text:
Propulsion and maneuvering underwater by flapping foil motion, optimized through years of evolution, is ubiquitous in nature, yet marine propulsors inspired by examples of highly maneuverable marine life or aquatic birds are not widely implemented in engineering. Performance data from flapping foils,moving in a rolling and pitching motion, are presented at high Reynolds numbers, Re=Uc/ν, or O(104), where U is the relative inflow velocity, c is the chord length of the foil, and νis the kinematic viscosity of the fluid, from water tunnel experiments using a foil actuator module designed after an aquatic penguin or turtle fin. The average thrust coefficients and efficiency measurements are recorded over a range of kinematic flapping amplitudes and frequencies. Results reveal a maximum thrust coefficient of 2.09, and for low values of angle of attack the thrust generally increases with Strouhal number, without much penalty to efficiency. Strouhal number is defined as St=2h0f/U, where f is the frequency of flapping, and 2h0 is the peak-to-peak amplitude of flapping. The thrust and efficiency contour plots also present a useful performance trend where, at low angles of attack, high thrust and efficiency can be gained at sufficiently high Strouhal numbers. Understanding the motion of aquatic penguins and turtle wings and emulating these motions mechanically can yield insight into the hydrodynamics of how these animals swim and also improve performance of biologically inspired propulsive devices.

The design of biologically inspired propulsion mechanisms for underwater vehicles continues to generate significant interest in the hydrodynamics of fish swimming. Fish and animals, such as aquatic penguins and turtles, are highly adapted to swimming in the ocean and serve as excellentmodels for developing novel propulsive devices for underwater vehicles. With the goal of designing compact, agile autonomous and unmanned underwater vehicles (AUVs and UUVs), engineers have turned to fish and their aquatic counterparts for inspiration (e.g. Bandyopadhyay,2005; Anderson and Chabra,2002; Triantafyllou et al.,2000; Anderson et al.,1998; Bandyopadhyay et al.,1997,). Biologists have studied the kinematics and morphology of swimming fish in great detail, revealing the superior agility of these creatures (e.g. Fish and Lauder,2006; Lauder,2000; Drucker and Lauder,1999; Videler,1993; Fish and Hui,1991). The undulatory nature of the fish body motion imparts flow control on the surrounding fluid generating a unique propulsive signature in the form of a reverse Kármán vortex street(Triantafyllou and Triantafyllou,1995). This simplified view of the wake structure does not account for three-dimensional effects that result from the geometry of the fish body and tail fin.

Flapping foils have been used as models for swimming fish fins to better understand fish swimming hydrodynamics. Heaving and pitching foils with moderate to high aspect ratios (Anderson,1996; Triantafyllou et al.,1993), and more three-dimensional motions of foils, modeling pectoral fins with lower aspect ratios, have also been investigated as alternative methods of propulsion for underwater vehicle technology(Bandyopadhyay, 2005; Lang et al., 2006; Kato, 2000). Extensive studies of the two-dimensional wake patterns behind flapping foils and live and robotic swimming fish have been performed both experimentally and numerically. Thorough reviews of this research on flapping foils can be found in the literature (Triantafyllou et al.,2004; Triantafyllou el al.,2005).

Typical wake patterns generated by the two-dimensional flapping foil motion consist of alternately rotating vortices organized to form a jet-like wake, or reverse Kármán street, which results in thrust generation over a range of Strouhal numbers and kinematic parameters. Strouhal number is defined as St=2h0f/U, where 2h0 is the peak-to-peak heave amplitude of the trailing edge, f is the flapping frequency (in Hz), and U is the incoming flow velocity relative to the foil. Optimal propulsive performance has been shown to be highly dependent on Strouhal number. Typically, flapping foils used for propulsion have high efficiencies in a range of Strouhal numbers 0.2–0.4 (Triantafyllou and Triantafyllou, 1995); higher thrust values can be achieved at higher values of Strouhal number but with less efficiency.

The unsteady flapping motion of a foil generates vorticity at the trailing edge and tips of the foil and also at the leading edge under certain conditions. These patterns vary with the amplitude and frequency of the motion as well as the shape of the kinematics employed (e.g. Hover et al., 2004; Anderson el al., 1998; Koochesfahani, 1989; Katz and Weihs, 1979). Experiments with heaving and pitching symmetric foils, with a maximum thickness that is 15% of the chord length (i.e. a NACA 0015 foil), at moderate Reynolds numbers [5000–12 000(Freymouth, 1990)] link the formation of a leading edge vortex, and its successive combination with the vorticity generated at the trailing edge, with high lift coefficients. Reynolds number (Re) is defined as the ratio of the inertial fluid forces to the viscous fluid forces: Re=(ρU2c–1)/(μUc–2)=ρUc/μ,where ρ is the density of the fluid, and μ the dynamic viscosity of the fluid. Re is often written in terms of kinematic viscosity:ν=μ/ρ; for water ν=10–6 m2s–1. This interaction is also important to achieve high efficiency for flapping foil swimming modes(Anderson et al., 1998).

A closer look at performance of finite aspect ratio, three-dimensional flapping foils is warranted at higher Reynolds numbers, in order to design vehicles to swim with optimal kinematics and maximizing thrust while optimizing efficiency. In an effort to understand this trade-off, tests were performed using a three-dimensional, linearly tapered hydrofoil forced to move in roll and pitching motions, and the measured force and hydrodynamic efficiency data are presented, discussed and compared with data reported from previous tests (Flores, 2003; McLetchie, 2004; Polidoro, 2003; Read, 2000).

### Three-dimensional flapping foil apparatus and kinematics

The three-dimensional (3D) flapping foil motion emulates that of an aquatic penguin wing or turtle fin. The 3D flapping foil actuator was constructed as a dual-canister design: two watertight canisters, one each for pitch and roll,house the motor and chain drive for each motion separately. The two canisters are coupled together and to the foil (see Fig. 1A). This apparatus is mounted in the Massachusetts Institute of Technology recirculating water tunnel, capable of flow speeds from 0.5 to 5.0 m s–1 with turbulence levels under 3%. The tunnel's square test section is 0.5 m×0.5 m×1.25 m long. Constrained to move in roll ϕ(t)and pitch θ(t) only, a diagram of the motion coordinate system is given in Fig. 1B. This type of flapping apparatus presents underwater vehicle designers and engineers a simpler design problem, e.g. fewer sealing issues, than the heaving/pitching foil. The thesis by Lim (Lim,2005) presents further details of the housing design and operation.

The kinematic motion of the foil is measured through potentiometers mounted to each rotating shaft, which map the pitch and roll motion, and also through the encoders mounted on each motor. The encoder data are used for redundancy and kinematics validation. A torque sensor is coupled to the roll motor shaft to calculate the power input from the motor end during actuation; the power input can be used to determine the mechanical efficiency of the flapping foil. The forces and torques acting directly on the foil are measured with a six-axis, waterproof strain gauge sensor, which measures the three force components (Fx, Fy, Fz) and the three moment components(Mx, My, Mz). Hydrodynamic efficiency is calculated using the torque measurements from the six-axis sensor to calculate power input to the fluid by the foil, thus bypassing the mechanical losses in the drive mechanism. Cross-coupling between the six channels is eliminated using the factory supplied matrix, which has been validated extensively (Lim,2005).

The average thrust and lift forces are found by applying a force rotation matrix to the force traces and averaging the data over 10–15 cycles. Average force data presented in subsequent sections are given in terms of the non-dimensional force coefficient, based on the planform area of the foil,instead of the swept area as would be used for a propeller. The hydrodynamic efficiency is determined by comparing the power output, calculated from the product of average thrust and velocity, to the sum of the power input, applied to the pitch and roll axes.

