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# Top arXiv papers

• Randomized benchmarking (RB) is an efficient and robust method to characterize gate errors in quantum circuits. Averaging over random sequences of gates leads to estimates of gate errors in terms of the average fidelity that are isolated from the state preparation and measurement errors that plague other methods like channel tomography and direct fidelity estimation. A decisive factor in the feasibility of randomized benchmarking is the number of samples required to obtain rigorous confidence intervals. Previous bounds were either prohibitively loose or required the number of sampled sequences to scale exponentially with the number of qubits in order to obtain a fixed confidence interval at a fixed error rate. Here we show that the number of sampled sequences required for a fixed confidence interval is dramatically smaller than could previously be justified. In particular, we show that the number of sampled sequences required is essentially independent of the number of qubits. We also show that the number of samples required with a single qubit is substantially smaller than previous rigorous results, especially in the limit of large sequence lengths. Our results bring rigorous randomized benchmarking on systems with many qubits into the realm of experimental feasibility.
• How many quantum queries are required to determine the coefficients of a degree-$d$ polynomial in $n$ variables? We present and analyze quantum algorithms for this multivariate polynomial interpolation problem over the fields $\mathbb{F}_q$, $\mathbb{R}$, and $\mathbb{C}$. We show that $k_{\mathbb{C}}$ and $2k_{\mathbb{C}}$ queries suffice to achieve probability $1$ for $\mathbb{C}$ and $\mathbb{R}$, respectively, where $k_{\mathbb{C}}=\smash{\lceil\frac{1}{n+1}{n+d\choose d}\rceil}$ except for $d=2$ and four other special cases. For $\mathbb{F}_q$, we show that $\smash{\lceil\frac{d}{n+d}{n+d\choose d}\rceil}$ queries suffice to achieve probability approaching $1$ for large field order $q$. The classical query complexity of this problem is $\smash{n+d\choose d}$, so our result provides a speedup by a factor of $n+1$, $\frac{n+1}{2}$, and $\frac{n+d}{d}$ for $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{F}_q$, respectively. Thus we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of $2$. For the case of $\mathbb{F}_q$, we conjecture that $2k_{\mathbb{C}}$ queries also suffice to achieve probability approaching $1$ for large field order $q$, although we leave this as an open problem.
• We study the consequences of 'super-quantum non-local correlations' as represented by the PR-box model of Popescu and Rohrlich, and show PR-boxes can enhance the capacity of noisy interference channels between two senders and two receivers. PR-box correlations violate Bell/CHSH inequalities and are thus stronger -- more non-local -- than quantum mechanics; yet weak enough to respect special relativity in prohibiting faster-than-light communication. Understanding their power will yield insight into the non-locality of quantum mechanics. We exhibit two proof-of-concept channels: first, we show a channel between two sender-receiver pairs where the senders are not allowed to communicate, for which a shared super-quantum bit (a PR-box) allows perfect communication. This feat is not achievable with the best classical (senders share no resources) or quantum entanglement-assisted (senders share entanglement) strategies. Second, we demonstrate a class of channels for which a tunable parameter achieves a double separation of capacities; for some range of \epsilon, the super-quantum assisted strategy does better than the entanglement-assisted strategy, which in turn does better than the classical one.
• A recent analysis by Pikovski et al. [Nat. Phys. 11, 668 (2015)] has triggered interest in the question of how to include relativistic corrections in the quantum dynamics governing many-particle systems in a gravitational field. Here we show how the center-of-mass motion of a quantum system subject to gravity can be derived more rigorously, addressing the ambiguous definition of relativistic center-of-mass coordinates. We further demonstrate that, contrary to the prediction by Pikovski et al., external forces play a crucial role in the relativistic coupling of internal and external degrees of freedom, resulting in a complete cancellation of the alleged coupling in Earth-bound laboratories for systems supported against gravity by an external force. We conclude that the proposed decoherence effect is an effect of relative acceleration between quantum system and measurement device, rather than a universal effect in gravitational fields.
• In real-world applications, observations are often constrained to a small fraction of a system. Such spatial subsampling can be caused by the inaccessibility or the sheer size of the system, and cannot be overcome by longer sampling. Spatial subsampling can strongly bias inferences about a system's aggregated properties. To overcome the bias, we derive analytically a subsampling scaling framework that is applicable to different observables, including distributions of neuronal avalanches, of number of people infected during an epidemic outbreak, and of node degrees. We demonstrate how to infer the correct distributions of the underlying full system, how to apply it to distinguish critical from subcritical systems, and how to disentangle subsampling and finite size effects. Lastly, we apply subsampling scaling to neuronal avalanche models and to recordings from developing neural networks. We show that only mature, but not young networks follow power-law scaling, indicating self-organization to criticality during development.
• Big Bang nucleosynthesis (BBN) theory predicts the abundances of the light elements D, $^3$He, $^4$He and $^7$Li produced in the early universe. The primordial abundances of D and $^4$He inferred from observational data are in good agreement with predictions, however, the BBN theory overestimates the primordial $^7$Li abundance by about a factor of three. This is the so-called "cosmological lithium problem". Solutions to this problem using conventional astrophysics and nuclear physics have not been successful over the past few decades, probably indicating the presence of new physics during the era of BBN. We have investigated the impact on BBN predictions of adopting a generalized distribution to describe the velocities of nucleons in the framework of Tsallis non-extensive statistics. This generalized velocity distribution is characterized by a parameter $q$, and reduces to the usually assumed Maxwell-Boltzmann distribution for $q$ = 1. We find excellent agreement between predicted and observed primordial abundances of D, $^4$He and $^7$Li for $1.069\leq q \leq 1.082$, suggesting a possible new solution to the cosmological lithium problem.
• Jan 17 2017 quant-ph arXiv:1701.04026v1
The Euclidean plane is certainly the simplest example of real Hilbert space. Viewed as a space of quantum states, it can be used as a nice introductive example in teaching quantum formalism. The pure states form the unit circle (actually one half of it), the mixed states form the unit disk (actually one half of it), and rotations in the plane rule time evolution through a Majorana-like equation involving only real quantities. The set of pure states or the set of mixed states at fixed "temperature" solve the identity and they are used for the integral quantization of functions on the unit circle and to give a semi-classical portrait of quantum observables. Interesting probabilistic aspects are developed. Since the tensor product of two planes, their direct sum, their cartesian product, are isomorphic (2 is the unique solution to x^x= x X x = x+x), and they are also isomorphic to C^2, and to the quaternion field H (as a vector space), we describe an interesting relation between entanglement of real states, one-half spin cat states, and unit-norm quaternions which form the group SU(2). Finally, we explain the most general form of the Hamiltonian in the real plane by considering the integral quantization of a magnetic-like interaction potential viewed as a classical observable on the unit 2-sphere.
• We use quantum energy teleportation in the light-matter interaction as an operational means to create quantum field states that violate energy conditions and have negative local stress-energy densities. We show that the protocol is optimal in the sense that it scales in a way that saturates the quantum interest conjecture.
• To investigate the performance of quantum information tasks on networks whose topology changes in time, we study the spatial search algorithm by continuous time quantum walk to find a marked node on a random temporal network. We consider a network of $n$ nodes constituted by a time-ordered sequence of Erdös-Rényi random graphs $G(n,p)$, where $p$ is the probability that any two given nodes are connected: after every time interval $\tau$, a new graph $G(n,p)$ replaces the previous one. We prove analytically that for any given $p$, there is always a range of values of $\tau$ for which the running time of the algorithm is optimal, i.e.\ $\mathcal{O}(\sqrt{n})$, even when search on the individual static graphs constituting the temporal network is sub-optimal. On the other hand, when the energy $1/\tau$ is close to the site energy of the marked node, we need sufficiently high connectivity to ensure optimality, namely the average degree of each node should be at least $\mathcal{O}(\sqrt{n})$. From this first study of quantum spatial search on a time-dependent network, it emerges that the interplay between temporality and connectivity is key to the algorithmic performance. Moreover, our study can be extended to establish high fidelity qubit transfer between any two nodes of the network. Overall, our findings show one can exploit temporality to achieve efficient quantum information tasks on dynamical random networks.
• Jan 17 2017 astro-ph.CO gr-qc hep-th arXiv:1701.04382v1
We argue that $isotropic$ scalar fluctuations in solid inflation are adiabatic in the super-horizon limit. During the solid phase this adiabatic mode has peculiar features: constant energy-density slices and comoving slices do not coincide, and their curvatures, parameterized respectively by $\zeta$ and $\mathcal R$, both evolve in time. The existence of this adiabatic mode implies that Maldacena's squeezed limit consistency relation holds after angular average over the long mode. The correlation functions of a long-wavelength spherical scalar mode with several short scalar or tensor modes is fixed by the scaling behavior of the correlators of short modes, independently of the solid inflation action or dynamics of reheating.
• Based on the convex least-squares estimator, we propose two different procedures for testing convexity of a probability mass function supported on N with an unknown finite support. The procedures are shown to be asymptotically calibrated.
• A fourth-order theory of gravity is considered which in terms of dynamics has the same degrees of freedom and number of constraints as those of scalar-tensor theories. In addition it admits a canonical point-like Lagrangian description. We study the critical points of the theory and we show that it can describe the matter epoch of the universe and that two accelerated phases can be recovered one of which describes a de Sitter universe. Finally for some models exact solutions are presented.
