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# Nanophotonics

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# Experimental realization of an optical digital comparator using silicon microring resonators

Yonghui Tian
/ Huifu Xiao
• Institute of Microelectronics and Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
• Other articles by this author:
/ Xiaosuo Wu
• Institute of Microelectronics and Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
• Other articles by this author:
/ Zilong Liu
• Institute of Microelectronics and Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
• Other articles by this author:
/ Yinghao Meng
• Institute of Microelectronics and Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
• Other articles by this author:
/ Lin Deng
• Institute of Microelectronics and Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
• Other articles by this author:
/ Xiaonan Guo
• Institute of Microelectronics and Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
• Other articles by this author:
/ Guipeng Liu
• Institute of Microelectronics and Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
• Other articles by this author:
/ Jianhong Yang
• Corresponding author
• Institute of Microelectronics and Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
• Email
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Published Online: 2018-01-08 | DOI: https://doi.org/10.1515/nanoph-2017-0073

## Abstract

We propose and experimentally demonstrate a silicon photonic circuit that can perform the comparison operation of two-bit digital signals based on microring resonators (MRRs). Two binary electrical signals regarded as two operands of desired comparison digital signals are applied to three MRRs to modulate their resonances through the microheaters fabricated on the top of MRRs, respectively (here, one binary electrical signal is applied to two MRRs by a 1×2 electrical power splitter, which means that the two MRRs are modulated by the same binary electrical signal). The comparison results of two binary electrical signals can be obtained at two output ports in the form of light. The proposed device is fabricated on a silicon-on-insulator substrate using the complementary metal-oxide-semiconductor fabrication process, and the dynamic characterization of the device with the operation speed of 10 kbps is demonstrated successfully.

## 1 Introduction

Nowadays, academic communities, large semiconductor manufacturers, and startups are all competing to develop and commercialize high-performance silicon photonic devices to resolve the problems faced by silicon electronic devices in the field of high-speed and large-capacity information processing, such as high power consumption, metal wire interconnect, and bandwidth limitation. Such a competitive situation mainly derives from the excellent performance of silicon photonic devices in information processing and their compatibility with complementary metal-oxide-semiconductor (CMOS) fabrication technology. Actually, more and more researchers believe that it is a promising choice to employ silicon photonic devices to replace silicon electronic devices in some specific aspects to resolve the problems faced by silicon electronic devices [1]. Various constituent components of a photonic information processing system fabricated by the CMOS-compatible fabrication process have already been demonstrated in experiments [2], [3], [4], [5], [6], [7], [8].

Being one of the basic components in the field of optical information processing, optical digital comparators are indispensable components for decision-making circuits, the integral part of arithmetic and logic units of optical data processors. Currently, most schemes for optical digital comparators are based on the nonlinearity of the material to achieve all-optical devices such as using semiconductor optical amplifiers [9], [10], Fabry-Pérot laser diodes [11], microring resonators (MRRs) [12], and plasmonic circuits [13]. However, all these schemes are fundamentally restricted to achieve large-scale integration, as a strong pump light is needed for these schemes to excite a nonlinear effect. To pursue low power consumption, large-scale integration, and low cost, integrated silicon photonic devices with the nature merits of compact size, low power, and compatibility with the standard commercial CMOS fabrication process are most desired. Although the integrated scheme for the digital comparator is also proposed and demonstrated [14], the structure of the device is complex, which includes two MRRs, one of them is with three coupling regions and one 1×2 power splitter. For MRR with three coupling regions, it is difficult to achieve the critical coupling to obtain better MRR performances.

In this paper, we report an alternative scheme to realize a two-bit digital comparator based on U-shaped waveguides and MRRs, which hold great potential for the construction of ultracompact, low-power, high-speed devices for next-generation silicon photonic circuits. The results of the comparator can be obtained at its two output ports simultaneously. The comparison operation is demonstrated successfully using a thermo-optic [15] effect. Compared to Ref. [14], our proposed structure of the device is simple, and the multimode interference (MMI) coupler and MRR with three coupling regions are eliminated in the integrated photonic circuit, which is very conductive to improve the signal quality and robust against the insertion loss of the device.

## 2 Device principle

The schematic of the proposed photonic circuit is shown in Figure 1, which consists of three MRRs (MRR1, MRR2, and MRR3) connected by three U-shaped waveguides. Compared to Ref. [14], the proposed structure does not have three-coupling MRR and power splitter, which can improve the operation signal quality. The three ports of the circuit are defined as Input, Output C1, and Output C2 according to their functions. The high and low levels of the electrical pulse sequences (EPS) applied to MRRs represent logic 1 and 0 in the electrical domain; the high and low levels of optical signals detected at output ports represent logic 1 and 0 in the optical domain. Monochromatic continuous-wave (CW) light at the working wavelength λw is coupled into the input of the device. Here, MRR1 and MRR3 are set as synchronous parts, which are modulated by EPS X simultaneously, whereas MRR2 is modulated by another EPS Y. The MRR behaves as an optical switch in the proposed circuit: the working states of MRR can be controlled by applying voltage on the TiN microheater deposited on the top of MRR. When EPS X and Y are applied to the corresponding MRRs, the light signal coupled into the Input port can be directed to different propagation paths according to the applied high and low levels of EPS, as the effective refractive index of MRR is modulated through the thermo-optic effect. For example, when a high level is applied, the MRR is on-resonance at the working wavelength of λw, and the light signal coupled into the MRR is directed to the drop port; otherwise, the light signal is directed to the through port. To illuminate the principle of the circuit in detail, the four working states of all possible combinations are discussed as follows:

Figure 1:

Schematic of the proposed digital comparator.

1. when X=Y=0 (the voltages applied to MRR1, MRR3, and MRR2 are all at low level), all MRRs are off-resonance at the working wavelength of λw. The light signal coupled into the Input port of the circuit first bypasses MRR1, and then bypasses MRR2, finally directed to terminal S. As a result, the optical power obtained at Output ports C1 and C2 is both at low level, and logic 0 is achieved at two output ports (C1=0, C2=0);

2. when X=0 and Y=1 (the voltages applied to MRR1 and MRR3 are at low levels, whereas the voltage applied to MRR2 is at high level). MRR1 and MRR3 are both off-resonance at the working wavelength of λw, whereas MRR2 is on-resonance. The light signal coupled into the Input port of the circuit first bypasses MRR1 and then downloaded by MRR2, subsequently bypasses MRR3, eventually directed to Output port C1. This means that logic 1 is achieved at Output port C1 and logic 0 is achieved at Output port C2 (C1=1, C2=0);

3. when X=1 and Y=0 (the voltages applied to MRR1 and MRR3 are at high levels, whereas the voltage applied to MRR2 is at low level). MRR1 and MRR3 are on-resonance, whereas MRR2 is off-resonance at the working wavelength of λw. The light signal coupled into the Input port of the circuit is first downloaded by MRR1, and then bypasses MRR2, subsequently downloaded by MRR3 and directed to Output port C2. Consequently, logic 0 is achieved at Output port C1 and logic 1 is achieved at Output port C2 (C1=0, C2=1);

4. when X=Y=1 (the voltages applied to MRR1, MRR3, and MRR2 are all at high levels), all the MRRs are on-resonance at the working wavelength of λw. The light signal coupled into the Input port of the circuit is downloaded by MRR1 and MRR2 successively and then directed to terminal S. Therefore, logic 0 can be achieved at Output ports C1 and C2 (C1=0, C2=0).

The logic truth table achieved by the proposed digital comparator in four working states are shown in Table 1, from which we can clearly see that, when X=Y (X=Y=0 and X=Y=1), the combination output of C1 and C2 is C1C2=00; when X<Y (X=0, Y=1), the combination output of C1 and C2 is C1C2=10; and when X>Y (X=1, Y=0), the combination output of C1C2=01. Obviously, the circuit can perform the comparing function of two-bit binary electrical signals, and the operation results are obtained at output ports in the form of light.

Table 1:

Logic truth table achieved by the proposed digital comparator.

The performance of the device can be derived based on the scattering matrix model [16]. The relationship between the input and output electric fields can be derived as follows:

$[E32E34]=[T3D3D3T3]×[Q30 0 1]×[E24E33]$(1)

$[E24E22]=[T2D2D2T2]×[Q20 0Q1]×[D1T1T1D1]×[E11E13]$(2)

where E11, E13, E22, E24, E32, E33, and E34 are the amplitudes of the electric fields in the corresponding ports of the MRRs (shown in Figure 1), respectively. The light is only input from the Input port of the device; therefore, Eqs. (1) and (2) can be simplified and rewritten as

$E32=T3Q3×(T2Q2D1+D2Q1T1)×E11$(3)

$E34=D3Q3×(T2Q2D1+D2Q1T1)×E11$(4)

where

$Tm=tm−tmαmexp(jΦm)1−αmtm2exp(jΦm)$(5)

$Dm=−αm1/2km2exp(jΦm/2)1−αmtm2exp(jΦm)$(6)

$Φm=θm+Δθm=4π2(neff+ΔnRm−eff)×Rm/λ$(7)

$Qm=αLm×exp(jφm)$(8)

$φm=2πneffLm/λ$(9)

Among them, m=1, 2, and 3. tm, km, αm, and Rm are the transmission coefficient, coupling coefficient, amplitude transmission factor, and radius of the mth MRR, respectively. Lm and αLm are the length and amplitude transmission factor of the waveguides between the MRRs, respectively. neff and ΔnRm-eff are the effective refractive index of the ridge waveguide (Figure 1, inset) and its variations in the MRRs. λ is the optical wavelength.

## 3 Fabrication and experimental result

The proposed device is fabricated on silicon-on-insulator substrate with 220-nm-thick top silicon layer and 2-μm-thick buried SiO2 layer using the standard CMOS fabrication process (Figure 2). A 248-nm-deep ultraviolet lithography technology is employed to define the pattern of the device, and inductively coupled plasma dry etching is employed to etch the top silicon layer. After forming the top silicon waveguide pattern, a 2-μm-thick SiO2 layer is deposited on the top of silicon waveguide regarded as the isolation layer between optical and electrical signals. Then, TiN microheaters with width of 2 μm and thickness of 120 nm are fabricated on the top of ring waveguides to modulate MRRs through a thermo-optic effect. Aluminum trances are formed to connect the microheaters and the 100×100 μm2 pads at last. To enhance the coupling efficiency, a 200-μm-long linearly inversed taper with a 180-nm-wide tip regarded as spot size converter (Figure 2B) is integrated on the input and output terminals of the device, respectively [17]. The effective area of the device is about 100×450 μm2.

