This paper seeks conditions that ensure that the attractor of a graph directed iterated function system (GD-IFS) cannot be realised as the attractor of a standard iterated function system (IFS). For a strongly connected directed graph, it is known that, if all directed circuits go through a vertex, then for any GD-IFS of similarities on $\mathbb{R}$ based on the graph and satisfying the convex open set condition (COSC), its attractor associated with this vertex is also the attractor of a (COSC) standard IFS. In this paper we show the following complementary result. If a directed circuit does not go through a vertex, then there exists a GD-IFS based on the graph such that the attractor associated with this vertex is not the attractor of any standard IFS of similarities. Indeed, we give algebraic conditions for such GD-IFS attractors not to be attractors of standard IFSs, and thus show that `almost-all' COSC GD-IFSs based on the graph have attractors associated with this vertex that are not the attractors of any COSC standard IFS.

We give estimates on the Kodaira dimension for fibrations over abelian varieties, and give some applications. One of the results strengthens the subadditivity of Kodaira dimension of fibrations over abelian varieties.

Extensive empirical studies show that the long distribution tail of travel time and the corresponding unexpected delay can have much more serious consequences than expected or moderate delay. However, the unexpected delay due to the distribution tail of travel time has received limited attention in recent studies of the valuation of travel time variability. As a complement to current valuation research, this paper proposes the concept of the value of travel time distribution tail, which quantifies the value that travelers place on reducing the unexpected delay for hedging against travel time variability. Methodologically, we define the summation of all unexpected delays as the unreliability area to quantify travel time distribution tail and show that it is a key element of two well-defined measures accounting for unreliable aspects of travel time. We then formally derive the value of distribution tail, show that it is distinct from the more established value of reliability (VOR), and combine it and the VOR in an overall value of travel time variability (VOV). We prove theoretically that the VOV exhibits diminishing marginal benefit in terms of the traveler's punctuality requirements under a validity condition. This implies that it may be economically inefficient for travelers to blindly pursue a higher probability of not being late. We then proceed to develop the concept of the travel time variability ratio, which gives the implicit cost of the punctuality requirement imposed on any given trip. Numerical examples reveal that the cost of travel time distribution tail can account for more than 10% of the trip cost, such that its omission could introduce non-trivial bias into route choice models and transportation appraisal more generally.

We provide an analytical characterization of the model flexibility of the synthetic control method (SCM) in the familiar form of degrees of freedom. We obtain estimable information criteria, which may be used to circumvent cross-validation when selecting either the tuning parameter in penalized variants of SCM or the weighting matrix in the SCM with covariates. We assess the impact of car license rationing in Tianjin; while a natural match is available, both it and other donors are noisy, inviting the use of SCM to average over approximately matching donors. The very large number of candidate donors calls for penalized variants of SCM and we observe that model selection using information criteria outperforms that based on cross-validation.

We show that the quotient of any bounded homogeneous domain by a unipotent discrete group of automorphisms is holomorphically separable. Then we give a necessary condition for the quotient to be Stein and prove that in some cases this condition is also sufficient.

We study the restriction of depth index statistic on the set of perfect matchings. In particular, we provide additional combinatorial description of the statistic for perfect matchings and calculate the generating polynomial. The main result of the present short paper is that the depth index on perfect matchings is equidistributed with the rank function of the Bruhat order.

In conventional particle beam microscopy, knowledge of the beam current is essential for accurate micrograph formation and sample milling. This generally necessitates offline calibration of the instrument. In this work, we establish that beam current can be estimated online, from the same secondary electron count data that is used to form micrographs. Our methods depend on the recently introduced time-resolved measurement concept, which combines multiple short measurements at a single pixel and has previously been shown to partially mitigate the effect of beam current variation on micrograph accuracy. We analyze the problem of jointly estimating beam current and secondary electron yield using the Cramer-Rao bound. Joint estimators operating at a single pixel and estimators that exploit models for inter-pixel correlation and Markov beam current variation are proposed and tested on synthetic microscopy data. Our estimates of secondary electron yield that incorporate explicit beam current estimation beat state-of-the-art methods, resulting in micrograph accuracy nearly indistinguishable from what is obtained with perfect beam current knowledge. Our novel beam current estimation could help improve milling outcomes, prevent sample damage, and enable online instrument diagnostics.

