Lesson two, deductive reasoning. Then I have blank blank. I'm going to tell you what goes in the blank right now is the words deductive reasoning. Deductive reasoning, which Jeremiah is also sometimes called logical reasoning, is the logical process of using true statements to arrive at a conclusion that you know is true. The one says write a logical conclusion which can be deduced from the following statements. All integers are rational numbers, that statement number one. Negative five is an integer, that statement number two. What can you conclude from that logically, reasonably? Sorry, we're doing A. You're on B, and you're right, but let's get there. All integers are rational numbers, negative five is an integer, conclusion. Boston, you're right, say it louder. Negative five is a rational number. How do I know? Well, all integers are rational numbers, and it says that negative five belongs to the integers, which means it automatically belongs to the rational numbers. We call this, you don't need to know this word, but I love logic and reasoning, so this is the nerd within me. We call this a syllogism, where you have two statements and you arrive at a third conclusion. Water freezes below zero degrees Celsius. The temperature is positive six degrees Celsius. Conclusion, Jordan. All triathletes have run a marathon. Mr. Dewick has never run a marathon. Mr. Dewick is over, oh hang on, no, no, no, no, no, let's do that. Not a triathlete. Mr. Dewick, if I had run a marathon, could you assume that I was a triathlete, or is it possible that I could have run a marathon and never done a triathlon? So this one does not work in reverse. You can't say, oh, he ran a marathon, therefore he is a triathlete. Maybe I can't swim. I signed up, let's do this for a second, B, or sorry, D. Now this next one is a very, very famous syllogism, but it's abstract. It says this, if A equals B and B equals C, what else do we know? Taylor, A equals C. Think about that one, ponder that one for a bit. You see it? If A equals B and B equals C, well, if they both equal B, they both equal each other. In fact, this one is used so often in math, it has a fancy name. We'll write it down because we're nerds. It's called transitivity. E, Jordan is taller than Joe. Joe is taller than Boston. Conclusion. Should we model it just to find out? Can you use the names that were given there instead of saying Boston is the smallest? Because I don't know if that's the, I would say that Boston is shorter than, if C times D gave me a positive number, and I know that C was negative, D has to be negative. Look up. I'd like to show you shorter ways to write things, so don't write this down. Don't write this down. That's way too much writing. Do you know how we abbreviate is negative Jeremiah in math? We go like this, is smaller than zero, which is way shorter to write than is negative. But if it's smaller than zero, doesn't that mean that it has to be negative? Well, that's actually a logical conclusion. Deductive reasoning there. You can write is negative. I'm going to do that for the rest of the year because I don't feel like writing out lots of stuff. Given the following, now we're going to try and use this deductive reasoning for a multi-step proof.  This is where if you get a bit lost, all it means is you're normal. You're looking at proofs. And I got to tell you, as someone with a math degree, I found proofs very, very difficult to wrap my brain around. But I want you to commit to at least trying to follow what's going on. So it's given the following. One plus two plus three, well, that's six. Two plus three plus four is nine. Three plus four plus five is 12. What was the pattern that we noticed if you're adding consecutive integers? What was the three consecutive integers? What was the shortcut for getting the answer? Yeah. Okay, so three at 15, 33, three times negative 24. Is that okay, Jeremiah? Let's make that a negative 24. Sorry? Second one from the bottom? Oh, but are those consecutive? Jeremiah, I did a typo when I was typing in a hurry. Thank you, Jeremiah, for catching two candies for you, but later on, absolutely. Jeremiah is saying, look, this is actually 11 plus 12 plus 13. If I'd done that on purpose, that would have been phenomenal teachings. Sadly, I didn't. You just caught a mistake. So you're saying the answer is actually 36, which it is. Nice catch. But our conjecture is, if we add three consecutive integers, the sum is three times the middle. Prove it. Now, to do a, because it's worked for these ones, but remember, we saw an example a few minutes ago in last day's homework, where that circle pattern worked for the first five, didn't work for the sixth one suddenly. How would I go about proving this? Here is where we're going to do a little bit of algebra. I would let the numbers be those. X minus one, X and X plus one. Now think, first of all, write that down, and then let's think about this for a while. What's the middle number in this group here, Joe? Just read it to me. X, what's one less than X? X minus one, is that how you would write it? Yeah. What's one more than X? In other words, are those three consecutive numbers? They are, I don't know what X is, but I'm telling you, whatever I plug in for X, those are three consecutive integers. Okay? I'm trying to do a general proof that works for any number. So I'll use X's or N's or whatever variable. Usually I use X's, if you like that. Can you read to me? Excuse me. Adam, can you read to me? Oh, I'm missing the word add. That was dumb. If we add three consecutive, can you read to me that first sentence? If we what? Stop. How would I write these three added together? How would I add these together? What would I write algebraically? Wouldn't I write? That's the sum. That's the, if we add three consecutive integers. So far, so good. The weirds so far, but. Oh, I think I can gather like terms here. For instance, I'm noticing X plus X plus X. What is 1X plus 1X plus 1X? And what is negative one plus one? What? They cancel? Why have I, by that statement there, by those two lines, why have I proved our conjecture? What does that last line say? Matt, read me the last line here. No, no, the very last line that I wrote here. Read me the last line right here. 3X equals sum. Oh, what was X? Joe, what did you tell me X was? It was the middle number. Yes? So here's what we've said. If we add three consecutive numbers, that's how I would translate that into math. When I gather like terms, it turns out the sum ends up being three times the middle. Not X, but what was X in this case? The middle. Did I just prove this? Yes. I showed algebraically that if you add any three consecutive integers, should I, any three? When you add them together, it simplifies to three times the middle one. How do I know this is the middle one? Well, it has to be because this one's one less and this one's one more. That must be in the middle. What do three dots mean? You've seen that symbol before? What do three dots mean? Boston, you're going to open your eyes?  Seriously.  That's why you're falling asleep. You've wrapped yourself in a blanket. I'm not mad at you. If you take your jacket off in about five minutes, your body temperature will go down, your heart rate will go up, and you'll be more awake. Give it time. What do these three dots mean? And you guys have you seen this in science yet? Three dots like that means, therefore, in conclusion, I'm going to use that because I'm a nerd. Therefore, our conjecture, why is it true, Boston? This conjecture was if we add three consecutive integers, so I algebraically, I wrote the algebraic version of adding any three consecutive integers, and when I simplified it, I got 3x, which was the second half of the statement. The sum is three times the middle number. There's three times the middle number. Turn the page. Are we next page over? I think we are, yes? I have to tell some of you that. Okay. Here's some notation for you. To symbolize an even number, I will write 2n. Why does that number have to be even no matter what? Is that an even number? If I put an even number in there or an odd, you know what? Because 2 goes into it no matter what, which makes it even. To symbolize an odd number, I'll either go 2n plus one, or sometimes you'll see the textbook go like this. Jordan, what we're saying is one more than an even number is automatically odd. Or conversely, Devin, we're saying one less than an even number is also automatically odd. Now, does that make sense? If you're an even number and you go up by one, are you automatically an odd number? No? Yes? The following case? Right? Even plus one is odd. Okay. We're going to use that notation. So if you ever start talking about an even number, just think 2n or 2x or 2y or whatever, two times something. If they ever say we start talking about an odd number, think two times something plus one. Or minus one.  Conjecture. If I add two odd numbers, the sum, the answer, must always be is always, well, if you add two numbers, what can you tell? Two odd numbers. What can you tell me about the answer? You're right. Even? Is that correct? Now, don't write this down. I would probably go, well, let's see. Five plus seven is 12. 