The prescribed roll and pitch motions are simple sinusoidal harmonics with the same circular frequency ω (rad s–1). The roll motion of the foil is given by:
$\ {\phi}(t)={\phi}_{0}\mathrm{sin}({\omega}t),$
(1)
where ϕ0 is the roll amplitude in radians; the pitch motion of the foil is defined as:
$\ {\theta}(t)={\theta}_{0}\mathrm{sin}({\omega}t+c)+{\theta}_{\mathrm{bias}},$
(2)
where θ0 is the pitch amplitude and Ψ is the phase angle between pitch and roll, both in radians. The static pitch biasθ bias is an optional parameter that introduces a non-zero average pitch angle to the foil; for these tests it is set to zero. The phase angle Ψ for the tests is set to π/2, as recommended(Read et al., 2003).
Fig. 1.

(A) Double canister flapping foil actuator design with the foil and the inline AMTI six-axis force sensor assembly between the pitch canister and the foil. The distance from center of roll axis to root of foil is indicated by r0; S is the foil span. (B) Foil motion coordinate system. The freestream flow is in the Ycdirection and Zc is oriented in the opposite direction from gravity. Roll, ϕ(t), is constrained to the XcZc plane and pitch,θ(t), is an angular motion about the foil shaft.

Fig. 1.

(A) Double canister flapping foil actuator design with the foil and the inline AMTI six-axis force sensor assembly between the pitch canister and the foil. The distance from center of roll axis to root of foil is indicated by r0; S is the foil span. (B) Foil motion coordinate system. The freestream flow is in the Ycdirection and Zc is oriented in the opposite direction from gravity. Roll, ϕ(t), is constrained to the XcZc plane and pitch,θ(t), is an angular motion about the foil shaft.

The angle of attack profile varies along the span of the three-dimensional flapping foil as it rolls and pitches. To simplify the kinematics, the motion can be decomposed to 2D heaving and pitching, versus rolling and pitching, at any span location on the foil. Although McLetchie's results(McLetchie, 2004) show varying centers of force, the 70% span location is selected to be consistent with conventional propeller notations and for easy comparison with past flapping foils experiments. This location is defined as:
$\ r_{0.7}=r_{0}+0.7S,$
(3)
where r0 is the distance from the center of roll axis to the root of the foil, and S is the span of the foil, as shown in Fig. 1A. The heave position is defined as:
$\ h(t)=h_{0}\mathrm{sin}({\omega}t),$
(4)
where h0 is the amplitude of the heave motion at r0.7; h0 is defined as:
$\ h_{0}=r_{0.7}{\phi}_{0}.$
(5)
The angle of attack at one span location can be found from the instantaneous pitch position of the foil and the ratio of the heave to forward velocity. Fig. 2 shows the vector diagram of the velocity components.
The true angle of attack profile can be calculated mathematically. Since a foil with a positive pitch produces a smaller angle of attack, one needs to subtract the angle of attack due to pitch from that due to roll to find the overall angle of attack profile:
$\ {\alpha}(t)=\mathrm{arctan}\left(\frac{{\dot{h}}(t)}{U}\right)-{\theta}(t),$
(6)
where U is the forward speed of the actuator and is the heave velocity. From Eqn 4, we can express the heave velocity as:
$\ {\dot{h}}(t)={\omega}r_{0.7}{\phi}_{0}\mathrm{cos}({\omega}t).$
(7)
Substituting Eqn 7 into Eqn 6, we get the expression for angle of attack in 3D kinematics:
$\ {\alpha}(t)=\mathrm{arctan}\left(\frac{{\omega}r_{\mathrm{0.7}}{\phi}_{\mathrm{0}}\mathrm{cos}({\omega}t)}{U}\right)-{\theta}_{0}\mathrm{cos}({\omega}t)-{\theta}_{\mathrm{bias}}.$
(8)
The maximum angle of attack, αmax, is calculated using Eqn 8 at r0.7; αmax is given in degrees throughout this paper.
Again, the Strouhal number can be used to describe the foil motion kinematics based on the roll amplitude and r0.7:
$\ St=\frac{2h_{0}f}{U}=\frac{2r_{0.7}{\phi}_{0}f}{U}.$
(9)
An estimate of the total width of the wake produced by the flapping foil is 2h0. This is essentially the peak-to-peak amplitude of the foil taken at the 0.7 chord for a 90° phase offset between heave and pitch. The roll amplitude is non-dimensionalized by converting this to 2D heave amplitude at 0.7 span and dividing by the chord length, h0.7/c. Finally, Reynolds number, Re=Uc0.7 /ν, is calculated based on the chord at 70% of the span, c0.7, and kinematic viscosity of the fluid,ν=10–6 m2 s–1.

The foil used in this experiment is fabricated with a NACA 0012 cross section. The NACA 0012 is a standard, symmetric foil profile with a maximum thickness that is 12% of the chord length(Abbott and von Doenhoff,1959). The test foil has a span of 24.6 cm from root to tip, and an average chord length, , of 5.5 cm from leading edge to trailing edge. The foil has a linearly tapering trailing edge profile. Tests were performed over a range of kinematic parameters:heave/chord ratio, h0.7/c={1.0, 1.5, 2.0};Strouhal number, St={0.2:0.1:0.6}; and maximum angle of attack,α max={15:5:45}°. The experiments were conducted at Reynolds numbers 27 000 to 55 000. The lower speed was used to achieve higher Strouhal numbers. The speed was monitored at all times by a Laser Doppler Velocimetry system and maintained to within ±0.01 m s–1.

Fig. 2.

Vector diagram for velocity components relative to heaving and pitching motions and incoming fluid velocity.

Fig. 2.

Vector diagram for velocity components relative to heaving and pitching motions and incoming fluid velocity.

### Three-dimensional flapping foil performance

The 3D flapping foil motion (roll/pitch) presents underwater vehicle designers with a more straightforward design problem compared to the heaving/pitching mechanism design and yields further insight into the performance of swimming animals such as aquatic penguins and turtles. To evaluate the performance of flapping foils sufficiently, accurate force and efficiency measurements are paramount. Force data are presented here for the 3D flapping foil mechanism for induced 2D heave-to-chord ratios(Eqn 4) h0.7/c=1.0, 1.5 and 2.0, over a range ofα max from 15° to 45°. Typical force traces obtained by the six-axis force sensor are presented in Fig. 3.

Using data acquired with the six-axis sensor, the average thrust of the foil is calculated by:
$\ \overline{F}_{x_{0}}=\frac{1}{T}{{\int}_{0}^{T}}\overline{F}_{x_{0}}(t)\mathrm{d}t.$
(10)
$$\overline{F}_{x_{0}}$$
is the x-force component translated into the reference frame of the tunnel, averaged over one flapping cycle. By axes convention, x is positive upstream. A non-dimensional thrust coefficient can be defined as:
$\ C_{\mathrm{T}}=\frac{-2\overline{F}_{x_{0}}}{{\rho}U^{2}\overline{c}S},$
(11)
where ρ is the fluid density, U is the relative flow velocity, S is the foil span and is the average chord length.

