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# Research

### Adaptive Nonparametric Regression via Trend Filtering

Trend filtering is a new class of nonparametric regression methods with the property of being locally adaptive. In the univariate case, it behaves exactly the same as the locally adaptive smoothing splines [Mammen and Van de Geer, 1997], i.e., minimax optimal for the classes of functions with hetereogeneous amoothness; computationally it is way faster than its spline cousin.

Graph trend filtering (GTF) is designed to filter signals obsesrved on the vertices of a graph with hetereogeneous smoothness. The above illustrations are examples of running GTF with k=0, k=1 and k=2, on 2D lattice grids, which induces piecewise constant, piecewise linear and piecewise quadratic structures on the filtered signal over a graph.

The figure above shows an application of a variant of GTF to detecting events in Manhattan using Taxi data, and it produces a detection that is significantly sharper and much less false positives than Laplacian smoothing. Code is becoming available on GitHub project: glmgen.

### Theoretical Foundation for Noisy Sparse Subspace Clustering

Left:Noiseless SSC illustration. Right:Noisy SSC illustration.

This is an geometric condition for the exact separation recovery of Lasso-SSC when data is noisy, or arbitrarily corrupted as long as the Frobenious norm of the additive error is bounded. We showed that the geometric gap of incoherence and inradius, which underpins the noiseless recovery guarantee in Soltanolkotabi and Candes's geometric analysis on SSC , is essentially proportional to the margin of error lassoSSC is able to tolerate. The results justified SSC's state-of-the-art performance in a number of real applications and reveal insight into what kind of problem can be robustly solved by Lasso-SSC.

In the JMLR extension of the work, we improved the bound and now we know that in the deterministic setting one can tolerate subgaussian noise of magnitude in the order of $\sqrt{d}$, where $d$ is the ambient dimension of the datapoint.

### Matrix factorization with missing data

Matrix factorization is a fundamental building blocks of many popular machine learning algorithms for regression, dimension reduction, factor analysis, clustering and etc. While this approach is widely adopted and empirically sound, it is rather ungrounded in terms of theoretical analysis.

Dr. Xu and I produced a stability analysis to the formulation of factorization methods. The theory naturally explains the long observed robustness of factorization methods in collaborative recommendation system. To start from here, we hope to progress towards a comprehensive theory in the behaviors of factorization methods then design provably robust algorithms for collaborative filtering.

An actively updating blog of matrix factorization literature is here.

### Practical matrix completion and matrix recovery

The recent theory of Emmanuel Candes, Terrence Tao, Ma Yi et. al about the tractable convex optimization to solve matrix completion and recovery are absolutely beautiful. Nevertheless, the real data matrix usually does not satisfy the assumptions which include incoherence, random support (of either observation mask or corruptions) and zero-noise. As an illustration, see the three feature trajectories of Oxford dinosaur sequence below:

Left: Input data with incomplete trajectories; Middle: Results of nuclear-norm based convex matrix completion; Right: Results of PARSuMi, our non-convex matrix completion algorithm.

Common practical challenges are:

1. Diagonal-band-shaped data matrix
2. Deterministic support of sampling or corruption
3. Near degenerate subspace (diminishing singular values)
4. Co-existence of noise, corruptions and missing data.

I hope to design algorithms (convex or non-convex) that works in practice by introducing additional constraints specific to the problem, such as exact rank, orthogonality, box constraint, non-negativity and smoothness.

### Subspace clustering and motion segmentation

The intensive research on low-rank matrix recent years has come to a point that its subspace structure and behaviors are (almost) thoroughly understood. People have started to look at more complicated structure such as a union of multiple subspaces.

A motivating application is the segmentation of feature trajectories of multiple rigidly moving object. Such motion segmentation task is equivalent to subspace clustering because each motion is at most rank-4. The seminal SSC paper of Vidal et. al suggests that the key of segmentation is the so-called "self-expressiveness", namely, the vectors should be most easily represented by vectors from the same subspace group.

The theory of subspace clustering sees some great progress lately, featuring Soltanolkotabi and Candes' geometric analysis on SSC, Liu Guangcan's analysis of LRR and Elhamifar and Vidal's new PAMI version SSC.

However, there are still some persisting problems/open questions:

1. General missing data, partial observations
2. Random sparse corruptions
3. Overlapping subspace
4. Graph connectivity
5. Stability results under noisy scenario

A recent paper titled "High-rank matrix completion" attempts to show that by assuming union-of-subspace model, the missing data of a matrix can be filled-in even if the matrix is full-rank! This is a head start towards a solution to Problem 1, but their local approach largely relies on assumption of the overwhelming number of column samples, which is by no means realistic.

### Change detection for practical surveilance camera

Segmentation of meaningful foreground from background is a first step in many CV applications. When the camera is static, such detection is rather simple. In practice, however, there are graduate change of illumination, dynamic/stochastic background motion (tree branches, waves, rain, fog and etc), jittering of camera which causes severe motion blur. All these make the problem not that trivial at first glance.

We designed an algorithm that solves jittering, motion blur, and image inpainting simultaneously with a key step being 'matrix completion', the following is an illustration of its results on 'traffic' sequence:

From Left to Right: 1. aligned jittering/blurry input; 2. recovered background; 3. recovered foreground; 4. missing data used for matrix completion (detected by a run of RPCA on small images). Note that motion blur is removed and black band on top of the 3rd row (caused by image alignment) is recovered perfectly.


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### Home > ACC7 > Chapter 7 Unit 4 > Lesson CC2: 7.1.5 > Problem7-62

7-62.

Simplify each expression.

1. $3.3(x+4)$

Use the Distributive Property.

$3.3(x)+3.3(4)$

Simplify.

$3.3x+13.2$

1. $9-5\frac{1}{2}x-3(-2x+1)$

Remember to follow the Order of Operations.

$\frac{1}{2}\textit{x} + 6$

1. $2.2(3.1+x)+3x$

Use the Distributive Property before trying to solve.

format: plain


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## Blog entry by Dillon Brousseau

Anyone in the world

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# Alfvén wave

An Alfvén wave, named after Hannes Alfvén, is a type of magnetohydrodynamic wave.

### Additional recommended knowledge

An Alfvén wave in a plasma is a traveling oscillation of the ions and the magnetic field. The ion mass density provides the inertia and the magnetic field line tension provides the restoring force. The wave propagates in the direction of the magnetic field, although waves exist at oblique incidence and smoothly change into the magnetosonic wave when the propagation is perpendicular to the magnetic field. The motion of the ions and the perturbation of the magnetic field are in the same direction and transverse to the direction of propagation. The wave is dispersionless with a speed of

$v_A = B/(\mu_0 n_im_i)^{1/2}\,$

$.\qquad \ = (2.18\times10^{11}\,\mbox{cm/s})\,(m_i/m_p)^{-1/2}\,(n_i/{\rm cm}^{-3})^{-1/2}\,(B/{\rm gauss})$

where $v_A\,$ is the velocity of the Alfvén wave, B is the magnetic field strength, $\mu_0\,$ is the permeability of the plasma, $n_i\,$ is the ion number density, and $m_i\,$ is the ion mass.

