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MÖBIUS INVARIANT FUNCTION SPACES and DIRICHLET SPACES with SUPERHARMONIC WEIGHTS

Guanlong Bao, Javad Mashreghi, Stamatis Pouliasis, Hasi Wulan

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Let Dμ be Dirichlet spaces with superharmonic weights induced by positive Borel measures μ on the open unit disk. Denote by M (Dμ) Möbius invariant function spaces generated by Dμ. In this paper, we investigate the relation among Dμ, M (Dμ) and some Möbius invariant function spaces, such as the space BMOA of analytic functions on the open unit disk with boundary values of bounded mean oscillation and the Dirichlet space. Applying the relation between BMOA and M (Dμ), under the assumption that the weight function K is concave, we characterize the function K such that QK = BMOA . We also describe inner functions in M (Dμ) spaces.

Original language English 1-18 18 Journal of the Australian Mathematical Society 106 1 https://doi.org/10.1017/S1446788718000022 Published - Feb 1 2019

Keywords

• Dirichlet spaces with superharmonic weights
• Möbius invariant function spaces
• inner functions

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# Inverse function

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Let f(x) = 7x + 5 and g(x) = x + 1. If h(x) = f(g(x)) , then what is the inverse of h(x)?

Jun 10, 2021

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+505
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f(x) = 7x + 5

and g(x) = x + 1

⇒ $$h(x)=f(g(x))$$

$$=f(x+1)$$

$$=7x+7+5$$

$$=7x+12$$

In order to find  $$h^{-1}(x)$$

Let  $$y=h(x)$$

$$y=7x+12$$

$$x= {y-12\over 7}$$

⇒ $$h^{-1}(x) = {x-12 \over 7}$$

Thus, inverse of h(x) is $${x-12 \over 7}$$

~Amy

Jun 10, 2021
edited by amygdaleon305  Jun 10, 2021

format: plain


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# Why is dot product attention faster than additive attention?

In section 3.2.1 of Attention Is All You Need the claim is made that:

Dot-product attention is identical to our algorithm, except for the scaling factor of $$\frac{1}{\sqrt{d_k}}$$. Additive attention computes the compatibility function using a feed-forward network with a single hidden layer. While the two are similar in theoretical complexity, dot-product attention is much faster and more space-efficient in practice, since it can be implemented using highly optimized matrix multiplication code.

It does not make sense why dot product attention would be faster. Additive attention is nearly identical computation wise; the main difference is $$Q + K$$ instead of $$Q K^T$$ in dot product attention. $$Q K^T$$ requires at least as many addition operations as $$Q + K$$, so how can it possibly be faster?


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wbCalib -- Wide-band Interferometric Visibility Calibration

wbCalib -- Wide-band Interferometric Visibility Calibration

Next: , Previous: (dir), Up: (dir)

wbCalib – Wide-band Optical Interferometric Visibility Amplitude Calibration

wbCalib v1.4.4 – Wide-band Interferometric Visibility Calibration

wbCalib is a product of the NASA Exoplanet Science Institute.

--- The Detailed Node Listing ---

General Description and Features

Algorithm, Internals, and Dependencies

Ancillary Scripts and Programs

Next: , Previous: Top, Up: Top

1 General Description and Features

wbCalib is a wide-band interferometric (squared) visibility amplitude (modulus) calibration application: it correlates observations on one or more science targets with observations on one or more calibration sources, estimates a model of the system visibility (see Algorithm below) at the times of the target scan, applies this system visibility estimate to estimate the calibrated target visibility amplitude at time, and optionally computes ancillary geometric information (i.e. u-v coordinates, delays, hour angles). wbCalib was developed in the context of PTI, where it reads the interferometric data from one or more vis-output "sum" files (L-1 data). All wbCalib input comes in two (or more) input ASCII files; all data output is ASCII and goes to stdout (with error messages to stderr).

The canonical use case for wbCalib is:

     wbCalib [options] xxx.calScript xxx.sum [yyy.sum...] [> xxx.calData]

where xxx.calScript is a script file that contains object designation and astrometry information in a standard format, and xxx.sum (and any number of additional input data files yyy.sum) is the standard L-1 V^2 reduced data product file. Options for wbCalib are summarized below in Arguments.

The reader unfamilier with the use of terms like interferometric visibility and interferometric resolution is refered to any standard interferometric textbook such as Thompson, Moran, and Swenson, the NRAO Imaging Summer School Notes, or lecture notes from the Michelson Summer School. In particular it is important (particularly for radio interferometrists) to note that wbCalib deals with the calibration of normalized squared visibility modulus, which optical interferometrists notationally describe as V^2, and often call squared visibilty amplitude, or visibility amplitude, or sometimes (unfortunately) just visibility. V^2 is a dimensionless quantity – unlike the corresponding currency in radio astronomy.

Next: , Previous: Description, Up: Description

1.1 Inputs and Outputs

wbCalib takes no fewer than two inputs, a calibration script that defines the targets and calibrators of interest in this run, and one or more input data file specifications. Output from wbCalib is written to stdout (with major warnings and errors also written to stderr):

     wbCalib xxx.calScript xxx.sum [yyy.sum...] [zzz.fits] [> xxx.calData]

Various command-line arguments that control the behavior of wbCalib are also available (see Arguments).

1.1.1 Calibration Script File

The ASCII wbCalib script file contains both science target and calibrator object designations, astrometric information, and calibrator diameter information:

     # Simbad Search HD 9939: Type: High proper-motion Star  K0IV V=6.99
HDC9939
01 37 25.107   +25 10 03.969 -0.211 -0.208 0.0238 # K0IV  V = 7.0, K = 5.0
---
2
# Simbad Search HD 7034: Type: Star  F0V V=5.160
##  7 HD7034--F0V      7306 +/-   0    1.79  11    25.82 +/- 0.66  0.52 +/- 0.01  XX
HDC7034   01 11 06.768   +31 25 29.051  -0.009 -0.012 0.00582 0.50 0.05   # F0V   V = 5.2, K = 4.5
# Simbad Search HD 7964: Type: Star  A3V V=4.752
## 9 HD7964--A3V      8686 +/-   0    9.18  20    47.44 +/- 1.85  0.50 +/- 0.01  XX         XXXX
HDC7964   01 19 27.993   +27 15 50.611   0.026 -0.012 0.01049 0.47 0.05   # A3V   V = 4.7, K = 4.6

Lines begining with the pound (hash – #) sign are comments and are ignored by wbCalib. The astrometry format is a standard J2000 position (hh mm ss dd mm ss), proper motion (arcsec/yr), plx (arcsec), ancillary info PTI-standard format. Targets to be calibrated are designated in the first section of the file, in this example a single target designated HDC9939. On a separate line the astrometry for the target is given. Then separated by a line separator (—), the calibrator section follows. The number of calibrators to use is given, followed by single-line calibrator specifications, indicating the calibrator designation, astrometry (J2000 position, proper motion, and parallax; units are RA: hh mm ss.s of time, Dec: dd mm ss.s of arc, pm: arcsec/yr, parallax arcsec), estimated angular diameter and standard error in such (units of millarcseconds, mas). Take note of the calibrator angular diameter and error fields – they are the critical parameters that determine the scale of the calibrated visibilities.

While the script file can be composed (or edited) by hand, a template calibration script file can be composed by the getCal experiment planning tool suite.

1.1.2 L-1 Wide-band Visibility File

The xxx.sum file is the standard L-1 data product from the PTI/KI V^2 processing code (Vis/Kvis). It's format looks like:

     ##   date      ObjDesig     UT (hrs)   delay (m)    WlN    V2Inc  V2Coh  SpecN  V2Coh  V2Inc  Jitt  Ratio
SUM: 08/18/100 HDC156164    4.628728   14.0533981   393.0  0.134  0.115   93.4  0.096  0.109  0.91  -0.23  -0.23    85.0   490.3    39.0   85.1  179.5    0.0  2.184   0.74   6  0  OK 17.25053024 24.83921432 100
SUM: 08/18/100 HDC156164    4.636042   14.0771360   412.3  0.180  0.156   95.9  0.122  0.134  0.81  -0.23  -0.23    85.0   490.3    20.5   85.1  179.5    0.0  2.186   0.94   1  0  OK 17.25053024 24.83921432 100

wbCalib requires at least one input sum file, but will in fact process any (large) number of additional sum file specifications on the command line (e.g.):

     wbCalib xxx.calScript xxx.sum yyy.sum zzz.sum aaa.sum ... [> xxx.calData]

This capability allows a single wbCalib invocation to calibrate large amounts of visibility data stored separately in individual sum files; you can calibrate nights individually, or calibrate 3 years worth of data.

L1 data files can also be used as input in FITS format, with no restrictions on file name or extension. Please refer to L1 FITS specification for details on the KI V2 L1 FITS format.

1.1.3 wbCalib Output

Depending on the user choice of wbCalib verbosity (see Arguments), default output from wbCalib looks like:

     (gnomad:27) wbCalib cs.hd9939 100343.sum -verbose
# Simbad Search HD 9939: Type: High proper-motion Star  K0IV V=6.99
# >> Calibrating star HDC9939  at 01 37 25.107 +25 10 03.969

# >> with respect to 2 calibrators:
# Simbad Search HD 7034: Type: Star  F0V V=5.160
##  7 HD7034--F0V      9021 +/- 232    0.23  11    27.97 +/- 0.40  0.36 +/- 0.02  XX
##  7 HD7034--F0V      7306 +/-   0    1.79  11    25.82 +/- 0.66  0.52 +/- 0.01  XX
# >>  star HDC7034  at 01 11 06.768 +31 25 29.051   size 0.5  error 0.05  8.5 degrees away from target
# Simbad Search HD 7964: Type: Star  A3V V=4.752
## 9 HD7964--A3V     12760 +/- 559    1.25  20    76.81 +/- 2.14  0.29 +/- 0.02
# >>  star HDC7964  at 01 19 27.993 +27 15 50.611   size 0.47  error 0.05  4.5 degrees away from target

# Wideband calibration code settings for this run:
# > Calibrating all wavebands
# > Calibrating all frame rates
# > Calibrating incoherent spectrometer V2 data
# > Applying jitter correction, coefficient 0.04
# > Using calibration scans within +/- 1 hr of the target scan
# > Weighting calibration scans WRT temporal proximity, variance doubles in 1 hr
# > Weighting calibration scans WRT sky proximity, variance doubles in 15 deg
# > No minimum calibrated scan uncertainty
# > Using 150 s scan time threshold
# > Flagging calibration warnings when multiple calibrators disagree by more than 2 sigma
# > Rejecting the scan when multiple calibrators disagree by more than 3 sigma

## reading baseline model from /home/bode/pti/data/100/100343.bline (PTI_NS)
# 128 data lines read from /home/bode/pti/data/100/100343.sum
# Calibration data format:
# name	#VisPts	MJD             m/d/y/h:m:s		UT	Delay(m)	WL(um)	CalVis       Err	RawVis	   Err	SysVis     Err	#Cal	U(m)	V(m)	HA(hr)	FrameRate	BLName

# Target HDC9939:
HDC9939	4	51886.14193	12/8/2000/03:24:22	3.40629	-3.83132	2.2265	0.525698	0.0567865	0.413766	0.0422374	0.787079        0.027807	1	-49.510653	-97.560135	-0.862746	100	PTI_NS
HDC9939	2	51886.14371	12/8/2000/03:26:56	3.44908	-3.33059	2.229	0.639641	0.114043	0.503215	0.0879654	0.786715        0.0275965	1	-48.948938	-97.795364	-0.819838	100	PTI_NS
HDC9939	5	51886.15861	12/8/2000/03:48:23	3.80653	 0.619862	2.2276	0.542716	0.0219756	0.428892	0.00918903	0.79027         0.027153	1	-44.022431	-99.652127	-0.461409	100	PTI_NS
HDC9939	5	51886.17455	12/8/2000/04:11:21	4.18931	 4.36715	2.225	0.571089	0.079679	0.447928	0.0606237	0.784341	0.0265822	1	-38.318529	-101.413309	-0.0775812	100	PTI_NS
HDC9939	5	51886.18987	12/8/2000/04:33:25	4.55698	 7.46152	2.2236	0.630103	0.0255842	0.492882	0.00842851	0.782225	0.0288067	1	-32.475094	-102.867623	0.291092	100	PTI_NS
# 5 calibrated scans output under name HDC9939

# 5 total calibrated scans output on target HDC9939

The first section of the output is an extended preamble (comment lines indicated by # symbols) containing the following pieces:

