Welcome traders to this week's live market and trade analysis session with me, Patrick Munnally, before we get going here. As always, I want to adhere to the risk disclaimer, notably that the views and opinions expressed by me during this presentation are solely mine. They're not indicative or representative of those held by Tick Mill UK or Tick Mill Europe Limited. Okay, we are going to get going here in just a minute. We're going to run through the trade plan for the New York session. If you have any questions during this presentation, feel free to drop them into the chat. I will come back to them as and when we have time depending upon the market action. So let's review the trade plan. I would also direct your attention to the Tick Mill Futures and Options Group where on a daily basis I share my trade plans for the London and New York sessions. We also have a trader chat function that you're free to join. Through the trader chat, I share institutional research and insights from Tier 1 investment banks and trading desks. As retail traders, essentially what we're trying to do is write the co-tails of these bigger players in the market. So it's useful to know how they're positioned and where they see opportunities developing in the markets and where the flow and sentiment is because these are the guys moving the big blocks of money around that we want to try and capture some of those moves. The trader chat, I also share real-time trading updates, my personal P&L trades that I'm taking with a very keen focus on risk management. As I tend to really want to drive the attention to risk management as opposed to profit targets and trade outcomes because it really is in risk management. I'm focusing on that. I'm focusing on discipline trading, a consistent approach to the markets that will ensure your long-term success as a trader. So feel free to request access to the group. You can see the email address. I'll just post it into the chat there one second. You can request access. All you have to do is share some contact information to avoid bots joining the group and flooding it with useless information. Let's take a look like I say at today's trade plan before we get going here. We've just had data out. We've got Lagarde speaking at the moment. She's standing by what she said on raise. They're basically on whole data dependence and that's put a little bit under the dollar here. We have seen an uptick in terms of the futures from the lows we saw. We've got PC inflation data out tomorrow. My current trading position, my current trading book, I'm short from yesterday. I've got an average fill from 49.25. I managed to scale off that position on the set off last night. So I've essentially got a risk-free short running from 49.35. I also have a long position running. I managed to scale some of that this morning. So I'm essentially running a risk-free long position from 48. That should read 82, not 92. The short position is potentially a swing position. I'll talk you through my rationale for that shortly. Moving into the cash session now. My lien is long against the Daily Ball Bear Zone. The Daily Ball Bear Zone reference down here is 48.90 to 48.82. So whilst that remains supportive, I'm looking for potentially further upside here. Noted that participation and range in terms of the overnight session is on the low side. We're currently trading above the overnight volume point of control, which is 49.01. But today's volume point of control is below yesterday's, which is 49.17. We have an expected range today of 49.27 on the upside and 48.70 on the downside. These are key reference points that I use on a daily basis. Two Sigma volatility events above and below last night's closing price are 48.42 to the downside and 49.55 to the upside. We also have weekly projected range resistance 49.66. So what that's telling me is any move up into that 49.50, 49.60 area would be a stretch and 95% of the time on a daily basis, the market will close above or below the two Sigma volatility support resistance. And as I said, we've got that added confidence of 49.66 being a weekly projected range resistance. So if today we hold our current lows and extend to the upside, I would pay very close attention to how price trades into that 55 to 66 area has potential meaningful resistance for the week. And we are getting somewhat late in the week, Thursday now, I think it's Friday. So if we test into those areas, that could cap the near term upside for these markets. So let's pull up the multi-time frame charts here. And you'll see the reason for the swing trades in terms of the futures here. We've traded into this weekly trend line resistance. So here on the weekly chart, we tagged that yesterday, and that's where we sold off from. And if we hold that, you can see the last test up into the weekly projected range resistance, you can see the similarities in terms of the price structure as we got into that area. We tested it, got a sell-off, then consolidated within that range for two or three days before getting a pretty meaty move to the downside. Now, I'm not suggesting that that's what's definitely going to happen here, but I'm positioned to potentially take advantage of that if that's the case. So if we are to replicate, as I say in history, Ryan's, if not repeat, so if we replicate that type of price action, we'd look for the highs to hold consolidation and then a potential rollover post that consolidation. But that's on a daily time frame, and today we're focused on intraday opportunities. So here on the four-hour time frame, this is a chart I share via the futures group. I've got the levels mapped out in terms of the two-sigma volatility resistance up there at that 49.55. It's also weekly R2, 49.58. We have the expected range resistance 49.27. We've got our daily bull bear zone, daily range support, and then that two-sigma support. Now, that two-sigma support to the downside, that 48.44 coincides with, sorry, 48.42 coincides with our weekly bull bear zone. So if we do get a pullback into that area again, I'll be paying attention, intraday looking for divergences on the intraday charts. I'm going to pull those up in a minute. Walk you through how I use those and the tools I use to execute trades on an intraday time frame, but certainly pay attention. If we do get a pullback 48.44, 48.42, 48.34, our key levels I'm paying attention to on the downside. So let's pull up the intraday charts and we can see we are trading above yesterday's midpoint, which is 49.11. 