## History

• 1942: Alfvén theorises the existence of electromagnetic-hydromagnetic waves in a paper published in Nature.
• 1949: Laboratory experiments by S. Lundquist produce such waves in magnetised mercury, with a velocity that approximated Alfvén's formula.
• 1949: Enrico Fermi uses Alfvén waves in his theory of cosmic rays. According to Alex Dessler in a 1970 Science journal article, Fermi had heard a lecture at the University of Chicago, Fermi nodded his head exclaiming "of course" and the next day, the physics world said "of course".
• 1950: Alfvén publishes the first edition of his book, Cosmical Electrodynamics, detailing hydromagnetic waves, and discussing their application to both laboratory and space plasmas.
• 1952: Additional confirmation appears in experiments by Winston Bostick and Morton Levine with ionized helium
• 1954: Bo Lehnert produces Alfvén waves in liquid sodium
• 1958: Eugene Parker suggests hydromagnetic waves in the interstellar medium
• 1958: Berthold, Harris, and Hope detect Alfvén waves in the ionosphere after the Argus nuclear test, generated by the explosion, and travelling at speeds predicted by Alfvén formula.
• 1958: Eugene Parker suggests hydromagnetic waves in the Solar corona extending into the Solar wind.
• 1959: D. F. Jephcott produces Alfvén waves in a gas discharge
• 1960: Coleman, et al, report the measurement of Alfvén waves by the magnetometer aboard the Pioneer and Explorer satellites
• 1960: Sugiura suggests evidence of hydromagnetic waves in the Earth's magnetic field
• 1966: R.O.Motz generates and observes Alfven waves in mercury
• 1970 Hannes Alfvén wins the 1970 Nobel Prize in physics for "for fundamental work and discoveries in magneto-hydrodynamics with fruitful applications in different parts of plasma physics"
• 1973: Eugene Parker suggests hydromagnetic waves in the intergalactic medium
• 1974: Hollweg suggests the existence of Hydromagnetic waves in interplanetary space
• 1974: Ip and Mendis suggests the existence of Hydromagnetic waves in the coma of Comet Kohoutek.
• 2007: Tomczyk, et al, report the detection of Alfvén waves in images of the solar corona with the Coronal Multi-Channel Polarimeter (CoMP) instrument at the National Solar Observatory, New Mexico.
• 2007: Alfvén wave discoveries appear in articles by Jonathan Cirtain and colleagues, Takenori J. Okamoto and colleagues, and Bart De Pontieu and colleagues. De Pontieu’s team also shows that the energy associated with the waves is sufficient to heat the corona and accelerate the solar wind. These results appear in a special collection of 10 articles, by scientists in Japan, Europe and the United States, in the 7 December issue of the journal Science.

## Further reading

• Alfvén, H. "Cosmic Plasma". Holland. 1981.
• Alfvén, H. "Existence of electromagnetic-hydrodynamic waves", Nature (1942) Vol. 150, pp. 405
• Berthold, W. K.; Harris, A. K.; Hope, H. J., "World-Wide Effects of Hydromagnetic Waves Due to Argus" (1960), Journal of Geophysical Research, Vol. 65, p.2233
• Bostick, Winston H.; Levine, Morton A., "Experimental Demonstration in the Laboratory of the Existence of Magneto-Hydrodynamic Waves in Ionized Helium", Physical Review (1952), vol. 87, Issue 4, pp. 671-671
• Coleman, P. J., Jr.; Sonett, C. P.; Judge, D. L.; Smith, E. J., "Some Preliminary Results of the Pioneer V Magnetometer Experiment", Journal of Geophysical Research (1960), Vol. 65, p.1856
• Dessler, A. J., "Swedish iconoclast recognized after many years of rejection and obscurity," Science (1970) , vol. 170, p. 604
• Fermi, E., "On the Origin of the Cosmic Radiation", Physical Review (1949), vol. 75, Issue 8, pp. 1169-1174
• Hollweg, J. V., "Hydromagnetic waves in interplanetary space", Astronomical Society of the Pacific, Publications (1974), vol. 86, Oct. 1974, p. 561-594.
• Ip, W.-H.; Mendis, D. A., "The cometary magnetic field and its associated electric currents", Icarus (1975), vol. 26, Dec. 1975, p. 457-461.
• Jephcott, D.F., "Alfvén waves in a gas discharge", Nature, (1959) vol.183, p.1653
• Lehnert, Bo, "Magneto-Hydrodynamic Waves in Liquid Sodium", Physical Review (1954), vol. 94, Issue 4, pp. 815-824
• Lundquist, S., "Experimental Investigations of Magneto-Hydrodynamic Waves", Physical Review (1949), vol. 76, Issue 12, pp. 1805-1809
• Otani, N. F., "Application of Nonlinear Dynamical Invariants in a Single Electromagnetic Wave to the Study of the Alfvén-Ion-Cyclotron Instability", Physics of Fluids 31, 1456-1464 (1988).
• Parker, E. N.,
• "Suprathermal Particle Generation in the Solar Corona", Astrophysical Journal (1958), vol. 128, p.677
• "Hydromagnetic Waves and the Acceleration of Cosmic Rays", Physical Review (1955), vol. 99, Issue 1, pp. 241-253
• "Extragalactic Cosmic Rays and the Galactic Magnetic Field", Astrophysics and Space Science (1973), Vol. 24, p.279
• Silberstein, M., and N. F. Otani, "Computer simulation of Alfvén waves and double layers along auroral magnetic field lines", Journal of Geophysical Research 99, 6351-6365 (1994). (PDF)
• Sugiura, Masahisa, "Some Evidence of Hydromagnetic Waves in the Earth's Magnetic Field", Physical Review Letters (1961), vol. 6, Issue 6, pp. 255-257
• Cramer, N. F., and S. V. Vladimirov, "Alfvén Waves in Dusty Interstellar Clouds". PASA, 14 (2).
• Otani, N. F., "The Alfvén ion-cyclotron instability, simulation theory and techniques". Journal of Computational Physics 78, 251-277 (1988).
• Falceta-Gonçalves, D. and Jatenco-Pereira, V., "The Effects of Alfvén Waves and Radiation Pressure in Dust Winds of Late-Type Stars". Astrophysical Journal, 576, 976 (2002).
• Motz, R.O., "Alfven Wave Generation in a Spherical System", Physics of Fluids, 9, 411-412, (1966)
• S. Tomczyk, S. W. McIntosh, S. L. Keil, P. G. Judge, T. Schad, D. H. Seeley and J. Edmondson, "Waves in the Solar Corona", Science Magazine, Vol. 317. no. 5842, pp. 1192-1196, (2007)
• EurekAlert! notification of 7th December 2007 Science special issue

