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# ListDensityPlot with discrete bins [duplicate]

We consider a table of the form

data = Flatten[Table[{i,j,f[i,j]},{i,1,ni},{j,1,nj}],1]

The function f[i_,j_] has been made so that 0 ≤ f[i,j] ≤ 1. I can plot this map using

ListDensityPlot[data, ColorFunction -> "Rainbow", InterpolationOrder -> 0]

However, the colors attributed to the values of f[i,j] are continuous and vary smoothly. I would like to have only a finite number of colors. For example : 0 ≤ f[i,j] ≤ 1/2 --> Red and 1/2 ≤ f[i,j] ≤ 1 --> Blue (Of course, my real bins will be different, but this is the idea.)

How could I proceed to make this plot so that only a finite number of colors are present on the plot with custom ranges ? I guess it has something to do with the ColorFunction, but I can't get the syntax correct...

-

## marked as duplicate by xzczd, Kuba, rasher, Sjoerd C. de Vries, m_goldbergMay 13 at 10:22

@Kuba I took out the definitions in order to get a minimum working example as simple as possible. Your suggestion definitely works using ColorFunction -> (If[# < .5, Red, Blue] &), ColorFunctionScaling -> False. However, I still don't understand this syntax. I would prefer to define a function fcolor[value_] which would give me the color associated to a value f[i_,j_]. What would then be the syntax for ColorFunction -> ... ? –  jibe May 13 at 8:15
Simple: ColorFunction -> fcolor –  Sjoerd C. de Vries May 13 at 9:30

Take a look at ColorFunction, ColorFunctionScaling, and InterpolationOrder, e.g.:

ListDensityPlot[{{0, 0, 0}, {1, 0, 1}, {0, 1, 2}, {1, 1, 3}},
ColorFunctionScaling -> None, InterpolationOrder -> 0,
ColorFunction ->
Function[arg,
Which[0 <= arg < 1, Red, 1 <= arg < 2, Blue, 2 <= arg < 3, Green, True, Orange]]]

You can use similar constructs with MatrixPlot and ArrayPlot, which may be better fits for your needs.

Per your comment, if you want to externalize the color function, e.g.:

fcolor[arg_] :=
Which[0 <= arg < 1, Red, 1 <= arg < 2, Blue, 2 <= arg < 3, Green, True, Orange]

ListDensityPlot[{{0, 0, 0}, {1, 0, 1}, {0, 1, 2}, {1, 1, 3}},
ColorFunctionScaling -> None, InterpolationOrder -> 0,
ColorFunction -> fcolor]

-
Thank you. As I commented above to Kuba. Would you which syntax I could use to define the function Function[arg, Which[0 <= arg < 1, Red, 1 <= arg < 2, Blue, 2 <= arg < 3, Green, True, Orange]] out of the ListDensityPlot ? With such a function fcolor[value_], what would become the part ColorFunction -> .. ? (This is an additional question, your approach works perfectly.) –  jibe May 13 at 8:18
@jibe: see edit. Read docs re: details on the functionality of the various settings (I did the examples with what I think you had in mind). Also, use of Interval might be useful to you in defining ranges. Also, I'm a doofus - xzczd references the very question in which I show these methods - take a peek at it... and don't take it personally if this question is closed as a duplicate, helps keep things tight. –  rasher May 13 at 8:22
This is exactly what I had in my mind. My next step will now be to get a nice PlotLegends on the side, but I will first try on my own. ^^ –  jibe May 13 at 8:24
Don't worry, I am not at all offended ! I am trying to use your answer to the duplicate, but the shorthened syntax using #, @ and & is not obvious for a beginner like me... In any case, thanks again. –  jibe May 13 at 8:32

format: plain


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# NEET Physics Thermodynamics Questions Solved

P-V diagram of a cyclic process ABCA is as shown in figure. Choose the correct statement

(1) $\Delta {Q}_{A\to B}$ = negative

(2) $\Delta {U}_{B\to C}$ = positive

(3) $\Delta {W}_{CAB}$ = negative

(4) All of these

(4) During process A to B, pressure and volume both are decreasing. Therefore, temperature and hence, internal energy of the gas will decrease (TPV) or $\Delta {U}_{A\to B}=$ negative. Further $\Delta {W}_{A\to B}$ is also negative as the volume of the gas is decreasing. Thus $\Delta {Q}_{A\to B}$ is negative.

In process B to C pressure of the gas is constant while volume is increasing. Hence temperature should increase or $\Delta {U}_{B\to C}$ = positive. During C to A volume is constant while pressure is increasing. Therefore, temperature and hence, internal energy of the gas should increase or $\Delta {U}_{C\to A}$ = positive. During process CAB volume of the gas is decreasing. Hence, work done by the gas is negative.

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text:
0

IN THIS ISSUE

### Research Papers

J Biomech Eng. 2010;132(8):081001-081001-8. doi:10.1115/1.4001027.

A millimeter-for-millimeter relation between an increase in length of an anterior cruciate ligament graft construct and an increase in anterior laxity has been demonstrated in multiple in vitro studies. Based on this relation, a 3 mm increase in length of the graft construct following surgery could manifest as a 3 mm increase in anterior laxity in vivo, which is considered clinically unstable. Hence, the two primary objectives were to determine whether the millimeter-for-millimeter relation exists in vivo for slippage-resistant fixation of a soft-tissue graft and, if it does not exist, then to what extent the increase in stiffness caused by biologic healing of the graft to the bone tunnel offsets the potential increase in anterior laxity resulting from lengthening at the sites of fixation. Sixteen subjects were treated with a fresh-frozen, nonirradiated, nonchemically processed tibialis allograft. Tantalum markers were injected into the graft, fixation devices, and bones. On the day of surgery and at 1, 2, 3, and 4 months, Roentgen stereophotogrammetric analysis was used to compute anterior laxity at 150 N of anterior force and the total slippage from both sites of fixation. A simple linear regression was performed to determine whether the millimeter-for-millimeter relation existed and a springs-in-series model of the graft construct was used to determine the extent to which the increase in stiffness caused by biological healing of the graft to the bone tunnel offset the increase in anterior laxity resulting from lengthening at the sites of fixation. There was no correlation between lengthening at the sites of fixation and the increase in anterior laxity at 1 month ($R2=0.0$, $slope=0.2$). Also, the increase in stiffness of the graft construct caused by biologic healing of the graft to the bone tunnel offset 0.7 mm of the 1.5 mm potential increase in anterior laxity resulting from lengthening at the sites of fixation. This relatively large offset of nearly 50% occurred because lengthening at the sites of fixation was small.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081002-081002-10. doi:10.1115/1.4001679.

In this paper, we develop structured tree outflow boundary conditions for modeling the airflow in patient specific human lungs. The utilized structured tree is used to represent the nonimageable vessels beyond the 3D domain. The coupling of the two different scales (1D and 3D) employs a Dirichlet–Neumann approach. The simulations are performed under a variety of conditions such as light breathing and constant flow ventilation (which is characterized by very rapid acceleration and deceleration). All results show that the peripheral vessels significantly impact the pressure, however, the flow is relatively unaffected, reinforcing the fact that the majority of the lung impedance is due to the lower generations rather than the peripheral vessels. Furthermore, simulations of a hypothetical diseased lung (restricted flow in the superior left lobe) under mechanical ventilation show that the mean pressure at the outlets of the 3D domain is about 28% higher. This hypothetical model illustrates potential causes of volutrauma in the human lung and furthermore demonstrates how different clinical scenarios can be studied without the need to assume the unknown flow distribution into the downstream region.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081003-081003-10. doi:10.1115/1.4001680.

Functional electrical stimulation (FES) has the capacity to regenerate motion for individuals with spinal cord injuries. However, it is not straightforward to determine the stimulation parameters to generate a coordinated movement. Musculoskeletal models can provide a noninvasive simulation environment to estimate muscle force and activation timing sequences for a variety of tasks. Therefore, the purpose of this study was to develop a musculoskeletal model of the feline hindlimb for simulations to determine stimulation parameters for intrafascicular multielectrode stimulation (a method of FES). Additionally, we aimed to explore the differences in modeling neuromuscular compartments compared with representing these muscles as a single line of action. When comparing the modeled neuromuscular compartments of biceps femoris, sartorius, and semimembranosus to representations of these muscles as a single line of action, we observed that modeling the neuromuscular compartments of these three muscles generated different force and moment generating capacities when compared with single muscle representations. Differences as large as $4 N m$ ($∼400%$ in biceps femoris) were computed between the summed moments of the neuromuscular compartments and the single muscle representations. Therefore, modeling neuromuscular compartments may be necessary to represent physiologically reasonable force and moment generating capacities of the feline hindlimb.

Topics: Force , Modeling , Muscle , Knee , Motion
Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081004-081004-9. doi:10.1115/1.4001682.

Interest in developing durable mitral valve repair methods is growing, underscoring the need to better understand the native mitral valve mechanics. In this study, the authors investigate the dynamic deformation of the mitral valve strut chordae-to-anterior leaflet transition zone using a novel stretch mapping method and report the complex mechanics of this region for the first time. Eight structurally normal porcine mitral valves were studied in a pulsatile left heart simulator under physiological hemodynamic conditions −120 mm peak transvalvular pressure, 5 l/min cardiac output at 70 bpm. The chordal insertion region was marked with a structured array of 31 miniature markers, and their motions throughout the cardiac cycle were tracked using two high speed cameras. 3D marker coordinates were calculated using direct linear transformation, and a second order continuous surface was fit to the marker cloud at each time frame. Average areal stretch, principal stretch magnitudes and directions, and stretch rates were computed, and temporal changes in each parameter were mapped over the insertion region. Stretch distribution was heterogeneous over the entire strut chordae insertion region, with the highest magnitudes along the edges of the chordal insertion region and the least along the axis of the strut chordae. At early systole, radial stretch was predominant, but by mid systole, significant stretch was observed in both radial and circumferential directions. The compressive stretches measured during systole indicate a strong coupling between the two principal directions, explaining the small magnitude of the systolic areal stretch. This study for the first time provides the dynamic kinematics of the strut chordae insertion region in the functioning mitral valve. A heterogeneous stretch pattern was measured, with the mechanics of this region governed by the complex underlying collagen architecture. The insertion region seemed to be under stretch during both systole and diastole, indicating a transfer of forces from the leaflets to the chordae and vice versa throughout the cardiac cycle, and demonstrating its role in optimal valve function.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081005-081005-7. doi:10.1115/1.4001739.

