text:
# Dimension too large from matlab2tikz, and aligning graph axis

I have a MATLAB generated plot that looks as follows:

Generated with this code:

plot(fitGoodnessNRMSE,'b','LineWidth',1);
axis([1 511 0.5 1]);
ylabel('Normalized RMS error'); xlabel('q (um^-1)');

I then export it as a tikz using matlab2tikz(). And include it in my latex document:

\documentclass{standalone}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{pgfplots}
\usepackage{amsmath}
\begin{document}
\input{test.tikz}
\end{document}

The latex file will not compile due to Error: Dimension too large. <recently read> \pgf@yy l.537 \end{axis}.

Inside the .tikz document is as follows:

\begin{tikzpicture}
\begin{axis}[%
width=4.52083333333333in,
height=3.565625in,
scale only axis,
xmin=1, xmax=511, xlabel={$\text{q (um}^\text{-}\text{1)}$},
ymin=0.5, ymax=1, ylabel={Normalized RMS error}]
table[row sep=crcr]{
1    -57.943745    \\
2    -14.11968    \\
3    -5.535387    \\
4    -2.4488078    \\
5    -1.0804409    \\
6    -0.30200709    \\
7    0.053585582    \\
8    0.32392563    \\
9    0.44304278    \\
.
.
511    0.90342708    \\
};
\end{axis}
\end{tikzpicture}%

Note the data hangs around 0.8 or 0.9 most of the points.

I found if I change ymin from 0.5 to 0, it compiles fine. Help as to why it does not work in its current state is greatly appreciated. I need the y axis to only display 0.5 to 1 like it does in MATLAB.

Secondly, I have been playing around with aligning plots in a table, my current attempt is shown below. I am using largely this example: Insert image and list inside a table, but cant figure out two things:

• How do I center the text under heading 'A'?

• The vertical edges of the middle graph don't quite line up with the top, as shown by the red arrows in the image. Is this because the middle graph has brackets in one of the label titles that need to push it a bit to the right. How do I go about making them line up?

If possible I would like to do this by not having to alter my .tikz file once it has been created by matlab2tikz, but I understand if this is simply not possible.

Thanks

• A comment for anyone with the same first problem (Dimension too large). If you run the cleanfigure command that is provided in the matlab2tikz submission, then it will discard any points not actually visible in the plot. This got rid of the troublesome first points which were stuffing up my image, and it worked fine after. – Steve Hatcher Mar 9 '14 at 11:04
• For what it's worth, I had a Dimension too large problem, and cleanfigure didn't fix it. I eventually realized I was setting an xtick that was way outside the axis limits, and that was causing the problem. – Andrew D. King Sep 18 '15 at 22:44

The problem with "Dimension too large" is a weakness in pgfplots: it has to map all coordinates into TeX's limited number range eventually, and it fails to detect that it cannot do so without discarding coordinates.

In your case, the input range of the y coordinates is much greater than the actually displayed portion. I'd suggest to add restrict y to domain=-5:4 - this will discard any coordinates below -5 and above 4 (adopt as needed). I suppose this will assist pgfplots such that it can map all coordinates properly.

Regarding the alignment stuff: The general advice would be to

• control the bounding boxes of your pictures. A good idea could be to use \begin{tikzpicture}[trim axis right]; this will exclude everything right of the right axis line from the bounding box. Then, configure your table such that the "B" column is right-aligned.
• another alternative would be to configure both ylabel style and yticklabel style such that they have a minimum height; perhaps that has the same effect.
• study section "Alignment Options" in the pgfplots manual carefully; it has lots of input on both horizontal and vertical alignment of multiple axes.

A more detailed answer would be possible if you post a minimal working example which demonstrates what you have so far.

Please note that you should, in general, pose only one question per post - it is considerably simpler to address issues separately rather than mixed.

I would suggest to pose a new question for the alignment stuff if needed; editing this will probably make it difficult to separate answers from the dimension too large issue.

format: plain


text:
Article

# Network-based models as tools hinting at nonevident protein functionality.

• ##### Osman Burak Okan
Faculty of Engineering and Natural Sciences, Sabanci University, 34956 Istanbul, Turkey.
Annual Review of Biophysics (Impact Factor: 12.63). 02/2012; 41:205-25. DOI: 10.1146/annurev-biophys-050511-102305
Source: PubMed

ABSTRACT Network-based models of proteins are popular tools employed to determine dynamic features related to the folded structure. They encompass all topological and geometric computational approaches idealizing proteins as directly interacting nodes. Topology makes use of neighborhood information of residues, and geometry includes relative placement of neighbors. Coarse-grained approaches efficiently predict alternative conformations because of inherent collectivity in the protein structure. Such collectivity is moderated by topological characteristics that also tune neighborhood structure: That rich residues have richer neighbors secures robustness toward random loss of interactions/nodes due to environmental fluctuations/mutations. Geometry conveys the additional information of force balance to network models, establishing the local shape of the energy landscape. Here, residue and/or bond perturbations are critically evaluated to suggest new experiments, as network-based computational techniques prove useful in capturing domain movements and conformational shifts resulting from environmental alterations. Evolutionarily conserved residues are optimally connected, defining a subnetwork that may be utilized for further coarsening.

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110 Views
• Source
##### Article: Designing molecular dynamics simulations to shift populations of the conformational States of calmodulin.
[Hide abstract]
ABSTRACT: We elucidate the mechanisms that lead to population shifts in the conformational states of calcium-loaded calmodulin (Ca(2+)-CaM). We design extensive molecular dynamics simulations to classify the effects that are responsible for adopting occupied conformations available in the ensemble of NMR structures. Electrostatic interactions amongst the different regions of the protein and with its vicinal water are herein mediated by lowering the ionic strength or the pH. Amino acid E31, which is one of the few charged residues whose ionization state is highly sensitive to pH differences in the physiological range, proves to be distinctive in its control of population shifts. E31A mutation at low ionic strength results in a distinct change from an extended to a compact Ca(2+)-CaM conformation within tens of nanoseconds, that otherwise occur on the time scales of microseconds. The kinked linker found in this particular compact form is observed in many of the target-bound forms of Ca(2+)-CaM, increasing the binding affinity. This mutation is unique in controlling C-lobe dynamics by affecting the fluctuations between the EF-hand motif helices. We also monitor the effect of the ionic strength on the conformational multiplicity of Ca(2+)-CaM. By lowering the ionic strength, the tendency of nonspecific anions in water to accumulate near the protein surface increases, especially in the vicinity of the linker. The change in the distribution of ions in the vicinal layer of water allows N- and C- lobes to span a wide variety of relative orientations that are otherwise not observed at physiological ionic strength. E31 protonation restores the conformations associated with physiological environmental conditions even at low ionic strength.
PLoS Computational Biology 12/2013; 9(12):e1003366. · 4.87 Impact Factor
• Source
##### Article: Effective harmonic potentials: insights into the internal cooperativity and sequence-specificity of protein dynamics.
[Hide abstract]
ABSTRACT: The proper biological functioning of proteins often relies on the occurrence of coordinated fluctuations around their native structure, or on their ability to perform wider and sometimes highly elaborated motions. Hence, there is considerable interest in the definition of accurate coarse-grained descriptions of protein dynamics, as an alternative to more computationally expensive approaches. In particular, the elastic network model, in which residue motions are subjected to pairwise harmonic potentials, is known to capture essential aspects of conformational dynamics in proteins, but has so far remained mostly phenomenological, and unable to account for the chemical specificities of amino acids. We propose, for the first time, a method to derive residue- and distance-specific effective harmonic potentials from the statistical analysis of an extensive dataset of NMR conformational ensembles. These potentials constitute dynamical counterparts to the mean-force statistical potentials commonly used for static analyses of protein structures. In the context of the elastic network model, they yield a strongly improved description of the cooperative aspects of residue motions, and give the opportunity to systematically explore the influence of sequence details on protein dynamics.
PLoS Computational Biology 08/2013; 9(8):e1003209. · 4.87 Impact Factor
• Source
##### Article: Protonation States of Remote Residues Affect Binding-Release Dynamics of the Ligand but not the Conformation of apo Ferric Binding Protein
[Hide abstract]
ABSTRACT: We have studied the apo (Fe3+ free) form of periplasmic ferric binding protein (FbpA) under different conditions and we have monitored the changes in the binding and release dynamics of H2PO4- that acts as a synergistic anion in the presence of Fe3+. Our simulations predict a dissociation constant of 2.2$\pm$0.2 mM which is in remarkable agreement with the experimentally measured value of 2.3$\pm$0.3 mM under the same ionization strength and pH conditions. We apply perturbations relevant for changes in environmental conditions as (i) different values of ionic strength (IS), and (ii) protonation of a group of residues to mimic a different pH environment. Local perturbations are also studied by protonation or mutation of a site distal to the binding region that is known to mechanically manipulate the hinge-like motions of FbpA. We find that while the average conformation of the protein is intact in all simulations, the H2PO4- dynamics may be substantially altered by the changing conditions. In particular, the bound fraction which is 20$\%$ for the wild type system is increased to 50$\%$ with a D52A mutation/protonation and further to over 90$\%$ at the protonation conditions mimicking those at pH 5.5. The change in the dynamics is traced to the altered electrostatic distribution on the surface of the protein which in turn affects hydrogen bonding patterns at the active site. The observations are quantified by rigorous free energy calculations. Our results lend clues as to how the environment versus single residue perturbations may be utilized for regulation of binding modes in hFbpA systems in the absence of conformational changes.
The journal of physical chemistry. B. 02/2014;


text:
# Is this problem in P: Finding a common key for a collection of systems of equations?

Let $B=\{b_1=g_1,\cdots,b_n=g_n\}$ be a set of binary variables $b_i$ and their corresponding values $g_i \in \{0,1\}$. Let $M=\{\sum_{e \in A}e \;:\; A \subset B\}$, i.e., $M$ is the set of all possible linear combinations of the equations in $B$.

Given $S_i \subset B$ for $i=1,\cdots,m$, is that possible to compute, in polynomial time, a $K \subset M$ with minimum size such that $S_i \cup K$ is a full rank system of equations (i.e., the values of all of the variables can be obtained by solving $S_i \cup K$)?

An example: Let $B=\{b_1=1,b_2=0,b_3=1\}$, $S_1=\{b_1=1,b_2=0\}$, and $S_2=\{b_2=0,b_3=1\}$. $K=\{b_1+b_3=0\}$ is the solution because both $S_1\cup K$ and $S_2 \cup K$ can be solved uniquely and $K$ has the minimum size 1.


text:
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# On-chip wavefront shaping with dielectric metasurface

## Abstract

Metasurfaces can be programmed for a spatial transformation of the wavefront, thus allowing parallel optical signal processing on-chip within an ultracompact dimension. On-chip metasurfaces have been implemented with two-dimensional periodic structures, however, their inherent scattering loss limits their large-scale implementation. The scattering can be minimized in single layer high-contrast transmitarray (HCTA) metasurface. Here we demonstrate a one-dimensional HCTA based lens defined on a standard silicon-on-insulator substrate, with its high transmission (<1 dB loss) maintained over a 200 nm bandwidth. Three layers of the HCTAs are cascaded for demonstrating meta-system functionalities of Fourier transformation and differentiation. The meta-system design holds potential for realizing on-chip transformation optics, mathematical operations and spectrometers, with applications in areas of imaging, sensing and quantum information processing.