• Based on the Effective Field Theory (EFT) of cosmological perturbations, we revisit the nonsingular cosmologices, without using the integral inequality. We clarify the pathology in nonsingular cubic Galileon models and show how to cure it in EFT with new insights into this issue. With a new application of $R^{(3)}\delta g^{00}$ operator, we build a model with a Genesis phase followed by slow-roll inflation. The spectrum of primordial perturbation may be simulated numerically, which shows itself a large-scale cutoff, as hinted by the CMB large scale anomalies.
• Briët et al. showed that an efficient communication protocol implies a reliable XOR game protocol. In this work, we improve this relationship, and obtain a nontrivial lower bound $2\log3\approx 3.1699$ of XOR-amortized communication complexity of the equality function. The proof uses an elegant idea of Pawłowski et al. in a paper on information causality. Although the improvement of a lower bound of a communication complexity is at most a factor 2, all arguments and proofs in this work are quite simple and intuitive.
• We consider the dynamics of message passing for spatially coupled codes and, in particular, the set of density evolution equations that tracks the profile of decoding errors along the spatial direction of coupling. It is known that, for suitable boundary conditions and after a transient phase, the error profile exhibits a "solitonic behavior". Namely, a uniquely-shaped wavelike solution develops, that propagates with constant velocity. Under this assumption we derive an analytical formula for the velocity in the framework of a continuum limit of the spatially coupled system. The general formalism is developed for spatially coupled low-density parity-check codes on general binary memoryless symmetric channels which form the main system of interest in this work. We apply the formula for special channels and illustrate that it matches the direct numerical evaluation of the velocity for a wide range of noise values. A possible application of the velocity formula to the evaluation of finite size scaling law parameters is also discussed. We conduct a similar analysis for general scalar systems and illustrate the findings with applications to compressive sensing and generalized low-density parity-check codes on the binary erasure or binary symmetric channels.
• A long-time quantum memory capable of storing and measuring quantum information at the single-qubit level is an essential ingredient for practical quantum computation and com-munication. Recently, there have been remarkable progresses of increasing coherence time for ensemble-based quantum memories of trapped ions, nuclear spins of ionized donors or nuclear spins in a solid. Until now, however, the record of coherence time of a single qubit is on the order of a few tens of seconds demonstrated in trapped ion systems. The qubit coherence time in a trapped ion is mainly limited by the increasing magnetic field fluctuation and the decreasing state-detection efficiency associated with the motional heating of the ion without laser cooling. Here we report the coherence time of a single qubit over $10$ minutes in the hyperfine states of a \Yb ion sympathetically cooled by a \Ba ion in the same Paul trap, which eliminates the heating of the qubit ion even at room temperature. To reach such coherence time, we apply a few thousands of dynamical decoupling pulses to suppress the field fluctuation noise. A long-time quantum memory demonstrated in this experiment makes an important step for construction of the memory zone in scalable quantum computer architectures or for ion-trap-based quantum networks. With further improvement of the coherence time by techniques such as magnetic field shielding and increase of the number of qubits in the quantum memory, our demonstration also makes a basis for other applications including quantum money.
• In this article we present a Bernstein inequality for sums of random variables which are defined on a graphical network whose nodes grow at an exponential rate. The inequality can be used to derive concentration inequalities in highly-connected networks. It can be useful to obtain consistency properties for nonparametric estimators of conditional expectation functions which are derived from such networks.
• We present a practical classification scheme for the four-partite entangled states under stochastic local operations and classical communication (SLOCC). By transforming a four-partite state into a triple-state set composed of two tripartite and one bipartite states, the entanglement classification is reduced to that of only tripartite and bipartite entanglements. This reduction method has the merit of being extendable to the classification of any multipartite entangled state, meanwhile provides an insight to the entanglement character of subsystem.
• Experimental demonstration of entanglement needs to have a precise control of experimentalist over the system on which the measurements are performed as prescribed by an appropriate entanglement witness. To avoid such trust problem, recently device-independent entanglement witnesses (\emphDIEWs) for genuine tripartite entanglement have been proposed where witnesses are capable of testing genuine entanglement without precise description of Hilbert space dimension and measured operators i.e apparatus are treated as black boxes. Here we design a protocol for enhancing the possibility of identifying genuine tripartite entanglement in a device independent manner. We consider three mixed tripartite quantum states none of whose genuine entanglement can be detected by applying standard \emphDIEWs, but their genuine tripartite entanglement can be detected by applying the same when distributed in some suitable entanglement swapping network.
• In a recent publication [J. Chem. Phys. 142, 034115 (2015)] we have derived a hierarchy of coupled differential equations in time domain to calculate the linear optical properties of molecular aggregates. Here we provide details about issues concerning the numerical implementation. In addition we present the corresponding hierarchy in frequency domain.
• We establish the existence of $1$-parameter families of $\epsilon$-dependent solutions to the Einstein-Euler equations with a positive cosmological constant $\Lambda >0$ and a linear equation of state $p=\epsilon^2 K \rho$, $0<K\leq 1/3$, for the parameter values $0<\epsilon < \epsilon_0$. These solutions exist globally to the future, converge as $\epsilon \searrow 0$ to solutions of the cosmological Poison-Euler equations of Newtonian gravity, and are inhomogeneous non-linear perturbations of FLRW fluid solutions.
• The hypothesis is made that, at large scales where General Relativity may be applied, the empty space is scale invariant. This establishes a relation between the cosmological constant and the scale factor of the scale invariant framework. This relation brings major simplifications in the scale invariant equations for cosmology, which now contain a new term, depending on the derivative of the scale factor, that opposes to gravity and produces an accelerated expansion. The displacements due to the acceleration term make a high contribution Omega_l to the energy-density of the Universe, satisfying an equation of the form Omega_m+\Omega_k+Omega_l = 1. The models do not demand the existence of unknown particles. There is a family of flat models with different density parameters Omega_m < 1. Numerical integrations of the cosmological equations for different values of the curvature and density parameter k and Omega_m are performed. The presence of even tiny amounts of matter in the Universe tends to kill scale invariance. The point is that for Omega_m = 0.3 the effect is not yet completely killed. The models with non-zero density start explosively with first a braking phase followed by a continuously accelerating expansion. Several observational properties are examined, in particular the distances, the m--z diagram, the Omega_m vs. lambda plot. Comparisons with observations are also performed for the Hubble constant H_0 vs. Omega_m, for the expansion history in the plot H(z)/(z+1) vs. redshift z and for the transition redshift from braking to acceleration. These first dynamical tests are satisfied by the scale invariant models, which thus deserve further studies.
• Jan 17 2017 quant-ph arXiv:1701.03859v1
In the paper, we give a characterization of arbitrary $n$-mode Gaussian coherence breaking channels (GCBCs) and show the tensor product of a GCBC and arbitrary a Gaussian channel maps all input states into product states. The inclusion relation among GCBCs, Gaussian positive partial transpose channels (GPPTC), entanglement breaking channels (GEBC), Gaussian classical-quantum channels (GCQC) and Gaussian quantum-classical channels (GQCC) is displayed. Finally, we prove that Gaussian classical ($\chi$-) capacity for the tensor products of GCBCs and arbitrary Gaussian channels is additive.
• In time domain astronomy, recurrent transients present a special problem: how to infer total populations from limited observations. Monitoring observations may give a biassed view of the underlying population due to limitations on observing time, visibility and instrumental sensitivity. A similar problem exists in the life sciences, where animal populations (such as migratory birds) or disease prevalence, must be estimated from sparse and incomplete data. The class of methods termed Capture-Recapture is used to reconstruct population estimates from time-series records of encounters with the study population. This paper investigates the performance of Capture-Recapture methods in astronomy via a series of numerical simulations. The Blackbirds code simulates monitoring of populations of transients, in this case accreting binary stars (neutron star or black hole accreting from a stellar companion) under a range of observing strategies. We first generate realistic light-curves for populations of binaries with contrasting orbital period distributions. These models are then randomly sampled at observing cadences typical of existing and planned monitoring surveys. The classical capture-recapture methods, Lincoln-Peterson, Schnabel estimators, related techniques, and newer methods implemented in the Rcapture package are compared. A general exponential model based on the radioactive decay law is introduced, and demonstrated to recover (at 95% confidence) the underlying population abundance and duty cycle, in a fraction of the observing visits (10-50%) required to discover all the sources in the simulation. Capture-Recapture is a promising addition to the toolbox of time domain astronomy, and methods implemented in R by the biostats community can be readily called from within Python.
• Decoherence due to charge noise is one of the central challenges in using spin qubits in semiconductor quantum dots as a platform for quantum information processing. Recently, it has been experimentally demonstrated in both Si and GaAs singlet-triplet qubits that the effects of charge noise can be suppressed if qubit operations are implemented using symmetric barrier control instead of the standard tilt control. Here, we investigate the key issue of whether the benefits of barrier control persist over the entire set of single-qubit gates by performing randomized benchmarking simulations. We find the surprising result that the improvement afforded by barrier control depends sensitively on the amount of spin noise: for the minimal nuclear spin noise levels present in Si, the coherence time improves by more than two orders of magnitude whereas in GaAs, by contrast the coherence time is essentially the same for barrier and tilt control. However, we establish that barrier control becomes beneficial if qubit operations are performed using a new family of composite pulses that reduce gate times by up to 90%. With these optimized pulses, barrier control is the best way to achieve high-fidelity quantum gates in singlet-triplet qubits.
