Area Theorem Research Articles

PROFILE OF STUDENTS’ MATHEMATICAL CONNECTION ABILITY IN SOLVING MATHEMATICS PROBLEMS BASED ON VISUALIZER AND VERBALIZER COGNITIVE STYLE

Mathematical connection ability is the ability of students to connect mathematical ideas and concepts in a structured way to solve various problems both inside and outside mathematics. Mathematical connection ability plays an important role in the process of solving mathematical problems. Cognitive style is one of the factors that effect mathematical connection abilities. This research is a qualitative descriptive study that aims to describe the students' mathematical connection skills with visualizer and verbalizer cognitive styles in solving mathematics problems. The research subjects consisted of two grade IX students who each had visualizer and verbalizer cognitive styles. The instrument used was the researcher herself, VVQ (visualizer verbalizer questionnaire), a mathematical connection ability test, and interview guidelines. In this study, the material of plane area and Pythagorean theorem are used for test mathematical connection skills. The results obtained are students with visualizer cognitive style get a good category for their mathematical connection ability in solving mathematics problems because they meet seven good indicators and one sufficient indicator from the mathematical connection ability in solving problems indicator, while students with verbalizer cognitive style get sufficient categories because they meet three good indicators, four sufficient indicators, and one less indicator from mathematical connection ability in solving problems indicator. Therefore, teachers are expected to be able to train students with questions in the context of everyday life that have a higher level of mathematical connection so that they can improve their connection skills and also train with variety exercices presentation so student with each cognitive style can be trained to understand a given problem.

Efficient high-energy Raman soliton generation in a Tm:doped large mode area fiber amplifier.

We report the Tm-doped all-fiber MOPA based on a LMA active fiber generating Raman solitons tunable in the range 1970-2300 nm directly from the LMA fiber. By tuning the chirp of the input pulse we reached more than 90 % energy transfer efficiency to Raman soliton. Solitons with 125 fs duration and up to 24 nJ energy are demonstrated in LMA fiber amplifier. We show experimentally that Raman solitons experience both amplification and absorption in active fiber components of the laser system and that the energy of a Raman soliton generated in an LMA fiber amplifier is limited by the soliton area theorem.

Holographic observables at large d

We study holographically non-local observables in field theories at finite temperature and in the large $d$ limit. These include the Wilson loop, the entanglement entropy, as well as an extension to various dual extremal surfaces of arbitrary codimension. The large $d$ limit creates a localized potential in the near horizon regime resulting in a simplification of the analysis for the non-local observables, while at the same time retaining their qualitative physical properties. Moreover, we study the monotonicity of the coefficient $\alpha$ of the entanglement's area term, the so called area theorem. We find that the difference between the UV and IR of the $\alpha$-values, normalized with the thermal entropy, converges at large $d$ to a constant value which is obtained analytically. Therefore, the large $d$ limit may be used as tool for the study and (in)validation of the renormalization group monotonicity theorems. All the expectation values of the observables under study show rapid convergence to certain values as $d$ increases. The extrapolation of the large $d$ limit to low and intermediate dimensions shows good quantitative agreement with the numerical analysis of the observables.

Area Theorem of Infection Curves for Large Basic Reproduction Number

A remarkable theorem is established in the SIR model in Epidemiology, that is, when the basic reproduction number α is larger than 5, the area of any infection curve is constant given by 1/c, where c is the removed ratio. From this theorem we see that as α gets bigger, the infection curve raises sharply, but its half width decreases inversely. Accordingly, the wave of Omicron will end with a shorter life-time than that of the wave of Delta, because α for Omicron is larger than Delta’s which is regarded as 5<α<9. A rough estimation of the 5-wave area of infection in Japan tells us that the 6-wave will reach a peak on early in Feb., then it will be controlled around early in March.