Characterization of high-intensity focused ultrasound (HIFU) systems using ex vivo tissues is an important part of the preclinical testing for new HIFU devices. In ex vivo characterization, the lesion volume produced by the absorption of HIFU energy is quantified as operational parameters are varied. This paper examines the three methods used for lesion-volume quantification: histology, magnetic resonance (MR) imaging, and numerical calculations. The methods were studied in the context of a clinically relevant problem for HIFU procedures—that of quantifying the change in the lesion volume with changing sonication time. The lesion volumes of sonicated samples of porcine liver were determined using the three methods, at focal intensities ranging from $800 W/cm2$ to $1700 W/cm2$ and sonication times between 20 s and 40 s. It was found that histology consistently yielded lower lesion volumes than the other two methods, and the calculated values were below magnetic resonance imaging (MRI) at high applied energies. Still, the three methods agreed with each other to within a $±10%$ difference for all of the experiments. Increasing the sonication time produced much larger changes in the lesion volume than increasing the acoustic intensity, for the same total energy expenditure, at lower energy (less than 1000 J) levels. At higher energy levels, (around 1500 J), increasing the sonication time and increasing the intensity produced roughly the same change in the lesion volume for the same total energy expenditure.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081006-081006-6. doi:10.1115/1.4001257.

Elastography, which is the imaging of soft tissues on the basis of elastic modulus (or, more generally, stiffness) has become increasingly popular in the last decades and holds promise for application in many medical areas. Most of the attention has focused on inhomogeneous materials that are locally isotropic, the intent being to detect a (stiff) tumor within a (compliant) tissue. Many tissues of mechanical interest, however, are anisotropic, so a method capable of determining material anisotropy would be attractive. We present here an approach to determine the mechanical anisotropy of inhomogeneous, anisotropic tissues, by directly solving the finite element representation of the Cauchy stress balance in the tissue. The method divides the sample domain into subdomains assumed to have uniform properties and solves for the material constants in each subdomain. Two-dimensional simulated experiments on linear anisotropic inhomogeneous systems demonstrate the ability of the method, and simulated experiments on a nonlinear model demonstrate the ability of the method to capture anisotropy qualitatively even though only a linear model is used in the inverse problem. As with any inverse problem, ill-posedness is a serious concern, and multiple tests may need to be done on the same sample to determine the properties with confidence.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081007-081007-8. doi:10.1115/1.4001681.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081008-081008-10. doi:10.1115/1.4001358.

A finite element model was developed for numerical simulations of nanoindentation tests on cortical bone. The model allows for anisotropic elastic and post-yield behavior of the tissue. The material model for the post-yield behavior was obtained through a suitable linear transformation of the stress tensor components to define the properties of the real anisotropic material in terms of a fictitious isotropic solid. A tension-compression yield stress mismatch and a direction-dependent yield stress are allowed for. The constitutive parameters are determined on the basis of literature experimental data. Indentation experiments along the axial (the longitudinal direction of long bones) and transverse directions have been simulated with the purpose to calculate the indentation moduli and the tissue hardness in both the indentation directions. The results have shown that the transverse to axial mismatch of indentation moduli was correctly simulated regardless of the constitutive parameters used to describe the post-yield behavior. The axial to transverse hardness mismatch observed in experimental studies (see, for example, Rho [1999, “Elastic Properties of Microstructural Components of Human Bone Tissue as Measured by Nanoindentation,” J. Biomed. Mater. Res., 45, pp. 48–54] for results on human tibial cortical bone) can be correctly simulated through an anisotropic yield constitutive model. Furthermore, previous experimental results have shown that cortical bone tissue subject to nanoindentation does not exhibit piling-up. The numerical model presented in this paper shows that the probe tip-tissue friction and the post-yield deformation modes play a relevant role in this respect; in particular, a small dilatation angle, ruling the volumetric inelastic strain, is required to approach the experimental findings.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081009-081009-6. doi:10.1115/1.4001883.

The fracture of bone due to indentation with a hard, sharp object is of significance in surgical procedures and certain trauma situations. In the study described below, the fracture of bovine bone under indentation was measured experimentally and predicted using the theory of critical distances (TCDs), a theory, which predicts failure due to cracking in the vicinity of stress concentrations. The estimated indentation fracture force was compared with the experimental results in three different cutting directions. Under indentation, the material experiences high levels of compression and shear, causing cracks to form and grow. The direction of crack growth was highly dependent on the bone’s microstructure: major cracks grew in the weakest possible structural direction. Using a single value of the critical distance $(L=320 μm)$, combined with a multiaxial failure criterion, it was possible to predict the experimental failure loads with less than 30% errors. Some differences are expected between the behavior of human bone and the bovine bone studied here, owing to its plexiform microstructure.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081010-081010-6. doi:10.1115/1.4001678.

Evaluating total knee replacement kinematics and contact pressure distributions is an important element of preclinical assessment of implant designs. Although physical testing is essential in the evaluation process, validated computational models can augment these experiments and efficiently evaluate perturbations of the design or surgical variables. The objective of the present study was to perform an initial kinematic verification of a dynamic finite element model of the Kansas knee simulator by comparing predicted tibio- and patellofemoral kinematics with experimental measurements during force-controlled gait simulation. A current semiconstrained, cruciate-retaining, fixed-bearing implant mounted in aluminum fixtures was utilized. An explicit finite element model of the simulator was developed from measured physical properties of the machine, and loading conditions were created from the measured experimental feedback data. The explicit finite element model allows both rigid body and fully deformable solutions to be chosen based on the application of interest. Six degrees-of-freedom kinematics were compared for both tibio- and patellofemoral joints during gait loading, with an average root mean square (rms) translational error of 1.1 mm and rotational rms error of 1.3 deg. Model sensitivity to interface friction and damping present in the experimental joints was also evaluated and served as a secondary goal of this paper. Modifying the metal-polyethylene coefficient of friction from 0.1 to 0.01 varied the patellar flexion-extension and tibiofemoral anterior-posterior predictions by 7 deg and 2 mm, respectively, while other kinematic outputs were largely insensitive.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081011-081011-7. doi:10.1115/1.4001159.

Insufficient lateralization of the femoral component coupled with exposed reamed cancellous bone has been speculated to predispose to femoral neck fracture. The current study examined the effect of mediolateral implant position and exposed cancellous bone on the strength of the resurfaced proximal femur. Composite femurs were prepared in three configurations: (1) partial, with the implant placed at the native femoral head offset of the femur, partially exposing reamed cancellous bone; (2) proud, with a medialized implant exposing a circumferential ring of cancellous bone; and (3) complete, with a lateralized implant covering all reamed cancellous bone. Specimens were loaded to failure in axial compression. A finite element model was used to further explore the effect of exposed cancellous bone, cement mantle thickness, and relative valgus orientation on the strain distributions in the resurfaced femur. The proud group (2063 N) was significantly weaker than both the partial (2974 N, $p=0.004$) and complete groups (5899 N, $p=0.001$) when tested to failure. The partial group was also significantly weaker than the complete group when tested to failure $(p=0.001)$. The finite element model demonstrated increasing levels of strain in the superior reamed cortical-cancellous bone interface with increasing degree of exposed cancellous bone. The condition of the femoral component medialized as the result of a thick cement mantle had the greatest detrimental impact on strain level in the superior reamed cancellous bone while a valgus oriented implant provided a protective effect. This study provides biomechanical evidence that exposed reamed cancellous bone significantly reduces the load-to-failure and increases maximum strains in the resurfaced proximal femur. The perceived benefit of reconstructing the femur to its native geometry may inherently weaken the proximal femur and increase femoral neck fracture risk if the femoral component is not sufficiently lateralized to cover all unsupported reamed cancellous bone. Relative valgus orientation of the implant may help to minimize the risk of neck fracture if reamed cancellous bone remains exposed following implant impaction.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081012-081012-7. doi:10.1115/1.4000086.

In spinal vertebral burst fractures, the dynamic properties of the trabecular centrum, which is the central region of porous bone inside the vertebra, can play an important role in determining the failure mode. If the failure occurs in the posterior portion of the vertebral body, spinal canal occlusion can occur and ejected trabecular bone can impact the spinal cord resulting in serious injury. About 15% of all spinal cord injuries are caused by such burst fractures. Unfortunately, due to the uniqueness of burst fracture injuries, postinjury investigation cannot always accurately assess the degree of damage caused by these fractures. This research makes an effort to begin understanding the governing effects in this important bone fracture event. Measurements of the dynamic deformation response of bovine trabecular bone with the marrow intact and marrow removed using a modified split-Hopkinson pressure bar apparatus are reported and compared with quasistatic deformation response results. Because trabecular bone is more compliant and lower in strength than cortical bone, typical Hopkinson pressure bar experimental techniques used for high strain rate testing of harder materials cannot be applied. Instead, a quartz-crystal-embedded, split-Hopkinson pressure bar developed for testing compliant, low strength materials is used. Care is taken into account for the orthotropic properties in the bone by testing only along the principle material axes, determined through microcomputed tomography. In addition, shaping of the stress wave pulse is used to ensure a constant strain rate and homogeneous specimen deformation. Results indicate that the strength of trabecular bone increases by a factor of approximately 2–3 when the strain rate increases from $10−3 s−1$ to $500 s−1$ and that the bone fractures beyond a critical strain.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):081013-081013-10. doi:10.1115/1.4001891.

Endothelial cells are known to respond to hemodynamic forces. Their phenotype has been suggested to differ between atheroprone and atheroprotective regions of the vasculature, which are characterized by the local hemodynamic environment. Once an atherosclerotic plaque has formed in a vessel, the obstruction creates complex spatial gradients in wall shear stress. Endothelial cell response to wall shear stress may be linked to the stability of coronary plaques. Unfortunately, in vitro studies of the endothelial cell involvement in plaque stability have been limited by unrealistic and simplified geometries, which cannot reproduce accurately the hemodynamics created by a coronary stenosis. Hence, in an attempt to better replicate the spatial wall shear stress gradient patterns in an atherosclerotic region, a three dimensional asymmetric stenosis model was created. Human abdominal aortic endothelial cells were exposed to steady flow ($Re=50$, 100, and 200 and $τ=4.5 dyn/cm2$, $9 dyn/cm2$, and $18 dyn/cm2$) in idealized 50% asymmetric stenosis and straight/tubular in vitro models. Local morphological changes that occur due to magnitude, duration, and spatial gradients were quantified to identify differences in cell response. In the one dimensional flow regions, where flow is fully developed and uniform wall shear stress is observed, cells aligned in flow direction and had a spindlelike shape when compared with static controls. Morphological changes were progressive and a function of time and magnitude in these regions. Cells were more randomly oriented and had a more cobblestone shape in regions of spatial wall shear stress gradients. These regions were present, both proximal and distal, at the stenosis and on the wall opposite to the stenosis. The response of endothelial cells to spatial wall shear stress gradients both in regions of acceleration and deceleration and without flow recirculation has not been previously reported. This study shows the dependence of endothelial cell morphology on spatial wall shear stress gradients and demonstrates that care must be taken to account for altered phenotype due to geometric features. These results may help explain plaque stability, as cells in shoulder regions near an atherosclerotic plaque had a cobblestone morphology indicating that they may be more permeable to subendothelial transport and express prothrombotic factors, which would increase the risk of atherothrombosis.

