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# DSP Processor, Amps, Console

#### macsound

##### Well-Known Member
More of a constant wonder and thought than anything else. Not purchasing any equipment, just want people's opinion.
Also worth stating, I understand how all the components in the signal chain work, but maybe not subtle nuances that may differ. Also worth the conversation.

Q Part 1 - If I was using a modern digital console, what benefit would an audio processor's DSP have over the EQ and delay built into the desk.
Q Part 2 - In the same vein - would DSP in a qualified name brand amp be considered the same, better or worse than using a separate audio processor feeding the amp.

#### MNicolai

##### Well-Known Member
Fight Leukemia
DSP is preferable for system tuning/balancing. Console's get wiped. Firmware gets reloaded, files get factory defaulted. Someone may have spent 3 days tuning the system properly with Smaart or SIM time aligning the cabinets, verifying polarity, setting limiters and EQ and gains, and one person can quickly undo all that work permanently by screwing around in a console.

Artistically, it's also more intuitive to send the mixes you want to a DSP and let the DSP parse them out. That's includes to back of house, front of house, delays, side fills, hearing assist, etc. If you try and set up discrete mixes for all of those different destinations in a console you'll burn up your mix and matrix busses in no time.

Other functions like hearing assist or a house mic feed backstage and into the lobby need to be operational even if the console is turned off. A DSP can facilitate this.

Logistically, I prefer all of the tuning/balancing in a DSP. You will get much of the same functionality in amplifier-provided settings but it doesn't scale well if you have lots of speakers. You burn up a lot of time making minor adjustments. Depending on the product line, for some configurations you end up doing some EQ/balancing in the DSP and then some in the amplifier. Puts you in a position where if you're not careful it's easy to fight yourself and murder your gain structure.

Other things I like to use DSP's for. I'll stick a touch panel in the booth and on-stage with 2 mix modes. One is for "Quick Mix", which is the house mic to backstage and lobbies, and a few handheld and lapel wireless mic's and some aux inputs on either side of the stage. All of the EQ and gains are preset in the DSP. Someone wants to run a quick talking-head event with just a couple mic's or run a rehearsal with playback -- you can give them a few faders and mute buttons on a touch panel and they don't have to go anywhere near the mix console or learn how to run a full sound system. The touch panel also offers Mix Console Mode, which kills the Quick Mix inputs and gives you much more control of the system from the mix console, while also offering you various options for things like booth monitors. Do you want to your booth monitors to be cue monitors that work off your solo buss, or do you want them to be just off of the house mic for a show that's largely acoustic? You have an option for that.

The DSP also does things like provides the house mic feed to the hearing assist and FOH/BOH feeds, while allowing you to inject vocal mixes that duck the house mic and ride on top. This gives you strong speech intelligibility where you need it, while still letting you hear the music. But in the areas this mix serves they really need speech intelligibility above all else.

A properly tuned system through a DSP will give you much more bang for the buck than trying to replicate this through your console or through your amps. If you spend $150,000 on your sound system, you should get$150,000 worth of value out of it. If you only spend $145,000 and skip the DSP, you're liable to get a sound system that sounds like you spent$5,000 on it.

In short, it lets you focus on the art of that specific event by offering you a solid baseline to drive you system from. Try juggling all of that in your console and every soundcheck becomes simply trying to get proof of life out of all of your different mixes and you end up not having time to make the show actually sound good.