• A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. We compute the Wilson Loop observable using a path integral with an Einstein-Hilbert action. Using axial-gauge fixing, we can write this path integral as the limit of a sequence of Chern-Simons integrals, studied earlier in our previous work on the Chern-Simons path integrals in $\mathbb{R}^3$. We will show that the Wilson Loop observable can be computed from a link diagram of a hyperlink, projected on a plane. Only crossings in the diagram will contribute to the path integral. Furthermore, we will show that it is invariant under an equivalence relation defined on the set of hyperlinks.
• Searches for stellar companions to hot Jupiters (HJs) have revealed that planetary systems hosting a HJ are approximately three times more likely to have a stellar companion with a semimajor axis between 50 and 2000 AU, compared to field stars. This correlation suggests that HJ formation is affected by the stellar binary companion. A potential model is high-eccentricity migration, in which the binary companion induces high-eccentricity Lidov-Kozai (LK) oscillations in the proto-HJ orbit, triggering orbital migration driven by tides. A pitfall of this `binary-LK' model is that the observed stellar binaries hosting HJs are typically too wide to produce HJs in sufficient numbers, because of suppression by short-range forces. We propose a modification to the binary-LK model in which there is a second giant planet orbiting the proto-HJ at a semimajor axis of several tens of AU. Such companions are currently hidden to observations, but their presence could be manifested by a propagation of the perturbation of the stellar binary companion inwards to the proto-HJ, thereby overcoming the barrier imposed by short-range forces. Our model does not require the planetary companion orbit to be eccentric and/or inclined with respect to the proto-HJ, but its semimajor axis should lie in a specific range given the planetary mass and binary semimajor axis, and the inclination with respect to the binary should be near $40^\circ$ or $140^\circ$. Our prediction for planetary companions to HJs in stellar binaries should be testable by future observations.
• A two-phase description of the quark-nuclear matter hybrid equation of state that takes into account the effect of excluded volume in both the hadronic and the quark-matter phases is introduced. The nuclear phase manifests a reduction of the available volume as density increases, leading to a stiffening of the matter. The quark-matter phase displays a reduction of the effective string-tension in the confining density-functional from available volume contributions. The nuclear equation of state is based upon the relativistic density functional model DD2 with excluded volume. The quark-matter is based upon a mean-field modification to the free fermi gas and will be discussed in greater detail. The interactions are decomposed into mean scalar and vector components. The scalar interaction is motivated by a string potential between quarks, whereas the vector interaction potential is motivated by higher-order interactions of quarks leading to an increased stiffening at high densities. As an application, we consider matter under compact star constraints of electric neutrality and $\beta$-equilibrium. We obtain mass-radius relations for hybrid stars that form a third family, disconnected from the purely hadronic star branch, and fulfill the $2M_\odot$ constraint.
• We present a demonstration of delensing the observed cosmic microwave background (CMB) B-mode polarization anisotropy. This process of reducing the gravitational-lensing generated B-mode component will become increasingly important for improving searches for the B modes produced by primordial gravitational waves. In this work, we delens B-mode maps constructed from multi-frequency SPTpol observations of a 90 deg$^2$ patch of sky by subtracting a B-mode template constructed from two inputs: SPTpol E-mode maps and a lensing potential map estimated from the $\textit{Herschel}$ $500\,\mu m$ map of the CIB. We find that our delensing procedure reduces the measured B-mode power spectrum by 28% in the multipole range $300 < \ell < 2300$; this is shown to be consistent with expectations from theory and simulations and to be robust against systematics. The null hypothesis of no delensing is rejected at $6.9 \sigma$. Furthermore, we build and use a suite of realistic simulations to study the general properties of the delensing process and find that the delensing efficiency achieved in this work is limited primarily by the noise in the lensing potential map. We demonstrate the importance of including realistic experimental non-idealities in the delensing forecasts used to inform instrument and survey-strategy planning of upcoming lower-noise experiments, such as CMB-S4.
• With the increased application of model-based whole-body control methods in legged robots, there has been an resurgence of interest in whole-body system identification strategies and adaptive control. An important class of methods relates to the identification of inertial parameters for rigid-body systems. That is, the mass, first mass moment (related to center of mass location), and 3D rotational inertia tensor for each link. It is known that the standard manipulator equations of a rigid-body system are linear in these inertial parameters, enabling many methods for offline model identification and online adaptive control. Recent work has shown that previous methods can lead to parameter estimates that are non-physical (i.e. can not possibly represent any physical rigid-body system), due to the violation of certain triangle inequalities on the principal moments of inertia. The main contribution of this paper is to formulate these added constraints as Linear Matrix Inequalities (LMIs). Matrix inequalities are developed not in terms of the inertia tensor, but in terms of the density-weighted covariance of each body, suggesting a statistical interpretation of the rigid body for inertia identification problems. Enforcing these extra conditions on physical realizability surprisingly results in an LMI of lower dimension in comparison to previous work. Through this insight, semi-definite programming approaches can be used to solve inertia parameter identification problems to global optimality
• A general procedure for constructing Yetter-Drinfeld modules from quantum principal bundles is introduced. As an application a Yetter-Drinfeld structure is put on the cotangent space of the Heckenberger-Kolb calculi of the quantum Grassmannians. For the special case of quantum projective space the associated braiding is shown to be non-diagonal and of Hecke type. Moreover, its Nichols algebra is shown to be finite-dimensional and equal to the anti-holomorphic part of the total differential calculus.
• Resonant activation in temporally driven spin-boson systems subjected to strong dissipation is researched by means of both, analytical and extensive numerical investigations. The phenomenon of resonant activation emerges in the presence of either fluctuating or periodically varying driving fields. Addressing the incoherent regime, a characteristic minimum emerges in the mean first passage time (MFPT) to reach an absorbing neighboring state whenever the intrinsic time scale of the modulation matches the characteristic time scale of the system dynamics. For the case of deterministic periodic driving the first passage time statistics displays a complex, multi-peaked first passage time probability density, which depends crucially on the details of the initial driving phase, the driving frequency and its strength. As an interesting feature we find that the MFPT enters the resonant activation regime at a critical frequency $\nu^*$ which depends very weakly on the strength of the driving.
• Congruence closure procedures are used extensively in automated reasoning and are a core component of most satisfiability modulo theories solvers. However, no known congruence closure algorithms can support any of the expressive logics based on intensional type theory (ITT), which form the basis of many interactive theorem provers. The main source of expressiveness in these logics is dependent types, and yet existing congruence closure procedures found in interactive theorem provers based on ITT do not handle dependent types at all and only work on the simply-typed subsets of the logics. Here we present an efficient and proof-producing congruence closure procedure that applies to every function in ITT no matter how many dependencies exist among its arguments, and that only relies on the commonly assumed uniqueness of identity proofs axiom. We demonstrate its usefulness by solving interesting verification problems involving functions with dependent types.
• The ability of a feed-forward neural network to learn and classify different states of polymer configurations is systematically explored. Performing numerical experiments, we find that a simple network model can, after adequate training, recognize multiple structures, including gas-like coil, liquid-like globular, and crystalline anti-Mackay and Mackay structures. The network can be trained to identify the transition points between various states, which compare well with those identified by independent specific-heat calculations. Our study demonstrates that neural network provides an unconventional tool to study the phase transitions in polymeric systems.
• Though distribution system operators have been adding more sensors to their networks, they still often lack an accurate real-time picture of the behavior of distributed energy resources such as demand responsive electric loads and residential solar generation. Such information could improve system reliability, economic efficiency, and environmental impact. Rather than installing additional, costly sensing and communication infrastructure to obtain additional real-time information, it may be possible to use existing sensing capabilities and leverage knowledge about the system to reduce the need for new infrastructure. In this paper, we disaggregate a distribution feeder's demand measurements into two components: 1) the demand of a population of air conditioners, and 2) the demand of the remaining loads connected to the feeder. We use an online learning algorithm, Dynamic Fixed Share (DFS), that uses the real-time distribution feeder measurements as well as models generated from historical building- and device-level data. We develop two implementations of the algorithm and conduct simulations using real demand data from households and commercial buildings to investigate the effectiveness of the algorithm. Case studies demonstrate that DFS can effectively perform online disaggregation and the choice and construction of models included in the algorithm affects its accuracy, which is comparable to that of a set of Kalman filters.
• In this paper, we introduce the new generalization of contraction mapping by a new control function and an altering distance . We establish some existence results of fixed point for such mappings. Our results reproduce several old and new results in the literature.