Commentary by Dr. Valentin Fuster

### Technical Briefs

J Biomech Eng. 2010;132(8):084501-084501-4. doi:10.1115/1.4001888.

A continuum model based on the contact mechanics theory was developed and used for evaluating virus indentation forces at the early stage of membrane invagination, as well as the work of the virus indentation forces and virus-cell contact pressures in a receptor-mediated endocytosis, depending on the virus size and virus/cell stiffnesses. The model indicated that early virus indentation forces are in the order of 1–10 pN and for a given extent of virus engulfment, they increase linearly with the elastic modulus of the host cell and also with the square of the virus radius. The work of invagination at the initial phase of virus endocytosis is in the order of tens of zeptojoules and peak virus-cell contact pressures at this stage are in the order of hundreds of Pascals to several kPa. For a given extent of virus engulfment, peak and average contact pressures increase linearly with the elastic modulus of the host cell but interestingly, they are negligibly affected by the virus size. The present model may be useful in the fields of cellular biomechanics, virology and nanodrug delivery to evaluate mechanical factors during the early phase of membrane invagination.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 2010;132(8):084502-084502-4. doi:10.1115/1.4001893.

This study aimed at measuring the static coefficient of friction $(μ)$ between bone and skeletal muscle tissues in order to support finite element (FE) modeling in orthopaedic and rehabilitation research, where such contact conditions need to be defined. A custom-made friction meter (FM) that employs the load cell and motion-controlled loading arm of a materials testing machine was designed for this study. The FM was used to measure $μ$ between fresh ulna bones and extensor muscles surrounding the ulna, which were harvested from five young adult pigs. Mean bone-muscle $μ$ were between 0.36 and 0.29, decreased with the increase in loads applied on the bone $(p<0.05)$ and plateaued at a mean $∼0.3$ for loads exceeding 4 N. Hence, for FE modeling of bone-muscle contacts through which loads with magnitudes of kgs to 10s-of-kgs are transferred, assuming $μ$ of $∼0.3$ appears to be appropriate.

Commentary by Dr. Valentin Fuster


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## Tuesday, April 13, 2010 ... //

### Desertec: Club of Rome to spend USD 550 billion

Desertec is an ambitious project initiated by the Club of Rome and TREC, i.e. Trans-Mediterranean Renewable Energy Corporation of Germany. They want to spend more than half a trillion dollars and build solar power plants on area equal to that of a "130 km x 130 km" square around Sahara that would cover 15% of the energy consumption of Europe.

The two "advantages" of the project are that the solar energy will be about twice more expensive than the energy based on the fossil fuels. Also, 15% of the dependence on Russia will be replaced by the dependence on the regimes led by bashi-bazooks in Africa. What a win-win situation. ;-)

The media disagree whether this project - that is only conceivable assuming huge subsidies justified by the global warming panic - will materialize. At any rate, there's still a chance that the project will make some sense: planning will continue through 2012 and the construction could end by 2050. By that time, the solar energy may become more economical.

As Chris Horner pointed out at Planet Gore, Ms Polly Higgins who recently urged the United Nations to adopt a new "ecocide" policy to execute employees of oil companies, climate skeptics, and other sane and productive people is on the board of Desertec. That doesn't sound terribly promising for the company.

#### snail feedback (1) :

reader ErkDemon (Eric Baird) said...

According to the link, the proposed "ecocide" policy is not about killing people who oppose the environmental movement, it's about inventing a new criminal offence called "ecocide" that relates to damaging the environment. Destroying forests by mass logging, poisoning rivers, desertifying land by over-irrigating with brackish water, oil spills, that sort of thing.

The idea is that it'd allow company executives to be prosecuted for damaging collective resources without having to prove that specific people's interests were harmed.

It's an interesting idea, but it'd need a lot of work (and a new name). For instance, if we plastered solar panels over the Sahara and reduced local temperatures, and created shade, and the structures encouraged condensation, new vegetation might start springing up around the solar installations. If we accidentally "greened" the Sahara, would this count as damaging a unique natural environment?

And if we're talking about criminalising the destruction of resources, then perhaps the UK government's irrevocable decision to shut down large sections of the UK's coalmining industry would be considered an offence, too ... that decision eliminated a big chunk of the then-EU's strategic reserves of high-quality coal, and I know that there were supposed to have been a lot of Germans who would have loved to have had those reserves (most of Germany's coal is supposed to be nasty, state-subsidised open-cast stuff), and probably considered the UK decision to be criminal damage to Europe's collective resources.

I guess it'd also mean that in warring regions, occupying forces would have to think twice before deliberately poisoning wells or rivers, or setting light to oilfields as acts of revenge.

I don't know how you'd quantify these things or decide penalties, though.

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# Why can one product in an equilibrium system not be produced without the other? [closed]

I performed an experiment where $\ce{CuSO4 (aq)}$ reacted with $\ce{NH3 (aq)}$ in one test tube, and in the other, $\ce{CuSO4 (aq)}$ reacted with $\ce{NH3 (aq)}$ and $\ce{NaOH (aq)}$. I know two products were formed, one being the complex ion $\ce{[Cu(NH3)4]^{2+}}$.

The goal of the experiment was to see if one of the products could be produced without the other, and I know that is not possible due to the solutions being at equilibrium, but why is this?

## closed as unclear what you're asking by paracetamol, porphyrin, Buttonwood, ron, M.A.R.May 5 '17 at 15:24

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Your question is not quite clear. What is the "one" product, what is the "other", what where your observations, and what is unclear about all that? – Ivan Neretin Nov 11 '16 at 19:33

In the first experiment, the dissolved $\ce{CuSO4}$ reacted with $\ce{NH3(aq)}$ to form the soluble, deep blue/purple complex via equation 1, $\ce{[Cu(NH3)4]^{2+}}$, just as you said.

1) $\ce{Cu^{2+} + 4NH3 <--> [Cu(NH3)4]^{2+}}$

Depending on the concentration of $\ce{NH3}$, you would also have precipitated some amount of insoluble $\ce{Cu(OH)2}$ from the hydroxide formed by the ammonia reacting with water as in equation 2:

2) $\ce{NH3 + H2O <--> NH4+ + OH-}$

The equilibrium of equation 2 lies far to the left, so when you add more ammonia, you aren't adding an equal amount of hydroxide.

The difference in the second experiment is that you explicitly added excess hydroxide. In this case the high hydroxide concentration would have pushed the dominant product to $\ce{Cu(OH)2}$ via equation 3.

3) $\ce{Cu^{2+} + 2OH- <--> Cu(OH)2}$ (solid)

Regarding the statement "The goal of the experiment was to see if one of the products could be produced without the other":

The products of both experiments were the same, just to different degrees; in other words, under different pH conditions, different products predominated. In the first experiment, you had far more ammonia than hydroxide, so equation 1 dominated; lots of $\ce{[Cu(NH3)4]^{2+}}$ and a small amount of $\ce{Cu(OH)2}$. In the second experiment, you added excess hydroxide so lots of $\ce{Cu(OH)2}$ was formed an very little $\ce{[Cu(NH3)4]^{2+}}$.

So the upshot is that these reactions are in equilibrium, and because of equation 2 you always had both $\ce{NH3}$ and $\ce{OH-}$ in your reaction mixture so that both equations 1 and 3 always produced some small amount of their respective products, $\ce{[Cu(NH3)4]^{2+}}$ and $\ce{Cu(OH)2}$, respectively.


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# transfer kernels and the Schur multiplier

Let $\Gamma$ be a finite $2$-group, and let $G$ be any subgroup of index $2$. Moreover, let Ver$: \Gamma/\Gamma' \to G/G'$ denote the group theoretical transfer, and let $M(\Gamma)$ be the Schur multiplier of $\Gamma$.

Is it true that $$| \text{ker}\ {\rm Ver}_{\Gamma/\Gamma' \to G/G'} | \le 2 | M(\Gamma) | \quad ?$$

The answer is positive for a couple of groups with small multiplier, such as cyclic $2$-groups, which have trivial Schur multiplier, the groups of order $8$, or dihedral and quaternion groups. Does current computer technology allow finding a counterexample by going through 2-groups of moderate size?

Edit. Here's a little bit of background. Let $K$ be a quadratic number field whose 2-class group has type (2,2). It is known that the Hilbert 2-class field $K^1$ has cyclic 2-class group, and that the Galois group $\Gamma$ of the second Hilbert 2-class field is either (2,2) itself, a dihedral, quaternion, or semi-dihedral 2-group. Let $K_j/K$ (j=1, 2, 3) denote the three unramified quadratic extensions inside $K^1$, and let $G_j$ be the Galois groups of $K^2/K_j$. Then $\Gamma$ is quaternion or semi-dihedral if and only if in each of the extensions $K_j/K$, exactly one ideal class of order 2 capitulates (i.e., becomes principal). Now these are exactly the groups among the possible Galois groups with trivial Schur multiplier. On the other hand, the transfer of ideal classes corresponds, via Artin's reciprocity law, to the transfer map (Verlagerung) from the abelianization of $\Gamma$ to that of the $G_j$. Thus in this case, we find that the order of the Schur multiplier is equal to one half of the maximal order of the capitulation kernels.

-
There are no counterexamples of order up to and including 128. 256 seems like it would take about a day, and 512 is probably just silly. If you had more description of what the counterexample would look like, it might be possible to greatly speed up the search. –  Jack Schmidt Jan 30 '11 at 5:15
I can now confirm that there are no counterexamples of order 256. But it would nice to have some suggestions as to why we might expect such a result to be true. –  Derek Holt Jan 30 '11 at 11:05
The whole thing came from computations of capitulation kernels in subfields of the Hilbert class fields of quadratic number fields almost 20 years ago. An almost classical observation in this area is the fact that for quaternion extensions, the Schur multiplier is trivial and in each quadratic extension exactly one nontrivial class capitulates. Another conjecture from back then was shot down by Boston and Leedham-Green, so I figured I'd have just about the same luck in this case. I'd be deeply grateful if one of you could write up a description of the calculation (in sage/GAP?) as an answer –  Franz Lemmermeyer Jan 30 '11 at 18:19

This is just a fairly literal translation of your question into GAP code.

If H is a subgroup of index 2 in G, and t is an element of G not in H, then the transfer map from G to H/[H,H] takes a particularly simple form:

If h in H, then h⋅1 = 1⋅h and h⋅t = t⋅ht, so tr(h) = h⋅ht.