13 plus 21 is 34. I would try a few odd number examples and I would see if I could spot the pattern. But I think you just reasoned it out. You said, look, two odd numbers when I add them together, they have to be even. I agree. Prove it. Okay. I'm going to let the numbers be 2x plus one and 2y plus one. Write that down and we'll talk about why I just did that. So here's the first number. Here's the second number. Why did I just do that? Well, is that an odd number no matter what? Yeah. Is that an odd number no matter what? And I used an x here and a y here because I want different answers to any two odd numbers. So far so good. A little weird. You with me? I'm right by last year. You with me? Okay. Taylor, can you read to me the conjecture again? Conjecture. Stop. Say that again. Okay. I'm going to write this algebraically. If I want to add two odd numbers, add means is going to be a plus sign. Here's my first odd number. Here's my second odd number. There is adding two odd numbers. Somehow I want to see if I can do some arithmetic, some algebra and end up with an answer that satisfies this conjecture. End up with an answer that I know has to be even. No matter what x or y I put in there. Hmm. Oh. Can I gather like terms? Courtney, see the little positive one right there? See the little positive one right there? What's positive one plus one? So here's what I have. What can you tell me about this? Hmm. Hmm. Can this be odd? Why not Jeremiah? You're right. I agree with you. You know what? I agree with you. Here's how I would do one more step. I would factor. Remember factoring from last year? There is a GCF in all three of these terms. What's the greatest common factor in all three of these terms? I would write this as two and then it's going to be bracket x plus y plus one. And believe it or not, Sydney, we're done. You may not realize it, but we are. Sydney, at the top of the page, I wrote an even number can be represented by. What did we say that we could represent an even number by at the very, very top of the page there? Louder. Two times something. I'm going to replace. You know what? I'm going to replace the letter n with time something. So what's any even number? Two times something. Look what I have here. Don't I have two times something? You know how I know? Because there's a two in front of everything. Doesn't that mean it has to be even? This is Jeremiah. This is how I would prove what you were saying. Look, I can see it. I don't know how to prove it. I would say, well, if I define every even number as two times something, why don't I try writing what I think is even as two times something? And even though Sydney, this is an ugly something compared to what I wrote at the top of the page, I'm still going to argue that it matches what we wrote at the top of the page. It's two times something. That has to be even since it's multiplied by two. That has to be even. If you're tough, if you're finding these awkward, if you're finding this weird, you're normal. Don't feel bad, but just don't shut down. Odd plus odd has to be even. Why? Because if I do a generic odd number and another generic odd number and I add them together algebraically, as it turns out, when I simplify, I get a two in front of everything, which means it's got to be even because two goes into everything. Let's try this one. You need your calculators for this. Get your calculators out. If you don't have a calculator, now's the time to fess up because I'm going to make fun of you if you don't have one. Get your calculator. It says choose a number. So all of you right down, right now, pick a number. Pick a number on your calculator. I'm going to pick 17. You all have different numbers on your calculator? It says add five. So plus five and then press equals. Then it says double the result times by two. Press equals. Then it says subtract four minus four and press equals. Divide by two. Subtract the number that I started with. I started with 17. I got three as an answer. What did you guys get? What? Oh, how could you do something different? What? Huh? You got three? You got three? Seven? And you followed the instructions and hit equals after each step? And did you get three? Oh, you got three. Oh, I thought you said you got something different. Sorry? 23. What did you start out with? 20. And you hit equals. Enter after each step? Okay. Yes. And if you don't do that, it's going to do it properly with bed mass. I don't think they want you to do it properly with bed mass. They want you to do it all in this order. Okay. Let's try seven. 20 plus five. Enter equals times two equals minus four equals. Divided by two equals minus my original number. Yeah, I'm getting three back.  About six inches from your nose there. That, those ones? Prove it! What did you use as your number? Yeah, you. 562. Ah, she's trying to see if she can game the system with a big number. What did you use? Okay. 566. What did you use? Have we proved it for every single number out there? How many numbers are there out there? Think Buzz Lightyear. How many numbers are there out there? Infinity. Can I try them all? No. You know what? Instead, I'm going to try it with one number. And you know what I'm going to let that number be? What have we done for our generic any number so far? X. Jordan, write this down. I'm going to let the number be X. Can I regate the word number with a number sign? Is that okay? I'm going to. And Sidney, what we're going to do is we're going to walk through this step by step by step, except instead of doing it on our calculator to a number, we're going to do it to the X. So, step one says choose a number. I did. Step two says add five. How would I add five to X? How would I write that algebraically? Five X? Is that adding or is that timesing? That's timesing. How would I add five? Even easier. Yeah. Five plus X. I like the variable to come first. All right. What's step three? We know what does the step say. Step three. Step three. Step three. Double the result. That would look like this. Two times X plus five. Except I can now front door, get rid of brackets. What does that simplify to? Two X plus 10? What's step four? Math now. Okay. So how would I do that? I would take my number, which is plus 10, and I would subtract four. When I gather like terms, what does two X plus 10 minus four simplify to? I'll give you a hand. Two X and a positive six. Okay. What was step five, Boston? Okay. Step five is take this two X plus six and divide it by two. Now, Boston, what that really means is divide the two X by two and divide the six by two. Yeah. What was step six, Boston? I've scrolled down. Ah, so I'm at X plus three. Subtract X. What is X plus three? Take away X. What does the answer have to be no matter what infinity of numbers you plug in? Five hundred and seventy works just fine. A trillion works just fine. What's the answer simplify to? Three. There's our proof. I think something like this might be an okay test question in my mind. First of all, because Corny, I kind of like these number puzzles where it's like, oh, what's the answer? My birthday. Yay. I think asking you to walk through a proof like that because it is kind of step by step algebraic. This I would feel would be okay on a quiz or a test. Prove why the answer is the answer that you got. Is that okay? There are way more famous ones and tougher ones and bizarre ones than just this one. But there's our proof. We're going to finish with example five, I think. No, example six and example five are the same thing. You know what? We may even finish the whole lesson. I won't give you any homework today, though. Example five says, see if you can spot the pattern. I'd like you to try not to use your calculator. Put your calculators down and away. Yes, Boston, you too. It should. You know what? I would argue I don't need to. I trust my algebra. I mean, I've convinced myself. You ready? This is kind of a series of questions. And if you pay attention, if you spot the pattern, you may actually be able to predict the upcoming questions. So what I'd like you to do first of all is what's six times six? Write down the answer. And I'm going to freeze the screen. And then right across from six times six, what's five times seven? And write down the answer. Write across from it. Yes? What's nine times nine? And then write across from it. What's eight times 10? Yeah, keep it to yourself. What's four times four? And then what's three times five? Have you spotted yet how these two, each line is similar? If you spotted it, if I said to you D, seven times seven, first of all, what's seven times seven? Can you predict what this question would be over here? Have you spotted the pattern yet? Shay, except I've gone small number, big number. And what's the answer? Have you noticed any pattern between your answers yet? Oh, if I told you the second question was nine times 11, and the answer was 99, what's the first question? Yeah? And what's the answer? So if I tell you this without a calculator, if 50 times 50 is 2,500, what's 49 times 51? 24.99, is it? Are we right, Shay? Sorry, you reached for your correct double check. Yes, you are. Okay. Oh, proof. Sorry. I want to prove that this pattern always works. You know what I really want to prove? Let the first number be X. So instead of a six times six, X times X. How would I write X times X? I heard it. X squared.  This column here is X squared. Where'd the five and the seven come from based on the six? How did I get the five and the seven? Where'd the eight and the ten come from based on the nine? How did I get an eight and a nine? How did I get the eight? Okay, so if I do that with an X, how would I write this? Is that one minus our generic number? And what I write this is that one plus our generic number? In other words, if our number is X squared, one minus one plus, you said yes? Here's our number. One less, one more. Foil that out. We've done foil. Foil it out. X minus one times X plus one. When you foil it out. Remember foil? We did this at the beginning of the year of the review. Boom, boom, boom. I don't want to put the red lines on this one because we're going to clutter things up. If you foil that out though, what do you get? You don't get X squared. You don't get 2X. Don't you get X squared minus one when you foil it out? Look up. Here's the English to this math statement that we've wrote. But once you've written that down, look up. Jeremiah, here's what it's saying. If you square a number, one less than the number times one more than the number is one less than the number squared. Is that actually what was happening? Isn't that how we did 49 times 51? We said it's the square minus one. It's the difference of squares. You called it, I think, last year in grade 10. How much more am I planning on doing here? We're going to pause for a second.


