How to calculate multiple solutions from big to small with FindInstance funtion

I want to find the nearest 50(or 80, even 100)solutions under 1.4628 from big to small, I tried FindInstance function in MMA, but still cannot get the desired results I need.Here is my code:

Clear["Global'*"]; $  RecursionLimit = Infinity; nco = 1.4681; ncl = 1.4628; nair = 1; rco = 4.2*10^3; rcl =   62.5*10^3; \[CapitalDelta]n = ncl - nair; wl = 1000; u = 2*\[Pi]*rcl*((ncl^2 - neffcl^2)^((1/2))/wl); w = 2*\[Pi]*rcl*((neffcl^2 - nair^2)^((1/2))/wl); J0 = BesselJ[0, u]; J1 = BesselJ[1, u]; K0 = BesselK[0, w]; K1 = BesselK[1, w]; N[  FindInstance[   J1/(u*J0) == (1 - 2*\[CapitalDelta]n)*(K1/(w*K0)) &&     1.45 < neffcl < 1.463, neffcl, PositiveReals, 50]  ] TM Plot[{J1/(u*J0), (1 - 2*\[CapitalDelta]n)*(K1/(w*K0))}, {neffcl, -2,    2}] Plot[{J1/(u*J0), (1 - 2*\[CapitalDelta]n)*(K1/(w*K0))}, {neffcl,    1.453, 1.463}, PlotRange -> {-0.005, 0.005}] 

I plotted two pictures, the first one represent the solutions of the unsolved function, and in the 2nd one, I plotted the formula with neffcl from 1.453 to 1.463.

As for the given results, we can see that MMA only returns 15 solutions, not 50 I set, but we can see both in the first and 2nd picture that there is absolutely more than 15 solutions for my formula, here is what MMA returns:

{{neffcl -> 1.45092}, {neffcl -> 1.45193}, {neffcl ->  1.45289}, {neffcl -> 1.4538}, {neffcl -> 1.45467}, {neffcl ->     1.45549}, {neffcl -> 1.45628}, {neffcl -> 1.45701}, {neffcl ->     1.45835}, {neffcl -> 1.45951}, {neffcl -> 1.4622}, {neffcl ->     1.46241}, {neffcl -> 1.46257}, {neffcl -> 1.46269}, {neffcl ->     1.46277}} 

enter image description here picture1 enter image description here Picture2

Somebody can tell me what is wrong with my code and how can I make it work? Thanks in advance!

Mini excavators: Small size, big popularity

Mini excavators are one of the quickest growing equipment types, with the machine’s popularity seemingly ever-increasing. According to data from Off-Highway Research, global sales for the mini excavator were at their highest point ever last year, at over 300,000 units.

The major markets for mini excavators have traditionally been developed countries, such as Japan and those in Western Europe, but the last decade has seen their popularity rise in many emerging economies. Most notable of these is China, which is now by far the biggest mini excavator market in the world.

Considering that mini excavators essentially replace manual labour, this is perhaps a surprising turnaround in the most populous country in the world where there is certainly no shortage of workers. Although all is perhaps not as it seems in the Chinese market – see the box out ‘China and mini excavators’ for more details.

One of the reasons for the mini excavator’s popularity is that it is easier to power a smaller and more compact machine with electricity rather than the traditional diesel power. It is the case that, especially in city centres of developed economies, there are often strict regulations regarding noise and emissions pollution.

There is no shortage of OEMs that are currently working on, or have released electric mini excavators – back in January 2019 Volvo Construction Equipment (Volvo CE) announced that, by mid-2020, it will begin to launch a range of electric compact excavators (EC15 to EC27) and wheeled loaders (L20 to L28) and stop new diesel engine-based development of these models.

Another OEM looking at electric power for this equipment segment is JCB, with the company’s 19C-1E electric mini excavators. The JCB 19C-1E is powered by four lithium-ion batteries, providing 20kWh of energy storage. This is enough for a full working shift for the majority of mini excavator customers on a single charge. The 19C-1E itself is a powerful, compact model with zero exhaust emissions at point of use and one that is considerably quieter than a standard machine.

Small machine is a relatively light and fast mechanical equipment. Like hand-held power tools, concrete vibrator, frog tamper, mini transporter, woodworking machinery, steel machinery, mini dumper, motorized dump truck, welding, etc. are all small machines.
In addition to large machinery and equipment in construction should pay attention to the safe use, to prevent injury, a variety of medium and small machines also have different degrees of danger, must be used in accordance with the safety requirements.