• Indications of any option flags that were passed to wbCalib
• What sources will be calibrated with respect to what calibrators (echoing the relevant astrometric and diameter parameters
• A summary of how the data is to be calibrated – a summary of the wbCalib settings for this run
• A summary of where the input data is being found
• A template line giving the format of the data to follow

The second section of the file containes a list of calibrated output data. For each specified target/pseudonym there is a designation header, and then a sequence of tab-delimited data lines. wbCalib output data lines have 15 – 19 fields (depending on whether target astrometry has been provided as part of the calibration script). The contents of these data lines is described by the following table:

 Field Item Description 1 Object Object Designation (string) 2 NSumRecords The number of sum records averaged into this target scan (integer) 3 MJD Modified Julian Date (day) (float) 4 UTC Date & Time UTC Date/time in mm/dd/yy/hh:mm:ss format 5 UTC UTC time (decimal hours) (float) 6 Delay Average delay (m) (float) 7 WL SNR-weighted mean wavelength for scan (microns) (float) 8 CalVis Calibrated wide-band visibility (dimensionless) 9 ErrVis Std Error in calibrated visibility (dimensionless) 10 RawVis Raw (L-1) wide-band visibility (dimensionless) 11 ErrRawVis Std Error in raw wide-band visibility (dimensionless) 12 SysVis Estimated system visibility (dimensionless) 13 ErrSysVis Std Error in estimated system visibility (dimensionless) 14 NCal Number of calibrators used in calibrating this scan (integer) 15 uCoord u * lambda – RA baseline projection (m) (float) 16 vCoord v * lambda – dec baseline projection (m) (float) 17 HRangle Hour angle on target for this scan (decimal hours) (float) 18 Frame Rate Frame rate for this scan (Hz) (integer) 19 Baseline Baseline designation for this scan (e.g. PTI_NS) (string)

The wbCalib output can also be in FITS format, if the -fits option is used. An optional argument may be used to specify the output file name (e.g. -fits=myFITSfile). If no argument is used, the default file name wbCalib-out.fits is used. The L2 FITS format conforms to the IAU working group standard for optical/infrared calibrated visibilities, please refer to IAU standard page for details. Please note that the use of the -fits and -doCalibrators options are mutually exclusive in the current release.

Previous: Inputs_and_Outputs, Up: Description

A wbCalib invocation with no arguments generates a short usage reminder.

A wbCalib invocation with argument -help (or -h) outputs the online help.

Additional information and discussion on wbCalib command-line options can be found in Arguments.

Next: , Previous: Description, Up: Top

2 Algorithm, Internals, and Dependencies

The fundamental purpose of wbCalib is to estimate the squared visibility modulus (herein refered to a visibility amplitude or visibility) on a target as measured by an idealized interferometer V^2_target based on the visibility amplitude measurement made by the interferometer V^2_meas. The wbCalib algorithm for doing this is very simple; following Mozurkewich91 (see References), we model the visibility amplitude performance of the interferometer as linearly degraded by a multiplicative factor refered to as the system visibility V^2_sys:

(2.1) V^2_meas = V^2_target * V^2_sys

so of course the idealized target visibility is estimated by inverting Eq. 2.1:

(2.2) V^2_target = V^2_meas / V^2_sys

Eq. 2.1 illustrates the physical interpretation of V^2_sys: since the ideal visibility amplitude for an unresolved point source is 1, V^2_sys represents the point source response of the interferometer. This observation further suggests the methodology for calibrating the system visibility – namely to periodically look at calibration point sources with the interferometer to quantify the interferometer response (duh)! Therefore, the job of wbCalib is to estimate the system visibility at the time of the target scan by analyzing available calibration measurements, and then to apply (in the sense of Eq 2.2) this system visibility model it to the measured target visibility at epoch to estimate the idealized target visibility at epoch.

This prescription sounds straightforward, but it leaves open (at least) two important questions. The first and most fundamental of these questions is to consider why the response of the interferometer isn't perfect (quantitatively, why isn't V^2_sys = 1)? (It is in fact the essence of the experimental art to appreciate and correct for the shortcomings in one's measurement apparatus.) A detailed answer to the question of interferometer performance imperfections is well beyond the scope of this manual; for our purposes suffice it to say that visibility degradations arise both from imperfections in the interferometer itself, and because the interferometer finds itself in an imperfect environment (namely a turbulent and opaque atmosphere). Both interferometer and enviromental imperfections can be both temporally and spatially (with sky position) varying; therefore calibration observations must be made on both a timescale and a spatial scale which capture these variations, whatever they may be. It is the job of the experiment designer to design an observing strategy that can capture the important variations in system performance while efficiently utilizing observing resources. Described more fully below, the system visibility estimation algorithms used by wbCalib are designed to characterize spatial and temporal variations in system performance under a varieity of calibration circumstances.

The second important question to consider is where does one go to find an unresolved source? The point of interferometers is high angular resolution. Since all astronomical sources are of finite surface temperature, depending on the part of sensitivity/spatial frequency space that one is working in, finding an unresolved source at any spatial proximity to your target may or may not be a straightforward exercise. (A related issue is the degree to which one can be certain that a given stellar source is even single; stellar multiplicity is a pervasive phenomenon.) Finite surface temperatures and spatial variations in interferometer and atmospheric conditions unfortunately can lead one to consider calibration sources that are partially resolved by the interferometer. Therefore, wbCalib's calibrator observation modeling can account for resolved calibrators with finite angular size (with the associated requirement of calculating baseline projection effects). The associated planning suite getCal can assist the user in selecting potential calibrators, estimating their angular sizes, retrieving available ancillary information, and preparing wbCalib input. In the event that calibrators in practice turn out to be less than ideal, wbCalib supports the use (and more to the point, the intercomparison of data from) multiple calibrators to refine the system visibility estimate and facilitate the identification of troublesome calibrators.

Finally, users find it useful for their calibration application to perform geometric projection (in the parlance, u-v coordinate) calculations. As these geometrical calculations are required to properly account for resolved calibrators, it is straightforward to have wbCalib compute the measurement u-v coordinates if the target astrometry is provided (see Astrometric_Processing).

Next: , Previous: Algorithm, Up: Algorithm

2.1 Algorithm Details

wbCalib has four main algorithmic components: Data_Management and averaging, System_Visibility_Estimation and application, Error_Estimation, and Astrometric_Processing. Each of these is discussed in turn.

2.2 Data Management

The majority of the wbCalib code is devoted to getting the right visibility measurements to the right place at the right time. This is straightforwardly and tediously implemented in wbCalib by creating and traversing a data base of visibility measurements contained in the input visibility (sum) file set. wbCalib implements this as a doubly-linked list data structure; a list over the object designations used in the measurement process, and hanging from the nodes of this first list a time sorted list of visibility measurements on the particular object designation. All of the memory allocation for the database is done dynamically as the structure is constructed in the course of reading the data files specified on the command line. Consequently there is no pre-set size limit as to the maximum amount of data wbCalib can process. A typical year's (2000) worth of L-1 PTI data is typically on the order of six Mbytes in size, so wbCalib can comfortably handle several years worth of such visibility measurement loads on modern computer hardware without really breaking a sweat.

The L-1 data product coming into wbCalib is usually two or more visibility averages over a block of (typically) 25 seconds, resulting in a scan that is typically 90 – 130 seconds in length. This is illustrated by the following segment from a typical L-1 (sum) visibility file (this data format was discussed above):

     SUM: 10/05/100 HDC182488    2.078131  -10.7919001   185.0  0.368  0.342    8.9  0.650  0.767  0.66   1.00   1.00    75.3   272.2   210.7   86.6   95.3   91.1  2.219   0.79   7  0  OK 19.39278221 33.22196579 100
SUM: 10/05/100 HDC182488    2.085947  -10.7238601   183.5  0.387  0.362    8.1  0.643  0.705  0.60   1.05   1.05    75.3   272.2   210.7   86.6   95.3   91.1  2.219   0.92   1  0  OK 19.39278221 33.22196579 100
SUM: 10/05/100 HDC182488    2.092986  -10.6627497   191.4  0.415  0.388    8.6  0.693  0.768  0.59   1.04   1.04    75.3   272.2   210.7   86.6   95.3   91.1  2.218   0.98   0  0  OK 19.39278221 33.22196579 100
SUM: 10/05/100 HDC182488    2.099931  -10.6026391   186.5  0.379  0.350    8.5  0.650  0.735  0.64   1.05   1.04    75.3   272.2   210.7   86.6   95.3   91.1  2.220   0.98   1  0  OK 19.39278221 33.22196579 100
SUM: 10/05/100 HDC182488    2.106878  -10.5426376   183.7  0.357  0.332    8.5  0.652  0.757  0.66   1.05   1.04    75.3   272.2   210.7   86.6   95.3   91.1  2.220   0.84   5  0  OK 19.39278221 33.22196579 100
SUM: 10/05/100 HDC174881    2.164375    2.0741479   433.5  0.211  0.201   20.8  0.299  0.330  0.53   1.13   1.12    75.3   508.7   336.8   86.5  104.3   98.4  2.220   0.94   2  0  OK 18.85997009 28.78367043 100
SUM: 10/05/100 HDC174881    2.171444    2.1225823   429.2  0.182  0.171   21.4  0.279  0.309  0.55   1.13   1.13    75.3   508.7   336.8   86.5  104.3   98.4  2.220   0.86   4  0  OK 18.85997009 28.78367043 100
SUM: 10/05/100 HDC174881    2.178542    2.1710274   425.4  0.193  0.182   19.2  0.305  0.322  0.51   1.13   1.13    75.3   508.7   336.8   86.5  104.3   98.4  2.222   0.98   0  0  OK 18.85997009 28.78367043 100
SUM: 10/05/100 HDC174881    2.185486    2.2182318   417.7  0.191  0.179   19.3  0.294  0.324  0.56   1.13   1.13    75.3   508.7   336.8   86.5  104.3   98.4  2.221   0.94   2  0  OK 18.85997009 28.78367043 100
SUM: 10/05/100 HDC174881    2.192431    2.2652693   427.4  0.169  0.159   20.6  0.300  0.325  0.56   1.13   1.13    75.3   508.7   336.8   86.5  104.3   98.4  2.221   0.94   2  0  OK 18.85997009 28.78367043 100
SUM: 10/05/100 HDC199763    2.256900  -21.3263380   276.1  0.350  0.319   10.4  0.603  0.662  0.71   1.56   1.56    75.3   351.6   266.4   86.4   96.3   94.3  2.224   0.92   2  0  OK 20.97099304 30.39593887 100
SUM: 10/05/100 HDC199763    2.264236  -21.2728906   272.2  0.316  0.286   11.0  0.573  0.629  0.72   1.56   1.56    75.3   351.6   266.4   86.4   96.3   94.3  2.220   0.94   0  0  OK 20.97099304 30.39593887 100
SUM: 10/05/100 HDC199763    2.271181  -21.1396865   272.4  0.350  0.322   11.4  0.605  0.666  0.71   1.56   1.56    75.3   351.6   266.4   86.4   96.3   94.3  2.220   0.94   1  0  OK 20.97099304 30.39593887 100
SUM: 10/05/100 HDC199763    2.278158  -21.0486798   275.4  0.260  0.234   10.9  0.571  0.665  0.80   1.56   1.56    75.3   351.6   266.4   86.4   96.3   94.3  2.222   0.88   4  0  OK 20.97099304 30.39593887 100
SUM: 10/05/100 HDC199763    2.285208  -20.9568405   266.0  0.377  0.351   10.7  0.647  0.688  0.60   1.56   1.56    75.3   351.6   266.4   86.4   96.3   94.3  2.221   0.98   0  0  OK 20.97099304 30.39593887 100

Here three sequential 130-second visibility scans on a calibrator (HDC182488), target (HDC174881), calibrator (HDC199763) sequence are represented by multiple (five) 25-second block averages for each of the visibility scan on the objects. wbCalib produces calibrated visibility estimates for these visibility data scans, and so it is responsible for consolidating the constituent blocks into scan averages. The time window over which these blocks are averaged is by default 150 sec, but this can be controlled by the command-line argument scanThreshold (see Arguments). This process is straightforward, and results in scans being represented by unweighted averages and sample standard deviations in the visibility data. wbCalib performs all subsequent visibility arithmetic in terms of these scan averages and standard deviations.