Welcome to the session of Expansion of Functions part 4. This is Swati Nikam, assistant professor, Department of Humanities and Sciences, Balchan Institute of Technology, Sulapur. At the end of this session, students can create the expansion of function about any point with the help of standard series which is obtained by using Metallorean series which we have seen in earlier video. Now, let us see some standard series which we are going to use in this video. First one is exponential series e raised to x which is equal to its expansion is 1 plus x plus x square by 2 factorial plus x cube by 3 factorial plus x raised to 4 by 4 factorial plus x raised to 5 by 5 factorial plus and so on. And second series is logarithmic series log of 1 plus x is equal to x minus x square by 2 plus x cube by 3 minus x raised to 4 by 4 plus x raised to 5 by 5 plus and so on. Now, let us see the use of these standard series to solve few examples. Example number one, expand function f of x is equal to log of 1 plus x whole raised to 2 in ascending powers of x up to x raised to 4. Answer, we know that series expansion of log of 1 plus x is, the expansion I have explained in earlier side for log of 1 plus x is written over here. Then log of 1 plus x whole square can be written as, so write down the bracket term which we have explained over here x minus x square by 2 plus x cube by 3 minus x raised to 4 by 4 and so on whole raised to 2. Now, in order to do this expansion, let me rewrite the given expression as first term x taking minus sign common and writing all other terms in one bracket as x minus x square by 2 minus x cube by 3 plus x raised to 4 by 4 minus x raised to 5 by 5 and so on. Whole raised to 2. We have really arranged the terms. Now, we have done this rearrangement to use the standard result a minus b whole square which is a square minus 2ab plus b square. So, here a is x and b is this bracket term. And therefore, log of 1 plus x whole square now becomes a square is x square minus 2 into a is x into b is bracket plus bracket square. Now, we will expand this first term x square as it is minus. So, this is twice x into bracket terms x cube by 2 minus and minus plus twice x raised to 4 by 3 next term plus and so on. Now, remember in our example it is as that we have to do the expansion up to x raised to 4 only. So, let us leave all other higher power terms and hence we will put dash dash for all other higher power terms. From this third expansion with plus sign we will rewrite this as now see in this expansion again we will use the same formula and rearrange the terms as a minus b whole square. But first term is x square itself. So, first term square is x square whole square is x raised to 4 by 4. So, no need to consider further expansion terms as they are greater than power 4. So, we will keep them as dash dash dash. Therefore, log of 1 plus x whole square is equal to x square minus x cube plus x raised to 4 into bracket since there are two terms. So, I have collected the coefficients 2 by 3 plus 1 by 4 plus all other terms dash dash dash dash and therefore finally the expression for log of 1 plus x whole square is equal to x square minus x cube plus x raised to 4 into bracket 11 by 12 plus and so on. Now friends before we proceed further by using laws of logarithm state the following first one log of a by b and second log of a into b. I hope you have written your answer. So, by using laws of logarithm we have log of a by b is equal to log a minus log b and log of a into b is equal to log a plus log b. Now, let us solve one more example prove that log of big bracket numerator x into e raised to x upon denominator e raised to x minus 1 big bracket complete is equal to x minus 2 minus x square by 24 plus x raised to 4 upon 2880 minus dash dash dash. Now in answer we will use the laws of logarithm which we have stated earlier log of a by b is equal to log a minus log b. So, log of numerator x e raised to x upon e to the power x minus 1 becomes log of a that is numerator x into e raised to x minus log of b that is denominator e raised to x minus 1. Once again this is log of a into b. So, which is log a that is log x plus log of e raised to x minus log of e to the power x minus 1. Here we have to use that is a expansion of e raised to x as 1 plus x plus x square by 2 factorial plus x cube by 3 factorial plus x raised to 4 by 4 factorial and so on and minus this one. Friends put the brackets properly. Therefore LHS becomes here it is log x log x as it is plus log of e to the power x now becomes x minus log of. So, see here if you open the bracket then first term is 1 and last term is minus 1 which it cancels and so all other terms are in ascending powers of x as x plus x square by 2 factorial plus x cube by 3 factorial plus x raised to 4 by 4 factorial and so on. Now which is equal to log x plus x log of since in this bracket all other terms contains x and hence let us have x common. So, writing all the terms in one bracket taking x common it is 1 plus x upon 2 factorial plus x square upon 3 factorial plus x cube upon 4 factorial and so on which is equal to log x plus x. Now see here it is of the form log of a into b where a is x and b is this bracket. So, log of a into b is log of a plus log of b but because of this minus sign we will rewrite it as minus log of x minus log of b which is bracket term. Now log x and minus log x cancels out and therefore it is equal to x minus log of in big bracket let me rearrange it as again first term 1 plus all other terms in one bracket as x by 2 factorial plus x square by 3 factorial plus x cube by 4 factorial and so on. So, we have done this arrangement in order to use the standard series log of 1 plus x. As I have said let us call the bracket term as capital X and so by using the result log of 1 plus capital X is equal to x minus x square by 2 plus x cube by 3 minus x raise to 4 by 4 plus and so on. And therefore log of x into e to the power x divided by e raise to x minus 1 becomes x minus log of 1 plus capital X remember capital X this big bracket now which is equal to x minus inside the big bracket capital X that is bracket minus capital X square bracket square divided by 2 plus capital X cube that is bracket cube divided by 3 minus capital X that is bracket minus 4 upon 4 plus and so on. Now since capital X is this bracket now in this expansion if you write down the terms directly in the expanded form the expression becomes very lengthy and confusing also. So, let me calculate the terms of powers of x separately as x square is equal to so for writing this bracket as x upon 2 factorial plus all the other terms in second bracket x square by 3 factorial plus x cube by 4 factorial and so on whole square. We have done this arrangement because we want to use the result a plus b whole square which is equal to a square that is first term square x by 2 factorial whole square plus 2 times a b that is 2 into x upon 2 factorial into bracket plus b square that is bracket square. So, it is a plus b whole square similarly to write down x cube we will again rewrite it as first term a x by 2 factorial plus second term b as bracket. So, a plus b whole cube is it is equal to inside the big bracket a cube that is x by 2 factorial whole cube plus 3 times a square that is x by 2 factorial square into b which is bracket plus 3 into a x by 2 factorial into b is bracket whole square plus b cube that is bracket cube. Similarly, x raised to 4 is a is x by 2 factorial plus b is bracket whole raised to 4 is equal to first term raised to 4 that is x by 2 factorial raised to 4 plus and so on. Now, all these expanded terms we will write down and get the expression as LHS log of x into e raised to x divided by e raised to x minus 1 is equal to x minus x by 2 factorial minus x square by 3 factorial minus x cube by 4 factorial minus x raised to 5 by 5 factorial plus dash dash dash plus x square upon 2 into 2 factorial square plus 2 into x cube upon 2 into 2 factorial into 3 factorial plus 2 into x raised to 4 upon 2 into 2 factorial into 4 factorial plus dash dash dash plus x raised to 4 upon 2 into 3 factorial square plus and so on minus x cube upon 3 into 2 factorial cube minus 3 into x raised to 4 upon 3 into 2 factorial square into 3 factorial dash dash dash plus x raised to 4 upon 4 into 2 factorial whole raised to 4 plus and so on. After calculating that is which is finally equal to x by 2 minus x square upon 24 here the term of x cube get cancelled plus x raised to 4 upon 2880 minus dash dash dash.


So this forum is excellent. It allows for real life challenges and experience to be at the forefront of the discussions for sharing of best practices. So resolution of shared issues isn't always easy. We need consistent collaboration between posts and customs to deliver our respective mandates and facilitate trade while we deal with the challenges around emerging regulations and opportunities for growing our international business. Ensuring we have aligned regulatory approaches for all WCO members ensures we can more easily meet our customs obligations. The development of the UPGlobal Postal Model was the foundation to ensure that all countries could move forward in a unified manner. One of the biggest challenges, of course, is quality data and the data itself. So once you change your processes to be automated, the absence of data or the lack of data or data quality certainly puts pressure and has a heavy impact on our operational processes. So we realize in Canada that we will always use and need a manual process to take those situations and still continue to move the mail. The CDS Customs Declaration System and the UPU solution enables posts and customs to work together to do these operational processes to receive risk assessment of duties and taxes and security. The lack of data or system changes certainly impact our business and our challenge. What's important to know is that we need to ensure we have the flexibility to react and joint processes in place to continue to move the mail.