• Several genetic alterations are involved in the genesis and development of cancers. The determination of whether and how each genetic alterations contributes to cancer development is fundamental for a complete understanding of the human cancer etiology. Loss of heterozygosity (LOH) is one of such genetic phenomenon linked to a variate of diseases and characterized by the change from heterozygosity (the presence of both alleles of a gene) to to homozygosity (presence of only one type of allele) in a particular DNA locus. Thus identification of DNA regions where LOH has taken place is a important issue in the health sciences. In this article we formulate the LOH detection as the identification of change-points in the parameters of a mixture model and present a detection algorithm based on the cumulative sums (CUSUM) method. We found that even under mild contamination our proposal is a fast and reliable method.
• It is known that for an isolated dielectric cylinder waveguide there exists the cutoff frequency $\omega_\ast$ below which there is no guided mode. It is shown in the paper that the infinite plane periodic array of such waveguides possesses the guided modes in the frequency domain which is below the frequency $\omega_\ast$. In the case of a finite array, the modes in this frequency domain are weakly radiating ones, but their quality factor $Q$ increases with the number of waveguides $N$ as $Q(N)\sim N^3$. This dependence is obtained both numerically, using the multiple scattering formalism, and is justified with a simple analytical model.
• Jan 17 2017 math.AG arXiv:1701.04385v1
The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus $g$. For each such graph $\Gamma$, the associated canonical linear system $\vert K_\Gamma\vert$ has the structure of a polyhedral complex. In this article we propose a tropical analogue of the Hodge bundle on $M_g^{trop}$ and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.
• Proteins are biological polymers that underlie all cellular functions. The first high-resolution protein structures were determined by x-ray crystallography in the 1960s. Since then, there has been continued interest in understanding and predicting protein structure and stability. It is well-established that a large contribution to protein stability originates from the sequestration from solvent of hydrophobic residues in the protein core. How are such hydrophobic residues arranged in the core? And how can one best model the packing of these residues? Here we show that to properly model the packing of residues in protein cores it is essential that amino acids are represented by appropriately calibrated atom sizes, and that hydrogen atoms are explicitly included. We show that protein cores possess a packing fraction of $\phi \approx 0.56$, which is significantly less than the typically quoted value of 0.74 obtained using the extended atom representation. We also compare the results for the packing of amino acids in protein cores to results obtained for jammed packings from disrete element simulations composed of spheres, elongated particles, and particles with bumpy surfaces. We show that amino acids in protein cores pack as densely as disordered jammed packings of particles with similar values for the aspect ratio and bumpiness as found for amino acids. Knowing the structural properties of protein cores is of both fundamental and practical importance. Practically, it enables the assessment of changes in the structure and stability of proteins arising from amino acid mutations (such as those identified as a result of the massive human genome sequencing efforts) and the design of new folded, stable proteins and protein-protein interactions with tunable specificity and affinity.
• In this paper, a new approach to solve the cubic B-spline curve fitting problem is presented based on a meta-heuristic algorithm called " dolphin echolocation ". The method minimizes the proximity error value of the selected nodes that measured using the least squares method and the Euclidean distance method of the new curve generated by the reverse engineering. The results of the proposed method are compared with the genetic algorithm. As a result, this new method seems to be successful.
• Observationally measuring the location of the H$_{2}$O snowline is crucial for understanding the planetesimal and planet formation processes, and the origin of water on Earth. In disks around Herbig Ae stars ($T_{\mathrm{*}}\sim$ 10,000K, $M_{\mathrm{*}}\gtrsim$ 2.5$M_{\bigodot}$), the position of the H$_{2}$O snowline is further from the central star compared with that around cooler, and less massive T Tauri stars. Thus, the H$_{2}$O emission line fluxes from the region within the H$_{2}$O snowline are expected to be stronger. In this paper, we calculate the chemical composition of a Herbig Ae disk using chemical kinetics. Next, we calculate the H$_{2}$O emission line profiles, and investigate the properties of candidate water lines across a wide range of wavelengths (from mid-infrared to sub-millimeter) that can locate the position of the H$_{2}$O snowline. Those line identified have small Einstein $A$ coefficients ($\sim 10^{-6} -10^{-3}$ s$^{-1}$) and relatively high upper state energies ($\sim$ 1000K). The total fluxes tend to increase with decreasing wavelengths. We investigate the possibility of future observations (e.g., ALMA, SPICA/SMI-HRS) to locate the position of the H$_{2}$O snowline. Since the fluxes of those identified lines from Herbig Ae disks are stronger than those from T Tauri disks, the possibility of a successful detection is expected to increase for a Herbig Ae disk.
• The purpose of this paper is to collect and make explicit the results of Gel'fand, Graev and Piatetski-Shapiro and Miyazaki for the $GL(3)$ cusp forms which are non-trivial on $SO(3,\mathbb{R})$. We give new descriptions of the spaces of cusp forms of minimal $K$-type and from the Fourier-Whittaker expansions of such forms give a complete and completely explicit spectral expansion for $L^2(SL(3,\mathbb{Z})\backslash PSL(3,\mathbb{R}))$, accounting for multiplicities, in the style of Duke, Friedlander and Iwaniec's paper on Artin $L$-functions. We directly compute the Jacquet integral for the Whittaker functions at the minimal $K$-type, improving Miyazaki's computation. The primary tool will be the study of the differential operators coming from the Lie algebra on vector-valued cusp forms.
• Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into $\mathbb{R}^n$ if the prescription of values is proper, and one-to-one if $n\ge 5$ and the prescription of values is one-to-one. We may also prescribe the flux map of the examples. We also show analogous results for a large family of directed holomorphic immersions $M\to\mathbb{C}^n$, including null curves.
• A general quantum thermodynamics network is composed of thermal devices connected to the environments through quantum wires. The coupling between the devices and the wires may introduce additional decay channels which modify the system performance with respect to the directly-coupled device. We analyze this effect in a quantum three-level device connected to a heat bath or to a work source through a two-level wire. The steady state heat currents are decomposed into the contributions of the set of simple circuits in the graph representing the master equation. Each circuit is associated with a mechanism in the device operation and the system performance can be described by a small number of circuit representatives of those mechanisms. Although in the limit of weak coupling between the device and the wire the new irreversible contributions can become small, they prevent the system from reaching the Carnot efficiency.
• We consider a nonlinear representation of a Lie algebra which is regular on an abelian ideal, we define a normal form which generalizes that defined in [D. Arnal, M. Ben Ammar, M. Selmi, \rm Normalisation d'une représentation non linéaire d'une algèbre de Lie. \it Annales de la Faculté des Sciences de Toulouse, 5e série, tome 9, No 3, 1988, p 355--379.]
• We report the first detection of the prebiotic complex organic molecule CH$_3$NCO in a solar-type protostar, IRAS16293-2422 B. This species is one of the most abundant complex organic molecule detected on the surface of the comet 67P/Churyumov-Gerasimenko, and in the insterstellar medium it has only been found in hot cores around high-mass protostars. We have used multi-frequency ALMA observations from 90 GHz to 350 GHz covering 11 unblended transitions of CH$_3$NCO and 40 more transitions that appear blended with emission from other molecular species. Our Local Thermodynamic Equilibrium analysis provides an excitation temperature of 232$\pm$41 K and a column density of (7.9$\pm$1.7)$\times$10$^{15}$ cm$^{-2}$, which implies an abundance of (7$\pm$2)$\times$10$^{-10}$ with respect to molecular hydrogen. The derived column density ratios CH$_3$NCO/HNCO, CH$_3$NCO/NH$_2$CHO, and CH$_3$NCO/CH$_3$CN are $\sim$0.3, $\sim$0.8, and $\sim$0.2, respectively, which are different from those measured in hot cores and in comet 67P/Churyumov-Gerasimenko. Our chemical modelling of CH$_3$NCO reproduces well the abundances and column density ratios CH$_3$NCO/HNCO and CH$_3$NCO/NH$_2$CHO measured in IRAS16293-2422 B, suggesting that the production of CH$_3$NCO could occur mostly via gas-phase chemistry after the evaporation of HNCO from dust grains.
• Jan 17 2017 cs.DM arXiv:1701.04375v1
A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing per edge. We characterize embeddings of maximal NIC-planar graphs in terms of generalized planar dual graphs. The characterization is used to derive tight bounds on the density of maximal NIC-planar graphs which ranges between 3.2(n-2) and 3.6(n-2). Further, we show that optimal NIC-planar graphs with 3.6(n-2) edges have a unique embedding and can be recognized in linear time, whereas the recognition problem of NIC-planar graphs is NP-complete. In addition, we show that there are NIC-planar graphs that do not admit right angle crossing drawings, which distinguishes NIC-planar from IC-planar graphs.
• The degree of commutativity of a finite group $F$, defined as the probability that two randomly chosen elements in $F$ commute, has been studied extensively. Recently this definition was generalised to all finitely generated groups. In this paper the degree of commutativity is computed for right-angled Artin groups with respect to their natural generating set. An additional result concerning sphere sizes of groups with rational spherical growth series (with respect to some finite generating set) is obtained.
• The interplay between quantum fluctuations and disorder is investigated in a spin-glass model, in the presence of a uniform transverse field $\Gamma$, and a longitudinal random field following a Gaussian distribution with width $\Delta$. The model is studied through the replica formalism. This study is motivated by experimental investigations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility $\chi_3$, which have led to a considerable experimental and theoretical debate. We analyzed two situations, namely, $\Delta$ and $\Gamma$ considered as independent, as well as these two quantities related as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature $T_f$; moreover, $T_f$ decreases by increasing $\Gamma$ towards a quantum critical point at zero temperature. The situation where $\Delta$ and $\Gamma$ are related appears to reproduce better the experimental observations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility. In this later case, by increasing $\Gamma$, $\chi_3$ becomes progressively rounded, presenting a maximum at a temperature $T^*$ ($T^*>T_f$). Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature $T^*$, one for $T_f<T<T^*$, and another one for $T>T^*$. It is argued that the conventional paramagnetic state corresponds to $T>T^*$, whereas the temperature region $T_f<T<T^*$ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.