Next: , Previous: Data_Management, Up: Algorithm

2.3 System Visibility Estimation

As described above in Algorithm, the system visibility estimate V^2_sys is the point source response of an imperfect interferometer in an imperfect environment. We generally consider V^2_sys to be a function of both time and sky location, so the problem at hand in calibrating a given target scan is estimating V^2_sys at the time of the target scan, and sky location of the target. Once that estimate has been made, it is simple to apply Eq. 2.2 to estimate the ideal target visibility amplitude. We shall describe the algorithm used by wbCalib to make this estimate in a series of steps.

First consider an individual scan at time t_i on an unresolved calibrator (one for whom the expected visibility amplitude V^2_cal is one). After computing a scan average, the estimation of V^2_sys at time t_i and the sky location of the calibrator of course follows straightforwardly from inverting Eq. 2.1:

(2.3) V^2_sys = V^2_meas / V^2_cal => V^2_meas

However, if this calibrator has the potential to be resolved by the interferometer V^2_cal is no longer necessarily one, and the denominator in Eq. 2.3 becomes nontrivial. wbCalib treats potentially resolved calibrators a uniformly bright disk sources of a given (in fact user specified) angular diameter theta_cal. It is straightforward to compute V^2_cal for the uniform disk of diameter theta_cal (a tedious derivation is given in http://olbin.jpl.nasa.gov/iss1999/coursenotes/chapts1and2.pdf):

(2.4) V^2_cal = (2 J_1(pi * theta_cal * B_proj / lambda ) / (pi * theta_cal * B_proj / lambda))^2

With B_proj the projected baseline on the calibrator object (see Astrometric_Processing), lambda the operating wavelength, and J_1 the first order Bessel function of the first kind. With such a V^2_cal, the last equality in Eq. 2.3 falls away, and we are left with V^2_sys = V^2_meas / V^2_cal as our estimate for V^2_sys at t_i and the calibrator sky position.

As our fundamental goal is to estimate (and apply) the system visibility at the time t_target and sky location of a given target scan, we can use our knowledge of V^2_sys at t_i as the basis for that estimate. In general the system visibility V^2_sys evolves with time and sky position; we are left with the problem of formulating an estimator for V^2_sys at t_target and sky location of the target given some set of calibrator observations taken at t_i. Speaking momentarily of just a single calibrator, it is simple to imagine a weighted averaging scheme that would take a set of calibrator observations at times t_i and estimate the system visibility at time t_target:

(2.5) V^2_sys(t_target) = Sum(V^2_sys(t_i) w_i) / Sum(w_i)

but this begs the question of what are the weights w_i? A simple weighted averaging scheme (derived by a minimum variance optimality criterion) uses weights as the inverse variance of the quantity estimate:

(2.6) w_i = [sigma^2_V^2_sys(t_i)]^-1

The quantity sigma^2_V^2_sys(t_i) can be straightforwardly estimated from the scan averaging on and variance of the calibrator scan at t_i (see discussion in Data_Management). A second, related question is to ask what about Eq. 2.5 makes it specific to time t_target? There are two possible answers to that question – both related to the postulate that the instrumental and environmental conditions tend to be more correlated on small timescales than on long timescales. The first answer is to sample calibrator scans over a time interval symmetric around the science target scan time – that way any temporal variability in the system visibility (V^2_sys(t)) is at least sampled in an approximately unbiased way, and a weighted averaging process (Eq. 2.5) estimating the system visibility will similarly be unbiased. wbCalib always constructs a symmetric time window around the science scan time in order to select calibrator scans. The user can control the size of this time window; the default value is one hour.

The second variant on the weighted averaging scheme used by wbCalib is to manipulate the weights (Eq. 2.6) used in the averaging process so as to more heavily weight calibrator scans temporarily near the science scan time. Our experience at PTI indicates that a scheme that doubles the error variance in one hour works well; this is easily accomplished with a slight modification to Eq. 2.6:

w_i = [sigma^2_V^2_sys(t_i) * (1 + (t_i - t_target)^2 / T_C^2)]^-1

Here T_C is the by default the one hour scan time window, but it is under the control of the the user, so wbCalib can supply a continuum of weighting schemes (including a nearly flat weighting scheme where T_C is large). This time weighting scheme can be turned off with the -notimeWeighting argument (see Arguments).

As we argued above, as well as being a function of time the system visibility can be a function of sky location. Thus one is motivated to chose calibration sources that are close to the science target; this is the primary operating model for getCal. Like the temporal argument, wbCalib provides a variance weighting scheme to emphasize spatially-near calibrators over those more distant:

(2.7) w_i = [sigma^2_V^2_sys(t_i) * (1 + (t_i - t_target)^2 / T_C^2) * (1 + angBetween(s_target,s_i) / Ang_C]^-1

By default, Ang_C is set to 15 degrees, but like T_C, Ang_C is controled by the user, so a nearly flat weighting scheme is again possible with Ang_C large. As before, this angle weighting scheme can be turned off with the -noangleProxWeighting switch (see Arguments).

2.4 Error Estimation

Error estimation in wbCalib follows from standard error propagation calculations. From Eq. 2.2 the estimated calibrated scan variance is given by:

(2.8) sigma^2_V^2_target = (V^2_meas / V^2_sys)^2 sigma^2_V^2_meas + ((V^2_meas / (V^2_sys)^2)^2 sigma^2_V^2_sys

The target visibility measurement variance is simply that calculated in the scan averaging process described above in Data_Management. The system visibility variance is compute by the canonical weighted average recipe adopted in Eq. 2.5 (with weights as computed in Eq. 2.7):

(2.9) sigma^2_V^2_sys = [Sum(1 / w_i)]^-1

= [Sum([sigma^2_V^2_sys(t_i) * (1 + (t_i - t_target)^2 / T_C^2) * (1 + angBetween(s_target,s_i) / Ang_C]^-1)]^-1

In our experience with PTI the errors computed by wbCalib are correct relative to each other, and approximately correct in an absolute sense (in the sense that models fit to the calibrated data typically have chi-squareds close to one. In particular we are NOT by default weighting the computed scan errors by root-Nsamples in the input data – however this option is available if the user desires with the -useRootNWeighting command-line argument (see Arguments).

Previous: Error_Estimation, Up: Algorithm

2.5 Astrometric Processing

The estimated calibrator visibility calculations for wbCalib (Eq. 2.4) call for the projection of the interferometer baseline on the calibrator direction unit vector s_hat:

(2.10) B_proj = B \dot s_hat

where B is the instantaneous baseline 3-vector. wbCalib uses the Navy NOVAS astrometric subroutine package to evaluate target astrometry (see Dependencies), and it is convenient to evaluate Eq. 2.10 in true equitorial intertial coordinates. That implies that we have to evaluate B – fixed to the rotating earth. wbCalib uses the MSC baseline class library to evaluate the instantaneous baseline vector (see Dependencies).

Further, wbCalib users find it useful to have projected interferometer baseline coordinates (u-v coordinates times the operating wavelength) on the calibration target packaged with along the calibrated target scan – this feature is more a desirement rather than a requirement. So (in instances where users provide target astrometry in their calibration script files – in all the examples provided herein) wbCalib computes u-v coordinates for calibrated target scans. This calculation is straightforward:

(2.11) u = B \dot ra_hat

v = B \dot dec_hat

with ra_hat and dec_hat as unit vectors in the instaneous directions of increasing right ascention and declination at the target location respectively:

ra_hat = (z_hat \cross s_hat) / | z_hat \cross s_hat |

dec_hat = s_hat \cross ra_hat

and z_hat a unit vector in the direction of the Earth's angular momentum vector (e.g. (0,0,1) in equitorial intertial coordinates).

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3 Dependencies

wbCalib uses (requires) the libraries below for its build. Beginning with version 1.0, the libraries are packaged and distributed with wbCalib.

• NOVAS – Naval Observatory Vector Astrometry Subroutines – C version (http://riemann.usno.navy.mil/AA/software/novas/novas_info.html)
• libcoords.a – C++ NOVAS wrapper
• libbaseline.a – C++ NOVAS wrapper
• libmv.a – C++ 3-vector/rotation matrix library
• libtimeUtil.a – C++ time utilities
• GNU getopt – Enhanced-functionality getopt command-line options processor

The optional FITS interface requires external C and Perl libraries for FITS file manipulation. These are available from:

Next: , Previous: Dependencies, Up: Top

4 wbCalib Command-Line Arguments

The command line arguments to wbCalib are:

• -help: print a short help message and exit. No argument, overrides all other options when used.
• -Kband|Hband|allband: select data from specific bands to process. Default is all data in the input data.
• -frameRate argument: calibrate data at specific frame rates (Hz). At PTI the valid values for the frame rate argument are 50 and 100 Hz (20 ms and 10 ms frame times respectively). An argument value of zero (0) uses all available frame times, and this is the default setting.
• -defaultFile argument: read calibration default values from a file. String argument required when used.
• -[no]jitterCorrection[=argument]: set the [non]use of jitter correction in the visibility calibration. The form of the jitter correction is:

V^2 *= exp(jitterCoeff * jitter^2)

where jitter is the rms frame-to-frame phase first difference. The default is to use jitter correction with default jitterCoeff = 0.04. jitterCoeff can optionally be set with the form:

-jitterCorrection=jitterCoeff

The negative form of the command (-nojitterCorrection) takes no argument when used.

• -[no]ratioCorrection: set the [non]use of intensity ratio correction in the visibility calibration. The default is to not to use ratio correction. No argument when used. The form of the ratio correction is:

V^2 *= RC

where RC = (1+r)^2/(4r), and r is the ratio of fluxes contributed by each telescope.

• -fluxBiasCorrection: set the use of flux bias correction to the visibilities. The default is no flux bias correction.

The flux bias correction can only be used for Keck Interferometer data. The form of the correction is:

V^2 *= 1 - BiasCoeff * log10( (Flux + 20 DN) / ( 1020 DN) )

where BiasCoeff is 0.063 for wideband and 0.16 for spectrometer.

• -WL: calibrate visibilities from the "white light" (wide-band) side of the beam splitter. No argument when used,
• -Spectrometer: calibrate composite wide-band visibilities from the spectrometer side of the beam splitter. This is the default setting. No argument when used.
• -[in]coherentV2: calibrate [in]coherent visibilities. Default is to use the incoherent visibilities. No argument when used.
• -calTimeWindow argument: set the calibration scan acceptance time window size (hr). Default value is 1 hr. Argument required when used.
• -[no]timeWeighting: set the [non] use of time proximity weighting of calibration scans. Default is to use time weighting. No argument when used.
• -timeErrorVal argument: if timeWeighting, set calibration scan time weighting value (hr), the time over which the calibrator visibility measurement variance is assumed to double. Default value is 1 hr. Argument required when used.
• -[no]angleProxWeighting: set the [non] use of sky proximity weighting of calibration scans. Default is to use sky proximity weighting. No argument when used.
• -angleErrorVal argument: if angleProxWeighting, set target/calibrator sky separation weighting value (deg), the angle over which the calibrator visibility measurement variance is assumed to double. Default value is 15 deg. Argument required when used.
• -[no]minScanUncertainty argument: sets the [non] use of a minimum floor to the calibrated visibility uncertainty (V2 units). Default is to have no artificial minimum uncertainty. Argument required when used to set, no argument required when not using minimum.
• -[no]rootNWeighting: set the [non] use of root-Nsample weighting when computing composite scan uncertainties. Default is to not use root-Nsamples weighting. No argument when used.
• -scanThreshold argument: sets the time window used for sample consolidation into an interferometric "scan". Default is 150 sec, argument required when used (units of seconds).
• -chiWarning argument: set multiple calibrator system visibility inconsistency threshold for triggering a warning (sigmas – dimensionless). Default is to report system visibility estimate inconsistencies at or greater than the two (2) sigma level (assuming that reporting is on, see verbosity options below). Argument required when used.
• -chiReject argument: set multiple calibrator system visibility inconsistency threshold for rejecting a target scan (sigmas – dimensionless). Default is to reject the science scan if multiple system visibility estimates are inconsistent at or greater than the three (3) sigma level. Argument required when used.
• -delayCheck[=argument]: check delay values for targets and calibrators, and report warnings when data delay value is not in agreement with model calculation. Default is to check delay values. Optional floating point argument specifies delay check tolerance (units of meters) – default value is 0.1 meters (approximately 3 arcmin on a 100 m baseline).
• -nodelayCheck: do not check delay values for targets and calibrators. Default is to check delay values.
• -doCalibrators: causes the calibrator observations to also be calibrated. This option cannot be used if the -fits options is used.
• -fits[=argument]: causes wbCalib output to be in FITS format. Optional argument is output file name. If no argument is provided, default file name will be used, wbCalib-out.fits. The KI L2 FITS format conforms to the IAU working group standard for optical/infrared calibrated visibilities. This option cannot be used if the -doCalibrators option is used. This option requires that the CFITSIO library and the Astro::FITS::CFITSIO Perl extension be installed in your system.
• -baseline <file>: Use the baseline supplied in <file>. Normal behavior is to look for a yyyyddd.baseline file for each yyyyddd.sum file given on the command line. If that fails wbCalib looks for yyyyddd.bline. If neither is found wbCalib uses a built-in default baseline. By using the -baseline option the users can override this search behavior by providing the baseline or bline filename directly. Note that when multiple SUM files are given in combination with this option, all of them will use the baseline supplied in <file>.
• -quiet: set wbCalib to supress all informational messages and all but the most serious warnings and errors. No argument when used.
• -shutUp: the same as quiet.
• -normal: set wbCalib to report nominal status, warnings, and errors. This is the default. No argument when used.
• -verbose: set wbCalib to excessively report on status, warnings, and errors. (This should be used when the user runs into difficulty with wbCalib operation.) No argument when used.

note: wbCalib command-line options may be used in any order, interspersed with non-options (filename arguments) and abbreviated to uniqueness (e.g. -h, -ver, -jitter, -K).