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Alfvén_wave". A list of authors is available in Wikipedia.
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It can be shown that the complex numbers satisfy many useful and familiar properties, which are similar to properties of the real numbers. Polar form. Properties of Modulus,Argand diagramcomplex analysis applications, complex analysis problems and solutions, complex analysis lecture notes, complex Your IP: 185.230.184.20 Modulus of a complex number gives the distance of the complex number from the origin in the argand plane, whereas the conjugate of a complex number gives the reflection of the complex number about the real axis in the argand plane. It is important to recall that sometimes when adding or multiplying two complex numbers the result might be a real number as shown in the third part of the previous example! Then the non negative square root of (x^2 + y^2) is called the modulus or absolute value of z (or x + iy). Ex: Find the modulus of z = 3 – 4i. Complex analysis. 1) 7 − i 2) −5 − 5i 3) −2 + 4i 4) 3 − 6i 5) 10 − 2i 6) −4 − 8i 7) −4 − 3i 8) 8 − 3i 9) 1 − 8i 10) −4 + 10 i Graph each number in the complex plane. For any two complex numbers z1 and z2, we have |z1 + z2| ≤ |z1| + |z2|. The modulus of a complex number The product of a complex number with its complex conjugate is a real, positive number: zz = (x+ iy)(x iy) = x2+ y2(3) and is often written zz = jzj2= x + y2(4) where jzj= p x2+ y2(5) is known as the modulus of z. Complex Number : Basic Concepts , Modulus and Argument of a Complex Number 2.Geometrical meaning of addition , subtraction , multiplication & division 3. In the above figure, is equal to the distance between the point and origin in argand plane. Cloudflare Ray ID: 613aa34168f51ce6 Then, the modulus of a complex number z, denoted by |z|, is defined to be the non-negative real number. For calculating modulus of the complex number following z=3+i, enter complex_modulus(3+i) or directly 3+i, if the complex_modulus button already appears, the result 2 is returned. Covid-19 has led the world to go through a phenomenal transition . Geometrically |z| represents the distance of point P from the origin, i.e. Share on Facebook Share on Twitter. Proof ⇒ |z 1 + z 2 | 2 ≤ (|z 1 | + |z 2 |) 2 ⇒ |z 1 + z 2 | ≤ |z 1 | + |z 2 | Geometrical interpretation. Basic Algebraic Properties of Complex Numbers, Exercise 2.3: Properties of Complex Numbers, Exercise 2.4: Conjugate of a Complex Number, Modulus of a Complex Number: Solved Example Problems, Exercise 2.5: Modulus of a Complex Number, Exercise 2.6: Geometry and Locus of Complex Numbers. as vertices of a It is denoted by z. Now consider the triangle shown in figure with vertices O, z1  or z2 , and z1 + z2. Given an arbitrary complex number , we define its complex conjugate to be . In the above result Θ 1 + Θ 2 or Θ 1 – Θ 2 are not necessarily the principle values of the argument of corresponding complex numbers. E-learning is the future today. Their are two important data points to calculate, based on complex numbers. Modulus of complex number properties Property 1 : The modules of sum of two complex numbers is always less than or equal to the sum of their moduli. Then, the modulus of a complex number z, denoted by |z|, is defined to be the non-negative real number. finite number of terms: |z1 z2 z3 ….. zn| = |z1| |z2| |z3| … … |zn|. E-learning is the future today. Modulus of a Complex Number. VII given any two real numbers a,b, either a = b or a < b or b < a. what you'll learn... Overview. Viewed 4 times -1 $\begingroup$ How can i Proved ... properties of complex modulus question. We know from geometry Let us prove some of the properties. Since a and b are real, the modulus of the complex number will also be real. CBSE Class 11 Maths Notes: Complex Number – Properties of Modulus and Properties of Arguments. Free math tutorial and lessons. This leads to the polar form of complex numbers. If the corresponding complex number is known as unimodular complex number. That is the modulus value of a product of complex numbers is equal SHARES. Let z = a + ib be a complex number. Active today. Property Triangle inequality. They are the Modulus and Conjugate. Thus, the modulus of any complex number is equal to the positive square root of the product of the complex number and its conjugate complex number. Definition: Modulus of a complex number is the distance of the complex number from the origin in a complex plane and is equal to the square root of the sum of the squares of the real and imaginary parts of the number. If x, y ∈ R, then an ordered pair (x, y) = x + iy is called a complex number. Complex numbers tutorial. Modulus of a Complex Number. 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modulus of complex number properties 2021


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# Realizing multi-functional all-optical data processing on nanoscale SiC waveguides

## Abstract

All-optical logics are realized on nanoscale SiC waveguides with add-drop micro-ring functionality, including the TE/TM polarized data decoding, the dual-port Kerr switching and gating beyond 12 Gbit/s. With employing the C-C bond enriched SiC thin film upon thermal oxide, the nonlinear refractive index of up to 2.44 × 10−12 cm2/W enables the asymmetric waveguide with polarization distinguishable transmission, which provides a polarization-selectivity to discreminate the TE/TM polarized data decoding with an nearly 9-dB extinction ratio. The TE/TM polarized decoding performance is comparable with a state-of-the-art fiberized in-line polarizer. The complementary transmission in the bus waveguide port facilitates the dual-port Kerr switching for data format conversion/inversion in both add/drop channels. Owing to the TE/TM polarization discriminated throughput, the asymmetric add-drop waveguide micro-ring also permits all-optical AND logic gating functions, where the ON-state outputs only if the pump bit is set at ON state and the probe bit with matched polarization. These results reveal the multi-functionality of the nanoscale SiC add-drop micro-ring waveguide for future photonic logics on chip.

