBs) was used to track attention to moving AOIs. AOIs were coded for the presence of static salience variables (size, brightness, colour and clutter) and dynamic attention theory dimensions (imminence, motivational relevance, task relevance and stability).
Findings
Static salience variables contributed 90% of explained variance in fixation and 57% in fixation duration. However, the data further supported the three-way interaction uniquely predicted by dynamic attention theory: between imminence (central vs peripheral), relevance (motivational or task relevant vs not) and stability (fleeting vs stable). The findings of this study indicate that viewers treat dynamic stimuli like real life, paying less attention to central, relevant and stable AOIs, which are available across time and space in the environment and so do not need to be memorised.
Research limitations/implications
Despite the limitations of small samples of consumers and video ads, the results of this study demonstrate the potential of two relatively recent innovations, which have received limited emphasis in the marketing literature: dynamic attention theory and IBBs.
Practical implications
This study documents what does and does not attract attention to video advertising. What gets attention according to salience theory (e.g. central location) may not always get attention in dynamic advertising because of the effects of relevance and stability. To better understand how to execute video advertising to direct and retain attention to important AOIs, advertisers and advertising researchers are encouraged to use IBBs.
Originality/value
This study makes two original contributions: to marketing theory, by showing how dynamic attention theory can predict attention to video advertising better than salience theory, and to marketing research, showing the utility of tracking visual attention to moving objects in video advertising with IBBs, which appear underutilised in advertising research
Cosmological particle production and the precision of the WKB approximation
Particle production by slow-changing gravitational fields is usually
described using quantum field theory in curved spacetime. Calculations require
a definition of the vacuum state, which can be given using the adiabatic (WKB)
approximation. I investigate the best attainable precision of the resulting
approximate definition of the particle number. The standard WKB ansatz yields a
divergent asymptotic series in the adiabatic parameter. I derive a novel
formula for the optimal number of terms in that series and demonstrate that the
error of the optimally truncated WKB series is exponentially small. This
precision is still insufficient to describe particle production from vacuum,
which is typically also exponentially small. An adequately precise
approximation can be found by improving the WKB ansatz through perturbation
theory. I show quantitatively that the fundamentally unavoidable imprecision in
the definition of particle number in a time-dependent background is equal to
the particle production expected to occur during that epoch. The results are
illustrated by analytic and numerical examples.Comment: 14 pages, RevTeX, 5 figures; minor changes, a clarification in Sec.
II
Multiple-copy state discrimination: Thinking globally, acting locally
We theoretically investigate schemes to discriminate between two
nonorthogonal quantum states given multiple copies. We consider a number of
state discrimination schemes as applied to nonorthogonal, mixed states of a
qubit. In particular, we examine the difference that local and global
optimization of local measurements makes to the probability of obtaining an
erroneous result, in the regime of finite numbers of copies , and in the
asymptotic limit as . Five schemes are considered:
optimal collective measurements over all copies, locally optimal local
measurements in a fixed single-qubit measurement basis, globally optimal fixed
local measurements, locally optimal adaptive local measurements, and globally
optimal adaptive local measurements. Here, adaptive measurements are those for
which the measurement basis can depend on prior measurement results. For each
of these measurement schemes we determine the probability of error (for finite
) and scaling of this error in the asymptotic limit. In the asymptotic
limit, adaptive schemes have no advantage over the optimal fixed local scheme,
and except for states with less than 2% mixture, the most naive scheme (locally
optimal fixed local measurements) is as good as any noncollective scheme. For
finite , however, the most sophisticated local scheme (globally optimal
adaptive local measurements) is better than any other noncollective scheme, for
any degree of mixture.Comment: 11 pages, 14 figure
Non-equilibrium dynamics of stochastic point processes with refractoriness
Stochastic point processes with refractoriness appear frequently in the
quantitative analysis of physical and biological systems, such as the
generation of action potentials by nerve cells, the release and reuptake of
vesicles at a synapse, and the counting of particles by detector devices. Here
we present an extension of renewal theory to describe ensembles of point
processes with time varying input. This is made possible by a representation in
terms of occupation numbers of two states: Active and refractory. The dynamics
of these occupation numbers follows a distributed delay differential equation.
In particular, our theory enables us to uncover the effect of refractoriness on
the time-dependent rate of an ensemble of encoding point processes in response
to modulation of the input. We present exact solutions that demonstrate generic
features, such as stochastic transients and oscillations in the step response
as well as resonances, phase jumps and frequency doubling in the transfer of
periodic signals. We show that a large class of renewal processes can indeed be
regarded as special cases of the model we analyze. Hence our approach
represents a widely applicable framework to define and analyze non-stationary
renewal processes.Comment: 8 pages, 4 figure
Jacobi Theta-functions and Discrete Fourier Transforms
Properties of the Jacobi Theta3-function and its derivatives under discrete
Fourier transforms are investigated, and several interesting results are
obtained. The role of modulo N equivalence classes in the theory of
Theta-functions is stressed. An important conjecture is studied.Comment: 10 pages, to appear on the Journal of Mathematical Physics. V3 with
additional reference
Teachers developing assessment for learning: impact on student achievement
While it is generally acknowledged that increased use of formative assessment (or assessment for learning) leads to higher quality learning, it is often claimed that the pressure in schools to improve the results achieved by students in externally-set tests and examinations precludes its use. This paper reports on the achievement of secondary school students who worked in classrooms where teachers made time to develop formative assessment strategies. A total of 24 teachers (2 science and 2 mathematics teachers, in each of six schools in two LEAs) were supported over a six-month period in exploring and planning their approach to formative assessment, and then, beginning in September 1999, the teachers put these plans into action with selected classes. In order to compute effect sizes, a measure of prior attainment and at least one comparison group was established for each class (typically either an equivalent class taught in the previous year by the same teacher, or a parallel class taught by another teacher). The mean effect size was 0.32
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