Let $Π$ be a regular algebraic cuspidal automorphic representation (RACAR) of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$. When $Π$ is $p$-nearly-ordinary for the maximal standard parabolic with Levi $\mathrm{GL}_1 \times \mathrm{GL}_2$, we construct a $p$-adic $L$-function for $Π$. More precisely, we construct a (single) bounded measure $L_p(Π)$ on $\mathbb{Z}_p^\times$ attached to $Π$, and show it interpolates all the critical values $L(Π\timesη,-j)$ at $p$ in the left-half of the critical strip for $Π$ (for varying $η$ and $j$). This proves conjectures of Coates-Perrin-Riou and Panchishkin in this case. We also prove a corresponding result in the right half of the critical strip, assuming near-ordinarity for the other maximal standard parabolic. Our construction uses the theory of spherical varieties to build a "Betti Euler system", a norm-compatible system of classes in the Betti cohomology of a locally symmetric space for $\mathrm{GL}_3$. We work in arbitrary cohomological weight, allow arbitrary ramification at $p$ along the Levi factor of the standard parabolic, and make no self-duality assumption. We thus give the first constructions of $p$-adic $L$-functions for RACARs of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ of 'general type' (i.e., those that do not arise as functorial lifts) for any $n > 2$.

We give a proof of Lusztig's conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type quantized enveloping algebra at the $\ell$-th root of unity, where $\ell$ is an odd prime power satisfying certain reasonable conditions.

In the context of the infrared triangle there have been recent discussions on the existence and the role of dual charges. We present a new viewpoint on dual magnetic charges in $p$-form theories, and argue that they can be inherited from the charges of a first order formulation as a topological BF theory with potential. This happens because, depending on the spacetime dimension and on the form degree, the so-called translational gauge symmetries of BF theory become reducible and therefore admit zero-modes. Although such zero-modes lead to trivial symmetries of the $p$-form theory, they are associated with non-trivial charges. These turn out to be precisely the dual magnetic charges. The centrally-extended current algebra of electric and magnetic charges in the $p$-form theory then descends naturally from that of BF theory. This is an effort towards finding an existence criterion for dual charges.

Structured light harnessing multiple degrees of freedom has become a powerful approach to use complex states of light in fundamental studies and applications. Here, we investigate the light field of an ultrafast laser beam with a wavelength-depended polarization state, a beam we term spectral vector beam. We demonstrate a simple technique to generate and tune such structured beams and demonstrate their spectroscopic capabilities. By only measuring the polarization state using fast photodetectors, it is possible to track pulse-to-pulse changes in the frequency spectrum caused by, e.g. narrowband transmission or absorption. In our experiments, we reach read-out rates of around 6 MHz, which is limited by our technical ability to modulate the spectrum and can in principle reach GHz read-out rates. In simulations we extend the spectral range to more than 1000 nm by using a supercontinuum light source, thereby paving the way to various applications requiring high-speed spectroscopic measurements.

This paper examines how shocks to currency volatilities predict exchange rates. Using option-implied volatilities, we construct a dynamic, directed network of volatility connections. Currencies that transmit more volatility shocks, which control for common correlation, earn lower excess returns. Buying the weakest and selling the strongest transmitters delivers high risk-adjusted performance, driven by spot exchange rate movements and not explained by standard factors. A general equilibrium model shows that volatility transmission related to idiosyncratic shocks proxies for priced country-specific risk. Assuming a monotonic amplification of domestic idiosyncratic risk, volatility transmission forecasts negatively future excess returns, consistent with the empirical evidence.

In this paper, we introduce and develop the notion of simple closed-curve magnetization. We provide an application to the Bellman lost in the forest problem by assuming special geometric conditions between the hiker and the boundary of the forest.

Operadic tangent cohomology generalizes the existing cohomology theories of Chevalley--Eilenberg, Hochschild, and Harrison to address the deformation theory of general types of algebras through gadgets known as deformation complexes. The cohomology of these is in general very non-trivial to compute, and in this paper we complement the existing computational techniques by producing a spectral sequence that converges to the operadic cohomology of a fixed algebra. Our main technical tool is that of filtrations arising from towers of cofibrations of algebras, which play the same role cell attaching maps and skeletal filtrations do for topological spaces. As an application, we consider the rational Adams--Hilton construction on topological spaces, where our spectral sequence gives rise to a seemingly new and completely algebraic description of the Serre spectral sequence, which we also show is multiplicative and converges to the Chas--Sullivan loop product. We also consider relative Sullivan--de Rham models of a fibration $p$, where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber-homotopy equivalences of $p$.