## Introduction

In past few years, the photonic gating functionality has received more attention to meet urgent requirement of high-speed all-optical data transmission and processing on chip in the coming era1. Numerous literatures related to the research on silicon photonic integrated circuits also thrives owing to its superior compatibility with currently available CMOS fabrication technology2,3,4. Among all demonstrated device schemes, the ultrafast optical switches and modulators have attracted more interests1. To realize large-scale photonic integrated circuits on chip, the polarization-handling is an important issue as most of the optoelectronic components are polarization-sensitive5. Typically, the on-chip polarizer can be made by a planar waveguide with high-aspect-ratio6,7,8; however, these waveguide structures exhibit high birefringence and different confinements for TE and TM modes so as to result in different propagation and bending losses. For such a high-aspect-ratio waveguide based polarizer, the coupling of light into the flat planar waveguide becomes more difficult than that into the channel waveguide. Alternatively, splitting the TE/TM-mode was implemented with asymmetrical coupling between a hybrid plasmonic waveguide and a strip dielectric waveguide9. Due to the high birefringence of the hybrid plasmonic waveguide, only the TM-mode dependent light in a strip dielectric waveguide can achieve the phase matching in the hybrid plasmonic waveguide. Previously, Almeida et al. demonstrate different switching mechanisms in Si waveguide by launching changing the pumping power at ~1 GW/cm2 or lower to induce Kerr effect for phase shift up to 180°, whereas the FCA effect is induced to decay the modulation speed down to 1 GHz by enlarging the pumping power above 30 GW/cm2 10. The on-off extinction ratio can be achieved to 13 dB with using the phase modulation to implement the Si waveguide Mach-Zehnder interferometer10. Concurrently, Almeida et al. achieved the all-optical switching in Si waveguide resonator by intensively pumping the waveguide to cause a refractive index change of 4.8 × 10−4 with associated modulation depth enhancing up to 94%11. In addition, Lin et al. performed the SiO2/SiOx/SiO2 strip-loaded waveguide to demonstrate optical amplification with a small-signal power gain as high as 9.5 dB12. Huang et al. observed the FCA effect in SiC micro-ring for implementing the all-optical data switching, which provides a signal-to-noise ratio (SNR) can be achieved as high ae 5 dB under the 1-Gbit/s PRZ-OOK data transmission13. In addition, the polarization modulation can also be applied with versatile approaches. Huang et al. used a multi-mode interfered waveguide coupler on silicon-on-insulator (SOI) to achieve highly polarized extinction ratios of 20 dB for TE-mode and 15 dB for TM-mode14. Xu et al. applied the asymmetrical directional couplers with bending and tapering waveguide to demonstrate a TE-selective polarizer with 30-dB extinction by detuning the phase mismatch between TE/TM polarizations15. In addition, Xie et al. employed the four-wave mixing effect to demonstrate the polarization-multiplexed non-return-to-zero amplitude-shift keying with a data rate of 20 Gbit/s16. On the other hand, Kerr effect is another method to perform all-optical switching. Ikeda et al. employed the SiNx add-drop micro-ring with a nonlinear refractive index of 2.4 × 10−15 cm2/W to demonstrate on-off-keying (OOK) data transmission at 1 Gbit/s17. Huang et al. increased the graphite-like C-C bonds in C-rich SiCx to enhance its nonlinear refractive index up to 6.6 × 10−13 cm2/W for performing the pulsed return-to-zero OOK (PRZ-OOK) data transmission at 12 Gbit/s18. Moreover, all-optical wavelength and format converted functionalities have also emerged to develop silicon photonic integrated circuits. Dekker et al. utilized the 1.55-μm femtosecond pulse to induce the Kerr effect in SOI waveguides for the wavelength up- and down-conversion19. Xu et al. employed the free-carrier dispersion in Si micro-ring waveguide to perform all-optical wavelength converter, and the modulated data rate can be achieved to 0.9 Gbit/s under a CW control power of 4.5 mW20. Astar et al. employed both cross-phase modulation and four-wave mixing to perform all-optical data conversion from non-return-to-zero OOK (NRZ-OOK) to RZ-OOK with a receiving sensitivity gain of 2.5 dB21. In addition, Huang et al. also employed the FCA effect in SiGeC waveguide to achieve the PRZ-OOK data format inversion up to 6 Gbit/s22. The all-optical logic gate is necessary to play a key role for ultrafast data processing in optical quantum computing and communication, which was mainly implemented with semiconductor optical amplifier (SOA) or periodically poled lithium niobate (PPLN) devices23,24. In particular, the SOA-based all-optical logic gates has achieved multi-functional OR, NOT and logics by utilizing cross-phase modulation (XPM), cross-gain modulation (XGM) and four-wave-mixing (FWM) effects23, and the PPLN-based logics including AND and NAND functions have also been demonstrated lately24. More recently, silicon waveguide based logics have also been demonstrated by utilizing versatile device structures including cascaded micro-rings or Mach-Zehnder interferometers25. Among these logic devices, different optical nonlinear effects such as FWM, two photon absorption (TPA) and plasmas dispersion based logics have been employed to demonstrate faster operation speed than electro-optic effect25,26,27,28,29. Nevertheless, the Kerr switching logic based on the add-drop micro-ring structure was seldom discussed for potential application in high-speed operation.

Unfortunately, using bulk silicon substrate for device fabrication has caused some drawbacks to limit its application in high-speed data processing. Remarkably, the bulk silicon inherently exhibits long minority carrier lifetime ranged from µs to ms, which restricts the operation bandwidth to only few MHz. Many researches were proposed to release such a bandwidth limitation by either increasing the drift velocity or shortening the carrier lifetime30,31,32,33. Among different mechanisms, the Si-QD embedded optoelectronic devices with strong quantum confinement effect have the most potential for signal processing beyond GHz. However, the bulk silicon also exhibits the two-photon absorption induced free-carrier absorption (FCA) effect34,35,36, which inevitably results in a recombination (or dissipation) time of ~1 ns to further limit the modulation speed. To meet this demand, nonlinear Kerr switching is proposed as an alternative to the FCA switching for implementing the ultrafast all-optical modulation3,37. Nevertheless, the nonlinear refractive index of bulk silicon is only 4 × 10−14 cm2/W at 1550 nm3, which is insufficient to realize significant nonlinear optical switching. Recently, the carbon-rich silicon carbide (C-C bond enriched SiC) with abundant graphene-like C-C sp2 bonds to provide strong optical nonlinearity was proposed38.

In this work, an asymmetric C-C bond enriched SiC add-drop micro-ring is demonstrated to implement TE/TM polarized data decoding, wavelength conversion, format inversion and all-optical logic operations. To perform the TE/TM polarized decoding, the polarization-selective property of the TE- and TM-mode related transmission spectra is utilized to attenuate a specific TE- or TM-mode intensity for polarization selective data receiving. The add-drop micro-ring can serve as an input polarization-angle dependent TE/TM switchable decoder. The add-drop micro-ring related TE/TM polarized decoding performance at data rates of 4, 6 and 12 Gbit/s is comparable with a commercial in-line polarizer. The C-C bond enriched SiC add-drop micro-ring based Kerr switch also enables the wavelength conversion at date rates of 1.2, 6 and 12 Gbit/s, owing to its transient nonlinear refractive index change with incoming data. Benefiting from the complementary transmission spectra at bus and micro-ring waveguide ports, and the Kerr switching effect enables the simultaneous TE/TM mode selection and data format inversion in the C-C bond enriched SiC micro-ring port. Only when the input probe polarization matches the mode dependent spectral dip, the probe can be cross-wavelength switched by pump. By controlling the polarization and the on/off status of pump and probe, the output of the asymmetric add-drop micro-ring waveguide would implement the functionality of all-optical AND logic gate. Such a C-C bond enriched SiC add-drop micro-ring can thus serve as a multi-function at data processor for photonic integrated circuit to be developed in next generation.