It is a pretty fun night for pitching tonight and MLB DFS because you got some really good names littering the top end of the pitching board We got Musgrove burns Ray Bueller Valdez Sandevolt guys like that and it does make things pretty fun from a DFS perspective And I think the good thing is that for tonight I'm not as worried about jamming in high salaried bats that was last night and obviously the Phillies didn't play out very well So maybe that's part of the reason we're looking down a hitter a bit for today But I do think it's a fun slate. It's a me slate where I can kind of choose for the pitchers I like decide who I want to prioritize and feel good about that not feel like I'm missing out at hitter So let's dive on in and get you said for this Tuesday main slate welcome on into the solo shot That's right here on the fan dual podcast network and number fire comm My name is Jim Sonnis. I am a senior writer and analyst for number fire comm You're gonna break down Tuesday's tenant game main slate with locks up for 7 10 p.m Eastern for this evening the second game of the double headers for the Nats and down the backs and the Giants and Mets are not on the main slate so Typically in the past we've had double headers on main slates, you know back before COVID It was a nightmare because you had to decide okay. Well, this guy play what time of this game starts stuff like that Don't worry about that for today. You're good to go So ignore the second half of those double headers and feel hopefully a lot more at ease as we get towards lock for tonight Couple weather there notes for today all temperature related and not towards rains That's a good thing for sure. Just 46 degrees in Boston for the blue jays and red socks I would downgrade hitters in that one. It is 43 degrees in Chicago for the Cubs in the rays I would downgrade hitters there as well 54 degrees in Kansas City for the Royals and Twins down great hitters there and finally it's 56 degrees in Oakland for the A's and Orioles slight downgrade there as well So a lot of places that are cooler for today some spots with some retractable rooms which may help a bit But overall that cooler than it has been the past couple of weeks I should play a factor in our stacks We'll still talk about a couple of those spots in the stacking section before or for today before we get there though Got a quick remind you to make sure you are subscribed to the number fire daily fantasy podcast feed Wherever you get your podcast because we have not just the solo shot of a weekday got NHL from Tom Vecchio USC PGA NASCAR all in the same place at the exact same feed So hit subscribe get those podcasts as they go up and maximize your research time after you listen Before the slate locks to give yourself more time to decide whether or not you agree with our takes on stuff Also, we had Dan Zimborsky a fan grass on covering the spread yesterday breaking down his thoughts on Which releases numbers to trust which ones be skeptical of Some teens he thinks the market is undervalued right now So if you like Dan's work or a fan grass check that out by searching for covering the spread wherever you get your podcasts Hey sports fans fandal is now offering an exciting twist the beloved same game parlay now We're producing same game parlay plus which allows you to combine same game parlets across multiple games All you have to do is head over to the fandal sportsbook And navigate to the same game parlay tab of the first game you are interested in select multiple bets in the first game And then plus it up now you can add more bets from other same game parlay labeled games Head over to fandal sportsbook today and out into the same game parlay plus must be 21 plus and present in eligible states gambling problem Call 1-800 gambler was the fandal comm slash RG in Arizona call 1-800 next step or text next step to 533 42 in Connecticut 1-888-789-7777 visit ccpg.org slash chat in Indiana 1-800 9 with it in Louisiana 1-877-770 stop in New York 1-877-8 hope and why our text hope and why in Tennessee call the red line at 1-800-899-789 or in West Virginia Visit 1-800 gambler net pitching preview for this Tuesday main slave Joe Musgrove Is the highest salary pitcher on fandal checking in at $11,000 followed by Corbin burns at 10 a we got Robbie Ray at $9,900 Walker Bueller's 97 from Breval des is 95 we have Patrick Corbin Kyle Gibson and Justin Steele as the others at $8,000 are higher and I think when you're looking at the pitching for today I was talking about there are good options and there are good options But despite that I still think that the top option for today is very clear And that to me is Corbin burns at the top I think you put them at the top and you forget it from there with course being less attractive for today I am okay with burns being definitive the top guy at 10 8 He's facing the pirates at home and the pirates Predictably struggling this year. They are last on the slate in both WRC plus and isolated Slugging against ratings based on their current active roster since the start of last year So a large sample strikeout rate about average at 23% but those low WRT plus and ISO numbers about what you expect So the situation for burns is very good and he seems pretty close to the guy. He was He was last year to open this year strikeouts are down a bit and the bad of all numbers have gotten worse But most of that was from opening day when he just seemed a bit off I was on him against the Cubs didn't seem great there last time out They bounced back a an 18.6% swinging strike rate in his second start He had eight strikeouts across seven innings and he also threw 97 pitches So pitch count no longer concerned the Corbin burns the swinging strike rate through two starts combined is now 16.1% which is very much in line with where he was last year The velocity is good to despite pitching and some colder weather which does play a factor in velocity so 10 8 is a lot to pay for Corbin burns, but I Was kind of expecting like almost 12k for today So, you know, I'll take it for sure when I don't need to splurge as much a hitter He's very worth it So I feel very comfortable making burns my top play of the day the second pitching slot behind burns depends on your comfort level with Robbie Ray and His velocity is down big and it's enough where it is a pretty massive red flag We just have to decide if it's enough to fully push us off ray for today I'm on the fence about it personally the velocity is legitimately rough because last year he averaged 94.6 miles per hour on a sports team fastball that is down two ticks this year to 92.6 his sliders down 1.7 miles per hour and Those decreases in velocity are translated to decreases in key metrics Specifically his swinging strike rate against him is 10.9 percent down from 15.5 percent last year So I think the concerns are legit There are a couple caveats worth mentioning though The first one is that those games are Minneapolis in Chicago and those games are very cold and velocity is typically down When it is colder, he was also facing the twins and white socks on the road And those two teams have a lot of guys who can bang lefties So does that fully erase the concerns not for me I think that's why he's not the top arm for me But I do think there is a chance that the dips for ray were due to match up in weather Those things will not be as big of a factor for tonight. You see some the Rangers They are definitely an improved offense from what they were last year, but they're still not an elite This active roster is a 24% strikeout rate against lefty since the start of last year where they 7% walk rate So it could be rough for Ray and if you want more safety Violence now how to get to Joe Musgrove, but I think if you're just gunning for stealing You know, you got burns locked in for cash games. You want to go for a ceiling in tournaments I think I like raise more at $9,900 Despite what I think are pretty legitimate concerns around his first couple of starts So to me Robbie Ray is number two behind Corbin Burns Despite the fact I do have legitimate concerns around his velocity and his start to the season As far as the value play goes I think you have two options both of whom are in terrible match ups Those guys are Nathan Ivaldi and Max Freed Freed gets a Dodgers Evaldi gets the Blue Jays and Evaldi Really struggling with hard contact. That's not something I'd want to do against the Jays So I think I'm gonna go with Freed here But the Evaldi does have more strikeout upside and is at least an option for today As for free the bad of all suppression is why I am here His slider velocity is up this year and it's something that started with him last year and over his past 11 starts with that Slider velocity being up. He's allowed just a 24% fly ball rates with a 36% hard hit rate those are very good numbers and I think we can buy into them because Freed over his career has been good at suppressing hard contact Not surprisingly this has led to a 2.2 to ERA for Freed across his 11 starts now His first two starts this year were not crazy good and they came against Lester competition But I'm betting he'll be good over the long run this year The swinging strike rate last time out for Freed was 14.1% for him. That is a very high mark The Dodgers not surprisingly very good against lefties So it's not an ideal situation but what we're getting with Freed is a good real-world pitcher for $7,400 and There may be some extra upside in him this year with the slider potentially Being a bit different. So I will take that upside and roll with it Again, though I would at least look into E of Aldi if you want another lower salary option at pitcher But I think to me Freed is my preferred guy because the bat of all suppression I think is better and sustainably so with regards to him Let's move now to the stacking section I talked about burns and the pitching section and if we're gonna get there to 10A We need some value batters to make that work And I think we can actually get those from Burns's teammates They're facing JT brew baker brew baker has done some interesting stuff since the start of last year and It's a kind of interesting You know, I think I think that he's shown some spurts of competency But he struggled to open this year which is interesting because he's lowered his foreseeing fastball usage quite a bit from where it was last year and Hasn't worked so far because he has met as many walks of strikeouts. He has a 5.8. He'll interact a VRA He's allowed a 52% fly ball rate That's pretty rough and he's also not getting a lot of widths his swinging strike rate is 10.7 percent. So That's rough so far, which means that brew baker could go back to what he did last year But and I would expect that as well But even even in last year, he had a four or a 5.36 era He's not a big round ball guy did let us some hard contact So the two routes for brew baker are a he can stay doing what he's been doing his first two starts Or B can go back to what he was last year And I think that either of those would make it okay to stack the