The general trend of the thrust results(Fig. 4) compares well with those for the 2D case (Read,2000) and 3D case (McLetchie,2004). The thrust coefficients decrease with decreasing Strouhal numbers and αmax values, and increases with increasing values of both Strouhal number and αmax. At heave- to- chord ratio of 1.0 (h0.7/c=1.0) a peak thrust coefficient of 1.6 is recorded at St=0.5 and αmax=30°. Extrapolating up to St=0.6, it appears that a higher peak thrust coefficient could be achieved. Such an increase is substantiated by McLetchie(McLetchie, 2004).

For the case of h0.7/c=1.5, peak thrust coefficients were measured for St=0.6 and angle of attacks of 30°and 35°. At h0.7/c=2.0, the maximum thrust coefficients now occur at the same Strouhal numbers (St=0.6) but have lower magnitudes at αmax=30° and 35°(Lim, 2005). Fig. 4 shows that for a given Strouhal number the thrust coefficient increases with the maximum angle of attack, but beyond a critical αmax value, the thrust coefficient is expected to decrease. This is similar to the data obtained for heaving/pitching foils at Re=750 and 1000 (Read, 2006). Increases in roll amplitude tend to have similar effects. The maximum thrust coefficient for these runs was recorded at h0.7/c=1.5. At the higher roll amplitude, h0.7/c=2.0, the peak thrust coefficient is lower, thus it is possible that an optimal roll amplitude can be found, between the heave-to-chord ratio of 1.5–2.0, for which the thrust coefficient can be maximized.

Fig. 3.

Processed data recorded for St=0.2, h0.7/c=1.5, αmax=20°. The horizontal axis is the time (in s). The lift and thrust are in the tunnel reference frame. The forces are calibrated using the AMTI sensitivity matrix.

Fig. 3.

Processed data recorded for St=0.2, h0.7/c=1.5, αmax=20°. The horizontal axis is the time (in s). The lift and thrust are in the tunnel reference frame. The forces are calibrated using the AMTI sensitivity matrix.

The left and bottom `borders' of the parametric space represent the boundaries for which the angle of attack profiles tend to corrupt, i.e. some regions of the foil would encounter negative angles of attack, resulting in drag instead of thrust production. Here, the results at St=0.2 show very low thrust production for all αmax values. This low thrust boundary represent the transition in the wake from drag- to thrust-producing vortices (Flores,2003). At high αmax, the transition to thrust does not occur until a Strouhal number of about 0.4. At experimental points for which low thrust values are measured (St⩽0.3;15°⩽αmax⩽25°), there are large errors associated with the results (>10% in certain cases). A more thorough analysis of the error from these tests is given in Lim(Lim, 2005). In addition, the thrust coefficients presented above were evaluated based on planform area. A more appropriate normalization, in keeping with that used in propeller performance analysis, might have been to use the projected swept area. This formulation would have produced numerically smaller values of CT (Techet el al., 2005).

Fig. 4.

Contour plots of thrust coefficient for h0/c=1.0 (A) and h0/c=1.5(B); αmax is given in degrees. Red circles represent location of each run performed; no data exist for the region shadowed in grey for h0/c=1.0.

Fig. 4.

Contour plots of thrust coefficient for h0/c=1.0 (A) and h0/c=1.5(B); αmax is given in degrees. Red circles represent location of each run performed; no data exist for the region shadowed in grey for h0/c=1.0.

The hydrodynamic efficiency of the foil is defined as the ratio of power output Pout over power input Pin to the fluid:
$\ {\eta}=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}.$
(12)
Power output is the product of the time-averaged thrust and flow velocity:
$\ P_{\mathrm{out}}=TU=-\overline{F}_{x_{0}}U.$
(13)
The power input Pin is the power input to the fluid,calculated from torque measurements with the six-axis sensor mounted to the foil shaft. While the torque sensor attached to the roll motor could measure the direct power input from the electric motor, a significant amount of the power is actually used to move the inertial mass of the pitch canister, and the losses due to backlash in the motors can be quite large. The motor power input is found to be in the order of ten greater than the power transmitted to the flow. Thus a more useful approach is to use the power input measured by the AMTI sensor to calculate hydrodynamic efficiency of the foil, such that the hydrodynamic efficiency of the foil can be compared directly with other foil designs and aquatic animals in future research.

In a reverse fashion to the thrust coefficient results, efficiency peaks at the lower end of Strouhal and αmax values and then decreases with increasing St and αmax values(Fig. 5). This is as expected since, in low thrust regions, minimal energy is lost as a result of kinetic energy being imparted to the flow. The maximum efficiency recorded is about 0.7, centered at St=0.3 and αmax=20° for h0.7/c=2.0. Its location corresponds with that reported by McLetchie (McLetchie,2004), although the magnitudes of efficiency differ. Read also shares the same peak location with measured efficiencies not exceeding 0.7,for 2D flapping foils (Read,2000).

Higher roll amplitudes result in greater energy expended in moving the foil through the large oscillations. It is only at higher frequencies where greater thrust is generated such that the efficiency appears to be improving. Figs 4 and 5 show that for lowα max values, thrust generally increases with Strouhal number without significant penalty to efficiency. This is good news with respect to designing mechanisms for underwater vehicles, as an optimal point can be identified for relatively high thrust production without sacrificing efficiency – it is also good news for aquatic swimmers, as they can produce significant propulsive forces with minimal effort. For example, at h0.7/c=1.5, we see relatively high efficiencies occurring at St=0.5, and αmax of 15° and 20°. This offers a good design point where efficiencies of more than 0.6 can be achieved with thrust coefficients ranging from 0.7 to 1.2.

The average range of thrust coefficient CT is found to be in the order of ±0.026, or ±0.044 N in terms of absolute thrust; as such, the percentage error, which is taken as the fraction of standard deviation over the mean, is naturally higher at the lower thrust runs(Lim, 2005). In the regions where the thrust is low (St⩽0.3;15°⩽αmax⩽25°), the percentage errors associated with the measurements still remain significant and thus the efficiency measurements also require careful consideration and further validation. Likewise a larger percentage error for efficiency η also arises for these cases, since the error for η is a combination of errors from power input and thrust measurements.

In summary, the 3D flapping foil, similar to a pectoral fin, holds the most promise for implementation in an underwater vehicle due to the reduced complexity over the heaving/pitching mechanism. To further extend our understanding of flapping foil performance at higher Reynolds numbers the 3D rolling/pitching foil apparatus was studied in a water tunnel at Reynolds numbers on the order of Re∼104. These tests, using force sensors to measure thrust and efficiency, with the 3D flapping foils are highly valuable from a design point of view.

Clearly indicated in the data presented here is that for a fixed Strouhal number, there is a critical maximum angle of attack value beyond which the thrust coefficient will start to decrease and performance is reduced. A peak planform area thrust coefficient of 2.09 was measured at h0.7/c=1.5, St=0.6 andα max=30°. Increasing the heave-to-chord ratio from 1.5 to 2.0 does not appear to improve CT values, but further iterations are still warranted. Combined with data from McLetchie(McLetchie, 2004), the data presented herein for the 3D foil suggests a useful performance trend where,for low αmax, high thrust and high efficiency can be gained at sufficiently high Strouhal numbers (St=0.6 and possibly higher). In particular, for higher roll amplitudes, these large oscillations produce high thrust with relatively little power loss. Higher thrust and efficiencies can be achieved through pre-shaping algorithms applied to the heaving/pitching foil motions for 2D flapping (Hover et al., 2004). Further investigation is warranted to determine whether pre-shaping would improve the performance of the 3D foil, but data suggests that this is a reasonable assumption.

Fig. 5.