## Results

### Characteristic Analysis and Structural Design of C-C bond enriched SiC Based Add-drop Micro-ring

First of all, the Raman scattering spectroscopy of a 350-nm SiC film deposited on a thin quartz plate was performed as shown in Fig. 1(a). Two peaks at around 795 and 920 cm−1 are observed, which respectively corresponds to the transverse optical (TO) and the longitudinal optical (LO) modes of Si-C bonds. These measured Raman peaks are broadened because of nearly amorphous structure of the as-grown SiC film. Later on, the X-ray photoelectron spectroscopy (XPS) was performed to analyze the composition of the SiC film, which provides Si2p and C1s orbital electron related peaks at 99.8 and 281.9 eV, indicating that the atomic ratios of Si and C are 40% and 51%, respectively, as shown in Fig. 1(b). More C atoms than Si atoms exist in such a C-C bond enriched SiC film, which form more graphene-like C-C bonds to enhance its optical nonlinearity. The dispersion can be observed from the wavelength dependent refractive index spectrum of the C-C bond enriched SiC film shown in Fig. 1(c), which reveals a refractive index of 1.938 at a wavelength of 1550 nm. The geometric structure of designed bus/ring waveguide with height of 350 nm and width of 500 nm is shown in Fig. 1(d), in which the diameter of ring resonator is 300 µm and the gap spacing between bus/ring waveguide (or ring/L-shaped waveguide) is 1400 nm. To increase the coupling efficiency, an inversed taper with its width increasing from 200 to 500 nm was utilized. The right part of Fig. 1(d) shows the surface images of SiC add-drop micro-ring. The RSoft commercial tool with a simulation based on beam propagation method was employed to guarantee the single-mode operation of the design. Figure 1(e) also shows the simulated field profiles for TE0 and TM0 modes. Due to the asymmetric waveguide structure, the designed waveguide prefers the TE0 mode to exhibit better modal confinement than the TM0 mode. Figure 1(f) shows the SEM image of the designed add-drop micro ring. After measurement, the ring diameter, the waveguide width and gap spacing are determined as 297 µm, 562 nm and 1375 nm, respectively. Figure 1(g) is the photograph illustrating the add-drop micro-ring aligned with the lensed fibers firmed on translation stages. The total insertion loss is about 6.13 dB, after measuring the coupling loss induced by lensed fibers, and the propagation loss of the add-drop micro-ring. The cut-back analysis indicates that the propagation loss is only 0.43 dB/mm in addition to the lensed-fiber-to-waveguide coupling loss of ~3 dB/facet39. Therefore, the propagation loss in C-C bond enriched SiC based waveguide can be ignored.

### TE- and TM-mode related Transmission Spectra of the Add-drop Micro-ring

The experimental setup for measuring the TE- and TM-mode related transmission spectra caused by the add-drop micro-ring is shown in Fig. 2(a). Initially, the throughput of incoming ASE signal passes through an in-line polarizer to form a unipolarized light, and a pure TE- (or TM-mode) light was obtained after properly tuning the polarization status, which launches into the SiC based add-drop micro-ring resonator for measuring the transmission spectrum, as shown in Fig. 2(a). The notched dips observed from the output port of the bus waveguide result from the light at specific wavelengths trapped in the micro-ring resonator. The trapped light can be partially coupled out from the micro-ring resonator, and two independent transmission spectra for TE- and TM-mode were observed during the input polarization adjustment. In principle, the normalized transmission spectra obtained from bus and ring ports of an add-drop micro-ring can be simulated by the following equations3,40,41,

$${T}_{bus}=\frac{{r}_{2}^{2}{a}^{2}-2{r}_{1}{r}_{2}a\,\cos (\varphi +{\varphi }_{{\rm{0}}})+{r}_{1}^{2}}{1-2{r}_{1}{r}_{2}a\,\cos (\varphi +{\varphi }_{{\rm{0}}})+{({r}_{1}{r}_{2}a)}^{2}},$$
(1)
$${T}_{ring}=\frac{(1-{r}_{1}^{2})(1-{r}_{2}^{2})a}{1-2{r}_{1}{r}_{2}a\,\cos (\varphi +{\varphi }_{{\rm{0}}})+{({r}_{1}{r}_{2}a)}^{2}},$$
(2)

In Eqs (1) and (2), Tbus and Tring respectively denote the transmittances of bus and ring ports, r1 (or r2) the self-coupling coefficient in the bus port (or ring port), k1 (or k2) the cross-coupling coefficient with the relationship between self- and cross-coupling coefficients of r12 + k12 = 1 (or r22 + k22 = 1) when ignoring the coupling loss. In addition, ϕ is the run-trip phase shift correlated to group index Ng by ϕ = βL = 2πNgL/λ, where β denotes the propagation constant, L = 2πR the cavity length with R representing the radius of the resonator and λ the input wavelength. ϕ0 is a random parameter denoting the initial phase shift, α is the amplitude attenuation expressed as α2 = exp(−αL) in the resonator with an attenuation coefficient of α. For convenience, the detailed parameters of the SiC based add-drop micro-ring resonator for simulation are listed in Table 1. Typically, the asymmetric waveguide induces different TE- and TM-mode group indices in the micro-ring to split their transmission comb spectra. If the waveguide design can be precisely detuned to achieve the symmetric structure, the transmission comb can ideally overlap with each other in piratical case.

As a result, the transmission spectra contributed by TE and TM modes can be simulated by using different group indices with NgTE = 1.705 and NgTM = 1.645, respectively, as also shown in Fig. 2(b,c). Note that the SiC based add-drop micro-ring resonator favors the TE0 mode more than the TM0 mode because of its asymmetric waveguide structure, which indicates a FWHM of 0.1727 nm, a Q-factor of 7919 and the modal extinction ratios (MERs) of 8.76 dB and 0.6 dB for the TE0 mode in bus and ring ports, respectively. In contrast, the TM0 mode incidence only shows Q-factor of 5398 and MER of 5.86 and 0.56 dB for the bus and ring ports, respectively. Note that a micro-ring resonator with extremely high Q-factor is unsuitable for performing such kind of wavelength-converted data processing at high switching speed. The relationship between the Q-factor and switching speed has been discussed in previous work. The photon lifetime (τp) in a micro-ring resonator can be described4.

$${\tau }_{p}\approx \frac{Q{\lambda }_{0}}{2\pi c},$$
(3)

where Q denotes the quality factor of the SiC based micro-ring add-drop, λ0 the wavelength, c the light speed. From the abovementioned formula, the photon lifetime is increased with enlarging the Q-factor of SiC based micro-ring add-drop. Typically, the modulation speed is inversely proportional to the photon lifetime in the SiC, as the data delivered by circulated photons keeps superimposed each other to induce the data waveform distortion for error bit receiving. Therefore, the higher Q factor for the SiC based add-drop micro-ring add-drop is unsuitable for performing the high-speed all-optical data switching.

### TE- and TM-mode related Transmission Spectra of the Add-drop Micro-ring

The two sets of TE- and TM-mode related transmission spectra in the add-drop micro-ring are separated with each other due to their different group refractive indices and deviated free spectral ranges (FSRs). Therefore, the add-drop micro-ring provides two distinct groups of transmission spectral combs for enabling the TE/TM polarized data decoding. To demonstrate, a CW light with 45°-incident angle is encoded by a phase modulator to form the encoded data stream for optical TE/TM polarized data generation, as shown in the upper row of Fig. 3(a).