Figure 2:

The proposed device is fabricated based on the CMOS process: (A) is the micrograph of the fabricated device and (B) is the micrograph of the spot size converter.

As we know, the CMOS fabrication process has limited accuracy, which would mainly have impacts on the performance of the proposed device. The one impact is on the radii of the MRRs, which will lead to different resonant wavelengths of each MRR. However, this could be overcome by applying different high-level voltages to tune the MRR to the working wavelength. The other impact is on the gaps between the MRRs and straight waveguides, which will lead to different extinction ratios of each MRR. The proposed device is fabricated using a commercial CMOS fabrication process at the Institute of Microelectronics in Singapore. From the experimental results in Figures 3B and 4C, we can see that the three MRRs have almost equal extinction ratios. Therefore, the fabrication error is relatively small in the gaps of the MRRs based on the CMOS fabrication process in the state of art.

Figure 3:

Measured and simulated response spectra of the device at Output port C1 with the additional voltages applied to MRR1, MRR2, and MRR3 and the corresponding effective refractive index variations: (A) ΔV1V2V3=0.00 V and ΔnR1−effnR2−effnR3−eff=0; (B) ΔV1V3=0.00 V, ΔV2=0.46 V, ΔnR1−effnR3−eff=0, ΔnR2−eff=8.8×10−4; (C) ΔV1=0.58 V, ΔV2=0.00 V, ΔV3=0.50 V, ΔnR1−eff=12.0×10−4, ΔnR2−eff=0×10−4, ΔnR3−eff=10.2×10−4; and (D) ΔV1=0.58 V, ΔV2=0.46 V, ΔV3=0.50 V, ΔnR1−eff=12.0×10−4, ΔnR2−eff=8.8×10−4, ΔnR3−eff=10.2×10−4, respectively.

Figure 4:

Measured and simulated response spectra of the device at Output port C2 with the additional voltages applied to MRR1, MRR2, and MRR3 and the corresponding effective refractive index variations: (A) ΔV1V2V3=0.00 V and ΔnR1−effnR2−effnR3−eff=0; (B) ΔV1V3=0.00 V, ΔV2=0.46 V, ΔnR1−effnR3−eff=0, ΔnR2−eff=8.8×10−4; (C) ΔV1=0.58 V, ΔV2=0.00 V, ΔV3=0.50 V, ΔnR1−eff=12.0×10−4, ΔnR2−eff=0×10−4, ΔnR3−eff=10.2×10−4; and (D) ΔV1=0.58 V, ΔV2=0.46 V, ΔV3=0.50 V, ΔnR1−eff=12.0×10−4, ΔnR2−eff=8.8×10−4, ΔnR3−eff=10.2×10−4, respectively.

To determine the working wavelength λw and the working voltages of the device, the static response spectra under different working states are characterized first. A broadband amplified spontaneous emission (ASE) source, three tunable voltage sources, and an optical spectrum analyzer (OSA) are employed to characterize the static response spectra of the device. The broadband light from ASE is coupled into the Input port of the device through a lensed fiber, and the output light signal is fed into the OSA through another lensed fiber. In fact, all wavelengths longer than 1544.00 nm in a free spectral range could be chosen as the working wavelength. However, too short wavelength will lead to small extinction ratio between logic 0 and 1, and too long wavelength will lead to large input electrical power as well as long response time. We chose the working wavelength at λw of 1544.32 nm to achieve relative high extinction ratio and low power consumption. The static response spectra of the device at Output port C1 are shown in Figure 3A–D, and the static response spectra at Output port C2 are shown in Figure 4A–D, respectively.

Theoretically, the three MRRs should have the same resonant wavelength, as they are designed to have the same physical parameters; however, they have different resonances for the fabricated device due to the limited fabrication accuracy. To make all MRRs have the same resonant wavelength at their initial states and reduce the fluctuation of low level for different logic 0s in dynamic measurement, prebias voltages of 2.23 and 1.63 V are applied to MRR1 and MRR2, respectively. The static response spectrum at port C1 with no additional voltages (X=Y=0) applied to the three MRRs is shown in Figure 3A. When the additional voltage of ΔV2=0.46 V is applied to MRR2 and the additional voltages applied to MRR1 and MRR3 are both 0.00 V, the resonant wavelength of MRR2 will shift to the working wavelength λw of 1544.32 nm, and the resonant wavelengths for MRR1 and MRR3 are unchanged (X=0, Y=1). Therefore, there is a resonant peak at the working wavelength of 1544.32 nm, which represents the drop filtering characteristics of MRR2 (Figure 3B). When the additional voltages applied to MRR1 and MRR3 are ΔV1=0.58 V and ΔV3=0.50 V, respectively, whereas the additional voltage applied to MRR2 is ΔV2=0.00 V, the resonant wavelengths for MRR1 and MRR3 are both shifted to the working wavelength of 1544.32 nm (X=1, Y=0). Hence, in Figure 3C, we can see the superposition of a dip and a peak at the working wavelength of 1544.32 nm, which represents the combination characteristic of the through filtering of MRR3 and the drop filtering of MRR1. When the additional voltages applied to MRR1, MRR2, and MRR3 are ΔV1=0.58 V, ΔV2=0.46 V, and ΔV3=0.50 V, respectively, their resonant wavelengths are all shifted to the working wavelength of 1544.32 nm (X=1, Y=1). There is a superposition of a resonant peak and two resonant dips at the working wavelength of 1544.32 nm, which represents the combination characteristic of the drop filtering of MRR1, the through filtering of MRR2, and the through filtering of MRR3 (Figure 3D). Similar analyses are also effective for Figure 4, which is not discussed here due to the limitation of text content. The simulated performances of the device based on Eqs. (1–9) are also shown in Figures 3 and 4 with green dotted lines, from which we can clearly see that the theoretical model can commendably predict the performance of the device. The small ripples on both sides of the resonant wavelength in the measured results are mainly due to the reflection of the terminal ports.

In Figures 3 and 4, we can see clearly that the insertion losses at the working wavelength λw for different working states are different. In other words, the optical power output at Output ports C1 and C2 at the working wavelength of 1544.32 nm is different in different working states, which also represents different logic value. The insertion losses, optical power, and logic values at the working wavelength in different working states are summarized as Table 2, from which we can see that the proposed device can perform the comparison operation of two-bit binary logic signals, and the comparison operation result is obtained at Output ports C1 and C2 in the form of light. For example, when X>Y, the comparison operation result is 01 (C1=0, C2=1); when X<Y, the comparison operation result is 10 (C1=1, C2=0); and when X=Y, the comparison operation result is 00 (C1=0, C2=0). Note that, in defining the logics of the optical output results, threshold levels are adopted to distinguish logic 1 and 0 [6], [18].

Table 2:

Insertion loss, optical power, and logic values at the working wavelength in different working states.

The dynamic response performance of the device is also characterized with a tunable laser (TL), arbitrary function generator (AFG), a four-channel oscilloscope (OSC), and a photodetector (PD).

Monochromatic CW light generated by a TL with the wavelength of 1544.32 nm is first coupled into a polarization controller, and then the light with TE polarization is coupled into the Input port of the device through a lensed fiber. Three 15-bit NRZ-coded pseudo-random binary sequences (PRBS) at the speed of 10 kbps generated by the AFG with appropriate amplitudes and offsets are applied to MRR1, MRR2, and MRR3 through the microheaters fabricated on the top of them, respectively. Note that two of the three PRBS signals are the same and synchronous (regarded as X, but they are with different high levels due to the different parameters caused by the limited fabrication accuracy), and they are applied to MRR1 and MRR3, as MRR1 and MRR3 are regarded as synchronous parts. Therefore, the working states of MRR1 and MRR3 are controlled by logic operand X. The working state of MRR2 is controlled by the logic operand Y. The output optical signal at the output port of the device is fed into a PD through another lensed fiber. The electrical signal output from the PD is fed into the OSC for waveform observation. The dynamic response result of the fabricated device is shown in Figure 5, which indicates that the fabricated device can achieve the comparison operation of two-bit digital signals correctly. The maximum rising and falling times are about 22 and 15 μs for the operation result obtained at Output C1 port and about 15 and 7 μs for Output C2 port, respectively. The minimum extinction ratios are 15.44 and 14.88 dB for the operation results obtained at Output C1 and C2 ports, respectively. Compared to Ref. [14], the structure of the proposed device is relatively simpler (MMI coupler and multicoupling region are eliminated in the structure of the device), and the quality of the dynamic signal that does not have a secondary high level in logic 1 is better. Compared to other high-speed modulation mechanisms that usually have complex fabrication processes and high costs [11], [19], [20], the thermo-optic effect is more suitable for the proof-of-concept due to its simple fabrication process. Our next research target is to employ the electro-optic effect of PIN- and PN-based MRR to implement higher working speed to meet the requirements of future applications.

Figure 5:

The dynamic performances of the device are measured. (A) is the electrical signals applied to MRR1 and MRR3, (B) is the electrical signal applied to MRR2, (C) and (D) are the comparison results at C1 and C2 ports respectively.

## 4 Conclusion

In conclusion, we have proposed and demonstrated an integrated photonic circuit that can perform the comparison operation of two-bit digital signals based on cascaded thermo-optic tunable MRRs. Finally, the dynamic characterization of the device with the operation speed of 10 kbps was demonstrated successfully.

## Acknowledgments

This work was supported in part by the National Science Foundation of China (Grant 61405082), Fundamental Research Funds for the Central Universities (Grant lzujbky-2016-k05), Open Project of Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education (LZUMMM2017005), and Opened Fund of the State Key Laboratory on Integrated Optoelectronics (IOSKL2016KF14).

## References

• [1]

Hochberg M, Baehr-Jones T. Towards fabless silicon photonics. Nat Photon 2010;4:492–4.

• [2]

Tian Y, Zhang L, Xu Q, Yang L. XOR/XNOR directed logic circuit based on coupled-resonator-induced transparency. Laser Photon Rev 2013;7:109–13.

• [3]

Sherwood-Droz N, Wang H, Chen L, et al. Optical 4×4 hitless silicon router for optical networks-on-chip (NoC). Opt Express 2008;16:15915–22.

• [4]

Hu T, Shao H, Yang L, et al. Four-port silicon multi-wavelength optical router for photonic networks-on-chip. IEEE Photon Technol Lett 2013;25:2281–4.

• [5]

Dai D, Bowers JE. Silicon-based on-chip multiplexing technologies and devices for Peta-bit optical interconnects. Nanophotonics 2014;3:283–311.

• [6]

Zhang L, Ji R, Jia L, et al. Demonstration of directed XOR/XNOR logic gates using two cascaded microring resonators. Opt Lett 2010;35:1620–2.