## Introduction

Metasurfaces are arrays of subwavelength structures capable of imposing a localized and spatially varying phase shift onto the transmitted or reflected electromagnetic (EM) wave1,2,3. Gradient variations of nanostructures in a subwavelength thin layer are capable of manipulating an out-of-plane EM wave in free space, leading to numerous applications from simple components of miniaturized flat lenses4,5,6,7,8 and holograms9,10,11, to more complicated systems of analog12,13 signal processing and spectrometers14,15. Metamaterials have also been used for manipulating in-plane waves16,17,18,19,20,21,22,23,24,25,26,27,28. Periodic structures, such as dielectric photonic crystals29 and metallic hyperbolic metasurfaces30, control the delay and momentum of in-plane light propagation. The inverse design method significantly reduces the footprint of functional simple components19,20. Optimization toward functions with multi-input/output for parallel signal processing can be of a significant computational cost, as the inverse design is numerically driven. Gradient varying on-chip metasurface based on plasmonics23,24 or dielectric metamaterials16,17,18,25,26,27,28 have been demonstrated for the on-chip lens. Compared with the two-dimensional metasurface-based image processors12, the on-chip meta-system can operate without an alignment step, but limited to process one-dimensional (1D) data represented through the wavefront in xy plane16,23,31,32. The insertion loss of reported on-chip metasurfaces (ranges from a few dB to tens of dB)16,23 is not suitable to meet the requirements for standard passive optical components (<1 dB), and their small critical dimensions (<100 nm) makes them incompatible with foundry process and more vulnerable to fabrication related geometric distortions (Supplementary Table 1).

As an analogy of electronic circuits, current photonic microsystem’s complexity and capability33,34,35 will be eventually limited by the individual components’ size and energy consumption. In this work, we explore parallel signal processing through on-chip 1D high-contrast transmitarray (HCTA). By slightly adjusting the dimension of the void slots defined in the device layer, we can control the coherent inference of the parallelly transmitted wavefront with low loss. We design, fabricate and characterize the on-chip metasurface lens (metalens) made of HCTA. The 1D HCTAs maintain less than 1 dB insertion loss across the S, C, and L telecommunication bands (>200 nm bandwidth) while maintaining a good tolerance to fabrication variation. The HCTA exhibits low insertion loss, which is feasible for parallel and multi-stage on-chip signal processing. Based on the 1D HCTA design, we experimentally demonstrate ultra-short, low loss and broadband mode size convertors and metasystems performing Fourier transform and spatial differentiation. With minimal feature size of 140 nm, the 1D HCTA is compatible with current deep UV photolithography technique used in silicon photonics foundry, and thus feasible for large-scale silicon photonic computational chips operating at the speed of light.

## Results

### Low loss 1D on-chip HCTAs

The designed HCTAs are 1D rectangular etched slot arrays defined in the silicon-on-insulator (SOI) substrate (Fig. 1a). Here we fix the lattice constant (a) of the HCTAs to be less than half of the wavelength (500 nm), and sweep the length (L) and width (w) to get the transmission and phase shift of the HCTA. The large refractive index contrast between silicon and silicon dioxide (>2) allows for a 2π phase shift to be achieved with high transmission36. The phase shift is introduced through the wave number differences in the slot and slab waveguides37 (Supplementary Figure 1). The optical mode profiles of the transmitted transverse electric (TE) wave tunneling through the slot are illustrated in Fig. 1b, c. The transmission and the phase shift can be controlled by varying the width and length of the slots (Fig. 1d, e). A phase shift from 0 to 2π combined with a transmission larger than 94% can be simultaneously achieved by fixing the slot width to be 0.14 µm while varying the slot length from 0.2 to 2.5 μm (Fig. 1f). It is noted that with a wider slot, a 2π phase shift can be achieved with a smaller slot length variation, but the transmission drops dramatically (Fig. 1d).

### Compact HCTA metalens

Here we use a gradient varying HCTA for on-chip wavefront control. The 1D HCTA along the y direction imposes a space-dependent phase shift on the impinging light (TE polarized) along the x direction. The phase shift of the transmitted wave is defined in the following equation for achieving the on-chip wave focusing:

$$\phi \left( y \right) = \frac{{2\pi }}{{\lambda _{\mathrm{d}}}}n_{{\mathrm{eff}}}\left( {f - \sqrt {f^2 + y^2} } \right)$$
(1)

where λd is the design wavelength in free space, neff is the effective refractive index of the guided light confined in the silicon slab, and f is the focal length. Here we fix the lattice constant (a = 500 nm) and slot width (w = 140 nm) for ensuring high transmission, and vary the slot length along the y direction for achieving the desired wavefront.

The analytically designed metalens is then numerically evaluated (see method) (http://www.lumerical.com/tcad-products/fdtd/). Figure 2a shows the optical intensity distribution at the xy plane in the middle of the 250-nm-thick silicon slab (z = 125 nm). The input light is centered at the wavelength of 1550 nm along the +x direction. The designed metalens is 11 µm wide in the y direction, with a focusing length of 25 µm and a spot size of 1.07 μm. The spot size is marked as the full width half maximum (FWHM) in the cross section of the mode profile (inset of Fig. 2a). Figure 2b shows the detailed in-plane electric field distribution of Ey across the metalens. With a gradient varying phase shift, the interference between the transmitted wave ‘bends’ the off-axis light toward the central axis. The focusing length of the metalens can be controlled by varying the gradience of the slot length. The simulated focal length is 6.7% shorter than the analytically design, due to the finite phase shift between adjacent slots. The focusing lengths can be adjusted from a few µm to sub-mm by controlling the step size of slot length in y direction. Figure 2c plots the focusing efficiency, transmission, and spot size versus focal lengths, indicating a trade-off between the focusing efficiency and spot size. At a fixed lattice constant, a short focusing length demands a larger gradient of the slot length, which increases the deviation of the transmitted wavefront compared with that of the ideal lens. As the focusing length reduces from 28 to 8 µm, the focusing efficiency reduces from 80% to 40%. The focusing efficiency is defined as the fraction of the incident light that passes through a rectangular aperture at the focal plane, with its width equals to three times the spot size and a height of 0.5 μm. For metalens with focusing lengths longer than 25 µm, a maximum focusing efficiency of 79% can be achieved (Fig. 2c). The spot size can be reduced to 0.4 µm at a shorter focusing length of 8 µm.

We also verify the broadband operation of the on-chip metalens. The focusing efficiency is above 74% and transmission is above 88% within a 200 nm bandwidth centered at 1550 nm (Fig. 2d). The focusing efficiency and transmission of the metalens varies less than 3% across the whole wavelength range. The spot size is about 1.08 µm in the wavelength range from 1550 to 1650 nm, and slightly increases (~0.05 µm) at shorter wavelengths (near 1450 nm). The strong light confinement in the slots minimizes the material dispersion in the HCTA (Fig. 1b, c), as silicon dioxide’s chromatic dispersion is an order of magnitude smaller than silicon. The broadband low dispersion (Supplementary Figs 2 and  3) is attributed to the geometric dispersion (with details discussed in Supplementary note 3). At oblique incidence, the metalens retains its transmission efficiency of 90% and focusing efficiency of 82% as the tilting angle is tuned from −5o to 5o. The focusing efficiency then gradually decreases down to 65% at 20o incidence angle, along with an increase in spot size (Fig. 2e).

The numerically evaluated performance of the on-chip metalens is then experimentally verified. The on-chip mode conversion is mapped in both the x and y directions. In the y direction, 11 waveguides are parallelly placed on the output plane to obtain the optical intensity distribution. Their center-to-center distances are set at 1 µm to avoid interference. The optical field variation in the x direction is sampled by 67 devices with their metalens and output plane distance (Δx) arranged from 3 to 18.9 μm. Figure 3a shows a scanning electron microscope (SEM) image of a device. The focal length f of the metalens under test is designed to be 8 μm. A 44-µm-long exponential taper guides the input light from a single mode waveguide (0.5 µm width) to the metasurface with a diameter of 11 µm. The metalens converges the beam width down to a spot size of 0.75 μm (FWHM) at the image distance of $$\Delta x = 10.1\;{\mathrm{\mu m}}$$ (Fig. 3b–c). The simulated and measured optical intensity distribution along the x-axis at different y positions are compared in Fig. 3b. The measured cross-sectional light intensity profiles (dots in Fig. 3c) align well with the simulated profiles (curves in Fig. 3c). The beam FWHM decreases to its minimum at $${\mathrm{\Delta }}x = 10.1\;{\mathrm{\mu m}}$$, where the light intensity increases an order of magnitude compared with the input wave.

With metalens assisted light focusing, a 13.7-μm-long linear taper can efficiently convert the mode from an 11-μm width down to a 0.5 µm width (Inset of Fig. 3d). Figure 3d compares the transmission spectra of the 13.7-μm-long taper with and without the metalens. The metalens is placed beside the grating coupler, as shown in the SEM image of the device (inset of Fig. 3d). The measured insertion loss of the metalens is less than 0.8 dB in the C band and increases to 2 dB in the S-band, which is comparable to the simulated results (transmission above 79% in the wavelength band from 1480 to 1580 nm). An extra 18 dB loss is observed for the same device design without a metalens (gray curve in Fig. 3d). This result experimentally demonstrates the broadband high transmission of the metalens, a critical feature required for multi-stage system integration.

### Mathematical operation with cascaded meta-system

Compared with conventional free-space optical information processing metasystems, the advanced lithography technique allows for easy alignment of the on-chip metasurface for a cascaded multilayer system with precise spacings. Here we use the designed metalens for implementing spatial Fourier transformations (FT). For a proof-of-concept demonstration, a device with 2 input ports and 11 output ports are fabricated on an SOI substrate (Supplementary Fig. 4a, b). The input and output planes are placed on the focal plane of the metalens (Inset of Fig. 4a). On the input port, light is coupled from single mode waveguides to the slab (profile is shown in Fig. 4a). On the output plane, the optical field distribution is sampled by densely spaced single mode waveguides with 1 μm center-to-center spacing. The transmitted power from those ports is measured and compared with the numerical and analytical results (Fig. 4b). The output intensity distribution can be directly converted to the FT of input intensity profile, by converting the y axis to spatial frequency $$\left( {\xi = y/(\lambda f)} \right)$$. Here, λ is the wavelength in the slab waveguide and f is the focal length. The design principle of the FT system is introduced in Supplementary Note 4.