If g in G∖H, then g⋅1 = t⋅(t−1g) and g⋅t = 1 ⋅(gt), so tr(g) = gt t−1g = g2.

In the GAP language, one creates the homomorphism with:

tr := function(g,h)
local t, k, gen, img;
t := First( g, t -> not t in h );
k := MaximalAbelianQuotient(h);
gen := GeneratorsOfGroup(g);
img := List( gen, function(x)
if x in h
then return Image( k, x*x^t );
else return Image( k, x*x );
fi;
end );
return GroupHomomorphismByImages( g, Image(k), gen, img );
end;

We can test the conjecture on 2-groups with the following GAP function:

isCounter := g -> not ForAll( MaximalSubgroupClassReps( g ),
h -> Index( Kernel( tr( g, h ) ), DerivedSubgroup( g ) )
<= 2 * Product( AbelianInvariantsMultiplier( g ) ) );

We then look for examples of small orders with the following GAP function:

OneSmallGroup( [2,4,8,16,32], isCounter ); # 5 seconds

which simply returns fail since no such counterexample of order 2, 4, 8, 16, or 32 exists. To test the higher orders one uses:

OneSmallGroup( [ 64], isCounter ); # 30 seconds
OneSmallGroup( [128], isCounter ); # 10 minutes

For larger orders, one needs to be more careful calculating the Schur multiplier, or applying a little theory since the naive method is dealing with larger index subgroups and then larger integer matrices.

-
Thank you very much! –  Franz Lemmermeyer Jan 31 '11 at 9:44

Take the short exact sequence of modules $\mathbb{Z}\hookrightarrow Ind^\Gamma_G\mathbb{Z}\twoheadrightarrow\mathbb{Z}$ and apply $H_i(\Gamma,-)$. Note that this sequence is exact because $|\Gamma:G|=2$. You obtain the long exact sequence, noting that $H_i(\Gamma,Ind^\Gamma_H\mathbb{Z})\cong H_iG$ by Shapiro's Lemma. As I remarked in the comment attached to this post, the coefficient module from the latter $\mathbb{Z}$ in the short exact sequence has nontrivial $\Gamma$-action (I will denote this coefficient by $\tilde{\mathbb{Z}}$, where the action is multiplication by $-1$ via elements of the nontrivial coset of $\Gamma/G$). Keeping that in mind, we have in particular:

$H_2(\Gamma,\tilde{\mathbb{Z}})\stackrel{\delta}{\rightarrow}H_1\Gamma\stackrel{tr}{\rightarrow}H_1G\rightarrow H_1(\Gamma,\tilde{\mathbb{Z}})$.

Exactness implies $Ker(tr)=Im(\delta)=H_2(\Gamma,\tilde{\mathbb{Z}})/Ker(\delta)$, so that $|Ker(tr)|\le |H_2(\Gamma,\tilde{\mathbb{Z}})|$. And we know that $tr=Ver$ and $H_2\Gamma=M(\Gamma)$.

I want to claim that $|H_2(\Gamma,\tilde{\mathbb{Z}})|\le 2\cdot|H_2\Gamma|$ (the latter homology has $\mathbb{Z}$-coefficient with trivial action), but at this moment I am unsure how to prove it. Hatcher's Algebraic Topology textbook gives a long exact sequence for general coefficient systems (pg330), with $H_3G\stackrel{res}{\rightarrow}H_3\Gamma\rightarrow H_2(\Gamma,\tilde{\mathbb{Z}})\rightarrow H_2G\stackrel{res}{\rightarrow}H_2\Gamma$, so this could be of use.

• Chris Gerig
-
But $\lvert \ker \operatorname{Ver}_{\Gamma\to G/G'}\rvert= 2$ and $M(\Gamma)=1$ for cyclic groups. So something must be wrong here, or am I missing something? –  Frieder Ladisch Jan 31 '11 at 13:06
Ah there is an error: the $\Gamma$-action on the latter $\mathbb{Z}$ in the short exact sequence is not trivial, it acts as $\gamma\cdot z = -z$ for $\gamma$ not in $G$. I will make the fix, and hopefully this brings in the desired factor of 2. –  Chris Gerig Jan 31 '11 at 18:41


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# Multiplying by Powers of Ten (B)

##### This Multiplying by Powers of Ten (B) worksheet also includes:

Fifth graders multiply five to six-digit numbers by powers of ten containing negative two exponents. This four-page activity contains approximately 40 problems, with exponents.

Concepts

Resource Details


text:
# Fractional part always diverge right?

Calculus Level 4

Let $$f_n(x)$$ be an $$n$$-th degree polynomial with all real coefficients and a positive leading coefficient.

For a fixed $$n$$, if $$\displaystyle \lim_{x\to\infty} \left\{ \sqrt[n]{f_n(x)} \right\}$$ converge, what can we say about the leading coefficient of $$f_n$$?

Details and Assumptions:

• $$\{x\}$$ denote the fractional part of $$x$$.

Bonus: Denote the polynomial as $$f_n(x) = a_n x^n +a_{n-1} x^{n-1} + \ldots + a_0$$. Prove that we can find the limit in terms of $$a_n$$ and $$a_{n-1}$$.

×


text:
### Book Store

Currently only available for.
CBSE

# JEE Physics 2012 Exam Questions

#### Multiple Choice Questions

1.

A cylindrical tube, open at both ends, has a fundamental frequency, f, in the air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now

• m1r1:m2r2

• m1 :m2

• r1 :r2

• 1:1

C.

r1 :r2

As their period of revolution is same, so its angular speed is also same. Centripetal acceleration is circular path,
a= ω2r
Thus,

211 Views

2.

A liquid in a beaker has temperature θ(t) at time t and θ0 is temperature of surroundings, then according to Newton's law of cooling the correct graph between loge (θ – θ0) and t is

A.

According to Newton's law of cooling.

Hence the plot of ln(θ – θ0) vs t should be a straight line with negative slope.

473 Views

3.

A thin liquid film formed between a U-shaped wire and a light slider supports a weight of 1.5 x10–2N (see figure). The length of the slider is 30 cm and its weight negligible. The surface tension of the liquid film is

• 0.0125 Nm-1

• 0.1 Nm-1

• 0.05 Nm-1

• 0.025 Nm-1

D.

0.025 Nm-1

The force of surface tension acting on the slider balances the force due to the weight.

⇒F = 2Tl = w
⇒2T(0.3) = 1.5 x 10–2
⇒T = 2.5 x 10–2 N/m

369 Views

4.

A particle of mass m is at rest at the origin at time t = 0. It is subjected to a force F (t) = F0e–bt in the x direction. Its speed v(t) is depicted by which of the following curves?

C.

So, velocity increases continuously and attains a maximum value of v= F/mb as t ∞
614 Views

5.

The mass of a spaceship is 1000 kg. It is to be launched from the earth's surface out into free space. The value of 'g'and 'R'(radius of earth) are 10 m/s2 and 6400km respectively. The required energy for this work will be;

• 6.4 x1011 J

• 6.4 x108 J

• 6.4 x109 J

• 6.4 x109 J

D.

6.4 x109 J

Potential energy on the earth's surface is -mgR while in free space, it is zero. So, to free the spaceship minimum required energy is
E =mgR
= 103 x 10 x 6400 x 103 J
= 6.4 x 1010 J

621 Views

6.

A Carnot engine, whose efficiency is 40%, takes in heat from a source maintained at a temperature of 500 K It is desired to have an engine of efficiency 60%. Then, the intake temperature for the same exhaust (sink) temperature must be

• the efficiency of Carnot engine cannot be made larger than 50%

• 1200 K

• 750 K

• 600 K

C.

750 K

Efficiency

690 Views

7.

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.
If two springs S1 and S2 of force constants k1 and k2, respectively, are stretched by the same force, it is found that more work is done on spring S1 than on spring S2.

Statement 1: If stretched by the same amount, work done on S1, will be more than that on S2
Statement 2 : k1 < k2

• Statement 1 is false, Statement 2 is true

• Statement 1 is true, Statement 2 is false

• Statement 1 is true, Statement 2 is the correct explanation for statement 1

• Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for statement 1.

A.

Statement 1 is false, Statement 2 is true

As no relation between k1 and k2 is given in the question, that is why nothing can be predicted about the statement I. But as in Statment II. k1<k2
Then, for same force

698 Views

8.

A spectrometer gives the following reading when used to measure the angle of a prism.
Given that 1 division on the main scale corresponds to 0.5 degree. Total divisions on the vernier scale is 30 and match with 29 divisions of the main scale. The angle of the prism from the above data

• 58.59 degree

• 58.77 degree

• 58.65 degree

• 59 degree

C.

58.65 degree

1035 Views

9.

A diatomic molecule is made of two masses m1 and m2 which are separated by a distance r. If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by (n is an integer)

D.

Rotational kinetic energy of the two body system rotating about their centre of mass is

864 Views

10.

A boy can throw a stone up to a maximum height of 10 m. The maximum horizontal distance that the boy can throw the same stone up to will be

• m
• 10 m

•  m
• 20 m

D.

20 m

Maximum speed with which the boy can throw stone is

Range is maximum when projectile is thrown at angle of 45o, Thus,

254 Views

Do a good deed today
Refer a friend to Zigya


text:
Review question

# Can we write $\sqrt{2}+\sqrt{2}+\sqrt{2}+\sqrt{2}$ as a power of $2$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9624

## Solution

For what value of $x$ is $\sqrt{2}+\sqrt{2}+\sqrt{2}+\sqrt{2}=2^x$ true?

A)$\quad \frac{1}{2}\qquad$ B)$\quad 1\frac{1}{2}\qquad$ C)$\quad 2\frac{1}{2}\qquad$ D)$\quad 3\frac{1}{2}\qquad$ E)$\quad4\frac{1}{2}$

We have that $\sqrt{2}+\sqrt{2}+\sqrt{2}+\sqrt{2}=4\sqrt{2}=2^2\times2^{1/2}=2^{2.5}.$

Thus $x = 2\frac{1}{2}$, and the answer is (C).


text:
Posted by: atri | August 30, 2010

WordPress help

If you do not have a wordpress account, you can get one here.

If you have not used wordpress before, you might need a bit of time to familiarize yourself with the process. (Or maybe not..)

Feel free to start a dummy blog and trying out posts. WordPress support can answer almost any question you might have.

(To come: a quick primer on LaTeX for your math formatting needs.)

A good place to start for formatting LaTeX will be the wordpress LaTeX support page.

If you know LaTeX, you can consider using Luca Trevisan‘s  LaTeX to wordpress (html) convertor.

If you do not know LaTeX, this page has pretty much all the information about math symbols that you will need for this course.