Going electric

JCB recently sold two models to London-based J Coffey Plant, with Coffey Plant Division Operations Manager Tim Rayner commenting, “The major benefit is no emissions at point of use. Our workers are not subjected to diesel emissions when using the 19C-1E. Confined areas are now clearer and safer to work in too, as there is no longer a need for emissions control equipment such as extraction units and ducting. The JCB electric minis bring value to the business and the industry as a whole.”

Another OEM looking at electric power is Kubota. “In recent years mini excavators powered by alternative fuel sources – such as electric – have seen a surge in popularity,” says Glen Hampson, business development manager construction at Kubota UK.

“The main driver behind this is that electric equipment gives the operator the ability to work in regulated low emission zones. Electric machinery can also enable work to be carried out in confined spaces underground without producing harmful emissions. It’s reduced noise output also makes it great for construction work in urban or heavily populated environments.”

Kubota launched a prototype electric compact mini excavator in Kyoto City, Japan, at the start of the year and Hampson adds, “At Kubota, our main priority will always be the development of machines that match the needs of our customers – the development of electric machines will allow us to do just this.”

Bobcat recently announced that it was launching a new R-Series mini excavators from 2-4 tonne with a new range of five compact excavator models: the E26, E27z, E27, E34 and E35z. One of the standout features for this range is said by the company to be the Cylinder-Inside-Boom (CIB) design concept.

According to Miroslav Konas, product manager, Bobcat Excavators Europe, Middle East and Africa (EMEA), “The CIB system aims to overcome what has been one of the weakest points in any mini-excavator – the vulnerability of the boom cylinder to damage such as that caused by colliding with the sides of trucks and other vehicles when loading waste and building materials.

“It does this by enclosing the hydraulic cylinder inside an extended boom structure, thus avoiding collisions with the top of the blade and the sides of vehicles. In fact, the boom structure protects the hydraulic boom cylinder in any position of its movement.”

Wood chipper is a kind of special equipment for producing wood chips, wood chipper is also called wood slicer, which is one of the wood processing series equipment.

Trenching machine, such as mini trencher, is one of the main types of construction machinery, a kind of trenching machinery used in earthwork construction, widely used in agricultural water conservancy construction, laying of communication cables and petroleum pipelines, municipal construction and military engineering, etc.

Useful tools and machines in home agriculture also include log splitter, stump grinder, etc.

Operator comfort

With the lack of skilled operators in the industry, keeping those behind the stick happy has never been more important. Volvo CE claim that the new 6-tonne ECR58 F generation compact excavator has the most spacious cab in the industry.

Operator well-being, confidence, and safety are supported through a simplified workstation and user-friendly experience. The seat-to-joystick position has been revised and improved, while still being suspended together – a technic that Volvo CE says it introduced to the industry.

Designed to offer the highest levels of operator convenience, the cab features soundproofing, numerous storage areas, and 12V and USB ports. A fully opening front window and slide side window contribute to all-around visibility and operators have an automotive style jog wheel, five inch colour display and easy-to-navigate menus.

Operator comfort is indeed important, but another reason for the general popularity of the mini excavator segment is the ever-increasing range of attachments on offer. For instance, Volvo CE’s ECR58 has a wide range of attachments which are easy to switch over, including buckets, breakers, thumbs, and the new Tilt Quick Coupler.

Talking about the mini excavator’s rise in popularity, Chris Sleight, managing director, Off-Highways Research, highlights attachments, saying, “At the lighter end, the range of attachments available mean it [a mini excavator] is often favoured over workers using hand-held air-powered tools. This is partly because this can be helpful in reducing workers’ exposure to noise and vibration, and also because it removes the worker from the immediate vicinity of the tool.”

Sleight also adds that, “In Europe and even North America the mini excavator is replacing other types of equipment. At the top end of the scale, its smaller footprint and ability to slew through 360 degrees means it is often now favoured over backhoe loaders.”

Bobcat’s Konas agrees with the importance of attachments, saying, “The various types of bucket we offer are still the main ‘tool’ from among the 25 different families of attachments we offer for our mini excavators, but we see a trend developing with more advanced hydraulic attachments growing in popularity. That’s why we developed our A-SAC system, which together with up to five independent auxiliary circuits available on our machines, we believe makes Bobcat the most advanced brand on the market to operate such complex attachments.

“Combining the arm-mounted hydraulic auxiliary lines and the optional A-SAC technology together enables a wide choice of machine customisation options to match any attachment requirement, further enhancing the role of these excavators as excellent tool carriers.”

Remote monitoring as a growing trend?