Based on the FT element, the spatial differentiation system is then developed to perform the spatial differentiation on the input signal (Supplementary Note 4). The differentiation meta-system is composed of three layers (Insets of Fig. 4c). The first and third layers of the meta-system are metalens, and the second layer is an HCTA mask layer with a specific transmission coefficient $$t\left( y \right) \propto \left[ {jy \cdot {\mathrm{exp}}\left( { - j\pi \frac{{y^2}}{{\lambda f}}} \right)} \right]$$ (Supplementary Fig. 4c, d). The spatially varying transmission coefficient is programed onto the middle HCTA mask layer by varying the width and the length of the slot (Fig. 1d, e). As shown in the left inset of Fig. 4c, the three stage system only has a footprint of 15 µm by 45 µm. A 2.5 µm wide waveguide guides light to the input plane of the metasystem. On the output plane, 11 waveguides are parallelly connected with a 1-µm center-to-center distance (Fig. 4c). Combining results from two identical devices with output waveguides arrays shifted 0.5 µm long the y direction, a spatial resolution of 0.5 µm can be achieved (red squares in Fig. 4b, d). The measured spatial spectra align with the numerically simulated and analytical results (Fig. 4d). We should note that the output has reversed coordinates, as the FT is applied twice. More results of Fourier transform and spatial differentiation can be found in Supplementary Figs. 68. It is noted that the focusing efficiency and focal distance vary less than 9% at incident angle from −16o to 16o (Fig. 1e and Supplementary Fig. 9). In our meta-systems (Fig. 4), the incident angle is confined within ±16°, resulting in accurate FT (Supplementary Note 5 and Supplementary Fig. 10).

With lithography defined metasurface orientation and spacing, the on-chip meta-system possesses a small footprint and high stability. We also verify the scalability and Foundry compatibility of our metasystem. Supplementary Note 4 includes results of the AIM photonics manufactured metasystems, including FT and spatial differentiation functions (Supplementary Fig. 5).

## Discussion

Here we presented a foundry fabrication compatible, ultracompact metasurface designed to achieve on-chip wavefront transformation with low insertion loss and broadband operation. An HCTA based metalens is analytically designed, numerically verified and experimentally demonstrated. The 1D metalens has a numerical aperture up to 2.14, which can focus light to within 10 µm with less than 1 dB loss. It can significantly shrink the length of tapers for mode size conversion. The spatial FT of the input signal is achieved by placing the input and output on the focal plane of a metalens. More complicated computational tasks based on FT can be performed by cascading multiple layers of HCTA. As an example, we demonstrate a three-layer metasystem for performing spatial differentiation.

The experimental demonstration of the functional HCTAs opens new directions for on-chip diffractive optical systems, which is distinguished from conventional waveguide based integrated photonic devices. The on-chip metasurface can be integrated with multimode waveguides38,39, to perform mode transformation in mode division multiplexing systems. Also, the 1D metasurface design enables many novel on-chip systems with low power consumption and ultracompact dimension, including diffractive optical computational circuits, on-chip spectrometers and light detection and ranging devices.

## Methods

### Device fabrication

The on-chip metasurface is fabricated on an SOI (100) substrate from Soitec, with the 250 nm device layer on 3 μm silicon dioxide layer. The designed patterns (e.g., HCTA, waveguides, grating couplers) are defined in CSAR 6200.09 positive resist using a Vistec EBPG5200 electron beam lithography system with 100 kV acceleration voltage, followed by resist development and a single step dry etch procedures. A 300-nm-thick silicon dioxide protection layer is deposited on the device layer by plasma-enhanced chemical vapor deposition. Part of the metasystem structure is manufactured through a multi-project wafer run at The American Institute for Manufacturing Integrated Photonics (AIM Photonics).

### Optical measurements

Continuous-wave tunable semiconductor lasers (AQ4321D) with a polarization controller is used for launching light onto the chip. The fiber-grating coupler loss is optimized to be 5 dB per input/output facet. The propagation loss in the channel waveguides is less than 1 dB. The output is monitored by a Newport InGaAs photodiode (818-IG-L-FC/DB) and an optical power meter (1830-R-GPIB).

### Numerical simulations

A three-dimensional finite-difference-time-domain method (http://www.lumerical.com/tcad-products/fdtd/) is used to simulate the optical field distribution and transmission spectra of the periodic HCTA, metalens, and the meta-systems. The conformal mesh with spatial resolution less than 1/10 of the smallest feature size is applied. For calculating the transmission of HCTAs, a y–z plane monitor is placed at Δx = 3 μm to collect the transmitted power. Δx is the distance to the left aligned side of the HCTA.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We acknowledge inspiring discussions with Prof. Dennis Prather and Prof. Gonzalo Arce. Z.W. and T.L. are supported by the National Aeronautics and Space Administration Early Career Faculty Award (NASA ECF). D.M., T.K., and T.G. are partially supported by Air Force Young Investigator Research Program (AFOSR YIP). T.L. also acknowledges the support from State Key Laboratory of Advanced Optical Communication Systems and Networks, China. Some the devices are fabricated in University of Delaware Nanofabrication Facility (UDNF) with inputs from Dr Kevin Lister.

## Author information

Authors

### Contributions

Z.W. and T.G. conceived the idea. Z.W. developed the design principle and performed numerical simulations. T.L. and A.S. developed the fabrication recipe. Z.W. and T.L. fabricated the samples. D.M. and T.K. built the experimental set-up. Z.W. and T.L. performed the measurements and analyzed the data. Z.W. and T.G. wrote the paper with inputs from all authors.

### Corresponding author

Correspondence to Tingyi Gu.

## Ethics declarations

### Competing interests

The authors declare no competing interest.

Peer review information: Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work.

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

## Rights and permissions

Reprints and Permissions

Wang, Z., Li, T., Soman, A. et al. On-chip wavefront shaping with dielectric metasurface. Nat Commun 10, 3547 (2019). https://doi.org/10.1038/s41467-019-11578-y

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41467-019-11578-y

• ### Integrated photonic metasystem for image classifications at telecommunication wavelength

• Zi Wang
• Lorry Chang
• Tingyi Gu

Nature Communications (2022)

• ### Metasurface wavefront control for high-performance user-natural augmented reality waveguide glasses

• Hyunpil Boo
• Yoo Seung Lee
• Chee Wei Wong

Scientific Reports (2022)

• ### High-performance gallium nitride dielectric metalenses for imaging in the visible

• Meng-Hsin Chen
• Wei-Ning Chou
• Hoang Yan Lin

Scientific Reports (2021)


text:
# Common Digraphs¶

All digraphs in Sage can be built through the digraphs object. In order to build a circuit on 15 elements, one can do:

sage: g = digraphs.Circuit(15)

To get a circulant graph on 10 vertices in which a vertex $$i$$ has $$i+2$$ and $$i+3$$ as outneighbors:

sage: p = digraphs.Circulant(10,[2,3])

More interestingly, one can get the list of all digraphs that Sage knows how to build by typing digraphs. in Sage and then hitting tab.

 ButterflyGraph() Returns a n-dimensional butterfly graph. Circuit() Returns the circuit on $$n$$ vertices. Circulant() Returns a circulant digraph on $$n$$ vertices from a set of integers. DeBruijn() Returns the De Bruijn digraph with parameters $$k,n$$. GeneralizedDeBruijn() Returns the generalized de Bruijn digraph of order $$n$$ and degree $$d$$. ImaseItoh() Returns the digraph of Imase and Itoh of order $$n$$ and degree $$d$$. Kautz() Returns the Kautz digraph of degree $$d$$ and diameter $$D$$. Path() Returns a directed path on $$n$$ vertices. RandomDirectedGNC() Returns a random GNC (growing network with copying) digraph with $$n$$ vertices. RandomDirectedGNM() Returns a random labelled digraph on $$n$$ nodes and $$m$$ arcs. RandomDirectedGNP() Returns a random digraph on $$n$$ nodes. RandomDirectedGN() Returns a random GN (growing network) digraph with $$n$$ vertices. RandomDirectedGNR() Returns a random GNR (growing network with redirection) digraph. RandomTournament() Returns a random tournament on $$n$$ vertices. TransitiveTournament() Returns a transitive tournament on $$n$$ vertices. tournaments_nauty() Returns all tournaments on $$n$$ vertices using Nauty.

AUTHORS:

• Robert L. Miller (2006)
• Emily A. Kirkman (2006)
• Michael C. Yurko (2009)
• David Coudert (2012)

## Functions and methods¶

class sage.graphs.digraph_generators.DiGraphGenerators

A class consisting of constructors for several common digraphs, including orderly generation of isomorphism class representatives.

A list of all graphs and graph structures in this database is available via tab completion. Type “digraphs.” and then hit tab to see which graphs are available.

The docstrings include educational information about each named digraph with the hopes that this class can be used as a reference.

The constructors currently in this class include:

Random Directed Graphs:
- RandomDirectedGN
- RandomDirectedGNC
- RandomDirectedGNP
- RandomDirectedGNM
- RandomDirectedGNR

Families of Graphs:
- DeBruijn
- GeneralizedDeBruijn
- Kautz
- Path
- ImaseItoh
- RandomTournament
- TransitiveTournament
- tournaments_nauty

ORDERLY GENERATION: digraphs(vertices, property=lambda x: True, augment=’edges’, size=None)

Accesses the generator of isomorphism class representatives. Iterates over distinct, exhaustive representatives.