49.17 is yesterday's volume point of control. So positioned currently looking at the long side, we are trading in the upper half of yesterday's range now. When we're trading within the range, obviously within the prior days range, there is the potential for chop. We prefer to trade away from ranges as opposed to within them, but we've got some key reference points as we get going here. The value area high from the overnight session is 49.09. This blue line is the full session volume waste average price. So when the future is open last night in New York, this is telling us the volume waste average price for the session. So I've been looking initially at any pullbacks back into this 49.04, 49.07. This gold line is giving us the midpoint for the current trading session. So 49.07, 49.04, the value area high 49.09. So any pullbacks into this area, I'm watching for bullish reversal patterns to put a look on the long side, targeting move up into test the prior yesterday's highs at 49.33. And if we get through there, then we have the further upside targets to play for in terms of the 49.50s to 49.60s. So that's the bullish scenario that I'll be watching. The bearish scenario would be that we fall through the overnight volume point of control. So 49.00. It's going to be a key reference point as we head into the cash session there. If we get through there, then we're looking at a retest of the Glovex lows, which are 48.93, and back into yesterday's lows, 48.89. And if we start to really lose support there, then I'll be tracking a potential equality objective to the downside. So let me just go back to the arrow in the chart here. So we have the ascending trend channel support coming in at that 48.89. So that's going to be a key reference point if we move down into that area. Let me just remove these for you for now. So if we hold the current highs and we start to see responsive selling into the open of the cash session here, let me just show you another downside target that would be in play. So an equality objective versus the swing highs that we have at 49.20. We'll actually put us back down right into the lows of the weak here into that 49.75 area. And potentially from there, we could see a pop, but then another leg to the downside. So it's going to be key to see whether or not we get some early bids here in the market. If we don't, we have some reference points that we're paying attention to on the downside. So we are two minutes out from the open. I'm just going to grab some water. And like I say, if you have any questions, feel free to drop them into the chat box and I'll cover them off as and when I can. I will alert you to any new positions that I initiate during the this opening 30 minutes or so of the cash session in New York. Okay, 30 seconds to the cash open. In terms of the indicators, the indicator pains here, this is the Delta, the lower pain. Delta is essentially a measurement of buying and selling pressure that I used for confirmation and non-confirmation potential entries. I'll walk you through that in real time if we get an opportunity. The pain above once it populates is the tick. The tick is a measurement in every woman's iteration of how many stocks are ticking up versus how many stocks ticking down on New York Stock Exchange. I use tick extremes as potential entries again on how we are set up. Though we have bullish a advanced decline line, this pain at the top here references the amount of stocks advancing versus the amount of stocks declining on the New York Stock Exchange. When it's above zero, that's bullish and when it's below zero, it's bearish. That's essentially giving you a look under the hood of what's happening below the index level with the constituent stocks in the market. We also have the next pain below is the breadth. That's the amount of money flow, the volume going into stocks moving up versus stocks moving down. Again, if that's above or below zero, that's going to give us a confirmation for bullish or bearish price action. If we can make new highs here above the Globex highs with confirmation from the Delta, I'm going to look at adding to my long position currently long from 48.92s here. I'm going to look at adding and seeing if we can get a run up to yesterday's highs, all-time highs, that 49.33 area. Let's just see how this sets up. Okay, just letting these opening rotations come through. Like I say, if we hold support here at the value area high, so the value area high represents the top of 70% of the volume that's traded in the current session, so it's actually the support at the moment. If we can take out the Globex highs here and we get a new high in terms of Delta, then I'm going to look like I say to add longs here. We've got positive AD line, positive breadth. That's going to be the play here out of the gate. And in terms of stop, I initially used a 10-point stop on any trade I take, which is equal to 1% of my account's equity. And more often than not, what I'm looking to do in terms of trade management is scale out half of my position, plus five or minus five, and then hold the rest. So that gives you an idea in terms of how I manage my positions. There's a few who intend to follow along here. So we're seeing a new high in the Delta. So if I can get in here, we're looking for a move through 21, and then initial stop will be 11. Delta pushing higher. You'll see that confirmed on the break with new highs. And then I will add a long exposure here. I guess so we're looking for a push higher then through the 33, looking for 45, then 50, 55, and 66 as weekly projected range resistance. So we're going to risk 10 with the potential of at least three or four to one risk reward developing here. But what we'd like to see as well here to support the upside is the volume point of control to rotate up into the upper distribution of the overline. You can clearly see we have two distributions are lower and an upper distribution. Support on the upside would see that volume point of control shift into the upper distribution. That would add support to the bullish thesis here. So the bullish consolidation, can we break the highs here with Delta confirmation to go long through 4921? Is the current plan running? If you see it, you're going to see that. That's the first one. Because obviously, the DCB is making the difference. trading above the developing value area high, which is now up at $49.14. Like I said, we really like to see a rotation higher in terms of the volume point of control as additional confirmation. So we have a double top in place now, which looks vulnerable. I'm not seeing a sharp rejection currently. I did see some initial selling, but we're consolidating within that opening swing there in terms of the Mag7 bullish Tesla, obviously taking a bath