Hitachi Construction Machinery (Europe) has published a white paper on the future of the compact equipment segment in Europe. In it they point out that 70% of the mini excavators sold in Europe are under three tonnes, helped by the fact that it is relatively straightforward to obtain a licence to tow one of these models on a trailer with a regular driving licence.

The White Paper predicts that remote monitoring will play an increasingly important role in the compact construction equipment market, of which mini excavators are an important part. The report says, “Tracking the location of compact equipment is particularly important, given that it moves frequently from one job site to another.

“Location and working hours data can therefore help owners, especially rental companies, with planning, enhancing efficiency and scheduling maintenance. Accurate location information is also vital from a security point of view – it’s much easier to steal a smaller machine than a larger model, and theft of compact equipment is therefore more commonplace.”

Different manufacturers offer various telematics packages with their mini excavators and mini roller; there is no industry standard. Hitachi mini excavators are connected to its remote monitoring system, Global e-Service, and the data can also be accessed via smartphone.

While location and working hours are key bits of information, the report speculates that the next-generation of equipment owners will want to view more detailed data. “Owners want access to more data from manufacturers. One reason for this is the influx of a younger, more tech-savvy generation of customers, who can better understand and analyse data to improve productivity and efficiency.”


There is no denying that electric scooters will be the future of urban commute, at least for short distances! They make it easier to maneuver through traffic and find parking while helping you reduce your carbon footprint – now that is all a win-win isn’t it? Now, what if your electric scooter could also be your adventure buddy and conquer all kinds of terrains with you? Because it actually can – say hello to Phat Scooter’s latest HD electric scooter! Think of it as the child of your electric bike and an electric scooter ATV.
This compact scooter is a discreet mean machine and can travel up to 20MPH. It can also go 30+ miles on a single charge cycle so it is perfect for those looking for an alternative to public transport post-pandemic but are not yet ready for an electric car. The HD electric scooter comes equipped with a powerful motor that can take on hills with ease, and the front + rear suspensions system provides a smooth ride at any speed as well as on any terrain. The tall handlebars are a smart design element because they enable you to stand up with adequate balance and support while you ride. The seat also has extra support to make the ride on tricky terrains as comfortable as possible. The most impressive and unique part about this scooter is its wide wheels which increase the stability and give the rider better balance. Unlike many other small scooters, the E-scooter electric scooter lets you ride anywhere – cement, asphalt, grass, dirt, or sand. Now only if it had snow tires, every New Yorker would get this!
Scooters have also fitted it with a new and improved battery which charges in four to six hours with a standard outlet. You can pick from a range of colors and customization options to truly make this scooter a part of your everyday life – I guess it is safe to say that this scooter will become your ride or die!

Generally speaking, e-bikes are bicycles with a battery-powered “assist” that comes via pedaling and, in some cases, a throttle. When you push the pedals on a pedal-assist e-bike, a small motor engages and gives you a boost, so you can zip up hills and cruise over tough terrain without gassing yourself. 
The harder you pedal, the bigger the boost, the faster you’ll ride—to a point. Electric bicycles let you hum along at a brisk clip, but they aren’t motorcycles. You’ll never hammer down the road at 45 mph. The motor is governed to stop propelling you further when you hit 20 to 28 miles per hour, depending on the bike. So you’ll save time on your commute (I shave about three minutes off a five-mile trip) but still enjoy the scenery.
Getting an electric motorbike can dramatically increase how often you ride, according to a survey of nearly 1,800 e-bike owners in North America. Beforehand, 55 percent of respondents said they rode daily or weekly. After buying an e-bike, that number soared to 91 percent. It makes sense: Even if you’re super fit, you still get tired (likely from training or racing) and remounting your bike can feel like a chore. If you have an e-bike, you can continue riding while giving your knackered legs a bit of a break. You can also go faster, which makes biking for longer trips more attractive, even when you’re pressed for time.
For those who aren’t frequent riders, e-bikes open up a whole new world. While you may not be conditioned to ride 5-10 miles at a time, you can cover those distances easily with an electric assist, which is a great way to build endurance and confidence. That same survey found that 94 percent of non-cyclists rode daily or weekly after getting an e-bike.
Name a type of riding, and there’s an e-bike for that. If you have zero interest in an electric road bike, you may find yourself head over heels for a high-capacity e-cargo bike that can haul 400 pounds of stuff while still cruising at a cool 15 mph. E-bikes are available in fat, cargo, commuter, recreational, hardtail, full-suspension mountain, and even performance road bike styles. For proof, here are a dozen e-bikes we love for every type of cyclist.
“People are buying electric bicycles as a way to reduce car trips,” Benjamin says. The data backs him up: 28 percent of survey respondents said they bought an e-bike specifically to replace driving a car. And many other reasons buyers listed for wanting an e-bike—including carrying cargo and kids, avoiding parking and traffic, and environmental concerns—also indicate a desire to get out from behind the wheel. Plus, you don’t need to change clothes or clean up when you arrive at your destination, because you don’t have to work up as much of a sweat.
Consider, too, that more than half of all driving trips are shorter than 10 miles, with some surveys reporting that the average single trip amounts to just 5.95 miles. That’s a no-brainer distance to cover by e-bike. In fact, the survey found that owners replaced 46 percent of their car commutes and 30 percent of their driving errands with e-bike rides. All you need is a great commuter bag to carry your stuff, and you’re set.