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5 wbCalib Build/Install

wbCalib uses the GNU autotools configuration system for configuration, build, and install. wbCalib requires the set of libraries contained in mscLib (see Dependencies). Previously it was necessary to install this package seperately and then define the environment variables MSCLIB (MSCPATH/lib) and MSCINCLUDE (MSCPATH/include) to point at the libraries and headers for mscLib. With V2Calib version 1.0, nbCalib, wbCalib, and mscLib are packaged and built together. It is no longer necessary to define the environment variables or use the prescribed mscSoftware tree. Users who previously used the tree may continue to do so by defining the environment variables as they have before; for more information see http://nexsci.caltech.edu/software/mscSoftwareTree.

     > ./configure
> make
> make check (recommended)
> make install

If configure finds gnuplot in your PATH it will automatically build KvisPlot (package to create standard set of plots form KI data). To stop configure from building and installing KvisPlot, pass it the –enable-kvisplot=no option:

     > ./configure --enable-kvisplot=no

I don't like to put NExScI software in into standard /usr/local/bin (among other things this requires root access), so new in wbCalib v0.58dev the configure script attempts to autodetect the root of the MSC software tree (MSCPATH – /home/bode/mscSoftware in my installation), and to use this location as the installation prefix (if MSCPATH is not found the configuation reverts to the default /usr/local). As always, either default prefix can by overridden with a prefix option (/usr/myLocal in the following example) to the wbCalib configure script, as in:

     > ./configure --prefix=/usr/myLocal
> make
> make check (recommended)
> make install

Either way, wbCalib executables are placed in $prefix/bin, man pages in $prefix/man, info files in $prefix/info. Ancillary build targets are (courtesy of GNU autotools):  > make check (automated test) > make doc > make clean > make dist > make distclean  Finally, several forms of the documentation are available. The default part of the build procedure should make the info format documentation (wbCalib.info), but the following additional documentation types (targets) are also available:  > cd doc > make html > make ps > make pdf  NOTE: The L2 FITS output capability requires that the CFITSIO library and the Astro::FITS::CFITSIO Perl extension be installed in your system. During installation, the configure process will determine whether or not this is the case, and if it is not, the installation will not be affected, and all other features will run. Please see http://heasarc.gsfc.nasa.gov/docs/software/fitsio/fitsio.html for information on the CFITSIO library and http://search.cpan.org/~pratzlaff/ for information on the Perl extension module. Next: , Previous: Build, Up: Top 6 wbCalib Test Cases wbCalib comes distributed with the required input for a test case (the test subdirectory). This test is designed to be run with the the make check target as part of the installation sequence (see Build). The test scripts run wbCalib against a pre-packaged calibration script and L-1 data in the test directory and then run the output through diff to check consistency with reference output. What if "make check" fails? We have noted two common types of installation problem. In the first "make check" fails due to roundoff differences between the packaging platform and your installation platform. This is usually the case when diff identified only 1 or 2 lines of non-matching output for each test. You may proceed with installation ("make install") and use. With the second problem "make check" fails and diff seems to identify all lines from the reference test output files. The likely cause is that the runtime linker could not resolve library dependencies for the executable. We have seen this behavior with gcc3 on Solaris; One of the missing libraries is libstdc++.so. The root cause lies with the installation configuration of your gcc3. You can check for this linking error on Solaris by typing  > ldd <path/to/your/V2Calib>/Calib/wbcEngine  If you see one or more libraries missing you have two options: 1) set LD_RUN_PATH then "make clean" and "make" again, or 2) set your LD_LIBRARY_PATH when executing In order for either of these options to work, you will need the path to gcc3 libraries on your system. It some cases the gcc3 libraries and binaries will be installed subdirectories of a gcc3 package location (for example /usr/local/gcc3 might be the package location, with binaries found under /usr/local/gcc3/bin and libraries under /usr/local/gcc3/lib). In other cases binaries and libraries may be installed in generic locations (e.g. /opt/bin and /opt/lib). If you cannot find your gcc3 libraries, contact your local system administrator for help. We recommend option 1) because it can be done once at installation and no further environment settings are required. If you choose option 2) you must set your LD_LIBRARY_PATH each time you run nbCalib or wbCalib (or you may set it in your .cshrc). Assuming that you have found your gcc3 library path, option 1) proceeds as  > setenv LD_RUN_PATH <your/gcc3/libraries> # for c-shell, OR > LD_RUN_PATH="/usr/local/gcc3/lib"; export LD_RUN_PATH # for bash > make clean > make > make check > make install  Option 2) is  > setenv LD_LIBRARY_PATH <your/gcc3/libraries>:${LD_LIBRARY_PATH}    # for c-shell, OR
> LD_LIBRARY_PATH="<your/gcc3/libraries>:\$LD_LIRBRARY_PATH"; export LD_LIBRARY_PATH # for bash

Next: , Previous: Test, Up: Top

7 Common Errors & Troubleshooting

Here we provide a list of common wbCalib error messages, their interpretation, and suggestions about their correction.

• Calibration Script Parsing Errors

There are a variety of parsing errors that can occur when trying to compose a calibration script. It got tedious to the point that I decided to move most of the work to getCal. Use this feature if you can – it'll save you a lot of headaches.

Most of the parsing errors give some sort of diagnostic message, like:

          Fatal Error -- Can't open calibration script file bogus

or

          Fatal Error -- premature eof encountered in bo

Fixing those should be straightforward.

Outside of the drop-dead obvious, there's no real magic bullet here – if you're having parsing problems compare your cal script against the example given in Inputs_and_Outputs.

• Delay Calculation Errors

An uncommon error is reporting of delay mismatches in the data:

          Warning: at 54234.654, 10.2 m measured/expected delay mismatch on target HDC9939

This has the potential of being serious, because it usually means there's some problem with the baseline model that's being applied, and at a minimum you're guaranteed that the u-v projections that wbCalib is calculating will be wrong. Less frequently it is also possible that there is some intrinsic problem with the data like a glitch in the delayline metrology card, or some mistake was made in composing the observing schedule, and target designations and astrometry got mixed up (just such an occurance was the initial impetus for writing getCal).

The first thing to do is to re-run wbCalib with the -verbose switch turned on. This will enable all the baseline inforamational messages, and you can check to see whether there are any irregularities in the baseline information:

          ## Error detected in /home/bode/pti/data/100/100335.bline; reading baseline model from /home/bode/pti/data/100/100335.baseline (PTI_NS)

The obvious thing to do in this circumstance is to make sure that the baseline model that gets assigned (e.g. PTI_NS) is what you think it should be. If not, you should start looking at the measurement database and see what the deal is with getting the baseline into wbCalib.

• Multiple Calibrator Inconsistency Warnings & Errors

When using multiple calibrators, wbCalib checks for system visibility estimate consistency among the calibrators. Consequently, probably the most common type of warning and error messages you'll see from wbCalib will be of the flavor:

          # Warning: at 51724.1534 sysVis estimates between calibrators HDC121107 and HDC128167 are inconsistent at the 2.7 sigma level

and

          # Reject: at 51724.1680 system visibility estimates between calibrators HDC121107 and HDC128167 are inconsistent at the 4.5 sigma level
# No suitable calibration found for 51724.1680

As discussed in Arguments, the user can control the thresholds at which these warnings and errors are reported. However, it's not an accident that we put this feature in wbCalib, and strong inconsistencies between calibrators can be a sign of either problems with the data or one of your calibrators. Old fashion detective work is about the only general advice we can provide here.

Next: , Previous: Common_Errors, Up: Top

8 Ancillary Scripts and Programs

wbCalib comes with several "helper" applications that make it a bit more civiilized. These can be built at the users option (see Build instructions above).

Previous: Ancillary, Up: Ancillary

badScans is a Perl script that allows the user to select certain scans for exclusion in subsequent processing and analyses. badScans is a very general filter for text files: it scans a stdin input text file, and looks for lines that contain one of an input list of exclusion keys. When it finds a line that contains an exclusion key, it prepends the line with a comment character (pound/hash sign – #). The last output from badScans lists the number of exclusions found.

or for instance in the context of a wbCalib invocation:

Previous: Ancillary, Up: Ancillary

8.2 makeCalScript

makeCalScript is a Perl script that allows the user compose a calibration script given a target name and a schedule file.

The usage of makeCalScript is:

> makeCalScript target catfile > outfile

Please note that the NExScI package getCal, also contains an utility for automatic generation of calibration scripts (csAdhoc). Please refer to the getCal documentation (http://nexsci.caltech.edu/software/getCal/index.html#gcCalScript).

Previous: Ancillary, Up: Ancillary

8.3 KvisPlot

KvisPlot is a Perl package which generates a standard set of plots for KI data. KvisPlot will be automatically built and installed if gnuplot is found in your PATH during configuration.

The usage of KvisPlot is:

     > KvisPlot -help
usage: KvisPlot filename
options:  -all (creates series of plots from sumfile)
-one (creates one plot of specified columns from any Kvis file)
-gif produces gif output files
-png produces png output files
-hc produces postscript output files
-spec (creates series of plots from specfile)
-anc ancfile (creates ancilliary plots, used only with all)
-bt btfile (creates block trending plots, used only with all)
-stdin (takes input data from stdin)
-help  (prints this message)

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wbCalib v1.4.4

Copyright 2007 California Institute of Technology. For questions or comments about this software, please fill out the contact form at http://nexsci.caltech.edu/contact/help_desk.html. Permission to use, copy and distribute this software and its documentation for academic and/or non-profit research purposes, without fee and without a written agreement is hereby granted, provided that the above copyright notice, this paragraph and the following three paragraphs appear in all copies. Reuse of all or part of the software contained here for commercial purposes is strictly prohibited.

The California Institute of Technology makes no proprietary claims to the results, prototypes, or systems supporting and/or necessary for the use of the research, results and/or prototypes for academic and/or non-profit research uses only. However, to the extent that any software and system built in collaboration with industry partners may incorporate proprietary designs of the industry partners, it is possible that certain restrictions may be imposed on the proprietary information.

In no event shall California Institute of Technology be liable to any party for direct, indirect, special, incidental or consequential damages, including lost profits, arising out of the use of this software and its documentation, even if the California Institute of Technology has been advised of the possibility of such damage.