after it missed earnings last night, a bit of a car crash in terms of the earnings call with Elon throwing his choice out of the pram, talking about walking away, so Tesla remains under pressure. But the others, Apple rotating around the flatline and video Microsoft Google or positive Amazon, a little bit weak meta up. So they're supportive and we've got yields pulling back here, across the curve to 10s and 30s are down. So we are set to look for a push higher here. We're looking to break these highs until I get 45 as the next upside objective. And it's not a start to the economy, it's a standard interest in one bit. But in the next week, we're gonna have a price increase, and as I said, and in the end, it's a little slightly larger, not as nice, but a little larger than itself. That's what it looks like to the emotion. Right. So it's very up, but they're slow and they're too one-wise, and it's a daily chaos. We are having an existing age of surf, training, but now it's not so much contributing to the economy. So it's a little non-settling, it's a coconut Okay, so we are looking for 49-21 to engage on the long side, I was saying yesterday's highs at 33, then on to 45s, positive breadth, positive AD line, ticks are, we haven't had a negative tick yet, so plenty of pressure on the upsides, one minute, M1 deltas are pushing highs. Justifications here, once we wait for this potential long to set up and confirm the growth, just a hundred points, a thousand questions as we sort of have a little bit of time. So seeing a little bit more consolidation, we're still looking constructive here for now, we're trading above the full session, mid full session, VWAP, those would be the first warning signs of weakness if we start to see prints below this area here, which is the view lines, the full session, volumetric storage base, the goal line full session midpoint, so that would be the first warning sign that the long set up was invalid, but really like I said, once we hold the overnight value area, sorry the overnight volume point of control which comes in at 4900, I'm still looking for an ultimate push higher, I don't really want to initiate at the volume point of control because we're right back into the majority of the trades from overnight, that's likely to be a chop zone, so I'm waiting for that to push higher with the deltas confirming to take this one higher if that trade plays out. Nothing to do on the short side until we take out yesterday's low, and then what I've been looking for would be a move down that then retests and rejects yesterday's low, and the Globex, I've used that as my reference point, so we'd have 4889 to 4893, so I'd look for a breakdown and pull back into that area that gets rejected, and then looking to play the break of the lows with the deltas making new lows as confirmation, and certainly at that stage I'd also want to see either the AD line or breadth turning negative and as an additional confirmation that we've exhausted the near-term upside and need to see some further price discovery to the downside, and as I referenced earlier we have that 74 as an equality objective if we hold the 4920s, but the downside scenario isn't in play at the moment whilst we hold those reference levels. Any questions? So anything coming through at the moment? I must be doing a reasonable job of explaining what I'm saying, which is good. We're holding above yesterday's mid, a midpoint of yesterday's trading range, which is the 4911, you might see this volume point of control rotates up into the 4916s, as I said the other situation obviously is that we're trading within yesterday's range, which more often than not will make make progress a little bit slower. Okay there isn't anything else to add at this stage, I don't see any questions coming through, so I won't just hold on here for the sake of it. The trade that I'm looking for is to engage on the long side through 4921, I'm going to see the deltas make new highs, confirming that move, we've got positive ad line, positive breath, the ad line is declining somewhat, but we're at elevated levels, tick is now rotating around the midpoint, which is indicative of this chop that we're seeing, that's my trade on the long side, so if we get through 4921 I want to engage on the long side, the alternative scenario is a breakthrough, yesterday's lows and a pullback to act as resistance to engage on the short side, targets are going to move down to 4970s would be the bearish trade, okay I said I'm running shorts from yesterday, potential swing trade versus yesterday's high, and I have lows from the overnight session which are risk-free as well, I'm just looking to add that long exposure for upside targets, I'll just reiterate those, there'd be a retest of yesterday's highs at the 33, then the 45, then the 55, and 66 would be a test of weekly projected range resistance, the 55 is a two sigma volatility resistance, so I'd be certainly looking to lock in profits on a 55 test and potentially play a countertrend short from there, if we get divergences developing on any test of the 55, we're just ticking up a bit here, so I'll hold see if we get to call this live here on top of the hour, so let's see if we can get this one confirmed the other thing we don't want to see is we don't want to see a ticket string print on the entry because that more often than not, programmatic trading models will use that as a profit-taking signal, so we want to see price making a new high without a new high ticket of the day I'm excited that we can make people share a new new plan because we want to remember that time we used to, it was always, it was always when I looked at it, I actually was sitting there sitting there speaking to each other on the side of the speakers, they're not much doing this because of the patience of the people that we have, they need to know how to get out there, but we don't want the taxing still to fall on the future, obviously it's not going to be used to that point, it's not going to be used now, plus that, I want them to learn how to get anywhere. Okay, the solution continues, okay look I'm going to wrap up the live stream here, you have the plan for the session, you know what the trade levels are and the confirmations I'm looking for, so best of luck with that for those of you who are in the TickMill Facebook group, I will be updating the trades via the TraderChat through the initial balance, which is the opening hour of trade in New York, so it takes 3 to 3.30 UK time. And that concludes this week's stream. As always, trade the plan and most importantly manage your risk. Until next week, thanks very much.