Contour plots of thrust coefficient and efficiency for h0/c=1.0 (A) and h0/c=1.5 (B); αmax is given in degrees. Blue diamonds represent location of each run targeted; no data exist for the region shadowed in grey for h0/c=1.0.

Fig. 5.

Contour plots of thrust coefficient and efficiency for h0/c=1.0 (A) and h0/c=1.5 (B); αmax is given in degrees. Blue diamonds represent location of each run targeted; no data exist for the region shadowed in grey for h0/c=1.0.

Further comprehensive and systematic investigations of foil geometry on flapping foil performance are warranted; for example, it was suggested that an optimal aspect ratio is nearer to 4.0 for flapping wings(Polidoro, 2003). The effect of chordwise or spanwise flexibility on the performance of flapping foils is also still not well understood. Live fish employ active flexure of their fins,especially their pectoral fins, suggesting that they can optimize their fin shape for maximized performance (Fish and Lauder, 2006). Further experiments in tandem with numerical simulations are necessary to further augment our understanding of flapping foil performance as seen in nature.

List of symbols and abbreviations

• c

foil chord

•
• mean chord

•
• CT

thrust coefficient

•
• f

flapping frequency (Hz)

•
• Fx, Fy,Fz

force components

•
• x0

x-force component translated into reference frame of test section

•
• heave velocity(=dh/dt)

•
• h0

heave amplitude

•
• h0.7

heave amplitude at 0.7 span location

•
• h(t)

heave position

•
• Mx, My,Mz

moment components

•
• r0

•
• r0.7

radius from base of foil shaft to 0.7 span

•
• Pin

power input

•
• Pout

power output

•
• Re

Reynolds number

•
• S

foil span

•
• St

Strouhal number

•
• t

time

•
• T

thrust

•
• U

free stream velocity

•
• α

angle of attack

•
• αmax

maximum angle of attack

•
• Ψ

phase angle between pitch and roll

•
• ϕ(t)

roll angle

•
• ϕ0

roll amplitude

•
• η

efficiency

•
• μ

dynamic viscosity of the fluid (kg m–1s–1)

•
• ν

kinematic viscosity of the fluid [ν=μ/ρ (kg m–1 s–1)]

•
• ρ

fluid density (kg m–3)

•
• θ(t)

pitch angle

•
• θ0

pitch amplitude

•
• θbias

pitch bias angle

•
• ω

The author would like to thank K. Lim for his assistance with experiments and Dr Franz Hover for his insightful discussions related to data processing and angle of attack effects. The development of the flapping foil mechanism was made possible through the support of Admiral Paul Sullivan, NAVSEA, and the MIT Sea Grant program (Grant NA16RG2255).

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Triantafyllou, M. S., Hover, F. S., Techet, A. H. and Yue, D. K. P. (
2005
). Review of hydrodynamic scaling laws in aquatic locomotion and fishlike swimming.
Appl. Mech. Rev.
58
,
226
-237.
Videler, J. (
1993
).
Fish Swimming
. London: Chapman & Hall.


text:
## Inkscape Extension ‘svg2tikz’ revisited.

In this earlier posting, I had written about the low-performing 3rd-party Inkscape Extension known as ‘svg2tikz’. Nevertheless, this extension may prove useful to some users, who wish to import an arbitrary document-type into Inkscape – preferably vector-based – and who wish to convert that into LaTeX in some way. And it seems that, even though this project was abandoned some time ago, work has slowly begun to resume on its source-code. And so, I should also fine-tune some of the earlier commentary I had made about this extension.

First off, there is an important detail about how to compile and install this extension, which its devs fail to point out anywhere. It needs to be built and installed, using Python 3, while many Linux computers still default to Python 2.7. Therefore, the commands to build and install it are:

$python3 setup.py build$ su
(...)
# python3 setup.py install

If one neglects this detail, then Unicode support is left out, and usually, SVG Files etc., will contain some Unicode characters. Further, as the Github comment states, while the importing of raster-based images is now supported, their import as Base-64 encoded, inline data is not. Therefore, within Inkscape, for example if a PDF File is being imported, the option needs to be unchecked, to ‘Embed’ graphics. And when Saving a Copy to TiKz Format, the option should also be unchecked, to ‘Indent Groups’.

But this last detail leads me to an important, additional observation. I have always known that the export of Text with the Figure has been dodgy. But lately, either because I’ve become more observant, or, because the behaviour of the latest version of the extension has improved, I’ve noticed what, exactly, goes wrong with Exporting Text along with the Figure.

(Updated 2/11/2020, 1h05 … )

## Low-Performing InkScape Extension, for TikZ Export.

One of the facts which I’ve been writing about is, that the application ‘InkScape’ supports exporting its graphics to the ‘PSTricks’ flavour of .TEX-File. It does this out-of-the-box and without apparent error. But I have also written that A flavour of .TEX-File exists, which is the PGF/TikZ form. And Linux has few graphical, interactive editors, to be able to create this file-format, because again, the file that is being sought is a text file. It would be nice of InkScape could additionally output to this format, since InkScape is a powerful, GUI-based graphics application.

Well as it happens, It’s possible to add some meagre extensions to InkScape, in user-space, or with elevated privileges. One such extension is called ‘svg2tikz‘, and so at first glance it would seem that users can benefit fully. But as is often the case, there are caveats with using this one extension.

(Updated 3/10/2018, 7h30 … )

format: plain


text:
Volume 414 - 41st International Conference on High Energy physics (ICHEP2022) - Detectors for Future Facilities, R&D, novel techniques
Detector Challenges of the strong-field QED experiment LUXE at the European XFEL
O. Borysov
Full text: pdf
Pre-published on: November 21, 2022
Published on:
Abstract
The LUXE experiment aims at studying high-field QED in electron-laser and photon-laser interactions, with the 16.5 GeV electron beam of the European XFEL and a laser beam with power of up to 350 TW. The experiment will measure the spectra of electrons, positrons and photons in expected rates from $10^{-3}$ to $10^{9}$ particles per 1 Hz bunch crossing, depending on the laser power and focus. These measurements have to be performed in the presence of low-energy high radiation background. To meet these challenges, for high-rate electron and photon fluxes, the experiment will use Cherenkov radiation detectors, scintillator screens, sapphire sensors as well as lead-glass monitors for backscattering off the beam dump. A four-layer silicon-pixel tracker and a compact electromagnetic tungsten calorimeter with silicon sensors will be used to measure the positron spectra. The layout of the experiment and the expected performance under the harsh radiation conditions are discussed. Beam tests results for the Cherenkov detector and the electromagnetic calorimeter prototypes are presented.
DOI: https://doi.org/10.22323/1.414.0321
How to cite

Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.