brewers for today I think it's okay to take the discounts here and be high on the brewers from a stacking perspective Making it a bit easier for you to use a Corbin burns as your pitcher for today So looking at this Milwaukee team again We're talking about all with the place with cold temperatures for today Milwaukee will not be one of them I'm guessing the roof will be closed which would make it 70 degrees there It would make it just one of three games with the temperature at 70 or higher So I do think that grades that well for Milwaukee overall for today I'd specifically be highest in the lefties here and I was looking you know researching for this late yesterday and before They played their game last night. I was like, yeah, you know, Yellich is hitting the ball hard but not lofting it And I was like, okay, that's at least something I wasn't like fully on board and then he hits like a bomb like 6,000 feet to center field So I'm like, okay, we're we're back in a Christian Yellich. Obviously. He's there for sure Colton Wong Omar Narváez, they're all high on our list and I will use those guys here as a Stacking focal point when trying to jam in Corbin burns as far as our second stack, you know, we get the good weather with Milwaukee We will not get that for our second and third stacks But there aren't enough good options in good weather spots at least for me to be fully on board with going there I do like the twins despite the weather in Kansas City If he's in Carlos Hernandez and he's gotten good results the starter He has a 3.56 era Since he moved into the rotation last year the peripheral zone not quite there in 13 appearances Hernandez has a 5.31 skill interactive era His strikeout rate is 16% which means he's letting up a lot of balls and play the batted ball numbers are fine 37% heart rate 39% fly ball rate those are okay and They're more okay when you're getting strikeouts and limiting the balls and play if you're letting up a lot of balls and play It can become a pretty big issue We saw the downsides of that is most recent start where Hernandez was facing Cleveland He led up four runs across four to third innings He has just two total strikeouts across his two starts and that's not something you expect to translate into good Long-term results. So I think eventually we'll see regression from Hernandez with this sub four era As a starter and that could happen tonight against this twins offense There's a chance that Byron bucks in his back with the twins for today He is scheduled to travel with the team and he said Monday that he was optimistic about his knee So wouldn't be shocked he's back in there I wouldn't necessarily expect it based on the initial diagnosed about a week But could happen for sure one guy I find super interesting here though is Louisa rise He has never hit for power, which means I've never liked a rise for DFS He is lofty in the ball a bit more this year though So although not expecting a huge power binge. I'm becoming more open to him I have very rarely used a rise in my DFS playing life But I'm open to at least considering it here based on a slight uptick in his fly ball rate He doesn't hit the ball like super soft or anything So not typically my cup of tea, but I'm more receptive to him now that I have been previously with that fly ball Potentially ticking up at least a bit as far as the third stack goes I've liked you say kikuchi for a couple separate times throughout his career He always looks like you love a Velo spike at the beginning of the year and you can get some strikeouts You know, I'll take those two things for sure But there are some lingering issues that are still there with kikuchi And I think I will stack the red socks against him for today I will note before we dive too much into this though the red socks might be in the midst of a COVID outbreak So if they lose guys like, you know, Xander Bogart, Shady Martinez, Trevor Storey And they lose like the guys we turn into against lefties downgrade them for sure I don't know who might be out for today, but it sounds like there are some weird stuff going on there So just keeping on that with that Specifically for kikuchi the big issue he has had is hard contact because if you look back for the full season last year Kikuchi lit up a 47% hard hit rates That is a crazy high number and it led to a 4.41 ERA Despite having the good aforementioned played discipline numbers and it got worse as the season went along But now kikuchi goes to a new team and that can sometimes lead to changes They'll unlock something fresh set of eyes stuff like that So I was curious how kikuchi would look in his blue jays debut But it didn't change that one key area He let up 12 balls and play in his first starts and nine of those had an exit below of at least 75 miles per hour His fly ball rate allowed was 50% We think he could she let up just four or just 10 home runs across 10 starts or 13 starts the more change up So the homers haven't been a huge issue so far, but now He's typically playing in more hitter friendly parks This will not be that for tonight given the weather, but I think it's a good profile for stacking I will roll at the Red Sox here and see if they can you see if kikuchi can turn things around if he can't the Red Sox could really knock him around the up and When doing so, I'm gonna be high on Bobby Dalbeck He is off to a brutal start this year the results are bad But a lot of that seems to come down to bad luck He has a 250 actual Woba like his actual was 250 but his expected Woba baseball savant is 355 so Dalbeck is still snacking the ball his strikeout rate is actually down to 30% and He torches lefties, so I know I will not be alone in doing this but Bobby Dalbeck will be a key focus for me tonight at $2,600 and Not saying he's gonna double-dunk, but I wouldn't be shocked if he did so Bobby Dalbeck Bobby Dungbeck a focal point for me for tonight the Red Sox and keep an eye on Who is not available for today because it could be kind of bad based on the initial reports of the Red Sox But at least as of right now, I am on board with stacking them for today Things to watch for this late. We do have a course field slate for today. The Phillies are at course again today I obviously don't mind them, but it's not as good of a spot against Kyle Freeland He has struggled to open the season, but the bat a ball numbers has still been very good with a lot of ground balls So I'm on board the Phillies for sure, but I do want to lower them due to a low fly ball matchup I think it's a good flop like situation like that. I was on them last night. They were terrible They come back here at a worse matchup and do well So maybe maybe this is a good spot to buy back in and people are off them But just straight up I'd rank them fourth for stacking for today Talked about the Red Sox and their COVID situation the A's very similar. They're dead They were down a lot of guys yesterday I think they had seven guys placed in the COVID IL that includes Jet Lowry who would have been a stacking option They're facing Chris Ellis today most likely and Ellis struggled the big leagues last year with a 52% hard hit rate allowed So I'm okay with them broadly the A's for today just keep in mind that they are pretty depleted and not necessarily in the best spot from a DFS perspective due to some bodies. They have lost recently Finally, I'm not opposed to some Royals stacked today They're facing Chris Archer and Archer threw four shutout innings in his twins debut That's good, but his swing and strike were with 7.9% let up a lot of hard contact So it was similar issues to what he showed in that short spurt as a starter last year The Royals salaries are very low So I'm okay with dabbling in them here and seeing what happens again weather in Kansas City not ideal But I think that for the most part we're gonna have to take some constellations the weather I think the one key exception is the Brewers for today So the Brewers are the place to stack in good weather and you'll have to make some constellations elsewhere for your batters for today Let's finish up here with some home run calls for this late. I was gonna go With JD Martinez is my boring call and that is boring But I think I'm gonna go Bobby Dahlbeck is my boring call and like I just feel really good about him for today basing off against Kikuchi Kikuchi can get some strikeouts that could be a concern with Dahlbeck, but I just loved like the the The all-out profile Bobby Dahlbeck's the boring home run calls a guy at $2,600. I'll go with it Bobby Dahlbeck It's the boring a dinger call for today. The fun one. I'm gonna go Omar Narvaez I have like a soft spot for him with regards to DFS because like people hate using catches on Fandall for a good reason for the most part but Narvaez actually has power and In a good matchup. He's playing likely indoors for today I think it's a good spot. So the home run calls officially for today Bobby Dahlbeck and Omar Narvaez Maybe we'll get to out of Dahlbeck for today as well That is all that we have here for today on the solo shot But again, make sure you are subscribed to the number fire daily fantasy podcast feed to get these podcasts right as they go up Each and every day to maximize that research time before you are locking in your line-ups in case you decide you don't agree With what we say here on the show if you've got questions for me I am on Twitter at Jim Sonnis J-i-m-s-a-n-n-e-s you can also follow the fan to a podcast network at fan to a podcast big Thank you to everyone for tuning in for today. Good luck to you with your DFS line-ups. It's nice We'll talk to you once again on Wednesday or another slate. This has been the solo shot right here on the fan dual podcast network


We have a gathering of people from around the world, most of whom are water practitioners. Many of them run utilities, they supply people with water, or they supply the people that supply people with water. So the business dimension is absolutely central. And we can see that business globally is worried, desperately worried about water. I work in South Africa and Southern Africa mainly, and it's really fascinating to hear about the challenges that countries like Jordan and others globally are facing and what they're doing to solve those. I'm pleased to see the minister, his excellency, speak at the end and really wrap things up on some of the main points that we were actually discussing in the working groups on refugees and the impact of the refugees on the water sector. We in the Arab countries believe that we have good partners internationally. They can't depend on us and we can approach them once we need help. That we are here to support Jordan, to support the region, to support the crisis as well to make sure that we have a response to this crisis that is happening in the Middle East. What is interesting is that you don't only learn what works, but in discussion with people you also learn what doesn't. The sessions that we already organized were very interesting. Tomorrow we have another one, so I think it was worthwhile to come.