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Let us continue our qualitative study of the wave equation and today we are going to look at what is called causality principle and finite speed of propagation a property which is exclusive to hyperbolic equations we will look at them through examples. This is these are the two sides of domain of dependence and domain of influence of the concepts. Causality principle, finite speed of propagation. Causality means cause and effect. What are the reasons in the past that are responsible for the current state? What will be the future events for which the current state is responsible for our influences? These questions were answered in lecture 5.2 using the explicit formulae for solutions to the Cauchy problem. In this lecture we attempt to answer the same questions without using the formulae for solutions. This kind of justifies the use of the word qualitative analysis in the title of this chapter because we are not using any quantitative formula for the solutions. The analysis presented in this lecture is a typical illustration of an a priori analysis. A priori means done before. Conclusions can be drawn on solutions despite zero knowledge on their existence. We may not be even knowing whether solution exists or not. Still we can conclude certain things about the solution of course if they exist that we do not know. So in this discussion once again like in lecture 5.2 we are going to switch off the non-homogeneous term and we have already explained reasons for that in lecture 5.2. So let us state in the form of a theorem. The hypothesis is let you from Rd cross 0 infinity to R be a classical solution of the Cauchy problem for homogeneous wave equation that is this is the problem where phi and psi are given of course in the Cauchy problem and use a solution to this Cauchy problem. Conclusion for X0 t0 a point in the space time the value of u of X0 t0 depends only on the values of phi and psi in the closure of this open ball with center X0 and radius ct0. Closure of this open ball is nothing but the closed ball with the same radius and center. Recall that B of X0, ct0 is the open ball with center at X0 in Rd and having a radius of ct0 lying in Rd. So maybe we just briefly write what that is. So those elements in Rd whose distance the Euclidean distance from the point X is less than ct0. This is the open ball and the closed unit ball we do not use but then let me introduce good notation is that those elements in Rd such that y minus X is less than or equal to ct0. Proof of this theorem it follows immediately from the formulae for solution to the Cauchy problem namely D'Alembert formula for D equal to 1, Poisson-Kirchhoff formulae for D equal to 2 and 3. This was done in lecture 5.2. A direct proof is presented in this lecture without using the explicit formulae for solutions. Of course we have to use something that may be some experience. Okay we prove the theorem for D equal to 1 only why is that because the proof for D equal to 2 and 3 are similar but for the obvious modification that needs to be done to the proof of D equal to 1. To avoid a lot of repetition we do not do the proof for D equal to 2 comma 3 but mention one important inequality there and how that can be derived just the idea we will present. So proof of Cauchy principle for D equal to 1. Let X0 t0 be a point and r cross 0 infinity. Consider the characteristic triangle which is a triangle formed by the characteristics lines through X0 t0 on the x axis that is this is the point X0 t0 this point is X0 minus ct0 is the point X0 plus ct0. This is called the characteristic triangle. What are these lines? This is X minus ct equal to X0 minus ct0 and this line is X plus ct equal to X0 plus ct0. Of course this line is nothing but t equal to 0 x axis. So fix a t such that t lies between 0 and t0 and draw the line t equal to t we will get a trapezium recall point X0 t0 X0 minus ct0 X0 plus ct0 draw this line t equal to t. So we get a trapezium which is formed by the intersection of all these lines that is this part let us call it f. So multiply the wave equation this is a trick the wave equation you have to multiply with ut you get this and re-organize the terms you get this if you expand this it will reduce to this. Now this is in a good shape because here derivative utt into ut is there u x x into ut is there whereas here some t derivative of some quantity x derivative some quantity is there it is like the divergence theorem we are in a good shape to apply divergence theorem therefore this is a good arrangement. So in fact as you see the divergence with x and t here of this quantity is 0 that is precisely this equation this equal to 0 means this. Now integrate this equation on the trapezium region that we have indicated on the previous slide you get this now we are ready to do integration by parts or apply Green's theorem and divergence will be converted to something else. So what we get is that this divergence of course it is equal to 0 is now converted to an integral on the boundary of f f is the trapezium region boundary of f is actually the trapezium and that this becomes the integrand dot n d sigma n is the normal outward unit normal to the points of boundary of f of course that varies from point to point on the boundary of f. So the boundary of the trapezium consists of four lines actually the boundary of the trapezium region is the trapezium itself it consists of four lines they are the base of the trapezium denoted by b given by the equation t equal to 0 a part of the characteristic denoted by k 1 given by the equation x plus ct equal to x naught plus ct naught upper part of the trapezium denoted by t t for top given by the equation t equal to t and a part of the