Measurement of the optical transmission matrix (TM) of an opaque material is an advanced form of space-variant aberration correction. Beyond imaging, TM-based methods are emerging in a range of fields including optical communications, optical micro-manipulation, and optical computing. In many cases the TM is very sensitive to perturbations in the configuration of the scattering medium it represents. Therefore applications often require an up-to-the-minute characterisation of the fragile TM, typically entailing hundreds to thousands of probe measurements. In this work we explore how these measurement requirements can be relaxed using the framework of compressive sensing: incorporation of prior information enables accurate estimation from fewer measurements than the dimensionality of the TM we aim to reconstruct. Examples of such priors include knowledge of a memory effect linking input and output fields, an approximate model of the optical system, or a recent but degraded TM measurement. We demonstrate this concept by reconstructing a full-size TM of a multimode fibre supporting 754 modes at compression ratios down to ~5% with good fidelity. The level of compression achievable is dependent upon the strength of our priors. We show in this case that imaging is still possible using TMs reconstructed at compression ratios down to ~1% (8 probe measurements). This compressive TM sampling strategy is quite general and may be applied to any form of scattering system about which we have some prior knowledge, including diffusers, thin layers of tissue, fibre optics of any known refractive profile, and reflections from opaque walls. These approaches offer a route to measurement of high-dimensional TMs quickly or with access to limited numbers of measurements.

In this paper, we introduce and develop the notion of spanning of integers along functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to a class of problems that requires to determine if the equations of the form $tf(n)=n-k$ has a solution $n\in \mathbb{N}$ for a fixed $k\in \mathbb{N}$ and some $t\in \mathbb{N}$. In particular, we show that \begin{align} \# \{n\leq s~|~tφ(n)+1=n,~\mathbf{for~some}~t\in \mathbb{N}\}\geq \frac{s}{2\log s}\prod \limits_{p | \lfloor s\rfloor }(1-\frac{1}{p})^{-1}-\frac{3}{2}e^γ\nonumber \end{align} as $s\longrightarrow \infty$, where $φ$ is the Euler totient function and $γ=0.5772\cdots$ is the Euler-Macheroni constant.

A combinatorial neural code $\mathscr C\subseteq 2^{[n]}$ is convex if it arises as the intersection pattern of convex open subsets of $\mathbb R^d$. We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to geometry and computational complexity. On the categorical side, we show that the map taking an acyclic oriented matroid to the code of positive parts of its topes is a faithful functor. We adapt the oriented matroid ideal introduced by Novik, Postnikov, and Sturmfels into a functor from the category of oriented matroids to the category of rings; then, we show that the resulting ring maps naturally to the neural ring of the matroid's neural code. For geometry and computational complexity, we show that a code has a realization with convex polytopes if and only if it lies below the code of a representable oriented matroid in the partial order of codes introduced by Jeffs. We show that previously published examples of non-convex codes do not lie below any oriented matroids, and we construct examples of non-convex codes lying below non-representable oriented matroids. By way of this construction, we can apply Mnëv-Sturmfels universality to show that deciding whether a combinatorial code is convex is NP-hard.

We study a gas of point particles with hard-core repulsion in one dimension where the particles move freely in-between elastic collisions. We prepare the system with a uniform density on the infinite line. The velocities $\{v_i; i \in \mathbb{Z} \}$ of the particles are chosen independently from a thermal distribution. Using a mapping to the non-interacting gas, we analytically compute the equilibrium spatio-temporal correlations $\langle v_i^m(t) v_j^n(0)\rangle$ for arbitrary integers $m,n$. The analytical results are verified with microscopic simulations of the Hamiltonian dynamics. The correlation functions have ballistic scaling, as expected in an integrable model.

In this paper, we study the distribution of the boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition that at some time $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $i=1,2\ldots,k-2$, then at that time we have $$ ||\mathcal{S}_{i+1}-\mathcal{S}_i||>\frac{\mathcal{D}(n)π}{k-1} $$ for all $i=1,\ldots,k-1$ and where $1>\mathcal{D}(n)>0$ is a constant depending on the degree of a certain polynomial of degree $n$. In particular, we show that given at most eight $\mathcal{S}_i$~($i=1,2,\ldots, 8$) runners running around a unit circular track with distinct constant speed and the additional condition $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $1\leq i\leq 6$ at some time $s>1$, then at that time their mutual distance must satisfy the lower bound $$ ||\mathcal{S}_{i}-\mathcal{S}_{i+1}||>\frac{Cπ}{7} $$ for some constant $1>C>0$ for all $1\leq i\leq 7$.