Zoltán Zimborás Jan 12 2017 20:38 UTC

Here is a nice description, with additional links, about the importance of this work if it turns out to be flawless (thanks a lot to Martin Schwarz for this link): [dichotomy conjecture][1].

[1]: http://processalgebra.blogspot.com/2017/01/has-feder-vardi-dichotomy-conjecture.html

Noon van der Silk Jan 05 2017 04:51 UTC

This is a cool paper!

Māris Ozols Dec 27 2016 19:34 UTC

Māris Ozols Dec 16 2016 15:38 UTC

Indeed, Schur complement is the answer to the ultimate question!

J. Smith Dec 14 2016 17:43 UTC

Very good Insight on android security problems and malware. Nice Work !

Stefano Pirandola Nov 30 2016 06:45 UTC

Dear Mark, thx for your comment. There are indeed missing citations to previous works by Rafal, Janek and Lorenzo that we forgot to add. Regarding your paper, I did not read it in detail but I have two main comments:

1- What you are using is completely equivalent to the tool of "quantum simulatio

...(continued)
Mark M. Wilde Nov 30 2016 02:18 UTC

An update http://arxiv.org/abs/1609.02160v2 of this paper has appeared, one day after the arXiv post http://arxiv.org/abs/1611.09165 . The paper http://arxiv.org/abs/1609.02160v2 now includes (without citation) some results for bosonic Gaussian channels found independently in http://arxiv.org/abs/16

...(continued)
Felix Leditzky Nov 29 2016 16:34 UTC

Thank you very much for the reply!

Martin Schwarz Nov 24 2016 13:53 UTC

Oded Regev writes [here][1]:

"Dear all,

Yesterday Lior Eldar and I found a flaw in the algorithm proposed
in the arXiv preprint. I do not see how to salvage anything from
the algorithm. The security of lattice-based cryptography against
quantum attacks therefore remains intact and uncha

...(continued)


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 [MSE] Shape of orbit of iterations with base b on Shell-Thron-region Gottfried Ultimate Fellow Posts: 766 Threads: 119 Joined: Aug 2007 08/17/2019, 02:08 PM (This post was last modified: 08/17/2019, 02:11 PM by Gottfried.) In MSE I've started a discussion providing also some pictures on the properties of shapes of the orbits 0 \to 1 \to b \to b^b \to ... and in which way there occurs "divergence". Such "divergence" has been proven for classes of bases b on boundary of the Shell-Thron-region by I.N.Baker & Rippon in 1983 and according to Sheldonison has been thoroughly        investigated by J.C.Yoccoz .                       See here: https://math.stackexchange.com/q/3323851                 Possibly I can take some pictures here, but the transfer of the mathjax-formulae is a visual horror for me... Conclusions I might transfer to this place if the discussion has some sufficient finishing.                      Gottfried Additional material:                  http://go.helms-net.de/math/tetdocs/_equ...quator.pdf       an earlier article of mine as first approximation to the problem                       https://math.stackexchange.com/q/1820410/1714                                         https://math.stackexchange.com/questions...it-chaotic Gottfried Helms, Kassel sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 08/17/2019, 02:28 PM (This post was last modified: 08/17/2019, 02:38 PM by sheldonison.) (08/17/2019, 02:08 PM)Gottfried Wrote: In MSE I've started a discussion providing also some pictures on the properties of shapes of the orbits 0 \to 1 \to b \to b^b \to ... and in which way there occurs "divergence". Such "divergence" has been proven for classes of bases b on boundary of the Shell-Thron-region by I.N.Baker & Rippon in 1983 and according to Sheldonison has been thoroughly        investigated by J.C.Yoccoz .                      Hey Gottfried, The unbounded values in the orbits of $\exp_b^{[\circ n]}(0)$ for these values of b in your MSE post is really cool. $\phi=\frac{\sqrt{5}+1}{2};\;\;\;\;\lambda=\exp\left(\frac{2\pi i}{\phi}\right);\;\;\;b=\exp\left(\lambda\exp(-\lambda)\right);\;\;\;l=\exp(\lambda);\;\;\;l=b^L$ Just a clarification on $\lambda=\frac{2\pi i}{c}$ multiplies with c real, and/or rational.  I think Yoccoz's work primarily involved iterating $z \mapsto \lambda z + z^2$ and his proof of the sharp convergence; non-convergence of the Schroeder $\Psi(f(z))=\lambda\Psi(z)$ when c is an irrational number.  Yoccoz proved that if c has a continued fraction that doesn't misbehave super badly, the the $\Psi(z)$ series converges, and that $\Psi(z)$ doesn't converge if the continued fraction misbehaves, see https://en.wikipedia.org/wiki/Brjuno_number  I don't know if Yoccoz's proof has been extended to proof it also applies to iterating exponentials, but the conjecture would be that it applies.  So when you find Siegel disc's pictures, they typically like to use a value of $c=\phi=\frac{\sqrt{5}+1}{2}$ since the golden ratio has the most ideally behaved continued fraction so it tends to show nice easily computable fractal behavior. I think $\Psi^{-1}(z)$ is the ideal tool to understand the behavior, but perhaps Yoccoz's work isn't quite as directly applicable as the general work on https://en.wikipedia.org/wiki/Siegel_disc - Sheldon Gottfried Ultimate Fellow Posts: 766 Threads: 119 Joined: Aug 2007 08/18/2019, 08:17 AM (08/17/2019, 02:28 PM)sheldonison Wrote: I think $\Psi^{-1}(z)$ is the ideal tool to understand the behavior, but perhaps Yoccoz's work isn't quite as directly applicable as the general work on https://en.wikipedia.org/wiki/Siegel_discHi Sheldon - in the meantime I've aggregated more data for plots based on c=silver constant, and I've to recompute accordingly data for the other types of c - just to apply the new insights to the small paper that I'd linked to, and hopefully extend that small paper towards becoming a better "catalogue".             However, I need a break for a couple of days and surely cannot be much productive in this matter. A question which is coming up at the moment is how to characterize the interior of the $z_0=1$ fractal shape, say the set of curves produced by $0.5 < z_0 < 0.99$, somehow like a gradient-field, displaying little arrows instead of dots, perhaps including directions of fractional iteration-height (as far as this might be meaningfully applicable). That's just the desire to embed the observation into known phenomena in other areas, hopefully of the physical world.            Moreover, I think I've to meditate first a bit about that Yoccoz-work and Siegel-discs which you have directed me to and what this shall give me for the understanding of the whole phenomen. Cordially -   Gottfried Gottfried Helms, Kassel sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 08/18/2019, 02:22 PM (08/18/2019, 08:17 AM)Gottfried Wrote: (08/17/2019, 02:28 PM)sheldonison Wrote: I think $\Psi^{-1}(z)$ is the ideal tool to understand the behavior, but perhaps Yoccoz's work isn't quite as directly applicable as the general work on https://en.wikipedia.org/wiki/Siegel_discHi Sheldon - in the meantime I've aggregated more data for plots based on c=silver constant, and I've to recompute accordingly data for the other types of c - just to apply the new insights to the small paper that I'd linked to, and hopefully extend that small paper towards becoming a better "catalogue".             However, I need a break for a couple of days and surely cannot be much productive in this matter. A question which is coming up at the moment is how to characterize the interior of the $z_0=1$ fractal shape, say the set of curves produced by  $0.5 < z_0 < 0.99$, somehow like a gradient-field, displaying little arrows instead of dots, perhaps including directions of fractional iteration-height (as far as this might be meaningfully applicable). That's just the desire to embed the observation into known phenomena in other areas, hopefully of the physical world.            Moreover, I think I've to meditate first a bit about that Yoccoz-work and Siegel-discs which you have directed me to and what this shall give me for the understanding of the whole phenomen. Cordially -   Gottfried Gottfried, I just read your equater.pdf; very nice.  There you also cover the dynamics for the rational case as well.  Since the Schroeder function doesn't exist when the multiplier at the fixed point is 1, or a rational root of 1, then one can use Ecalle's solution for the Abel function.  Considering what happens as the continued fraction becomes less well behaved is more difficult, as the irrational number gets closer and closer to behaving like a rational number.  Most cases with an irrational multiplier still have a Schroeder function so there is still an infinite number of copies where $z_n$ gets arbitrarily close to zero and there is a logarithmic singularity at zero so $z_{n-1}$ gets arbitrarily large so the fractal should still be unbounded, but it might take an uncountable number of iterations to show that behavior...  The nice thing about using a multiplier of the golden ratio is that one can actually compute the Schroeder function and get good numerical results for the orbits for  $0.5 < z_0 < 0.99$, which gives the same gradient curves as iterating $z\mapsto b^z$.   The Schroder function has a 1-1 mapping from the unbounded fractal to a circle.  One can also study the Julia set for these iteration mappings, but I haven't done it.  There are an infinite number of other pre-images of the fractal since the $b^z$ exponential function has a period of $\frac{\2\pi i}{\ln b}$.   Thanks for posting a delightful topic, both on MSE, and here. - Sheldon Ember Edison Junior Fellow Posts: 38 Threads: 5 Joined: May 2019 10/30/2019, 04:58 PM (This post was last modified: 10/30/2019, 05:17 PM by Ember Edison.) (08/18/2019, 02:22 PM)sheldonison Wrote: There you also cover the dynamics for the rational case as well.  Since the Schroeder function doesn't exist when the multiplier at the fixed point is 1, or a rational root of 1, then one can use Ecalle's solution for the Abel function.   Hi I want to use fatou.gp.ecalle() evaluate Shell-Thron-region rational base, but my series input isn't work. can you give me some code for examples?  And...I want to evaluate derivative for sexp/slog/pent/sroot, Can I just use deriv(sexptaylor()), deriv(slogtaylor()), deriv(kecalle), deriv(pentaylor()), deriv(gm)?  I see the ihex_deriv() you use are not simple. sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 10/31/2019, 09:57 PM (This post was last modified: 11/01/2019, 06:00 AM by sheldonison.) (10/30/2019, 04:58 PM)Ember Edison Wrote: (08/18/2019, 02:22 PM)sheldonison Wrote: There you also cover the dynamics for the rational case as well.  Since the Schroeder function doesn't exist when the multiplier at the fixed point is 1, or a rational root of 1, then one can use Ecalle's solution for the Abel function.   Hi I want to use fatou.gp.ecalle() evaluate Shell-Thron-region rational base, but my series input isn't work. can you give me some code for examples?  And...I want to evaluate derivative for sexp/slog/pent/sroot, Can I just use deriv(sexptaylor()), deriv(slogtaylor()), deriv(kecalle), deriv(pentaylor()), deriv(gm)?  I see the ihex_deriv() you use are not simple. l=1/exp(1); /* the fixed point */ b=exp(-exp(1)); /* the base */ f=b^b^(l+x)-l; /* this is the f(f(x); a function with a derivative=1 and a fixed point of ~zero */ f=strip0fx(f); /* strip off the x^0 term which was only approximately zero; it is required to be exactly zero */ ecalle(f,16); /* initialize formal series for ecalle assymptotic; 16 positive terms; two neg terms; one log term */ z1=0.1; z2=b^b^(z1+l)-l; /* test example; z1=0.1; z2=f(f(0.1)); */ ecalleu(z1) /* [abel_function,abel_function_derivative]; [44.892671576806680805639501168174, */ ecalleu(z2) /* [abel_function,abel_function_derivative]; [45.892671576806680693723673529703, */ the closer z1 and z2 are to zero, the more accurate the asymptotic series; in this case the abel function is accurate to 16 decimal digits.  Notice that f has fixed point of zero, derivative of 1, and no x^2 term f = x*  1.0000000000000000000000000000000 +x^ 2* -4.324402048806904993 E-39   /* ecalle smart enough to ignore x^2 */ +x^ 3* -1.2315093498217750378717379100958 +x^ 4*  0.83689737179948615587202206894090 +x^ 5*  0.90996916721907065130183768671435 This leads to an ecalle assymptotic form for f with two neg terms; a_2/x^2 + a_1/x; a log term, and we calculated 16 terms of the assymptotic above. - Sheldon - Sheldon Daniel Fellow Posts: 70 Threads: 25 Joined: Aug 2007 11/08/2019, 01:34 AM Hello, I was interested in Siegal disks in the exponential map. Dr. Robert Devaney told me in no uncertain terms that the exponential map produced no Siegal disks because the exponential function is an entire function and entire functions can't produce Siegal disks. I'm sorry that I have no personal insight into the matter. Daniel Ember Edison Junior Fellow Posts: 38 Threads: 5 Joined: May 2019 11/08/2019, 04:55 AM (10/31/2019, 09:57 PM)sheldonison Wrote: (10/30/2019, 04:58 PM)Ember Edison Wrote: (08/18/2019, 02:22 PM)sheldonison Wrote: There you also cover the dynamics for the rational case as well.  Since the Schroeder function doesn't exist when the multiplier at the fixed point is 1, or a rational root of 1, then one can use Ecalle's solution for the Abel function.   Hi I want to use fatou.gp.ecalle() evaluate Shell-Thron-region rational base, but my series input isn't work. can you give me some code for examples?  And...I want to evaluate derivative for sexp/slog/pent/sroot, Can I just use deriv(sexptaylor()), deriv(slogtaylor()), deriv(kecalle), deriv(pentaylor()), deriv(gm)?  I see the ihex_deriv() you use are not simple. ecalleu(z1) /* [abel_function,abel_function_derivative]; [44.892671576806680805639501168174, */ ecalleu(z2) /* [abel_function,abel_function_derivative]; [45.892671576806680693723673529703, */ - Sheldon 1.Is the sexpeta() merged upper & low superfunction? 2.Can the cheta/incheta/sexpeta/slogeta normal work when base isn't eta? sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 11/08/2019, 03:20 PM (This post was last modified: 11/08/2019, 03:38 PM by sheldonison.) (11/08/2019, 04:55 AM)Ember Edison Wrote: 1.Is the sexpeta() merged upper & low superfunction? 2.Can the cheta/incheta/sexpeta/slogeta normal work when base isn't eta? 1. no sexpeta; and cheta are both generated from the Ecalle assymptotic series solution for $f(z)=\exp(z)-1$, which has two sectors.  The Ecalle assymptotic generates two different solutions, depending on whether you approach the fixed point of zero from z>0 by iterating $f^{-1}(z)=\log(z+1)$ before evaluating the series, which generates the cheta upper superfunction, or whether you approach the fixed point from z<0 by iterating f(z) before evaluating the series which generates the sexpeta lower superfunction.  That two different analytic functions can be generated from the same assymptotic series makes sense when you realize Ecalle's solution is a divergent assymptotic series, and you need to iterate f(z) or f^-1(z) enough times so that |z| of z is small enough to generate an accurate result. The conjecture is that the limit of Kneser, as the base approaches eta from above would be sexpeta; I don't know if the conjecture has been proven. 2. no, the code for cheta/eta are only for exp(z)-1.   For any given assymptotic Ecalle series, you pretty much need to handcode evaluating that series, taking into account the required error terms and how much precision you want, and how many terms of the series to generate, and which of the 2n sectors of the f^n function you are interested in.  For the superfunction, you need approximations of the inverse of ecalle; the approximations I use are not generic. Also, it might get pretty tricky as the denominator of the derivative at the fixed point grows beyond small single digits ... - Sheldon Ember Edison Junior Fellow Posts: 38 Threads: 5 Joined: May 2019 11/08/2019, 05:55 PM (This post was last modified: 11/08/2019, 06:40 PM by Ember Edison.) (11/08/2019, 03:20 PM)sheldonison Wrote: 1. no sexpeta; and cheta are both generated from the Ecalle assymptotic series solution for $f(z)=\exp(z)-1$, which has two sectors.  The Ecalle assymptotic generates two different solutions, depending on whether you approach the fixed point of zero from z>0 by iterating $f^{-1}(z)=\log(z+1)$ before evaluating the series, which generates the cheta upper superfunction, or whether you approach the fixed point from z<0 by iterating f(z) before evaluating the series which generates the sexpeta lower superfunction.  That two different analytic functions can be generated from the same assymptotic series makes sense when you realize Ecalle's solution is a divergent assymptotic series, and you need to iterate f(z) or f^-1(z) enough times so that |z| of z is small enough to generate an accurate result. The conjecture is that the limit of Kneser, as the base approaches eta from above would be sexpeta; I don't know if the conjecture has been proven. 1.What base can't merged? Is the all Shell-Thron-region rational base and Singularity base can't merged, or just Singularity base, or just eta? 2.What does "limit" mean? Is the ${\lim_{\delta \to 0^+}}fatou.gp.sexp_{\eta+I*\delta}(z)$ is merge superfunction? upper? lower? « Next Oldest | Next Newest »