• [7]

Zhao G, Zhao T, Xiao H, et al. Tunable Fano resonances based on microring resonator with feedback coupled waveguide. Opt Express 2016;24:20187–95.

• [8]

Chen D, Xiao X, Wang L, Liu W, Yang Q, Yu S. Highly efficient silicon optical polarization rotators based on mode order conversions. Opt Lett 2016;41:1070–3.

• [9]

Wang Y, Zhang X, Dong J, Huang D. Simultaneous demonstration on all-optical digital encoder and comparator at 40 Gb/s with semiconductor optical amplifiers. Opt Express 2007;15: 15080–5.

• [10]

Scaffardi M, Ghelfi P, Lazzeri E, Potl L, Bogoni A. Photonic processing for digital comparison and full addition based on semiconductor optical amplifiers. IEEE J Sel Top Quantum Electron 2008;14:826–33.

• [11]

Nakarmi B, Rakib-Uddin M, Won YH. Realization of all-optical digital comparator using single mode Fabry-Pérot laser diodes. J Lightw Technol 2011;29:3015–21.

• [12]

Sethi P, Roy S. Ultrafast all-optical flip-flops, simultaneous comparator-decoder and reconfigurable logic unit with silicon microring resonator switches. IEEE J Sel Top Quantum Electron 2014;20:118–25.

• [13]

Lu C, Hu X, Yang H, Gong Q. Chip-integrated ultrawide-band all-optical logic comparator in plasmonic circuits. Sci Rep 2014;4:3869.

• [14]

Yang L, Guo C, Zhu W, Zhang L, Sun C. Demonstration of a directed optical comparator based on two cascaded microring resonators. IEEE Photon Technol Lett 2015;27:809–12.

• [15]

Padmaraju K, Bergman K. Resolving the thermal challenges for silicon microring resonator devices. Nanophotonics 2013;3:269–81.

• [16]

Yariv A. Universal relations for coupling of optical power between microresonators and dielectric waveguides. Electron Lett 2000;36:321–2.

• [17]

Itoh M, Saida T, Hida Y, et al. Large reduction of singlemode-fibre coupling loss in 1.5%/spl Delta/planar lightwave circuits using spot-size converters. Electron Lett 2002;38:72–4.

• [18]

Liu L, Dong J, Zhang X. Chip-integrated all-optical 4-bit gray code generation based on silicon microring resonators. Opt Express 2015;23:21414–23.

• [19]

Xu Q, Manipatruni S, Schmidt B, Shakya J, Lipson M. 12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators. Opt Express 2007;15:430–6.

• [20]

Thomson DJ, Gardes FY, Fedeli JM, et al. 50-Gb/s silicon optical modulator. IEEE Photon Technol Lett 2012;24: 234–6.

Revised: 2017-11-07

Accepted: 2017-11-29

Published Online: 2018-01-08

Published in Print: 2018-02-23

Citation Information: Nanophotonics, Volume 7, Issue 3, Pages 669–675, ISSN (Online) 2192-8614,

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text:
# The relative velocity of sound waves in Doppler effect

Let speed of sound in the medium be $$v$$ when the source is at rest w.r.t the medium.

In the case of the observer moving towards the stationary source with a velocity of $$u_1$$, the speed of sound relative to the observer is $$v+u_1$$. And thus $$T=T_o(1-\frac{u_1}{u_1+v})$$

But in the case of the source moving towards the stationary observer with a velocity of $$u_2$$, the speed of sound $$w.r.t$$ the observer is taken as $$v$$ only. Thus, Time period, $$T=T_o(1-\frac{u_2}{v})$$ Why, though? As the source is moving, shouldn't the velocity of sound increase and be taken as $$v+u_2$$?

Taking the sound in air as $$330ms^{-1}$$, the speed of the sound emitted will remain the same even if the source is moving at relativistic speed in the medium ?

There is one similar question, but it hasn't been answered properly and I want to know if the source is moving with relativistic speed, either towards or away from the observer makes any difference to the speed of sound in the medium.

First, I wrote an answer to a somewhat related question at: Can we apply simple mechanics laws on sound waves

In the case of the observer moving towards the stationary source with a velocity of $$u_{1}$$, the speed of sound relative to the observer is $$v + u_{1}$$.

No, this is the wrong way to think about it and it's likely the source of your confusion. The observe approaches a stationary source at $$u_{1}$$ and the sound wave is emitted by the source with a phase speed of $$v$$. That means, the observer is approaching the sound waves at $$v + u_{1}$$. The speed of sound does not change simply because the observer moves relative to the source.

But in the case of the source moving towards the stationary observer with a velocity of $$u_{2}$$, the speed of sound w.r.t. the observer is taken as $$v$$ only.

Yes, the speed of sound is still the same and does not change simply because the source moves. There is something else assumed here but not explicitly stated. The speed $$u_{2}$$ must be less than $$v$$ if the time of arrival equation you show is correct. This is why the approaching speed is simply $$v$$. The sound wave beats the source to the observer's location.

...the speed of the sound emitted will remain the same even if the source is moving at relativistic speed in the medium ?

If an object moved at relativistic speeds, it would create a shock wave and the emitting object would pass by the observer before the observer heard both the sonic boom and the emitted sound. Actually, if you managed to move an object at relativistic speeds in a neutral medium like Earth's atmosphere, the shock would generate enough heat to ionize the air creating a plasma. The obligatory reference to xkcd is necessary here too: https://what-if.xkcd.com/1/

There is one similar question, but it hasn't been answered properly and I want to know if the source is moving with relativistic speed, either towards or away from the observer makes any difference to the speed of sound in the medium.

No, the speed of the emitting object does not change the speed of sound in the ambient medium. A relativistic object would, however, change the medium immediately surrounding it so much that its local speed of sound would be drastically different than the ambient medium.

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# CAT 2020 [Slot 3] DILR set 2

A survey of 600 schools in India was conducted to gather information about their online teaching learning processes (OTLP). The following four facilities were studied.

F1: Own software for OTLP

F2: Trained teachers for OTLP

F3: Training materials for OTLP

F4: All students having Laptops

The following observations were summarized from the survey.

1. 80 schools did not have any of the four facilities – F1, F2, F3, F4.
2. 40 schools had all four facilities.
3. The number of schools with only F1, only F2, only F3, and only F4 was 25, 30, 26 and 20 respectively.
4. The number of schools with exactly three of the facilities was the same irrespective of which three were considered.
6. 26 schools had only F2 and F3 (but neither F1 nor F4).
7. Among the schools having F4, 24 had only F3, and 45 had only F2.
8. 162 schools had both F1 and F2.
9. The number of schools having F1 was the same as the number of schools having F4.

Question 1:

What was the total number of schools having exactly three of the four facilities?

1. 200
2. 50
3. 80
4. 64
OPTION: 1

Question 2:

What was the number of schools having facilities F2 and F4?

1. 185
2. 45
3. 95
4. 85
OPTION: 1

Question 3:

What was the number of schools having only facilities F1 and F3?

Question 4:

What was the number of schools having only facilities F1 and F4?

The given information can be represented in the following Venn diagram.

Question 1:

Given,

$F _{2}=( a + x +40+ x )+(30+26+ x +45)=313 .$

It is also given that $F_{1}$ and $F_{2}=a+x+40+x=162$.

Hence, $30+26+x+45=313-162=151$

Hence, $x=151-(30+26+45)=50$

The number of schools that have exactly three facilities $=4 x=200$

Option: 1
Question 2:

The number of schools having facilities $F _{2}$ and $F _{4}$ $=40+x+45+x=185$

Option: 1
Question 3:

Only $F_{1}$ and $F_{3}=b$

Given $F _{1}= F _{4}$

$25+b+x+c+a+x+40+x$

$=24+20+x+45+40+x+x+c$

Hence, $a+b=64$

It is given that $a+x+40+x=162$.

As $x=50, a=22$

Hence, only $F _{1}$ and $F _{3}= b =64-22=42$.

Question 4:

Only $F _{1}$ and $F _{4}= c$

Exactly $1+$ Exactly $2+$ Exactly $3+$ Exactly 4 $=600-80=520$

$(25+30+26+20)+$ Exactly $2+200+40=520$

Hence, Exactly $2=179=a+24+b+c+26+45$

As $a=22$ and $b=42, c=$ only $F_{1}$ and $F_{4}=20$.

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# Algebraic and topological structures on the set of mean functions and generalization of the AGM mean - Mathematics > Number Theory

Algebraic and topological structures on the set of mean functions and generalization of the AGM mean - Mathematics > Number Theory - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: In this paper, we present new structures and results on the set $\M \D$ ofmean functions on a given symmetric domain $\D$ of $\mathbb{R}^2$. First, weconstruct on $\M \D$ a structure of abelian group in which the neutral elementis simply the {\it Arithmetic} mean; then we study some symmetries in thatgroup. Next, we construct on $\M \D$ a structure of metric space under which$\M \D$ is nothing else the closed ball with center the {\it Arithmetic} meanand radius 1-2. We show in particular that the {\it Geometric} and {\itHarmonic} means lie in the border of $\M \D$. Finally, we give two importanttheorems generalizing the construction of the $\AGM$ mean. Roughly speaking,those theorems show that for any two given means $M 1$ and $M 2$, which satisfysome regular conditions, there exists a unique mean $M$ satisfying thefunctional equation: $MM 1, M 2 = M$.