INPUT:

• vertices - natural number
• property - any property to be tested on digraphs before generation.
• augment - choices:
• 'vertices' - augments by adding a vertex, and edges incident to that vertex. In this case, all digraphs on up to n=vertices are generated. If for any digraph G satisfying the property, every subgraph, obtained from G by deleting one vertex and only edges incident to that vertex, satisfies the property, then this will generate all digraphs with that property. If this does not hold, then all the digraphs generated will satisfy the property, but there will be some missing.
• 'edges' - augments a fixed number of vertices by adding one edge In this case, all digraphs on exactly n=vertices are generated. If for any graph G satisfying the property, every subgraph, obtained from G by deleting one edge but not the vertices incident to that edge, satisfies the property, then this will generate all digraphs with that property. If this does not hold, then all the digraphs generated will satisfy the property, but there will be some missing.
• implementation - which underlying implementation to use (see DiGraph?)
• sparse - ignored if implementation is not c_graph

EXAMPLES: Print digraphs on 2 or less vertices.

sage: for D in digraphs(2, augment='vertices'):
...    print D
...
Digraph on 0 vertices
Digraph on 1 vertex
Digraph on 2 vertices
Digraph on 2 vertices
Digraph on 2 vertices

Note that we can also get digraphs with underlying Cython implementation:

sage: for D in digraphs(2, augment='vertices', implementation='c_graph'):
...    print D
...
Digraph on 0 vertices
Digraph on 1 vertex
Digraph on 2 vertices
Digraph on 2 vertices
Digraph on 2 vertices

Print digraphs on 3 vertices.

sage: for D in digraphs(3):
...    print D
Digraph on 3 vertices
Digraph on 3 vertices
...
Digraph on 3 vertices
Digraph on 3 vertices

Generate all digraphs with 4 vertices and 3 edges.

sage: L = digraphs(4, size=3)
sage: len(list(L))
13

Generate all digraphs with 4 vertices and up to 3 edges.

sage: L = list(digraphs(4, lambda G: G.size() <= 3))
sage: len(L)
20
sage: graphs_list.show_graphs(L)  # long time

Generate all digraphs with degree at most 2, up to 5 vertices.

sage: property = lambda G: ( max([G.degree(v) for v in G] + [0]) <= 2 )
sage: L = list(digraphs(5, property, augment='vertices'))
sage: len(L)
75

Generate digraphs on the fly: (see http://oeis.org/classic/A000273)

sage: for i in range(0, 5):
...    print len(list(digraphs(i)))
1
1
3
16
218

REFERENCE:

• Brendan D. McKay, Isomorph-Free Exhaustive generation. Journal of Algorithms Volume 26, Issue 2, February 1998, pages 306-324.
ButterflyGraph(n, vertices='strings')

Returns a n-dimensional butterfly graph. The vertices consist of pairs (v,i), where v is an n-dimensional tuple (vector) with binary entries (or a string representation of such) and i is an integer in [0..n]. A directed edge goes from (v,i) to (w,i+1) if v and w are identical except for possibly v[i] != w[i].

A butterfly graph has $$(2^n)(n+1)$$ vertices and $$n2^{n+1}$$ edges.

INPUT:

• vertices - ‘strings’ (default) or ‘vectors’, specifying whether the vertices are zero-one strings or actually tuples over GF(2).

EXAMPLES:

sage: digraphs.ButterflyGraph(2).edges(labels=False)
[(('00', 0), ('00', 1)),
(('00', 0), ('10', 1)),
(('00', 1), ('00', 2)),
(('00', 1), ('01', 2)),
(('01', 0), ('01', 1)),
(('01', 0), ('11', 1)),
(('01', 1), ('00', 2)),
(('01', 1), ('01', 2)),
(('10', 0), ('00', 1)),
(('10', 0), ('10', 1)),
(('10', 1), ('10', 2)),
(('10', 1), ('11', 2)),
(('11', 0), ('01', 1)),
(('11', 0), ('11', 1)),
(('11', 1), ('10', 2)),
(('11', 1), ('11', 2))]
sage: digraphs.ButterflyGraph(2,vertices='vectors').edges(labels=False)
[(((0, 0), 0), ((0, 0), 1)),
(((0, 0), 0), ((1, 0), 1)),
(((0, 0), 1), ((0, 0), 2)),
(((0, 0), 1), ((0, 1), 2)),
(((0, 1), 0), ((0, 1), 1)),
(((0, 1), 0), ((1, 1), 1)),
(((0, 1), 1), ((0, 0), 2)),
(((0, 1), 1), ((0, 1), 2)),
(((1, 0), 0), ((0, 0), 1)),
(((1, 0), 0), ((1, 0), 1)),
(((1, 0), 1), ((1, 0), 2)),
(((1, 0), 1), ((1, 1), 2)),
(((1, 1), 0), ((0, 1), 1)),
(((1, 1), 0), ((1, 1), 1)),
(((1, 1), 1), ((1, 0), 2)),
(((1, 1), 1), ((1, 1), 2))]

Circuit(n)

Returns the circuit on $$n$$ vertices

The circuit is an oriented CycleGraph

EXAMPLE:

A circuit is the smallest strongly connected digraph:

sage: circuit = digraphs.Circuit(15)
sage: len(circuit.strongly_connected_components()) == 1
True

Circulant(n, integers)

Returns a circulant digraph on $$n$$ vertices from a set of integers.

INPUT:

• n (integer) – number of vertices.
• integers – the list of integers such that there is an edge from $$i$$ to $$j$$ if and only if (j-i)%n in integers.

EXAMPLE:

sage: digraphs.Circulant(13,[3,5,7])
Circulant graph ([3, 5, 7]): Digraph on 13 vertices

TESTS:

sage: digraphs.Circulant(13,[3,5,7,"hey"])
Traceback (most recent call last):
...
ValueError: The list must contain only relative integers.
sage: digraphs.Circulant(3,[3,5,7,3.4])
Traceback (most recent call last):
...
ValueError: The list must contain only relative integers.

DeBruijn(k, n, vertices='strings')

Returns the De Bruijn digraph with parameters $$k,n$$.

The De Bruijn digraph with parameters $$k,n$$ is built upon a set of vertices equal to the set of words of length $$n$$ from a dictionary of $$k$$ letters.

In this digraph, there is an arc $$w_1w_2$$ if $$w_2$$ can be obtained from $$w_1$$ by removing the leftmost letter and adding a new letter at its right end. For more information, see the Wikipedia article on De Bruijn graph.

INPUT:

• k – Two possibilities for this parameter :
• An integer equal to the cardinality of the alphabet to use, that is the degree of the digraph to be produced.
• An iterable object to be used as the set of letters. The degree of the resulting digraph is the cardinality of the set of letters.
• n – An integer equal to the length of words in the De Bruijn digraph when vertices == 'strings', and also to the diameter of the digraph.

• vertices – ‘strings’ (default) or ‘integers’, specifying whether the vertices are words build upon an alphabet or integers.

EXAMPLES:

sage: db=digraphs.DeBruijn(2,2); db
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
sage: db.order()
4
sage: db.size()
8

TESTS:

sage: digraphs.DeBruijn(5,0)
De Bruijn digraph (k=5, n=0): Looped multi-digraph on 1 vertex
sage: digraphs.DeBruijn(0,0)
De Bruijn digraph (k=0, n=0): Looped multi-digraph on 0 vertices

GeneralizedDeBruijn(n, d)

Returns the generalized de Bruijn digraph of order $$n$$ and degree $$d$$.

The generalized de Bruijn digraph has been defined in [RPK80] [RPK83]. It has vertex set $$V=\{0, 1,..., n-1\}$$ and there is an arc from vertex $$u \in V$$ to all vertices $$v \in V$$ such that $$v \equiv (u*d + a) \mod{n}$$ with $$0 \leq a < d$$.

When $$n = d^{D}$$, the generalized de Bruijn digraph is isomorphic to the de Bruijn digraph of degree $$d$$ and diameter $$D$$.

INPUTS:

• n – is the number of vertices of the digraph
• d – is the degree of the digraph

EXAMPLE:

sage: GB = digraphs.GeneralizedDeBruijn(8, 2)
sage: GB.is_isomorphic(digraphs.DeBruijn(2, 3), certify = True)
(True, {0: '000', 1: '001', 2: '010', 3: '011', 4: '100', 5: '101', 6: '110', 7: '111'})

TESTS:

An exception is raised when the degree is less than one:

sage: G = digraphs.GeneralizedDeBruijn(2, 0)
Traceback (most recent call last):
...
ValueError: The generalized de Bruijn digraph is defined for degree at least one.

An exception is raised when the order of the graph is less than one:

sage: G = digraphs.GeneralizedDeBruijn(0, 2)
Traceback (most recent call last):
...
ValueError: The generalized de Bruijn digraph is defined for at least one vertex.

REFERENCES:

 [RPK80] S. M. Reddy, D. K. Pradhan, and J. Kuhl. Directed graphs with minimal diameter and maximal connectivity, School Eng., Oakland Univ., Rochester MI, Tech. Rep., July 1980.
 [RPK83] S. Reddy, P. Raghavan, and J. Kuhl. A Class of Graphs for Processor Interconnection. IEEE International Conference on Parallel Processing, pages 154-157, Los Alamitos, Ca., USA, August 1983.
ImaseItoh(n, d)

Returns the digraph of Imase and Itoh of order $$n$$ and degree $$d$$.

The digraph of Imase and Itoh has been defined in [II83]. It has vertex set $$V=\{0, 1,..., n-1\}$$ and there is an arc from vertex $$u \in V$$ to all vertices $$v \in V$$ such that $$v \equiv (-u*d-a-1) \mod{n}$$ with $$0 \leq a < d$$.

When $$n = d^{D}$$, the digraph of Imase and Itoh is isomorphic to the de Bruijn digraph of degree $$d$$ and diameter $$D$$. When $$n = d^{D-1}(d+1)$$, the digraph of Imase and Itoh is isomorphic to the Kautz digraph [Kautz68] of degree $$d$$ and diameter $$D$$.

INPUTS:

• n – is the number of vertices of the digraph
• d – is the degree of the digraph

EXAMPLES:

sage: II = digraphs.ImaseItoh(8, 2)
sage: II.is_isomorphic(digraphs.DeBruijn(2, 3), certify = True)
(True, {0: '010', 1: '011', 2: '000', 3: '001', 4: '110', 5: '111', 6: '100', 7: '101'})

sage: II = digraphs.ImaseItoh(12, 2)
sage: II.is_isomorphic(digraphs.Kautz(2, 3), certify = True)
(True, {0: '010', 1: '012', 2: '021', 3: '020', 4: '202', 5: '201', 6: '210', 7: '212', 8: '121', 9: '120', 10: '102', 11: '101'})

TESTS:

An exception is raised when the degree is less than one:

sage: G = digraphs.ImaseItoh(2, 0)
Traceback (most recent call last):
...
ValueError: The digraph of Imase and Itoh is defined for degree at least one.

An exception is raised when the order of the graph is less than two:

sage: G = digraphs.ImaseItoh(1, 2)
Traceback (most recent call last):
...
ValueError: The digraph of Imase and Itoh is defined for at least two vertices.

REFERENCE:

 [II83] (1, 2) M. Imase and M. Itoh. A design for directed graphs with minimum diameter, IEEE Trans. Comput., vol. C-32, pp. 782-784, 1983.
Kautz(k, D, vertices='strings')

Returns the Kautz digraph of degree $$d$$ and diameter $$D$$.

The Kautz digraph has been defined in [Kautz68]. The Kautz digraph of degree $$d$$ and diameter $$D$$ has $$d^{D-1}(d+1)$$ vertices. This digraph is build upon a set of vertices equal to the set of words of length $$D$$ from an alphabet of $$d+1$$ letters such that consecutive letters are differents. There is an arc from vertex $$u$$ to vertex $$v$$ if $$v$$ can be obtained from $$u$$ by removing the leftmost letter and adding a new letter, distinct from the rightmost letter of $$u$$, at the right end.