A particle entering a scattering and absorbing medium executes a random walk through a sequence of scattering events. The particle ultimately achieves first-passage, leaving the medium or it is absorbed. The Kubelka-Munk model describes a flux of particles moving perpendicular to the surface of a plane-parallel medium. The particle path alternates between the positive direction into the medium and the negative direction back towards the surface. Backscattering events from the positive to the negative direction occur at local maxima or peaks, while backscattering from the negative to the positive direction occur at local minima or valleys. The probability of a particle avoiding absorption as it follows its path decreases exponentially with the path-length \(λ\). The reflectance of a semi-infinite slab is therefore the Laplace transform of the distribution of path-length that ends with a first-passage out of the medium. In the case of a constant scattering rate the random walk is a Poisson process. We verify our results with two iterative calculations, one using the properties of iterated convolution with a symmetric kernel and the other via direct calculation with an exponential step-length distribution. We present a novel demonstration, based on fluctuation theory of sums of random variables, that the first-passage probability as a function of the number of peaks in the alternating path is a step-length distribution-free combinatoric expression. Counting paths with backscattering on the real half-line results in the same Catalan number coefficients as Dyck paths on the whole numbers. Including a separate forward-scattering Poisson process results in an expression related to counting Motzkin paths. We therefore connect walks on the real line to discrete path combinatorics.

Focused ion beam (FIB) microscopy suffers from source shot noise - random variation in the number of incident ions in any fixed dwell time - along with random variation in the number of detected secondary electrons per incident ion. This multiplicity of sources of randomness increases the variance of the measurements and thus worsens the trade-off between incident ion dose and image accuracy. Time-resolved sensing combined with maximum likelihood estimation from the resulting sets of measurements greatly reduces the effect of source shot noise. Through Fisher information analysis and Monte Carlo simulations, the reduction in mean-squared error or reduction in required dose is shown to be by a factor approximately equal to the secondary electron yield. Experiments with a helium ion microscope (HIM) are consistent with the analyses and suggest accuracy improvement for a fixed source dose, or reduced source dose for a desired imaging accuracy, by a factor of about 3.

In this paper, we introduce a particular class of matrices. We study the concept of a matrix to be \emph{balanced}. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix statistics in this setting. The crux will be to understand the determinants and the eigenvalues of balanced matrices. It turns out that there exist a direct communication among the leading entry, the trace, determinants and, hence, the eigenvalues of these matrices of order $2\times 2$. These matrices have an interesting property that allows us to predict their quadratic forms using their spectrum, without an information about their entries.

The living organism is considered as an open system, whereas Prigogine's approach to the thermodynamics of such systems is used. The approach allows one to formulate the law of individual growth and development (ontogenesis) of the living organism, whereas it has taken into account that the development and functioning of the system are occurring under the special internal program. The thermodynamic equation of growth is followed a method of estimation of the specific entropy of organism. The theory is compared with experimental data, whereas estimates of the specific entropy of some species were done; it shows a reduction of specific entropy in the evolutionary row: yeast - insects - reptiles - birds.

In many studies of environmental change of the past few centuries, 210Pb dating is used to obtain chronologies for sedimentary sequences. One of the most commonly used approaches to estimate the ages of depths in a sequence is to assume a constant rate of supply (CRS) or influx of `unsupported' 210Pb from the atmosphere, together with a constant or varying amount of `supported' 210Pb. Current 210Pb dating models do not use a proper statistical framework and thus provide poor estimates of errors. Here we develop a new model for 210Pb dating, where both ages and values of supported and unsupported 210Pb form part of the parameters. We apply our model to a case study from Canada as well as to some simulated examples. Our model can extend beyond the current CRS approach, deal with asymmetric errors and mix 210Pb with other types of dating, thus obtaining more robust, realistic and statistically better defined estimates.

The systematic study of fission fragment yields under different initial conditions provides a valuable experimental benchmark for fission models that aim to understand this complex decay channel and to predict reaction product yields. Inverse kinematics coupled to the use of a high-resolution spectrometer is shown to be a powerful tool to identify and measure the inclusive isotopic yields of fission fragments. In-flight fusion fission was used to produce secondary beams of neutron-rich isotopes in the collision of a 238U beam at 24 MeV/u with 9Be and 12C targets at GANIL using the LISE3 fragment-separator. Unique A,Z,q identification of fission products was attained with the dE-TKE-Brho-ToF measurement technique. Mass, and atomic number distributions are reported for the two reactions that show the importance of different reaction mechanisms for these two targets.