The California Institute of Technology specifically disclaims any warranties, including the implied warranties or merchantability and fitness for a particular purpose. The software and documentation provided hereunder is on an "as is" basis, and the California Institute of Technology has no obligations to provide maintenance, support, updates, enhancements or modifications.


text:
## Traldi, Lorenzo

Compute Distance To:
 Author ID: traldi.lorenzo Published as: Traldi, Lorenzo; Traldi, L. Homepage: http://sites.lafayette.edu/traldil/ External Links: MGP · ORCID
 Documents Indexed: 88 Publications since 1981 Reviewing Activity: 127 Reviews Co-Authors: 30 Co-Authors with 30 Joint Publications 492 Co-Co-Authors
all top 5

### Co-Authors

 57 single-authored 6 Brijder, Robert 3 Gordon, Gary 3 Smith, Derek A. 3 Zulli, Louis 2 Backman, Spencer 2 Ella, W. 2 Follett, Michael 2 Gilson, E. 2 Hopkins, Samuel Francis 2 Massey, William S. 2 Rowe, J. P. 1 Andersen, R. E. 1 Balan, A. O. 1 Benashski, Jessica K. 1 Blasiak, Jonah 1 Cutler, Jonathan 1 Dinitz, Michael H. 1 Ellis-Monaghan, Joanna A. 1 Gold, Jerrold M. 1 Hoogeboom, Hendrik Jan 1 Kronenthal, Brian G. 1 Martin, Ryan R. 1 Moore, Justin T. 1 Radcliffe, Andrew John 1 Sakuma, Makoto 1 Sharkey, Thomas C. 1 Shtylla, Blerta 1 Silver, Daniel S. 1 Williams, Susan G. 1 Yacobi, Oded
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### Serials

 14 Journal of Knot Theory and its Ramifications 10 Congressus Numerantium 9 Discrete Mathematics 5 European Journal of Combinatorics 5 The Electronic Journal of Combinatorics 4 JCMCC. The Journal of Combinatorial Mathematics and Combinatorial Computing 3 Pacific Journal of Mathematics 3 Transactions of the American Mathematical Society 3 Linear Algebra and its Applications 3 Combinatorics, Probability and Computing 2 Ars Combinatoria 2 Mathematics Seminar Notes 2 Networks 2 Proceedings of the American Mathematical Society 2 Topology and its Applications 2 Advances in Applied Mathematics 2 Kobe Journal of Mathematics 2 Bulletin of the Institute of Combinatorics and its Applications 1 Discrete Applied Mathematics 1 Houston Journal of Mathematics 1 Algebra Universalis 1 Journal of Combinatorial Theory. Series B 1 Mathematische Zeitschrift 1 Osaka Journal of Mathematics 1 Social Choice and Welfare 1 Discussiones Mathematicae. Graph Theory 1 Contributions to Discrete Mathematics 1 Applied Mathematical and Computational Sciences
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### Fields

 54 Combinatorics (05-XX) 31 Manifolds and cell complexes (57-XX) 6 Convex and discrete geometry (52-XX) 4 Commutative algebra (13-XX) 4 Group theory and generalizations (20-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Computer science (68-XX) 2 Operations research, mathematical programming (90-XX) 2 Information and communication theory, circuits (94-XX) 1 Associative rings and algebras (16-XX) 1 Probability theory and stochastic processes (60-XX)

### Citations contained in zbMATH Open

59 Publications have been cited 276 times in 156 Documents Cited by Year
A dichromatic polynomial for weighted graphs and link polynomials. Zbl 0713.57003
Traldi, Lorenzo
1989
Generalized activities and the Tutte polynomial. Zbl 0742.05022
Gordon, Gary; Traldi, Lorenzo
1990
Milnor’s invariants and the completions of link modules. Zbl 0525.57004
Traldi, Lorenzo
1984
Binary nullity, Euler circuits and interlace polynomials. Zbl 1227.05168
Traldi, Lorenzo
2011
The transition matroid of a 4-regular graph: an introduction. Zbl 1319.05034
Traldi, Lorenzo
2015
A bracket polynomial for graphs. I. Zbl 1204.57009
Traldi, Lorenzo; Zulli, Louis
2009
On the Goertiz matrix of a link. Zbl 0536.57003
Traldi, Lorenzo
1985
Binary matroids and local complementation. Zbl 1304.05015
Traldi, Lorenzo
2015
A bracket polynomial for graphs. II: Links, Euler circuits and marked graphs. Zbl 1192.57008
Traldi, Lorenzo
2010
Chain polynomials and Tutte polynomials. Zbl 0997.05038
Traldi, Lorenzo
2002
On the colored Tutte polynomial of a graph of bounded treewidth. Zbl 1091.05027
Traldi, Lorenzo
2006
Parametrized Tutte polynomials of graphs and matroids. Zbl 1108.05024
Ellis-Monaghan, Joanna A.; Traldi, Lorenzo
2006
The adjacency matroid of a graph. Zbl 1298.05210
Traldi, Lorenzo; Brijder, Robert; Hoogeboom, Hendrik Jan
2013
Fourientation activities and the Tutte polynomial. Zbl 1371.05133
Backman, Spencer; Hopkins, Sam; Traldi, Lorenzo
2018
On the $$\beta$$-invariant for graphs. Zbl 0905.05079
Benashski, Jessica K.; Martin, Ryan R.; Moore, Justin T.; Traldi, Lorenzo
1995
Parallel connections and coloured Tutte polynomials. Zbl 1069.05021
Traldi, Lorenzo
2005
On the interlace polynomials. Zbl 1257.05070
Traldi, Lorenzo
2013
On the realization of double occurrence words. Zbl 1205.05006
Shtylla, B.; Traldi, L.; Zulli, L.
2009
On a conjecture of K. Murasugi. Zbl 0563.57003
Massey, William S.; Traldi, Lorenzo
1986
Series and parallel reductions for the Tutte polynomial. Zbl 0995.05031
Traldi, Lorenzo
2000
Isotropic matroids. I: Multimatroids and neighborhoods. Zbl 1351.05043
Brijder, Robert; Traldi, Lorenzo
2016
Isotropic matroids. II: Circle graphs. Zbl 1351.05044
Brijder, Robert; Traldi, Lorenzo
2016
Interlacement in 4-regular graphs: a new approach using nonsymmetric matrices. Zbl 1317.05092
Traldi, Lorenzo
2014
Goeritz and Seifert matrices from Dehn presentations. Zbl 1448.57010
Silver, Daniel S.; Traldi, Lorenzo; Williams, Susan G.
2020
Conway’s potential function and its Taylor series. Zbl 0732.57007
Traldi, Lorenzo
1988
A generalization of Torres’ second relation. Zbl 0491.57004
Traldi, Lorenzo
1982
Polynomials for directed graphs. Zbl 0801.05030
Gordon, Gary; Traldi, Lorenzo
1993
Weighted interlace polynomials. Zbl 1216.05059
Traldi, Lorenzo
2010
On the linear algebra of local complementation. Zbl 1236.05107
Traldi, Lorenzo
2012
A subset expansion of the coloured Tutte polynomial. Zbl 1049.05024
Traldi, Lorenzo
2004
A bracket polynomial for graphs. III: Vertex weights. Zbl 1218.57010
Traldi, Lorenzo
2011
Multivariate Alexander quandles. III. Sublinks. Zbl 1436.57011
Traldi, Lorenzo
2019
Multivariate Alexander quandles. I: The module sequence of a link. Zbl 1440.57011
Traldi, Lorenzo
2020
Tied dice. II. Some asymptotic results. Zbl 1305.05003
Cooley, C.; Ella, W.; Follett, M.; Gilson, E.; Traldi, L.
2014
Clutters and circuits. Zbl 0868.05016
Traldi, Lorenzo
1997
$$(K,j)$$-domination and $$(K,j)$$-reliability. Zbl 0888.90077
Rodriguez, Jose; Traldi, Lorenzo
1997
Generalized activities and reliability. Zbl 1067.05019
Traldi, Lorenzo
2005
Traldi, Lorenzo; Sakuma, Makoto
1983
A note on reliability and expected value. Zbl 0955.05023
Traldi, Lorenzo
1998
The determinantal ideals of link modules. I. Zbl 0489.57001
Traldi, Lorenzo
1982
Multivariate Alexander colorings. Zbl 1405.57018
Traldi, Lorenzo
2018
Notes on a theorem of Naji. Zbl 1347.05096
Traldi, Lorenzo
2017
Link colorings and the Goeritz matrix. Zbl 1372.57023
Traldi, Lorenzo
2017
Isotropic matroids. III: Connectivity. Zbl 1366.05062
Brijder, Robert; Traldi, Lorenzo
2017
Erratum to: “Multivariate Alexander quandles. II. The involutory medial quandle of a link”. Zbl 1460.57007
Traldi, Lorenzo
2020
Duality and the signed Laplacian matrix of a graph. Zbl 1385.05052
Smith, Derek A.; Traldi, Lorenzo; Watkins, William
2018
On the star-delta transformation in network reliability. Zbl 0777.90015
Traldi, Lorenzo
1993
Clutters and circuits. II. Zbl 0911.05025
Traldi, Lorenzo
1998
Addendum to: “Polynomials for directed graphs”. Zbl 0835.05031
Gordon, Gary; Traldi, Lorenzo
1994
Dice games and Arrow’s theorem. Zbl 1129.91302
Traldi, Lorenzo
2006
Clutters and circuits. III. Zbl 1090.05014
Traldi, Lorenzo
2003
A bracket polynomial for graphs. IV: Undirected Euler circuits, graph-links and multiply marked graphs. Zbl 1234.57011
Traldi, Lorenzo
2011
Generalized activities and $$K$$-terminal reliability. Zbl 0752.05054
Traldi, Lorenzo
1991
The determinantal ideals of link modules. II. Zbl 0479.57003
Traldi, Lorenzo
1983
Traldi, Lorenzo
1983
Linking numbers and Chen groups. Zbl 0677.57007
Traldi, Lorenzo
1989
On the interlace polynomials of forests. Zbl 1229.05119
Anderson, C.; Cutler, J.; Radcliffe, A. J.; Traldi, L.
2010
Splitting cubic circle graphs. Zbl 1339.05327
Traldi, Lorenzo
2016
A characterization of circle graphs in terms of multimatroid representations. Zbl 1431.05032
Brijder, Robert; Traldi, Lorenzo
2020
Goeritz and Seifert matrices from Dehn presentations. Zbl 1448.57010
Silver, Daniel S.; Traldi, Lorenzo; Williams, Susan G.
2020
Multivariate Alexander quandles. I: The module sequence of a link. Zbl 1440.57011
Traldi, Lorenzo
2020
Erratum to: “Multivariate Alexander quandles. II. The involutory medial quandle of a link”. Zbl 1460.57007
Traldi, Lorenzo
2020
A characterization of circle graphs in terms of multimatroid representations. Zbl 1431.05032
Brijder, Robert; Traldi, Lorenzo
2020
Multivariate Alexander quandles. III. Sublinks. Zbl 1436.57011
Traldi, Lorenzo
2019
Fourientation activities and the Tutte polynomial. Zbl 1371.05133
Backman, Spencer; Hopkins, Sam; Traldi, Lorenzo
2018
Multivariate Alexander colorings. Zbl 1405.57018
Traldi, Lorenzo
2018
Duality and the signed Laplacian matrix of a graph. Zbl 1385.05052
Smith, Derek A.; Traldi, Lorenzo; Watkins, William
2018
Notes on a theorem of Naji. Zbl 1347.05096
Traldi, Lorenzo
2017
Link colorings and the Goeritz matrix. Zbl 1372.57023
Traldi, Lorenzo
2017
Isotropic matroids. III: Connectivity. Zbl 1366.05062
Brijder, Robert; Traldi, Lorenzo
2017
Isotropic matroids. I: Multimatroids and neighborhoods. Zbl 1351.05043
Brijder, Robert; Traldi, Lorenzo
2016
Isotropic matroids. II: Circle graphs. Zbl 1351.05044
Brijder, Robert; Traldi, Lorenzo
2016
Splitting cubic circle graphs. Zbl 1339.05327
Traldi, Lorenzo
2016
The transition matroid of a 4-regular graph: an introduction. Zbl 1319.05034
Traldi, Lorenzo
2015
Binary matroids and local complementation. Zbl 1304.05015
Traldi, Lorenzo
2015
Interlacement in 4-regular graphs: a new approach using nonsymmetric matrices. Zbl 1317.05092
Traldi, Lorenzo
2014
Tied dice. II. Some asymptotic results. Zbl 1305.05003
Cooley, C.; Ella, W.; Follett, M.; Gilson, E.; Traldi, L.
2014
The adjacency matroid of a graph. Zbl 1298.05210
Traldi, Lorenzo; Brijder, Robert; Hoogeboom, Hendrik Jan
2013
On the interlace polynomials. Zbl 1257.05070
Traldi, Lorenzo
2013
On the linear algebra of local complementation. Zbl 1236.05107
Traldi, Lorenzo
2012
Binary nullity, Euler circuits and interlace polynomials. Zbl 1227.05168
Traldi, Lorenzo
2011
A bracket polynomial for graphs. III: Vertex weights. Zbl 1218.57010
Traldi, Lorenzo
2011
A bracket polynomial for graphs. IV: Undirected Euler circuits, graph-links and multiply marked graphs. Zbl 1234.57011
Traldi, Lorenzo
2011
A bracket polynomial for graphs. II: Links, Euler circuits and marked graphs. Zbl 1192.57008
Traldi, Lorenzo
2010
Weighted interlace polynomials. Zbl 1216.05059
Traldi, Lorenzo
2010
On the interlace polynomials of forests. Zbl 1229.05119
Anderson, C.; Cutler, J.; Radcliffe, A. J.; Traldi, L.
2010
A bracket polynomial for graphs. I. Zbl 1204.57009
Traldi, Lorenzo; Zulli, Louis
2009
On the realization of double occurrence words. Zbl 1205.05006
Shtylla, B.; Traldi, L.; Zulli, L.
2009
On the colored Tutte polynomial of a graph of bounded treewidth. Zbl 1091.05027
Traldi, Lorenzo
2006
Parametrized Tutte polynomials of graphs and matroids. Zbl 1108.05024
Ellis-Monaghan, Joanna A.; Traldi, Lorenzo
2006
Dice games and Arrow’s theorem. Zbl 1129.91302
Traldi, Lorenzo
2006
Parallel connections and coloured Tutte polynomials. Zbl 1069.05021
Traldi, Lorenzo
2005
Generalized activities and reliability. Zbl 1067.05019
Traldi, Lorenzo
2005
A subset expansion of the coloured Tutte polynomial. Zbl 1049.05024
Traldi, Lorenzo
2004
Clutters and circuits. III. Zbl 1090.05014
Traldi, Lorenzo
2003
Chain polynomials and Tutte polynomials. Zbl 0997.05038
Traldi, Lorenzo
2002
Series and parallel reductions for the Tutte polynomial. Zbl 0995.05031
Traldi, Lorenzo
2000
A note on reliability and expected value. Zbl 0955.05023
Traldi, Lorenzo
1998
Clutters and circuits. II. Zbl 0911.05025
Traldi, Lorenzo
1998
Clutters and circuits. Zbl 0868.05016
Traldi, Lorenzo
1997
$$(K,j)$$-domination and $$(K,j)$$-reliability. Zbl 0888.90077
Rodriguez, Jose; Traldi, Lorenzo
1997
On the $$\beta$$-invariant for graphs. Zbl 0905.05079
Benashski, Jessica K.; Martin, Ryan R.; Moore, Justin T.; Traldi, Lorenzo
1995
Addendum to: “Polynomials for directed graphs”. Zbl 0835.05031
Gordon, Gary; Traldi, Lorenzo
1994
Polynomials for directed graphs. Zbl 0801.05030
Gordon, Gary; Traldi, Lorenzo
1993
On the star-delta transformation in network reliability. Zbl 0777.90015
Traldi, Lorenzo
1993
Generalized activities and $$K$$-terminal reliability. Zbl 0752.05054
Traldi, Lorenzo
1991
Generalized activities and the Tutte polynomial. Zbl 0742.05022
Gordon, Gary; Traldi, Lorenzo
1990
A dichromatic polynomial for weighted graphs and link polynomials. Zbl 0713.57003
Traldi, Lorenzo
1989
Linking numbers and Chen groups. Zbl 0677.57007
Traldi, Lorenzo
1989
Conway’s potential function and its Taylor series. Zbl 0732.57007
Traldi, Lorenzo
1988
On a conjecture of K. Murasugi. Zbl 0563.57003
Massey, William S.; Traldi, Lorenzo
1986
On the Goertiz matrix of a link. Zbl 0536.57003
Traldi, Lorenzo
1985
Milnor’s invariants and the completions of link modules. Zbl 0525.57004
Traldi, Lorenzo
1984
Traldi, Lorenzo; Sakuma, Makoto
1983
The determinantal ideals of link modules. II. Zbl 0479.57003
Traldi, Lorenzo
1983
Traldi, Lorenzo
1983
A generalization of Torres’ second relation. Zbl 0491.57004
Traldi, Lorenzo
1982
The determinantal ideals of link modules. I. Zbl 0489.57001
Traldi, Lorenzo
1982
all top 5