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# What is the frequency of f(theta)= sin t - cos t ?

Jun 24, 2016

$2 \pi$

#### Explanation:

Period of f(t) = cos t - sin t --> $2 \pi$
Period of f(t) is the least common multiple of $2 \pi$ and $2 \pi$

Jun 24, 2016

$\frac{1}{2 \pi}$

#### Explanation:

The separate periods of sin t and cost are the same, $2 \pi$.

So, the period for the compounded oscillation f(t)=sin t-cos t is $2 \pi$.

The frequency is the reciprocal of the period and is $\frac{1}{2 \pi}$.


text:
In [1]:
import scanpy as sc
import bbknn

In [2]:
sc.settings.verbosity = 3             # verbosity: errors (0), warnings (1), info (2), hints (3)
sc.settings.set_figure_params(dpi=80)

scanpy==1.6.0 anndata==0.7.4 umap==0.4.6 numpy==1.18.1 scipy==1.4.1 pandas==1.0.1 scikit-learn==0.22.2.post1 statsmodels==0.11.1 python-igraph==0.8.2 leidenalg==0.8.3

Import some pancreas data, pooled from multiple experiments. The data features a heavy batch effect.

The object's creation is detailed here.

In [3]:
adata = sc.read('pancreas.h5ad', backup_url='https://www.dropbox.com/s/qj1jlm9w10wmt0u/pancreas.h5ad?dl=1')

/usr/local/lib/python3.7/site-packages/anndata/compat/__init__.py:161: FutureWarning: Moving element from .uns['neighbors']['distances'] to .obsp['distances'].

This is where adjacency matrices should go now.
FutureWarning,
/usr/local/lib/python3.7/site-packages/anndata/compat/__init__.py:161: FutureWarning: Moving element from .uns['neighbors']['connectivities'] to .obsp['connectivities'].

This is where adjacency matrices should go now.
FutureWarning,

Run BBKNN, which computes a batch balanced neighbourhood graph, which can then be used as the basis for a UMAP. This performs batch correction at the neighbourhood graph stage, and the resulting manifold has the experiments integrated better.

Running BBKNN requires a dimensionality reduction (PCA by default) present in .obsm of your object, and some sort of batch variable as a column in .obs. The documentation discusses all of the available parameters. This notebook shows the effect of some parameter choices on the data from In[17].

In [4]:
bbknn.bbknn(adata, batch_key='batch')

computing batch balanced neighbors
finished: added to .uns['neighbors']
.obsp['distances'], distances for each pair of neighbors
.obsp['connectivities'], weighted adjacency matrix (0:00:04)
computing UMAP

Park et al. 2020 improved batch mixing by introducing ridge regression into the workflow. Regressing out technical effect while including a biological grouping into the procedure is insufficient to remove batch effect from a UMAP based on a regular neighbour graph, but helps mix batches better when used as input for BBKNN. One may not be armed in a biological grouping at the time of analysis, in which case a coarse clustering (aiming to feature as many batches as possible per cluster) can be used in its place. This results in the following analysis flow:

bbknn
clustering
ridge regression
pca
bbknn

Let's generate and visualise a post-BBKNN clustering.

In [5]:
sc.tl.leiden(adata, resolution=0.4)

running Leiden clustering


text:
# State Any Four Characteristics of a Homologous Series - Science and Technology 2

State any four characteristics of a homologous series

State two characteristics of a homologous series.