Re: Processing amps. I run into these a lot and end up ignoring their onboard processing if I have a DSP in the system. Even in a 700,000 d&b rig I'll use the amps only for the Array Processing feature and keep all of the EQ, delay, and so forth in the DSP. Makes programming easier, and reduces the number of moving parts. I brought a system online not long ago where they had 24 Y-series boxes, 2 Renkus sticks for side fills and ceiling speakers for delays. They had 7 different array configurations with and without front fills, with the line arrays flown out high for theater, or hanging way down low for rock n roll, with L/R subs when they need a thrust and with a 6-sub array for everything else. Tuning had to happen at the same time the control system to flip between all these presets was going on. We encountered issues with the Crestron programmer reloading code and recalling presets that wiped our settings. Our philosophy quickly went away from using presets in amplifiers or even in the DSP to using routers in the DSP to flip signal chains accordingly. Made it much easier to guarantee tuning efforts would not be compromised inadvertently by an accidental preset recall before the latest changes were saved. Laundry list of reasons to keep system settings out of amplifiers, but especially out of consoles. Last edited: #### NickVon ##### Well-Known Member I think @MNicolai has explained this very well. That said... some further thoughts, He's coming from experience in what appears to be fully fledged installed sound systems. Though this all holds true for a room which might just have L+R on sticks as the only PA. Many of these Poor man system or band/event touring systems might not have a "Designed" sound system. You could be working with some Powered QSC K12, Subs, 1 pair of Delay's Maybe a second set of remote PA mains in a side ballroom. In these circumstances using a digital console to carve out matrix's for each of these speaker setups, and locations can be a lot quicker and maybe the only way to do some baseline room EQ, or some LPF tuning on the subs that fit the new ballroom your in, simply because no system processors are on hand. Still it's great if the gear is available for small touring, and one-an-done setups to have system processors that can be used, but many do not. And having some level of that ability to do that functionality in a console is great. But MNicolai has pretty clearly stated why when both options are available we use System processors over consoles/and Amps. #### TimMc ##### Well-Known Member Premium Member I had a reply in my head the moment I saw the topic, then read Mike's reply and it turned into "yeah, what he said." I have a couple of minor disagreements but nothing serious enough to justify a reply to him. Nick's, though, points out the way that *portable* systems, frequently used in different venues, locations and places provides a "quick and dirty" way to accomplish *some* of what can be done in a dedicated system processor and for bar bands or doing children's theater in old shopping malls, it's probably a decent compromise... and I've been known to do it myself for one-off performances. That said, all but the most sophisticated digital mixers fall short of the capabilities of a system DSP. I've stirred the pot many times over at the ProSound Web forums about the impropriety of using a mixer's output processing in lieu of DSP and faced incredible push back from folks who want to eliminate a 1 rack space box that handles so much processing and replace it with tools that are up to the task in only the simplest of situations. Sort of like the guys/gals that want to get rid of the analog multipair snake and replace it with CAT5e, only to discover they still need to run power, coms and talkback lines and cue lights to FOH, so they saved nothing in labor and replaced a reasonably robust cable with a relatively fragile cable. Sometimes the "new ways" are just that, only new, without representing an improvement in quality or reliability. Installed systems or systems that routinely are utilized in the same venues should have dedicated SYSTEM DSP. This frees up the console's matrix outputs and processing for creative use rather than system control. #### macsound ##### Well-Known Member Thanks Tim, Nick and MNicolai The assistive listening with a mic in the room is something I've never thought of before. In a perfect world, I can see the benefit of the crestron running the different DSP parameters depending on the style of the presentation, but in a more simplistic setting, offloading that processing as needed. What about everyone's preferences on signal flow. For me, I tend to always run L+R of the stereo out and Subs, front fill and mono delay as Auxs. So in that scenario, would you send all those feeds to the same DSP? Also, working in corporate environments, there's commonly DSP for the installed sound systems to make it match the fidelity of the canned music or installed mics. But once you patch in and try to do your own EQ for best feedback avoidance or use as fills with a smaller added system, you're fighting the DSP that was designed for a completely different purpose than was intended. How do you commonly deal with this? Thanks for the chatter #### TimMc ##### Well-Known Member Premium Member Most of anything you'd "fight" in a ballroom DSP is lack of fidelity (eq'd to death for feedback control) or system protective limiting. If either of those make the system unsuitable, why are you using it? As for signal/work flow, "it depends..." mostly on how the system was designed. Since I provide transportable systems to Bands You've Head Of® we'll configure DSP however the band mixerperson wants it. Typically Main L, Main R, Subs, Front fills, Side Hangs; we've done 'stereo subs' (for proprietary processing) and discreetly driven side hangs (stereo-ish), balcony/under balc fills, plus lobby/green room feeds, feeds to the LD for IEMs. DSP makes it possible, for all zones that get the same mix, to be driven from 1 signal but distributed to each zone with its own EQ, level, compression or limiting, independent of one another. #### MNicolai ##### Well-Known Member Premium Member Fight Leukemia Thanks Tim, Nick and MNicolai What about everyone's preferences on signal flow. For me, I tend to always run L+R of the stereo out and Subs, front fill and mono delay as Auxs. So in that scenario, would you send all those feeds to the same DSP? In a presentation or education venue I'll typically do L/R out of the console, downmix to the subs within the DSP, and set aside a different console mix buss for the FF's. FF's typically you have enough instrumental stage volume that all you want to drive out of these speakers are vocals and sound effects so it's not always ideal to give them a downmix from L/R. There is an exception to every rule though. In a professional theater environment I will plan for a Qlab injection into the system. In this case I'll set up the usual output flows from the console and DSP, but I'll also give a Dante pathway into every speaker output that the console can drive off its direct outputs. Then you patch Qlab into the mix console via Dante Virtual Soundcard and can get a fader for all of your Qlab outputs during a show for any surround or immersive effects you may want to do. Also, working in corporate environments, there's commonly DSP for the installed sound systems to make it match the fidelity of the canned music or installed mics. But once you patch in and try to do your own EQ for best feedback avoidance or use as fills with a smaller added system, you're fighting the DSP that was designed for a completely different purpose than was intended. How do you commonly deal with this? Thanks for the chatter As a systems designer, I typically designate inputs for this purpose or give touch panel control to override the EQ on these inputs. Usually if the output EQ for the speakers is good, you should not need much input EQ for something like an SM58 that you would be fighting. There are many examples of bad DSP programmers out there who break this form. Lectern gooseneck mic's are usually where you get into a little trouble because they tend to need a little more EQ to dampen down their brightness, and they need phantom power. If you leave phantom power engaged on these inputs all of the time you're likely to blow up someone's phone when they jack in on 1/8" to XLR adapter instead of through a DI, and anytime someone plugs in a mic which phantom engaged you'll get that notorious popping sound through the whole PA. This why I try to be a little church and state about how I lay these systems out for the different use cases. #### mbrown3039 ##### Well-Known Member What Mike said, and here's my simplified version: in all of the live/production systems we design/install, we endeavor to have four (yes, 4!) layers of processing: at the amp, to tune each amp channel to the specific speaker/component it's driving (crossover, roll offs, boost/cut, Q, etc.); a pro-grade DSP platform (Atlas, Biamp, BSS, Symetrix, etc.) to tune the room (delays, filters, crossover points, etc); the console itself, where individual settings for each performer/instrument are made/adjusted from night to night; and, in front of the console, a good 31-band analog EQ where the FOH Engineer can quickly grab a particular frequency in the case of feedback. Best wishes, (the other) mike PS - IMO, use Crestron for control or video switching but not for audio DSP -- their product line just isn't quite there yet. m #### macsound ##### Well-Known Member What Mike said, and here's my simplified version: in all of the live/production systems we design/install, we endeavor to have four (yes, 4!) layers of processing: at the amp, to tune each amp channel to the specific speaker/component it's driving (crossover, roll offs, boost/cut, Q, etc.); a pro-grade DSP platform (Atlas, Biamp, BSS, Symetrix, etc.) to tune the room (delays, filters, crossover points, etc); the console itself, where individual settings for each performer/instrument are made/adjusted from night to night; and, in front of the console, a good 31-band analog EQ where the FOH Engineer can quickly grab a particular frequency in the case of feedback. Best wishes, (the other) mike PS - IMO, use Crestron for control or video switching but not for audio DSP -- their product line just isn't quite there yet. m I think you've hit what I've been thinking. 1 Speaker EQ, 2 PA Tuning, 3 Room tuning, 4 application compensation like speech vs live music vs playback. And which of these falls in each level of DSP. I'd probably say Speaker EQ would be done with amp DSP, PA and Room tuning with soundweb or whatever and application compensation on the console. But then like other posters have mentioned, how much are you fighting each other with each layer of DSP #### MNicolai ##### Well-Known Member Premium Member Fight Leukemia PS - IMO, use Crestron for control or video switching but not for audio DSP -- their product line just isn't quite there yet. m Re: Crestron DSP's -- Agreed. Crestron's product not only isn't there, they've made a conscientious decision to go with a fixed platform at no cost advantage. For the exact same cost of a Q-Sys 110f you get a fraction of the processing power and flexibility. I have a Crestron DSP-1282 on my home office and a DSP-1283 in my secondary Smaart rig because I got a smokin' deal on them but I wouldn't put them in a project. I find the programming/tuning software to be pretty user-hostile, and you have to stick a control system on top of it at additional cost. If I go with Q-Sys, I have a powerful control system baked in already I can control through a touch panel or a web GUI, and if I want an even better GUI I can stick a Crestron touch panel and controller on it if I absolutely have to. Crestron just can't compete offering a subpar product at the same price point as a Q-Sys 110f. I was asked to look at a city hall project recently involving broadcast and they wanted to handle the dais and all the broadcast feeds through a pair of 1283's. I told them a condition of mine for taking on the project would be to flip the DSP's to Q-Sys. The programming time would be faster, the signal flows would make it sound better and be more customized to what they were trying to achieve, and I had much more control for remote monitoring and troubleshooting if my phone rung during a broadcast and they needed me to log in and triage anything mid-meeting. If they stuck with Crestron, my hands would be tied. I could remotely log in and see meters flickering and could double check mute statuses but that's about it. I wasn't going to put my company's or my own reputation on Crestron's DSP products. #### TimMc ##### Well-Known Member Premium Member I think you've hit what I've been thinking. 1 Speaker EQ, 2 PA Tuning, 3 Room tuning, 4 application compensation like speech vs live music vs playback. And which of these falls in each level of DSP. I'd probably say Speaker EQ would be done with amp DSP, PA and Room tuning with soundweb or whatever and application compensation on the console. But then like other posters have mentioned, how much are you fighting each other with each layer of DSP They don't have to be "fighting each other" if the signal flow is properly designed and @mbrown3039 is spot on. Starting with loudspeaker processing (whether in the amp or an external processor) and working back to the mixer, each "layer" does its thing predicated on the assumption that the prior processing is correct and properly implemented for its tasks; i.e. that making the loudspeakers work to the manufacturer's spec, then using a system controller to align coverage zones and EQ them, limit them and set appropriate levels relative to other zones; and the "creative" side of processing at the mixing desk. As for presets for "playback" and "speech presentation", those belong in the console (if there is an operator). The false economy of cutting out a qualified operator seems to be a Holy Grail of fixed installations, but I'd opt for a DSP implementation that assigns particular inputs to specific processing rather than reconfiguring a system DSP on the fly. Example: the microphone inputs should be sufficiently high-passed and not feed any subwoofers, whilst the "music" inputs would be full range and include a send to the subs. As for Crestron - I'm still left wondering how such an expensive and patently unfriendly system became so popular. Were they the ONLY thing available 15 years ago? #### mbrown3039 ##### Well-Known Member As for Crestron - I'm still left wondering how such an expensive and patently unfriendly system became so popular. Were they the ONLY thing available 15 years ago? Kinda' sort of, yeah....AMX had some egg on their face in the late 90s/early 00s after a major product line had not delivered as promised. Crestron was already in place and capitalized on AMX's missteps. Extron's control system came to some prominence at that time as well, as did several smaller (at the time) players like Aurora Multimedia, Xantech and Vantage. These days, Crestron is the GIANT elephant in the room (and throws their weight around accordingly), even though only ~20% of their business is control systems -- the bulk of their revenu these days is video switching/transmission/management. #### MNicolai ##### Well-Known Member Premium Member Fight Leukemia As for Crestron - I'm still left wondering how such an expensive and patently unfriendly system became so popular. Were they the ONLY thing available 15 years ago? They do what they do and they do it very well. They have some highly qualified audio people on staff but in terms of product development they've really targeted a very specific niche for their audio products and have no intention of pursuing other avenues. To their credit, they'll achieve more success by building DSP's that primarily play nicely with their own ecosystem than by trying to build DSP's that are intended to be standalone or integrated with other control systems. They would have a much harder time competing in those markets with Biamp/QSC/Symetrix. A lot of their market is corporate, and some of the key differentiators they have over other solutions are their enterprise grade security, scalability in deployment, and ease of maintenance (need to push firmware or an updated piece of code to 200 boxes? you can do it within a lunch break). Are you a Crestron dealer, programmer, or factory-trained installer and having problems getting something to work? You are fully availed of 24/7 factory support that one way or another will work with you through the bitter end of a project until the customer is satisfied. When you get into low-latency video transmission and switching, there's little better than what they offer. You pay a premium for the product, but the product is generally rock solid. Most horror stories are because of inexperienced programmers or from installations where the control functionality wasn't routinely updated as various parts and pieces needed replacing (i.e. the VHS player is now a DVD/VCR combo and none of the controls we have are compatible). We haven't seen as much turbulence in the market over the last 5-8 years with pretty much everything reliably being HDMI, DVI, or DisplayPort that is compatible with either HDMI or DVI, so in recent past the ability to keep existing systems in place and reliable has been fairly easy compared to the 10 years that came before it when we were diving around being SDI, composite, component, RGBHV, S-video, etc. and system shelf-life was much shorter. HDMI CEC control has helped kill off customized controls that would've otherwise been IR/RS232 with customized commands so replacing a TV doesn't mean you need to call your programmer and pay for a trip charge. It's like buying a major line array speaker system for1M. You don't do it just because you want d&b, L'acoustics, or Meyer. You do it because you can draw a straight line between the cost of those systems and value and return on investment you receive from them. If the acquisition is made independent of and unaligned with business goals then buyer's remorse is unavoidable. It's like choosing to buy ETC over Elation or Chauvet DJ. For some people, their expectations will be satisfied on the lower tier. For others, they will end up replacing all of their equipment 5 years earlier than if they had gone ETC from the outset.

From an integrator and consultant's perspective, it's the devil you know, and it's consistent, and it's scalable. If you cheap out on something for budget reasons, the solutions you reach for may work for this project but fall apart on the next one. You don't build up a groove or institutional momentum because you are forced to constantly be reinventing how you deploy these systems.

FWIW, I have more than a few horror stories about burned bridges, serviceability nightmares, and unreliability from the projects that tried to do a Crestron-type deployment with a product at half the cost. Half the cost often equates to additional service calls, more time spent futzing with firmware, calls to the factory, etc, and either the installer or the customer ends up eating these costs. Sometimes it means an integrator getting everything installed, finding out the product doesn't do what they were promised, and having to redo it all with a different product at their own expense. Installers bill out at $85-105/hr, and programmers can be$150/hr. Whoever ends up eating those costs will feel it.

