Abstract: We measure the clustering of dark matter halos in a large set of collisionless cosmological simulations of the flat LCDM cosmology. Halos are identified using the spherical overdensity algorithm, which finds the mass around isolated peaks in the density field such that the mean density is Delta times the background. We calibrate fitting functions for the large scale bias that are adaptable to any value of Delta we examine. We find a ~6% scatter about our best fit bias relation. Our fitting functions couple to the halo mass functions of Tinker et. al. (2008) such that bias of all dark matter is normalized to unity. We demonstrate that the bias of massive, rare halos is higher than that predicted in the modified ellipsoidal collapse model of Sheth, Mo, & Tormen (2001), and approaches the predictions of the spherical collapse model for the rarest halos. Halo bias results based on friends-of-friends halos identified with linking length 0.2 are systematically lower than for halos with the canonical Delta=200 overdensity by ~10%. In contrast to our previous results on the mass function, we find that the universal bias function evolves very weakly with redshift, if at all. We use our numerical results, both for the mass function and the bias relation, to test the peak-background split model for halo bias. We find that the peak-background split achieves a reasonable agreement with the numerical results, but ~20% residuals remain, both at high and low masses.

Abstract: Laplacian matrices play an important role in linear-consensus algorithms. This paper studies optimal linear-consensus algorithms for multivehicle systems with single-integrator dynamics in both continuous-time and discrete-time settings. We propose two global cost functions, namely, interaction-free and interaction-related cost functions. With the interaction-free cost function, we derive the optimal (nonsymmetric) Laplacian matrix by using a linear-quadratic-regulator-based method in both continuous-time and discrete-time settings. It is shown that the optimal (nonsymmetric) Laplacian matrix corresponds to a complete directed graph. In addition, we show that any symmetric Laplacian matrix is inverse optimal with respect to a properly chosen cost function. With the interaction-related cost function, we derive the optimal scaling factor for a prespecified symmetric Laplacian matrix associated with the interaction graph in both continuous-time and discrete-time settings. Illustrative examples are given as a proof of concept.

Abstract: By using the new fractional Taylor’s series of fractional order f ( x + h ) = E α ( h α D x α ) f ( x ) where E α ( . ) denotes the Mittag–Leffler function, and D x α is the so-called modified Riemann–Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration, one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a (usual) Gaussian white noise. One can then derive two new families of fractional Black–Scholes equations, and one shows how one can obtain their solutions. Merton’s optimal portfolio is once more considered and some new results are contributed, with respect to the modeling on one hand, and to the solution on the other hand. Finally, one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics.

Abstract: In many scientific disciplines complex computer models are used to understand the behaviour of large scale physical systems. An uncertainty anal- ysis of such a computer model known as Galform is presented. Galform models the creation and evolution of approximately one million galaxies from the begin- ning of the Universe until the current day, and is regarded as a state-of-the-art model within the cosmology community. It requires the specification of many in- put parameters in order to run the simulation, takes significant time to run, and provides various outputs that can be compared with real world data. A Bayes Linear approach is presented in order to identify the subset of the input space that could give rise to acceptable matches between model output and measured data. This approach takes account of the major sources of uncertainty in a consistent and unified manner, including input parameter uncertainty, function uncertainty, observational error, forcing function uncertainty and structural uncertainty. The approach is known as History Matching, and involves the use of an iterative suc- cession of emulators (stochastic belief specifications detailing beliefs about the Galform function), which are used to cut down the input parameter space. The analysis was successful in producing a large collection of model evaluations that exhibit good fits to the observed data.

Abstract: In the planar N=4 supersymmetric Yang-Mills theory, the conformal symmetry constrains multi-loop n-edged Wilson loops to be given in terms of the one-loop n-edged Wilson loop, augmented, for n greater than 6, by a function of conformally invariant cross ratios. That function is termed the remainder function. In a recent paper, we have displayed the first analytic computation of the two-loop six-edged Wilson loop, and thus of the corresponding remainder function. Although the calculation was performed in the quasi-multi-Regge kinematics of a pair along the ladder, the Regge exactness of the six-edged Wilson loop in those kinematics entails that the result is the same as in general kinematics. We show in detail how the most difficult of the integrals is computed, which contribute to the six-edged Wilson loop. Finally, the remainder function is given as a function of uniform transcendental weight four in terms of Goncharov polylogarithms. We consider also some asymptotic values of the remainder function, and the value when all the cross ratios are equal.