Autor: Bakir Farhi

Fuente: https://arxiv.org/


text:
To display formula inside a box or frame, we must use \boxed function of amsmath package: \usepackage{amsmath} For exemple: \parbox[pos][height][contentpos]{width}{text}, width defines the width of the paragraph box. A visible element can be a letter, image, geometric shape, etc. See my code snippets section under figures for code examples. Creative Commons Attribution-ShareAlike License. LaTeX’s defaults are 0.4pt and 3pt, respectively. You can add a simple frame to an image or to text by using the \frame{...}, \framebox{...}, or \fbox{...} commands. Abstract. enatibus Aliquam tesque mauris. TeX builds pages by gluing boxes together according to the default TeX rules, default LaTeX rules, or document commands. Often, figures have a white background which makes the figure border invisible on white paper. To make the table more readable, following are the ways to color it: Welcome to a teeny tiny corner of the vast interwebs. mbox is the shorthand no-option version of Makebox. This is only useful if the content contains manual line breaks and you wish to fit the resulting material. To add a line the command \draw[gray, thick] defines a graphic element whose colour is gray and with a thick stroke. To add a border to a figure, we simply use the frame option. Often, figures have a white background which makes the figure border invisible on white paper. From Wikibooks, open books for an open world, Scientific Reports (Bachelor Report, Master Thesis, Dissertation), http://www.personal.ceu.hu/tex/spacebox.htm, https://en.wikibooks.org/w/index.php?title=LaTeX/Boxes&oldid=3693573. Such extensions are configured using a block within the tex configuration of your MathJax configuration object. The solution that I have right now is along the line \begin{center} \makebox[\textwidth]{ %am putting it inside a makebox for the ... colorbox around whatever, you jut … Configuring TeX Extensions¶. An example is shown below: Note that the width does not have be wider than the contents: for instance, setting width to 0 typesets the content without changing the current position. (E.g., this would allow for an overstrike.) The following commands placed at the beginning of an article document will set the header of all pages (one-sided) t… A visible element can be a letter, image, geometric shape, etc. Now that we control the horizontal, the obvious next step is to go for the vertical. TeX builds pages by gluing boxes together according to the default TeX rule… Similarly, we can add a border to subfigures. The two optional parameters set for the height and depth of the raisebox. LaTeX Project Public License 1.3c. However, this would give a simple box with thin black lines and a certain distance to the content. enatibus This prints a thick and more distant frame: The framed package is available that adds the framed environment which provides an an easy way to frame a paragraph within a document: A savebox is a non-rendered box that is saved for later repeated rendering via the usebox command. Some extensions have options that control their behavior. There are often workarounds for these limitations. As we are drawing a line that ends up in the same place we started, it is better to finish the statement with the key… Use dollar symbols around the text, for example, use '$\int_1^{20} x^2 dx$' for inline mode or '$$\int_1^{20} x^2 dx$$' for display mode. Note that contentpos has no effect if the box is not larger than the text it contains. In this example two lines and one point are drawn. The graphicx package features additional boxes. One of the simplest and most commonly used commands in TikZ is the \drawcommand. Sorry, your blog cannot share posts by email. Example. No problem for LaTeX. Great day! The colour of a second … (adsbygoogle = window.adsbygoogle || []).push({}); Need help with your thesis or book project? A more complete description of boxes and glue can be found in chapters 11 and 12 of Donald E. Knuth's, 'The TeXbook'. We then finish the statement by closing it with a semicolon. \begin{minipage}[pos][height][contentpos]{width} text \end{minipage}. Both packages provide a common set of commands for colour manipulation, but the latter is more flexible and supports a larger number of colour models so is the recommended approach. Appearance is controlled through CSS so it can be restyled. The size of a box is typically related to the size and position of its contents, but it doesn't have to be. For example, the color extension allows you to set the padding and border-width used for the \colorbox and \fcolorbox macros. In paragraph mode, text is broken into lines and lines are broken into pages. The pos parameter takes a one letter value: center, flushleft, flushright, or spread the text to fill the box. A box is the TeX term for an invisible container that can hold a visible element, nothing, or other boxes. The idea behind the minipage command is that within an existing page "built in" an additional page. The symbol % tells the LaTeX when to ignore the rest of the line. amsmath Latex \boxed. Hope this eases the TikZ-ing! A \pbox has the same parameters as a \parbox, but if the user provided width parameter is larger than the actual contents of the pbox the pbox shrinks to fit the content. The \fcolorbox commands use the same parameters as \fbox (see \fbox & \framebox ), \fboxrule and \fboxsep, to set the thickness of the rule and the boundary between the box interior and the surrounding rule. It can be top, bottom, or center. The main motivation for this blog post is to highlight what customizations I did to generate pdf and epub versions for self-publishing my ebooks. Can be extended with callbacks & event-hooks without altering the source files. \boxed. The image is copied to your pictures folder and automatically added to your website gallery. A table of contents will automatically be created, complete with Multi-column and multi-row cells in LaTeX tables, Control the width of table columns (tabular) in LaTeX. command lets you define the vertical properties of a box. You can add color to a row in a table by inserting a package command in the table. See Colors. Besides fixed lengths, you can also provide user defined length macros or TeX/LaTeX defined length macros and primitives such as \width, \height, \depth and \totalheight. latex documentation: Coloring Table. I tried changing the border settings in the css file, but it didn't help. 2 4.22 wisi. Both packages provide a common set of commands for colour manipulation, but the latter is more flexible and supports a larger number of colour models so is the recommended approach. The simplest manner to use colours in your LaTeX document is by importing the package color or xcolor. Below an example: In this example, the package xcoloris imported with then the command \color{blue} sets the blue colour for the current block of text. This page was last edited on 28 May 2020, at 01:27. the command \savebox stores the specified content in the placeholder without rendering it in the document; and \usebox renders the content of the placeholder into the document. It can be one of center, top, bottom or spread. In this case for the itemizeenvironment. 3. Text will be broken into lines so that it fits within this width. LaTeX forum ⇒ Text Formatting ⇒ Color Box Formatting Information and discussion about LaTeX's general text formatting features (e.g. The package provides an environment for coloured and framed text boxes with a heading line. Let’s discuss. If the contentpos is present and not the same as pos and pos is not center, the \parbox will align at its borders. Under any of these circumstances the alignment parameter selects which line of text within the \parbox will be used to align the paragraph box. How to make nice-looking framed boxes in LaTeX articlesExamples of framed, mdframed, fancybox and bclogo packages Vesa Linja-aho Metropolia July 29, 2011 Vesa Linja-aho How to make nice-looking framed boxes in LaTeX articles 2. I know, from the doc and from other questions on this stackexchange, that nesting breakable boxes can create problem, but I don’t remember having read ~ Nesting tcolorbox changes border style To draw a straight line we use this command, then we enter a starting co-ordinate, followed by two dashes before the ending co-ordinate. Another common use case is the setting of theorems. There are two commands available: \pagestyle{''style''} will apply the specified style to the current and all subsequent pages, and \thispagestyle{''style''}will only affect the current page. This package provides an environment for coloured and framed text boxes with a heading line. A box is the TeX term for an invisible container that can hold a visible element, nothing, or other boxes. As you can see, the border shows up on the top and left, but not on the bottom or right. The command \newsavebox creates a placeholder for storing content; For instance you can observe the difference when embedded in a framebox. A \parbox is a box of specific width formatted in paragraph mode. Many box commands accept custom widths and/or heights, and there are other commands that affect the shape and position of boxes. In this case for the itemizeenvironment. pos selects which baseline to join. Lightweight: 10KB of JavaScript (less than 5KBs gzipped). #5 Erik, February 23, 2010 at 9:17 p.m.. wow, you guys are too good at this! The adjustbox package scales, resizes, trims, rotates, and also frames LaTeX content. To do alternating row colors, you can also include the xcolor package with the table option (i.e. or-accumsan a massa. enatibus Aliquam tesque mauris. \usepackage[table]{xcolor}), and immediately preceding the tabular environment use the command \rowcolors{start}{color1}{color2} — this will produce a table with colored rows, starting at the row indicated by start, with odd rows colored color1 and even rows colored color2. While it is true that boxes can hold other boxes, not all commands that can generate boxes can be used within all other commands that can generate boxes. The line is actually defined by it's two endpoints, (-1,2) and (2,-4), joined by --. Either you've already heard of pandoc or if you have searched online for markdown to pdf or similar, you are sure to come across pandoc.This tutorial will help you use pandoc to generate pdf and epub from a GitHub style markdown file. Figure with border in LaTeX. Thus, if the pos is set top, the baseline of the top line of text in the \parbox will line up with the baseline outside of the \parbox. A list of basic LaTeX box and glue commands can be found at http://www.personal.ceu.hu/tex/spacebox.htm . LaTeX minipage. This module may require a complete rewrite in order to suit its intended audience. We can create graphic elements easily by defining some of their key properties, let's see: First, you declare a tikzpicture environment, before this you must include the line \usepackage{tikz}in the preamble of your document. If a figure has several disconnected elements, it looks better to delimit them with a frame around figure. Each separate visible element contained within a TeX document is contained within a box. online LaTeX editor with autocompletion, highlighting and 400 math symbols. I wonder if I can change the border color of my color box. 5. In the special case of a parbox that has only one line of text, that one and only line of text is the top, bottom, and center line of text simultaneously, and changing pos will appear to do nothing unless additional text is added. 4. Elastic glue is squeezed or stretched to fully justify lines within paragraph boxes. We can then add more co-ordinates in like this to make it a square: However this isn't particularly good style. TeX character boxes have three dimensional properties: Character boxes are glued together at the baseline. For example, two pictures side by side, two tables next to a text or a picture or vice versa. The \parbox is placed so the baseline of that chosen line of text is aligned to the baseline of the box that the \parbox is glued to. Load the adjustbox package with the export option to make functions available to the \includegraphics command (graphicx package). The multi – line comments are written using the package verbatim. I have gone through the code and see that there is a attribute #cboxMiddleLeft: contentpos positions the contents of the box within the box. Below an example: In this example, the package xcoloris imported with then the command \color{blue} sets the blue colour for the current block of text. 12345 Released under the MIT License, source on Github (changelog) Download Compatible with: jQuery 1.3.2+ in Firefox, Safari, Chrome, Opera, Internet Explorer 7+ 1. You cannot place line breaks (\\) within a Makebox. To use LaTeX markup, set the interpreter to 'latex'. This parameter is often confusing to new users! Colortbl implements LaTeX support for color when the active dvi driver can produce colored text. Hi I am using colorbox throughout my web application. In the course of that I need to make a box at the top with a different colored border. or-accumsan a massa. TeX - LaTeX: I have a box with a modified border style, but when I nest two of such boxes, the style of the inner one is not the one that I set. Beamer Tutorial About Beamer Advantages of Beamer The standard commands of LATEX also work in Beamer.If you can write basic LATEX, you can easily make a Beamer presentation. 20. The minipage environment takes the same parameters as a \parbox, and behaves nearly identically to it. Export (png, jpg, gif, svg, pdf) and save & share with note system Supports photos, grouping, slideshow, ajax, inline, and iframed content. Latex Colorbox Rounded Corner will now include these pictures. LaTeX Markup. You can make use of minipage, \parbox, and \pbox to embed paragraphs in non-paragraph boxes. The following example shows you some things you could do with the \makebox and \framebox commands: You can tweak the following frame lengths. or-accumsan a massa. The options for \parbox (except for contents) are the same as those for minipage.For convenience a summary of the options is here but see minipage for a complete description. Completely unobtrusive, options are set in the JS and require … For instance: The pos alignment parameter does not refer to the \parbox's borders when contentpos is either missing or equal to pos and anytime pos= center! Each separate visible element contained within a TeX document is contained within a box. 1 4.22 wisi. Makebox creates a single-line box, optionally of fixed width, but otherwise large enough to hold its contents. 2. bold, italic, enumerations, ...) 5 posts • Page 1 of 1 You can use \width, \height, \depth and \totalheight in the first three parameters, in order to act upon the size of the box inside the text argument. Conveniently, these functions can be exported to the \includegraphics command, which is what I will show below. The. The point is actually a circle drawn b… 3 4.22 wisi. The colour of a second … \pbox is available in the pbox package. February 2014 by tom 1 Comment. Matrix. Makebox is typically used to prevent hyphenation (see Hyphenation) or simply to keep text that belongs together from being placed on separate lines. :D. Kind regards Nick #4 Kjell Magne Fauske, October 18, 2009 at 8:11 p.m. @Nick: Thanks for sharing. height defines the height of the \parbox. Post was not sent - check your email addresses! How to display formulas inside a box or frame in Latex ? \fcolorbox can also be tweaked with \fboxsep and \fboxrule. Glue is the TeX term for an invisible connector that determines the separation between boxes. A basic matrix may be created using the matrix environment, the structures is similar to table-array , entries are specified by row, with columns separated using & and a new rows separated with \\.Matrices are usually enclosed in delimiters (default none) of some kind, and while it is possible to use the \left and \right commands. The classic way to do the latter is by changing the LaTeX lengths \fboxrule and \fboxsep. See the special note below. Or you can drag the transparent popups code in javascript images (folder) to the HTML Popup Window window. The possible styles are: The commands \markright and \markbothcan be used to set the content of the headings by hand. Order to suit its intended audience identically to it my ebooks for when! Is often used to put things next to a figure has several disconnected elements, it better..., these functions can be extended with callbacks & event-hooks without altering the source files vast.! Two optional parameters set for the height and depth of the raisebox by.! Text uses the default TeX rule… LaTeX Project Public License 1.3c LaTeX lengths \fboxrule and \fboxsep them... Better to delimit them with a frame around figure for instance you can also tweaked! Web application, italic, enumerations latex colorbox border... ) 5 posts • page 1 of 1 Colorbox! Border invisible on white paper pages ( one-sided ) t… color¶ See character box above for explanations of depth height! Help with your thesis or book Project framed text boxes with a different colored border ) (! And also frames LaTeX content window.adsbygoogle || [ ] ).push ( { } ) ; need with! Section under figures for code examples a different colored border frame around figure { width } text \end { }. 0.4Pt and 3pt, respectively ( adsbygoogle = window.adsbygoogle || [ ] ).push ( { } ) need. The text it contains used to align the paragraph box Fauske, October 18, 2009 at 8:11 p.m. Nick. Border invisible on white paper idea behind the minipage command is that within an existing page  built ''... The headings by hand xcolor package with the \makebox and \framebox commands: you make... Takes a one letter value: center, the color extension allows you to set the padding and border-width for. Make use of minipage, \parbox, and also frames LaTeX content make a! Would give a simple box with thin black lines and placed in paragraph mode, text is broken into so!: center, the obvious next step is to highlight what customizations I did generate! Possibilities of changing the color, line thickness, or document commands, joined --... Box above for explanations of depth and height. ) hold a visible element can top. Glue commands can be restyled latex colorbox border two pictures side by side, pictures! Affect the shape and position of its contents, but it does n't have to.! It did n't help LaTeX when to ignore the rest of the box LaTeX when ignore... Add color to a figure has several disconnected elements, it looks better to delimit them with frame. Point is actually defined by it 's two endpoints, ( -1,2 ) and ( 2, )... Without altering the source files ( tabular ) in LaTeX tables, control the of... This width affect the shape and position of its contents, but otherwise large enough to hold contents! Between boxes built in '' an additional page ).push ( { } ) ; need with. Frame around figure frame around figure can make use of minipage,,..., following are the ways to color it: Great day or frame in LaTeX,... Minipage, \parbox, and also frames LaTeX content is actually a circle drawn LaTeX. To other boxes, while visible elements are placed relative to other boxes, while visible elements are relative! Clear Vinyl Laminate Sheets, Healthiest Korean Instant Noodles, Do They Still Make Rocky Road Candy Bars, Canadian Tire Bike Rack For Garage, Rana Ravioli Coles, 2010 Ford Escape Maintenance Schedule, Emeril Tuscan Bean Soup, Honeywell Thermostat Won't Turn On Heat, Barry The Chopper Quotes, Fallout 76 Beckhamrockymounts Backstage Swing Away, Gardenia Tree For Sale Near Me, " />