The key thing to remember is that, in general, Crestron solutions are not for the types of people and environments people here at CB regularly encounter. They're for the other people who need to use the room productively when a technician isn't in the room. These systems are at their prime when deployed for making technology-centric spaces accessible to people who have zero technical background and just want to get their laptop up on a screen, turn on a lectern mic, and dim the lights without hiring 2-3 board ops for an hour long 1-microphone event, however many events like that out of the year for the 7-10 years of the system's shelf-life before the next major overhaul. You can associate dollars and cents to the value that system offers.

#### macsound

##### Well-Known Member
They don't have to be "fighting each other" if the signal flow is properly designed and @mbrown3039 is spot on. Starting with loudspeaker processing (whether in the amp or an external processor) and working back to the mixer, each "layer" does its thing predicated on the assumption that the prior processing is correct and properly implemented for its tasks; i.e. that making the loudspeakers work to the manufacturer's spec, then using a system controller to align coverage zones and EQ them, limit them and set appropriate levels relative to other zones; and the "creative" side of processing at the mixing desk.

As for presets for "playback" and "speech presentation", those belong in the console (if there is an operator). The false economy of cutting out a qualified operator seems to be a Holy Grail of fixed installations, but I'd opt for a DSP implementation that assigns particular inputs to specific processing rather than reconfiguring a system DSP on the fly. Example: the microphone inputs should be sufficiently high-passed and not feed any subwoofers, whilst the "music" inputs would be full range and include a send to the subs.

As for Crestron - I'm still left wondering how such an expensive and patently unfriendly system became so popular. Were they the ONLY thing available 15 years ago?

The reasons I've experienced them fighting each other is because of installation choices in venues with existing systems that you're unable to adjust.
Definitely installs made for the purpose of eliminating operators and then the venue realizing that doesn't work but trying to squeeze the most out of a system designed to do something else.

Let me throw another question in here.
If I was designing a system that included the main venue, then overflow room, then 70v system, do you decrease the amount of time and $you spend tuning each system based on its fidelity and use case? I used to work in a hotel where the lobby system sounded amazing but the 5th floor system with the same gear sounded horrid. Part of me assumed this was because of time and money, part of me thinks its because the newbies did the "cheap" rooms. Same goes for places like Universal Studios. Huge Bose contract has surprisingly made the line array at waterworld sound incredible, Citywalk sound fairly good, ambient music in the park sound ok and the moaning murtle in the harry potter bathroom sound completely untuned. #### mbrown3039 ##### Well-Known Member There is no "one size fits all" solution for either speakers or processing, and different types of system can accommodate more/less tuning for a variety of reasons. 70V ceiling speakers will (typically) not have the dynamic range or fidelity of a high-quality performance speaker, for example. Signal processors will have different capabilities based on the type/quality of components inside of them (I once watched Sam Berkow -- co-developer of SMAART -- open up the processing module of a major manufacturer's DSP-enabled amplifier; looking over the motherboard, he pointed out three component changes they could make -- even naming the manufacturer who made the component he was recommending -- that would gain them 6 - 9 more dB of performance. The company made the changes and realized an 8dB increase in SPL.) So, don't expect a 70V system to sound the same as a line array -- although, a properly designed/tuned 70V system can sound very good. Similarly, a "speakers on sticks" system using an off-the-shelf processor (dbx DriverackPA, perhaps?) will sound significantly better than any portable system from 20 years ago but still won't match something using a pro-grade DSP. As with any tools, I always suggest buying the best you can afford, even if it means sandwiches for dinner for a couple of weeks. #### mbrown3039 ##### Well-Known Member They do what they do and they do it very well. They have some highly qualified audio people on staff but in terms of product development they've really targeted a very specific niche for their audio products and have no intention of pursuing other avenues. To their credit, they'll achieve more success by building DSP's that primarily play nicely with their own ecosystem than by trying to build DSP's that are intended to be standalone or integrated with other control systems. They would have a much harder time competing in those markets with Biamp/QSC/Symetrix. A lot of their market is corporate, and some of the key differentiators they have over other solutions are their enterprise grade security, scalability in deployment, and ease of maintenance (need to push firmware or an updated piece of code to 200 boxes? you can do it within a lunch break). Are you a Crestron dealer, programmer, or factory-trained installer and having problems getting something to work? You are fully availed of 24/7 factory support that one way or another will work with you through the bitter end of a project until the customer is satisfied. When you get into low-latency video transmission and switching, there's little better than what they offer. You pay a premium for the product, but the product is generally rock solid. Most horror stories are because of inexperienced programmers or from installations where the control functionality wasn't routinely updated as various parts and pieces needed replacing (i.e. the VHS player is now a DVD/VCR combo and none of the controls we have are compatible). We haven't seen as much turbulence in the market over the last 5-8 years with pretty much everything reliably being HDMI, DVI, or DisplayPort that is compatible with either HDMI or DVI, so in recent past the ability to keep existing systems in place and reliable has been fairly easy compared to the 10 years that came before it when we were diving around being SDI, composite, component, RGBHV, S-video, etc. and system shelf-life was much shorter. HDMI CEC control has helped kill off customized controls that would've otherwise been IR/RS232 with customized commands so replacing a TV doesn't mean you need to call your programmer and pay for a trip charge. It's like buying a major line array speaker system for$1M. You don't do it just because you want d&b, L'acoustics, or Meyer. You do it because you can draw a straight line between the cost of those systems and value and return on investment you receive from them. If the acquisition is made independent of and unaligned with business goals then buyer's remorse is unavoidable. It's like choosing to buy ETC over Elation or Chauvet DJ. For some people, their expectations will be satisfied on the lower tier. For others, they will end up replacing all of their equipment 5 years earlier than if they had gone ETC from the outset.

From an integrator and consultant's perspective, it's the devil you know, and it's consistent, and it's scalable. If you cheap out on something for budget reasons, the solutions you reach for may work for this project but fall apart on the next one. You don't build up a groove or institutional momentum because you are forced to constantly be reinventing how you deploy these systems.

FWIW, I have more than a few horror stories about burned bridges, serviceability nightmares, and unreliability from the projects that tried to do a Crestron-type deployment with a product at half the cost. Half the cost often equates to additional service calls, more time spent futzing with firmware, calls to the factory, etc, and either the installer or the customer ends up eating these costs. Sometimes it means an integrator getting everything installed, finding out the product doesn't do what they were promised, and having to redo it all with a different product at their own expense. Installers bill out at $85-105/hr, and programmers can be$150/hr. Whoever ends up eating those costs will feel it.

The key thing to remember is that, in general, Crestron solutions are not for the types of people and environments people here at CB regularly encounter. They're for the other people who need to use the room productively when a technician isn't in the room. These systems are at their prime when deployed for making technology-centric spaces accessible to people who have zero technical background and just want to get their laptop up on a screen, turn on a lectern mic, and dim the lights without hiring 2-3 board ops for an hour long 1-microphone event, however many events like that out of the year for the 7-10 years of the system's shelf-life before the next major overhaul. You can associate dollars and cents to the value that system offers.

Well, we must be blessed then: we consistently design and install systems other than Crestron for much lower cost and plenty of reliability and feature sets. And, for the record, DM is nothing but HDBt with a layer of Crestron control thrown on top of it. I can hook up any HDBt receiver to a DM system and it works just fine...m


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#### Je--yMellow

##### New member
can someone please provide steps and explain

#### Jomo

##### Elite Member
Hi,
Please read the posting guidelines and submit a post that follows these guidelines.

#### Jomo

##### Elite Member
I'll explain what one of the lines mean.
The lines that starts with $77,100 means that if your taxable income is between$77,100 and $160,850 then your tax is$15,698 plus 28% of the amount over $77,1000. Hopefully that will help you. #### Subhotosh Khan ##### Super Moderator Staff member can someone please provide steps and explain View attachment 29843 I'll do a similar but different problem for you, using the same table. What was the tax owed by a single taxpayer on a taxable income of$ 90,000?

The income is over 77100 but not over 160850. So we will use data 4 th row

90000 - 77100 = 12900

So the tax is 15698 +28% of 12900 = 15698 + 3612 = $19310 Follow the exact same procedure. If you don't follow something - tell us exactly where you are lost and we will provide you MapQuest and compass. #### Je--yMellow ##### New member I'll do a similar but different problem for you, using the same table. What was the tax owed by a single taxpayer on a taxable income of$ 90,000?

The income is over 77100 but not over 160850. So we will use data 4 th row

90000 - 77100 = 12900

So the tax is 15698 +28% of 12900 = 15698 + 3612 = \$19310

Follow the exact same procedure. If you don't follow something - tell us exactly where you are lost and we will provide you MapQuest and compass.
Thank you so much!! I understand it now!

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# Math Help - I need a hint abot how to solve a question...

1. ## I need a hint abot how to solve a question...

Hi guys I need a hint to find partial limits of a sequence.
I uploaded a pic of the sequence. Thank you

2. ## Re: I need a hint abot how to solve a question...

Originally Posted by davidciprut
Hi guys I need a hint to find partial limits of a sequence.
I uploaded a pic of the sequence. Thank you
I would like help, but I have no idea what "find partial limits of a sequence" means.

3. ## Re: I need a hint abot how to solve a question...

I am translating it exactly the same, I don't know how else I can put it. The sequence has partial limits, (subsequences that converges to different points I guess...) and I have to clarify what they (the limits) are. Is it more clear?


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## Props and Equipment for Paradigms Activities

### Spins OSP Software

The Spins software, an elegant and simple simulation of the Stern--Gerlach experiment, is used extensively in the Spins and Quantum Measurement Paradigm.