### Cited by 165 Authors

 38 Traldi, Lorenzo 11 Brijder, Robert 9 Moffatt, Iain 6 Gordon, Gary 5 Il’yutko, Denis Petrovich 5 Makowsky, Johann-Andreas 4 Backman, Spencer 4 Silver, Daniel S. 4 Williams, Susan G. 3 Britz, Thomas 3 Cochran, Thomas (Tim) Daniel 3 Ellis-Monaghan, Joanna A. 3 Hoogeboom, Hendrik Jan 3 Jin, Xian’an 3 Manturov, Vassiliĭ Olegovich 3 Noble, Steven Derek 3 Przytycki, Józef H. 3 Suciu, Alexander I. 3 Zhang, Fuji 2 Averbouch, Ilia 2 Bernardi, Olivier 2 Chun, Carolyn 2 Diao, Yuanan 2 Gioan, Emeric 2 Godlin, Benny 2 Hopkins, Samuel Francis 2 Huggett, Stephen A. 2 Kochol, Martin 2 Las Vergnas, Michel 2 Li, Guisong 2 McMahon, Elizabeth W. 2 Smith, Derek A. 2 Vignes-Tourneret, Fabien 2 Wang, He 2 Wu, Haidong 1 Algaba, Antonio 1 Allagan, Julian A. D. 1 Awan, Jordan 1 Bailey, Alison 1 Beaudin, Laura 1 Ben Geloun, Joseph 1 Benedetti, Riccardo 1 Bläser, Markus 1 Bollobás, Béla 1 Bonin, Joseph E. 1 Bordewich, Magnus 1 Brunat, Josep M. 1 Burgos, Juan Manuel 1 Butler, Clark 1 Caravelli, Francesco 1 Chan, Swee Hong 1 Chernyak, Arkady A. 1 Chmutov, Sergei V. 1 Chrisman, Micah Whitney 1 Cohen, Daniel C. 1 Conrey, John Brian 1 de Mier, Anna 1 Duan, Yinghua 1 Dupont, Clément 1 Duzhin, Sergeĭ Vasil’evich 1 Eisenbud, David 1 Ella, W. 1 Ernst, Claus 1 Esperet, Louis 1 Eubanks-Turner, Christina 1 Farr, Graham E. 1 Fernández-Sánchez, Fernando 1 Fink, Alex 1 Fischer, Eldar 1 Follett, Michael 1 Friedl, Stefan 1 Frigerio, Roberto 1 Funk, Daryl 1 Gabbard, James 1 Gibson, Andrew 1 Gillespie, Bryan R. 1 Gilson, E. 1 Gitler, Isidoro 1 Grant, Katie 1 Guedes de Oliveira, António 1 Hetyei, Gábor 1 Hoffmann, Christian 1 Huang, Aiping 1 Huseby, Arne Bang 1 Ikeda, Toru 1 Im, Young Ho 1 Ito, Noboru 1 Jonoska, Nataša 1 Kadokami, Teruhisa 1 Kaiser, Uwe 1 Kálmán, Tamás 1 Kang, Ross J. 1 Kato, Keiju 1 Kawauchi, Akio 1 Kim, Hwee 1 Kobayashi, Tsuyoshi 1 Komisarski, Andrzej 1 Lee, Kyeonghui 1 Lee, Sang Youl 1 Lee, Sooyeon ...and 65 more Authors
all top 5

### Cited in 48 Serials

 20 Journal of Knot Theory and its Ramifications 17 Discrete Mathematics 13 European Journal of Combinatorics 12 Advances in Applied Mathematics 7 Discrete Applied Mathematics 7 Transactions of the American Mathematical Society 7 Topology and its Applications 7 The Electronic Journal of Combinatorics 4 Mathematical Proceedings of the Cambridge Philosophical Society 3 Advances in Mathematics 3 Journal of Combinatorial Theory. Series B 3 Proceedings of the American Mathematical Society 3 Linear Algebra and its Applications 3 Combinatorics, Probability and Computing 3 Journal of Mathematical Sciences (New York) 2 Journal of Statistical Physics 2 Inventiones Mathematicae 2 Journal of Combinatorial Theory. Series A 2 Mathematische Zeitschrift 2 Theoretical Computer Science 2 Journal of Algebraic Combinatorics 2 Electronic Journal of Graph Theory and Applications 2 Arnold Mathematical Journal 1 American Mathematical Monthly 1 Mathematical Notes 1 Periodica Mathematica Hungarica 1 Physica A 1 Mathematics Magazine 1 Information Sciences 1 Journal of Geometry 1 Journal of Pure and Applied Algebra 1 Manuscripta Mathematica 1 Osaka Journal of Mathematics 1 Proceedings of the Japan Academy. Series A 1 Social Choice and Welfare 1 Algorithmica 1 International Journal of Foundations of Computer Science 1 Annales Mathématiques Blaise Pascal 1 The New York Journal of Mathematics 1 Journal of Mathematical Chemistry 1 Geometry & Topology 1 Annals of Combinatorics 1 AKCE International Journal of Graphs and Combinatorics 1 Mathematics in Computer Science 1 Science China. Mathematics 1 Forum of Mathematics, Sigma 1 Research in the Mathematical Sciences 1 Algebraic Combinatorics
all top 5

### Cited in 23 Fields

 102 Combinatorics (05-XX) 68 Manifolds and cell complexes (57-XX) 21 Convex and discrete geometry (52-XX) 12 Computer science (68-XX) 7 Statistical mechanics, structure of matter (82-XX) 5 Group theory and generalizations (20-XX) 4 Associative rings and algebras (16-XX) 4 Information and communication theory, circuits (94-XX) 3 Order, lattices, ordered algebraic structures (06-XX) 3 Algebraic geometry (14-XX) 3 Probability theory and stochastic processes (60-XX) 3 Biology and other natural sciences (92-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Algebraic topology (55-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 General and overarching topics; collections (00-XX) 1 Mathematical logic and foundations (03-XX) 1 Commutative algebra (13-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Nonassociative rings and algebras (17-XX) 1 Special functions (33-XX) 1 Ordinary differential equations (34-XX) 1 Operations research, mathematical programming (90-XX)


text:
# Compact lists with paralist

The space occupied by list environments in LaTeX (itemize, enumerate, etc.) is sometimes a bit too much. For example, when you need to save space in your document or simply because you don’t like the looks of the standard list environments. In this post, the paralist package will be used to create compact lists. This will be done by elimination of vertical spacing and by introducing multicolumn lists (with the help of the multicol package). In the end, you will be able to create great looking lists that do not take up an unnecessary amount of space in your document.

## Compact lists

Let’s start by creating a compact list. In order to do so, call the paralist package in your preamble:

\usepackage{paralist}

With the paralist package, a few new environments are introduced. The ones we will be using in this post are: compactitem and compactenum. As their name imply, the first creates a compact itemized list and the latter creates a compact numbered list. Run the following code to see the difference:

\documentclass{article} \usepackage{lipsum} \usepackage{paralist} \begin{document} \lipsum[1] \begin{itemize} \item This is entry number one \item This is entry number two \end{itemize} \lipsum[2] \begin{compactitem} \item This is entry number one, compact \item This is entry number two, compact \end{compactitem} \lipsum[3] \end{document}

In here, the lipsum package is used to create dummy text.

## Multicolumn lists

Let’s extend the knowledge from above to multicolumn lists. Multicolumn list can save a lot of space in your document and can look a lot nicer if the list entries are short. The idea is pretty simple: We’ll define a three column environment that contains the list environment. This can be done with the multicol package. By default, a multicolumn environment introduces additional spacing above and underneath. To remove this, implement the following in your preamble:

\usepackage{multicol} \setlength{\multicolsep}{0pt}

Now, a compact multicolumn list can be created with the following code:

\documentclass{article} \usepackage{lipsum} \usepackage{paralist} \usepackage{multicol} \setlength{\multicolsep}{0pt} \begin{document} \lipsum[1] \begin{multicols}{3} \begin{compactitem} \item One \item Two \item Three \item Four \item Five \item \ldots \end{compactitem} \end{multicols} \lipsum[2] \end{document}

That’s it!

## Alternatives

For compact list environments, the method described above is my preference. However, a lot of other options are available. As found here the popular enumitem package can also be used, as well as shortlst and mdwlist.

# 1 Comment

1. lorretta says:

Greetings! I’ve been following your web site for some time
now and finally got the courage to go ahead and give you a shout out from Lubbock Tx!
Just wanted to say keep up the fantastic work!


text:
Diffusion term

Discretisation of Diffusive Term

1. Description

A control volume in mesh is made up of set of faces enclosing it. The figure 1.1 shows a typical situation. Where A represent the magnitude of area of the face. And n represents the normal unit vector of the face under consideration.