Then, the optical TE/TM polarized data stream is sent into the add-drop micro-ring, as shown in Fig. 3(b). If the wavelength of encoded data stream coincides with one of the TE-mode related notched dips, only the TM-mode component can pass through the add-drop micro-ring and the TE-mode dependent light would be weakened. This leads to the decoding from the TE/TM polarized data to the on-off keying (OOK) data. Moreover, the throughput data format observed in the bus port is reversed with that in the ring port because they have complementary transmission spectra. That is, the add-drop micro-ring concurrently enables all-optical wavelength conversion and format inversion of data stream in bus and ring ports, respectively. Particularly, the asymmetric add-drop micro-ring also serves as a TE/TM switchable decoder with inserting and tuning a polarization controller, whether the format of the output signal can be preserved or inverted on the basis of the input polarization relative to the waveguide preference.

When operating the 12-Gbit/s TE/TM polarized decoding, the results of a typical asymmetric bus waveguide and the proposed add-drop micro-ring with a specific data stream of “01010010” at different input polarization angles are compared in Figs 3(c) and 3(d). Note that the decoding performance of the bus waveguide is almost independent from the polarization angle turning as the asymmetric waveguide induces different propagation losses for TE and TM modes. On the contrary, the add-drop micro-ring exhibits polarization sensitive behavior on TE/TM polarized decoding. When adjusting the wavelength at TE-mode notched dip and properly detuning the polarization angle to 30°, the converted data with maximal amplitude is observed from the add-drop micro-ring, as shown in the left part of Fig. 3(d). Increasing the input polarization angle weakens the decoded data amplitude because of the same TE and TM components. Continuously tuning the polarization angle to 90°, which becomes perpendicular with the preferred one in the add-drop micro-ring, inverts the format of decoded data with the maximal value, as shown in the right of Fig. 3(d). Such a polarization-selective add-drop micro-ring serves as the TE/TM switchable decoder to provide an OOK data output with an extinction ratio (ER) of 6.67 dB and an amplitude of 200 mV higher than those of 4.26 dB and 60 mV directly decoded from the bus waveguide.

Moreover, the TE/TM polarized decoding performances of the add-drop micro-ring at different data rates (4, 6 and 12 Gbit/s) are shown in Fig. 4. The first row of Fig. 4 shows the incoming TE/TM polarized signal with a specific data stream of “01010110” and a pulsewidth of 83 ps after decoded by a commercial 25-dB in-line polarizer. For comparison, the traces of format preserved and inverted data streams after TE/TM polarized decoding with the add-drop micro-ring are shown in the second and the third rows of Fig. 4, respectively. As the data rate of the input data stream increases from 4 to 12 Gbit/s, both of the commercial in-line polarizer and the add-drop micro-ring can successfully decode the TE/TM polarized data with standard CMOS logic qualified data criterion42. By lengthening the pattern length of the probe TE/TM polarized signal to 27-1, the eye diagrams of add-drop micro-ring converted 4-, 6, and 12-Gbit/s data are shown in the fourth rows of Fig. 4, and the related parameters are summarized in Table 2. These eye diagrams exhibits similar ER of ~7.5 dB, which are smaller than the MER maximum of 8.76 dB obtained at TE-mode dip because of the existence of DC offset. Note that the RMS jitter of the converted 4, 6, and 12-Gbit/s data are 9.0, 11.8 and 7.3 ps respectively.

### Cross-wavelength 10-Gbit/s Data Conversion and Inversion with Kerr Effect

The experimental setup for implementing the data switching in the C-C bond enriched SiC based add-drop micro-ring with enhanced Kerr nonlinearity is shown in Fig. 5(a). Both an intense pump and a CW probe light with properly controlled polarization via two in-line polarizers are 50/50 coupled into the polarization-sensitive add-drop micro-ring. Note that the intense pump pulse results in transient change on refractive index as well as the notch wavelength of the C-C bond enriched SiC based add-drop micro-ring, which consequently red-shifts the transmission combs so as to all-optically switch the probe. The initial wavelength of probe decides whether the data format is preserved or inverted as it sets within or out of the notched dip of the add-drop micro-ring4. With the bus and L-shaped dual output ports, the add-drop micro-ring simultaneously enables the optical data streams at different wavelength channels with preserved and inverted formats. To check the enhanced Kerr nonlinearity, the conversion performance in the bus and ring ports of the add-drop micro-ring was characterized by a single-bit pulse initially, as shown in Fig. 5(b). When both of the pulse and probe wavelengths are set at the same transmission notch of the add-drop micro-ring, the probe would be modulated by the pump to convert the data in the bus port and invert the data in the ring port, as shown in the insert of Fig. 5(b). By detuning the probe wavelength out of the transmission notch, the converted data format in bus and ring ports immediately exchanges each other, as plotted in Fig. 5(b).

From the wavelength spacing of 0.12 nm between format preserved maximum conversion and format inverted maximum inversion, the transmission spectral shift in the add-drop micro-ring owing to the pump induced Kerr nonlinearity change is observed, as shown in Fig. 5(c). The corresponding group refractive index change is ΔNg = 1.35 × 10−4, and the nonlinear refractive index (n2) of the C-C bond enriched SiC can be calculated by using n2 = ΔNg/(M × Ipeak), in which M denotes the field magnification factor of the add-drop micro-ring and Ipeak the pulse intensity coupled into the resonator. During experiment, M = 1.18 is obtained for the add-drop micro-ring. With the total loss of ~6.7 dB, the peak pump power of 1.078 W, and the waveguide effective area of 2.3 µm2, the nonlinear refractive index of the C-C bond enriched SiC is characterized as 2.44 × 10−12 cm2/W.

Subsequently, the high-speed data switching with a specific pattern of “01010010” at different data rates of 1.2, 6 and 12 Gbit/s are analyzed, as shown in Fig. 6(a,b). Note that the initially rising part of the inverted data trace is mainly contributed to interference from closely overlap between the TE and TM related dips. When increasing the pump date rate up to 12 Gbit/s, the switched probe output becomes distorted more than ever with degrading eye patterns failed to follow up the original ones as shown in Fig. 7. As a result, the eye diagram of converted 12-Gbit/s data presents an ER of 4.48 dB, a SNR of 5.56 dB, and a cross point of 42%, and the RMS jitter, rising and falling time are 8.12 ps, 59.1 ps and 44.8 ps, respectively. This results can obtain the corresponding BER of 7.49 × 10−5. From Eq. (3), this SiC based add-drop micro-ring with a Q-factor of 7919 can support the data transmission with a modulation bandwidth of 153 GHz. When decreasing the Q-factor, the modulation bandwidth can be increased but the coupling ratio is also increased to reduce the modulation depth. This trade-off between the modulation bandwidth and modulation depth is considered when trying to improve the bandwidth. In addition, one of the solutions to improve the operation speed is utilizing the new data formats with higher spectral usage efficiency, including the four-level pulse amplitude modulation (PAM-4) and the quadrature amplitude modulation orthogonal frequency division multiplexing (QAM-OFDM) data formats. For example, the PAM-4 data format can increase twice the transmission capacity over the NRZ-OOK data format. Under the finite modulation bandwidth of device, the spectral usage efficiency is an important issue for the silicon photonics in the future.