### The Partial Derivative Machine

The Partial Derivative Machine is used as a method of physically interpreting the concepts behind partial derivatives. Instructions for making your own can be found at Building the PDM: Instructions.

### Whiteboards

We use three different sizes of whiteboards for various activities:

#### Wall-Mounted Whiteboards

These whiteboards are large and vertically mounted on the walls (like the front blackboard). We use these for activities where student groups present their results/calculations to the rest of the class.

#### Tabletop Whiteboards

These whiteboards are about $2'\times3'$ and lie flat on the groups' table/desks. We use these for most small group activities as a workspace for students to share their ideas with other members of their group (and so that the instructor can easily monitor the group's progress!).

#### Individual Small Whiteboards

These whiteboards are ideally $12''\times18''$ and are for individual use. Students can each respond to a Socratic question and the instructor can collect these whiteboards for discussion with the whole class.

You should be able to purchase the whiteboard material at any home improvement store (e.g. Home Depot). It is technically melamine, but that word just confuses many employees; instead ask for “that shiny white shower board stuff.” It should come in $4'\times8'$ sheets, plywood-size, approximately \$10/sheet. Make sure that it is smooth and shiny and on a thin backing. Backing should NOT be plywood (it comes this way, but is too heavy). Thickness maybe $\frac{1}{4}"$. You can get five medium sized whiteboards ($2'\times3'$) out of one sheet (if you are clever) and 24 small white boards. Often the home improvement store will cut them for you at a small charge per cut.

If you buy framed whiteboards, ready-made for classroom use, they will be much more expensive. For large, wall-mounted whiteboards, you can build a frame yourself. Don't forget a “chalk” tray! But, if you are going to use the wall-mounted boards for many classes over many years, it is probably worth investing in professional enamel-on-steel boards available from classroom suppliers.

### Model Observer

We use a small model (about 10--15 cm) of a person to help students learn how to do proofs by contradiction when using Gauss' and Ampere's Laws. Index cards or cardboard cutouts allow the students to decorate them (which a large percentage do).

### Spheres and Cylinders

• Spheres: various size balls (e.g. basketball, soccer ball, tennis ball, ping pong ball, etc.).
• Cylinders: a piece of thin cardboard or cardstock can easily be taped or stapled into a cylinder.

### Butterfly Net

Make a "butterfly net" with a hanger or other wire loop and a cloth lingerie bag. The butterfly net is a used when discussing Stokes' Theorem. By changing the shape of the net, you can demonstrate how different surfaces might have the same closed boundary. By bending the wire, you can show what it means to change the boundary.

### Cube

Similar to the butterfly net, storage cubes can be used to help students visualize flux through a cube. By stacking the storage cubes together, it is possible to show how the flux out of one small cube is the negative of the flux out of the adjacent cube---this is the fundamental idea in the proof of the Divergence Theorem.

### Coordinate Axes

Students encounter a variety of orthogonal coordinate systems in their courses on electrodynamics. To illustrate orthogonal coordinates, the instructor can use a set of coordinate axes with which they may place different labeled unit vectors, allowing students to visualize the coordinate system they are working with.

A paddle wheel (in this case, made out of snap-together constructive toys) can be used to demonstrate the effect of a vector field on a physical system as part of a discussion of curl. Geometrically, the direction of the curl is the direction of the axis of the paddle wheel when the wheel is oriented in the plane of maximum circulation.

### Pumpkin Quarters

A pumpkin that has been sectioned into quarters can be used as an interactive physical representation of a spherical coordinate system.

### Pineapple Sections

Canned pineapple sections can be used as an interactive physical representation of a cylindrical coordinate system.

### Surfaces

We use plastic “surface” graphs of multivariable functions that are dry-erasable, transparent, and lightweight. The surfaces were created as part of the Raising Calculus to the Surface and Raising Physics to the Surface projects. Surface Kits include a surface, its corresponding laminated contour map, and a slope measurement tool.

### Contour Maps & Vector Maps

We print out contour maps or vector maps and put them in dry-erasable sleeves. Often, our maps correspond to plastic surfaces.

### Equipment Primarily Used In Labs

• Force Meters
• Oscilloscopes
• Coaxial Cable
• Thermometers
props equipment


text:
# My Favorite One-Liners: Part 113

I tried a new wisecrack when teaching my students about Euler’s formula. It worked gloriously.

# Euler’s Equation

This was hands-down my favorite variant of the “distracted boyfriend” meme that went around the internet last year.

# How I Impressed My Wife: Part 6a

This series was inspired by a question that my wife asked me: calculate

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

Originally, I multiplied the top and bottom of the integrand by $\tan^2 x$ and performed a substitution. However, as I’ve discussed in this series, there are four different ways that this integral can be evaluated.
Starting with today’s post, I’ll begin a fifth method. I really like this integral, as it illustrates so many different techniques of integration as well as the trigonometric tricks necessary for computing some integrals.
Since $Q$ is independent of $a$, I can substitute any convenient value of $a$ that I want without changing the value of $Q$. As shown in previous posts, substituting $a =0$ yields the following simplification:

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

$= \displaystyle \int_{0}^{2\pi} \frac{dx}{\cos^2 x + 2 \cdot 0 \cdot \sin x \cos x + (0^2 + b^2) \sin^2 x}$

$= \displaystyle \int_{0}^{2\pi} \frac{dx}{\cos^2 x + b^2 \sin^2 x}$

$= \displaystyle \int_{-\pi}^{\pi} \frac{dx}{\cos^2 x + b^2 \sin^2 x}$

$= \displaystyle \int_{-\infty}^{\infty} \frac{ 2(1+u^2) du}{u^4 + (4 b^2 - 2) u^2 + 1}$

Earlier, I evaluated this last integral using partial fractions, separating into the cases $|b| = 1$, $|b| > 1$, and $|b| < 1$. Now, I’ll calculate this same integral using contour integration. (See Wikipedia and Mathworld for more details.)

It turns out that $Q$ can be rewritten as

$Q = \displaystyle \lim_{R \to \infty} \oint_{C_R} \frac{ 2(1+z^2) dz}{z^4 + (4 b^2 - 2) z^2 + 1}$,

where $C_R$ is the contour in the complex plane shown above (graphic courtesy of Mathworld). That’s because

$\displaystyle \lim_{R \to \infty} \oint_{C_R} \frac{ 2(1+z^2) dz}{z^4 + (4 b^2 - 2) z^2 + 1}$

$= \displaystyle \lim_{R \to \infty} \int_{-R}^R \frac{ 2(1+z^2) dz}{z^4 + (4 b^2 - 2) z^2 + 1} + \lim_{R \to \infty} \int_{\gamma_R} \frac{ 2(1+z^2) dz}{z^4 + (4 b^2 - 2) z^2 + 1}$

$= \displaystyle \int_{-\infty}^{\infty} \frac{ 2(1+z^2) dz}{z^4 + (4 b^2 - 2) z^2 + 1} + \lim_{R \to \infty} \int_{\gamma_R} \frac{ 2(1+z^2) du}{z^4 + (4 b^2 - 2) z^2 + 1}$

$= Q + \displaystyle \lim_{R \to \infty} \int_{\gamma_R} \frac{ 2(1+z^2) dz}{z^4 + (4 b^2 - 2) z^2 + 1}$

To show that the limit of the last integral is equal to 0, I use the parameterization $z = R e^{i \theta}$, so that $dz = i R e^{i \theta}$:

$\displaystyle \lim_{R \to \infty} \left| \int_{\gamma_R} \frac{ 2(1+z^2) dz}{z^4 + (4 b^2 - 2) z^2 + 1} \right|$

$= \displaystyle \lim_{R \to \infty} \left| \int_0^{\pi} \frac{ 2R(1+R^2 e^{2i\theta}) d\theta}{R^4 e^{4 i\theta} + (4 b^2 - 2) R^2 e^{2i\theta} + 1} \right|$

$\le \displaystyle \lim_{R \to \infty} \pi \max_{0 \le \theta \le \pi} \left| \frac{ 2R(1+R^2 e^{2i\theta})}{R^4 e^{4 i\theta} + (4 b^2 - 2) R^2 e^{2i\theta} + 1} \right|$

$= \displaystyle \pi \max_{0 \le \theta \le \pi} \lim_{R \to \infty} \left| \frac{ 2R(1+R^2 e^{2i\theta})}{R^4 e^{4 i\theta} + (4 b^2 - 2) R^2 e^{2i\theta} + 1} \right|$

$= \displaystyle \pi \max_{0 \le \theta \le \pi} 0$

$= 0$.

The above limit is equal to zero because the numerator grows like $R^3$ while the denominator grows like $R^4$. (This can be more laboriously established using L’Hopital’s rule).

Therefore, I have shown that

$Q = \displaystyle \lim_{R \to \infty} \oint_{C_R} \frac{ 2(1+z^2) dz}{z^4 + (4 b^2 - 2) z^2 + 1}$,

and this contour integral can be computed using residues.

I’ll continue with this fifth evaluation of the integral, starting with the case $|b| = 1$, in tomorrow’s post.