Below are some LaTeX code that you might have to use. (If you have a request for something particular, leave a comment and I will include it below.) To be able to display the code, I have added a space between “\$” and “latex”– if you remove the space, you will get the displayed math that follows.

• Subscript: \$ latex a_b\$ gives $a_b$
• Superscript: \$ latex a^b\$ gives $a^b$


text:
Student
· CoimbatoreIndia
Edit
Delete
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Report
IDEA

I have an idea to create a source of energy(or electicity) in a manner that does not use any kind of fossil fuels to make the generator function…in the sense that the whole generator sustains itself to produce energy mechanically.

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DISCOVER
CHAT


text:
# Connecting to a DB2 database from R

January 16, 2010
By

(This article was first published on The Average Investor's Blog » R, and kindly contributed to R-bloggers)

Unlike RMySQL and RSQLite there is no RDB2. However, I found it pretty straightforward connecting to a DB2 database using the JDBC driver and the RJDBC package.

For all this to work, DB2 should be setup to use TCPIP, which is not used by default. You need to set the DB2COMM DB2 environment variable to TCPIP:

db2set DB2COMM=TCPIP
db2stop
db2start

You also need to make sure that the dbm configuration parameter SVCENAME is set, either to a port number like 50000 or to a string like db2c_DB2.

C:\installed\sqllib\BIN>db2 get dbm cfg | grep SVCENAME
TCP/IP Service name                          (SVCENAME) = db2c_DB2
SSL service name                         (SSL_SVCENAME) =

If SVCENAME (the line starting with “TCP/IP Service name” is not set you can set it to use port 50000, which is the default used by DB2.

db2 set dbm cfg using SVCENAME 50000
db2stop
db2start

You also need to install the RJDBC package.

Then you have to determine the location of the DB2 JDBC driver. If DB2 is installed locally, the driver is already installed and it is located under sqllib\java. For example, I installed DB2 in C:\installed\sqllib, thus the driver I am going to use is C:\installed\sqllib\java\db2jcc4.jar.

Now it’s time to startup R, once the GUI comes up, we can load the package:

library(JDBC)

Then we need to load the DB2 JDBC driver:

jcc = JDBC("com.ibm.db2.jcc.DB2Driver",
"c:/installed/sqllib/java/db2jcc4.jar")

At this point we can establish a database connection:

conn = dbConnect(jcc,
"jdbc:db2://localhost:50000/SAMPLE",

Let’s run a query (to select all records from the employee table) and store the results into a data frame:

rs = dbSendQuery(conn, "select * from employee")
df = fetch(rs, -1)

Until now we were assuming that DB2 is installed. The alternative is to connect to a server running DB2. In this case you only need to install the JDBC driver from the IBM web site – the following video outlines the steps.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...


text:
ASPECT
Changes between version 0.2 and version 0.3

This is the list of changes made after the release of ASPECT version 0.2 and before 0.3. All entries are signed with the names of the author.

1. Changed: To unify spelling, the environment variable that indicates the directory in which deal.II is located is now called DEAL_II_DIR instead of the old DEAL_DIR. The old spelling is still accepted, however.
(Wolfgang Bangerth, 2013/05/31)

2. Fixed: Fix a bug in adiabatic initial conditions introduced in r1599.
(Juliane Dannberg, 2013/05/08)

3. New: The doxygen generated documentation of ASPECT now automatically has references to all of the deal.II classes used.
(Wolfgang Bangerth, 2013/05/04)


text:
## February 27, 2005

### Cole Cr#%

Juan Cole is an indispensable source of information and insight into the situation in Iraq. He really knows the country, and collates an impressive amount of information from both the Western and Arabic Press.

His blog is compelling reading, but it would be that much more compelling, if it were not laced with tendentious crap like

AP reports that the one-day total for war-related violence in Iraq, including the police station bombing in Tikrit reported here yesterday morning, came to 30. That is about 11,000 persons a year if the rate were constant and extrapolated out.

Do you really want to take a one-day death count (and an anomalously high one, at that) and extrapolate that forward for a year? Do you think the resulting number has any meaning?

With a couple of minutes more work, you could take a 5-day running average death count (as reported on his blog and cross-referenced with iraqbodycount) and extrapolate that number forward. Isn’t 4745 deaths/year bad enough?

Given the inadequacies of the reportage, it’s likely to be an underestimate. But, at least, it won’t be instantly dismissed — by any reader with an ounce of sense — as a statistical fluctuation.

Posted by distler at 9:45 AM | Permalink | Followups (2)

### Conversations with Greg

Greg Moore was in town for a few days, and we had — as always — some very interesting discussions. Among the topics was his recent paper with Dabholkar, Denef and Pioline.

I’ve talked before about the Ooguri-Strominger-Vafa proposal relating the entropy of a charged $N=2$ black (which appears as a nontrivial solution of type-IIA strings compactified on some Calabi-Yau, $M$) to the topological string partition function for the same Calabi-Yau. Specifically,

(1)

$\Omega(p,q) = \int d\phi \left|e^{i\frac{\pi}{2}F(p+i\phi,256)}\right|^2 e^{\pi q\cdot\phi}$ where $(p^I,q_I)$, $I=0,\dots,h^{1,1}(M)$ are the electric and magnetic charges, $F(X^I,\lambda^2)$ is the holomorphic topological string free energy, and $\Omega$ is a microcanonical partition function — the number, or perhaps some index, of the number of states of charge $(p,q)$.

1. What contour of integration should be chosen (if one exists) so that the integral is well-defined?
2. Exactly what is $\Omega(p,q)$ counting?
3. Is the formula right?
Posted by distler at 1:59 AM | Permalink | Followups (1)

## February 23, 2005

### Reinvention

Back in second grade, I was dissatisfied with the algorithm we were being taught for doing subtraction. So I “invented” my own

363
– 185

5 is bigger than 3, so we subtract them in the opposite order (5–3=2) and take the tens-complement (8) of the result. As usual, we borrow from the 6 (which becomes a 5) and we repeat: 8–5=3 and take the tens-complement (7). Finally 2–1=1, so the answer is 178.

While only slightly different from the conventional algorithm, I felt this one to be an improvement because I never had to know how to subtract from numbers larger than 10 (e.g. who cares what 13–5 is?).

I haven’t thought much about this little juvenile act of rebellion until a couple of months ago, when I was going over my daughter’s 3rd grade math homework with her. She was doing similar subtraction problems. But, in keeping with the times, she was charged with explaining her methods for arriving at the answer.

Imagine my surprise when she explained her method to me. It was exactly the same “unconventional” algorithm that I had used when I was her age. It was not what the teacher had taught; she had figured it out on her own.

[Her method was the same, but her accuracy was not the greatest. So I taught her the other trick that I learned in that era: check your work by doing arithmetic modulo 9: 178=1+7+8= 7 mod 9, 185=1+8+5=5 mod 9. So 178+185=5+7=12=1+2 = 3 mod 9, which agrees with 363=3+6+3=3 mod 9.]

Now, I don’t know what this has to do with Larry Summers’ remarks on the dearth of women in the Hard Sciences (at least in this country). My personal experience echoes that of the AIP Study. An alarmingly large majority of the women who arrived at Harvard the year I did, intending to major in Physics, had decided by sophomore year to do something else. As a consequence, it was unsurprising that, by the time I started graduate school, there was only one woman in an entering class of 28. Sean Carroll takes on the thankless task of confronting Summers hypotheses with the data. I’m afraid I can’t muster the energy.

I’m much too busy trying to nurture that spark of creativity in my daughter, hoping that, a decade from now, she doesn’t face the stark choice that my classmates at Harvard/Radcliffe faced a generation ago.

Posted by distler at 4:28 PM | Permalink | Followups (6)

## February 17, 2005

The last straw was when I received a Korean trackback, encoded in euc-kr.

The Trackback Specification makes no mention of character encodings, and MovableType’s original implementation was blissfully ignorant of any such notion. The sender of a Trackback ping sent a string of bytes (which represented a string of characters in charset of his blog) and the recipient dutifully published that string of bytes on his blog. If the recipient’s charset happened not to be the same as that of the sender, well, then, the result was gibberish.

The most recent versions of MovableType convey the sender’s charset in the HTTP headers of the Trackback. But the recipient doesn’t actually do anything with the information.

As a result, I had a slowly increasing number of gibberish Trackbacks on my blog, with no end in sight.

If you want something done right …

Posted by distler at 1:04 PM | Permalink | Followups (18)

## February 16, 2005

### Coming Soon to a Hard Drive Near You

Once upon a time, a megabyte was a lot of data. In 1989, when Joanne Cohn first started emailing preprints to a couple of hundred colleagues, people quickly found themselves exceeding their mail quotas. And not everyone was interested in every paper. So why waste bandwidth and precious disk space on all that junk?

The idea of centralizing the storage, and sending people only the papers they requested, prompted Paul Ginsparg to start the hep-th archive in 1991.

Flash forward 14 years.

250 GB hard drives are cheap, and a laptop with less than 60 GB seems positively claustrophobic. Hep-th has grown tremendously. But even with over two hundred submissions a month, it’s still a puny amount of data by today’s standards. The entire archive from 1991 to the present (8GB of PDF files) fits easily on an iPod.

So, in reversal of history, Joanna Karczmarek is gearing up to distribute the whole shebang via bittorent. She’s currently offering 2004 (pdf papers and a plain-text list of abstracts) as a modest 850 MB torrent.

A year’s worth of physics for your iPod Shuffle.

Posted by distler at 3:27 PM | Permalink | Followups (6)

## February 13, 2005

### Words to Live By

“Don’t fire a gun while you’re driving a car.”
— my 9 year old daughter, admonishing her 4 year old brother

## February 11, 2005

### Berkovits Update

I’ve spent the past few days in a lengthy email correspondence with Nathan Berkovits about some of the questions raised in my previous post about his “pure spinor” approach to the superstring.

The main progress has been in clarifying his construction of the bilocal operators, $\hat{b}(y,z)$, which play the role of anti-ghosts in his theory. Recall that I was more than a little worried about the apparent $z$-dependence of the amplitudes.

## February 8, 2005

### MathML News

“Not another one!” I hear you groan.

Posted by distler at 10:34 PM | Permalink | Followups (6)

## February 5, 2005

### Who Do You Trust?

The debate over Google’s new rel="nofollow" attribute for “untrusted” links continues to simmer. I explained our (Musings and the String Coffee Table’s) policy a while back. Trackbacks and Comment-Author Links are innoculated with rel="nofollow".

Comment-Author Links of PGP-signed comments are exempt from the rel="nofollow" policy.
So … go generate yourself a PGP key, put your public key on your website (make sure it’s served right), start signing your comments here, and watch your PageRank soar.