characteristic denoted by k 2. So remember this is the point x 0 t 0 then we have drawn this. So this is x minus x naught minus ct naught x naught plus ct naught then we took this part. So this equation for this is t equal to t. So this we are calling a b for base this is k 1 p for top and k 2. So the integral is now on these lines. So we need to determine what is the normal to each of these sides. So base of the trapezium outward unit normally 0 minus 1 top the unit outward normally 0 1. So this is the base this is in the direction of negative t axis because these x axis this is t axis positive t axis direction. So this is in the negative direction so 0 minus 1 and here the top normally is in this direction outward unit normal it is in the direction of the positive t axis so 0 1. Then we have the sides of the trapezium which is k 1 and k 2 at this point because k 1 itself is straight line it is very easy it is constant on the direction is the same on all the points on k 1. Similarly the outward unit normal is in this direction and that is constant for all points on k 2. Therefore life is simpler so this integral on the boundary now we can split into 4 parts which is b union k 1 union t union k 2. So on b we have applied what is the normal similarly on k 1 k 2 1 t we have used the formula for the normal that we have written down on the previous slide we get this expression. So some of these 4 terms is 0. Now we do a trick the integral on k 1 can be expressed like this and this is an integral on k 1 of some non-negative quantity because of the presence of the square this is always greater than or equal to 0 and hence this integral is greater than or equal to 0 that is that is the property of d sigma non-negative functions internally will be non-negative similarly here remember this d sigma is nothing but the measure on the boundary which is coming from the domain trapezium. So this has all the nice properties if you integrate a non-negative function integral will be non-negative. So the integrals over k 1 and k 2 both of them are non-negative numbers okay. So here I have some of 4 quantities in that some of 2 of them is non-negative. So what can I say about the some of the other 2 it should be non-positive that means that these 2 terms together is less than or equal to 0 which means I have the integral of this quantity on the top is less than or equal to integral of this quantity on the bottom. If you notice from this inequality if u is 0 on the bottom u and ut are 0 on the bottom then this is 0 then they will be 0 on the top also this is the one which gives us uniqueness of solutions as we are going to see on the next slides. So this inequality is called domain of dependence inequality that means on the trapezium on the top portion the integral is less than or equal to integral on the bottom portion. So let u and v be solutions of the Cauchy problem defined w by the difference u and v that is w equal to u minus v note that w solves the wave equation due to linearity of the wave equation of course and the Cauchy data will be 0 because both u and v are solutions to the same Cauchy problem. Therefore both u and v will satisfy the same Cauchy data and hence the difference will be satisfying the 0 Cauchy data. Therefore w of x0 is 0 and wt of x0 is 0 on the bottom. Now we can apply the domain of dependence inequality and conclude that we have this right hand side has become 0 right hand side was the same integral on b that is 0 because on b both wt and wx are 0. Therefore this is what we have but if you look at this already non-negative quantity and we are saying that is less than or equal to 0. So this is always greater than or equal to 0. Therefore the only possibility is that integrand is 0 which means wt at what point x, t and wx at the point x, t is 0. Now t is arbitrary we will choose and capital T less than t0. Therefore we get wt of xt and wx of xt is 0 for every xt belonging to the characteristic triangle. This implies that w is a constant function but w is already 0 on b. Therefore it must be 0 everywhere. W is 0 everywhere is same as saying u is equal to v everywhere in the characteristic triangle. Therefore the solution is the same. u at x0, t0 will be same as v at x0, t0. So proved on the last slide w is a 0 function inside the characteristic triangle. In particular u of x0 equal to v of x0, t0 this finishes the proof of the theorem. This is another proof of uniqueness of solutions to the Cauchy problem. We already gave one proof of uniqueness earlier. Now this is another proof of uniqueness of solutions. One more proof we are going to see using the energy method. Pretty much the actors in that energy method we have already seen in the domain of dependence inequality. We will do that in the forthcoming lectures. Consequences of domain of dependence inequality. The above discussion shows that u of x0, t0 depends only on the values of the Cauchy data on the base of the characteristic triangle defined by x0, t0. Namely the interval on the x axis x0 minus ct0, x0 plus ct0. In other words the Cauchy data phi psi at a spatial point x0 can influence the solution only in the region enclosed by the two characteristics starting from x0, 0 on the x axis which is given by xt in the space time domain such that x minus ct is less than or equal to x0 and x plus ct is greater than or equal to x0. Let us look at the causality principle for d greater than or equal to 2 and characteristic cone. What is a characteristic cone? The characteristic cone also called light cone at a point x0 t0 in space time is defined as the set xt in rd cross 0 infinity such that norm x minus x0 equal to ct times mod t minus t0 where norm x minus x0 is a Euclidean distance in rd between x and x0. The characteristic