### All-optical logic gate application

To perform the all-optical AND gate with the C-C bond enriched SiC based add-drop micro-ring, an intense one-bit pulse with a specific polarization is launched into the add-drop micro-ring, and the pump wavelength is set at the TE-mode notched dip. Because of the TE/TM dependency of the add-drop micro-ring, the probe polarization must be controlled at purely TE or TM polarization (1TE or 1TM) by using an in-line polarizer and a polarization controller, as shown in Fig. 8(a). On the other hand, the probe light can also be filtered out by tuning its polarization, which could be denoted as 0TE or 0TM. For convenience, the TE and TM dependent truth tables of the add-drop micro-ring are listed in Tables 3 and 4.

If the probe wavelength is set at the TE-mode notched dip, only the TE-polarized probe can sense the spectral red-shift induced by pump pulse set at ON-state, i.e. pump = 1 and probe = 1TE. When probe or pump is set at OFF-state (pump = 0 or probe = 0TE), respectively, the output would transfer to OFF-state to implement the AND gate, as shown in Table 3. Simultaneously, when probe is set at TM-polarization or the probe is at OFF-state, the output also transfers to OFF-state. When the probe wavelength is controlled at the TM-mode related dip, the corresponding truth table of all-optical logic gate is listed in Table 4, which is similar to that of the TE case. As a result, the TE-mode and TM-mode dependent results of all-optical logic gate are shown in Fig. 8(b,c). As expected, only if the pumping pulse is set at ON-state with selected wavelength and the polarized probe matches to the TE (or TM) notch dependent wavelength, the output bit can transfer to ON-state.

## Methods

### Fabrication of SiC Based Add-drop Micro-ring

The SiC film was deposited on a 3-µm thermal oxide wafer by using a plasma enhanced chemical vapor deposition (PECVD)43. During synthesis, a gaseous mixture of 200-sccm methane (CH4) and 105-sccm Ar-diluted silane (8% SiH4 + 92% Ar) was sent into the chamber. The chamber pressure and substrate temperature were kept at 40 Pa and 550 °C, respectively. A 350-nm SiC film was deposited with a RF plasma power of 120 W and deposition time of 12 min. The waveguide structure was patterned by using e-beam lithography afterwards. Before reactive-ion etching (RIE), Cr hard mask was deposited on the patterned structure by using e-gun. An optimized gaseous mixture of CHF3 and O2 was applied during the RIE process. Furthermore, a 2-μm SiO2 layer was deposited on the waveguide structure, which finally delivers the SiC based add-drop micro-ring.

### Analytic setup of TE/TM polarized decoder, all-optical Kerr switching and all-optical logic in the C-C bond enriched SiC based add-drop micro-ring

For the following experiments, the experimental setup is mainly divided into three parts. The first orange block servers as a TE/TM polarized data decoded probe source, which consists of a tunable laser source (TL, Agilent, 8164 A), a polarization controller, a phase modulator and an arbitrary waveform generator (AWG, Tektronix, 7122B) delivered electrical data. A linearly polarized light with 45° incident angle was modulated by the phase modulator to deliver the optical TE/TM polarized data44. The optical TE/TM polarized data was then amplified by an EDFA. In addition, the second purple block shows the experimental setup of pulsed pumping source, and it consists of a tunable laser (TL, HP, 8168 F) source, a 40-GHz Mach-Zehnder modulator (MZM, JDSU, 10024180) and an AWG delivered electrical pulse-train with a pulsewidth of 83 ps. By null-biasing the MZM, the optical pumping pulse was obtained. To achieve sufficiently high peak power with minimal intensity noise for the pumping pulse, it was interactively amplified and filtered by using two-stage erbium-doped fiber amplifiers (EDFAs, JDSU, OAB1552 + 20FA6 & SDO, EFAH1B111NC02) and optical band-pass filters (OBPFs, SANTEC, OTF-910 & JDS, TB1500B). Finally, the blue block is composed of a tunable laser (TL, Agilent, 8164 A) and an EDFA, which acts as a continuous-wave (CW) probe source. For implementing the TE/TM switchable TE/TM polarized decoder, the optical TE/TM polarized data was sent into the polarization-selective add-drop micro-ring. Besides, for cross-wavelength data conversion and inversion with Kerr effect, both the pumping pulse and the CW probe are launched into the to the SiC add-drop micro-ring by using a 50/50 coupler. Moreover, an additional OBPF was employed to filter out the pumping pulse. For all-optical logic application, the CW probe with TE/TM polarization was selected by using an in-line polarizer. Finally, the results of bus- and ring-port outputs are received by using a photo diode (PD, Nortel, pp-10G) and analyzed by using a digital sampling oscilloscope (DSO, Agilent, 86100 A + 86109B).

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## Acknowledgements

This work was supported By the Ministry of Science and Technology, Taiwan, R.O.C., under grants, MOST-104-2221-E-002-117-MY3, MOST-106-2221-E-002-152-MY3, MOST-106-2218-E-005-001- and MOST-107-2218-E-992-304-.

## Author information

Authors

### Contributions

G.-R. Lin proposed the concept. G.-R. Lin and S.-C. Syu designed the experiment. S.-C. Syu fabricated the SiC waveguide. S.-C. Syu, C.-H. Cheng, H.-Y. Wang, Y.-C. Chi and C.-I. Wu carried out the experimental data. S.-C. Syu and G.-R. Lin analyzed the data. S.-C. Syu, C.-H. Cheng, Y.-C. Chi and G.-R. Lin contributed to the writing of the manuscript.

### Corresponding author

Correspondence to Gong-Ru Lin.