The all-or-nothing phenomenon in sparse linear regression

We study the problem of recovering a hidden binary k -sparse p -dimensional vector \beta from n noisy linear observations Y=X\beta+W , where X_{ij} are i.i.d. \mathcal{N}(0,1) and W_i are i.i.d. \mathcal{N}(0,\sigma^2) . A closely related hypothesis testing problem is to distinguish the pair (X,Y) generated from this structured model from a corresponding null model where (X,Y) consist of purely independent Gaussian entries. In the low sparsity k=o(\sqrt{p}) and high signal-to-noise ratio k/\sigma^2 \to \infty regime, we establish an “all-or-nothing” information-theoretic phase transition at a critical sample size n^{\ast}=2 k\log (p/k) /\log (1+k/\sigma^2) , resolving a conjecture of Gamarnik and Zadik (2017). Specifically, we show that if \liminf_{p\to \infty} n/n^{\ast}>1 , then the maximum likelihood estimator almost perfectly recovers the hidden vector with high probability and moreover the true hypothesis can be detected with a vanishing error probability. Conversely, if \limsup_{p\to \infty} n/n^{\ast}<1 , then it becomes information-theoretically impossible even to recover an arbitrarily small but fixed fraction of the hidden vector support, or to test hypotheses strictly better than random guess. Our proof of the impossibility result builds upon two key techniques, which could be of independent interest. First, we use a conditional second moment method to upper bound the Kullback–Leibler (KL) divergence between the structured and the null model. Second, inspired by the celebrated area theorem, we establish a lower bound to the minimum mean squared estimation error of the hidden vector in terms of the KL divergence between the two models. An extended abstract version of this work appeared at the Proceedings of the Conference on Learning Theory (COLT), 2019.

The Bohr inequality for certain harmonic mappings

Let HC(ϕ) and HCc(ϕ) denote the classes of sense-preserving harmonic mappings f=h+g¯ in D with the dilation g′(z)=αzh′(z) for |α|<1 such that h is Ma–Minda type convex function and Ma–Minda type convex function with respect to the conjugate points respectively. Here, the function ϕ:D→ℂ, called Ma–Minda function, is analytic and univalent in D such that ϕ(D) has positive real part, symmetric with respect to the real axis, starlike with respect to ϕ(0)=1 and ϕ′(0)>0. The classes HC(ϕ) and HCc(ϕ) are derived from the work of Sun et al. (2016) on the class M(α,β). We study the growth theorems for the functions in these classes. For general ϕ, the sharp coefficient bounds of the functions in the classes HC(ϕ) and HCc(ϕ) are not yet known. For the function f=h+g¯ of the form h(z)=z+∑n=2∞anzn and g(z)=∑n=2∞bnzn in D, using the Bohr phenomenon for subordination classes, we find the radius Rf<1 such that Bohr inequality |z|+∑n=2∞(|an|+|bn|)|z|n≤d(f(0),∂f(D))holds for |z|=r≤Rf for functions in the classes HC(ϕ) and HCc(ϕ) . As a consequence, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes. In addition, we discuss the area theorem for functions in the class HC(ϕ) and obtain an improved version of Bohr inequality for functions in HC(ϕ).

An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

Holographic bound on area of a compact binary merger remnant

Using concomitantly the generalized second law of black hole thermodynamics and the holographic Bekenstein entropy bound embellished by loop quantum gravity corrections to quantum black hole entropy, we show that the boundary cross sectional area of the postmerger remnant formed from the compact binary merger in gravitational wave detection experiments like GW150914 et seq., by the LIGO-VIRGO collaboration, is bounded from below. This lower bound is more general than the bound obtained from application of Hawking's classical area theorem for black holes, since it does not depend on whether the inspiralling compact binary pair or the postmerger remnant consists of black holes or other exotic compact objects. The derivation of the bound entails an estimate of the entropy of the gravitational waves emitted during the binary merger, which adapts to gravitational waves an extant formalism proposed originally for particle ensembles. The results for the minimal cross sectional area of the merger remnant due to binary compact mergers observed recently by the LIGO-VIRGO collaboration are discussed. While accurate measurement of the mass of the remnant for the binary neutron star merger GW170817 remains a challenge, we provide a proof of principle that for binary neutron star mergers our lower bound on the cross sectional area of the remnant provides an alternative approach to probe the validity of neutron star equations of state, independent of the tidal deformabilities of the components.

Testing the Black-Hole Area Law with GW150914.

We present observational confirmation of Hawking's black-hole area theorem based on data from GW150914, finding agreement with the prediction with 97% (95%) probability when we model the ringdown including (excluding) overtones of the quadrupolar mode. We obtain this result from a new time-domain analysis of the pre- and postmerger data. We also confirm that the inspiral and ringdown portions of the signal are consistent with the same remnant mass and spin, in agreement with general relativity.

