Recursos de colección

Caltech Authors (169.428 recursos)

Repository of works by Caltech published authors.

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Mostrando recursos 1 - 20 de 3.322

  1. On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms

    Luca, Florian; Radziwill, Maksym; Shparlinski, Igor E.
    We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato-Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ≤ n^(11/2) (log n)^((−1/2) + o(1)) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the...

  2. Limitations to mollifying ζ(s)

    Radziwiłł, Maksym
    We establish limitations to how well one can mollify the Riemann zeta-function on the critical line with mollifiers of arbitrary length. Our result gives a non-trivial lower bound for the contribution of the off-diagonal terms to mollified moments of ζ(s). On the Riemann Hypothesis, we establish a connection between the mollified moment and Montgomery's Pair Correlation Function.

  3. A converse to Halász's theorem

    Radziwiłł, Maksym
    We show that the distribution of large values of an additive function on the integers, and the distribution of values of the additive function on the primes are related to each other via a Levy Process. As a consequence we obtain a converse to an old theorem of Halasz. Halasz proved that if f is an strongly additive function with f (p) ∈ {0, 1}, then f is Poisson distributed on the integers. We prove, conversely, that if f is Poisson distributed on the integers then for most primes p, f(p) = o(1) or f(p) = 1 + o(1).

  4. A structure theorem in probabilistic number theory

    Radziwiłł, Maksym
    We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. This is a consequence of a structure theorem making clear the inter-relation between the distribution of an additive function on the integers, and its distribution on the primes.

  5. Large deviations in Selberg's central limit theorem

    Radziwiłł, Maksym
    Following Selberg it is known that as T → ∞, [formula] uniformly for Δ ≤ (log log log T)^((1/2) - ε). We extend the range of Δ to Δ « (log log T)^((1/10) - ε). We also speculate on the size of the largest Δ for which the above normal approximation can hold and on the correct approximation beyond this point.

  6. On large deviations of additive functions

    Radziwill, Maksym
    We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. The Kac-Kubilius model suggests that the distribution of values of a given additive function can be modeled by a sum of random variables. We show that the model is accurate (in a large deviation sense) when one is looking at values of the additive function around its mean, but fails, by a constant multiple, for large values of the additive function. We believe that this phenomenon arises, because the model breaks down for the values of...

  7. Discrepancy bounds for the distribution of the Riemann zeta-function and applications

    Lamzouri, Youness; Lester, Stephen; Radziwiłł, Maksym
    We investigate the distribution of the Riemann zeta-function on the line ℜ(s)=σ. For 1/2<σ≤1 we obtain an upper bound on the discrepancy between the distribution of ζ(s) and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of ζ(s) inside of the critical strip, strengthening a previous result of the first author. As an application of these results we obtain the first effective error term for the number of solutions to ζ(s)=a in a strip 1/2<σ1<σ2<1. Previously in the strip 1/2<σ<1 only an asymptotic estimate was available due to...

  8. Reconciling two notions of quantum operator disagreement: Entropic uncertainty relations and information scrambling, united through quasiprobabilities

    Yunger Halpern, Nicole; Bartolotta, Anthony; Pollack, Jason
    How violently do two quantum operators disagree? Different fields of physics feature different measures of incompatibility: (i) In quantum information theory, entropic uncertainty relations constrain measurement outcomes. (ii) In condensed matter and high-energy physics, the out-of-time-ordered correlator (OTOC) signals scrambling, the spread of information through many-body entanglement. We unite these measures, deriving entropic uncertainty relations for scrambling. The entropies are of distributions over weak and strong measurements' possible outcomes. Weakness causes the OTOC quasiprobability (a nonclassical generalization of a probability, in terms of which the OTOC decomposes) to govern terms in the uncertainty bound. Scrambling strengthens the bound, we show, in numerical simulations of a spin chain. Beyond scrambling, we derive...