The Kautz digraph of degree $$d$$ and diameter $$D$$ is isomorphic to the digraph of Imase and Itoh [II83] of degree $$d$$ and order $$d^{D-1}(d+1)$$.

INPUTS:

• k – Two possibilities for this parameter :
• An integer equal to the degree of the digraph to be produced, that is the cardinality minus one of the alphabet to use.
• An iterable object to be used as the set of letters. The degree of the resulting digraph is the cardinality of the set of letters minus one.
• D – An integer equal to the diameter of the digraph, and also to

the length of a vertex label when vertices == 'strings'.

• vertices – ‘strings’ (default) or ‘integers’, specifying whether

the vertices are words build upon an alphabet or integers.

EXAMPLES:

sage: K = digraphs.Kautz(2, 3)
sage: K.is_isomorphic(digraphs.ImaseItoh(12, 2), certify = True)
(True, {'201': 5, '120': 9, '202': 4, '212': 7, '210': 6, '010': 0, '121': 8, '012': 1, '021': 2, '020': 3, '102': 10, '101': 11})

sage: K = digraphs.Kautz([1,'a','B'], 2)
sage: K.edges()
[('1B', 'B1', '1'), ('1B', 'Ba', 'a'), ('1a', 'a1', '1'), ('1a', 'aB', 'B'), ('B1', '1B', 'B'), ('B1', '1a', 'a'), ('Ba', 'a1', '1'), ('Ba', 'aB', 'B'), ('a1', '1B', 'B'), ('a1', '1a', 'a'), ('aB', 'B1', '1'), ('aB', 'Ba', 'a')]

sage: K = digraphs.Kautz([1,'aA','BB'], 2)
sage: K.edges()
[('1,BB', 'BB,1', '1'), ('1,BB', 'BB,aA', 'aA'), ('1,aA', 'aA,1', '1'), ('1,aA', 'aA,BB', 'BB'), ('BB,1', '1,BB', 'BB'), ('BB,1', '1,aA', 'aA'), ('BB,aA', 'aA,1', '1'), ('BB,aA', 'aA,BB', 'BB'), ('aA,1', '1,BB', 'BB'), ('aA,1', '1,aA', 'aA'), ('aA,BB', 'BB,1', '1'), ('aA,BB', 'BB,aA', 'aA')]

TESTS:

An exception is raised when the degree is less than one:

sage: G = digraphs.Kautz(0, 2)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for degree at least one.

sage: G = digraphs.Kautz(['a'], 2)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for degree at least one.

An exception is raised when the diameter of the graph is less than one:

sage: G = digraphs.Kautz(2, 0)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for diameter at least one.

REFERENCE:

 [Kautz68] (1, 2) W. H. Kautz. Bounds on directed (d, k) graphs. Theory of cellular logic networks and machines, AFCRL-68-0668, SRI Project 7258, Final Rep., pp. 20-28, 1968.
Path(n)

Returns a directed path on $$n$$ vertices.

INPUT:

• n (integer) – number of vertices in the path.

EXAMPLES:

sage: g = digraphs.Path(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.size()
4
sage: g.automorphism_group().cardinality()
1

RandomDirectedGN(n, kernel=<function <lambda> at 0x7fb3108fa410>, seed=None)

Returns a random GN (growing network) digraph with n vertices.

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen with a preferential attachment model, i.e. probability is proportional to degree. The default attachment kernel is a linear function of degree. The digraph is always a tree, so in particular it is a directed acyclic graph.

INPUT:

• n - number of vertices.
• kernel - the attachment kernel
• seed - for the random number generator

EXAMPLE:

sage: D = digraphs.RandomDirectedGN(25)
sage: D.edges(labels=False)
[(1, 0), (2, 0), (3, 1), (4, 0), (5, 0), (6, 1), (7, 0), (8, 3), (9, 0), (10, 8), (11, 3), (12, 9), (13, 8), (14, 0), (15, 11), (16, 11), (17, 5), (18, 11), (19, 6), (20, 5), (21, 14), (22, 5), (23, 18), (24, 11)]
sage: D.show()  # long time

REFERENCE:

• [1] Krapivsky, P.L. and Redner, S. Organization of Growing Random Networks, Phys. Rev. E vol. 63 (2001), p. 066123.
RandomDirectedGNC(n, seed=None)

Returns a random GNC (growing network with copying) digraph with n vertices.

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen with a preferential attachment model, i.e. probability is proportional to degree. The new vertex is also linked to all of the previously added vertex’s successors.

INPUT:

• n - number of vertices.
• seed - for the random number generator

EXAMPLE:

sage: D = digraphs.RandomDirectedGNC(25)
sage: D.edges(labels=False)
[(1, 0), (2, 0), (2, 1), (3, 0), (4, 0), (4, 1), (5, 0), (5, 1), (5, 2), (6, 0), (6, 1), (7, 0), (7, 1), (7, 4), (8, 0), (9, 0), (9, 8), (10, 0), (10, 1), (10, 2), (10, 5), (11, 0), (11, 8), (11, 9), (12, 0), (12, 8), (12, 9), (13, 0), (13, 1), (14, 0), (14, 8), (14, 9), (14, 12), (15, 0), (15, 8), (15, 9), (15, 12), (16, 0), (16, 1), (16, 4), (16, 7), (17, 0), (17, 8), (17, 9), (17, 12), (18, 0), (18, 8), (19, 0), (19, 1), (19, 4), (19, 7), (20, 0), (20, 1), (20, 4), (20, 7), (20, 16), (21, 0), (21, 8), (22, 0), (22, 1), (22, 4), (22, 7), (22, 19), (23, 0), (23, 8), (23, 9), (23, 12), (23, 14), (24, 0), (24, 8), (24, 9), (24, 12), (24, 15)]
sage: D.show()  # long time

REFERENCE:

• [1] Krapivsky, P.L. and Redner, S. Network Growth by Copying, Phys. Rev. E vol. 71 (2005), p. 036118.
RandomDirectedGNM(n, m, loops=False)

Returns a random labelled digraph on $$n$$ nodes and $$m$$ arcs.

INPUT:

• n (integer) – number of vertices.
• m (integer) – number of edges.
• loops (boolean) – whether to allow loops (set to False by default).

PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

EXAMPLE:

sage: D = digraphs.RandomDirectedGNM(10, 5)
sage: D.num_verts()
10
sage: D.edges(labels=False)
[(0, 3), (1, 5), (5, 1), (7, 0), (8, 5)]

With loops:

sage: D = digraphs.RandomDirectedGNM(10, 100, loops = True)
sage: D.num_verts()
10
sage: D.loops()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None), (4, 4, None), (5, 5, None), (6, 6, None), (7, 7, None), (8, 8, None), (9, 9, None)]

TESTS:

sage: digraphs.RandomDirectedGNM(10,-3)
Traceback (most recent call last):
...
ValueError: The number of edges must satisfy 0<= m <= n(n-1) when no loops are allowed, and 0<= m <= n^2 otherwise.

sage: digraphs.RandomDirectedGNM(10,100)
Traceback (most recent call last):
...
ValueError: The number of edges must satisfy 0<= m <= n(n-1) when no loops are allowed, and 0<= m <= n^2 otherwise.

RandomDirectedGNP(n, p, loops=False, seed=None)

Returns a random digraph on $$n$$ nodes. Each edge is inserted independently with probability $$p$$.

INPUTS:

• n – number of nodes of the digraph
• p – probability of an edge
• loops – is a boolean set to True if the random digraph may have loops, and False (default) otherwise.
• seed – integer seed for random number generator (default=None).

REFERENCES:

 [1] P. Erdos and A. Renyi, On Random Graphs, Publ. Math. 6, 290 (1959).
 [2] Gilbert, Random Graphs, Ann. Math. Stat., 30, 1141 (1959).

PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

EXAMPLE:

sage: set_random_seed(0)
sage: D = digraphs.RandomDirectedGNP(10, .2)
sage: D.num_verts()
10
sage: D.edges(labels=False)
[(1, 0), (1, 2), (3, 6), (3, 7), (4, 5), (4, 7), (4, 8), (5, 2), (6, 0), (7, 2), (8, 1), (8, 9), (9, 4)]

RandomDirectedGNR(n, p, seed=None)

Returns a random GNR (growing network with redirection) digraph with n vertices and redirection probability p.

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen uniformly. With probability p, the arc is instead redirected to the successor vertex. The digraph is always a tree.

INPUT:

• n - number of vertices.
• p - redirection probability
• seed - for the random number generator.

EXAMPLE:

sage: D = digraphs.RandomDirectedGNR(25, .2)
sage: D.edges(labels=False)
[(1, 0), (2, 0), (2, 1), (3, 0), (4, 0), (4, 1), (5, 0), (5, 1), (5, 2), (6, 0), (6, 1), (7, 0), (7, 1), (7, 4), (8, 0), (9, 0), (9, 8), (10, 0), (10, 1), (10, 2), (10, 5), (11, 0), (11, 8), (11, 9), (12, 0), (12, 8), (12, 9), (13, 0), (13, 1), (14, 0), (14, 8), (14, 9), (14, 12), (15, 0), (15, 8), (15, 9), (15, 12), (16, 0), (16, 1), (16, 4), (16, 7), (17, 0), (17, 8), (17, 9), (17, 12), (18, 0), (18, 8), (19, 0), (19, 1), (19, 4), (19, 7), (20, 0), (20, 1), (20, 4), (20, 7), (20, 16), (21, 0), (21, 8), (22, 0), (22, 1), (22, 4), (22, 7), (22, 19), (23, 0), (23, 8), (23, 9), (23, 12), (23, 14), (24, 0), (24, 8), (24, 9), (24, 12), (24, 15)]
sage: D.show()  # long time

REFERENCE:

• [1] Krapivsky, P.L. and Redner, S. Organization of Growing Random Networks, Phys. Rev. E vol. 63 (2001), p. 066123.
RandomTournament(n)

Returns a random tournament on $$n$$ vertices.

For every pair of vertices, the tournament has an edge from $$i$$ to $$j$$ with probability $$1/2$$, otherwise it has an edge from $$j$$ to $$i$$.

INPUT:

• n (integer) – number of vertices.

EXAMPLES:

sage: T = digraphs.RandomTournament(10); T
Random Tournament: Digraph on 10 vertices
sage: T.size() == binomial(10, 2)
True
sage: digraphs.RandomTournament(-1)
Traceback (most recent call last):
...
ValueError: The number of vertices cannot be strictly negative!

TransitiveTournament(n)

Returns a transitive tournament on $$n$$ vertices.

In this tournament there is an edge from $$i$$ to $$j$$ if $$i<j$$.

INPUT:

• n (integer) – number of vertices in the tournament.