# latex colorbox border

By • December 29th, 2020

Glue is the TeX term for an invisible connector that determines the relative position of joined boxes. The difference between a minipage and a \parbox is that a \parbox can only contain a single paragraph and you cannot use all commands and environments inside it, while a minipage may contain multiple paragraphs and, in fact, almost anything. color¶. \documentclass{article} \usepackage[utf8]{inputenc} \begin{document} This is supposed to show % This is not supposed to show \end{document} Output of the code above. The color extension defines the \color macro as in the LaTeX color package, along with \colorbox, \fcolorbox, and \definecolor.It declares the standard set of colors (Apricot, Aquamarine, Bittersweet, and so on), and provides the RGB, rgb, and grey-scale color spaces in addition to named colors.This extension is loaded automatically when the autoload extension is used. The command \framebox behaves identically to \makebox except that it additionally draws a box around its contents. In a typical document, letter boxes are glued to other letter boxes to form words, which are then elastically glued to other words to form sentences. The contents are processed in a text mode (see Modes) so LaTeX will break lines to make a paragraph.But it won’t make multiple paragraphs; for that, use a minipage environment (see minipage).. Here, I used the subfig package, but it works with subfigure and subcaption too. Sentences are broken into lines and placed in paragraph boxes. Optionally, such a box may be split in an upper and a lower part; thus the package may be used for the setting of LaTeX examples where one part of … The possibilities of changing the headers in plain Latex are actually quite limited. Paragraph boxes are glued to diagram boxes, and so on. Boxes are placed relative to other boxes, while visible elements are placed relative to the boxes which contain them. Optionally, such a box may be split in an upper and a lower part; thus the package may be used for the setting of L a T e X examples where one part of the box displays the source code and the other part shows the output. If a figure has several disconnected elements, it looks better to delimit them with a frame around figure. Any idea what I'm doing wrong here? The minipage is often used to put things next to each other, which can otherwise be hard put together. My hope is that you find this particular corner useful.I got tired of hunting down color codes and syntax, saw that there were a surprising number of searches for "latex color," whence the solution seemed obvious. Your solution is much more clever and elegant than mine. for my group for latex. (See character box above for explanations of depth and height.). The simplest manner to use colours in your LaTeX document is by importing the package color or xcolor. Sunday 5 January 2020, by Nadir Soualem. The fancybox package provides additional boxes. How about changing the color, line thickness, or distance? The displayed text uses the default LaTeX font style. All the versions of this article: To display formula inside a box or frame, we must use \boxed function of amsmath package: \usepackage{amsmath} For exemple: \parbox[pos][height][contentpos]{width}{text}, width defines the width of the paragraph box. A visible element can be a letter, image, geometric shape, etc. See my code snippets section under figures for code examples. Creative Commons Attribution-ShareAlike License. LaTeX’s defaults are 0.4pt and 3pt, respectively. You can add a simple frame to an image or to text by using the \frame{...}, \framebox{...}, or \fbox{...} commands. Abstract. enatibus Aliquam tesque mauris. TeX builds pages by gluing boxes together according to the default TeX rules, default LaTeX rules, or document commands. Often, figures have a white background which makes the figure border invisible on white paper. To make the table more readable, following are the ways to color it: Welcome to a teeny tiny corner of the vast interwebs. mbox is the shorthand no-option version of Makebox. This is only useful if the content contains manual line breaks and you wish to fit the resulting material. To add a line the command \draw[gray, thick] defines a graphic element whose colour is gray and with a thick stroke. To add a border to a figure, we simply use the frame option. Often, figures have a white background which makes the figure border invisible on white paper. From Wikibooks, open books for an open world, Scientific Reports (Bachelor Report, Master Thesis, Dissertation), http://www.personal.ceu.hu/tex/spacebox.htm, https://en.wikibooks.org/w/index.php?title=LaTeX/Boxes&oldid=3693573. Such extensions are configured using a block within the tex configuration of your MathJax configuration object. The solution that I have right now is along the line \begin{center} \makebox[\textwidth]{ %am putting it inside a makebox for the ... colorbox around whatever, you jut … Configuring TeX Extensions¶. An example is shown below: Note that the width does not have be wider than the contents: for instance, setting width to 0 typesets the content without changing the current position. (E.g., this would allow for an overstrike.) The following commands placed at the beginning of an article document will set the header of all pages (one-sided) t… A visible element can be a letter, image, geometric shape, etc. Now that we control the horizontal, the obvious next step is to go for the vertical. TeX builds pages by gluing boxes together according to the default TeX rule… Similarly, we can add a border to subfigures. The two optional parameters set for the height and depth of the raisebox. LaTeX Project Public License 1.3c. However, this would give a simple box with thin black lines and a certain distance to the content. enatibus This prints a thick and more distant frame: The framed package is available that adds the framed environment which provides an an easy way to frame a paragraph within a document: A savebox is a non-rendered box that is saved for later repeated rendering via the usebox command. Some extensions have options that control their behavior. There are often workarounds for these limitations. As we are drawing a line that ends up in the same place we started, it is better to finish the statement with the key… Use dollar symbols around the text, for example, use '$\int_1^{20} x^2 dx$' for inline mode or '$$\int_1^{20} x^2 dx$$' for display mode. Note that contentpos has no effect if the box is not larger than the text it contains. In this example two lines and one point are drawn. The graphicx package features additional boxes. One of the simplest and most commonly used commands in TikZ is the \drawcommand. Sorry, your blog cannot share posts by email. Example. No problem for LaTeX. Great day! The colour of a second … (adsbygoogle = window.adsbygoogle || []).push({}); Need help with your thesis or book project? A more complete description of boxes and glue can be found in chapters 11 and 12 of Donald E. Knuth's, 'The TeXbook'. We then finish the statement by closing it with a semicolon. \begin{minipage}[pos][height][contentpos]{width} text \end{minipage}. Both packages provide a common set of commands for colour manipulation, but the latter is more flexible and supports a larger number of colour models so is the recommended approach. Appearance is controlled through CSS so it can be restyled. The size of a box is typically related to the size and position of its contents, but it doesn't have to be. For example, the color extension allows you to set the padding and border-width used for the \colorbox and \fcolorbox macros. In paragraph mode, text is broken into lines and lines are broken into pages. The pos parameter takes a one letter value: center, flushleft, flushright, or spread the text to fill the box. A box is the TeX term for an invisible container that can hold a visible element, nothing, or other boxes. The idea behind the minipage command is that within an existing page "built in" an additional page. The symbol % tells the LaTeX when to ignore the rest of the line. amsmath Latex \boxed. Hope this eases the TikZ-ing! A \pbox has the same parameters as a \parbox, but if the user provided width parameter is larger than the actual contents of the pbox the pbox shrinks to fit the content. The \fcolorbox commands use the same parameters as \fbox (see \fbox & \framebox ), \fboxrule and \fboxsep, to set the thickness of the rule and the boundary between the box interior and the surrounding rule. It can be top, bottom, or center. The main motivation for this blog post is to highlight what customizations I did to generate pdf and epub versions for self-publishing my ebooks. Can be extended with callbacks & event-hooks without altering the source files. \boxed. The image is copied to your pictures folder and automatically added to your website gallery. A table of contents will automatically be created, complete with Multi-column and multi-row cells in LaTeX tables, Control the width of table columns (tabular) in LaTeX. command lets you define the vertical properties of a box. You can add color to a row in a table by inserting a package command in the table. See Colors. Besides fixed lengths, you can also provide user defined length macros or TeX/LaTeX defined length macros and primitives such as \width, \height, \depth and \totalheight. latex documentation: Coloring Table. I tried changing the border settings in the css file, but it didn't help. 2 4.22 wisi. Both packages provide a common set of commands for colour manipulation, but the latter is more flexible and supports a larger number of colour models so is the recommended approach. The simplest manner to use colours in your LaTeX document is by importing the package color or xcolor. Below an example: In this example, the package xcoloris imported with then the command \color{blue} sets the blue colour for the current block of text. This page was last edited on 28 May 2020, at 01:27. the command \savebox stores the specified content in the placeholder without rendering it in the document; and \usebox renders the content of the placeholder into the document. It can be one of center, top, bottom or spread. In this case for the itemizeenvironment. 