Open Access


text:
2022-07-03

Having trouble with proving this binomial identity. I have taken it as far as I can go.
I have broken down this equation into factorials, but I'm unsure of where to go from here. This may not even be the right approach to solve this binomial transform. Any help would be appreciated.
Binomial transform identity:
$\begin{array}{rl}& \sum _{k=0}^{n}\left(-1{\right)}^{n-k}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{n+jk}{jk}\right)={j}^{n}\\ & \sum _{k=0}^{n}\left(-1{\right)}^{n-k}\frac{\left(jk+1\right)\left(jk+2\right)\dots \left(jk+n\right)}{\left(n-k\right)!\phantom{\rule{thinmathspace}{0ex}}k!}={j}^{n}\end{array}$

Kiley Hunter

Expert

Step 1
In trying to prove
$\sum _{k=0}^{n}\left(-1{\right)}^{n-k}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{n+jk}{jk}\right)={j}^{n}$
$\sum _{k=0}^{n}\left(-1{\right)}^{n-k}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{n+jk}{n}\right)\phantom{\rule{0ex}{0ex}}=\left[{z}^{n}\right]\left(1+z{\right)}^{n}\sum _{k=0}^{n}\left(-1{\right)}^{n-k}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(1+z{\right)}^{jk}\phantom{\rule{0ex}{0ex}}=\left[{z}^{n}\right]\left(1+z{\right)}^{n}\left(\left(1+z{\right)}^{j}-1{\right)}^{n}.$.
Step 2
Now $\left(\left(1+z{\right)}^{j}-1{\right)}^{n}=\left(jz+\cdots {\right)}^{n}$ so the only term that contributes to the coefficient extractor in z is ${j}^{n}{z}^{n}.$. (The n factors contribute at least z so for the power to be less than or equal to n we have to choose z from each factor. As soon as we choose just one factor to a power at least two the term produces a power that is larger than n and does not contribute to the coefficient extractor.) We get $\left[{z}^{n}\right]\left(1+z{\right)}^{n}{j}^{n}{z}^{n}={j}^{n}$ as claimed.

Alissa Hancock

Expert

Step 1
Recall that, if p(x) is a polynomial of degree n,
$\sum _{k=0}^{n}\left(-1{\right)}^{n-k}\left(\genfrac{}{}{0}{}{n}{k}\right)p\left(x+k\right)$ is n! times the leading coefficient of p(x) (this may be proven by induction). Now, select
$p\left(x\right)=\frac{\left(jx+n\right)\left(jx+n-1\right)\cdots \left(jx+1\right)}{n!},$,
which is a polynomial of degree n with leading coefficient ${j}^{n}/n!$. For integer k,
$p\left(k\right)=\frac{\left(jk+n\right)\left(jk+n-1\right)\cdots \left(jk+1\right)}{n!}=\left(\genfrac{}{}{0}{}{jk+n}{n}\right)=\left(\genfrac{}{}{0}{}{n+jk}{jk}\right).$.
Step 2
So, $\sum _{k=0}^{n}\left(-1{\right)}^{n-k}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{n+jk}{jk}\right)$ is n! times the leading coefficient of p(x), i.e. ${j}^{n}$.

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Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:58:15Z http://mathoverflow.net/feeds/question/28361 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Menny 2010-06-16T08:26:53Z 2010-06-16T13:20:10Z
My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k_v)$ can be compact (with respect to the $p$-adic topology)?

I more or less understand that if $G=SL_1(D)$ where $D$ is a division ring of dimension $n^2$ and of order $n$ in the Brauer group over $Q_p$ then $G(Q_p)$ is compact. I also understand that $Spin(q)$ when $q$ has more than $5$ variable cannot be compact over local non-archimedean fields.

Are there more examples? I think that one can classify all the examples but I don't manage to do it or find a reference for it.

Can someone outline a route to take in order to understand it thoroughly (for someone with basic understanding of algebraic groups and Galois cohomology)?

Thanks a lot!
 http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar/28364#28364 Answer by Charles Matthews for Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Charles Matthews 2010-06-16T09:00:47Z 2010-06-16T09:00:47Z
See http://eom.springer.de/a/a012530.htm and the concept of anisotropic group. As it says there, the classification involved is pretty much the classification of semisimple groups.
 http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar/28365#28365 Answer by Kevin Buzzard for Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Kevin Buzzard 2010-06-16T09:03:46Z 2010-06-16T09:03:46Z
I'm no expert, but I think the theorem is that (for $G$ reductive over a local field $F$) $G(F)$ is compact iff $G$ is $F$-anisotropic, that is, every $F$-torus in $G$ (or equivalently every maximal $F$-torus) has the property that its only $F$-character is the trivial character. This will be somewhere in Platonov-Rapinchuk but I don't have it to hand so can't give a more precise reference :-( . So, for example, in your $SL_1(D)$ example (presumably this means the norm 1 elements of $D$?) the maximal tori in $D^*$ are coming from fields $E/F$ embedding into $D$, so the maximal tori in $G$ are the norm 1 elements of $E^*$ and these have no rational characters (the norm is the only rational character of $E^*$ and you've just removed it).
 http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar/28373#28373 Answer by Victor Protsak for Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Victor Protsak 2010-06-16T10:50:33Z 2010-06-16T10:50:33Z
As Charles said, for a semisimple group over a local field compact $\iff$ anisotropic.

I confirm that Chapter 6 of Platonov-Rapinchuk contains the proof of (a) the vanishing of the Galois cohomology $H^1(K, G)$ for a connected simply connected semisimple group $G$ over a local field $K$ and (b) Classification theorem: if $G$ is anisotropic simple connected simply connected then $G=SL_1(D).$ However, the proofs are neither self-contained nor transparent and rely on an argument close to circular. The original proofs due to Kneser (Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern. II. Math. Z. 89 1965 250--272, MR0188219) also use case-by-case arguments. A uniform proof follows from Bruhat-Tits theory, but if I understand it right, it comes closer to the end of their series of papers.
 http://mathoverflow.net/questions/28361/compact-simple-simply-connected-algebraic-groups-over-q-p-or-other-local-non-ar/28377#28377 Answer by Boyarsky for Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields Boyarsky 2010-06-16T11:54:21Z 2010-06-16T13:20:10Z
Here are some remarks on the answers of Charles Matthews, Kevin Buzzard, and Victor Protsak. For justification of Kevin Buzzard's claim, see G. Prasad, "An elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits" from Bull. Soc. Math. France 110 (1982), pp. 197--202, for an incredibly elegant and short proof that over any henselian valued field $F$, a connected reductive $F$-group $G$ is $F$-anisotropic if and only if $G(F)$ is "bounded" (a property defined in terms of a choice of closed immersion of $G$ into an affine space over $F$, the choice of which doesn't matter; this is meaningful for any affine $F$-scheme of finite type and equivalent to compactness when $F$ is locally compact). Platanov-Rapinchuk has a universal assumption that all fields of characteristic 0 (except when they're finite), so unfortunately that reference is insufficient for uniform arguments over all non-archimedean local fields. I suppose (near?-)circularity (suggested by Victor Protsak) is a more serious issue. :)

There remains the matter of determining, for locally compact non-archimedean $F$ and connected reductive $F$-groups, precisely when the $F$-anisotropic case can actually occur. As Victor Protsak mentioned, via Bruhat-Tits theory one sees that for connected semisimple $F$-groups which are absolutely simple and simply connected over a non-archimedean local field $F$ (i.e., $G$ a simply connected $F$-form of a Chevalley group), such forms never exist away from type A, and in type A the $F$-anisotropic examples are precisely the $F$-groups of norm-1 units of central simple algebras over $F$. (Note the contrast with the case $F = \mathbb{R}$, for which there's always a "compact form" of any Chevalley type.)

Let me now briefly explain why this handles the general connected reductive case, by a standard kind of argument with central isogenies and separable Weil restriction. (This is explained also in the article [2] of Tits referenced in Charles Matthews' answer.) If $f:G' \rightarrow G$ is a (possibly inseparable) central $F$-isogeny between connected reductive $F$-groups then the preimage of an $F$-torus of $G$ is an $F$-torus of $G'$ (since maximal tori in $G'$ are their own functorial centralizers, so $\ker f$ is of multiplicative type, nothing funny happens when $f$ is not separable). Since $F$-anisotropcity of an $F$-torus is invariant under $F$-isogenies (as we see using the $F$-rational character group, or more direct arguments), it follows that $G$ is $F$-anisotropic if and only if $G'$ is. (This argument has the advantage of working over any field $F$, in contrast with a direct attack on the topology of rational points by using finiteness theorems for Galois cohomology of connected reductive groups.) Thus, by considering an arbitrary connected reductive $F$-group $G$ and letting $G'$ denote the product of its maximal central $F$-torus and the simply connected central cover of the derived group $D(G)$, we see that the problem comes down to the simply connected case.