We're looking at the second topical conversation here in 10, 15 minutes, but maybe this might just be a plus and a solution to all of the issues of lack of revenue and no funds. Niger and other African countries may benefit from a donor-assisted funding by OPEC for the international development. It may be a record that OPEC funded or fund head in July announced that its FEST financial bonds and the global markets amounting to $1 billion could be issued this year, speaking on the sidelines of Islamic Development Bank Group meeting recently. The director general of the OFID Abdulahi Makhafi was quoted as saying that the fund could issue low interest bonds following its 2021 high credit rating and is only waiting for international markets to stabilize. Now since its establishment, the fund has been providing development assistance in 125 countries with a total financial cost of $190 billion and a capital of about $7.5 billion. Now records show that about 50% of financing provided by funds in African countries, including the Arab countries in North Africa and Sudan, funds have also been made available to other Arab countries, the Caribbean, Eastern Europe and Asian regions. The fund targets the transport, health and education and energy sector, which are among the important sectors in target regions and countries in addition to the agriculture and water sectors. Now according to the OFID bus, more than $1 billion was spent in the health related sectors during the outbreak of coronavirus pandemic. Currently, the fund does not intend to increase its capital to members' subscription. We have Muktak Mohamed joining the conversation as a developmental economist. Muktak, thank you so much for being part of the show. Good morning. Yes, so with this development, African countries, including Nigeria, to benefit from that, do you think that this might just be a plus for us and for our economy, maybe going ahead to solve some of the issues that we might be faced with? Well, any money to our economy is a plus for us as a nation, especially with what we are going through economically. But again, it's wanting to have the monies on it and to deploy the money to do the right things that this money has been for. I think there's a challenge with Nigeria. If you look at it, it's not that countries, it's not that we're not having money. Sometimes the challenge has been how do we disperse this fund into critical sectors and put it in the structures that will make this result be visible to all, not the more you see, the less you understand. So for me, it's not having money. That's our issue. Are we going to really deploy this money into those sectors that opaque nations are trying to do? Like now at the moment, we have a big challenge, which we should be looking at technologically, way of solving it, even if the government are beginning to look at deploying militants to guide the pipeline. What technology can we use to guide a pipeline? Can this money be channeled towards this possible that we can increase our revenue? So those are part of the things that I think government should be looking at, not just really looking at all these funds are coming. Are we just going to put it there in our account, in our to boost our foreign reserve, or are we going to use it for infrastructure and development, which is key to the development of Nigeria economy? Well, so what can an economy as Niger that over time has been complained by OPEC itself saying, would you not meet the quota? And that has also been responsible for our earnings, shortfalls on our earnings. That's on the one hand. And also the issue of all theft is also another one. What are your thoughts now? Not meeting our quota is as regards of the oil theft that is happening in the Niger Delta. And like I said, the government have not been proactive before we used to say about militancy. Now we graduated from militancy to oil theft. And oil theft is not easy to be done without the help of the rich and mighty. And is it that the government don't have a political way? If you listen to what the Pengers said over the weekend, they said that oil theft is being collaborated by every sector of the Nigerian, the security apparatus, the oil companies. I mean, a lot of people are involved in oil theft. And so that has also bring down revenue. And you can see we're struggling to meet about 222 billion barret. Now we have moved to about 900. And it's sad, especially when you look at the price of food oil by our own pek. We pecked it at about $70 per barret. And it's doing about, now it's going to $90. I don't want to have it yet. We are not benefiting from it. Even the PC have not been able to limit anything to the fabrication account. So for me, we need to begin to look at ways to solve our problem. And one of those ways to use technology to fight oil theft, not by giving one person the contract to begin to guide all the pipeline. And again, when you give it in Delta State, you give all the militants in Niger, in Biasa, will you give all the militants in Riversdale? Already, Asari Dukubo is already crossing his tail with his own pipeline. He should not be guided by government. So we have a big issue at hand. But I think the challenge we have in the Nigerian government thus far is the government always look for a quick solution to challenges. Instead of looking at a long-term solution and taking the time to find these solutions. This is just the last one. We're looking at the relationship with OPEC and Nigeria here. Being that we're part of OPEC. It's also been reported that the organization of petroleum exporting countries and his ally had raised Nigeria's oil production quota to 1.830 million barrels per day. That's in September from 1.826 million barrels per day, which was in August. Do you think that this is actually really a help for us with the fact that we struggle? And do you think that Nigeria will be able to meet this increased quota? I see you smiling there. Well, if you are doing about 900, 900, 900 households for the last time and you are being raised to about one point. How do you meet it when you've not tackled the challenge that is there? OPEC know the challenge that we are going through the node. And again, this challenge also can also be helped by the international committee because this oil that has been stored in our waters has been sold into the international community. And there are people that are already buying that on the high sea. Is OPEC going to help us? It shouldn't just be a Nigerian problem alone because again, when you look at the quantity of those oil being stolen from Nigeria, that also increased the supply and also at the long run may affect the price of all OPEC countries. So it's not just a Nigerian problem. I think OPEC also should look at increasing our number of output does not solve our problem because the critical fundamental problem, which is oil theft, have not been solved yet. So we can we already just dance around the bush but we will not make any significant progress. I thought OPEC would be saying we have a technology. We have a way to help people. We are going to make sure that all stolen goods are not bought in the international market. I think that is the way we need OPEC, not by increasing our output because at the stance that Nigeria is going to be a very challenging period. Not because we cannot produce but because majority of what we produce have been stolen not just in Nigeria, in the high sea. So they need to do something. And the security apparatus, the government need to find a political way. I know it's an election year and governments are taking the backseat but this is critical to the to the survival of Nigerian economy. We need to do something fast and we need OPEC to help us, not just by increasing our output and to help us fight this oil theft, especially in the waters. Well, that's so much we can take now. And it's been great speaking with you this morning on this particular issue. Thank you so much, Mohamed Muktak for being part of the show. Can you say a good morning to him? Well, Mohamed Muktak is a developmental economist and we have been talking about OPEC and its funding plus the fact that our quota has been increased. It should have been a plus for us but of course we have other issues that we're grappling with. That's the size of our conversation. If you missed out on every part of it, it will be okay to follow us on Facebook, Twitter and Instagram and do subscribe to our YouTube channel. It's at Plus TV Africa and Plus TV Africa Live South. Many thanks for watching. I am Messi Popo.


My name is Jeff Thielman. I'm honored to serve on the Arlington School Committee and excited to be a candidate for reelection. The School Committee sets policies for the Arlington Public Schools, works with district leadership to develop the annual school department budget and hires and evaluates the superintendent. Arlington has a collaborative school committee with people of different views and talents and I enjoy working with each of them to serve our community. I'm running for reelection to finish the work the committee has started in recent years to support the implementation of the district's new strategic plan and to be a voice for groups in our community who may not feel fully represented in our school system. Since 2003, I've been part of several projects that improved our district. I served on the Thompson Building Committee where I gained detailed knowledge of how to access and use state and local funds for school building projects. I was a member of the committee that managed the reconfiguration of the Gibbs facility into a sixth grade school, helped lead the effort to secure funding to expand the Thompson and Hardy Schools. And since 2016, I've chaired the Arlington High School Building Committee which is overseeing the largest capital project in our town's history. One reason I'm running for another term is to see the high school project through to its completion. We have managed the project through the pandemic and a recent period of inflation staying on budget the entire time. Phases one and two are done and our students are enjoying classes in the new steam and humanities wings which include a new auditorium, library and cafeteria. The building is beautiful and students, faculty and staff love it. Work on phase three, our indoor athletic space and amphitheater is underway and later this spring will begin prep work on phase four, the outdoor facilities including new fields and a connection to the bike path. The project will be done in the fall of 2025. In my current term on the school committee, I served on one of the working groups for the strategic plan and one of my goals over the next three years is to be sure the plan is funded and implemented. The plan requires the district to ensure that all students have access to a rigorous and excellent education. It includes steps to attract and retain quality teachers and staff, improve the district's infrastructure and operations and enhance communication and partnerships with parents in the public. The plan has a particular focus on supporting students with individual education plans, English language learners, black and Latino students, students who identify as LGBTQIA plus and students from low income families. My entire professional career has been spent leading schools and nonprofit organizations that serve those who are often left behind in our society. Early in my career, I oversaw the startup of 24 high schools that serve economically disadvantaged students in urban communities across the United States. I served as president of a high school in the city of Boston for six years and for the past nine years, I've been the president and chief executive officer of a multi-state human services organization that serves more than 20,000 immigrants and refugees throughout New England. And in this role, I oversee a staff of more than 250 professionals. That professional experience will continue to guide my thinking on our town's school committee. I would be honored to have one of your two votes on April 6th and you also can vote by mail or vote in person at town hall prior to election day. Thank you for watching.