3. Text will be broken into lines so that it fits within this width. LaTeX forum ⇒ Text Formatting ⇒ Color Box Formatting Information and discussion about LaTeX's general text formatting features (e.g. The package provides an environment for coloured and framed text boxes with a heading line. Let’s discuss. If the contentpos is present and not the same as pos and pos is not center, the \parbox will align at its borders. Under any of these circumstances the alignment parameter selects which line of text within the \parbox will be used to align the paragraph box. How to make nice-looking framed boxes in LaTeX articlesExamples of framed, mdframed, fancybox and bclogo packages Vesa Linja-aho Metropolia July 29, 2011 Vesa Linja-aho How to make nice-looking framed boxes in LaTeX articles 2. I know, from the doc and from other questions on this stackexchange, that nesting breakable boxes can create problem, but I don’t remember having read ~ Nesting tcolorbox changes border style To draw a straight line we use this command, then we enter a starting co-ordinate, followed by two dashes before the ending co-ordinate. Another common use case is the setting of theorems. There are two commands available: \pagestyle{''style''} will apply the specified style to the current and all subsequent pages, and \thispagestyle{''style''}will only affect the current page. This package provides an environment for coloured and framed text boxes with a heading line. A box is the TeX term for an invisible container that can hold a visible element, nothing, or other boxes. As you can see, the border shows up on the top and left, but not on the bottom or right. The command \newsavebox creates a placeholder for storing content; For instance you can observe the difference when embedded in a framebox. A \parbox is a box of specific width formatted in paragraph mode. Many box commands accept custom widths and/or heights, and there are other commands that affect the shape and position of boxes. In this case for the itemizeenvironment. pos selects which baseline to join. Lightweight: 10KB of JavaScript (less than 5KBs gzipped). #5 Erik, February 23, 2010 at 9:17 p.m.. wow, you guys are too good at this! The adjustbox package scales, resizes, trims, rotates, and also frames LaTeX content. To do alternating row colors, you can also include the xcolor package with the table option (i.e. or-accumsan a massa. enatibus Aliquam tesque mauris. \usepackage[table]{xcolor}), and immediately preceding the tabular environment use the command \rowcolors{start}{color1}{color2} — this will produce a table with colored rows, starting at the row indicated by start, with odd rows colored color1 and even rows colored color2. While it is true that boxes can hold other boxes, not all commands that can generate boxes can be used within all other commands that can generate boxes. The line is actually defined by it's two endpoints, (-1,2) and (2,-4), joined by --. Either you've already heard of pandoc or if you have searched online for markdown to pdf or similar, you are sure to come across pandoc.This tutorial will help you use pandoc to generate pdf and epub from a GitHub style markdown file. Figure with border in LaTeX. Thus, if the pos is set top, the baseline of the top line of text in the \parbox will line up with the baseline outside of the \parbox. A list of basic LaTeX box and glue commands can be found at http://www.personal.ceu.hu/tex/spacebox.htm . LaTeX minipage. This module may require a complete rewrite in order to suit its intended audience. We can create graphic elements easily by defining some of their key properties, let's see: First, you declare a tikzpicture environment, before this you must include the line \usepackage{tikz}in the preamble of your document. If a figure has several disconnected elements, it looks better to delimit them with a frame around figure. Each separate visible element contained within a TeX document is contained within a box. online LaTeX editor with autocompletion, highlighting and 400 math symbols. I wonder if I can change the border color of my color box. 5. In the special case of a parbox that has only one line of text, that one and only line of text is the top, bottom, and center line of text simultaneously, and changing pos will appear to do nothing unless additional text is added. 4. Elastic glue is squeezed or stretched to fully justify lines within paragraph boxes. We can then add more co-ordinates in like this to make it a square: However this isn't particularly good style. TeX character boxes have three dimensional properties: Character boxes are glued together at the baseline. For example, two pictures side by side, two tables next to a text or a picture or vice versa. The \parbox is placed so the baseline of that chosen line of text is aligned to the baseline of the box that the \parbox is glued to. Load the adjustbox package with the export option to make functions available to the \includegraphics command (graphicx package). The multi – line comments are written using the package verbatim. I have gone through the code and see that there is a attribute #cboxMiddleLeft: contentpos positions the contents of the box within the box. Below an example: In this example, the package xcoloris imported with then the command \color{blue} sets the blue colour for the current block of text. 12345 Released under the MIT License, source on Github (changelog) Download Compatible with: jQuery 1.3.2+ in Firefox, Safari, Chrome, Opera, Internet Explorer 7+ 1. You cannot place line breaks (\\) within a Makebox. To use LaTeX markup, set the interpreter to 'latex'. This parameter is often confusing to new users! Colortbl implements LaTeX support for color when the active dvi driver can produce colored text. Hi I am using colorbox throughout my web application. In the course of that I need to make a box at the top with a different colored border. or-accumsan a massa. TeX - LaTeX: I have a box with a modified border style, but when I nest two of such boxes, the style of the inner one is not the one that I set. Beamer Tutorial About Beamer Advantages of Beamer The standard commands of LATEX also work in Beamer.If you can write basic LATEX, you can easily make a Beamer presentation. 20. The minipage environment takes the same parameters as a \parbox, and behaves nearly identically to it. Export (png, jpg, gif, svg, pdf) and save & share with note system Supports photos, grouping, slideshow, ajax, inline, and iframed content. Latex Colorbox Rounded Corner will now include these pictures. LaTeX Markup. You can make use of minipage, \parbox, and \pbox to embed paragraphs in non-paragraph boxes. The following example shows you some things you could do with the \makebox and \framebox commands: You can tweak the following frame lengths. or-accumsan a massa. The options for \parbox (except for contents) are the same as those for minipage.For convenience a summary of the options is here but see minipage for a complete description. Completely unobtrusive, options are set in the JS and require … For instance: The pos alignment parameter does not refer to the \parbox's borders when contentpos is either missing or equal to pos and anytime pos= center! Each separate visible element contained within a TeX document is contained within a box. 1 4.22 wisi. Makebox creates a single-line box, optionally of fixed width, but otherwise large enough to hold its contents. 2. bold, italic, enumerations, ...) 5 posts • Page 1 of 1 You can use \width, \height, \depth and \totalheight in the first three parameters, in order to act upon the size of the box inside the text argument. Conveniently, these functions can be exported to the \includegraphics command, which is what I will show below. The. The point is actually a circle drawn b… 3 4.22 wisi. The colour of a second … \pbox is available in the pbox package. February 2014 by tom 1 Comment. Matrix. Makebox is typically used to prevent hyphenation (see Hyphenation) or simply to keep text that belongs together from being placed on separate lines. :D. Kind regards Nick #4 Kjell Magne Fauske, October 18, 2009 at 8:11 p.m. @Nick: Thanks for sharing. height defines the height of the \parbox. Post was not sent - check your email addresses! How to display formulas inside a box or frame in Latex ? \fcolorbox can also be tweaked with \fboxsep and \fboxrule. Glue is the TeX term for an invisible connector that determines the separation between boxes. A basic matrix may be created using the matrix environment, the structures is similar to table-array , entries are specified by row, with columns separated using & and a new rows separated with \\.Matrices are usually enclosed in delimiters (default none) of some kind, and while it is possible to use the \left and \right commands. 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# Solving a recurrence relation with variable coefficients

I am trying to solve this recurrence inequality for $$a_n$$, hopefully in terms of $$a_0，a_1，c$$ and $$x$$,

$$(1-x)a_{n+1} \le xa_n + (1+x)a_{n-1} + c\quad\forall\ n\ge1$$

What I tried: RSolve. But RSolve expects an equality, and not an inequality. Any tips on how to proceed? $$a_i \in \mathbb{R}$$, $$x\in\mathbb{Q}$$, $$c > 0$$, $$0 < x < 1$$ and $$a_i$$ is not constant for all $$i$$.

Edit: I have some control over $$x$$ so I chose $$x=1/2$$ and used an equality instead of an inequality, then applied RSolve.

Simplify[RSolve[(1/2) a[n] == (1/2) a[n - 1] + (3/2)*a[-2 + n] + c, a[n], n]]

I got the result $$a(n)\to c_1 \left(\frac{1}{2} \left(1-\sqrt{13}\right)\right)^n+c_2 \left(\frac{1}{2} \left(\sqrt{13}+1\right)\right)^n-\frac{2 c}{3}.$$ However what I need is the inequality, and in terms of $$x$$. Do you know any other method in Mathematica available to do this?