Figure 1.1
$\vec r_{0}$ and $\vec r_{1}$ are position vector of centroids of cells cell 0 and cell 1 respectively.
${\rm{d\vec s}} = \vec r_{1} - \vec r_{0}$

We wish to approaximate $D_f = \Gamma _f \nabla \phi _f \bullet {\rm{\vec A}}$ at the face.

2. Approach 1

We define vector $\vec \alpha {\rm{ = }}\frac{{{\rm{\vec A}}}}{{{\rm{\vec A}} \bullet {\rm{d\vec s}}}}$

giving us the expression:

$D_f = \Gamma _f \nabla \phi _f \bullet {\rm{\vec A = }}\Gamma _{\rm{f}} \left[ {\left( {\phi _1 - \phi _0 } \right)\vec \alpha \bullet {\rm{\vec A + }}\bar \nabla \phi \bullet {\rm{\vec A - }}\left( {\bar \nabla \phi \bullet {\rm{d\vec s}}} \right)\vec \alpha \bullet {\rm{\vec A}}} \right]$

where $\bar \nabla \phi _f$ and $\Gamma _f$ are suitable face averages.


text:
# 4.5: The Eigenfunctions of Operators are Orthogonal

Consideration of the quantum mechanical description of the particle-in-a-box exposed two important properties of quantum mechanical systems. We saw that the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. We now examine the generality of these insights by stating and proving some fundamental theorems. These theorems use the Hermitian property of quantum mechanical operators, which is described first.

## Hermitian Operators

Since the eigenvalues of a quantum mechanical operator correspond to measurable quantities, the eigenvalues must be real, and consequently a quantum mechanical operator must be Hermitian. To prove this, we start with the premises that $$ψ$$ and $$φ$$ are functions, $$\int d\tau$$ represents integration over all coordinates, and the operator $$\hat {A}$$ is Hermitian by definition if

$\int \psi ^* \hat {A} \psi \,d\tau = \int (\hat {A} ^* \psi ^* ) \psi \,d\tau \label {4-37}$

This equation means that the complex conjugate of Â can operate on $$ψ^*$$ to produce the same result after integration as Â operating on $$φ$$, followed by integration. To prove that a quantum mechanical operator $$\hat {A}$$ is Hermitian, consider the eigenvalue equation and its complex conjugate.

$\hat {A} \psi = a \psi \label {4-38}$

$\hat {A}^* \psi ^* = a^* \psi ^* = a \psi ^* \label {4-39}$

Note that $$a^* = a$$ because the eigenvalue is real. Multiply Equation $$\ref{4-38}$$ and $$\ref{4-39}$$ from the left by $$ψ^*$$ and $$ψ$$, respectively, and integrate over the full range of all the coordinates. Note that $$ψ$$ is normalized. The results are

$\int \psi ^* \hat {A} \psi \,d\tau = a \int \psi ^* \psi \,d\tau = a \label {4-40}$

$\int \psi \hat {A}^* \psi ^* \,d \tau = a \int \psi \psi ^* \,d\tau = a \label {4-41}$

Since both integrals equal $$a$$, they must be equivalent.

$\int \psi ^* \hat {A} \psi \,d\tau = \int \psi \hat {A}^* \psi ^* \,d\tau \label {4-42}$

The operator acting on the function,

$\hat {A}^* \int \psi ^* \hat {A} \psi \,d\tau = \int \psi \hat {A} ^* \psi ^* \,d\tau_*$

produces a new function. Since functions commute, Equation $$\ref{4-42}$$ can be rewritten as

$\int \psi ^* \hat {A} \psi d\tau = \int (\hat {A}^*\psi ^*) \psi d\tau \label{4-43}$

This equality means that $$\hat {A}$$ is Hermitian.

Orthogonality Theorem

Eigenfunctions of a Hermitian operator are orthogonal if they have different eigenvalues. Because of this theorem, we can identify orthogonal functions easily without having to integrate or conduct an analysis based on symmetry or other considerations.

Proof

$$ψ$$ and $$φ$$ are two eigenfunctions of the operator Â with real eigenvalues $$a_1$$ and $$a_2$$, respectively. Since the eigenvalues are real, $$a_1^* = a_1$$ and $$a_2^* = a_2$$.

$\hat {A} \psi = a_1 \psi$

$\hat {A}^* \psi ^* = a_2 \psi ^* \label {4-44}$

Multiply the first equation by $$φ^*$$ and the second by $$ψ$$ and integrate.

$\int \psi ^* \hat {A} \psi \,d\tau = a_1 \int \psi ^* \psi \,d\tau$

$\int \psi \hat {A}^* \psi ^* \,d\tau = a_2 \int \psi \psi ^* \,d\tau \label {4-45}$

Subtract the two equations in Equation $$\ref{4-45}$$ to obtain

$\int \psi ^*\hat {A} \psi \,d\tau - \int \psi \hat {A} ^* \psi ^* \,d\tau = (a_1 - a_2) \int \psi ^* \psi \,d\tau \label {4-46}$

The left-hand side of Equation $$\ref{4-46}$$ is zero because $$\hat {A}$$ is Hermitian yielding

$0 = (a_1 - a_2 ) \int \psi ^* \psi \, d\tau \label {4-47}$

If $$a_1$$ and $$a_2$$ in Equation $$\ref{4-47}$$ are not equal, then the integral must be zero. This result proves that nondegenerate eigenfunctions of the same operator are orthogonal.

$$\square$$

Exercise $$\PageIndex{1}$$

Draw graphs and use them to show that the particle-in-a-box wavefunctions for $$n = 2$$ and $$n = 3$$ are orthogonal to each other.

Thus, the eigenstate $$\psi_a$$ is a state which is associated with a unique value of the dynamical variable corresponding to $$\hat{A}$$. This unique value is simply the associated eigenvalue. It is easily demonstrated that the eigenvalues of operators associated with experimental measurements are all real.

Two wavefunctions, $$\psi_1(x)$$ and $$\psi_2(x)$$, are said to be orthogonal if

$\int_{-\infty}^{\infty}\psi_1^\ast \psi_2 \,dx = 0. \label{4.5.1}$

Consider two eigenstates of $$\hat{A}$$, $$\psi_a(x)$$ and $$\psi_{a'}(x)$$, which correspond to the two different eigenvalues $$a$$ and $$a'$$, respectively. Thus,

$A\psi_a = a \psi_a \label{4.5.2}$

$A\psi_a' = a' \psi_a' \label{4.5.3}$

Multiplying the complex conjugate of the first equation by $$\psi_{a'}(x)$$, and the second equation by $$\psi^*_{a'}(x)$$, and then integrating over all $$x$$, we obtain

$\int_{-\infty}^\infty (A \psi_a)^\ast \psi_{a'} dx = a \int_{-\infty}^\infty\psi_a^\ast \psi_{a'} dx, \label{ 4.5.4}$

$\int_{-\infty}^\infty \psi_a^\ast (A \psi_{a'}) dx = a' \int_{-\infty}^{\infty}\psi_a^\ast \psi_{a'} dx. \label{4.5.5}$

However, from Equation $$\ref{4-46}$$, the left-hand sides of the above two equations are equal. Hence, we can write

$(a-a') \int_{-\infty}^\infty\psi_a^\ast \psi_{a'} dx = 0.$

By assumption, $$a \neq a'$$, yielding

$\int_{-\infty}^\infty\psi_a^\ast \psi_{a'} dx = 0.$

In other words, eigenstates of an Hermitian operator corresponding to different eigenvalues are automatically orthogonal.

The eigenvalues of operators associated with experimental measurements are all real.

## Orthogonality of Degenerate Eigenstates

Consider two eigenstates of $$\hat{A}$$, $$\psi_a$$ and $$\psi'_a$$, which correspond to the same eigenvalue, $$a$$. Such eigenstates are termed degenerate. The above proof of the orthogonality of different eigenstates fails for degenerate eigenstates. Note, however, that any linear combination of $$\psi_a$$ and $$\psi'_a$$ is also an eigenstate of $$\hat{A}$$ corresponding to the eigenvalue $$a$$. Thus, even if $$\psi_a$$ and $$\psi'_a$$ are not orthogonal, we can always choose two linear combinations of these eigenstates which are orthogonal. For instance, if $$\psi_a$$ and $$\psi'_a$$ are properly normalized, and

$\int_{-\infty}^\infty \psi_a^\ast \psi_a' dx = c,\label{ 4.5.10}$

then it is easily demonstrated that

$\psi_a'' = \frac{\vert c\vert}{\sqrt{1-\vert c\vert^2}}\left(\psi_a - c^{-1} \psi_a'\right) \label{4.5.11}$

is a properly normalized eigenstate of $$\hat{A}$$, corresponding to the eigenvalue $$a$$, which is orthogonal to $$\psi_a$$. It is straightforward to generalize the above argument to three or more degenerate eigenstates. Hence, we conclude that the eigenstates of an Hermitian operator are, or can be chosen to be, mutually orthogonal.

Theorem: Gram-Schmidt Orthogonalization

If the eigenvalues of two eigenfunctions are the same, then the functions are said to be degenerate, and linear combinations of the degenerate functions can be formed that will be orthogonal to each other. Since the two eigenfunctions have the same eigenvalues, the linear combination also will be an eigenfunction with the same eigenvalue. Degenerate eigenfunctions are not automatically orthogonal, but can be made so mathematically via the Gram-Schmidt Orthogonalization. The proof of this theorem shows us one way to produce orthogonal degenerate functions.

Proof

If $$ψ$$ and $$φ$$ are degenerate, but not orthogonal, define $$Φ = φ - Sψ$$ where $$S$$ is the overlap integral $$\int \psi ^* \psi d\tau$$, then $$ψ$$ and $$Φ$$ will be orthogonal.

\begin{align*} \int \psi ^* \phi d\tau &= \int \psi ^* (\varphi - S\psi ) d\tau \\[4pt] &= \int \psi ^* \psi d\tau - S \int \psi ^*\psi d\tau \label {4-48} \\[4pt] &= S - S = 0 \end{align*}

$$\square$$

Exercise $$\PageIndex{2}$$

Find $$N$$ that normalizes $$\psi$$ if $$\psi = N(φ_1 − Sφ_1)$$ where $$φ_1$$ and $$φ_2$$ are normalized wavefunctions and $$S$$ is their overlap integral.

$S= \langle φ_1 | φ_2 \rangle \nonumber$

We conclude that the eigenstates of an Hermitian operator are, or can be chosen to be, mutually orthogonal.


text:
# What is wrong with this naive approach to Hilbert's 10th problem?

Background: I have been studying some decidability results in number theory. In doing so, I have always assumed there was no need for me to study pedantic definitions of decidability using Turing machines or similar, since the naive idea of an algorithm ('something which you could program a computer to do') seemed to be enough to understand most proofs. I now notice that this intuitive understanding might lead me to false conclusions. /Background

Consider the following problem: to find an algorithm which - on input a polynomial with coefficients in $\mathbb{Z}$ and an arbitrary number of variables - outputs YES if and only if the polynomial has an integer root, NO otherwise (Hilbert's 10th problem).

By a theorem of Matiyasevich, this problem is unsolvable, in that such an algorithm does not exist. The following naive approach of mine would imply that such an algorithm does exist:

• Clearly if a polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ has a root $\bar{z} = (z_1, \ldots, z_n) \in \mathbb{Z}^n$ then there is a proof for this in the theory of arithmetic (axiomitized by the Peano axioms, for example). Indeed: just calculate $f(\bar{z})$ and note that this equates to zero. It follows by Gödel's Completeness Theorem that also if $f$ has no root in $\mathbb{Z}^n$, there is a proof of this fact in the first-order theory of arithmetic.
• Given a finite sequence of symbols from the language of arithmetic (logical symbols, $+, -, \cdot, 0, 1$ and a symbol $\downarrow$ indicating that a new line should begin), there exists an algorithm which checks whether this sequence is a valid proof in the theory of arithmetic.
• The number of finite sequences of symbols in the language of arithmetic is countable. Even stronger: a bijection $\varphi$ from $\mathbb{N}$ to this set can be constructed explicitly as the number of symbols is finite, so we can just list all sequences with only 1 symbol, then all sequences with 2 symbols and so on.
• Now define the following algorithm: for $m = 1,2,3, \ldots$, verify whether $\varphi(m)$ is a valid proof. If so, check if the last line spells "$f$ has a root in $\mathbb{Z}^n$" or "$f$ has no root in $\mathbb{Z}^n$. If so, output YES or NO accordingly. If not, continue with $m+1$ instead of $m$.
• By my first point, the above algorithm always terminates.