# How I Impressed My Wife: Part 4b

Previously in this series, I have used two different techniques to show that

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x} = \displaystyle \frac{2\pi}{|b|}.$

Originally, my wife had asked me to compute this integral by hand because Mathematica 4 and Mathematica 8 gave different answers. At the time, I eventually obtained the solution by multiplying the top and bottom of the integrand by $\sec^2 x$ and then employing the substitution $u = \tan x$ (after using trig identities to adjust the limits of integration).
But this wasn’t the only method I tried. Indeed, I tried two or three different methods before deciding they were too messy and trying something different. So, for the rest of this series, I’d like to explore different ways that the above integral can be computed.
Let me backtrack to a point in the middle of the previous solution:

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

$= \displaystyle \int_0^{2\pi} \frac{2 \, dx}{1+\cos 2x + 2 a \sin 2x + (a^2 + b^2)(1-\cos 2x)}$

$= 2 \displaystyle \int_0^{2\pi} \frac{d\theta}{(1+a^2+b^2) + 2 a \sin \theta + (1 - a^2 - b^2) \cos \theta}$

$= 2 \displaystyle \int_{0}^{2\pi} \frac{d\theta}{S + R \cos (\theta - \alpha)}$

$= 2 \displaystyle \int_{0}^{2\pi} \frac{d\phi}{S + R \cos \phi}$,

where $R = \sqrt{(2a)^2 + (1-a^2-b^2)^2}$ and $S = 1 + a^2 + b^2$ (and $\alpha$ is a certain angle that is now irrelevant at this point in the calculation).

Earlier in this series, I used the magic substitution $u = \tan \displaystyle \frac{\phi}{2}$ to evaluate this last integral. Now, I’ll instead use contour integration; see Wikipedia for more details. I will use Euler’s formula as a substitution (see here and here for more details):

$z = e^{i \phi} = \cos \phi + i \sin \phi$,

so that the integral $Q$ is transformed to a contour integral in the complex plane. Under this substitution, as discussed in yesterday’s post,

$\cos \phi = \displaystyle \frac{1}{2} \left[z + \displaystyle \frac{1}{z} \right]$

and

$d\phi = \displaystyle -\frac{i}{z} dz$

Employing this substitution, the region of integration changes from $0 \le \phi \le 2\pi$ to a the unit circle $C$, a closed counterclockwise contour in the complex plane:

$Q = 2 \displaystyle \oint_C \frac{\displaystyle -\frac{i}{z} dz}{S + \displaystyle \frac{R}{2} \left[z + \displaystyle \frac{1}{z} \right]}$

$= -4i \displaystyle \oint_C \frac{dz}{Rz^2 + 2Sz + R}$

$= \displaystyle -\frac{4i}{R} \oint_C \frac{dz}{z^2 + 2\frac{S}{R}z + 1}$

While this looks integral in the complex plane looks a lot more complicated than a regular integral, it’s actually a lot easier to compute using residues. I’ll discuss the computation of this contour integral in tomorrow’s post.

# How I Impressed My Wife: Part 4a

Previously in this series, I have used two different techniques to show that

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x} = \displaystyle \frac{2\pi}{|b|}.$

Originally, my wife had asked me to compute this integral by hand because Mathematica 4 and Mathematica 8 gave different answers. At the time, I eventually obtained the solution by multiplying the top and bottom of the integrand by $\sec^2 x$ and then employing the substitution $u = \tan x$ (after using trig identities to adjust the limits of integration).
But this wasn’t the only method I tried. Indeed, I tried two or three different methods before deciding they were too messy and trying something different. So, for the rest of this series, I’d like to explore different ways that the above integral can be computed.
Let me backtrack to a point in the middle of the previous solution:

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

$= \displaystyle \int_0^{2\pi} \frac{2 \, dx}{1+\cos 2x + 2 a \sin 2x + (a^2 + b^2)(1-\cos 2x)}$

$= 2 \displaystyle \int_0^{2\pi} \frac{d\theta}{(1+a^2+b^2) + 2 a \sin \theta + (1 - a^2 - b^2) \cos \theta}$

$= 2 \displaystyle \int_{0}^{2\pi} \frac{d\theta}{S + R \cos (\theta - \alpha)}$

$= 2 \displaystyle \int_{0}^{2\pi} \frac{d\phi}{S + R \cos \phi}$,

where $R = \sqrt{(2a)^2 + (1-a^2-b^2)^2}$ and $S = 1 + a^2 + b^2$ (and $\alpha$ is a certain angle that is now irrelevant at this point in the calculation).

In the previous solution, I used the “magic substitution” $u = \tan \displaystyle \frac{\phi}{2}$ to convert the last integrand to a simple rational function. Starting today, I’ll use a completely different technique to compute this last integral.

The technique that I’ll use is contour integration; see Wikipedia for more details. I will use Euler’s formula as a substitution (see here and here for more details):

$z = e^{i \phi} = \cos \phi + i \sin \phi$,

so that the integral $Q$ is transformed to a contour integral in the complex plane.

Under this substitution,

$\displaystyle \frac{1}{z} = e^{-i\phi} = \cos(-\phi) + i \sin(-\phi) = \cos \phi - i \sin \phi$

Using these last two equations, I can solve for $\cos \phi$ and $\sin \phi$ in terms of $z$ and $\displaystyle \frac{1}{z}$. I’ll begin with $\cos \phi$:

$z + \displaystyle \frac{1}{z} = \cos \phi + i \sin \phi + \cos \phi - i \sin \phi$

$z + \displaystyle \frac{1}{z} = 2 \cos \phi$

$\displaystyle \frac{1}{2} \left[z + \displaystyle \frac{1}{z} \right] = \cos \phi$

Though not necessary for this particular, let me solve for $\sin \phi$ for completeness:

$z - \displaystyle \frac{1}{z} = \cos \phi + i \sin \phi - [ \cos \phi - i \sin \phi]$

$z - \displaystyle \frac{1}{z} = 2i \sin \phi$

$\displaystyle \frac{1}{2i} \left[z - \displaystyle \frac{1}{z} \right] = \sin\phi$

Finally, let me solve for the differential $d\phi$:

$z = e^{i \phi}$

$dz = i e^{i \phi} d\phi$

$\displaystyle \frac{1}{i} e^{-i \phi} dz = d\phi$

$-i e^{-i \phi} dz = d\phi$

$\displaystyle -\frac{i}{z} dz = d\phi$

I’ll continue with this different method of evaluating this integral in tomorrow’s post.

# Full lesson plan: Modular multiplication and encryption

Over the summer, I occasionally teach a small summer math class for my daughter and her friends around my dining room table. Mostly to preserve the memory for future years… and to provide a resource to my friends who wonder what their children are learning… I’ll write up the best of these lesson plans in full detail.

In this lesson, the students practiced their skills with multiplication and division to create modular multiplication tables. Though this is a concept ordinarily first encountered in an undergraduate class in number theory or abstract algebra, there’s absolutely no reason why elementary students who’ve mastered multiplication can’t do this exercise. This exercise strengthens the notion of dividing with a remainder and leads to a fun application with encrypting and decrypting secret messages. Indeed, this activity made be viewed as a child-appropriate version of the RSA encryption algorithm that’s used every time we use our credit cards. This was mentioned in two past posts: https://meangreenmath.com/2013/10/17/engaging-students-finding-prime-factorizations and https://meangreenmath.com/2013/07/11/cryptography-as-a-teaching-tool

This lesson plan is written in a 5E format — engage, explore, explain, elaborate, evaluate — which promotes inquiry-based learning and fosters student engagement.

Lesson Plan: Kid RSA Lesson

Other Documents:

Vocabulary Sheet

Three Letter Words

RSA Numbers

Modular Multiplication Assessment

Modular Multiplcation Practice

Kid RSA

# Full lesson plan: Platonic solids

Over the summer, I occasionally teach a small summer math class for my daughter and her friends around my dining room table. Mostly to preserve the memory for future years… and to provide a resource to my friends who wonder what their children are learning… I’ll write up the best of these lesson plans in full detail.

This was the first lesson that I taught to this audience: constructing the five regular polyhedra and inductively deriving Euler’s formula. This lesson plan is written in a 5E format — engage, explore, explain, elaborate, evaluate — which promotes inquiry-based learning and fosters student engagement.

Platonic Solids Lesson

Post Assessment 1

Post Assessment 2

V-E-F Chart

Vocabulary Sheet

# Math T-shirts

The following are the three finalists for the T-shirt contest sponsored by the Mathematical Association of America.

# Reminding students about Taylor series (Part 6)

Sadly, at least at my university, Taylor series is the topic that is least retained by students years after taking Calculus II. They can remember the rules for integration and differentiation, but their command of Taylor series seems to slip through the cracks. In my opinion, the reason for this lack of retention is completely understandable from a student’s perspective: Taylor series is usually the last topic covered in a semester, and so students learn them quickly for the final and quickly forget about them as soon as the final is over.

Of course, when I need to use Taylor series in an advanced course but my students have completely forgotten this prerequisite knowledge, I have to get them up to speed as soon as possible. Here’s the sequence that I use to accomplish this task. Covering this sequence usually takes me about 30 minutes of class time.

I should emphasize that I present this sequence in an inquiry-based format: I ask leading questions of my students so that the answers of my students are driving the lecture. In other words, I don’t ask my students to simply take dictation. It’s a little hard to describe a question-and-answer format in a blog, but I’ll attempt to do this below.

In the previous posts, I described how I lead students to the definition of the Maclaurin series

$f(x) = \displaystyle \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} x^k$,

which converges to $f(x)$ within some radius of convergence for all functions that commonly appear in the secondary mathematics curriculum.

Step 7. Let’s now turn to trigonometric functions, starting with $f(x) = \sin x$.

What’s $f(0)$? Plugging in, we find $f(0) = \sin 0 = 0$.

As before, we continue until we find a pattern. Next, $f'(x) = \cos x$, so that $f'(0) = 1$.