Today, we are going to discuss the quantum mechanical treatment of various interactions of an atom and the atom is the simplest possible atom we can think of that is the hydrogen atom. Why you discuss this because the simplest system which is paramagnetic. So, to discuss any quantum mechanical properties we must start with Hamiltonian. So, what is the Hamiltonian of the hydrogen atom? As you all know Hamiltonian operator is a sum of this kinetic energy operator and the potential energy operator. What a hydrogen atom is T and B can be written as in the kinetic energy operator which is kinetic energy of the electron and the kinetic energy of the proton. So, this will be this is the kinetic energy operator for the electron and the kinetic energy operator of the proton. These are the Laplacian operator and the potential energy B is the electron the Coulomb attraction between the two. So, this is minus z e square by r where r is the distance between the electron and proton. So, in a particular coordinate system if I have x y and z let us say here the electron is sitting this proton is sitting then this distance r and for each of this coordinates here and here I have got the corresponding operators. Now, in magnetic resonance we really do not worry at all about these type of interactions. These are treated to be a constant energy that the system has. What we are interested in is over and above this interaction what are the magnetic interactions which are responsible for the magnetic resonance spectrum in general and EPR spectrum in particular. So, these terms are not used at all they are just giving rise to a constant energy of the system. So, what are the magnetic interactions that this atom has? So, if we put this hydrogen atom in a magnetic field then the interactions that we can think of is the Zeeman interaction here. So, this is the magnetic moment interacting with the magnetic field give rise to the corresponding operate energy. So, for electron the magnetic moment comes from the spin angle momentum and this is the relationship. So, its operator for Zeeman interaction is G e beta e s dot b. Similarly, for the nucleus it is G n beta n i is the magnetic moment of the nucleus. So, that gives the Hamiltonian for the nuclear Zeeman interaction which is G n beta n i dot b. In addition there is another interaction that is possible which comes from the magnetic dipole interaction. I have one let us say magnetic dipole coming from the proton another magnetic dipole coming from the electron. So, these two magnetic dipoles can interact and that is shown here this slide this is the let us say direction of the electric dipole moment this is the direction of the magnetic dipole moment in sudden x y z coordinate system and this is the distance between the two. So, this energy of these two magnetic dipoles depends on the orientation of these two dipoles. In fact, it is the relative orientation of them that matters the energy is given as this. So, it not only depends on the magnitude of the dipole, but also distance between them and the orientation of mu with respect to r mu with respect to r. So, therefore, this interaction is directional dependent and you call it anisotropic interactions. Now, what will be the corresponding operator for this we simply replace the magnetic moment of each of them by their corresponding operators in terms of the angular momentum. So, mu e is replaced by G e beta e s mu n is replaced by G e n beta n i in a similar fashion this is the expression for the dipole dipole Hamiltonian operator for electron nuclear interaction. Now, this dipole dipole interaction of is such that its average value turns out to be 0. So, when the species is tumbling rapidly let us say liquid or gas in their collisions of atoms then the average value of this energy becomes 0. How do you visualize that this energy of interaction becomes 0 it is not very obvious from this type of operator or this type of energy expression for that of course, one can calculate and turn and find out the indeed that is 0 if we take care of all possible orientations, but let us try to understand at least qualitatively whether this energy can be 0. So, for that this is the schematic representation of two magnets let us say mu 1 and mu 2 they are kept at a distance r and they are aligned such a way that mu 1 forms an angle phi with respect to r and mu 2 forms an angle theta. Then what we have here is mu 1 mu 2 and this is r 1 mu 2 mu 2 mu 2 mu vector. So, if I keep this in a magnetic field then both of them will get aligned along the magnetic field. So, you put it here then what will be the corresponding energy of interaction if their orientation changes and also go from let us say one alignment to other alignment. This is shown here this is the direction of the extra magnetic field and let us say one magnetic moment is fixed here and other one is moved from this arrangement to this arrangement. Here these two magnets are parallel to each other. So, north pole is point in north pole south pole is point in south pole. So, this energy will be repulsive. So, we call the energy is positive same is true if this magnet is brought here. On the other hand when second magnet is pointing in this direction and at the top of this first magnetic moment mu 1 then you see here the north and south here north south are pointing in such a way that there will be attractive interaction. So, this north is pointing to south. So, the energy of interaction will be negative here same is true here. So, approximately in this region the energy of interaction is negative and same is true here and in this region is positive and this energy is positive. So, it turns out that if we really look at the interaction operator in detail the angle of approximately 54 degree is the defining angle within which this rather within this cone of this interaction region where the energy is negative here also energy is negative and exactly in this along this region the energy of interaction becomes 0. So, the two regions are divided in such a way that somewhere energy is positive and somewhere energy of interaction is negative. So, here now if the system is tumbling very rapidly that all possible regions are covered equally then one could considerably assume that this total energy actually becomes 0 that is indeed the case if we do the calculation exactly for all possible orientations. So, in solution if your spectrum for example when the species is tumbling very rapidly then one can ignore this interaction but it cannot be ignored if let us say solution is viscous and tumbling is not very rapid or if the species is held in a solid state or in a frozen solution then the tumbling is restricted very severely. So, we cannot neglect this interaction but for our purpose right now let us neglect this. So, what other type of interaction this electron nuclear can have that is called electron nuclear hyperfine interaction it is a special type of interaction which does not depend on the direction it is called an isotropic interaction and also a special requirement is that the electron sort of seats on the magnetic nucleus. So, unlike here the dipole level interaction there here if the let us say this is the nucleus which is proton for hydrogen atom of course electron is moving all over this is the decided by the orbital of the electron. Now if there is some finite probability that electron actually occupies this place this is what I call sitting on the proton then that interaction is a special type of interaction that is called the electron nuclear hyperfine interaction it was proposed by Fermi the expression is given here. See just like dipole level interaction this also depends on the strength of the magnetic moment of electron and the magnetic moment nucleus but it also depends on this probability of finding the electron at the nucleus. So if the psi is the wave function of this electron which is describing its motion around the nucleus the psi 0 and the square of that gives the probability of finding the electron at the nucleus. Now for those orbitals which satisfies this requirement that this has to be non-zero then this expression shows that this isotropic electron nuclear is hyperfine interaction will be non-zero. So not all orbitals have this property we know from the property of hydrogen orbitals it is that s type of orbital for example 1 s, 2 s etcetera. So here if you plot the wave function square as a function of the distance from this nucleus r then this is psi r square. So this for 1 s orbital it goes down exponentially. So at r equal to 0 this is value is non-zero similarly for 2 s orbital it looks like something like this this could be 2 s orbital. So for all the s orbitals there is finite probability of finding the electron at the nucleus. In contrast if you take other type orbital let us say p orbital this is the 2 p orbital similarly d and other orbitals there here the at the nucleus the probability is exactly 0. So for these sort of orbitals there will not be any such electron nuclear hyperfine interaction but for these orbitals it will be. So for hydrogen atom this is of course the exact form but for other system let us say free radical species this will be the corresponding wave function of the electron which is producing the interaction and probability of finding the electron at the particular nuclear site has to be used here. So this whole of this term here is replaced by a constant called A is a parameter in fact. So this constant is called hyperfine coupling constant or hyperfine splitting constant. Now these 2 terms though they are used often synonymously they may not mean exactly the same thing. Let us right now let us understand this just a subtle difference what I mean by that in this Hamiltonian isotropic Hamiltonian I write A s dot i. So here this is an energy of interaction. So here A gives the strength of the interaction the A gives the strength of interaction. So here I will be calling it as hyperfine coupling constant with coupling interaction between the electron dipole and the nuclear dipole. So when it is measured in the units of energy this is the coupling constant. Now we will see that this will of course show its presence by splitting the EPR spectrum. So EPR spectrum is measured as a function of magnetic field generally. So we measure something from the experiment so which is a measure of the splitting in terms of the magnetic field. So this interaction causes splitting in the spectrum. So whatever property we measure from here the measure of the splitting that is of course related to this one but that splitting that we measure from the experiment would be called the hyperfine splitting constant and we will see how they are related later. So I have already said that psi 0 square gives the probability of finding the electron at the nucleus and this is non-zero only for electrons in S type of orbitals. So this Hamiltonian where we are dealing only with the magnetic interactions such as this electron Z1 interaction, nuclear Z1 interaction and the hyperfine interaction. So this type of Hamiltonian are called spin Hamiltonian where we see that we do not include the kinetic energy of the various constituent particles or potentiality that they have. Only this interaction energies are included in the spin Hamiltonian. Now if the magnetic field B is along the Z direction then this Hamilton offered a curvature as B0 SZ minus GE G electric nuclear Z1 SZ IZ. This can be written as the magnetic sum of two components plus this is one component plus another component. We will see why you do that in a moment. This term which involves the X and Y this mistake SX IX plus SY IY. So this can be written in terms of the raising and lowering of order S plus minus equal to AX plus minus on SY. So SX gives S plus plus I minus 2 and SY is equal to SY is equal to SY minus I minus Y 2I. Similarly we can write for the IX and IY. Then SX plus IX for example this becomes S plus plus S minus by 2 into S which gives SX plus IX plus. Similarly if we take the other one by putting this one then this term will cancel out. So what we see here that the simplification of this will turn out to be this way that the electron Z1 nuclear Z1 and this SZ as a plus this is the term that comes out to be S plus I minus plus S minus I plus by 2.  Here now let us see the relative magnitude of the various interactions. This is the electron Z1 term this is nuclear Z1. So we know that magnetic moment of electron is about 2000 times higher than this. So this will be the most dominant interaction. This will be very very small compared to that one and this magnitude of the hyperbolic constant is also not very large. So this will still be the dominant interaction. This is usually very small. So what we do now is that we treat this as the unperturbed Hamiltonian or main Hamiltonian plus this is the perturbation. So A0 is equal to this is the main Hamiltonian and perturbation Hamiltonian is A by 2 minus plus S minus I plus this one. With this approximation we will see how the energy level that appears for hydrogen atom in the next discussion. We stop here now.