EXAMPLES:

sage: g = digraphs.TransitiveTournament(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.size()
10
sage: g.automorphism_group().cardinality()
1

TESTS:

sage: digraphs.TransitiveTournament(-1)
Traceback (most recent call last):
...
ValueError: The number of vertices cannot be strictly negative!

tournaments_nauty(n, min_out_degree=None, max_out_degree=None, strongly_connected=False, debug=False, options='')

Returns all tournaments on $$n$$ vertices using Nauty.

INPUT:

• n (integer) – number of vertices.
• min_out_degree, max_out_degree (integers) – if set to None (default), then the min/max out-degree is not constrained.
• debug (boolean) – if True the first line of genbg’s output to standard error is captured and the first call to the generator’s next() function will return this line as a string. A line leading with “>A” indicates a successful initiation of the program with some information on the arguments, while a line beginning with “>E” indicates an error with the input.
• options (string) – anything else that should be forwarded as input to Nauty’s genbg. See its documentation for more information : http://cs.anu.edu.au/~bdm/nauty/.

Note

To use this method you must first install the Nauty spkg.

EXAMPLES:

sage: for g in digraphs.tournaments_nauty(4): # optional - nauty
....:    print g.edges(labels = False)        # optional - nauty
[(1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2)]
[(1, 0), (1, 3), (2, 0), (2, 1), (3, 0), (3, 2)]
[(0, 2), (1, 0), (2, 1), (3, 0), (3, 1), (3, 2)]
[(0, 2), (0, 3), (1, 0), (2, 1), (3, 1), (3, 2)]
sage: tournaments = digraphs.tournaments_nauty
sage: [len(list(tournaments(x))) for x in range(1,8)] # optional - nauty
[1, 1, 2, 4, 12, 56, 456]
sage: [len(list(tournaments(x, strongly_connected = True))) for x in range(1,9)] # optional - nauty
[1, 0, 1, 1, 6, 35, 353, 6008]


text:
# Scherk surface

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2010 Mathematics Subject Classification: Primary: 53A10 Secondary: 49Q05 [MSN][ZBL]

A minimal surface discovered by H. Scherk in 1834. It is defined by the equation $z = \ln \frac{\cos y}{\cos x} = \ln \cos y - \ln \cos x$ and it is the only minimal surface that can be represented as a translation surface of the form $z = f(x) + g (y)$. Scherk's surface and its modifications are used for the construction of auxiliary functions that allow one to find examples of the unsolvability of the Dirichlet problem for the Euler–Lagrange equation for minimal surfaces over non-convex domains.

Scherk's surface possesses a number of interesting properties:

• it is a complete surface of infinite genus containing a countable number of straight lines;
• its universal covering surface presents an example of a complete minimal surface of conformally-hyperbolic type;
• its Gauss map omits exactly four points.

The last property of Scherk's surface becomes evident from its Weierstrass representation. More precisely if we consider the meromorphic function $w \mapsto f(w) = \frac{2}{1-w^4}$ and the holomorphic function $f (w) =w$, Scherk's surface can be parameterized as $\begin{array}{lll} x (r, \phi) &= \mathcal{Re}\, \int_0^\omega f (1-g^2)\, dw &= \mathcal{Re}\, (ln (1+ i \omega) + \ln (1- i \omega))\\ y (r, \phi) &= \mathcal{Re}\, \int_0^\omega i f (1+g^2)\, dw & = \mathcal{Re}\, i ( \ln (1+\omega) + \ln (1-\omega))\\ z (r, \phi) &= \mathcal{Re}\, \int_0^\omega 2 f g\, dw & = \mathcal{Re}\, \left[ \ln \frac{1-\omega}{1+\omega} + i \ln \frac{1+i\omega}{1-i\omega}\right] \, , \end{array}$ where $\omega = r e^{i\phi}$. The four points missing in the image of the Gauss map corresponds then to the four poles $\{\pm 1, \pm i\}$ of $f$.

By analogy with this representation one may construct the generalized Scherk surfaces using $f (w) = [(w-w_1) (w-w_2) (w-w_3) (w-w_4)]^{-1}$ with arbitrary distinct points $w_1, w_2, w_3, w_4$ in the complex plane. The Gauss map of this minimal surface omits precisely $4$ arbitrarily assigned points on $\mathbb S^2$.

The existence of such minimal surfaces motivated the conjecture that the only complete minimal surfaces in $\mathbb R^3$ whose (oriented) normals omit more than four directions are the planes. After several improvements of a classical result of Osserman, the conjecture was finally proved in [Fu]; see also [Xa].

Scherk's surface belongs to the family of so-called periodic minimal surfaces; a picture of it, and other interesting properties, can be found in [TF].

#### References

 [DHKW] U. Dierkes, S. Hildebrandt, A. Küster, O. Wohlrab, "Minimal surfaces", vol. I, p. 108. Springer 1992. [DoC] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1975) Zbl 0733.53001 Zbl 0606.53002 Zbl 0326.53001 [Fu] H. Fujimoto, "On the number of exceptional values of the Gauss map of minimal surface." J. Math. Soc. Japan. 40 (1988) pp. 235-247 [Gr] Gray, A. "Minimal Surfaces via the Weierstrass Representation." Ch. 32 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 735-760, 1997. [Ni] J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) MR0448224 Zbl 0319.53003 [Os] R. Osserman., "A survey of minimal surfaces" , Dover (1986) [Sp] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish MR0532834 MR0532833 MR0532832 MR0532831 MR0532830 MR0394453 MR0394452 MR0372756 MR1537051 MR0271845 MR0267467 Zbl 1213.53001 Zbl 0439.53005 Zbl 0439.53004 Zbl 0439.53003 Zbl 0439.53002 Zbl 0439.53001 Zbl 0306.53003 Zbl 0306.53002 Zbl 0306.53001 Zbl 0202.52201 Zbl 0202.52001 [St] D.J. Struik, "Differential geometry" , Addison-Wesley (1957) MR0939369 MR1528072 MR0036551 MR1562092 Zbl 0697.53002 Zbl 0105.14707 Zbl 0041.48603 Zbl 0007.38806 Zbl 0001.16402 Zbl 59.0010.03 Zbl 57.0860.04 [TF] A.A. Tuzhilin, A.T. Fomenko, "Elements of the geometry and topology of minimal surfaces" , Moscow (1991) (In Russian) MR1162118 MR1134130 Zbl 0745.53001 Zbl 0781.53002 [Xa] F. Xavier, "The Gauss map of a complete non-flat minimal surface cannot omit 7 points of the sphere." Ann. of Math. 113 (1981) pp. 211-214.
How to Cite This Entry:
Scherk surface. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Scherk_surface&oldid=32370
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article


text:
$$\require{cancel}$$

34A: Pascal’s Principle, the Continuity Equation, and Bernoulli’s Principle

There are a couple of mistakes that tend to crop up with some regularity in the application of the Bernoulli equation $$P+\frac{1}{2}\varrho v^2 +\varrho gh =\mbox{constant}$$. Firstly, folks tend to forget to create a diagram in order to identify point 1 and point 2 in the diagram so that they can write the Bernoulli equation in its useful form: $$P_1+\frac{1}{2}\varrho v_1^2 +\varrho gh_1 =P_2+\frac{1}{2}\varrho v_2^2 +\varrho gh_2$$. Secondly, when both the velocities in Bernoulli’s equation are unknown, they forget that there is another equation that relates the velocities, namely, the continuity equation in the form $$A_1v_1 =A_2v_2$$ which states that the flow rate at position 1 is equal to the flow rate at position 2.

Pascal’s Principle

Experimentally, we find that if you increase the pressure by some given amount at one location in a fluid, the pressure increases by that same amount everywhere in the fluid. This experimental result is known as Pascal’s Principle.

We take advantage of Pascal’s principle every time we step on the brakes of our cars and trucks. The brake system is a hydraulic system. The fluid is oil that is called hydraulic fluid. When you depress the brake pedal you increase the pressure everywhere in the fluid in the hydraulic line. At the wheels, the increased pressure acting on pistons attached to the brake pads pushes them against disks or drums connected to the wheels.

Example 34-1

A simple hydraulic lift consists of two pistons, one larger than the other, in cylinders connected by a pipe. The cylinders and pipe are filled with water. In use, a person pushes down upon the smaller piston and the water pushes upward on the larger piston. The diameter of the smaller piston is 2.20 centimeters. The diameter of the larger piston is 21.0 centimeters. On top of the larger piston is a metal support and on top of that is a car. The combined mass of the support-plus-car is 998 kg. Find the force that the person must exert on the smaller piston to raise the car at a constant velocity. Neglect the masses of the pistons.

Solution

We start our solution with a sketch.

Now, let’s find the force $$R_N$$ exerted on the larger piston by the car support. By Newton’s 3rd Law, it is the same as the normal force $$F_N$$ exerted by the larger piston on the car support. We’ll draw and analyze the free body diagram of the car-plus-support to get that.

$\sum F_{\uparrow}=0$

$F_N-mg=0$

$F_N=mg$

$F_N=998kg\space (9.80\frac{\mbox{newtons}}{kg})$

$F_N=9780 \space \mbox{newtons}$

Table of Forces
Symbol Name  Agent  Victim
$$F_g=mg$$ Gravitational Force on Support-Plus Car  The Earth's Gravitational Field  The Support-Plus Car
$$F_N$$  Normal Force  The Large Piston  The Support-Plus Car

Now we analyze the equilibrium of the larger piston to determine what the pressure in the fluid must be in order for the fluid to exert enough force on the piston (with the car-plus-support on it) to keep it moving at constant velocity.

$\sum F_{\uparrow}=0$

$F_{PL}-R_N=0$

$PA_L-R_N=0$

$P=\frac{R_N}{A_L}\label{34-1}$

 Table of Forces Symbol Name Agent Victim $$R_N=F_N=9780 \space \mbox{newtons}$$ Interaction Partner to Normal Force (see above) Support (That part, of the hydraulic lift, that the car is on). Large Piston $$F_{PL}$$ Pressure-Related Force on Large Piston The Water Large Piston

$$A_L$$ is the area of the face of the larger piston. We can use the given larger piston diameter $$D_L= 0.210\space m$$ to determine the area of the face of the larger piston as follows:

$A_L=\pi r_L^2 \quad \mbox{where} \space r_L=\frac{D_L}{2} \space \mbox{is the radius of the larger piston}$

$A_L=\pi \Big ( \frac{D_L}{2} \Big ) ^2$

$A_L=\pi \Big ( \frac{0.210 \mbox{m}}{2} \Big ) ^2$

$A_L=0.03464 \space m^2$

Substituting this and the value $$R_N = F_N = 9780 \space \mbox{newtons}$$ into equation $$\ref{34-1}$$ above yields

$P=\frac{9780 \space \mbox{newtons}}{0.03464 \space \mbox{m}^2}$

$P=282333 \frac{N}{m^2}\quad \mbox{(We intentionally keep 3 too many significant figures in this intermediate result.)}$

Now we just have to analyze the equilibrium of the smaller piston to determine the force that the person must exert on the smaller piston.