Next, $f''(x) = -\sin x$, so that $f''(0) = 0$.

Next, $f'''(x) = -\cos x$, so that $f''(0) = -1$.

No pattern yet. Let’s keep going.

Next, $f^{(4)}(x) = \sin x$, so that $f^{(4)}(0) = 0$.

Next, $f^{(5)}(x) = \cos x$, so that $f^{(5)}(0) = 1$.

Next, $f^{(6)}(x) = -\sin x$, so that $f^{(6)}(0) = 0$.

Next, $f^{(7)}(x) = -\cos x$, so that $f^{(7)}(0) = -1$.

OK, it looks like we have a pattern… albeit more awkward than the patterns for $e^x$ and $\displaystyle \frac{1}{1-x}$. Plugging into the series, we find that

$\displaystyle \sin x= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \dots$

If we stare at the pattern of terms long enough, we can write this more succinctly as

$\sin x = \displaystyle \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}$

The $(-1)^n$ term accounts for the alternating signs (starting on positive with $n=0$), while the $2n+1$ is needed to ensure that each exponent and factorial is odd.

Let’s see… $\sin x$ has a Taylor expansion that only has odd exponents. In what other sense are the words “sine” and “odd” associated?

In Precalculus, a function $f(x)$ is called odd if $f(-x) = -f(x)$ for all numbers $x$. For example, $f(x) = x^9$ is odd since $f(-x) = (-x)^9 = -x^9$ since 9 is a (you guessed it) an odd number. Also, $\sin(-x) = -\sin x$, and so $\sin x$ is also an odd function. So we shouldn’t be that surprised to see only odd exponents in the Taylor expansion of $\sin x$.

A pedagogical note: In my opinion, it’s better (for review purposes) to avoid the $\displaystyle \sum$ notation and simply use the “dot, dot, dot” expression instead. The point of this exercise is to review a topic that’s been long forgotten so that these Taylor series can be used for other purposes. My experience is that the $\displaystyle \sum$ adds a layer of abstraction that students don’t need to overcome for the time being.

Step 8. Let’s now turn try $f(x) = \cos x$.

What’s $f(0)$? Plugging in, we find $f(0) = \cos 1 = 0$.

Next, $f'(x) = -\sin x$, so that $f'(0) = 0$.

Next, $f''(x) = -\cos x$, so that $f'(0) = -1$.

It looks like the same pattern of numbers as above, except shifted by one derivative. Let’s keep going.

Next, $f'''(x) = \sin x$, so that $f'''(0) = 0$.

Next, $f^{(4)}(x) = \cos x$, so that $f^{(4)}(0) = 1$.

Next, $f^{(5)}(x) = -\sin x$, so that $f^{(5)}(0) = 0$.

Next, $f^{(6)}(x) = -\cos x$, so that $f^{(6)}(0) = -1$.

OK, it looks like we have a pattern somewhat similar to that of $\sin x$, except only involving the even terms. I guess that shouldn’t be surprising since, from precalculus we know that $\cos x$ is an even function since $\cos(-x) = \cos x$ for all $x$.

Plugging into the series, we find that

$\displaystyle \cos x= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \dots$

If we stare at the pattern of terms long enough, we can write this more succinctly as

$\cos x = \displaystyle \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}$

As we saw with $e^x$, the above series converge quickest for values of $x$ near $0$. In the case of $\sin x$ and $\cos x$, this may be facilitated through the use of trigonometric identities, thus accelerating convergence.

For example, the series for $\cos 1000^o$ will converge quite slowly (after converting $1000^o$ into radians). However, we know that

$\cos 1000^o= \cos(1000^o - 720^o) =\cos 280^o$

using the periodicity of $\cos x$. Next, since $\latex 280^o$ is in the fourth quadrant, we can use the reference angle to find an equivalent angle in the first quadrant:

$\cos 1000^o = \cos 280^o = \cos(360^o - 80^o) = \cos 80^o$

Finally, using the cofunction identity $\cos x = \sin(90^o - x)$, we find

$\cos 1000^o = \cos 80^o = sin(90^o - 80^o) = \sin 10^o$.

In this way, the sine or cosine of any angle can be reduced to the sine or cosine of some angle between $0^o$ and $45^o = \pi/4$ radians. Since $\pi/4 < 1$, the above power series will converge reasonably rapidly.

Step 10. For the final part of this review, let’s take a second look at the Taylor series

$e^x = \displaystyle 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} + \frac{x^6}{6} + \frac{x^7}{7} + \dots$

Just to be silly — for no apparent reason whatsoever, let’s replace $x$ by $ix$ and see what happens:

$e^{ix} = \displaystyle 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \dots + i \left[\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \dots \right]$

after separating the terms that do and don’t have an $i$.

Hmmmm… looks familiar….

So it makes sense to define

$e^{ix} = \cos x + i \sin x$,

which is called Euler’s formula, thus proving an unexpected connected between $e^x$ and the trigonometric functions.


text:
Algebraic Expressions

Basic Concept

Coefficient, Variable, and Constant

• A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y
• A Coefficient is a number used to multiply a variable, $$8x$$ means $$8$$ times $$x$$, so $$8$$ is a coefficient
• A Constant is a number on its own

\begin{array}{cccc}
\huge \color{blue} -2x^3  && \huge \color{blue} + &&\huge \color {blue} 5\\\\
1^{\text{st}} \text{ term} && && 2^{\text{nd}} \text{ term}\\
\text{Coefficient : −2}&& &&\text{5 called constant}\\
\text{Exponent of variable : 3}&& &&\text{Exponent of variable : 0}
\end{array}

Polynomial

MonomialThe special names for polynomials with 1 term.

Example: $$3xy^2,\: 2x^5,\: 4y, \:5x$$

BinomialThe special names for polynomials with 2 terms.

Example: $$7x - 2, \:8x^2 - 3y, \:2x - 4y^2, \:9a - b^4$$

TrinomialThe special names for polynomials with 3 terms.

Example: $$2x^2 + 3y - 5z, \:5a^2 + b + c^2, \: 3i - 2j + 5k$$


text:
## Large-scale hierarchical networks: How to compute an optimal architecture?

Please always quote using this URN: urn:nbn:de:0297-zib-7799
• In this article, we present a mathematical model and an algorithm to support one of the central strategic planning decisions of network operators: How to organize a large number of locations into a hierarchical network? We propose a solution approach that is based on mixed-integer programming and Lagrangian relaxation techniques. As major advantage, our approach provides not only solutions but also worst-case quality guarantees. Real-world scenarios with more than 750 locations have been solved within 30 minutes to less than 1\% off optimality.

### Additional Services

Author: Andreas Bley, Thorsten Koch, Roland Wessäly ZIB-Report Hierarchical Networks ; Lagrangian Relaxation; Mixed-Integer Programming 68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area) / 68Mxx Computer system organization / 68M10 Network design and communication [See also 68R10, 90B18] 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C11 Mixed integer programming 2004/01/19 ZIB-Report (04-04) App. in Networks 2004: 11th International Telecomunications Network Strategy and Planning Symposium 2004, Vienna, Austria, pp. 429-434, H. Kaindl (ed.), VDE Verlag, ISBN 3-8007-2840-0