Now let's do a survey of the more popular languages in use today. The C language, created back in the 1970s, is a statically but weakly typed language. Like virtually all other high-level languages, C code is composed of expression statements, control flow statements like if and while, and functions. However, unlike most other high-level languages in use today, C does not automatically manage memory, so the programmer must manually allocate and deallocate heap memory. Partly for this reason, and partly because of its weak typing, code written in C can link with code written in assembly with little overhead. On the whole, C is a high-level language that uniquely maintains much of the manual control and efficiency of assembly code. Consequently, C is a good choice for writing low-level software, including operating systems. The Linux kernel, for instance, is written mostly in C. The code sample here is the Hello World program written in C. Notice the use of curly braces and semicolons, which several popular languages like JavaScript later imitated. The C language is not just one of the most successful languages in the history of programming, it's also one of the most influential. The language called C++, for instance, is basically the C language, but with more stuff added. In other words, C++ is a superset of C. It has everything C has but more. Consequently, a C++ compiler should be able to compile any code written in C. However, this isn't entirely the case because C++ is not a strict superset. Over the years, both C and C++ have evolved on their own paths, and while there's still very heavy overlap, if you look at the details, you'll find differences that will cause some C code to not compile properly with a C++ compiler. Today, it's best to think of C and C++ as really two very separate languages. The name C++ is a bit of a joke. In the C language, the operator, which is two adjacent plus signs, is used to increment the value of a variable by one. If you have a number variable called, say, X, and you see X++, that means take the value of X and increase it by one. So the joke is that C++ is like C, but one higher. The primary features which C++ adds to C are for facilitating object-oriented programming. C++ can be basically summed up as C with object-oriented features. To be clear, it's not impossible to program in an object-oriented style in regular C. It's just that C++ adds features that make programming in that style more convenient. Despite these added features, C++ maintains most of the same efficiency that C itself has, so when you want to code in an object-oriented style but also want the efficiencies of C, C++ is the primary choice. For two decades now, C++ has been the language of choice for most commercial applications like Photoshop or Microsoft Word, and for most commercial video games. Objective C was created about the same time as C++ for largely the same purpose. Objective C is like C, but adds features for object-oriented programming. The popularity of Objective C today mostly stems from Apple pushing it as the main application development language on OS X and iOS. Without this push, its doubtful Objective C would be much used at all. The Java language was created in the 1990s by the company Sun Microsystems. Like C++ and Objective C, Java is heavily biased towards object-oriented programming, and its syntax imitates the C syntax. However, Java is not at all a superset of C and has automatic memory management with garbage collection. Java also has strong rather than weak typing and a hybrid mix of static and dynamic typing, which we'll explain in the unit on Java. Java was the first widely used language which compiles its source code to an intermediate form of code and then runs this intermediate form via a virtual machine. By the late 90s, Java had become the most popular language in just a few short years, and so deciding that they needed something like Java, Microsoft created the C-Sharp language. C-Sharp started as a virtually verbatim imitation of Java with a few more features, though over the years it has added several major features to further distinguish itself. Microsoft makes the most popular C-Sharp virtual machine, which it calls the CLR, the Common Language Runtime. While the CLR only runs on Windows, Mono is a free open-source C-Sharp VM that runs on other platforms, including Mac and Linux. The Perl language, which became popular in the 1990s, especially on Linux, was the first example of a dynamic-interpreted language really making it big, especially on commodity PC hardware. Before Perl, dynamic languages weren't used much because of their lackluster performance. Perl didn't actually fix this problem, but by the 1990s, computers became powerful enough that, for many tasks, the slowness of a dynamic-interpreted language was worth the cost. Compared to the primary languages of the time, including C and C++, Perl is much more expressive, allowing the programmer to get more work done per line of code. The Perl Hello World program, for example, is just a single line that says print Hello World. Perl, though, is criticized by many for overly complicated syntax and semantics. This complexity clearly isn't reflected in the Hello World program, but it's actually quite easy to write lines of Perl code that look like line noise, like a bunch of random characters to the untrained eye. Perl suffers from heaping conveniences upon other conveniences, making Perl code often difficult to read and understand. In contrast, another dynamic-interpreted language, called Python, is widely praised for the clarity and comprehensibility of its code. Whereas Perl is criticized for its overly complex syntax, Python is praised for its notably clean and simple syntax. The clean appearance is achieved largely by virtue of Python using indentation to denote blocks of code, just like we saw in Pigeon. So, instead of, say, surrounding the statements of an if-body in curly braces, we just put those statements indented in one level underneath the if itself. Semantically, however, Perl and Python are really, really similar at their core, though I would say that Python has a much cleaner type system and is therefore better for object-oriented programming. Like Python, the Ruby language is pretty much like Perl semantically but with better syntax. However, I personally strongly prefer Python because Ruby has too many misguided convolutions like Perl. PHP is yet another Perl-like dynamic-interpreted language from the 1990s. PHP owes most of its popularity to the fact that, in the late 90s, it by far had the easiest learning curve if you wanted to get into web programming. Though PHP is heavily criticized by many programmers today for being ugly and badly designed in general, it's still commonly used for many large websites. You can't argue with good timing and PHP just came along at the right time. One more language that shares the basic semantics of Perl, Python, Ruby, and PHP is JavaScript. Despite the name, JavaScript really has no special relationship to Java. They're totally separate languages. JavaScript was created as a language to embed in the Netscape Navigator web browser, such that web pages could include JavaScript code to manipulate the contents of the page. This feature became popular, so other browsers added support for JavaScript, and today JavaScript is still the only language that runs in all major web browsers. The most unique thing about JavaScript semantics concern its approach to object-oriented programming. While most languages like C++, Java, and Python facilitate object-oriented programming with a feature called Classes, JavaScript gets most of the same effect with a feature called Prototypical Inheritance.


Hello everybody! This is a video about how to fix various errors and problems that may appear when you are using a USB flash drive. For example, the computer cannot see a flash drive, the flash drive became smaller, the flash drive is recognized in RAW format, or Windows cannot recognize its file system, or you see the message insert disk when the flash drive is already connected, and so on. I would like to note that these problems and the methods to solve them are also relevant for dealing with external hard disks and memory cards. For example, when trying to open a flash drive connected to the computer, Windows flashes the error insert disk. One quite easy and very effective way to solve such problems would be cleaning the disk and its partitions with the help of clean command and the disk part tool. It is very important to remember that this process will erase the entire disk, so make sure that you have chosen the right device, otherwise you risk deleting data from the wrong disk. So to begin with, right-click on the Start menu and click on Command Prompt Administrator. Before you continue, make sure that the USB drive, memory card, or external hard disk, which you are going to clean, are now connected to your computer. Then enter the command Disk Part, and then List Disk. Now we can see the list of all disks installed in this computer. In my case, I need to clean the flash drive marked as Disk 5. Be careful. If you choose the wrong disk number, it will be erased and important data may be lost. After that, enter Select Disk 5. That is, this command chooses the particular disk and all actions will now affect this disk. To clean the flash drive's partition table, type in the command Clean and press Enter. After the process is complete, we can see the report saying that disk cleanup was successful. Now enter Create Partition Primary. This way we create the main partition in the flash drive. And use the command Select Partition 1 to choose this partition to work with. Now we need to format the flash drive in the chosen file system, for example NTFS. To do it, enter the command Format FS equals NTFS Quick. That is, we perform quick drive format. If we take away the word Quick, then we will get regular formatting, which takes up much longer. For example, let us choose Quick Formatting, press Enter, and wait until the process is 100% completed. Then assign a letter for this flash drive, for example M, with the help of this command assign letter equals M. As you do that, finish the work of Disk Part with the command Exit. Close the command prompt, go to the Explorer, and now we can see that our flash drive is completely operable. As an example, I will create a folder named Test on this flash drive. As you can see, the drive works perfectly and there are no problems. If you liked this video, click the like button below and subscribe to our channel to see more. Thank you for your attention and good luck.