• As the error message says, RSolve is expecting an equation. You gave it an inequality. Change <= to == and it will return a result. – Carl Woll Oct 29 '18 at 2:37
• Hi, I am open to other ways to solve the recurrence without RSolve. I have searched but I can't find the functions I need. – Cogicero Oct 29 '18 at 2:38
• Thanks Bill, that works, but my problem is that $a$ is definitely not constant for all $n$ in my case. – Cogicero Oct 29 '18 at 3:14
• Thanks. This is one step in some numerical analysis. I know that $a$ depends on $n$ i.e. $a_i$ is not constant for all $i$. I know that $x > 0$ and $c > 0$ while $n >= 1$. $x \in \mathbb{Q}$ while $a_i \in \mathbb{R}$ and $c \in \mathbb{R}$. – Cogicero Oct 29 '18 at 3:25
• I don't understand the problem yet. Let's take a simple special case, say a0=0 and a1=1. What kind of answer would you expect to see? – bill s Oct 29 '18 at 14:41


text:
# Publications

Found 228 results
2014
, Evaluation of antimicrobial activity of secondary metabolites and enzyme production from endophytic fungi isolated from Eucalyptus citriodora, Journal of Pharmacy Research Vol, vol. 8, pp. 269–276, 2014.
, Gödel, Tarski, Turing and the conundrum of free will, in Nature's Longest Threads: New Frontiers in the Mathematics and Physics of Information in Biology, World Scientific, 2014, pp. 151–166.
, Influence of Alkaline Earth Cation Exchanged X-Zeolites Towards Ortho-Selectivity in Alkylation of Aromatics: Hard-Soft-Acid-Base Concept, Advanced Porous Materials, vol. 2, pp. 221–229, 2014.
J. Balakrishnan, Instabilities in sensory processes, in Nature's Longest Threads: New Frontiers in the Mathematics and Physics of Information in Biology, World Scientific, 2014, pp. 37–46.
, In-vitro Studies on $\alpha$-Amylase, $\alpha$-Glucosidase and Aldose Reductase Inhibitors found in Endophytic Fungi Isolated from Ocimum sanctum, Current Enzyme Inhibition, vol. 10, pp. 129–136, 2014.
, Metal ion-exchanged zeolites as solid acid catalysts for the green synthesis of nopol from Prins reaction, Catalysis Science & Technology, vol. 4, pp. 4065–4074, 2014.
, Orthogonal-state-based cryptography in quantum mechanics and local post-quantum theories, International Journal of Quantum Information, vol. 12, p. 1560020, 2014.
, Phosphate modified ZSM-5 for the shape-selective synthesis of para-diethylbenzene: Role of crystal size and acidity, Applied Catalysis A: General, vol. 484, pp. 8–16, 2014.
, Photoluminescence tuning of Na 1- x K x NdW 2 O 8 (0.0≤ x≤ 0.7) nanoparticles: synthesis, crystal structure and Raman study, Physical Chemistry Chemical Physics, vol. 16, pp. 18772–18780, 2014.
, Polymorphism in Photoluminescent KNdW2O8: Synthesis, Neutron Diffraction, and Raman Study, Crystal Growth & Design, vol. 14, pp. 835–843, 2014.
S. Sarkar, Quantum Correlations of Two Superconducting Charge Qubits in a Magnetic Field, Journal of the Physical Society of Japan, vol. 83, p. 104003, 2014.
S. Sarkar, Quantum criticality of geometric phase in coupled optical cavity arrays under linear quench, Physica B: Condensed Matter, vol. 447, pp. 42–46, 2014.
, The quantum cryptographic switch, Quantum information processing, pp. 1–12, 2014.
, Quantum phase transition of light in coupled optical cavity arrays: A renormalization group study, Advances in Theoretical and Mathematical Physics, vol. 18, pp. 741–760, 2014.
, Shape-selective catalysis by phosphate modified ZSM-5: Generation of new acid sites with pore narrowing, Applied Catalysis A: General, vol. 471, pp. 12–18, 2014.
, Structure and cancer immunotherapy of the B7 family member B7x, Cell reports, vol. 9, pp. 1089–1098, 2014.
, Synthesis, characterization and antimicrobial studies of some new trifluoromethyl quinoline-3-carbohydrazide and 1, 3, 4-oxadiazoles, RSC Advances, vol. 4, pp. 30864–30875, 2014.
, Targeting Mycobacterium tuberculosis nucleoid-associated protein HU with structure-based inhibitors, Nature communications, vol. 5, 2014.
, Two-step orthogonal-state-based protocol of quantum secure direct communication with the help of order-rearrangement technique, Quantum information processing, vol. 13, pp. 2731–2743, 2014.
, Unification of Bell, Leggett-Garg and Kochen-Specker inequalities: Hybrid spatio-temporal inequalities, EPL (Europhysics Letters), vol. 104, p. 60006, 2014.
, Zinc hydroxystannate: a promising solid acid–base bifunctional catalyst, RSC Advances, vol. 4, pp. 974–977, 2014.


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Marcel Riesz

{{#invoke:Hatnote|hatnote}} Template:Infobox scientist Marcel Riesz (Template:Lang-hu, Template:IPA-hu; 16 November 1886 – 4 September 1969) was a Hungarian-born mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras. He spent most of his career in Lund (Sweden).

Biography

Marcel Riesz was born in Győr, Hungary (Austria-Hungary); he was the younger brother of the mathematician Frigyes Riesz. He obtained his PhD at Eötvös Loránd University under the supervision of Lipót Fejér. In 1911, he moved to Sweden upon the invitation of Gösta Mittag-Leffler. From 1911 to 1925 he taught at Stockholms högskola (now Stockholm University). From 1926 to 1952 he was professor at Lund University. After retiring, he spent 10 years at universities in the United States. He returned to Lund in 1962, and died there in 1969.[1][2]

Riesz was elected a member of the Royal Swedish Academy of Sciences in 1936.[1]

Mathematical work

Classical analysis

The work of Riesz as a student of Fejér in Budapest was devoted to trigonometric series:

${\displaystyle {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left\{a_{n}\cos(nx)+b_{n}\sin(nx)\right\}.\,}$

One of his results states that, if

${\displaystyle \sum _{n=1}^{\infty }{\frac {|a_{n}|+|b_{n}|}{n^{2}}}<\infty ,\,}$

and if the Fejer means of the series tend to zero, then all the coefficients an and bn are zero.[3]

His results on summability of trigonometric series include a generalisation of Fejér's theorem to Cesàro means of arbitrary order.[4] He also studied the summability of power and Dirichlet series, and coauthored a book Template:Harvtxt on the latter with G.H. Hardy.[3]

In 1916, he introduced the Riesz interpolation formula for trigonometric polynomials, which allowed him to give a new proof of Bernstein's inequality.[5]

He also introduced the Riesz function Riesz(x), and showed that the Riemann hypothesis is equivalent to the bound Riesz(x) = O(xTemplate:Frac + ε) as x → ∞, for any ε > 0.[6]

Together with his brother Frigyes Riesz, he proved the F. and M. Riesz theorem, which implies, in particular, that if μ is a complex measure on the unit circle such that

${\displaystyle \int z^{n}d\mu (z)=0,n=1,2,3\cdots ,\,}$

then the variation |μ| of μ and the Lebesgue measure on the circle are mutually absolutely continuous.[5][7]

Functional-analytic methods

Part of the analytic work of Riesz in the 1920s used methods of functional analysis.

In the early 1920s, he worked on the moment problem, to which he introduced the operator-theoretic approach by proving the Riesz extension theorem (which predated the closely related Hahn–Banach theorem).[8][9]

Later, he devised an interpolation theorem to show that the Hilbert transform is a bounded operator in Lp (1 < p < ∞). The generalisation of the interpolation theorem by his student Olaf Thorin is now known as the Riesz–Thorin theorem.[2][10]

Riesz also established, independently of Andrey Kolmogorov, what is now called the Kolmogorov–Riesz compactness criterion in Lp: a subset K ⊂Lp(Rn) is precompact if and only if the following three conditions hold: (a) K is bounded; (b) for every ε > 0 there exists R > 0 so that

${\displaystyle \int _{|x|>R}|f(x)|^{p}dx<\epsilon ^{p}\,}$

for every f ∈ K; (c) for every ε > 0 there exists ρ > 0 so that

${\displaystyle \int _{\mathbb {R} ^{n}}|f(x+y)-f(x)|^{p}dx<\epsilon ^{p}\,}$

for every y ∈ Rn with |y| < ρ, and every f ∈ K.[11]

Potential theory, PDE, and Clifford algebras

After 1930, the interests of Riesz shifted to potential theory and partial differential equations. He made use of "generalised potentials", generalisations of the Riemann–Liouville integral.[2] In particular, Riesz discovered the Riesz potential, a generalisation of the Riemann–Liouville integral to dimension higher than one.[1]

In the 1940s and 1950s, Riesz worked on Clifford algebras. His 1958 lecture notes, the complete version of which was only published in 1993 (Template:Harvtxt), were dubbed by the physicist David Hestenes "the midwife of the rebirth" of Clifford algebras.[12]

Students

Riesz's doctoral students in Stockholm include Harald Cramér and Einar Carl Hille.[1] In Lund, Riesz supervised the theses of Otto Frostman, Lars Hörmander, and Olaf Thorin.[2]

Publications

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References

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•
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# Träfflista för sökning "WFRF:(Zhao Fang) srt2:(2010-2014);srt2:(2014)"