But in the simply connected semisimple case, the general structure of connected semisimple groups over fields (in terms of central isogenous quotient of direct product of commuting simple "factors") implies that $G$ is uniquely a direct product of commuting $F$-simple connected semisimple $F$-groups, each of which is simply connected, so we may assume $G$ is $F$-simple. Then by an elementary result of Borel and Tits (6.21 in "Groupes reductif", IHES), $G = {\rm{Res}}_ {F'/F}(G')$ for a finite separable extension $F'/F$ and a connected semisimple $F'$-group $G'$ that is absolutely simple and simply connected. By the good behavior of Weil restriction with respect to the formation of the topological group of rational points, it follows that the equality ${\rm{Res}}_ {F'/F}(G')(F) = G'(F')$ of abstract groups is a homeomorphism, so we can replace $(G,F)$ with $(G',F')$ to reduce to the case when $G$ is also absolutely simple, the case addressed by Victor Protsak above. (A more algebraic argument with Galois descent relating maximal $F'$-tori in $G'$ and maximal $F$-tori in its Weil restriction to $F$ shows the equivalence of anisotropicity for $G'$ and its Weil restriction through the finite separable $F'/F$, where $F$ can be taken to be any field at all.)

Conclusion: for non-archimedean local $F$, the $F$-anisotropic connected reductive $F$-groups are precisely the central quotients of products of an $F$-anisotropic torus and groups of norm-1 units of central division algebras over finite separable extensions of $F$.


text:
## El Mahjoub Echchaabaoui, Mohamed Laghdir, Strong subdifferential calculus for convex set-valued mappings and applications to set optimization

Full Text: PDF
DOI: 10.23952/asvao.4.2022.2.06
Volume 4, Issue 2, 1 August 2022, Pages 223-237

Abstract. The aim of this paper is to establish the strong vector subdifferential of the convex set-valued mapping $F + G\circ H$, where $F,$ $G$, and $H$ are convex set-valued mappings, and $G$ is nondecreasing. An application is given to deal with optimality conditions for a constrained convex set-valued optimization problem.


text:
# Problem with nomenclature

I am trying to get a list of abbreviations into my paper using the

\usepackage{nomencl}
\makenomenclature

package. However, I won't create a list in my document when I type

\printnomenclature

Can anyone please help? I am very new to Tex, so please help me out. The documentation says I have to run makeindex <filename>.nlo -s nomencl.ist -o <filename>.nls I have no idea what this means, nor do I know where to type this or what filename to enter.

My file looks like this, and I am typing in TexMaker:

\documentclass[12pt]{article}
\usepackage{nomencl}
\makenomenclature

\begin{document}

\printnomenclature

\nomenclature{AIMR}{Association for Investment Management and Research}

blablabla AIMR

\end{document}

• You have to run: makeindex.exe <path to your tex file>\yourtexfile.nlo -s nomencl.ist -o <path to your tex file>\yourtexfile.nls. Run pdflatex then above command and then pdflatex. – user11232 Dec 10 '13 at 11:40
• Does your TeXMaker offer the compilation method "arara"? If it does, just write % arara: pdflatex "Enter" % arara: nomencl "Enter" % arara: pdflatex on the very top of your main file. "Enter" means: press Enter, don't type it. Than you can compile with arara instead of PDFLaTeX and everything works fine. – LaRiFaRi Dec 10 '13 at 11:46
• My TexMaker does not offer arara. Also, concerning Harish's response, where do I have to run it? Do I have to create a text file? Do I open cmd? I literally have no idea what that means. Could you please provide step-by-step help? What programs do I use for running? – Thomas H. Dec 10 '13 at 13:06
• Also, there is no .nlo or .ist file in my folder. – Thomas H. Dec 10 '13 at 13:18
• So I managed to understandwhat "run" means. I entered makeindex.exe <path to your tex file>\yourtexfile.nlo -s nomencl.ist -o <path to your tex file>\yourtexfile.nls into cmd and it created a .nlo and a .ist file. However, I still can't get it to create a abbreviations list in my output file. – Thomas H. Dec 10 '13 at 13:41

This is going to be long...

# From command prompt:

Let us say that your tex file (file.tex) is in D:\my folder. Open file.tex in texmaker and choose pdflatex in the build list, compile. Now open command prompt (Start→ run → cmd). In the command prompt that opens type

makeindex.exe D:\my folder\file.nlo -s nomencl.ist -o D:\my folder\file.nls

Now go back to texmaker and again compile with pdflatex using build button.

# Using arara:

Download and install arara. Make sure that arara.exe is in system path. (while installing, the installer asks this option. Choose yes).

Now open texmaker. Using the menu go to UserUser CommandsEdit User Commands as in the picture.

Now the following window will open. Type Arara in Menu Item and arara % under Command as shown. Now on the left hand side Command 1: Arara should appear as below

Press OK. Now in the build list an entry with arara should appear as shown. Choose it.

Now add following lines to your file.tex:

%% this is file.tex. Following three lines need to be added.
% arara: pdflatex
% arara: nomencl
% arara: pdflatex

\documentclass{amsart}
\usepackage{nomencl}

\usepackage{etoolbox}
\makeatletter
% copy the formatting of section titles
\let\nomencl@section\section
% if the level is greater than 1000, then amsart
% doesn't include the section's title in the toc
\patchcmd{\nomencl@section}{{1}}{{1001}}{}{}
% patch \thenomenclature to call \nomencl@section
\patchcmd{\thenomenclature}
{\section*}
{\nomencl@section}
{}{}
\makeatother

\renewcommand{\nomname}{List of Notations}
\makenomenclature

\setcounter{tocdepth}{1}

\nomenclature[000]{$Hi$}{World}

\begin{document}
\tableofcontents

\printnomenclature[1in]

\section{world}
\end{document}

Now press Build button which reads 1: Arara as below:

Hope it is all done well. :-)

• It still won't work. I'm really in over my head here. I will just manually type all abbreviations, using the tabbing-environment. Thank you all for your effort. – Thomas H. Dec 10 '13 at 14:30
• @ThomasH. won't work doesn't help. Please give details. Typing by hands is bad. Also try to post your tex code. Does my code work? – user11232 Dec 10 '13 at 14:31
• @ThomasH. Don't give up ;-) – user11232 Dec 10 '13 at 22:16
• I would like to add this as a comment to Harish Kumar answer but is my first day here and I can't. I want to share something that worked for me (Windows 8 user) I did exactlye what Harish said about the command prompt but, being on the root folder of the project. So sumarizing the process that worked for me it would be(assuming that the tex file is in "C:\Myfolder\file.tex" - Open your 'file.tex' in TexMaker and compile it with pdflatex - Open de comand prompt in your root folder "C:\Myfolder" or open it anywhere and type "cd C:\Myfolder". - Once you are in that folder type "makeindex.exe file – Manuel Pena Oct 24 '15 at 22:48

The advantage of Texmaker is that it allows you to do without the command prompt. And it is not necessary to go through arara either.There are several ways to do this.