Well, OK, maybe not the last one. But this is a wee bit more incentive to do what you should be doing anyway.

Posted by distler at 12:41 AM | Permalink | Followups (11)

## February 4, 2005

### Multiloop Amplitudes

Surprising to say, at this late date, but there’s been considerable recent progress in multiloop string perturbation theory.

D’Hoker and Phong have a pair of new papers, looking at genus-2 scattering amplitudes. I’ve written about their previous work in some detail. The current papers extend their story to N-point functions at genus-2.

Meanwhile, prodded by Luboš who, in his weblog post and privately, has been championing Nathan Berkovits’s pure-spinor approach to the covariant Green-Schwarz superstring, I decided to take a closer look.

The action, in Nathan’s theory looks deceptively simple:

(1)$S = \int d^2z[-\textstyle{\frac{1}{2}} \partial x^m\overline{\partial} x_m - p_\alpha\overline{\partial}\theta^\alpha - \tilde{p}_\alpha\partial\tilde{\theta}^\alpha + w_\alpha\overline{\partial}\lambda^\alpha + \tilde{w}_\alpha\partial\tilde{\lambda}^\alpha ]$

The only wrinkle is that the commuting ghost fields $\lambda^\alpha$, $\tilde{\lambda}^\alpha$ obey a pure-spinor condition

(2)$\lambda^t \gamma^\mu \lambda = 0,\qquad \tilde{\lambda}^t \gamma^\mu \tilde{\lambda} = 0$

so, despite appearances, this is not a free field theory. For compatibility with the pure-spinor constraint, the antighosts, $w_\alpha$, have a gauge-invariance

(3)$w_\alpha \to w_\alpha + \Lambda^m(\gamma_m\lambda)_\alpha$

The “BRST operator” is

(4)\array{\arrayopts{\colalign{right left}} Q & =\oint \lambda^\alpha d_\alpha\\ d_\alpha & = p_\alpha -\frac{1}{2}\gamma^m_{\alpha\beta} \theta^\beta \partial x_m - \frac{1}{8}\gamma^m_{\alpha\beta}\gamma_{m\gamma\delta} \theta^\beta\theta^\gamma\partial\theta^\delta }

Because of the gauge-invariance, however, $w_\alpha$ can only appear in gauge-invariant combinations like

(5)$N_{m n} = \frac{1}{2} w_\alpha (\gamma_{mn})^\alpha_\beta \lambda^\beta,\qquad J= w_\alpha \lambda^\alpha$

and correlation functions involving these objects (and the $\lambda$s), says Nathan, can be computed using free fields. Unfortunately, there’s no candidate (composite) local operator, $b$, which satisfies $\{Q,b\}= T$. Instead, Nathan has a rather strange prescription to contruct a bilocal operator, of ghost number zero, which satisfies

(6)$\{Q,\hat{b}(y,z)\} = T(y)Z_B(z)$

where

(7)$Z_B = \frac{1}{2} B_{mn} \lambda\gamma^{mn}d \delta (B N)$

for some constant antisymmetric tensor $B$. Aside from the strange bilocality, we by construction break the Lorentz-invariance in our definition of the $\hat{b}$s.

Similarly, the dimension-(1,1) (integrated) vertex operators are not built from the dimension-0 BRST cohomology, $V$, by acting with $b$. Instead, they’re constructed in an ad-hoc way as ghost-number zero fields satisfying $[Q,U]= \partial V$. And, in order to define the amplitudes, one needs a plethora of further insertions of non-Lorentz-invariant “Picture-changing Operators” (of which $Z_B$ above was an example).

All of these various sources of non-Lorentz-invariance, says Nathan, only change the integrand by surface terms. And, if you use a certain prescription for integrating over the zero modes of the $\lambda$s (remember, it’s a nonlinear space), all will be OK.

As you can tell, I have many, many questions about this — very interesting — proposal. But I’ll close with four:

1. Is it true that

(8)$\partial_z \hat{b}(y,z) = [Q,\cdot]$

(as surely is required for a sensible amplitude)? The expression for $\hat{b}(y,z)$ is deucedly complicated, and I can’t see why this is true.

2. The usual relation that $b_{-1}\overline{b}_{-1}$ acting on the dimension-0 (fixed-location) vertex operator gives you the dimension-(1,1) (integrated) vertex operator is crucial to the proof of unitarity of multiloop amplitudes (so crucial, that we rarely think about it). What replaces that here?
3. The unphysical poles that one encounters in multiloop NSR amplitudes when one naïvely uses the picture-changing formalism is a consequence of the index theorem applied to the bosonic ghosts (not, as implied in footnote 12, of bosonizing those ghosts). One might worry that similar poles arise here.
4. Is it really true that the only legacy of the nonlinear nature of the pure-spinor constraint is in the zero-mode integration?
Posted by distler at 12:30 PM | Permalink | Followups (5)

## February 1, 2005

The group of spammers I blogged about previously, the ones using crapflooding techniques (multiple POSTs from behind anonymous proxies) for comment spam, finally returned, this time as Trackback spammers. Lotta people seem to have been hit hard.

Since it took the Crapflooders only a week or so to figure out that Trackback flooding was easier and more fun than Comment flooding, I was wondering when these spammers would come to the same realization. Last night, they finally did. Golem received several hundred trackback attempts, in two concerted waves.

When the crapflooders were at it, a throttle on the number of trackbacks in a given time-period was my main defence. That throttle is now built-into MT 3.1x.

Since then, I’ve wised up a bit, and block submissions (of Trackbacks and Comments) from open HTTP Proxies. Thanks to Brad Choate’s plugin, modified to use the opm.blitzed.org DNSBL list of open Proxies (instead of the irrelevant dsbl.org list of open SMTP servers), all of the hundreds of would-be spam Trackbacks were blocked.

The modifications to Brad’s plugin are easy,

--- plugins/dsbl_deny.pl.orig   Thu Nov 11 11:06:29 2004
+++ plugins/dsbl_deny.pl        Thu Nov 11 11:08:58 2004
@@ -12,9 +12,9 @@
my ($eh,$app, $comment) = @_; my$remote_ip = $app->remote_ip; my ($a, $b,$c, $d) = split /\./,$remote_ip;
-    if (checkdnsrr("$d.$c.$b.$a.list.dsbl.org")) {
+    if (checkdnsrr("$d.$c.$b.$a.opm.blitzed.org")) {
$app->log("Blocked comment post from known open proxy:$remote_ip");
-        my $url = "http://dsbl.org/listing?$remote_ip";
+        my $url = "http://opm.blitzed.org/proxy?ip=$remote_ip";

# we're forcing out the header here and exiting since I can't find
# a cleaner way to force a redirection to the DSBL.org site...

The only surprising thing was how well it performed.

#### Update (2/2/2005):

Zack is, alas, correct. My “internal working version” is a little more heavily hacked than I let on (or even remembered). So, pending Brad releasing a new version of his plugin, here’s a (slightly neatened-up) canned replacement which filters both Comments and Trackbacks.

#### Update (2/4/2005):

Ever the scientist, I decided to check whether the success of the opm.blitzed.org open proxy list in covering the particular proxies used by these spammers in their recent Trackback Spam runs was due to dumb luck or to genuine comprehensiveness. So I decided to look up a much larger sample of IP addresses, used in recent weeks by these lowlifes for referrer spam and their (feeble attempts at) comment spam. The result is that opm.blitzed.org lists only about half of those IP addresses1. In other words, I got lucky2.

So I’ve begun to deploy some other countermeasures against them, which I will surely write about anon. In the meantime, there’s a wee buglet in the plugin I posted the other day. If you downloaded it, please download it again.

1 Looking back at a weeks-old list of proxies may not be a reliable measure. Many of these may once have been open, but are now closed and delisted. So this surely understates the effectiveness of the Blitzed list. By how much is hard to tell.

2 In case you’re wondering about the Central Limit Theorem, a spam run seems to use about a dozen different proxies, but these are likely not uncorrelated. If the spammers use trojanned PCs for their spam run, none of them will show up on the Blitzed list. Conversely, if they happen to use open proxies, which have previously been used to connect to certain IRC channels (monitored by the BOPM), they all will appear.

Posted by distler at 5:57 PM | Permalink | Followups (16)


text:
Generates a SLiM script from a built-in slendr template, substituting all required parameters by values specified by the user. The user has an option to either run the script in batch mode, open it in SLiMgui, or simply save the generated SLiM script to a given location for later use.

slim(
model,
seq_length,
recomb_rate,
output_dir = model\$path,
output_prefix = "output",
save_locations = FALSE,
track_ancestry = FALSE,
ts_recording = FALSE,
sampling = NULL,
method,
verbose = TRUE,
include = NULL,
burnin = 0,
seed = NULL,
slim_path = NULL,
overwrite = FALSE
)

## Arguments

model Model object created by the compile function Total length of the simulated sequence (in base-pairs) Recombination rate of the simulated sequence (in recombinations per basepair per generation) A directory where to put simulation outputs (by default, all output files are placed in a model directory) A common prefix of output files (by default, all files will share a prefix "output") Save location of each individual throughout the simulation? Track ancestry proportion dynamics in all populations throughout the simulations (default FALSE)? If a non-zero integer is provided, ancestry will be tracked using the number number of neutral ancestry markers equal to this number. Turn on tree sequence recording during SLiM initialization? How to run the script? ("gui" - open in SLiMgui, "batch" - run on the command-line, "script" - simply return the script) Write the SLiM output log to the console (default FALSE)? Vector of paths to custom SLiM scripts which should be combined with the backend SLiM code Length of the burnin (in model's time units, i.e. years) Random seed (if missing, SLiM's own seed will be used) Optional way to specify path to an appropriate SLiM binary Overwrite the contents of the output directory (default FALSE)? A data frame of times at which a given number of individuals should be remembered in the tree-sequence (see sampling for a function that can generate the sampling schedule in the correct format). If missing, only individuals present at the end of the simulation will be recorded in the tree-sequence output file.


text:
# News for September 2014

It is unfortunate that in the month of september we had very few papers in theory that appeared in ECCC or Arxiv. And of them I could not find any paper that falls in the category of property testing / query complexity or sublinear algorithms.  It is possible I have overlooked some papers. Please do point out any paper that you may think relevant that appeared in the month of September.

# New for August 2014

We’ve got a nice collection of papers this August, ranging from graph property testing, locally testable codes, to new lower bound approaches.