#### Solution

Characteristics of a homologous series:

• Each member of the series differs from the preceding one by the addition of a -CH2 group and by 14 a.m.u.
• All members of a homologous series share the general formula.
Example: The general formula for alkane is CnH2n+2 and alkene is CnH2n.
• The physical properties of the members show a gradation in properties as the molecular mass increases.
• The chemical properties also show a gradient similarity.
Example: Methane and ethane react with chlorine to form methyl chloride and ethyl chloride, respectively.
CH4 + Cl2 → CH3Cl
C2H6 + Cl2 → C2H5Cl
• All members of a homologous series can be prepared by the same general method of preparation.
Example: Alcohols can be prepared from alkyl halides.

CH_3Br+KOH " Boil"/""CH_3OH+KBr

C_2H_5Br+KOH " Boil"/""C_2H_5OH+KBr

Concept: Homologous Series of Carbon Compound
Is there an error in this question or solution?

#### APPEARS IN

Lakhmir Singh Class 10 Chemistry (Science)
Chapter 4 Carbon And Its Compounds
Q 38.2 | Page 241


text:
# Vector Addition and Subtraction| Analytical Method| College Physics| Problem 3.18

### You drive 7.50 km in a straight line in a direction 15° east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.) (b) Show that you still arrive at the same point if the east and north legs are reversed in order.

SOLUTION:

Part A

Consider the following figure:

The east distance is the component in the horizontal direction.

$D_E=7.50\:km\:\cdot sin\:\left(15^{\circ} \right)$

$D_E=1.94\:km$

The north distance is the vertical component

$D_E=7.50\:km\cdot cos\left(15^{\circ} \right)$

$D_E=7.24\:km$

Part B

Based from the figure, we can easily see that the order is reversible in the addition of vectors. We say that $D_E+D_N=D_N+D_E$


text:
# zbMATH — the first resource for mathematics

Theorems on Apostol-Euler polynomials of higher order arising from Euler basis. (English) Zbl 1297.11008
Summary: Recently, D. S. Kim and T. Kim [Integral Transforms Spec. Funct. 24, No. 9, 734–738 (2013; Zbl 1360.11040)] have introduced an useful method to get interesting identities related to Bernoulli and Euler polynomials of higher order. S. Araci et al. [“Theorems on Genocchi polynomials of higher order arising from Genocchi basis”, Preprint] also obtained many interesting equalities related to Genocchi polynomials of higher order arising from Genocchi basis. In this paper, we consider Apostol-Euler polynomials of higher order arising from Euler basis by using the method of Kim. We also try to derive a numerous of interesting properties related to special functions.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials


text:
# Replicative Function of x minus Floor of x is Replicative

## Theorem

Let $f: \R \to \R$ be a real function.

Let $f$ be a replicative function.

Let $g: \R \to \R$ be the real function defined as:

$g \left({x}\right) = f \left({x - \left \lfloor{x}\right \rfloor}\right)$

Then $g$ is also a replicative function.

## Proof

 $\displaystyle \sum_{k \mathop = 0}^{n - 1} g \left({x + \frac k n}\right)$ $=$ $\displaystyle \sum_{k \mathop = 0}^{n - 1} \left({f \left({x + \frac k n - \left \lfloor{x + \frac k n}\right \rfloor}\right)}\right)$


text:
# Embedding vs one-hot encoding performance for char-level RNNs

Mathematically, embedding would be equivalent to one-hot encoding followed by a linear layer (which we may need or not). Though, for me the performance aspect is not clear. That is:

• embeddings are sparse representations:
• sparse lookups may take more time (do they?)
• one-hot encoding is a dense representation:
• more memory
• data transformation may be a bottleneck (e.g. one-hot encoding per batch)

While for word-level RNNs dense representations are unfeasible, for character-level (typically from 26 to ~100 dims, depending which characters you take) I saw both - networks with one-hot encoding and embeddings.

Code-wise and data-wise embeddings are better (cleaner code, no transformations). Though, is it at the cost of performance, and if so - how much? So:

• Are there any rule of thumb where to use one?
• Or are there any performance tests showing that for given conditions (number of characters, sequence length) one is way more performant that the other?


text:
# Difference in movement between the input value and the actual movement

-Robot: waffle Pi -SBC: Raspberry Pi 3

OS installed in SBC: Raspbian OS installed in Remote PC: Ubuntu 16.04 LTS (Xenial Xerus) ROS version: 1.12.14

Hello everyone, I wrote two codes to derive my robot one is for linear movement and the other is for the rotational movement. the problem is that the error is always increased by increasing the traveled distance (rotation /angle). That means if I enter 360 degrees rotation angle the robot will rotate almost 340 degrees (20 degrees error) and this error will higher when rotating for example 720 degrees and so on. When I apply the code using "fake node" there is no errors. Does anyone meet the same problem and have ideas to solve it? Best wishes Below is my code (for the rotation movement):

#!/usr/bin/env python
import rospy
from geometry_msgs.msg import Twist
#PI = 3.1415926535897   the basic position

def move():
speed = input("Input your speed (degrees/sec):")
angle = input("Type your distance (degrees):")
clockwise = input("Clockwise?: ")

PI = 3.1415926535897
angular_speed = speed*2*PI/360
relative_angle = angle*2*PI/360

#vel_msg.linear.x=0
#vel_msg.linear.y=0
#vel_msg.linear.z=0
#vel_msg.angular.x = 0
#vel_msg.angular.y = 0

if clockwise:
vel_msg.angular.z = -abs(angular_speed)
else:
vel_msg.angular.z = abs(angular_speed)

t0 = rospy.Time.now().to_sec()
#rospy.spin()  WRONG USE
r=rospy.Rate(100)
current_angle = 0
while(current_angle < relative_angle):
velocity_publisher.publish(vel_msg)
t1 = rospy.Time.now().to_sec()
current_angle = angular_speed*(t1-t0)
r.sleep()

vel_msg.angular.z = 0
velocity_publisher.publish(vel_msg)
#rospy.spin()   . if we uncomment tje rospy.spin the program will not stop automatically but only when pressing Ctrl+C

if __name__ == '__main__':
try:
rospy.init_node('move', anonymous=True)
velocity_publisher = rospy.Publisher('/cmd_vel', Twist, queue_size=10)
vel_msg = Twist()

move()
except rospy.ROSInterruptException:
pass

edit retag close merge delete

Generally speaking when you test things on "fake nodes", you are using an ideal environment, however the behavior will be slightly different when you try the same code on a real machine because encoders, sensors etc. have their own inherent error.

I think the problem arises when you compute the current_angle in that way. The error of the calculations will increase over time because is the natural behavior of such logic. Have you tried to listen between transforms, E.g.: base_link -> Odom and calculate the relative transform?

With the relative transform you will be able to compute with high precision the magnitude the robot steered and implement the logic to control when to stop.

However, you will have to take into account that this method also comes with errors, it will depends on the precision of your transform broadcaster.

Maybe this resource is usefull for your problem: Turtlebot Free Space Navigation

( 2019-10-16 04:25:17 -0500 )edit

Sort by » oldest newest most voted

@Weasfas, @stevemacenski, Thank you for the response today I will use your advice and try to solve the problem.

EDITED. The problem of the rotation error had solved I solved this problem by doing the next: I did some experiments and found that the error was almost (-13 degree) therefore in the code I added this error to the angle entered by the user. And because of this -13 degree was only for 360 degrees and it increases by increasing the angle I established a formula that increases this expected error 13 by increasing the angle: E.g. 13 for 360 26(132) for 720 (3602) , 39 (133) for 1080 (3603) and so on the formula that is being used: relative_angle = (angle+((angle/360)13))2PI/360 where the term ((angle/360)13) represents the increased error by increasing the angle

more

One issue is that this is open loop. @Weasfas comment represents one particular (very ROS-y) way of closing that loop, you need to have something that's measuring the actual agular distance traveled by your wheels so you can derive how far the wheels have actually moved vs what you commanded it to move (less wheel slippage).

This is generally what we refer to as "odometry", which in short is just the actual measurements from some sensors about relative motion. This could be from wheel encoders, signal strengths, visual feature matching, etc. One of the first things you might learn in a control theory class is response time, just because you command a motor to go somewhere, doesn't mean it can instantaneously.

20 degrees is about a 5% error, in my book that's pretty reasonable for open loop control of this nature

more


text:
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Jul 23

• How We Use Social Media for Informal Learning – From the article: "For every sales pitch you hear about social learning, for every shiny social network logo you see out there on the web, you should be looking for real use cases and stories of social media being used in a real world setting…"
• Ed-Tech Cheat Sheet – Nice reference info graphic (wish they had produced it as a two-page PDF…)

written by Jeff Kelly \\ tags: , , , , ,

Jul 16

written by Jeff Kelly \\ tags: , , , , , , , , , ,

Aug 06

written by Jeff Kelly \\ tags: , , , , , , , ,

Jan 14


text:
2010
Wed
14
Apr

# A Tale of Two Tai Tais

(4)
Posted at 9:06 pm

OH MY GOD! You haven’t changed at all since I last saw you years ago! Still look so young and pretty!

, exclaimed the well-dressed as she gave Tai-Tai B a big hug over a table at Starbucks.

Don’t flatter me lah. *Hiaz* My two kids have been giving me sleepless nights all this while. In fact, I’ve just seen the school principal after one of them was allegedly involved in a big fight in some rugby game *smacks forehead*

How’s life for you?

Same old routine everyday. Shopping, gym and spa in the day. Mahjong at night if I can gather the girlfriends. Plus the occasional weekend shopping trip to Japan and Korea. Life’s getting boring to be honest.