$Rev: 13581$


text:
# M Borisyak

Ontology type: schema:Person

NAME

M

SURNAME

Borisyak

### Publications in SciGraph latest 50 shown

• 2018-12 Search for lepton-flavour-violating decays of Higgs-like bosons in THE EUROPEAN PHYSICAL JOURNAL C
• 2018-12 Evidence for an ηc(1S)π- resonance in B0→ηc(1S)K+π- decays in THE EUROPEAN PHYSICAL JOURNAL C
• 2018-11 Measurement of the time-integrated CP asymmetry in D0 → KS0KS0 decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-11 Study of ϒ production in pPb collisions at sNN=8.16 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-10 Erratum to: Measurement of the CKM angle γ using B± → DK± with D → KS0π+π−, KS0K+K− decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-10 Search for beautiful tetraquarks in the ϒ(1S)μ+μ− invariant-mass spectrum in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-10 Central exclusive production of J/ψ and ψ(2S) mesons in pp collisions at s=13 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-09 Angular moments of the decay Λb0 → Λμ+μ− at low hadronic recoil in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-09 Search for a dimuon resonance in the ϒ mass region in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-09 Measurement of Z → τ+τ− production in proton-proton collisions at s=8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-08 Measurement of forward top pair production in the dilepton channel in pp collisions at s=13 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-08 Search for CP violation using triple product asymmetries in Λb0 → pK−π+π−, Λb0 → pK−K+K− and Ξb0 → pK−K−π+ decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-08 Measurement of the CKM angle γ using B± → DK± with D → KS0π+π−, KS0K+K− decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-08 Observation of the decay Λb0 → ψ(2S)pπ− in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-08 Measurement of Ds± production asymmetry in pp collisions at s=7 and 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-08 Observation of the decay B¯s0→χc2K+K− in the ϕ mass region in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-07 Measurement of ϒ production in pp collisions at s=13 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-07 Evidence for the decay BS0→K¯∗0μ+μ− in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-06 Measurement of the inelastic pp cross-section at a centre-of-mass energy of 13 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-06 Measurement of CP violation in B0 → D∓π± decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-06 Studies of the resonance structure in D0→K∓π±π±π∓ decays in THE EUROPEAN PHYSICAL JOURNAL C
• 2018-05 Erratum to: Measurement of CP observables in B± → DK*± decays using two- and four-body D final states in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-05 Measurement of the CP asymmetry in B− → Ds−D0 and B− → D−D0 decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-03 Measurements of the branching fractions of Λc+ → pπ−π+, Λc+ → pK−K+, and Λc+ → pπ−K+ in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-03 Measurement of CP asymmetry in Bs0 → Ds∓K± decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-03 A measurement of the CP asymmetry difference between Λc+ → pK−K+ and pπ−π+ decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-03 First measurement of the CP-violating phase ϕsdd¯ in Bs0 → (K+π−)(K−π+) decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-03 Search for the lepton-flavour violating decays B( s)0 → e±μ∓ in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-02 Measurement of branching fractions of charmless four-body Λb0 and Ξb0 decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-01 First observation of B+ → Ds+K+K− decays and a search for B+ → Ds+ϕ decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-01 Search for excited Bc+ states in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-12 Bose-Einstein correlations of same-sign charged pions in the forward region in pp collisions at s=7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-12 Measurement of the B± production cross-section in pp collisions at s=7 and 13 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-12 Measurement of the Y(nS) polarizations in pp collisions at s=7 and 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-12 Updated search for long-lived particles decaying to jet pairs in THE EUROPEAN PHYSICAL JOURNAL C
• 2017-11 Study of bb¯ correlations in high energy proton-proton collisions in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-11 Updated branching fraction measurements of B( s)0 → KS0h+h′ − decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-11 Measurement of CP observables in B± → DK*± decays using two- and four-body D final states in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-11 Measurement of CP violation in B0 → J/ψKS0 and B0 → ψ(2S)KS0 decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-10 Study of prompt D0 meson production in pPb collisions at sNN=5 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-10 Improved limit on the branching fraction of the rare decay KS0→μ+μ- in THE EUROPEAN PHYSICAL JOURNAL C
• 2017-10 Erratum to: Measurement of the J/ψ pair production cross-section in pp collisions at s=13 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-09 Study of charmonium production in b-hadron decays and first evidence for the decay Bs0→ϕϕϕ in THE EUROPEAN PHYSICAL JOURNAL C
• 2017-08 Test of lepton universality with B0 → K*0ℓ+ℓ− decays in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-08 Resonances and CP violation in Bs0 and B¯s0→J/ψK+K− decays in the mass region above the ϕ(1020) in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-07 Observation of the decay Bs0 → ηcϕ and evidence for Bs0 → ηcπ+π− in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-06 Observation of the decay Λb0 → pK−μ+μ− and a search for CP violation in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-06 Measurement of the J/ψ pair production cross-section in pp collisions at s=13 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-06 Measurements of prompt charm production cross-sections in pp collisions at s=5 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-05 Study of the D0p amplitude in Λb0 → D0pπ− decays in JOURNAL OF HIGH ENERGY PHYSICS
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text:
# Tamarkin’s paper, chapter 3

### One comment

1. chriskuo says:

Let $K$ be a compact set in a locally compact topological space $X$. In this chapter, Tamarkin defines $\mathbb{Z}_K$ by the homotopy limit of \v{C}ech complexes formed by its finite open covers. My question is in the case $K$ can be associated with some good stratification, for example, $[a,b] \subset \mathbb{R}^1$, is this object the same as the \v{C}ech complex associated to the stratification?


text:
## The convolution structure for Jacobi function expansions.(English)Zbl 0267.42009

### MSC:

 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 44A15 Special integral transforms (Legendre, Hilbert, etc.) 44A35 Convolution as an integral transform 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 43A90 Harmonic analysis and spherical functions
Full Text:

### References:

 [1] Askey, R., Jacobi polynomials, I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. To appear inSIAM J. Math. Anal. · Zbl 0242.33019 [2] Chébli, H., Sur la positivité des opérateurs de “translation généralisée{” associés à un operateur de Sturm-Liouville sur [0, .C. R. Acad. Sci. Paris 275 (1972), 601–604.} · Zbl 0241.47032 [3] Erdélyi, A., et al.,Higher transcendental functions vol. I. McGraw-Hill, New York, 1953. · Zbl 0051.30303 [4] Flensted-Jensen, M., Paley-Wiener type theorems for a differential operator connected with symmetric spaces.Arkiv för Matematik 10 (1972), 143–162. · Zbl 0233.42012 [5] Gasper, G., Positivity and the convolution structure for Jacobi series.Ann. of Math. 93 (1971), 112–118. · Zbl 0208.08101 [6] –, Banach algebras for Jacobi series and positivity of a kernel.Ann. of Math. 95 (1972), 261–280. · Zbl 0236.33013 [7] Helgason, S.,Differential geometry and symmetric spaces. Academic Press, New York, 1962. · Zbl 0111.18101 [8] Koornwinder, T. H., The addition formula for Jacobi polynomials, I. Summary of results.Nederl. Akad. Wetensch. Proc. Ser A 75=Indag. Math. 34 (1972), 188–191. · Zbl 0247.33017 [9] Koornwinder, T. H. Jacobi polyomials, II. An analytic proof of the product formula. To appear inSIAM J. Math. Anal. · Zbl 0242.33020 [10] Kunze, R. A. &Stein, E. M., Uniformly bounded representations and harmonic analysis of the 2{$$\times$$}2 real unimodular group.Amer. J. Math. 82 (1960), 1–62. · Zbl 0156.37104 [11] Tréves, F.,Topological vector spaces, distributions and kernels. Academic Press, New York, 1967. · Zbl 0171.10402 [12] Weinberger, A., A maximum property of the Cauchy problem.Ann. of Math. 64 (1956), 505–512. · Zbl 0072.10003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# A proof of the scaling/shift property of variance

I would like to show that $Var(aX + b) = a^2 Var(X)$ using a different proof from my book

Let $Y = aX + b$

$Var(Y) = E(Y^2) - E[Y]^2$

= $E[(aX + b)^2] - E(aX + b)^2$

= $E[(a^2 X^2 + b^2 + 2abX] - (a\mu+b)^2$

= $a^2E(X^2) + b^2 + 2abE[X] - (a^2\mu^2 + b^2 + 2ab\mu)$

= $a^2E(X^2) + b^2 + 2ab\mu - (a^2E[X]^2 + b^2 + 2ab\mu)$

= $a^2E[X^2] - a^2E[X]^2$

= $a^2(E[X^2]-E[X]^2)$

= $a^2 Var(X)$

My book does like three lines and they start out with $Var(aX + b) = E[(aX + b - (a\mu + b) ) ^2]$. I don't understand why they substract $(a\mu + b)$ instead of $\mu$ because $Var(X) = E[(X - \mu)^2]$

Is my proof correct?

-

As mentioned above, they subtract $a\mu + b$ because that is the mean of $Y$. Every time you find the variance of a random variable from the definition, you need to subtract its mean in the formula.
Note that $E[Y] = E[aX + b] = aE[X] + b = a\mu + b$.
And what you're trying to find is $Var(Y)$, not $Var(X)$. – Tunococ Oct 17 '12 at 6:45
I am not sure if you feel that I answered your question or not. I basically said the same thing as Ken Dunn. You asked why they subtract $a\mu + b$ instead of $\mu$. My answer is because you're computing $Var(Y)$, and $E[Y] = a\mu + b$. – Tunococ Oct 18 '12 at 9:59