Deconstructions of Reductions from Communication Complexity to Property Testing using Generalized Parity Decision Trees by Roei Tell (ECCC). Blais, Brody, and Matulef (BBM) introduced the beautiful and powerful method of using communication complexity to derive property testing lower bounds. This result is all about understanding this approach is greater depth, and really trying to get at the crux of the reductions. The BBM approach uses rather heavy hammers (communication complexity lower bounds), so could we get away with weaker hammers to get property testing lower bounds? The key insight is that less powerful models can be reduced to existing property testing problems, and these are equivalent to the BBM reductions. While this (as of now) does not lead to newer bounds, it does shed significant light on existing bounds. The weaker model is essentially that of linear decision trees. Such reductions were also discovered by Brushundi, Chakraborty, and Kulkarni, proving $$\Omega(k)$$ lower bounds for testing $$k$$-linearity.

Locally Correctable and Testable Codes Approaching the Singleton Bound by Or Meir (ECCC). Locally Testable Codes (LTCs) are codes that have a constant (or polylogarithmic) time tester. It is known that such codes with minor caveats cannot have a constant rate (a number of papers on this topic: Katz and Trevisan, Dinur and Kaufman, Ben-Sasson and Viderman). But when the query complexity can be $$n^\beta$$ (where $$n$$ is the codeword size), the rate can be arbitrarily close to $$1$$ (Viderman, and more refs in the paper). This result takes this fact to the very extreme. The Singleton Bound is a lower bound between the tradeoff between rate and relative distance of a code. Amazingly, this lower bound tradeoff can be achieved by an LTC (and also Locally Correctable Code) testable with $$n^\beta$$ queries!

Complexity of Nondeterministic Graph Parameter Testing by Marek Karpinksi and Ronald Markó (arXiv). It is easiest to think of nondeterminstic graph testing with the example of testing bipartiteness for dense graphs. Suppose we wish to test $$G$$ for bipartiteness. As a “certificate”, we are provided a candidate partition $$V = V_1 \cup V_2$$. There is a “simple” constant-time tester that can property testing if $$G$$ is bipartite with respect to this partition. (Just sample two uniform random vertices, and check if the edge between them respects the bipartition.) So we say that bipartiteness is non-determinstically testable: if $$G$$ is far from being bipartite, no certificate can convince the simple tester. The full definition is somewhat complex, but it is basically the same idea. (In this context, “testablility” always refers to constant time testable.) Lovász and Vestergombi proved the striking result that any non-deterministically testable property is also deterministically testable, but their proof did not give an explicit bound on the query complexity. Gishboliner and Shapira fixed that, and gave a tower-type bound for the query complexity. So if the query complexity of the underlying “simple” tester is $$q$$, then the non-deterministic property has a tester whose query complexity is a tower of height $$q$$. This results goes much further, and shows the final query complexity is at most triply exponential in $$q$$. This is quite significant, since it is uses a weaker regularity approach than previous results.

# News for July 2014

In this month’s post, we review papers that raise two questions that I find particularly interesting. First, can sublinear algorithms find applications in efficient approximation of CSPs? And second, can property testing be extended to give meaningful results in settings where Hamming distance is not the “right” measure of distance between functions?

Going for Speed: Sublinear Algorithms for Dense r-CSPs by Grigory Yaroslavtsev (arXiv). In their seminal paper, Goldreich, Goldwasser, and Ron considered the MAX-CUT problem. They showed that both the testing variant of the problem (where the algorithm must distinguish between graphs with a cut of size $$s$$ from those with maximum cut size at most $$s – \epsilon N^2$$) and the approximation variant (where the algorithm must find a cut that has at most $$\epsilon N^2$$ fewer edges than the largest cut in the graph) can be solved in sublinear time. This result inspired the very successful line of research on testing properties of graphs. We can also view this result as motivation for studying the approximability of general CSPs: given some constraint satisfaction problem, can we find an assignment that satisfies almost as many constraints as the maximally satisfying assignment in sublinear time? This current paper begins the exploration of this question with an improved bound on the query complexity of the MAX-CUT problem and with the first general result for approximating CSPs of arity greater than 2 (i.e., where the constraints are on more than 2 variables).

Lp testing by Piotr BermanSofya Raskhodnikova, and Grigory Yaroslavtsev (preprint). Almost all of the work in testing properties of functions is based on the Hamming distance: for a given property P, a tester is required to accept functions that have the property P and to reject the functions that differ from every function with the property P on an $$\epsilon$$ fraction of the inputs. This notion of distance is appropriate for algebraic properties of functions and for properties of boolean functions, but for other properties of real-valued functions, $$L_p$$ distances are sometimes more appropriate. This paper provides the first systematic study of property testing under these distances. An important high-level message from this work is that the testing problems can be very different under these distances than under Hamming distance. For example, the classic problem of $$\epsilon$$-testing whether $$f : [n] \to \mathbb{R}$$ is monotone requires $$\Theta(\frac{\log n}{\epsilon})$$ queries in the Hamming distance setting, but only $$\Theta(\frac1{\epsilon^p})$$ queries in the $$L_p$$ distance setting!

# News for June 2014

Unfortunately the month of June was a very dry month for property testing. We could not find any papers on property testing (in ECCC or ArXiv) during the month of June. Hopefully we will have lots of papers in the month of July.

# News for May 2014

May seems to be a slow month for property testing, with only one paper on locally testable codes. Nonetheless, we have unearthed a paper from April (missed despite our best efforts) on a related topic of hidden set approximations.

The Power of an Example: Hidden Set Size Approximation Using Group Queries and Conditional Sampling by Dana Ron and Gilad Tsur (arXiv). Consider a known universe $$U$$ and an unknown subset $$S$$. Our aim is to approximately determine $$|S|$$. We can perform subset queries: for a query subset $$T$$, an oracle returns whether $$T \cap S \neq \emptyset$$ or (a more powerful model) the oracle returns a uniform random element in $$T \cap S$$. This paper gives a detailed study of this problem under various settings. There is a fascinating array of upper and lower bounds, including situations where $$U$$ is ordered and $$T$$ must be an interval, the difference between adaptivity and non-adaptivity, arbitrary subset queries, and much more. The topic seems to be quite rich, and it should yield some nice sets of problems to study further. It appears that some lower bounds are quite open. Students, pay attention!

Limitations on Testable Affine-Invariant Codes in the High-Rate Regime by Venkatesan Guruswami, Madhu Sudan, Ameya Velingker, and Carol Wang (ECCC). Locally Testable Codes (LTCs) are codes that (typically) have a constant time tester. Interestingly, recent work on testing codes using a linear number of queries (some small constant fraction of the code length) has connections to the Small Set Expansion (SSE) problem, which in turn is related to the famous Unique Games Conjecture (UGC). A paper by Barak et al. gives a construction of a small set expander (a graph where all small sets have high conductance) where the Laplacian has many small eigenvalues (so it’s “difficult” to find a low conductance cut). This construction uses high-rate LTCs that are linear time testable. In some sense, the better the rate, the better construction one gets. This paper asks how far this idea can go. So the idea is to construct the best rate linear-time testable LTC. Reed-Muller codes fall in this category, but their rate is quite far from optimal. Recent work by Guo et al. gives a slightly better construction, but this is far from the (near) optimal rate achieved by BCH codes (which are not known to be testable). All in all, this paper shows that for affine-invariant codes, the Guo et al. construction is essentially the best testable high-rate code one can get. I have obviously done no justice to the depth of the connections, the intricacies of the parameters, or the overall coolness of this subject. So go read the paper!

# News for April 2014

As winter finally releases its grip here on the east coast, we celebrate with forests, with bounded-derivative properties, and with hypergraphs. (I believe these are the traditional ingredients for the beginning-of-spring celebrations in New England, but I could be wrong.)

Testing Forest-Isomorphism in the Adjacency List Model by Mitsuru Kusumoto and Yuichi Yoshida (arXiv). Which properties of sparse graphs can we test efficiently? To a large extent, this fundamental question remains open. The natural setting in which to study this question is the adjacency list query model. In this model, a testing algorithm can select any vertex $$v$$ and index $$i$$; it then receives the identity $$w$$ of the $$i$$th neighbor of $$v$$. Kusumoto and Yoshida show that in this model, we can test if two forests (i.e., collections of trees) on $$n$$ vertices are identical up to relabeling of the vertices with polylog($$n$$) queries. They also show that, remarkably, this tester can be used to test any property of forests in the adjacency list model with polylog($$n$$) queries.

Property Testing on Product Distributions: Optimal Testers for Bounded Derivative Properties by Deeparnab Chakrabarty, Kashyap Dixit, Madhav Jha, and C. Seshadhri (arXiv). Two basic properties of a function $$f : [n]^d \to \mathbb{R}$$ on the hypergrid are monotonicity and the Lipschitz property. These two properties are special cases of a more general class of properties called bounded-derivative properties. In this paper, the authors give optimal bounds on the number of queries required to test these properties over every product distribution on the hypergrid. These results are obtained by deriving new dimension-reduction results and, most interestingly, by establishing and exploiting a strong connection between bounded-derivative properties and binary search trees.

Any Monotone Property of 3-uniform Hypergraphs is Weakly Evasive by Raghav Kulkarni, Youming Qiao, and Xiaoming Sun (ECCC). What happens if we remove the distance promise in the definition of property testing? Or, equivalently, how efficient can testers be if they must distinguish objects with some property $$P$$ from every object that does not have the property $$P$$? The answer to this question seems completely obvious: no non-trivial property can be tested in this setting with a sublinear number of queries. We can easily verify that this answer is indeed correct for our favorite properties… but is it really true for every non-trivial property? As it turns out, this question is far from trivial, and in fact leads to the famous evasiveness conjectures and to the celebrated results of Rivest and Vuillemin and of Kahn, Saks, and Sturvevant. The current paper combines and extends ideas from both of these papers to show that deterministic testers for any monotone non-trivial property of 3-uniform hypergraphs indeed have linear query complexity.

# News for March 2014

On Learning and Testing Dynamic Environments by Oded Goldreich and Dana Ron (ECCC). Given a dynamic environment that is evolving according to a local rule, the goal is to query some of the states in the system at some point of time and determine if the system is evolving according to some fixed rule or is far from it. A harder problem is to learn the evolution of the environment. They give some upper and some lower bounds for various versions of these problems. In particular they show that using sub-linear queries a lot can be achieved even in this setting. This is the first of its kind result and possibly many more results in this area will follow soon.

# News for February 2014

So what did we see in February 2014?

Local Algorithms for Sparse Spanning Graphs by Reut Levi, Dana Ron, and Ronitt Rubinfeld (arXiv). Given a graph $$G = (V,E)$$, the aim is the find a small sparse spanning subgraph $$G’$$ in the local algorithm setting. So, given any edge $$(u,v) \in E$$, we must determine in sublinear time whether this edge exists in $$G’$$. The main result is an $$\Omega(|V|)$$ lower bound, and a matching algorithm when $$G$$ is either an expander or hyperfinite.