## Ethics declarations

### Competing Interests

The work of Prof. Lin and Mr. Syu has been funded by the research project grants from the Ministry of Science and Technology, Taiwan, under grants, MOST-104-2221-E-002-117-MY3, MOST-106-2221-E-002-152-MY3, MOST-106-2218-E-005-001- and MOST-107-2218-E-992-304-. National Taiwan University pays the salary to Prof. Lin. Other authors declare no competing interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Syu, SC., Cheng, CH., Wang, HY. et al. Realizing multi-functional all-optical data processing on nanoscale SiC waveguides. Sci Rep 8, 14859 (2018). https://doi.org/10.1038/s41598-018-33073-y

• Accepted:

• Published:

### Keywords

• Kerr Switch
• Nonlinear Refractive Index
• Asymmetric Waveguide Structure
• Extinction Ratio (ER)
• Hybrid Plasmonic Waveguide

• ### Third-order optical nonlinearity in silicon oxycarbide films

• Xiuru Yu
• , Luying Yin
• , Heng Lu
• , Yanyan Huo
• , Shouzhen Jiang
• , Lina Zhao
• , Baoyuan Man
•  & Tingyin Ning

Optical Materials (2020)

• ### Third-order optical nonlinearity in nonstoichiometric amorphous silicon carbide films

• Xiuru Yu
• , Baoyong Ding
• , Heng Lu
• , Yanyan Huo
• , Qianqian Peng
• , Xianwu Xiu
• , Chao Zhang
• , Cheng Yang
• , Shouzhen Jiang
• , Baoyuan Man
•  & Tingyin Ning

Journal of Alloys and Compounds (2019)


text:
## Dealing with Missing Data on the Frontend

March 21, 2011

What if you got an unexpected result from an ActiveRecord query? If it’s important for the current page, rendering a 500 is a good idea. If the data is being used for a non-vital section, it’s best to render the page without the section: @manager = Manager.find(params[:id]) rescue nil <div id=”sidebar”> <% if @manager.nil? %>… Read more

## Ode to Textual Identifiers

March 18, 2011

Entities of a domain model are distinct from all other objects in their longevity. These objects are typically stored in a database, or a similar form of long-term storage, and need portable identifiers which can shared within a system, and beyond. Identifiers are essentially “handles to entities”, which can be passed around, in the same… Read more

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March 12, 2011

This morning we had the pleasure to present at the Lean Startup Stage at SXSW. We presented our case study on lessons learned in Agile, continuous deployment and creating high performing teams. Wouldn’t you love to build and deploy your software at impressive speed? We’ve just heard Eric speaking about maximizing the time through the… Read more

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Like many teams we use a Subversion pre-commit hook to perform simple validation on code we’re about to commit. When we commit to our front-end repository, for example, the hook verifies that lines don’t have trailing whitespace and that all text files end with a newline character. Unfortunately, when a SVN pre-commit hook fails, the… Read more

## See us at Cloud Connect 2011 – Workshop on Continuous Delivery

March 03, 2011

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## What’s your stock portfolio return?

Consider the following scenario: you invest your money starting with $100K in a brokerage account, and buy some AAPL shares at$200. Then you transfer $5K more into the account and buy some AMZN shares at$120. Later on you sell your APPL shares when it reaches $300. In the end you transfer$10K out… Read more


text:
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The panic shopping at Australian supermarkets is getting out of control.

Perhaps 2020 will be remembered for its “great toilet roll crisis” as much as the year that novel coronavirus COVID-19 came to Australia.

I am honestly a bit bewildered at the extent of panic shopping that is now taking place in Australia. Unfortunately this is the sort of phenomenon that feeds upon itself; once a few people start, it’s hard not to become affected similarly.

I have tried to remain calm, but I cannot ignore the reality that there is currently a supply problem with quite a few common grocery items. Aside from toilet paper, it is becoming difficult to find infant formula, rice, flour and pasta.

At home, we have managed to keep a fairly well-stocked pantry in case we have to go into “lockdown”. Whether that happens or not, I cannot say but there is a definite sense of inevitability about that scenario. After the Victorian Chief Medical Officer, Dr. Brett Sutton, advised the public that they should try to obtain a “two-week supply of food and a 30-day supply of prescription medication“, things seem to have gone a bit mad.

What has struck me as most peculiar is that people have been stocking up on perishable foodstuffs such as eggs and vegetables. I am unsure what the point of this is, because they can’t be frozen. Australia has ample supply of each so there is no risk of actual food shortages.

I must say that I have been generally impressed with the ABC’s informative and responsible coverage of the COVID-19 pandemic. Leigh Sales’ Coronavirus Special on 7:30 last week was excellent, as was Rafael Epstein’s Drive programme today on ABC Radio Melbourne. Kudos to both.

Life is going to change in the short term. I had to visit the doctor’s earlier in the week and the surgery had had a makeover in an attempt to enforce “social distancing”.

I am also having to put some serious thought into the logistics of working from home. At least I have that option.

I hope the Commonwealth will manage this matter better than the bushfires crisis. Time will tell.

It’s easy to get spooked by all of this. Frankly, seeing the panic buying is becoming a bit upsetting. As always, its the poor and the disabled who pay the most in these scenarios. Stories of hoards in buses from Melbourne stripping rural supermarkets in Victoria just leaves me shaking my head, whilst Woolworths’ first attempt at helping out the needy didn’t work out as planned. Collectively, we can really do much better than this.

The truth is that the economic and social interruption will be significant. But in all likelihood, it will mostly be over in another six months and we can then start to get things rolling again. Australia experienced four years of interruption during World War II. That can’t have been easy and I am not even sure whether that’s a valid comparison to make but it does show that if we all pull together and help each other out, we can get through the tough times.

We just have to remain strong and appeal to our better selves.

One response to “Coronavirus and Mad Shoppers”

On 18 March 2020, Andrew wrote:

There are some curious shortages. Why would people stockpile tinned beetroot? We are nearly out of rice so I snapped up some today, only to find out it is precooked, but that is ok. We could have filled our car with all grocery items in Tasmania before we returned last Sunday, but we trusted Victorian people would have come to common sense. It seems not. Home hallways must be stacked with toilet paper and sacks of rice. While supermarkets are hiring extra staff, what will the economic impact be when they are laid off and people use up their stashes of toilet paper and rice.

    <q cite="">


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# Fighting wild fires with sub zero air

I understand this is probably too costly to be practical, but was intrigued how one might use the mechanism of fire combustion against itself. Fires produce their own retardants (water and carbon dioxide). If you could cool this with quantities of subzero air (e.g. from above) would the dense air attack the flow mechanism that feeds the fire (the oxygen intake at ground requires the upward escape of combustion products). If you had a mobile store of liquid nitrogen and a high-throughput turbine & heat exchanger, using the heat of vaporisation and heat capacity of liquid N2, could you 'hose' down a fire using subzero air?

• @JKB Nitrogen would put it out. So would any not-very-reactive gas. Fire needs three things: fuel, $O_2$ and ignition temp. You first have to cut off the oxygen to stop the combustion reaction and then cool the fuel below the ignition point so that it doesn't flare up again when the $O_2$ comes back. That's how $CO_2$ extinguishers work. – Oscar Bravo Feb 6 at 19:40


text:
# Chapter 3 - Section 3.1 - Congruent Triangles - Exercises: 15

Using method ASA for 2 congruent pairs of angles and one included pair of sides, we can prove that $\triangle ABC\cong\triangle DBC$.

#### Work Step by Step

Since $\vec{CB}$ bisects $\angle ACD$, two resulting angles of this bisect would be equal with each other. That means $\angle 3\cong\angle 4$. It is also mentioned that - $\angle A\cong\angle D$ - $\overline{AC}\cong\overline{CD}$ We now have 2 angles and the included side of $\triangle ABC$ are congruent with 2 angles and the included side of $\triangle DBC$. Therefore, according to method ASA, $\triangle ABC\cong\triangle DBC$.

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