cone at a point x0 t0 together with its interior together with its interior this is only the surface this is with the interior we are considering now that is called solid light cone that is the solid light cone is this set. So this is the picture this you can imagine r2 or r3 it is easy to imagine r2, r3 you cannot really imagine because the picture will be in four dimensions but imagine this is r2 then this is what is called forward cone or future cone this is the backward cone or past cone this is vertex which is x0 t0. So the definition of a characteristic cone is consistent with the definition of a characteristic hyper surface which we have introduced in lecture 3.7. Recall that the analytic characterization for a hyper surface given by phi of xt equal to 0 to be a characteristic surface is at grad phi dot a grad phi equal to 0 where a is the diagonal matrix minus c square minus c square minus c square d times and 1 for the wave equation in d space dimensions. So this equation takes this form when we expand this gradient is in xt and what is a is a diagonal matrix described in the last slide. So when we do that we get this expression it is phi t square minus c square into norm grad phi square equal to 0. So this equation is nothing but phi t square minus c square mod grad phi square equal to 0 or you can put norm grad v square if you want or phi t is equal to plus or minus c mod grad phi. Note that this function which is here is a solution to the last equation you can substitute and check. Now what is phi of xt equal to 0 represent it is nothing but the characteristic cone through the point x0 t0. So why the characteristic cone is also called light cone characteristic cone is a union of all light rays that emanate from the point x0 t0 which travel at the speed c that is mod dx by dt equal to c this is the speed expression for the speed. In other words this set which we have here is nothing but union of these sets what is this this is a line right passing through the point x0 t0 because when t equal to t0 in the direction v. So take all this direction or the vectors with this length c and then this is precisely that. So both the sets are same that is why the characteristic cone is also called light cone speed is c okay t cross sections of light cone what they are each t cross section of the solid light cone will be a solid sphere if you omit solid here light cone will be sphere. Solid sphere means the entire interior is included that is for each fixer t the intersection of light cone with the hyper plane t equal to t is a d dimensional sphere lying in the hyper plane t equal to t that is sphere lying in d dimensions not d dimensional sphere sphere will be one dimension less if you consider solid sphere yeah. So let us not discuss that it is a sphere lying in rd that is what this sentence means when the t sections are projected in the space rd they are the spheres with center x0 and radius c times mod t minus t0 clearly as t goes to infinity the spheres are expanding. So what is the idea of the proof of causality principle in d greater than or equal to 2 as in the case of d equal to 1 the main idea is to multiply the wave equation with ut and then integrate it on some part of the solid light cone and then perform integration by parts. The region of integration was a trapezium shaped region for d equal to 1 which would now correspond to first term of the cone that is the reason why we used the notation f to denote the trapezium shaped region in d equal to 1 first term of the cone. So imagine this is the kind of cone okay so you cut this so you have this so this is the first term of the cone. So you have a bottom portion you have a top portion and you have a lateral portion. So multiply the homogeneous wave equation with ut and rearrange exactly in one dimension we have done so we get this and then integrate on the first term of the cone defined by t equal to 0 this is the bottom portion t equal to t is the top portion. Now proceeding exactly as in d equal to 1 the domain of dependence inequality we obtain that is exactly the same integral over t is less than or equal to integral over b. From here the uniqueness of solutions to Cauchy problem follows again okay you take u and v to be solutions even for the non-homogeneous Cauchy problem subtract u minus v call it as w that will satisfy the homogeneous wave equation with the homogeneous Cauchy data which means that integral on b is 0 on bottom that function and the derivative with respect to t will be 0 therefore we have this is 0 and this is true for every arbitrary t and therefore in the first term both of them coincide and hence even at the point x naught t naught same proof. So with these modifications the proof given for d equal to 1 goes through for d equal to 2 comma 3 also we will see one more proof of uniqueness using energy method later on. So what are the consequences of domain of dependence inequality the solid backward cone is called the past history of the vertex x naught t naught. So if this is x naught t naught the past cone we said is this right so this is the past and this will be the future of t naught in other words the Cauchy data at a spatial point x naught can influence the solution only in the future cone with vertex at x comma 0 which is forward solid light cone emanating from x naught comma 0. The figure on the next slide depicts past and future cones located at a point x naught t naught. So we have already seen this picture. So finite speed of propagation finite speed of propagation is a common feature for wave equation in all dimensions we are going to study the speed of propagation of the Cauchy data thus the homogeneous wave equation is studied. We illustrate finite speed of propagation using examples