Coherent propagation of a half-cycle unipolar attosecond pulse in a resonant two-level medium

Here we numerically investigate the propagation of unipolar half-cycle attosecond pulses with a duration much shorter than the period of the proper oscillations T 0 of the medium and its relaxation times T 1 and T 2 in a two-level resonant absorbing and amplifying medium. We find that despite breaking down of the famous McCall and Hahn area theorem, some phenomena described by this theorem are still qualitatively preserved. In particular, it is shown that pulses with small initial electric pulse area become sign-alternating upon the propagation, similar to the traditional 0 π pulses; pulses with 4 π , 5 π , and 6 π area are unstable and split into lower-order pulses, exactly as in the long-pulse case. The electric pulse area conservation rule and limitations of unipolar pulse amplification in two-level gain media, which follows from this rule, are discussed.

Self-Induced Transparency in Warm and Strongly Interacting Rydberg Gases.

We study dispersive optical nonlinearities of short pulses propagating in high number density, warm atomic vapors where the laser resonantly excites atoms to Rydberg P states via a single-photon transition. Three different regimes of the light-atom interaction, dominated by either Doppler broadening, Rydberg atom interactions, or decay due to thermal collisions between ground state and Rydberg atoms, are found. We show that using fast Rabi flopping and strong Rydberg atom interactions, both in the order of gigahertz, can overcome the Doppler effect as well as collisional decay, leading to a sizable dispersive optical nonlinearity on nanosecond timescales. In this regime, self-induced transparency (SIT) emerges when areas of the nanosecond pulse are determined primarily by the Rydberg atom interaction, rather than the area theorem of interaction-free SIT. We identify, both numerically and analytically, the condition to realize Rydberg SIT. Our study contributes to efforts in achieving quantum information processing using glass cell technologies.

Light rays, singularities, and all that

This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald theorem. The article is based on lectures at the 2018 summer program Prospects in Theoretical Physics that was held at the IAS as well as the New Zealand Mathematical Research Institute summer school held in Nelson in January, 2020.

Spatially coupled turbo-coded continuous phase modulation: asymptotic analysis and optimization

For serially or parallel concatenated communication systems, spatial coupling techniques enable to improve the threshold of these systems under iterative decoding using belief propagation (BP). For the case of low-density parity-check (LDPC) codes, it has been shown that, under some asymptotic assumptions, spatially coupled ensembles have BP thresholds that approach the bitwise maximum a posteriori (MAP) threshold of the related uncoupled ensemble. This phenomenon is often referred to as threshold saturation, and it has sometimes very important consequences. For example, in the case of regular LDPC code ensembles, spatial coupling enables to achieve asymptotically the capacity for any class of binary memoryless symmetric channels. Since then, this threshold saturation has been conjectured or proved for several other types of concatenations. In this work, we consider a serially concatenated scheme which is the serial concatenation of a simple outer convolutional code and a continuous phase modulator (CPM) separated by an interleaver. Then, we propose a method to do the spatial coupling of several replicas of this serially concatenated scheme, aiming to improve the asymptotic convergence threshold. First, exploiting the specific structure of the proposed system, an original procedure is proposed in order to terminate the spatially coupled turbo-coded CPM scheme. In particular, the proposed procedure aims to ensure the continuity of the transmitted signal among spatially coupled replicas, enabling to keep one of the core characteristics and advantages of coded CPM schemes. Then, based on an asymptotic analysis, we show that the proposed scheme has very competitive thresholds when compared to carefully designed spatially coupled LDPC codes. Furthermore, it is shown how we can accelerate the convergence rate of the designed systems by optimizing the connection distributions in the coupling matrices. Finally, by investigating on different continuous phase modulation schemes, we corroborate the conjecture stating that spatially coupled turbo-coded CPM schemes saturate to a lower bound very close to the threshold given by the extrinsic information transfer (EXIT) area theorem.

Entropy production from quasinormal modes

Horizons of black branes have an associated entropy current with non-negative divergence. We compute this divergence in a late-time transseries expansion for an inhomogeneous system evolving towards a maximally symmetric asymptotically anti-de Sitter black brane. The horizon area equilibrates on half the time-scale set by the dominant quasinormal mode and we find a simple analytic expression for this evolution purely in terms of the background and the quasinormal mode frequencies. This computation does not require a gradient expansion and is thus non-perturbative in momenta. We generalize this to include scalar and gauge field matter in any number of dimensions. Restricting to homogeneous evolution we match and prove earlier numerical work showing that the apparent horizon entropy saturates the area theorem, in that its time derivative periodically vanishes. The same is true for spherically symmetric evolution towards the Schwarzschild black hole.