How to ‘clone’ a wp plugin to make small changes

I’m looking to make small changes to the Advanced Bank Payment Transfer Gateway plugin and was looking for guidance for this since reaching out to the developer didn’t help.

Basically this plugin works like I want it to, however I’m from a place disconnected from the international banking system, so the info we need to make a bank transfer is a bit different, the info it currently asks to fill in and shows to the clients is this:

standard format

but I need it to ask for something like this:


Can I somehow clone or copy the plugin and have a new plugin that works the way I need it?

Absolute noob here so any advice or guidance is very welcomed. Thanks for reading!

What happens when Wild Shape/Polymorph runs out in a space that’s too small?

So in a 5e campaign I’m running, the party Druid has recently been doing some scouting of the local monster caves in spider form to avoid notice. I’ve also described how the goblins and kobolds who live in these caves use a variety of tunnels too small for Medium sized creatures in order to get around. Since the party doesn’t have any halflings or gnomes, I didn’t really think the party would ever get into those tunnels. But obviously if a goblin can fit, so can a spider, so there is the possibility that the Druid might try to scout them out.

But what if something goes wrong? What if the Spiderdruid were to get lost and run out of time in Wild Shape, or encounter a hungry lizard and drop to 0 Spiderhealth?

Is there anything in the rules to suggest what should happen when somebody tries to revert to their normal size in a space where they normally wouldn’t fit?

With a space sized for a Small creature such as in my game, I might just rule that he’s stuck unless he makes difficult Dexterity checks or waits to get his Wild Shape back. However, what if it was an even smaller space, such as a burrow or a pipe or a deep crack in the rock? Someplace where his human body literally won’t fit?

If there aren’t any rules for this, I’m open to suggestions on how it can best be handled.

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Too small to represent and Chop in inaccessible algorithm


I want to apply a specified Chop function to every step in an function call in Mathematica (in my case LegendreP), especially when encountering a machine underflow error. The resulting function should be applicable to a dataframe.

The Setup

I am trying to calculate a function including an associated Legendre function with complex indices

function[t_] = 1/Sin[t] * LegendreP[-1/2 + V * I, l+1, Cos[t]]

where t is between 0 and Pi, V is of order of magnitude 10 and l is between 10 and 100. Beside the function I need its logarithmic derivative function'/function.

Ideally I want to do this by applying the function to a dataframe and appending the result in a separate column.

Append[#, "function" -> function[#timeframe]]

where timeframe is a column with all the t values.

The Problem

When I run this code for any l bigger than 12 and very small t~1e-5 values, the LegendreP Algorithm throws a machine underflow error General::munfl because it cannot execute a multiplication of extremly small complex numbers.

While for a single call or a plot, it seems to do some chopping or return Indeterminate, when I write it to a dataframe with

Append[#, "function" -> function[#timeframe]]

it just returns Failure and does not write anything to the dataframe.

What I have tried so far

I have tried to use Chop and Threshold, but this does not seem to apply to the single steps of the algorithm but only the final result.

The way I "solve" the problem at the moment is to catch the error and return 0 instead of my function. This is not ideal since the real or imaginary part of the step in question and the result might not be negligible while the other one is, or it might diverge instead of converge to 0.

Since the multiplication that raises the error lists numbers ~1e-300 or so, I doubt that the problem is solvable by increasing the precision.

My Goal

Ideally I’d like to call Chop, whenever Mathematica encounters a machine underflow. The behavior of Chop on complex numbers is exactly what I need. This way I should be able to preserve the real or imaginary part that does not vanish.

Is the error handling different, when applied to a dataframe as it relates to this question (for plots or even single evaluation points I don’t have the same issue) or can an indeterminate/NaN be written to a dataframe?

Is there a way to set a "global chop rule"?

Grateful for any hint 😀