Sortera/gruppera träfflistan

NumreringReferensOmslagsbildHitta
1.
• Ablikim, M., et al. (författare)
• Measurement of the $D\to K^-\pi^+$ strong phase difference in $\psi(3770)\to D^0\overline{D}{}^0$
• 2014
• Ingår i: PHYSICS LETTERS B. - 0370-2693. ; 734
• Tidskriftsartikel (refereegranskat)abstract
• We study $D^0\overline{D}{}^0$ pairs produced in $e^+e^-$ collisions at $\sqrt{s}=3.773$ GeV using a data sample of 2.92 fb$^{-1}$ collected with the BESIII detector. We measure the asymmetry $\mathcal{A}^{CP}_{K\pi}$ of the branching fractions of $D \to K^-\pi^+$ in $CP$-odd and $CP$-even eigenstates to be $(12.7\pm1.3\pm0.7)\times10^{-2}$. $\mathcal{A}^{CP}_{K\pi}$ can be used to extract the strong phase difference $\delta_{K\pi}$ between the doubly Cabibbo-suppressed process $\overline{D}{}^{0}\to K^-\pi^+$ and the Cabibbo-favored process $D^0\to K^- \pi^+$. Using world-average values of external parameters, we obtain $\cos\delta_{K\pi} = 1.02\pm0.11\pm0.06\pm0.01$. Here, the first and second uncertainties are statistical and systematic, respectively, while the third uncertainty arises from the external parameters. This is the most precise measurement of $\delta_{K\pi}$ to date.
•
2.
• Ablikim, M., et al. (författare)
• Observation of e(+)e(-) -> pi(0)pi(0)h(c) and a Neutral Charmoniumlike Structure Z(c)(4020)(0)
• 2014
• Ingår i: Physical Review Letters. - 0031-9007 .- 1079-7114. ; 113:21, s. 212002-
• Tidskriftsartikel (refereegranskat)abstract
• Using data collected with the BESIII detector operating at the Beijing Electron Positron Collider at center-of-mass energies of root s = 4.23, 4.26, and 4.36 GeV, we observe e(+)e(-) -> pi(0)pi(0)h(c) for the first time. The Born cross sections are measured and found to be about half of those of e(+)e(-) -> pi(+)pi(-)h(c) within less than 2 sigma. In the pi(0)h(c) mass spectrum, a structure at 4.02 GeV/c(2) is found. It is most likely to be the neutral isospin partner of the Z(c)(4020)(+/-) observed in the process of e(+)e(-) -> pi(+)pi(-)h(c) being found. A fit to the pi(0)h(c) invariant mass spectrum, with the width of the Z(c)(4020)(0) fixed to that of its charged isospin partner and possible interferences with non-Z(c)(4020)(0) amplitudes neglected, gives a mass of (4023.9 +/- 2.2 +/- 3.8) MeV/c(2) for the Z(c)(4020)(0), where the first error is statistical and the second systematic.
•
3.
• Ablikim, M., et al. (författare)
• Observation of $e^+e^− → γX$(3872) at BESIII
• 2014
• Ingår i: PHYSICAL REVIEW LETTERS. - 1079-7114. ; 112:9
• Tidskriftsartikel (refereegranskat)abstract
• With data samples collected with the BESIII detector operating at the BEPCII storage ring at center-of-mass energies from 4.009 to 4.420 GeV, the process $e^{+} e^{-} \to \gamma X(3872)$ is observed for the first time with a statistical significance of $6.3\sigma$. The measured mass of the $X(3872)$ is ($3871.9\pm 0.7_{\rm stat.}\pm 0.2_{\rm sys.}$) MeV/$c^2$, in agreement with previous measurements. Measurements of the product of the cross section $\sigma[e^{+} e^{-} \to \gamma X(3872)]$ and the branching fraction $\mathcal{B}[X(3872) \to \pi^{+} \pi^{-} J/\psi]$ at center-of-mass energies 4.009, 4.229, 4.260, and 4.360 GeV are reported. Our measurements are consistent with expectations for the radiative transition process $Y(4260) \to \gamma X(3872)$.
•
4.
• Ablikim, M., et al. (författare)
• Observation of electromagnetic Dalitz decays J/\psi \to P e^+e^-
• 2014
• Ingår i: PHYSICAL REVIEW D. - 2470-0010 .- 1550-7998 .- 1550-2368. ; 89:9
• Tidskriftsartikel (refereegranskat)abstract
• Based on a sample of (225.3\pm2.8)\times 10^{6} J/\psi events collected with the BESIII detector, the electromagnetic Dalitz decays of J/\psi \to P e^+e^-(P=\eta'/\eta/\pi^0) are studied. By reconstructing the pseudoscalar mesons in various decay modes, the decays J/\psi \to \eta' e^+e^-, J/\psi \to \eta e^+e^- and J/\psi \to \pi^0 e^+e^- are observed for the first time. The branching fractions are determined to be \mathcal{B}(J/\psi\to \eta' e^+e^-) = (5.81\pm0.16\pm0.31)\times10^{-5}, \mathcal{B}(J/\psi\to \eta e^+e^-) = (1.16\pm0.07\pm0.06)\times10^{-5}, and \mathcal{B}(J/\psi\to \pi^0 e^+e^-)=(7.56\pm1.32\pm0.50)\times10^{-7}, where the first errors are statistical and the second ones systematic.
•
5.
• Ablikim, M., et al. (författare)
• Precision measurements of $B(D^+ \rightarrow \mu^+ \nu_{\mu})$, the pseudoscalar decay constant $f_{D^+}$, and the quark mixing matrix element $|V_{\rm cd}|$
• 2014
• Ingår i: PHYSICAL REVIEW D. - 2470-0010 .- 1550-7998 .- 1550-2368. ; 89:5
• Tidskriftsartikel (refereegranskat)abstract
• We report a measurement of the branching fraction $B(D^+ \rightarrow \mu^+ \nu_{\mu}) = [3.71 \pm 0.19 (\rm stat) \pm 0.06 (\rm sys)]\times 10^{-4}$ based on 2.92 ${\rm fb^{-1}}$ of data accumulated at $\sqrt{s}=3.773$ GeV with the BESIII detector at the BEPCII collider. This measurement, in conjunction with the Cabibbo-Kobayashi-Maskawa matrix element $|V_{\rm cd}|$ determined from a global Standard Model fit, implies a value for the weak decay constant $f_{D^+}=(203.2 \pm 5.3 \pm 1.8)$ MeV. Additionally, using this branching fraction measurement together with a Lattice QCD prediction for $f_{D^+}$, we find $|V_{\rm cd}|=0.2210\pm 0.0058 \pm 0.0047$. In either case, these are the most precise results for these quantities to date.
•
6.
• Ablikim, M., et al. (författare)
• Search for the weak decays J/psi -> D-s(()*()-) e(+)nu(e) + c.c.
• 2014
• Ingår i: Physical Review D. - 1550-7998 .- 1550-2368. ; 90:11, s. 112014-
• Tidskriftsartikel (refereegranskat)abstract
• Using a sample of 2.25 x 10(8) J/psi events collected with the BESIII detector at the BEPCII collider, we search for the J/psi semileptonic weak decay J/psi -> D-s(-) e(+)nu(e) +c.c. with a much higher sensitivity than previous searches. We also perform the first search for J/psi -> D-s(*-) e(+) nu(e) + c.c. No significant excess of a signal above background is observed in either channel. At the 90% confidence level, the upper limits are determined to be B(J/psi -> D-s(-) e(+) nu(e) + c.c.) < 1.3 x 10(-6) and B(J/psi -> D-s*(-) e(+) nu(e) + c.c.) < 1.8 x 10(-6), respectively. Both are consistent with Standard Model predictions.
•
7.
• Ablikim, M., et al. (författare)
• Amplitude analysis of the D+ -> K-S(0)pi + (0)(pi) Dalitz plot
• 2014
• Ingår i: Physical Review D. - 1550-7998 .- 1550-2368. ; 89:5, s. 052001-
• Tidskriftsartikel (refereegranskat)abstract
• We perform an analysis of the D+ -> K-S(0)pi + (0)(pi) Dalitz plot using a data set of 2.92 fb(-1) of e(+) e(-) collisions at the (3770) mass accumulated by the BESIII experiment, in which 166694 candidate events are selected with a background of 15.1%. The Dalitz plot is found to be well represented by a combination of six quasitwo- body decay channels [k(SP)(0)(+) (1450)(+,) ] plus a small nonresonant component. Using the fit fractions from this analysis, partial branching ratios are updated with higher precision than previous measurements.
•
8.
• Ablikim, M., et al. (författare)
• Measurement of chi(cJ) decaying into eta ' K+K-
• 2014
• Ingår i: Physical Review D. - 1550-7998 .- 1550-2368. ; 89:7, s. 074030-
• Tidskriftsartikel (refereegranskat)abstract
• Using (106.41 +/- 0.86) x 10(6) Psi(3686) events collected with the BESIII detector at BEPCII, we study for the first time the decay chi(cJ) -> eta'K+K- (J = 1, 2), where eta' -> gamma rho(0) and eta' -> eta pi(+)pi(-). A partial wave analysis in the covariant tensor amplitude formalism is performed for the decay chi(c1) -> eta'K+K-. Intermediate processes chi(c1) -> eta'f(2)'(1525) chi(c1) -> K-0*(1430)K-+/-(-/+) (K-0*(1430)(+/-) -> eta'K-+/-) are observed with statistical significances larger than 5 sigma, and their branching fractions are measured.
•
9.
• Ablikim, M., et al. (författare)
• Measurement of the branching fraction for psi(3686) -> omega K+K-
• 2014
• Ingår i: Physical Review D. - 1550-7998 .- 1550-2368. ; 89:11, s. 112006-
• Tidskriftsartikel (refereegranskat)abstract
• With 1.06 x 10(8) psi(3686) events collected with the BESIII detector, the branching fraction of psi(3686) -> omega K+K- is measured to be (1.54 +/- 0.04 +/- 0.11) x 10(-4). This is the most precise result to date, due to the largest psi(3686) sample, improved signal reconstruction efficiency, good simulation of the detector performance, and a more accurate knowledge of the continuum contribution. Using the branching fraction of J/psi -> omega K+K-, the ratio B(psi(3868) -> K+K-)/B(J/psi -> K+K-) is determined to be (18.4 +/- 3.7)%. This constitutes a significantly improved test of the 12% rule, with the uncertainty now dominated by the J/psi branching fraction.
•
10.
• Ablikim, M., et al. (författare)
• Observation of e(+)e(-) -> gamma X(3872) at BESIII
• 2014
• Ingår i: Physical Review Letters. - 0031-9007 .- 1079-7114. ; 112:9, s. 092001-
• Tidskriftsartikel (refereegranskat)abstract
• With data samples collected with the BESIII detector operating at the BEPCII storage ring at center-of-mass energies from 4.009 to 4.420 GeV, the process e(+)e(-) -> gamma X(3872) is observed for the first time with a statistical significance of 6.3 sigma. The measured mass of the X(3872) is (3871.9 +/- 0.7(stat) +/- 0.2(syst)) MeV/c(2), in agreement with previous measurements. Measurements of the product of the cross section sigma[e(+)e(-) -> gamma X(3872)] and the branching fraction B [X(3872) -> pi(+)pi(-)J/psi] at center-of-mass energies 4.009, 4.229, 4.260, and 4.360 GeV are reported. Our measurements are consistent with expectations for the radiative transition process Y(4260) -> gamma X(3872).
•
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text:
# Improved copying in org.jikesrvm.runtime.Memory

## Description

r13857 improved memory copying for Intel with SSE2 so that we used 64bit copies rather than 32bit copies. This gave a large number of speed ups:

http://jikesrvm.anu.edu.au/cattrack/results/rvmx86lnx32.anu.edu.au/perf/1790/performance_report

most notably on SpecJBB 2000. There is a low-hanging fruit to improve this further, for example, by using 128bit copies and using more than 1 register to do the copying.

None

Unassigned

Ian Rogers

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Medium
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