Change the command of the MakeIndex button

If you use the button below to launch the compilers one by one you just need to change the MakeIndex button command.

To do this you must go to Options - Configure Texmaker - Commands. And must change the command corresponding to MakeIndex to :

makeindex %.nlo -s nomencl.ist -o %.nls -t %.nlg

All you have to do is use the button to compile once with PDFLaTeX, once with MakeIndex and once more with PDFLaTeX.

Create a customized Quick Build

If you prefer to use the Quick Build button (or press the F1 key) rather than pressing the buttons calling the compilers a lot of times you can edit what the Quick Build button do.

Go to Options - Configure Texmaker - Quick Build and create your own command with the wizard. For example, for operations PDFLaTeX + BibTeX + MakeIndex + PDFLaTeX x2 + View PDF I got the following command :

pdflatex -synctex=1 -interaction=nonstopmode %.tex|bibtex %.aux|makeindex %.idx|pdflatex -synctex=1 -interaction=nonstopmode %.tex|pdflatex -synctex=1 -interaction=nonstopmode %.tex|evince %.pdf

Warning, if you have modified the MakeIndex call command, you still need to modify this last command line. All you have to do is replace in the command line makeindex %.idx by makeindex %.nlo -s nomencl.ist -o %.nls -t %.nlg. Now your Quick Build will work with nomencl.

Create a customized command

If you prefer use customized command rather than Quick Build, go to User - User Commands - Edit User Commands and create your own command with the wizard ; then replace in the command line makeindex %.idx by makeindex %.nlo -s nomencl.ist -o %.nls -t %.nlg.


text:
# Marian Cucoanes' Inequality With Roots and Powers

### Proof 1

Denote $a^2=x,\,$ $b^2=y,\,$ $c^2=z.\,$ By Heron's formula in expanded form,

$4S=\sqrt{(2(xy+yz+zx)-(x^2+y^2+z^2)}.$

The required inequality is equivalent to

$\displaystyle \small{(x+y+z)^2\ge[(2(xy+yz+zx)-(x^2+y^2+z^2)]\left[\frac{(x+y+z)(xy+yz+zx)}{2xyz}-\frac{3}{2}\right]}.$

WLOG, $x+y+z=3,\,$ $xy+yz+zx=3(1-t^2),\,$ $0\le t\lt 1.\,$ And, since

$0\lt (2(xy+yz+zx)-(x^2+y^2+z^2)=3(1-4t^2),$

we have $\displaystyle t\lt\frac{1}{2}.\,$ Also, due to a result by Vo Quoc Ba Can, $xyz\ge (1+t^2)(1-2t).\,$ Thus, suffice it to show that

$\displaystyle 1\ge \frac{3(1-4t^2)(1-t^2)}{2(1+t)^2(1-2t)}-\frac{1-4t^2}{2},$

which is equivalent to

$\displaystyle 3-4t^2\ge \frac{3(1+2t)(1-t)}{(1+t)},$

i.e.,

$\displaystyle 2t^2(1-2t)\ge 0,$

which is true because $\displaystyle t\lt\frac{1}{2}.$

### Proof 2

We'll use Heron's formula in the expanded form:

$16S^2=2(a^2b^2+b^2c^2+c^2a^2)-(a^4+b^4+c^4)$

which implies

(1)

$\displaystyle\left(\frac{a^2+b^2+c^2}{4S}\right)^2-3=\frac{(a^2-b^2)^2+(b^2-c^2)^2+(c^2-a^2)^2}{8S^2}.$

Further,

(2)

$\displaystyle\small{(a^2+b^2+c^2)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)^2-9=\frac{(a^2-b^2)^2}{a^2b^2}+\frac{(b^2-c^2)^2}{b^2c^2}+\frac{(c^2-a^2)^2}{c^2-a^2}}.$

The required inequality is equivalent to

$\displaystyle\left(\frac{a^2+b^2+c^2}{4S}\right)^2-3\ge\frac{1}{2}\left(a^2+b^2+c^2)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)^2-9\right)$

which (using (1),(2)) is equivalent to

$\displaystyle\frac{1}{8S^2}\sum_{cycl}(a^2-b^2)\ge\frac{1}{2}\sum_{cycl}\frac{(a^2-b^2)^2}{a^2b^2}.$

This, in turn, is equivalent to

$\displaystyle\sum_{cycl}\frac{(a^2-b^2)}{8S^2a^2b^2}(a^2b^2-4S^2)\ge 0$

which is true because $4S^2=a^2b^2\sin^2C\le a^2b^2.$

### Acknowledgment

Leo Giugiuc has kindly posted at the CutTheKnotMath facebook page the above problem, due to Marian Cucoanes, with his solution (Proof 1) and later added the solution by the problem's author (Proof 2).


text:
# Proving that a node is the only child

Say I have a graph of messages like this:

   A
|
B
/ \
C   D

By including the parent hash in the message to be hashed I can show that B is a child of A, and C and D are children of B.

For example

Hash("This is message A") = ab3ed
Hash("This is message B + ab3ed") = e4c90
Hash("This is message C + e4c90") = 83e39
Hash("This is message D + e4c90") = 456aa

This way I can prove that C and D are children of B, and descendants of A.

Is there a way I can cryptographically prove that a certain node is the only child ie has no siblings?

• So, how do you prove that B doesn't have a sibling, that is, you don't have a node E with Hash("This is message E + ab3ed") = 31415 ? Oct 26 '15 at 17:06
• Sounds pretty similar to the double spending problem of digital money. Perhaps you need some kind of blockchain or a global ledger.
– user27950
Oct 26 '15 at 17:35
• Do you have all the data in advance or is it added dynamically? Oct 26 '15 at 17:48
• @poncho: Yes, indeed. As it is constructed in the example is impossible to prove that B doesn't have a sibling E. My question can be rephrased as: How can I change the example so that I can prove that B doesn't have a sibling. Oct 26 '15 at 18:04
• @mikeazo: It is added dynamically Oct 26 '15 at 18:05

Given only the information in the original question, it would be easy for an attacker to calculate generate an arbitrary message E, then calculate the Hash("This is message E + ab3ed") = 31415, and if the system can be tricked into trusting that 31415 is the hash of a valid message, there's no way to prove that E isn't really a valid sibling of B.

This may be why every hash tree implementation I've ever seen doesn't include the hash of the parent in the hash of the child. Instead, it includes the hash of the children in the hash of the parent.

Say I have a graph of messages like this:

   A
|
B
/ \
C   D

By including every child hash in the message to be hashed I can show that B is the only child of A, and C and D are the only children of B.

For example

Hash("This is message A; its child has hash e4c90") = ab3ed
Hash("This is message B; its children have hashes 83e39 and 456aa") = e4c90
Hash("This is message C") = 83e39
Hash("This is message D") = 456aa

Assuming a cryptographically secure hash, and assuming I somehow get the root hash ab3ed from a trusted source, This way I can prove that C and D are the only children of B, and B is the only child of A -- i.e., B has no siblings.

If dynamically added, then I assume you know of the siblings at the time a node needs to be added, and ideally some order to the siblings. If that's all true, then instead of adding the hash of the parent, add the hash of the last sibling. If there are no siblings at insertion time, then revert to adding the parent's hash. So then you'll be able to prove that a node has no siblings if and only if its hash can be recalculated correctly when referencing the parent node's hash.

That should work regardless of any later changes to the siblings' content, unless all siblings are later deleted. Then the function can then only tell you that there used to be siblings at the time a node was added.

I would like to extend Chrystographer answer.