text: Visually, the symbol is a squiggly equals sign. Symbol Description Shortcut ¶ paragraph sign ALT+0182 ± plus-or-minus sign ALT+ In the computing field, especially in Unix-based systems, the tilde indicates the user's home directory. The Approximately Equal symbol (≈) is a mathematical operator used to indicate that two expressions or statements are similar but not exactly equal to each other. by Helen Tupper and Sarah Ellis | Jan 2, 2020. For this reason it is sometimes recommended to avoid the == operator in JavaScript in favor of ===. Another approximation symbol is the double tilde ≈, meaning "approximately equal to". Equality of truth values (through bi-implication or logical equivalence), may be denoted by various symbols including =, ~, and в‡”. Account & Lists Account Returns & Orders. Hello, Sign in. ≈), or the single "squiggly" (~), if shorthand is approved. Embrace Opportunity and Carve Your Own Path Through the Squiggly World of Work. From Simple English Wikipedia, the free encyclopedia, https://simple.wikipedia.org/w/index.php?title=Equals_sign&oldid=7193673, Creative Commons Attribution/Share-Alike License, "â" (one tilde above two lines, often used in, "â" (two lines with a dot above them, often used for "is defined as"), "â¡" (three lines, often used for equivalence). It's either the "squiggly equals sign" (i.e. 1844 === (1800..1899) is false, since it is interpreted to mean Integer#=== rather than Range#===.[11]. 4. Note: you can see all of Name of the symbol combinations that you can use in the AutoCorrect Options. ", "Conventions for interlinear morpheme-by-morpheme glosses", "An International Perspective between Problem Types in Textbooks and Students' understanding of relational equality", Scientific Symbols, Icons, Mathematical Symbols, https://en.wikipedia.org/w/index.php?title=Equals_sign&oldid=995775224#Approximately_equal, Short description is different from Wikidata, Articles with unsourced statements from June 2013, Articles with unsourced statements from August 2020, Articles lacking reliable references from August 2020, Creative Commons Attribution-ShareAlike License, This page was last edited on 22 December 2020, at 20:47. The equals sign is also used in defining attributeвЂ“value pairs, in which an attribute is assigned a value. But = is used for equality and not assignment in the Pascal family, Ada, Eiffel, APL, and other languages. So what does 80 divided by 1.94 approximately equal? In PHP, the triple equals sign, ===, denotes value and type equality,[8] meaning that not only do the two expressions evaluate to equal values, but they are also of the same data type. The first important computer programming language to use the equals sign was the original version of Fortran, FORTRAN I, designed in 1954 and implemented in 1957. This symbol (in US English) informally means "approximately", "about", or "around", such as "~30 minutes before", meaning "approximately 30 minutes before". ﹦. COVID-19 Updates and Information Personal Income Tax Cybersecurity Passenger (Class D) Driver's Licenses Stop the Spread SNAP benefits (formerly food stamps) Learn more. Welcome to Useful Shortcuts, THE Alt Code resource!. In Fortran, = serves as an assignment operator: X = 2 sets the value of X to 2. Not a recommended shortcut if doing serious mathematics. In chemical formulas, the two parallel lines denoting a double bond are commonly rendered using an equals sign. If you want to see the big symbol, enter \bigcup: 3. [17], In Morse code, the equals sign is encoded by the letters B (-...) and T (-) run together (-...-). ＝. 0 == false is false. As well as Fortran, = is used for assignment in such languages as C, Perl, Python, awk, and their descendants. Use your numeric keypad with your NUM LOCK on and you will be good to go! squiggly definition: 1. consisting of a line or lines that curve and twist in a way that is not regular: 2. consisting…. [9], In Ruby, equality under == requires both operands to be of identical type, e.g. After entering the symbol, click the space; it changed entering a name to the appropriate symbol. The Section 8 housing choice voucher program is a federal government program for assisting very low-income families, the elderly, and the disabled to afford decent, safe, and sanitary housing in the private market. If you only occasionally need to type the does not equal sign (or other math symbols), the first section below has you covered (you’ll also improve your knowledge of Microsoft Word too). Fullwidth Equals Sign. The triple bar symbol в‰Ў (U+2261, LaTeX \equiv) is often used to indicate an identity, a definition (which can also be represented by U+225D ≝ EQUAL TO BY DEFINITION or U+2254 ≔ COLON EQUALS), or a congruence relation in modular arithmetic. Superscript Equals Sign. Computers display the equals sign with the Unicode or ASCII character 003D (in hexadecimal ). It looks like two parallel horizontal lines. Learn more. The symbol used to say when items are not equal is "â " (slashed equal sign).[1]. ?, and ( ) with Thesaurus.com. The etymology of the word "equal" is from the Latin word "æqualis", as meaning "uniform", "identical", or "equal", from aequus ("level", "even", or "just"). If you like keyboard shortcuts and use special characters, here are a few more for Microsoft Windows. The two squiggly lines is one of at least five symbols used for”approximately equal” in mathematics Symbols used to denote items that are approximately equal are wavy equals signs. Click Alt+= again to exit from the equitation. For example, the assignment X = X + 2 increases the value of X by 2. Some of these symbols include: Each of these symbols has more than one possible meaning, and are all used to state that two things are about equal (or equivalent in some way). JavaScript has the same semantics for ===, referred to as "equality without type coercion". Fortran did not have an equality operator (it was only possible to compare an expression to zero, using the arithmetic IF statement) until FORTRAN IV was released in 1962, since when it has used the four characters .EQ. [12][13] The Unicode character used for the tone letter (U+A78A)[14] is different from the mathematical symbol (U+003D). If you need help using alt codes find and note down the alt code you need then visit our instructions for using alt codes page. If this is the case then A, x and b are conformable. ≈ (Unicode 2248) ≃ (Unicode 2243) – used to indicate asymptotically equal to In LaTeX, this is done with the "\neq" command. The tilde is also used to indicate congruence of shapes by placing it over an = symbol, thus: ≅. ﹦. A possibly unique case of the equals sign of European usage in a person's name, specifically in a double-barreled name, was by pioneer aviator Alberto Santos-Dumont, as he is also known not only to have often used a double hyphen resembling an equals sign = between his two surnames in place of a hyphen, but also seems to have personally preferred that practice, to display equal respect for his father's French ethnicity and the Brazilian ethnicity of his mother.[15]. The equals sign is sometimes used incorrectly within a mathematical argument to connect math steps in a non-standard way, rather than to show equality (especially by early mathematics students). The symbol || was used by some and Г¦ (or Е“), from the Latin word aequalis meaning equal, was widely used into the 1700s" (History of Mathematics, University of St Andrews).[7]. Symbol Meaning Example In Words; Triangle: ABC has 3 equal sides: Triangle ABC has three equal sides: Angle: ABC is 45° The angle formed by ABC is 45 degrees. If interpreted strictly as it says, it would imply that: A correct version of the argument would be: This difficulty results from subtly different uses of the sign in education. [1], The symbol в‰… is often used to indicate isomorphic algebraic structures or congruent geometric figures.[1]. [citation needed]. The equals sign is placed between the things stated to be exactly equal or the same. A rival programming-language usage was pioneered by the original version of ALGOL, which was designed in 1958 and implemented in 1960. Paperback $20.41$ 20. WINDOWS: on computers with Windows operating system like Windows 8, Win 7, Vista, Windows XP, etc.. To get the letter, character, sign or symbol "=": ( Equals sign ) on computers with Windows operating system: 1) Press the "Alt" key on your keyboard, and do not let go. Starting in algebra courses, the sign takes on a relational meaning of equality between two calculations. In his book Recorde explains his design of the "Gemowe lines" (meaning twin lines, from the Latin gemellus[5], And to auoide the tediouЕїe repetition of theЕїe woordes : is equalle to : I will Еїette as I doe often in woorke vЕїe, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicauЕїe noe .2. thynges, can be moare equalle. This is a weaker statement than the other two. The symbol for approximately equal is a squiggly equals sign. The expression 0 == false is true, but 0 == undefined is false, even though both sides of the == act the same in Boolean context. In Unicode and ASCII, it has the code point 3D. Role. Some of the commonly used symbols: \infty - Infinity \leq - Less than or equal \geq - Greater than or equal \partial - Partial differential \sum - Summa \prod - Product Sign \subset - Contained in \in - Element of \cup - Union (if you want to see big symbol, enter \bigcup) \neq - Not equal to \approx - Almost equal to (asymptotic to) \equiv - Identical to (equivalent) Neither Greater-Than Nor Equal To. 3. only makes sense if the order of x equals the number of columns of A and the order of b equals the number of its rows. [citation needed] The letters BT stand for Break Text, and are put between paragraphs, or groups of paragraphs in messages sent via Telex,[citation needed] a standardised tele-typewriter. Symbol Name Code ÷ Division (Obelus) 246 × Multiplication 0215 ± Plus or minus 0177 ≈ Approximate 247 √ Square root 251 ⁿ Power n 252 ² Squared 253 ¼ Quarter 0188 ½ Half 0189 ¾ Three quarters 0190 ∞ Infinity 236 ≥ Greater than or equal 242 ≤ Less than or equal 243 π Pi 227 ° Degree 248 This is an irrational number ; when written in decimal form, it is nonterminating and nonrepeating. I want to add the symbol for almost equal, ie ≈, but haven’t been able to find it. Expressed to four significant digits, 2 1/2 1.414. For instance, the expression 0 == false is true, but 0 === false is not, because the number 0 is an integer value whereas false is a Boolean value. The equals sign is placed between the things stated to be exactly equal or the same. The equals sign is also used as a grammatical tone letter in the orthographies of Budu in the Congo-Kinshasa, in Krumen, Mwan and Dan in the Ivory Coast. In an equation, the equals sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value. (1800..1899) == 1844 is false, since the types are different (Range vs. Integer); however (1800..1899) === 1844 is true, since === on Range values means "inclusion in the range". The language B introduced the use of == with this meaning, which has been copied by its descendant C and most later languages where = means assignment. Following ALGOL, most languages that use = for equality use := for assignment, although APL, with its special character set, uses a left-pointing arrow. It can mean "similar to", including "of the same order of magnitude as", such as: "x ~ y" meaning that x and y are of the same order of magnitude. This means approximately equal. A few languages, such as BASIC and PL/I, have used the equals sign to mean both assignment and equality, distinguished by context. 4.6 out of 5 stars 202. to test for equality. Confusion between the two uses of the sign sometimes persists at the university level. You can also insert the symbol from , … So we wanna estimate what this is. 41. The equal sign, equals sign, or "=" is a mathematical symbol used to indicate equality. And so you can view that squiggly equal sign as being what does this roughly equal to? If you frequently need to insert math symbols like this, I recommend turning on your Math AutoCorrect feature as discussed in the second section below. There are several symbols that can be used to say items are "approximately the same," "similar to" or "about equal." [10] Under these semantics, === is non-symmetric; e.g. For example, a value of type Range is a range of integers, such as 1800..1899. - [Instructor] In this video, we're gonna get a little bit of practice estimating with multiplication. Both usages have remained common in different programming languages into the early 21st century. The symbol has been used since 1995 by the Human Rights Campaign, which lobbies for marriage equality, and subsequently by the United Nations Free & Equal, which promotes LGBT rights at the United Nations. It doesn't have to be exactly right. In mathematics, the equals sign can be used as a simple statement of fact in a specific case (x = 2), or to create definitions (let x = 2), conditional statements (if x = 2, then ...), or to express a universal equivalence ((x + 1)ВІ = xВІ + 2x + 1). The etymology of the word "equal" is from the Latin word "Г¦qualis",[3] as meaning "uniform", "identical", or "equal", from aequus ("level", "even", or "just"). 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