$\sum F_{\uparrow}=0$

$F_{PS}-F_{PERSON}=0$

$PA_S-F_{PERSON}=0$

$F_{PERSON}=PA_S$

The area $$A_S$$ of the face of the smaller piston is just $$\pi$$ times the square of the radius of the smaller piston where the radius is $$\frac{D_S}{2}$$, half the diameter of the smaller piston. So:

$F_{PERSON}=P\pi \Big(\frac{D_S}{2} \Big)^2$

$F_{PERSON}=282333 \frac{N}{m^2} \pi \Big(\frac{0.0220 m}{2} \Big)^2$

$F_{PERSON}=107N$

Fluid in Motion—the Continuity Principle

The Continuity Principle is a fancy name for something that common sense will tell you has to be the case. It is simply a statement of the fact that for any section of a single pipe, filled with an incompressible fluid (an idealization approached by liquids), through which the fluid with which the pipe is filled is flowing, the amount of fluid that goes in one end in any specified amount of time is equal to the amount that comes out the other end in the same amount of time. If we quantify the amount of fluid in terms of the mass, this is a statement of the conservation of mass. Having stipulated that the segment is filled with fluid, the incoming fluid has no room to expand in the segment. Having stipulated that the fluid is incompressible, the molecules making up the fluid cannot be packed closer together; that is, the density of the fluid cannot change. With these stipulations, the total mass of the fluid in the segment of pipe cannot change, so, any time a certain mass of the fluid flows in one end of the segment, the same mass of the fluid must flow out the other end.

This can only be the case if the mass flow rate, the number of kilograms-per-second passing a given position in the pipe, is the same at both ends of the pipe segment.

$\dot{m_1}=\dot{m_2}\label {34-2}$

An interesting consequence of the continuity principle is the fact that, in order for the mass flow rate (the number of kilograms per second passing a given position in the pipe) to be the same in a fat part of the pipe as it is in a skinny part of the pipe, the velocity of the fluid (i.e. the velocity of the molecules of the fluid) must be greater in the skinny part of the pipe. Let’s see why this is the case.

Here, we again depict a pipe in which an incompressible fluid is flowing.

Keeping in mind that the entire pipe is filled with the fluid, the shaded region on the left represents the fluid that will flow past position 1 in time $$\Delta t$$ and the shaded region on the right represents the fluid that will flow past position 2 in the same time $$\Delta t$$. In both cases, in order for the entire slug of fluid to cross the relevant position line, the slug must travel a distance equal to its length. Now the slug labeled $$\Delta m_2$$ has to be longer than the slug labeled $$\Delta m_1$$ since the pipe is skinnier at position 2 and by the continuity equation $$\Delta m_1=\Delta m_2$$ (the amount of fluid that flows into the segment of the pipe between position 1 and position 2 is equal to the amount of fluid that flows out of it). So, if the slug at position 2 is longer and it has to travel past the position line in the same amount of time as it takes for the slug at position 1 to travel past its position line, the fluid velocity at position 2 must be greater. The fluid velocity is greater at a skinnier position in the pipe.

Let’s get a quantitative relation between the velocity at position 1 and the velocity at position 2.

Starting with

$\Delta m_1=\Delta m_2$

we use the definition of density to replace each mass with the density of the fluid times the relevant volume:

$\varrho \Delta V_1=\varrho \Delta V_2$

Dividing both sides by the density tells us something you already know:

$\Delta V_1=\Delta V_2$

As an aside, we note that if you divide both sides by $$\Delta t$$ and take the limit as $$\Delta t$$ goes to zero, we have $$\dot{V_1}=\dot{V_2}$$ which is an expression of the continuity principle in terms of volume flow rate. The volume flow rate is typically referred to simply as the flow rate. While we use the SI units $$\frac{\mbox{m}^3}{\mbox{s}}$$ for flow rate, the reader may be more familiar with flow rate expressed in units of gallons per minute.

Now back to our goal of finding a mathematical relation between the velocities of the fluid at the two positions in the pipe. Here we copy the diagram of the pipe and add, to the copy, a depiction of the face of slug 1 of area $$A_1$$ and the face of slug 2 of area $$A_2$$.

We left off with the fact that $$\Delta V_1=\Delta V_2$$ . Each volume can be replaced with the area of the face of the corresponding slug times the length of that slug. So,

$A_1 \Delta x_1=A_2 \Delta x_2$

Recall that $$\Delta x_1$$ is not only the length of slug 1, it is also how far slug 1 must travel in order for the entire slug of fluid to get past the position 1 line. The same is true for slug 2 and position 2. Dividing both sides by the one time interval $$\Delta t$$ yields:

$A_1 \frac{\Delta x_1}{\Delta t}=A_2 \frac{\Delta x_2}{\Delta t}$

Taking the limit as $$\Delta t$$ goes to zero results in:

$A_1 v_1=A_2 v_2 \label{34-3}$

This is the relation, between the velocities, that we have been looking for. It applies to any pair of positions in a pipe completely filled with an incompressible fluid. It can be written as

$Av=\mbox{constant}\label{34-4}$

which means that the product of the cross-sectional area of the pipe and the velocity of the fluid at that cross section is the same for every position along the fluid-filled pipe. To take advantage of this fact, one typically writes, in equation form, that the product $$Av$$ at one location is equal tothe same product at another location. In other words, one writes equation $$\ref{34-3}$$.

Note that the expression $$Av$$, the product of the cross-sectional area of the pipe, at a particular position, and the velocity of the fluid at that same position, having been derived by dividing an expression for the volume of fluid $$\Delta V$$ that would flow past a given position of the pipe in time $$\Delta t$$, by $$\Delta t$$, and taking the limit as $$\Delta t$$ goes to zero, is none other than the flow rate (the volume flow rate) discussed in the aside above.

$\mbox {Flow Rate}=Av$

Further note that if we multiply the flow rate by the density of the fluid, we get the mass flow rate.

$\dot{m}=\varrho Av\label{34-5}$

Fluid in Motion—Bernoulli’s Principle

The derivation of Bernoulli’s Equation represents an elegant application of the Work-Energy Theorem. Here we discuss the conditions under which Bernoulli’s Equation applies and then simply state and discuss the result.

Bernoulli’s Equation applies to a fluid flowing through a full pipe. The degree to which Bernoulli’s Equation is accurate depends on the degree to which the following conditions are met:

1. The fluid must be experiencing steady state flow. This means that the flow rate at all positions in the pipe is not changing with time.
2. The fluid must be experiencing streamline flow. Pick any point in the fluid. The infinitesimal fluid element at that point, at an instant in time, traveled along a certain path to arrive at that point in the fluid. In the case of streamline flow, every infinitesimal element of fluid that ever finds itself at that same point traveled the same path. (Streamline flow is the opposite of turbulent flow.)
3. The fluid must be non-viscous. This means that the fluid has no tendency to “stick to” either the sides of the pipe or to itself. (Molasses has high viscosity. Alcohol has low viscosity.)

Consider a pipe full of a fluid that is flowing through the pipe. In the most general case, the cross-sectional area of the pipe is not the same at all positions along the pipe and different parts of the pipe are at different elevations relative to an arbitrary, but fixed, reference level.

Pick any two positions along the pipe, e.g. positions 1 and 2 in the diagram above. (You already know that, in accord with the continuity principle, $$A_1v_1 =A_2v_2$$.) Consider the following unnamed sum of terms:

$P+\frac{1}{2} \varrho \space v^2+\varrho gh$

where, at the position under consideration:

• $$P$$ is the pressure of the fluid,
• $$\varrho$$ (the Greek letter rho) is the density of the fluid,
• $$v$$ is the magnitude of the velocity of the fluid,
• $$g=9.80 \frac{N}{kg}$$ is the near-surface magnitude of the earth’s gravitational field, and
• $$h$$ is the elevation, relative to a fixed reference level, of the position in the pipe.

The Bernoulli Principle states that this unnamed sum of terms has the same value at each and every position along the pipe. Bernoulli’s equation is typically written:

$P+\frac{1}{2} \varrho \space v^2+\varrho gh=\mbox{constant}\label{34-6}$

but to use it, you have to pick two positions along the pipe and write an equation stating that the value of the unnamed sum of terms is the same at one of the positions as it is at the other.

$P_1+\frac{1}{2} \varrho \space v_1^2+\varrho gh_1=P_2+\frac{1}{2} \varrho \space v_2^2+\varrho gh_2 \label{34-7}$

A particularly interesting characteristic of fluids is incorporated in this equation. Suppose positions 1 and 2 are at one and the same elevation as depicted in the following diagram:

Then $$h_1 = h_2$$ in equation $$\ref{34-7}$$ and equation $$\ref{34-7}$$ becomes:

$P_1+\frac{1}{2} \varrho \space v_1^2=P_2+\frac{1}{2} \varrho \space v_2^2$

Check it out. If $$v_2 > v_1$$ then $$P_2$$ must be less than $$P_1$$ in order for the equality to hold. This equation is saying that, where the velocity of the fluid is high, the pressure is low.


text:
# derivative of 1/[sqrt[x]: proving with derivative definition

• Nov 6th 2009, 12:40 PM
Evan.Kimia
derivative of 1/[sqrt[x]: proving with derivative definition
i need to solve 1/sqrt[x] using the limit method. i checked it first using the various rules and got 1/2x^3/2. heres what i did:

placed the function in the definition

1/sqrt[(x+h)] - 1/sqrt[x]
----------------------------
h

[(x+h)^-.5] - [x^-.5]
------------------------
h

apply conjugate:
[(x+h)^-.5] - [x^-.5] * [(x+h)^-.5] + [x^-.5]
------------------------
h * [(x+h)^-.5] + [x^-.5]

im lost here on out.
• Nov 6th 2009, 01:09 PM
Defunkt
Quote:

Originally Posted by Evan.Kimia
i need to solve 1/sqrt[x] using the limit method. i checked it first using the various rules and got 1/2x^3/2. heres what i did:

placed the function in the definition

1/sqrt[(x+h)] - 1/sqrt[x]
----------------------------
h

[(x+h)^-.5] - [x^-.5]
------------------------
h

apply conjugate:
[(x+h)^-.5] - [x^-.5] * [(x+h)^-.5] + [x^-.5]
------------------------
h * [(x+h)^-.5] + [x^-.5]

im lost here on out.