If this is a part of a incoming or outgoing message, you cannot prove it. Anybody can write valid messages. You need Merkle tree and digital signature of the root's hash.


text:
# The formula for finding the percentage of isotopes in an element's atomic weight?

In my class my teacher showed us how to find it, with a method, but he didn't really state the formula. I conjured up a formula from what I observed and it works. Here it is:

$$\frac{i_1x + i_2y}{100} = A$$

$A$ = atomic mass
$i_1$ = First's isotope's atomic weight
$i_2$ = Second isotope's atomic weight
$x \text{ and } y$ = Percentages of isotopes, respectively
$x+y = 100$

Therefore: $$\frac{i_1x + i_2(100-x)}{100} = A$$

I'm just wondering, is there an official formula, or is this method it? Thank you in advance.

Ps. If I didn't word the title correctly, feel free to edit it to something that would make much more sense! thank you!

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Have you studied weighted averages before? It's exactly that type of mathematics. – Nicolau Saker Neto Mar 7 '14 at 22:08
In my class we just did three questions. My teacher would give us an element such as Rubidium, the isotope's atomic mass, and the percentage of one isotope, then we would have to find the atomic mass. Or he would give us enough information to find another variable in my equation. (When I see my equation I'm not saying I made it, it's probably out there somewhere, I just don't know where) – Samir Chahine Mar 7 '14 at 22:12

The average relative atomic mass of an element comprised of $n$ isotopes with relative atomic masses $A_i$and relative fractional abundances $p_i$ is given by:

$$A = p_1A_1 + p_2A_2 + ... + p_nA_n = \sum\limits_{i=1}^npiAi$$

Isotope Isotopic Mass × % Abundance

12C $12.000000 * 0.98892 = 11.867$

13C $13.003354 * 0.01108 = 00.144$

Since we have $p_1A_1$ and $p_2A_2$, we add those together to find A.

Chemical relative atomic mass of carbon: $$A = 00.144 + 11.867 = 12.0011$$

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I'm sorry to be a pain, but can you please use an example with $$Br - 79 \rightarrow{} 50.69%$$ $$Br - 81 \rightarrow{} 49.31$$ To find the atomic mass? – Samir Chahine Mar 7 '14 at 22:18
@SamirChahine I provided an example with carbon 12 and carbon 13. – anosdsd Mar 7 '14 at 22:25
Thank you so much, this helps a lot, I aprreciate it! I didn't quite understand how the whole concept worked but I now have a much more intricate understanding, thank you. – Samir Chahine Mar 7 '14 at 22:30
@SamirChahine My pleasure! – anosdsd Mar 7 '14 at 22:31

Once you know how it works, you never want to do that by hand again ;) For bromine:

isotopes = [79, 81]
abundances = [50.69, 49.31]

weighted = map(lambda x, y: x * y, isotopes, abundances)
# multiply isotope mass and abundance

weighted = sum(weighted)/100
# sum them up and divide by 100

print weighted
>>>> 79.9862

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Huh, I should of done that ahah! Python is really handy sometimes, thank you for the idea! – Samir Chahine Mar 8 '14 at 1:12


text:
# What's the easiest way to get a link directly to an answer?

I can get the link to a question by hovering the cursor over its title. What's the smart way to get a direct link to one of the answers? The only way I know is to go to the user's page and hunt for it there. So I am clearly missing something.

All posts (questions and answers) have a "share" link beneath them, in the same area where the "edit" link is. This brings up a pop-up with a short URL that links directly to the question or answer.

• Thanks! It was totally off my radar. – Organic Marble May 9 '19 at 18:20
• @OrganicMarble You might want to use it for questions too, since the link in the title of the question is the long form, which has a portion of the title embedded. – called2voyage May 9 '19 at 18:24
• Great tip, many thanks! – Organic Marble May 9 '19 at 18:28
• This is what I've been using in my posts. For what it's worth, you can also get links to comments. Place your mouse over the comment's time and date, right click (or your browser's equivalent), and choose "Copy Link Location" or similar item from the pop-up menu. – DrSheldon May 9 '19 at 23:29
• I wish there was a magic text-generating button that would construct a [title](url) string and put it in my copy/paste buffer. It is so time-consuming to make those by hand, but I think people would use them all the time, even in comments. I think that the addition of the title (or eye-candy) would increase the frequency that people will go ahead and click through, increasing awareness. – uhoh May 10 '19 at 3:59
• Also, using the share link allows you to gain points towards badges, whereas using the URL doesn't. – Rory Alsop May 10 '19 at 11:36
• @uhoh: If you simply do [url] with the url being a Space.SE question or answer, it automatically displays it as the question title. Not sure if it works with questions across different SE sites, though. Does not seem to work for comments. – DrSheldon May 10 '19 at 16:11
• @DrSheldon Didn't work in the comments at all. Was that what you were talking about? – called2voyage May 10 '19 at 16:17
• @DrSheldon Didn't work in an answer for me either. – called2voyage May 10 '19 at 16:18
• Just posted a sample below. I was wrong about the square brackets, just the bare URL. I do it all the time! I only know of it working from a question or answer (not a comment), to a question or answer (not a comment). I'm not sure if the different behavior regarding comments is intentional, or should be reported as a bug. – DrSheldon May 10 '19 at 16:31
• @DrSheldon Oh yeah, I do a bare URL in questions and answers too. I wish that worked in the comments. – called2voyage May 10 '19 at 16:32
• @DrSheldon Bare URL only works for links to questions though. – called2voyage May 10 '19 at 16:33
• @called2voyage: Ah, correct you are. I don't link to answers nearly as much as I do questions, and apparently when I do, I tend to write something like In [this answer](...). Whereas I do a lot of Related: followed by a bare question URL. I should have noticed the difference. – DrSheldon May 10 '19 at 16:47

This is a test of https://space.meta.stackexchange.com/q/1273/58

What's the easiest way to get a link directly to an answer?

Testing https://space.meta.stackexchange.com/a/1274/58

https://space.meta.stackexchange.com/a/1274/58

It does work, even with long titles. However, this is display feature (appearance only) and is subject to change. It's not the same as button that creats a real text string of the form: [title](url) which you can copy/paste anywhere without depending on the page to regenerate the appearance of the title.

Also, I'm not sure what happens to the title generation/appearance if the original post is deleted the displayed text reverts back to the bare url (per comments below, thanks folks!).

Testing https://space.meta.stackexchange.com/q/1128/12102

Slightly nonstandard questions from new users frequently getting the silent, drive-by insta-close votes, can something be done?

Testing https://space.meta.stackexchange.com/q/1112/12102

Is voting to close a question for "primarily opinion-based answers" while answering with fact exercising faulty logic, or even gamesmanship?

However it doesn't seem to work for questions from another site:

https://aviation.stackexchange.com/questions/22022/did-the-space-shuttle-boat-tail-used-in-the-transport-configuration-save-fuel

so here I've done it manually again. Here's how it looked before my edit (after I re-pasted the url just to make sure):

• If the original post is deleted, the title won't be resolved at all and showing the bare URL instead. – Andrew T. May 11 '19 at 6:17
• @AndrewT. I would guess as much as well, but how do you know? What makes you say that? – uhoh May 11 '19 at 13:15
• @uhoh I tested it--he's right. – called2voyage May 13 '19 at 19:26
• @called2voyage updated, thanks! – uhoh May 13 '19 at 23:34
• @called2voyage but it doesn't seem to work from another SE site? (see edit) – uhoh May 22 '19 at 0:19