, replied Tai-Tai A, as she twiddled her exquisitely manicured fingernails.

Wow enjoying life huh? I still have to take care of my two kids, and face my two parents-in-law every day as they are living with us in our HDB flat!

, protested Tai-Tai B.

Gosh you are still living with his parents??? My husband and I moved into our spanking new condo the day we got married. Till now no parents, no kids – just a new convertible sports car we bought last year Haha.

Tai-Tai A‘s fingers moved slowly to caress the 1-carat diamond on her ring.

Ooh I’m so envious! And to make things worse, I have to handle everything by myself every time my husband gets called up for those stupid NS stints!

, complained Tai-Tai B.

The you should have married a PR like what I’ve done! No NS! Hee hee …

It’s now Tai-Tai A‘s LV bag’s turn to be fondled.

Yes I just helped my husband calculate his income last night.

Oh so how much did your husband earn last year?

Errr … can don’t say or not? Embarassing lah especially when compared to yours! Let’s just say that his total tax payable is 4% of his total annual income.

Oh I see. We did ours last night too and my husband’s total tax payable is 6% of his total annual income. Oh well … earn more pay more … what to do? *Sigh*

As Tai-Tai B‘s face began turning green with jealousy, unbeknown to them, a kaypoh maths tutor had been busily eavesdropping on their entire conversation from the next table. And being unable to tolerate Tai-Tai A‘s hao lian bitchy attitude, she began to stroke the keys on her calculator in an attempt to find out exactly how much their husbands were earning …

As stated, Tai-Tai A‘s husband’s total tax payable is 6% of his total annual income and Tai-Tai B‘s husband’s total tax payable is 4% of his total annual income.

Given that both Tai-Tais are not working (obviously, else they won’t be Tai-Tais), and that their husbands are entitled to the following reliefs, where applicable (Source: IRAS 2009):

• Personal: \$1000
• Parent: \$7000 per parent staying with them / \$4500 per parent not staying with them
• Spouse: \$2000
• Children: \$4000 per child
• NSMan: \$3000 where applicable

Using information from their conversation above and IRAS 2009 tax rates below, calculate the total annual income of each husband to find out whose husband earns more.

Chargeable Income Rate (%) Gross Tax Payable (\$)
First \$20,000
Next \$10,000
0
3.50
0
350
First \$30,000
Next \$10,000
-
5.50
350
550
First \$40,000
Next \$40,000
-
8.50
900
3400
First \$80,000
Next \$80,000
-
14
4300
11200
First \$160,000
Next \$160,000
-
17
15500
27200
First \$320,000
Next \$320,000
-
20
42700

While it’s highly unlikely that you’ll be asked to read through such a lengthy bitchy conversation in an actual exam question on the minor topic of taxation, there’ll still usually be a wee bit of reading comprehension involving the basics of calculating taxes i.e.

• All mortals are taxed on their
Chargeable Income = Total/Gross Annual Income − Reliefs
• Find out which row in the given tax rates table the chargeable income falls into.

By the way, Tai-Tai A just asked again …

### Revision Exercise

To show that you have understood what Miss Loi just taught you, you must:

1. qwerty1106 says

2010
Jul
3
Sat
10:13pm

1

Hey the condo is sparkling not spanking.
Are you sure you can do this question? think about 45min le(include referencing to Discovering Mathematics 4B) and i cant even know how to start!!! need to do this le:orz and burn more joss sticks(or rather learn this topic 1st)

• 2010
Jul
10
Sat
11:18pm

1.1

Ha! Just caught this comment ...

As mentioned, this question is a little atypical so you won't discover any clues in Discovering Mathematics (pun not intended).

Though the process of obtaining the in this case is admittedly a wee bit involved, the essence of solving this type of local taxation problem (where we're shown our mighty IRAS tax table) is basically the same i.e. find which row does the chargeable income falls into.

As chargeable income = total income − reliefs, we first calculate the reliefs of each Tai Tai's husband

Tai Tai A's husband's total reliefs (read through the paras in red)
= \$1000 (personal) + \$2000 (Spouse) + 2 × \$4500 (Parents not staying - ok let's assume her husband is a filial only child with both parents still alive)
= \$12000

Tai Tai B's husband's total reliefs (read through the paras in green)
= \$1000 (personal) + \$2000 (Spouse) + 2 × \$7000 (Both parents staying with them) + 2 × \$4000 (two kids) + \$3000 (NSMan)
= \$28000

Now we don't know how much they earn, but we're given their tax payable as a % of their total income. To find out which row of the table they belong to, we can deduce from the table the maximum % of total income they are taxable at each row:

Chargeable Income Rate (%) Max Tax Payable (\$) Max % of A's total income
=max tax payable/(max chargeable income+ relief (\$12000))
Max % of B's total income
=max tax payable/(max chargeable income+relief (\$28000))
≤ \$20,000 0 0 0 0
≤ \$30,000 3.50 350 0.833 0.603
≤ \$40,000 5.50 900 1.731 1.324
≤ \$80,000 8.50 4300 4.674 3.981
≤ \$160,000 10.40 15500 9.012 8.245
≤ \$320,000 17 42700 12.86 12.27

So from their given percentages of 6% and 4% we can see from the table above that their total chargeable income is somewhere between \$80000 and \$160000.

⇒ Tax that each of them have to pay (t)
= \$4300 (for the first \$80000) + 14% of the part of the next \$80000 (= total chargeable income (i) − \$80000)

And since we know that
and

Can you solve for the unknowns and see whose husband actually has a higher total income?

### 2 Reactions

1. Miss Loi tweeted

2010
Apr
14
Wed
2:08pm
2. Miss Loi tweeted

2010
Apr
14
Wed
2:09pm

3

A Tale of Two Tai Tais: http://bit.ly/a0QLdF

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text:
Maximum of Chi-square RV's

1. Jan 14, 2010

m26k9

Hello,
I am trying to find the distribution of the maximum of a set of independed Chi-square RV's with 2-degrees of freedom. Actually I only want to find the MEAN value.
I am using the following formula to find the PDF.

$$f_{X_{\mathsf{max}}}(x) = NF_X(x)^{N-1}f_X(x)$$

Following PDF and CDF is used:
$$f_X(x)=\frac{1}{2}e^{-\frac{x}{2}}$$
$$F_X(x) = 1-e^{-\frac{x}{2}}$$

So what I want to find is: (Assuming N variables)
$$E[f_{X_{MAX}}(x)] =N \int_0^R xF_X(x)^{N-1}f_X(x)$$

I am stuck in a neverending integration by-parts.
If anybody know any solution to this or any method to find this, please let me know.

Cheer.s

Last edited: Jan 14, 2010
2. Jan 15, 2010

bpet

Did you mean to say $$E[X_{MAX}]$$.

Mathematica gives the answer 2*HarmonicNumber[n] but I'm not sure how to derive it.

3. Jan 15, 2010

m26k9

Thank you bpet.
Yes I want to find $$E[X_{MAX}]$$. Sorry for the mistake.
I will search for Harmonicanumber. Havent heard of it before.

Cheers.

4. Jan 24, 2010

EngWiPy

You need to calculate the mean of the maximum of independent Chi-square RVs, is it right? So:

$$E_{X^*}[X]=\,\int_0^{\infty}x\,f_{X^*}(x)\,dx$$

where $$f_{X^*}(x)$$ is the first equation you wrote, but the PDF and CDF must be of Chi-square not of exponentials. Then substiute these data into the integration and evaluate the integral.

5. Jan 24, 2010

m26k9

Thank you very much David.
I'm not sure what you meant by but the PDF and CDF must be of Chi-square not of exponentials?

Because of the exponentials I could not find a closed-from expression there is a pattern and I could find a recursive solution. Could you please explain a bit what you meant earlier?

Cheers.

6. Jan 24, 2010

EngWiPy

I mean the PDF and CDF in the first equation must be of a Chi-square random variable. To elaborate, suppose that we have n independent and identically distributed Chi-square random variables: $$X_1,\,X_2,\ldots,\,X_n$$ with $$f_X(x)$$ and $$F_X(x)$$ as the PDF and CDF, respectively. Arrange them in ascending order as: $$X_{(1)}\leq X_{(2)}\leq X_{(3)} \leq \cdots\leq X_{(n)}$$. Then the PDF of $$X_{(n)}$$ (the maximum RV) is:

$$f_{X_{(n)}}(x)=\,n\left[F_X(x)]^{n-1}\,f_X(x)$$

You can find the distributions of a Chi-square RV from any probability book.

7. Jan 24, 2010

bpet

m26k9's expression for the distribution is correct.

8. Jan 24, 2010

m26k9

Thank you very much David and Bpet.

The problem I had was to evaluate the integral and was thinking if there is a commonly known closed-form expression for $$f_{X_{(n)}}(x)$$. I guess I have to do with the recursive one.

Thank you again guys.

9. Jan 25, 2010

EngWiPy

Yes, bpet is right, I am sorry, where Chi-square distributions with two degrees of freedom become as m26k9 wrote with the assumption that the variance is unity. So, the PDF of the maximum RV becomes:

$$f_{X_{(n)}}(x)=\,n\left[F_X(x)\right]^{n-1}\,f_X(x)=\,\frac{n}{2}\left[1-\text{e}^{-x/2}\right]^{n-1}\,\text{e}^{-x/2}$$.

Where using binomial expansion:

$$\left[1-\text{e}^{-x/2}\right]^{n-1}=\sum_{k=0}^{n-1}(-1)^k{n-1\choose k}\text{e}^{-kx/2}$$

After simple manipulation, use the table of integral to solve the resulting integral.

Good luck