for d equal to 1 to 3. So let us consider d equal to 1 let the initial data phi and psi be 0 outside this interval minus 3 comma minus 2 therefore u of 0 0 is 0 x equal to 0 t equal to 0 that is actually phi of 0 0 is not in this interval therefore phi is 0. Let us now study the behavior of u of 0 t that means I am standing at x naught equal to 0 I want to study what happens for t positive. For t positive such that minus ct is bigger than minus 2 that is t less than 2 by c u of 0 t will be 0 we will see this in a picture it will be very easy. So that is the information at t equal to 0 has not reached the point x naught equal to 0 till this time t1 equal to 1 by c from which time the information will be received at this point. Thus it took a time of 2 by c to travel a distance of 2 why the distance of 2 this interval is at a distance of 2 from the point x naught equal to 0 and thus the speed is c u of 0 t remains 0 till t equal to 2 and illustration of this example is given on the next slide. So here we are standing at this point x naught equal to 0 right now at time t equal to 0 because this is t equal to 0 the information is only here in this interval minus 3 comma 2 outside that phi and psi are 0 therefore u is 0 here let us consider this instant time instant 1 then also if you see from our formulae this does not intersect the interval minus 3 2 therefore u is still 0 here when you go to 2 that is when you pick up some information from here from the interval it has reached this point 0 at time t equal to 2 because it is hitting this point possibly phi is nonzero here who knows but because the support of phi and psi is 0 it will be 0 only here phi of 0 phi of minus 2 psi of minus 2 will be 0.  So you may pick up some information from here that means information from this interval minus 3 comma minus 2 where the support of phi psi lies is reaching the point x not equal to 0 which is at a distance of 2 distance is 2 it takes time 2 time 2 units. So this example is with c equal to 1 so speed is 1 distance is 2 so you take 2 units of time to reach information so after 2 the information starts coming for example if you are at 3 as you see here domain of dependence for u of 0 comma 3 contains minus 3 comma minus 2 intersection non-empty in fact in this example it contains. Now if you are at this point 4 time 4 at this point you see that this is the interval definitely in this you have the support of phi and psi so information has reached there. So in d equal to 2 comma 3 let us look at consider a ball b of 0 r center 0 origin at the origin and radius r it let it denote open ball of radius r with center at origin support the suppose that the Cauchy data is supported inside this ball we know that u of x t depends on the values of the Cauchy data on this b this is a closed ball x c t intersection b 0 r only this is non-empty we have non-zero solution otherwise it will be 0. So if the above intersection is empty u will be 0 in other words for each fixed t positive the support of the function x going to u of x t is contained in union over y in this ball of radius r center origin of the closed ball with center as y and radius c t which is nothing but this ball ball center 0 radius r plus c t. Thus for each fixed t the support of the solution is a compact set if the Cauchy data is compactly supported we have observed this in d equal to 1 and illustrated with the picture also in other words the support spreads with finite speed let y be not in this ball of radius r with center 0 what will happen then not only u of y comma 0 is 0 this is the initial time right phi and psi are concentrated inside b 0 or outside that phi and psi are 0 y is a point which is outside that therefore u of y 0 is what it is phi of y and y is not in this ball therefore this is 0 but also u of y t will continue to be 0 for all times up to norm y minus norm y minus r by c this is referred to as finite speed of propagation so what we have done in this lecture is we have derived the domain of dependence in equality for d equal to 1 for d equal to 2 3 we just give the idea so using domain of dependence in equality we proved uniqueness of solutions to the Cauchy problem the second proof of uniqueness and we revisited domains of dependence and influence with past future cause effect points of view thank you


Hi and welcome to this session. I'm Kanika and I'm going to help you to solve the following question. The question says express each of the complex number given in the exercises 1 to 10 in the form a plus iota b. Ninth part is whole cube of 1 by 3 plus 3 iota.  This is the key idea in this question. Proceed on with our solution. Given complex number is whole cube of 1 by 3 plus 3 iota. Right. We know the formula of whole cube of a plus iota b. Now here a is equal to 1 by 3 and b is equal to 3. Now using this formula this expression is equal to 1 by 3 whole cube minus 3 into 3 square into 1 by 3 plus iota into 3 into square of 1 by 3 into 3 minus iota into cube of 3. And this is equal to 1 by 27 minus 3 into 9 into 1 by 3 plus iota into 3 into 1 by 9 into 3 minus iota into 27. And this is equal to 1 by 27. We can cancel 3 from both numerator and denominator. So we are left with minus 9 plus 3 into 3 is 9. So we have 9 by 9. 9 by 9 is 1. So we are left with only iota minus 27 iota. And this is equal to. Now solve the real part and imaginary part separately. 1 by 27 minus 9 is equal to minus 242 upon 27 iota minus iota 27 is equal to minus 26 iota. Hence our required solution is minus 242 upon 27 minus 26 iota. So we have expressed the given complex number in the form a plus iota b where a is equal to minus 242 by 27 and b is equal to minus 26. This is our required answer. Bye and take care. Hope you have enjoyed the session.