$\frac{\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}}{h} = \frac{\sqrt{x} - \sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}} = \frac{x-x-h}{h\sqrt{x}\sqrt{x+h}(\sqrt{x} + \sqrt{x+h})}$ $= \frac{-1}{\sqrt{x}\sqrt{x+h}(\sqrt{x} + \sqrt{x+h})} \to \frac{-1}{\sqrt{x}\sqrt{x}(\sqrt{x} + \sqrt{x})} = \frac{-1}{2x^{\frac{3}{2}}}$
• Nov 6th 2009, 02:51 PM
tonio
Quote:

Originally Posted by Evan.Kimia
i need to solve 1/sqrt[x] using the limit method. i checked it first using the various rules and got 1/2x^3/2. heres what i did:

placed the function in the definition

1/sqrt[(x+h)] - 1/sqrt[x]
----------------------------
h

[(x+h)^-.5] - [x^-.5]
------------------------
h

apply conjugate:
[(x+h)^-.5] - [x^-.5] * [(x+h)^-.5] + [x^-.5]
------------------------
h * [(x+h)^-.5] + [x^-.5]

im lost here on out.

Good idea to use the conjugate but you have to push forward a little more:

$\lim_{h\to 0}\,\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}=\lim_{h\to 0}\,\frac{\frac{1}{x+h}-\frac{1}{x}}{h\left(\frac{1}{\sqrt{x+h}}+\frac{1}{ \sqrt{x}}\right)}$ $=\lim_{h\to 0}\,\frac{-h}{h\left(\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}\ right)x(x+h)}=\lim_{h\to 0}\,\frac{-1}{\left(\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}\r ight)x(x+h)}$ $=-\frac{1}{\frac{2}{\sqrt{x}}x^2}=-\frac{1}{2x^{3/2}}=-\frac{1}{2x\sqrt{x}}$

Tonio


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# What's application of Bernstein Set?

Bernstein Set: A subset of the real line that meets every uncountable closed subset of the real line but that contains none of them. It's from wiki.

My question is this: How to construct a Bernstein set? And what's its application in mathematics?

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Why do you ask people in the comments to "vote you"? –  Asaf Karagila Jul 12 '12 at 6:46

One uses the axiom of choice in some form to construct a Bernstein set. The most straightforward way is by transfinite recursion. There are $2^\omega$ uncountable closed subsets of $\Bbb R$, so we can list them as $\{F_\xi:\xi<2^\omega\}$, and it can be proved that each of them has cardinality $2^\omega$.

Now suppose that $\eta<2^\omega$, and for each $\xi<\eta$ you’ve chosen points $x_\xi,y_\xi\in F_\xi$ so that all of these points are distinct. Let $X_\eta=\{x_\xi:\xi<\eta\}$ and $Y_\eta=\{y_\xi:\xi<\eta\}$. Then $|X_\eta\cup Y_\eta|<2^\omega$, so $F_\eta\setminus(X_\eta\cup Y_\eta)$ is infinite, and we can choose distinct $x_\eta,y_\eta\in F_\eta\setminus(X_\eta\cup Y_\eta)$ to continue the construction.

Now let $X=\bigcup_{\xi<2^\omega}X_\xi$ and $Y=\bigcup_{\xi<2^\omega}Y_\xi$; by construction $X$ and $Y$ are disjoint sets meeting each uncountable closed subset of $\Bbb R$, so both are Bernstein sets.

Added: In my experiences they are most useful as a tool for constructing (counter)examples. This post in Dan Ma’s Topology Blog is a good example of such use.

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Very appreciatd your construction! (Please vote me:)) –  Paul Jul 12 '12 at 1:00

Since a Bernstein set is nonmeasurable, you can't do it without some form of the Axiom of Choice: there's no "explicit" construction.

An example of an application: show that every measurable set of positive measure contains a nonmeasurable set.

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A nice application! (please vote me also:) –  Paul Jul 12 '12 at 1:01

Notice that the usual construction of the Bernstein set can also yield $\mathfrak c$ disjoint Bernstein sets in an arbitrary uncountable Polish space by working diagonally, similarly to how you construct a bijection between $\mathbf N$ and $\mathbf N^2$: you take one point from the first closed set into the first Bernstein set, two points from what remains of the second closed set to first two Bernstein sets etc, resulting in a sequence of distjoint sets, each of which intersects all but possibly some $\kappa<\mathfrak c$ uncountable closed sets. Then you notice that every (uncountable) closed set contains $\mathfrak c$ other (uncountable) closed subsets, so each of those actually intersects ALL uncountable closed sets.

In many ways, Bernstein sets are "as pathological as possible" for a subset of a Polish space, and as such are often good source of examples of pathological behavior.

Bernstein sets are not only non-measurable (and extremely so: the outer measure of a Bernstein set with respect to any continuous Borel measure is full, while the inner is zero!), but also do not have Baire property.

They are also quite interesting as topological measure spaces. Every compact subset of the Bernstein set is countable, so it's impossible to define a (nontrivial) continuous Radon measure on Bernstein set, which makes them sort of the opposite of Polish spaces -- in a Polish space, any finite Borel measure is Radon.

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"The usual construction of the Bernstein set can also yield c disjoint Bernstein sets in an arbitrary uncountable Polish space by working diagonally." It's nice:) Vote me! –  Paul Jul 12 '12 at 2:11

This is exactly the Proposition 3.1 in Handbook of Game Theory, Volume 1, Chapter 3, Jan Mycielski.

There is a payoff set such that the Gale-Stewart game is not determined.

Proof: Let the payoff set be the Bernstein set. Given an arbitary strategy, the set of all possible outcomes generated by it is a perfect set(see Brian M. Scott's answer), so for any strategy of one player, there is pair of opponent's strategies that generate two outcomes that are in the payoff set and its complement respectively.

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Here's one: If $\omega_1$ injects into reals then there is an uncountable set of reals without perfect subset. Hence the statement that every set of reals has the perfect set property implies that $\omega_1$ is inaccessible in $L$ (constructible universe).

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Here's an "application": Let P be a partition of reals induced by the equivalence relation $x \sim y$ iff $x - y$ is rational. A choice set for P is called a Vitali set. Show that for each real $m > 0$ there is a Vitali set of outer measure $m$.

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# How do you graph -2x + 3y < -6 ?

Jul 20, 2015

Graph line: - 2x + 3y < - 6 (1)

#### Explanation:

Bring the line equation to standard form: -2x + 3y + 6 < 0

First, graph the line: -2x + 3y + 6 = 0 by its 2 intercepts.
Make x = 0 -> y = -2. Make y = 0 -> x = 3.
To find the solution set, use the origin O as test point. Replace x = 0 and y = 0 into inequality (1). We get: 0 < -6. Not true. Then, the solution set is the area that doesn't contain O. Color or shade it. That is the answer.
graph{-2x + 3y + 6 = 0 [-10, 10, -5, 5]}


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# Solving equations with e

For example, let's say that f(x) = e^(-9t)
Once I've found the second derivative, I equate it to zero in order to determine concavity, but I don't know how to solve it after that

I'm just having a hard time solving with the ln or whatever you have to do..

Hootenanny
Staff Emeritus
Gold Member
Are you trying to solve the equation to find t=something? Or are you tring to do something else? Is t a function of x? Could you clarify your problem please because at the moment it makes no sense whatsoever.

Last edited:
Redbelly98
Staff Emeritus
Homework Helper
If you're trying to determine concavity, you need to figure out when the 2nd derivative is positive, negative, or zero. Just setting it to zero will tell you where the inflection points are (neither concave up nor down), and if there are no such points then you'll get "no solution" when you do this.

sorry, what i meant was set the first derivative to zero, and solve for t. Then use the t value to solve for the second derivative.

(The function should have been f(t) not f(x))

Redbelly98
Staff Emeritus
Homework Helper
Okay.

Plug the value of t you have (from setting 1st derivative = 0) into the 2nd derivative.

Is the result positive, negative, or zero?

Hootenanny
Staff Emeritus
Gold Member
Plug the value of t you have (from setting 1st derivative = 0) into the 2nd derivative.
Err, just one problem with that: et has no roots. A purely exponential function has no stationary points and therefore it's first derviative is never zero. However, functions involving expoentials can have stationary points. For example,

$$y(x) = e^x-x$$

$$y^\prime(x) = e^x-1$$

Hence, y(x) has a stationary point when,

$$e^x = 1 \Rightarrow x=0$$

Last edited:
HallsofIvy
Homework Helper
For example, let's say that f(x) = e^(-9t)
Once I've found the second derivative, I equate it to zero in order to determine concavity, but I don't know how to solve it after that

I'm just having a hard time solving with the ln or whatever you have to do..
No, you don't "equate the second derivative to zero in order to determin concavity".

The concavity of f(t) at a particular t depends upon the sign of f" at that particular t.

The second derivative of e^(-9t) is 81e^(-9t) and , as pointed out by others, that is never equal to 0. At t= 0, it is 81> 0. Since 81e^(-9t) is continuous and never equal to 0, it is always positive and so f(t)= e^(-9t) is concave upward for all t.

Thanks for all your help everyone


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# Classifying acid-base reaction between aluminium hydride and lithium hydride

So I was dealing with a reaction the other day, which is also briefly written in the title:

$$\ce{AlH3(s) + LiH(aq) -> Li+(aq) + AlH4-(aq)}$$

The strange thing is:

• Arrhenius definition cannot explain the reaction, since there is no $\ce{OH-}$ in the reaction formula.

• Also Brønsted–Lowry definition cannot explain the reaction, because the stuff which $\ce{LiH}$ gives to $\ce{AlH3}$ is not a proton (hydrogenium, $\ce{H+}$) but a protide (hydride, $\ce{H-}$).

• And as far as I know, this is the exact definition of Lewis acid and base written in my chemistry textbook:

• Lewis acid: A substance which can receive unshared electron pair
• Lewis base: A substance which can give unshared electron pair

leading to a conclusion that the above reaction also cannot be explained through the Lewis definition. Below are my questions:

• Is there anything wrong in my thought?
• Is that really an acid-base reaction? If it is, what kind of acid-base reaction is it?
• "(aq)" really? Sep 28 '17 at 14:04
• @Mithoron Yes, that is how it was written. Sep 29 '17 at 7:51
• Then better get a new book, 'cause you'd sooner get a nuke in your basement then aqueous LiH. Sep 29 '17 at 14:06

The hydride ion has a lone pair which is, in general, not bound to anything. The aluminium atom has at least one unpopulated orbital (typically taken to be a $\mathrm p_z$ orbital in a planar $\ce{AlX3}$ environment). This makes hydride a Lewis base and aluminium a Lewis acid. The association of hydride to aluminium is thus a Lewis acid-base reaction.


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# American Institute of Mathematical Sciences

May  2009, 8(3): 1067-1072. doi: 10.3934/cpaa.2009.8.1067

## On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$

 1 Department of Mathematics, Shanghai Finance University, Shanghai 201209

Received  April 2008 Revised  October 2008 Published  February 2009

In this paper we prove the global regularity of classical solutions to the incompressible Navier-Stokes equations in $\mathbf R^3$ for a family of large initial data with finite energy.
Citation: Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067
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