  9. A Characterization of "Phelpsian" Statistical Discrimination

    Chambers, Christopher P.; Echenique, Federico
    We establish that statistical discrimination is possible if and only if it is impossible to uniquely identify the signal structure observed by an employer from a realized empirical distribution of skills. The impossibility of statistical discrimination is shown to be equivalent to the existence of a fair, skill-dependent remuneration for every set of tasks every signal-dependent optimal assignment of workers to tasks. Finally, we connect this literature to Bayesian persuasion, establishing that if the possibility of discrimination is absent, then the optimal signalling problem results in a linear payoff function (as well as a kind of converse).

  10. Understanding the Role of Adaptivity in Machine Teaching: The Case of Version Space Learners

    Chen, Yuxin; Singla, Adish; Mac Aodha, Oisin; Perona, Pietro; Yue, Yisong
    In real-world applications of education, an effective teacher adaptively chooses the next example to teach based on the learner's current state. However, most existing work in algorithmic machine teaching focuses on the batch setting, where adaptivity plays no role. In this paper, we study the case of teaching consistent, version space learners in an interactive setting. At any time step, the teacher provides an example, the learner performs an update, and the teacher observes the learner's new state. We highlight that adaptivity does not speed up the teaching process when considering existing models of version space learners, such as "worst-case"...

  11. Teaching categories to human learners with visual explanations

    Mac Aodha, Oisin; Su, Shihan; Perona, Pietro; Yue, Yisong
    We study the problem of computer-assisted teaching with explanations. Conventional approaches for machine teaching typically only provide feedback at the instance level e.g., the category or label of the instance. However, it is intuitive that clear explanations from a knowledgeable teacher can significantly improve a student's ability to learn a new concept. To address these existing limitations, we propose a teaching framework that provides interpretable explanations as feedback and models how the learner incorporates this additional information. In the case of images, we show that we can automatically generate explanations that highlight the parts of the image that are responsible...

  12. Fast Conditional Independence Test for Vector Variables with Large Sample Sizes

    Chalupka, Krzysztof; Perona, Pietro; Eberhardt, Frederick
    We present and evaluate the Fast (conditional) Independence Test (FIT) -- a nonparametric conditional independence test. The test is based on the idea that when P(X∣Y,Z)=P(X∣Y), Z is not useful as a feature to predict X, as long as Y is also a regressor. On the contrary, if P(X∣Y,Z)≠P(X∣Y), Z might improve prediction results. FIT applies to thousand-dimensional random variables with a hundred thousand samples in a fraction of the time required by alternative methods. We provide an extensive evaluation that compares FIT to six extant nonparametric independence tests. The evaluation shows that FIT has low probability of making both...

  13. It's all Relative: Monocular 3D Human Pose Estimation from Weakly Supervised Data

    Ronchi, Matteo Ruggero; Mac Aodha, Oisin; Eng, Robert; Perona, Pietro
    We address the problem of 3D human pose estimation from 2D input images using only weakly supervised training data. Despite showing considerable success for 2D pose estimation, the application of supervised machine learning to 3D pose estimation in real world images is currently hampered by the lack of varied training images with associated 3D poses. Existing 3D pose estimation algorithms train on data that has either been collected in carefully controlled studio settings or has been generated synthetically. Instead, we take a different approach, and propose a 3D human pose estimation algorithm that only requires relative estimates of depth at training time. Such...

  14. Teaching Multiple Concepts to Forgetful Learners

    Hunziker, Anette; Chen, Yuxin; Mac Aodha, Oisin; Gomez Rodriguez, Manuel; Krause, Andreas; Perona, Pietro; Yue, Yisong; Singla, Adish
    How can we help a forgetful learner learn multiple concepts within a limited time frame? For long-term learning, it is crucial to devise teaching strategies that leverage the underlying forgetting mechanisms of the learners. In this paper, we cast the problem of adaptively teaching a forgetful learner as a novel discrete optimization problem, where we seek to optimize a natural objective function that characterizes the learner's expected performance throughout the teaching session. We then propose a simple greedy teaching strategy and derive strong performance guarantees based on two intuitive data-dependent parameters, which characterize the degree of diminishing returns of teaching...