Achievable Rate Region for Iterative Multi-User Detection via Low-Cost Gaussian Approximation

We establish a multiuser extrinsic information transfer (EXIT) chart area theorem for the interleave-division multiple access (IDMA) scheme, a special form of superposition coding, in multiple access channels (MACs). A low-cost multi-user detection (MUD) based on the Gaussian approximation (GA) is assumed. The evolution of mean-square errors (MSE) of the GA-based MUD during iterative processing is studied. We show that the $K$ -dimensional tuples formed by the MSEs of $K$ users constitute a conservative vector field. The achievable rate is a potential function of this conservative field, so it is the integral along any path in the field with value of the achievable rate solely determined by the two path terminals. Optimized error correcting codes can be found given the integration paths in the MSE fields by matching EXIT type functions. The above findings imply that i) low-cost GA detection can provide MAC capacity-approaching performance, ii) the sum-rate capacity can be achieved independently of the integration path in the MSE fields; and iii) the integration path determining achievable rate tuples of all users can be an extra degree of freedom for code design.

Photon echoes in optically dense media

The authors derive analytical expressions for secondary echoes formed under the two-pulse excitation of the optically dense media developing McCall-Hahn area theorem for echo effects. They find that a series of self-reviving echo signals with a total area of 2$\ensuremath{\pi}$ or 0$\ensuremath{\pi}$ is excited and propagates deep in the media demonstrating strong non-linear nature of light-matter interaction at small changes of second pulse area. The pulse area approach paves the way for precise coherent spectroscopy, the study of different photon echoes and quantum control of light pulses in the optically dense media.

Two-pulse photon echo area theorem in an optically dense medium.

We perform a theoretical and experimental study of the two-pulse photon echo area conservation law in an optically dense medium. The experimental properties of the echo signal are studied at 4K on the optical transition 3H6(1)→3H4(1) (793 nm) of Tm3+ in a YAG crystal for a wide range of pulse areas of the two incoming light pulses, up to θ 1 r o x4π and θ 2≈7π respectively, with optical depth 1.5. We analyze the experimental data by using the analytic solution of the photon echo area theorem for plane waves. We find that the transverse Gaussian spatial profile of the beam leads to an attenuation of the echo area nutation as function of θ1 and θ2. Additional spatial filtering of the photon echo beam allows to recover this nutation. The experimental data are in good agreement with the solution of photon echo pulse area theorem for weak incoming pulse areas θ 1,2≲π. However at higher pulse areas, the observations diverge from the analytic solution requiring further theoretical and experimental studies.

All-optical generation of quantum entangled states with strictly constrained ultrafast laser pulses

We present a quantum optimal control theory study combined with theoretical analysis to determine a pulsed laser field, capable of generating a maximally entangled state in two trapped two-level atoms. By expanding the time-dependent unitary operator to the first leading term of Magnus expansion, we reexamine the pulse area theorem for the trapped atoms driven by an arbitrarily temporary field. Due to the dipole-dipole interaction blockade, we find that the two trapped atoms described by a three-level ladder system can be reduced into an equivalent two-level model by using narrow-bandwidth pulses, leading to an analytical solution for generating the maximally entangled state. We also solve a highly constrained optimization problem to search for optimal laser pulses with broad bandwidths. A zero pulse-area constraint is employed to remove the dc offset of the optimized laser pulses, and a fixed fluence limitation combined with a constant pulse-area constraint at the resonant frequency of the equivalent two-level system are utilized to restrict the unitary evolution of quantum systems by the first leading term of Magnus expansion. This work provides a potentially useful approach to find all-optical control schemes for generating the maximally entangled state by using ultrafast laser pulses while satisfying multiple strict limitations.

Counting Parallel Segments: New Variants of Pick\u2019s Area Theorem

Counting Parallel Segments: New Variants of Pick\u2019s Area Theorem

Fast quantum state transfer in hybrid quantum dot-metal nanoparticle systems by shaping ultrafast laser pulses

Semiconductor quantum dots (SQDs) have relatively short decoherence times, but have the potential for extremely fast operations implemented by femto-second laser pulses, making them a possible candidate for very high speed quantum information processing. These operations are enhanced by positioning a metal nano-particle (MNP) near to the SQD which significantly increases the strength of electric fields applied to it. Here we consider an SQD with three-levels in a Lambda configuration, and explore the requirements for implementing a transfer from one ground state to the other via the upper level. A recently developed optimization method allows us to fix the energy of the laser pulse while satisfying the requirement that the pulse has no DC component. We show that the use of carefully shaped pulses significantly reduces the required laser power. We also determine the dependence of the pulse power on the distance between the SQD and the MNP, and explore the degree to which the process follows the pulse area theorem for first-order perturbative processes.