Abstract: We consider a generalization of stochastic bandits where the set of arms, $\cX$, is allowed to be a generic measurable space and the mean-payoff function is "locally Lipschitz" with respect to a dissimilarity function that is known to the decision maker. Under this condition we construct an arm selection policy, called HOO (hierarchical optimistic optimization), with improved regret bounds compared to previous results for a large class of problems. In particular, our results imply that if $\cX$ is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally continuous with a known smoothness degree, then the expected regret of HOO is bounded up to a logarithmic factor by $\sqrt{n}$, i.e., the rate of growth of the regret is independent of the dimension of the space. We also prove the minimax optimality of our algorithm when the dissimilarity is a metric. Our basic strategy has quadratic computational complexity as a function of the number of time steps and does not rely on the doubling trick. We also introduce a modified strategy, which relies on the doubling trick but runs in linearithmic time. Both results are improvements with respect to previous approaches.

Abstract: We consider the problem of selecting an optimal set of sensors, as determined, for example, by the predictive accuracy of the resulting sensor network. Given an underlying metric between pairs of set elements, we introduce a natural metric between sets of sensors for this task. Using this metric, we can construct covariance functions over sets, and thereby perform Gaussian process inference over a function whose domain is a power set. If the function has additional inputs, our covariances can be readily extended to incorporate them---allowing us to consider, for example, functions over both sets and time. These functions can then be optimized using Gaussian process global optimization (GPGO). We use the root mean squared error (RMSE) of the predictions made using a set of sensors at a particular time as an example of such a function to be optimized; the optimal point specifies the best choice of sensor locations. We demonstrate the resulting method by dynamically selecting the best subset of a given set of weather sensors for the prediction of the air temperature across the United Kingdom.

Abstract: We measure and study the evolution of the UV galaxy Luminosity Function (LF) at z=3-5 from the largest high-redshift survey to date, the Deep part of the CFHT Legacy Survey. We also give accurate estimates of the SFR density at these redshifts. We consider ~100,000 Lyman-break galaxies at z~3.1, 3.8 & 4.8 selected from very deep ugriz images of this data set and estimate their rest-frame 1600A luminosity function. Due to the large survey volume, cosmic variance plays a negligible role. Furthermore, we measure the bright end of the LF with unprecedented statistical accuracy. Contamination fractions from stars and low-z galaxy interlopers are estimated from simulations. To correct for incompleteness, we study the detection rate of simulated galaxies injected to the images as a function of magnitude and redshift. We estimate the contribution of several systematic effects in the analysis to test the robustness of our results. We find the bright end of the LF of our u-dropout sample to deviate significantly from a Schechter function. If we modify the function by a recently proposed magnification model, the fit improves. For the first time in an LBG sample, we can measure down to the density regime where magnification affects the shape of the observed LF because of the very bright and rare galaxies we are able to probe with this data set. We find an increase in the normalisation, $\phi^{*}$, of the LF by a factor of 2.5 between z~5 and z~3. The faint-end slope of the LF does not evolve significantly between z~5 and z~3. We do not find a significant evolution of the characteristic magnitude in the studied redshift interval. The SFR density is found to increase by a factor of ~2 from z~5 to z~4. The evolution from z~4 to z~3 is less eminent.

Abstract: We consider two variables that are related to each other by an invertible function. While it has previously been shown that the dependence structure of the noise can provide hints to determine which of the two variables is the cause, we presently show that even in the deterministic (noise-free) case, there are asymmetries that can be exploited for causal inference. Our method is based on the idea that if the function and the probability density of the cause are chosen independently, then the distribution of the effect will, in a certain sense, depend on the function. We provide a theoretical analysis of this method, showing that it also works in the low noise regime, and link it to information geometry. We report strong empirical results on various real-world data sets from different domains.

Abstract: Truncated multipliers compute the n most-significant bits of the n × n bits product. This paper focuses on variable-correction truncated multipliers, where some partial-products are discarded, to reduce complexity, and a suitable compensation function is added to partly compensate the introduced error. The optimal compensation function, that minimizes the mean square error, is obtained in this paper in closed-form for the first time. A sub optimal compensation function, best suited for hardware implementation, is introduced. Efficient multipliers implementation based on sub-optimal function is discussed. Proposed truncated multipliers are extensively compared with previously proposed circuits. Experimental results, for a 0.18 μm technology, are also presented.

Abstract: .We present new advances in the spectral extraction of pointlike sources adapted to the Infrared Spectrograph (IRS) on board the Spitzer Space Telescope. For the first time, we created a supersampled point-spread function of the low-resolution modules. We describe how to use the point-spread function to perform optimal extraction of a single source and of multiple sources within the slit. We also examine the case of the optimal extraction of one or several sources with a complex background. The new algorithms are gathered in a plug-in called AdOpt which is part of the SMART data analysis software.