A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems by Yuichi Yoshida (arXiv). This result gives a complete characterization of (two-sided) locally testable affine-invariant properties. These are based on existing decomposition theorems for functions into structured and pseudorandom parts. This result is analogous to that of Alon, Fischer, Newman, and Shapira (link) on characterizations for testability over dense graphs.

Testing probability distributions underlying aggregated data by Clément Canonne and Ronitt Rubinfeld (arXiv). There has been much recent work on distribution testing, and this work discusses stronger models where better algorithms can be obtained. Given a distribution $$D$$ over $$[n]$$, we can get samples from this distribution, and also query the probability $$D(i)$$ of any $$i \in [n]$$. With these “dual” queries, one gets distribution testers for a variety of problems surpassing previous lower bounds. Interestingly, this is stronger than the conditional model introduced recently (here and here).

Strong Locally Testable Codes with Relaxed Local Decoders by Oded Goldreich, Tom Gur, and Ilan Komargodski (ECCC). For a locally testable code (LTC), the property of being a codeword is efficiently (constant time) testable. A locally decodable code (LDC) is a code from which individual bits of the original message can be recovered in constant time. A relaxed local decoder is allowed to declare failure for recovered a small fraction of message bits. This allows for bypassing the difficult (and prominent) open problem of constructing small LDCs. This result gives a construction of near-linear LTCs with a relaxed local decoder. This result leads to a property that is hard to test but easy to verify (in the MA proofs of proximity sense, refer to last month’s open problem).

# Open problem for February 2014: Better approximations for the distance to monotonicity

Slightly delayed, Feb’s open problem is by one of the three possible “yours truly”s, Sesh. Looking for more reader participation, hint, hint.

Basic setting: Consider $$f:\{0,1\}^n \rightarrow R$$, where $$R$$ is some ordered range. There is a natural coordinate-wise partial order denoted by $$\prec$$. The function is monotone if for all $$x \prec y$$, $$f(x) \leq f(y)$$. The distance to monotonicity, $$\epsilon_f$$ is the minimum fraction of values that must be changed to make $$f$$ monotone. This is an old problem in property testing.

Open problem: Is there an efficient constant factor approximation algorithm for $$\epsilon_f$$? In other words, is there a $$poly(n/\epsilon_f)$$ time procedure that can output a value $$\epsilon’ = \Theta(\epsilon_f)$$?

State of the art: Existing monotonicity testers give an $$O(n)$$-approximation for $$\epsilon_f$$, so there is much much room for improvement. (I’d be happy to see a $$O(\log n)$$-approximation.) Basically, it is known that the number of edges of $$\{0,1\}^n$$ that violate monotonicity is at least $$\epsilon_f 2^{n-1}$$ [GGLRS00], [CS13]. A simple exercise (given below) shows that there are at most $$n\epsilon_f 2^n$$ violated edges. So just estimate the number of violated edges for an $$O(n)$$-approximation.
(Consider $$S \subseteq \{0,1\}^n$$ such that modifying $$f$$ on $$S$$ makes it monotone. Every violated edge must have an endpoint in $$S$$.)

Related work: Fattal and Ron [FR10] is a great place to look at various related problems, especially for hypergrid domains.

References

[CS13] D. Chakrabarty and C. Seshadhri. Optimal Bounds for Monotonicity and Lipschitz Testing over Hypercubes and Hypergrids. Symposium on Theory of Computing, 2013.

[FR10] S. Fattal and D. Ron. Approximating the distance to monotonicity in high dimensions . ACM Transactions on Algorithms, 2010.

[GGL+00] O. Goldreich, S. Goldwasser, E. Lehman, D. Ron, and A. Samorodnitsky. Testing Monotonicity . Combinatorica, 2000.


text:
# Cardinal characteristics

Assuming Continuum hypothesis is not true, How many cardinals $k$ exist which are $\aleph_1 < k < \mathfrak c$? Can I assume that there is a finite number of these cardinals or is there an infinite number (like ordinals). Is there a common topology on these cardinals (like, for example, the order topology on ordinals)?

-
I don't see why this is about topology. If this is simply a question "For what values of $\alpha$, it is consistent to have $2^{\aleph_0}=\aleph_\alpha$?", then we have an answer for that already. If you are trying to understand the diagram, note that hardly any inequality in the diagram is strict. So you might be asking how many of these can have different values, or you might be asking if it is consistent that each node in the diagram you added has a different value. But the text of the question still asks the very simple question about the possible values of $\frak c$. – Asaf Karagila Sep 1 '14 at 14:23

The number of cardinals between $\aleph_1$ and $\mathfrak c$ can be anything. In particular, it can (consistently with ZFC) be any finite number you like, or it can be countably infinite, or it can be uncountably infinite. It can even be $\mathfrak c$ itself. These facts come from a paper of Solovay, "$2^{\aleph_0}$ can be anything it ought to be," written very soon after Cohen proved the independence of CH.


text:
# Trapezoid rule-esque integration with unevenly spaced n-D grid

I have a function (or rather many functions) defined over an n-dimensional grid. The grid points will often but not always be evenly spaced. I want to compute integrals of these functions numerically.

My question now is what the best approach to this is. In 1-D I can always works with the trapezoid rule, e.g.:

chapeaux[
grid_List?(VectorQ[#, InternalRealValuedNumericQ] &),
f_List?(VectorQ[#, InternalRealValuedNumericQ] &)
] :=
Module[
{
diffs = Differences[grid],
means = MovingAverage[f, 2]
},
diffs.means
];

And here are some simple tests:

gigi = Range[-10., 10., .1];
grigri = Sort@BlockRandom[RandomReal[{-10. , 10.}, 100*Length@gigi]];
sinf = Sin@gigi;
sinfr = Sin@grigri;

NIntegrate[Sin[x], {x, -10, 10}] // RepeatedTiming

{0.0049, 0.}

chapeaux[gigi, sinf] // RepeatedTiming

{0.000066, -8.32667*10^-17}

chapeaux[grigri, sinfr] // RepeatedTiming

{0.00070, 0.00164231}

harm = gigi^2;
harmr = grigri^2;

NIntegrate[x^2, {x, -10, 10}] // RepeatedTiming

{0.0034, 666.667}

chapeaux[gigi, harm] // RepeatedTiming

{0.000068, 666.7}

chapeaux[grigri, harmr] // RepeatedTiming

{0.000694, 666.281}

And sure in the unstructured case I need about 100X as many points to even get okay-ish accuracy, but that's not the end of the world, so now the question is how best to do this or something similar in n-dimensions. Of course, the functions I'm using have no analytic analog and so the best I can do is NIntegrate the Interpolation, but I'm hoping to do this faster and cleaner.

At some level this is mostly a math question, of course, but I figured it's worth asking here before implementing as the accumulated math knowledge here is much greater than my own.

• Using NIntegrate over an ElementMesh as done here, does a simple integration rule without adaptive refinement. – Michael E2 Jan 7 '19 at 22:40
• @MichaelE2 any idea how fast that can be? I'd be totally down to just let NIntegrate do it all for me but I might need to this to be fast in the future. – b3m2a1 Jan 7 '19 at 22:42
• @MichaelE2 follow up: can that be done without Interpolation or is that basically a necessity in methods like this? – b3m2a1 Jan 7 '19 at 22:43
• @b3m2a1 By "grid", do you mean a tensor product grid? – Henrik Schumacher Jan 7 '19 at 22:49
• @HenrikSchumacher unfortunately no. I have points in space that won't necessarily have come from a tensor product...although that is my dominant use case (although with different mesh spacings in different dimensions) so if you have something clean for that I'd be happy to see that too. – b3m2a1 Jan 7 '19 at 22:51

I would suggest Tai's method! ;)

Okay, there are two issues here with the trapezoidal rule:

1. It is crucial that the interval endpoints are included in the grid.
2. Trapezoidal rule with uniform subdivision is exact of all odd functions just because the quadrature is symmetric. Unfortunately, you tested with Sin which is an odd function. This made the trapezoidal rule look way more potent than it is.

This is a fairer comparison:

f = Function[x, Sin[x] + x^2 + x/2]
gigi = Range[-10., 10., .1];
grigri = Join[{-10.}, Sort@BlockRandom[RandomReal[{-10., 10.}, Length@gigi - 2]], {10.}];
sinf = f@gigi;
sinfr = f@grigri;

a = NIntegrate[f[x], {x, -10, 10}];
b = chapeaux[gigi, sinf];
c = chapeaux[grigri, sinfr];

Abs[a - b]/Abs[a]
Abs[a - c]/Abs[a]

0.00005

0.000259904

The error of the trapezoidal ruled for the grid points $$x_0,\dotsc,x_n$$ should be be bounded by

$$\sup_{x\in I} |f''(x)| \, \cdot \sum_{i=1}^n |x_{i}-x_{i-1}|^2.$$

So uniform grids optimize this bound by minimizing the sum of squares of the subintervals.

• I see. Any hope if I can't have a uniform grid? – b3m2a1 Jan 7 '19 at 23:14
• If you know that data belongs to a function with bounded 4th derivative, you may obtain better accuracy with Simpson's 3/8 rule. – Henrik Schumacher Jan 7 '19 at 23:17
• For unstructured grid upto, 3 dimension, one may employ also a "trapezoidal rule" by (i) computing DelaunayMesh and employing the finite-element mass matrix for piecewise-linear functions on that mesh. But I am not aware of any integration rules with higher precision on totally unstructured grids. And for unstructured grids of dimension $d \geq 4$, I am totally clueless. – Henrik Schumacher Jan 7 '19 at 23:33


text:
## Gibbs Free Energy Ionization

$\Delta G^{\circ}= \Delta H^{\circ} - T \Delta S^{\circ}$

$\Delta G^{\circ}= -RT\ln K$

$\Delta G^{\circ}= \sum \Delta G_{f}^{\circ}(products) - \sum \Delta G_{f}^{\circ}(reactants)$

EllieSchmidtke_4I
Posts: 51
Joined: Wed Feb 20, 2019 12:16 am

### Gibbs Free Energy Ionization

How is calculating the standard gibbs free energy of ionization different than just calculating regular standard gibbs free energy?

Ashley Nguyen 2L
Posts: 103
Joined: Sat Aug 17, 2019 12:18 am
Been upvoted: 1 time

### Re: Gibbs Free Energy Ionization

I don't think so. I calculated Gibbs free energy ionization the same way as standard gibbs free energy.

Rhea Shah 2F
Posts: 97
Joined: Thu Jul 25, 2019 12:17 am

### Re: Gibbs Free Energy Ionization

You just use the ionization constant and plug that into the -RTlnK equation. It's the same as calculating the regular gibbs free energy.