In genome-wide association studies (GWASs) of common diseases/traits, we often analyze multiple GWASs with the same phenotype together to discover associated genetic variants with higher power. Since it is difficult to access data with detailed individual measurements, summary-statistics-based meta-analysis methods have become popular to jointly analyze data sets from multiple GWASs. In this paper, we propose a novel summary-statistics-based joint analysis method based on controlling the joint local false discovery rate (Jlfdr). We prove that our method is the most powerful summary-statistics-based joint analysis method when controlling the false discovery rate at a certain level. In particular, the Jlfdr-based method achieves higher power than commonly used meta-analysis methods when analyzing heterogeneous data sets from multiple GWASs. Simulation experiments demonstrate the superior power of our method over meta-analysis methods. Also, our method discovers more associations than meta-analysis methods from empirical data sets of four phenotypes. The R-package is available at: http://bioinformatics.ust.hk/Jlfdr.html.

Assuming that Bogoliubov's theory of weakly interacting dilute Bose gas defines a self-consistent model Hamiltonian, we investigate its thermodynamic limit as we take the volume to infinity. The infinite volume is taken via a sequence of scaled convex regions with piecewise smooth boundary and the volumes staying proportional to the cube of the diameter of the region. To get a strict bound on the behavior of the thermodynamic limit, we use the recent formulation of Bogoliubov's theory of condensation in terms of heat kernels for a given domain as well as an estimate of the difference of traces between the heat kernel with Neumann boundary conditions on this domain and the infinite space result. We cannot control the limiting process by the area term; however, we can come arbitrarily close to it.

Genome-wide association studies (GWAS) are widely used to discover genetic variants associated with diseases. To control false positives, all findings from GWAS need to be verified with additional evidences, even for associations discovered from a high power study. Replication study is a common verification method by using independent samples. An association is regarded as true positive with a high confidence when it can be identified in both primary study and replication study. Currently, there is no systematic study on the behavior of positives in the replication study when the positive results of primary study are considered as the prior information. In this paper, two probabilistic measures named Reproducibility Rate (RR) and False Irreproducibility Rate (FIR) are proposed to quantitatively describe the behavior of primary positive associations (i.e. positive associations identified in the primary study) in the replication study. RR is a conditional probability measuring how likely a primary positive association will also be positive in the replication study. This can be used to guide the design of replication study, and to check the consistency between the results of primary study and those of replication study. FIR, on the contrary, measures how likely a primary positive association may still be a true positive even when it is negative in the replication study. This can be used to generate a list of potentially true associations in the irreproducible findings for further scrutiny. The estimation methods of these two measures are given. Simulation results and real experiments show that our estimation methods have high accuracy and good prediction performance.

Capturing depth and reflectivity images at low light levels from active illumination of a scene has wide-ranging applications. Conventionally, even with single-photon detectors, hundreds of photon detections are needed at each pixel to mitigate Poisson noise. We develop a robust method for estimating depth and reflectivity using on the order of 1 detected photon per pixel averaged over the scene. Our computational imager combines physically accurate single-photon counting statistics with exploitation of the spatial correlations present in real-world reflectivity and 3D structure. Experiments conducted in the presence of strong background light demonstrate that our computational imager is able to accurately recover scene depth and reflectivity, while traditional maximum-likelihood based imaging methods lead to estimates that are highly noisy. Our framework increases photon efficiency 100-fold over traditional processing and also improves, somewhat, upon first-photon imaging under a total acquisition time constraint in raster-scanned operation. Thus our new imager will be useful for rapid, low-power, and noise-tolerant active optical imaging, and its fixed dwell time will facilitate parallelization through use of a detector array.

We show that it can be suboptimal for Bayesian decision-making agents employing social learning to use correct prior probabilities as their initial beliefs. We consider sequential Bayesian binary hypothesis testing where each individual agent makes a binary decision based on an initial belief, a private signal, and the decisions of all earlier-acting agents---with the actions of precedent agents causing updates of the initial belief. Each agent acts to minimize Bayes risk, with all agents sharing the same Bayes costs for Type I (false alarm) and Type II (missed detection) errors. The effect of the set of initial beliefs on the decision-making performance of the last agent is studied. The last agent makes the best decision when the initial beliefs are inaccurate. When the private signals are described by Gaussian likelihoods, the optimal initial beliefs are not haphazard but rather follow a systematic pattern: the earlier-acting agents should act as if the prior probability is larger than it is in reality when the true prior probability is small, and vice versa. We interpret this as being open minded toward the unlikely hypothesis. The early-acting agents face a trade-off between making a correct decision and being maximally informative to the later-acting agents.