Abstract: A basic question in the theory of functional equations is as follows: When is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation? If the problem accepts a unique solution, we say the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam in 1940 and affirmatively solved by Hyers. In 1978 Th.M. Rassias generalized the Hyers result to approximately linear mappings. In this book, we present the general solutions of several functional equations, and we investigate the stability of these functional equations.

Abstract: Multiple-test procedures are increasingly important as technology in- creases scientists' ability to make large numbers of multiple measurements, as they do in genome scans. Multiple-test procedures were originally defined to input a vector of input p-values and an uncorrected critical p-value, interpreted as a fami- lywise error rate or a false discovery rate, and to output a corrected critical p-value and a discovery set, defined as the subset of input p-values that are at or below the corrected critical p-value. A range of multiple-test procedures is implemented us- ing the smileplot package in Stata (Newson and the ALSPAC Study Team 2003, Stata Journal 3: 109-132; 2010, Stata Journal 10: 691-692). The qqvalue com- mand uses an alternative formulation of multiple-test procedures, which is also used by the R function p.adjust. qqvalue inputs a variable of p-values and out- puts a variable of q-values that are equal in each observation to the minimum familywise error rate or false discovery rate that would result in the inclusion of the corresponding p-value in the discovery set if the specified multiple-test pro- cedure was applied to the full set of input p-values. Formulas and examples are presented.

Abstract: In this article we prove a maximal $L^p$-regularity result for stochastic convolutions, which extends Krylov's basic mixed $L^p(L^q)$-inequality for the Laplace operator on ${\mathbb{R}}^d$ to large classes of elliptic operators, both on ${\mathbb{R}}^d$ and on bounded domains in ${\mathbb{R}}^d$ with various boundary conditions. Our method of proof is based on McIntosh's $H^{\infty}$-functional calculus, $R$-boundedness techniques and sharp $L^p(L^q)$-square function estimates for stochastic integrals in $L^q$-spaces. Under an additional invertibility assumption on $A$, a maximal space--time $L^p$-regularity result is obtained as well.

Abstract: In excursion set theory, the computation of the halo mass function is mapped into a first-passage time process in the presence of a barrier, which in the spherical collapse model is a constant and in the ellipsoidal collapse model is a fixed function of the variance of the smoothed density field. However, N-body simulations show that dark matter halos grow through a mixture of smooth accretion, violent encounters, and fragmentations, and modeling halo collapse as spherical, or even as ellipsoidal, is a significant oversimplification. In addition, the very definition of what is a dark matter halo, both in N-body simulations and observationally, is a difficult problem. We propose that some of the physical complications inherent to a realistic description of halo formation can be included in the excursion set theory framework, at least at an effective level, by taking into account that the critical value for collapse is not a fixed constant δ c , as in the spherical collapse model, nor a fixed function of the variance σ of the smoothed density field, as in the ellipsoidal collapse model, but rather is itself a stochastic variable, whose scatter reflects a number of complicated aspects of the underlying dynamics. Solving the first-passage time problem in the presence of a diffusing barrier we find that the exponential factor in the Press-Schechter mass function changes from exp{–δ2 c /2σ2} to exp{–aδ2 c /2σ2}, where a = 1/(1 + DB ) and DB is the diffusion coefficient of the barrier. The numerical value of DB , and therefore the corresponding value of a, depends among other things on the algorithm used for identifying halos. We discuss the physical origin of the stochasticity of the barrier and, from recent N-body simulations that studied the properties of the collapse barrier, we deduce a value DB 0.25. Our model then predicts a 0.80, in excellent agreement with the exponential fall off of the mass function found in N-body simulations, for the same halo definition. Combining this result with the non-Markovian corrections computed in Paper I of this series, we derive an analytic expression for the halo mass function for Gaussian fluctuations and we compare it with N-body simulations.

Abstract: In this paper, the function projective synchronization is investigated in coupled partially linear chaotic systems. By Lyapunov stability theory, a control law is derived to make the state vectors asymptotically synchronized up to a desired scaling function. Furthermore, based on function projective synchronization, a scheme for secure communication is presented in theory. The corresponding numerical simulations are performed to verify and illustrate the analytical results.