  15. Asymmetric Information, Reputation, and Welfare in Online Credit Markets

    Xin, Yi
    This paper studies the impact of reputation/feedback systems on the operation of online credit markets using data from Prosper.com. The ability of lenders to recover their loans is one of the main concerns in these markets, where the problems of asymmetric information are two-fold. On the one hand, borrowers differ in their inherent risks; on the other hand, additional incentives are necessary to motivate borrowers to exert effort. In this paper, I investigate the channels through which reputation/feedback systems improve the total welfare of market participants when both adverse selection and moral hazard are present. A finite-horizon dynamic model of...

  16. Young Accreting Compact Objects in M31: The Combined Power of NuSTAR, Chandra, and Hubble

    Lazzarini, M.; Hornschemeier, A. E.; Williams, B. F.; Wik, D.; Vulic, N.; Yukita, M.; Zezas, A.; Lewis, A. R.; Durbin, M.; Ptak, A.; Bodaghee, A.; Lehmer, B. D.; Antoniou, V.; Maccarone, T.
    We present 15 high mass X-ray binary (HMXB) candidates in the disk of M31 for which we are able to infer compact object type, spectral type of the donor star, and age using multiwavelength observations from NuSTAR, Chandra, and the Hubble Space Telescope (HST). The hard X-ray colors and luminosities from NuSTAR permit the tentative classification of accreting X-ray binary systems by compact object type, distinguishing black hole from neutron star systems. We find hard state black holes, pulsars, and non-magnetized neutron stars associated with optical point source counterparts with similar frequency. We also find nine non-magnetized neutron stars coincident with globular clusters and an equal number of pulsars...

  17. A note on the Liouville function in short intervals

    Matomäki, Kaisa; Radziwiłł, Maksym
    In this note we give a short and self-contained proof that, for any δ > 0, ∑_(x≤n≤x+x^δ)λ(n) = o(x^δ) for almost all x ∈ [X,2X]. We also sketch a proof of a generalization of such a result to general real-valued multiplicative functions. Both results are special cases of results in our more involved and lengthy recent pre-print.

  18. Quantum Unique Ergodicity for half-integral weight automorphic forms

    Lester, Stephen; Radziwiłł, Maksym
    We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic Hecke cusp forms for Γ_0(4) lying in Kohnen's plus subspace and for half-integral weight Hecke Maaβ cusp forms for Γ_0(4) lying in Kohnen's plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp equidistribute with respect to hyperbolic measure on Γ_0(4)∖H as the weight tends to infinity.

  19. Simulating the classical XY model with a laser network

    Tamate, Shuhei; Yamamoto, Yoshihisa; Marandi, Alireza; McMahon, Peter; Utsunomiya, Shoko
    Drawing fair samples from the Boltzmann distribution of a statistical model is a challenging task for modern digital computers. We propose a physical implementation of a Boltzmann sampler for the classical XY model by using a laser network. The XY spins are mapped onto the phases of multiple laser pulses in a fiber ring cavity and the steady-state distribution of phases naturally realizes the Boltzmann distribution of the corresponding XY model. We experimentally implement the laser network by using an actively mode-locked fiber laser with optical delay lines, and demonstrate Boltzmann sampling for a one-dimensional XY ring.

  20. Constructing and Constraining Wave Functions for Identical Quantum Particles

    Sebens, Charles T.
    I address the problem of explaining why wave functions for identical particles must be either symmetric or antisymmetric (the symmetry dichotomy) within two interpretations of quantum mechanics which include particles following definite trajectories in addition to, or in lieu of, the wave function: Bohmian mechanics and Newtonian quantum mechanics (a.k.a. many interacting worlds). In both cases I argue that, if the interpretation is formulated properly, the symmetry dichotomy can be derived and need not be postulated.

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