Where did I make a mistake, i.e. what part of my argument violates the strict definition of what an algorithm is? Unless I am overlooking something, it seems like something I could make a computer do if my programming skills would allow it.

• Can you elaborate on this for a second: "It follows by Gödel's Completeness Theorem that also if ff has no root in Zn, there is a proof of this fact in the first-order theory of arithmetic." Nov 12 '17 at 0:48
• Just remember that Godel's completeness theorem says that a sentence that is satisfied by every model is provable. In your case, you in fact only looked at one model, namely the (conventionally called) standard one. Nov 12 '17 at 6:15

The problem is that it's possible $f$ has no integer roots, but there is no proof of this fact (in whatever theory of arithmetic you are using). You're right that if $f$ does have a root, then you can prove it by just plugging in that root. But if $f$ does not have a root, that fact need not be provable. In that case, your algorithm will never halt.
• @Bib-lost Similarly, there are models of PA which "think" that PA is inconsistent, in the sense that there is a "natural number" coding a proof of $0=1$ from the PA-axioms; but this "natural number" is in fact nonstandard. Nov 11 '17 at 18:37


text:
# calculation threshold for minimum risk classifier?

Suppose Two Class $C_1$ and $C_2$ has an attribute $x$ and has distribution $\cal{N} (0, 0.5)$ and $\cal{N} (1, 0.5)$. if we have equal prior $P(C_1)=P(C_2)=0.5$ for following cost matrix:

$L= \begin{bmatrix} 0 & 0.5 \\ 1 & 0 \end{bmatrix}$

why, $x_0 < 0.5$ is the threshold for minimum risk (cost) classifier?

This is my note example that I misunderstand, (i.e, how this threshold is reached? )

Edit 1: I think for thresholds of likelihood ratio we can use P(C1) / P(C2).

Edit 2: I add from Duda Book on Pattern some text about threshold.

For a cost matrix $$L= \begin{bmatrix} 0 & 0.5 \\ 1 & 0 \end{bmatrix} \begin{matrix} c_1 \\ c_2 \end{matrix} \;\text{prediction} \\ \hspace{-1.9cm} \begin{matrix} c_1 & c_2 \end{matrix} \\ \hspace{-1.9cm}\text{truth}$$

the loss of predicting class $c_1$ when the truth is class $c_2$ is $L_{12} = 0.5$, and the cost of predicting class $c_2$ when the truth is class $c_1$ is $L_{21} = 1$. There is no cost for correct predictions, $L_{11} = L_{22} = 0$. The conditional risk $R$ for predicting either class $k$ is then

\begin{align} R(c_1|x) &= L_{11} \Pr (c_1|x) + L_{12} \Pr (c_2|x) = L_{12} \Pr (c_2|x) \\ R(c_2|x) &= L_{22} \Pr (c_2|x) + L_{21} \Pr (c_1|x) = L_{21} \Pr (c_1|x) \end{align} For a reference see these notes on page 15.

In order to minimize the risk/loss you predict $c_1$ if the cost from the mistake of doing so (that's the loss of the wrong prediction times the posterior probability that the prediction is wrong $L_{12} \Pr (c_2|x)$) is smaller than the cost of wrongfully predicting the alternative,

\begin{align} L_{12} \Pr (c_2|x) &< L_{21} \Pr (c_1|x) \\ L_{12} \Pr (x|c_2) \Pr (c_2) &< L_{21} \Pr (x|c_1) \Pr (c_1) \\ \frac{L_{12} \Pr (c_2)}{L_{21} \Pr (c_1)} &< \frac{\Pr (x|c_1)}{ \Pr (x|c_2)} \end{align} where the second line uses Bayes' rule $\Pr (c_2|x) \propto \Pr (x|c_2) \Pr (c_2)$. Given equal prior probabilities $\Pr (c_1) = \Pr (c_2) = 0.5$ you get $$\frac{1}{2} < \frac{\Pr (x|c_1)}{ \Pr (x|c_2)}$$

so you choose to classify an observation as $c_1$ is the likelihood ratio exceeds this threshold. Now it is not clear to me whether you wanted to know the "best threshold" in terms of the likelihood ratios or in terms of the attribute $x$. The answer changes according to the cost function. Using the Gaussian in the inequality with $\sigma_1 = \sigma_2 = \sigma$ and $\mu_1 = 0$, $\mu_2 = 1$, \begin{align} \frac{1}{2} &< \frac{\frac{1}{\sqrt{2\pi}\sigma}\exp \left[ -\frac{1}{2\sigma^2}(x-\mu_1)^2 \right]}{\frac{1}{\sqrt{2\pi}\sigma}\exp \left[ -\frac{1}{2\sigma^2}(x-\mu_2)^2 \right]} \\ \log \left(\frac{1}{2}\right) &< \log \left(\frac{1}{\sqrt{2\pi}\sigma}\right) -\frac{1}{2\sigma^2}(x-0)^2 - \left[ \log \left(\frac{1}{\sqrt{2\pi}\sigma}\right) -\frac{1}{2\sigma^2}(x-1)^2 \right] \\ \log \left(\frac{1}{2}\right) &< -\frac{x^2}{2\sigma^2} + \frac{x^2}{2\sigma^2} - \frac{2x}{2\sigma^2} + \frac{1}{2\sigma^2} \\ \frac{x}{\sigma^2} &< \frac{1}{2\sigma^2} - \log \left(\frac{1}{2}\right) \\ x &< \frac{1}{2} - \log \left(\frac{1}{2}\right) \sigma^2 \end{align} so a prediction threshold in terms of $x$ as you search for can only be achieved if the losses from false predictions are the same, i.e. $L_{12} = L_{21}$ because only then can you have $\log \left( \frac{L_{12}}{L_{21}} \right) = \log (1) = 0$ and you get the $x_0 < \frac{1}{2}$.

• Nice Answer, but confused me! if you want to choose $x_0=0.5$ or $x_0<0.5$, which one is correct? – user153695 Apr 1 '15 at 14:47
• So right on the decision boundary $x_0=0.5$ you can't tell exactly if an observation should be in class one or two (because it's exactly on the boundary). So choosing whether observation $i$ should be in class 1 if $x_0 \leq 0.5$ or $x_0 < 0.5$ is up to you. With large enough samples this should happen for very few observations so at the margin it will matter litter for your result. – Andy Apr 1 '15 at 14:55
• all of my problem that set bounty to it thtat my prof. calculated $x_0<0.5$ and not accept $x_0=0.5$ please see my edit in question, I thin threshold should be $x_0<0.5$ . – user153695 Apr 1 '15 at 15:05
• maybe 0.5-ln :) – user153695 Apr 1 '15 at 15:08
• @whuber thanks, I completely missed that so I started from a completely wrong end. – Andy Apr 1 '15 at 20:19


text:
# Category:Composite Sinusoidal Modeling

### References

• Shigeki Sagayama. Composite Sinusoidal Modeling (CSM) (1979). Contributions - Speech Analysis and Synthesis. Sagayama/Moriya/Kameoka/(Ono) Laboratory, University of Tokyo. "Proposed modeling speech by a sum of ${\displaystyle n\,}$ sinusoids and equate autocorrelations of the speech signal and the model at lowest  ${\displaystyle 2n-1\,}$ points. The model frequencies were proven equivalent to Line Spectrum Pair frequencies. Applied to Yamaha's best selling sound IC, the CSM Speech Synthesis patent earned best among all NTT patents for several years."
• Fumitada Itakura ((Please provide a date))“Linear Statistical Modeling of Speech and its Applications -- Over 36 year history of LPC --”, in Proceedings of the 18th International Congress on Acoustics (ICA 2004), We3.D, Kyoto, Japan, Apr. 2004.[1], volume vol.3, pages III-2077-2082: “6. Composite Sinusoidal Modeling(CSM) In 1975, Itakura proposed the line spectrum representation (LSR) concept and its algorithm to obtain a set of parameters for new speech spectrum representation. Independently from this, Sagayama developed a composite sinusoidal modeling (CSM) concept which is equivalent to LSR but give a quite different formulation, solving algorithm and synthesis scheme. Sagayama clarified the duality of LPC and CSM and provided the unified view covering LPC, PARCOR, LSR, LSP and CSM, CSM is not only an new concept of speech spectrum analysis but also a key idea to understand the linear prediction from a unified point of view. ...
• “Duality theory of composite sinusoidal modeling and linear prediction”, in Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP '86., volume 11, Apr 1986, DOI:10.1109/ICASSP.1986.1168815, pages 1261-1264
• US patent application 4815135, Tetsu Taguchi, "Speech signal processor", published 1989-03-21 , assigned to Nec Corporation

## Subcategories

This category has only the following subcategory.

## Media in category "Composite Sinusoidal Modeling"

The following 4 files are in this category, out of 4 total.


text:
Function numbering in algorithms

Algorithms are getting numbered automatically but unable to get same for the functions.

In the following example, Algorithms are numbered as Algorithm 1 and Algorithm 2 by themselves but Functions didn't. Please advice.

\documentclass{ucbthesis}
\usepackage{graphicx}
\usepackage{subfig}
\usepackage{epstopdf}
\usepackage[ruled,vlined,linesnumbered]{algorithm2e}
\usepackage{algpseudocode}
\usepackage{tikz,pgfplots,pgfplotstable}
\usepackage{tikz-qtree}
\usepackage{forest}
\usepackage{lettrine}
\usepackage{mathtools}
\usepackage{lscape}
\usepackage{pgf-pie}
\usepackage{caption}
\usepackage{dashbox}
\usepackage{mathptmx}
\usepackage{xcolor}
\usepackage{hyperref}
\begin{document}
\begin{algorithm}
\caption{First Algorithm}
\For{$j \in V$}{
}
\end{algorithm}

\begin{algorithm}
\caption{Second Algorithm}
\For{$j \in V$}{
}
\end{algorithm}

\begin{function}
\label{function1}
\caption{First Function ()}
\For{$k \in R$}{

}
\end{function}

\begin{function}
\label{function2}
\caption{Second Function ()}
\For{$k \in R$}{

}
\end{function}

I am refering to function \ref{function1} and function \ref{function2}

\end{document}

What you need is the package option procnumbered, which “makes the procedure and function to be numbered as algorithm”. See Page 19 of the algorithm2e manual.

Added: In order to get the cross-reference right, you need to place \label after \caption as @egreg mentioned in the comments below. See also this short explanation by @PhilMiller and this longer explanation by @anon.

\documentclass{ucbthesis}
\usepackage[ruled,vlined,linesnumbered,procnumbered]{algorithm2e}
\usepackage{newtxtext}
\usepackage{newtxmath}

\begin{document}
\begin{algorithm}
\caption{First Algorithm}
\For{$j \in V$}{
}
\end{algorithm}

\begin{algorithm}
\caption{Second Algorithm}
\For{$j \in V$}{
}
\end{algorithm}

\begin{function}
\caption{First Function ()}
\label{function1}
\For{$k \in R$}{
}
\end{function}

\begin{function}
\caption{Second Function ()}
\label{function2}
\For{$k \in R$}{
}
\end{function}

I am refering to Function~\ref{function1} and Function~\ref{function2}.

\end{document}

In case you want your functions to be numbered independently, then you need to set up a new counter for the function environment:

\documentclass{ucbthesis}
\usepackage[ruled,vlined,linesnumbered,procnumbered]{algorithm2e}
\usepackage{newtxtext}
\usepackage{newtxmath}
% Set up a new counter for function
\newcounter{function}
\usepackage{etoolbox}
% Change the original counter to the new counter
\makeatletter
\AtBeginEnvironment{function}{%
\let\c@algocf\c@function
}
\makeatother

\begin{document}
\begin{algorithm}
\caption{First Algorithm}
\For{$j \in V$}{
}
\end{algorithm}

\begin{algorithm}
\caption{Second Algorithm}
\For{$j \in V$}{
}
\end{algorithm}

\begin{function}
\caption{First Function ()}
\label{function1}
\For{$k \in R$}{
}
\end{function}

\begin{function}
\caption{Second Function ()}
\label{function2}
\For{$k \in R$}{
}
\end{function}

I am refering to Function~\ref{function1} and Function~\ref{function2}.

\end{document}

Added: Please use non-breaking space ~ between a label name and its number.
• @YogeshSharma \label should go after \caption. – egreg Jul 23 '18 at 7:34
• @YogeshSharma Please follow @egreg’s comment. I updated my answer to include a few explanations. Also, please use ~ between a label name and its number. – Ruixi Zhang Jul 23 '18 at 12:47