Abstract: Hardness amplification is the fundamental task of converting a $\delta$-hard function $f:\{0,1\}^n\to\{0,1\}$ into a $(1/2-\epsilon)$-hard function $\mathit{Amp}(f)$, where $f$ is $\gamma$-hard if small circuits fail to compute $f$ on at least a $\gamma$ fraction of the inputs. In this paper we study the complexity of black-box proofs of hardness amplification. A class of circuits $\mathcal{D}$ proves a hardness amplification result if for any function $h$ that agrees with $\mathit{Amp}(f)$ on a $1/2+\epsilon$ fraction of the inputs there exists an oracle circuit $D\in\mathcal{D}$ such that $D^h$ agrees with $f$ on a $1-\delta$ fraction of the inputs. We focus on the case where every $D\in\mathcal{D}$ makes nonadaptive queries to $h$. This setting captures most hardness amplification techniques. We prove two main results: (1) The circuits in $\mathcal{D}$ “can be used” to compute the majority function on $1/\epsilon$ bits. In particular, when $\epsilon\leq1/\log^{\omega(1)}n$, $\mathcal{D}$ cannot consist of oracle circuits that have unbounded fan-in, size $\mathrm{poly}(n)$, and depth $O(1)$. (2) The circuits in $\mathcal{D}$ must make $\Omega\left(\log(1/\delta)/\epsilon^2\right)$ oracle queries. Both our bounds on the depth and on the number of queries are tight up to constant factors. Our results explain why hardness amplification techniques have failed to transform known lower bounds against constant-depth circuit classes into strong average-case lower bounds. Our results reveal a contrast between Yao's XOR lemma ($\mathit{Amp}(f):=f(x_1)\oplus\cdots\oplus f(x_t)\in\{0,1\}$) and the direct-product lemma ($\mathit{Amp}(f):=f(x_1)\circ\cdots\circ f(x_t)\in\{0,1\}^t$; here $\mathit{Amp}(f)$ is non-Boolean). Our results (1) and (2) apply to Yao's XOR lemma, whereas known proofs of the direct-product lemma violate both (1) and (2). One of our contributions is a new technique for handling “nonuniform” reductions, i.e., the case when $\mathcal{D}$ contains many circuits.

Abstract: The goal of our work is to develop an algorithm for automatic and robust detection of global intrinsic symmetries in 3D surface meshes. Our approach is based on two core observations. First, symmetry invariant point sets can be detected robustly using critical points of the Average Geodesic Distance (AGD) function. Second, intrinsic symmetries are self-isometries of surfaces and as such are contained in the low dimensional group of Mobius transformations. Based on these observations, we propose an algorithm that: 1) generates a set of symmetric points by detecting critical points of the AGD function, 2) enumerates small subsets of those feature points to generate candidate Mobius transformations,and 3) selects among those candidate Mobius transformations the one(s) that best map the surface onto itself. The main advantages of this algorithm stem from the stability of the AGD in predicting potential symmetric point features and the low dimensionality of the Mobius group for enumerating potential self-mappings. During experiments with a benchmark set of meshes augmented with human-specified symmetric correspondences, we find that the algorithm is able to find intrinsic symmetries for a wide variety of object types with moderate deviations from perfect symmetry.

Abstract: We investigate the possibility of finding satisfying assignments to Boolean formulae and testing validity of quantified Boolean formulae (QBF) asymptotically faster than a brute force search. Our first main result is a simple deterministic algorithm running in time $2^{n - \Omega(n)}$ for satisfiability of formulae of linear size in $n$, where $n$ is the number of variables in the formula. This algorithm extends to exactly counting the number of satisfying assignments, within the same time bound. Our second main result is a deterministic algorithm running in time $2^{n - \Omega(n/\log(n))}$ for solving QBFs in which the number of occurrences of any variable is bounded by a constant. For instances which are ``structured'', in a certain precise sense, the algorithm can be modified to run in time $2^{n - \Omega(n)}$. To the best of our knowledge, no non-trivial algorithms were known for these problems before. As a byproduct of the technique used to establish our first main result, we show that every function computable by linear-size formulae can be represented by decision trees of size $2^{n - \Omega(n)}$. As a consequence, we get strong super linear {\it average-case} formula size lower bounds for the Parity function.

Abstract: We investigate how a niching based evolutionary algorithm fares on the BBOB function test set, knowing that most problems are not very well suited to this algorithm class. However, as the CMA-ES is included as basic local search algorithm, the niching approach still performs fairly well, with some potential to improve. Basin identification is done via the heuristic nearest-best clustering scheme.

Abstract: In this paper, we construct trigonometric functions in the form of a sum T p (h, k) which is referred to as a Dedekind-type DC-(Dahee and Changhee) sum. We establish analytic properties of this sum, find its trigonometric representations, and prove a reciprocity theorem for these sums. Furthermore, we obtain relationships between the Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums, and the sum T p (h, k). We also give some applications related to these sums and functions.