Hey guys, hey guys, happy to have everybody on board tonight. We have a very special guest with us, Tana down. I think she's below me. Will she be below? Yeah, hopefully she's below. And we've been working with Tana for, I don't know, the last two, three weeks, a month now. And we just thought it was a great opportunity this evening just to get her on board for a genuine chat. So Tana, thank you for being with us this evening. Do you want to talk about your challenges throughout your life with, I don't know, maybe dieting and how you found the continued diet? Yeah, guys for inviting me. So, like, I've had weight challenges pretty much off and on since I was about 10 or 11. So, you know, as you can imagine, I've probably done every like fad diet out there from, you know, Jenny Craig, weight watchers, all of them. So I kind of stumbled upon the ketogenic diet during the Atkins kind of like craze. But I think a lot of people misunderstood like Dr. Atkins. And I did a lot of reading on my own and kind of informing myself and realize that he wasn't really saying you could eat like, you know, a pound of bacon and, you know, a half a dozen eggs. There was a lot to the actually the science behind it. And if you really listened to him, there was a lot of good science behind it. So I think that his idea kind of evolved into the whole ketogenic lifestyle that we like know today. So I kind of like have been like on and off ketogenic for about the last 10 or 15 years. And I know we were just talking before we jumped on live there and you were saying that you have had some challenges in the past there with, you know, like being able to get into like a good keto system. I mean, you know, at this stage, you want to talk a little bit about that and then Dustin, maybe if you can tell us like how that happens and why that happens, not just for Tana, but for people in general.  Yeah, I actually one of the first things that one of my people that I was talking with told me to do is go buy an inexpensive meter. Like the where you could test your ketones and I was, you know, doing 20 grams of carbohydrates or less per day and was only poking like a 0.3. So that's actually a blood with my blood glucose meter. I do have like, you know, a sluggish thyroid and they do have like, but I don't have any other health issues.  And now I'm 56. I know that like weight, weight and age and all that stuff play into a lot of it. Well, I like to hop in here. And first off, I don't, I was looking for my blood meter, but basically it's like a glucose meter, but it's for ketones.  And 0.3 ketones is not therapeutic. It's a small level of ketone. So people that are trying to elevate their ketone levels, they, you know, they are going to be 0.3 at parts of the day, but you want to keep them elevated to more of a therapeutic above 0.5, 1, 1.5 or more ideal. And just so everybody knows, I'm not a huge fan of doing that consistently. As long as you have the right formulating results, if you're not getting results, we may have to check in from time to time. But it was interesting, Tanya, I know that after you drank ketones, you drank this amazing product. What did your ketone levels go to after you did that? It went from 0.3 to 1.9. So that's pretty amazing. And I was just kind of messing with it, you know, because you want, as they say, my grandma used to say, the perfect proof is in the pudding. So I was going to like see, like if it worked and I drank my drink in about an hour later, I did my stick test and I went from like 0.3 to 1.9. Wow. So why this matters so much, y'all, is the, what's funny, Liam, I saw somebody was coming in my room and it was your room. I was like, wait a minute, my wife's not here. But what's funny about that is, or what's important about that is that when you get to that 1.9 or above 1 or 0.5, what happens is that it has a huge impact on your cognition, how your muscle preservation is, how your heart functions, how your body functions.  So having that higher number is really, really important. But there's something I want to, I want to mention, Tanya, is this, is that, did you, let me ask you this question. When you saw the 0.3, did you feel discouraged at all or feel like? Yeah, it's definitely discouraging when you're sticking to a 20 carb or lower per day. So you're giving up, you know, some things that you like. Obviously having done keto for quite off and on for quite a long time, I stopped eating a lot of this stuff. There's obviously some things that I would like to have, like I like baked potatoes and stuff like that. So you're giving up things to switch to a different lifestyle, but you're not really seeing results. So yeah, you do start to get kind of frustrated with it. Right. And then what happens is people then quit and they stop because they were like, what's the point? Why do I keep doing this? And it's not going to result. So I want to take everything really quick in the chat if you can put down on the chat, if you had one of those sticking moments where you feel like things aren't working the way they should be. And I'm not going to get too nerdy here, but just a little bit. Yes, you're 55. Yes, your hormones have changed. All those things do matter. They have an impact, right? But there's something that's really important that I expressed. And there's some new research that's come out where, because what I'm going to share with you might have started when you were 10 years old. So it's like almost not even your fault anymore. I mean, if it happens to us when we're kids, because our parents' choices, like my parents' choices were not great for me when I was a kid. And some of the challenges started when I was 10 or eight. So I'm trying to correct them now, but I'm holding myself this as a high standard and not getting results. So there's a thing called the glute 4 transporter. And it's a mechanism that if it's not activated or active, you can't metabolize body fat even on a perfectly formulated ketogenic diet. So the best diet in the world for you to get healthier or burned fat, you may not be able to get long-term results on it because this thing isn't activated. So they studied our product. They study a lot of things, but one of them was our products. Well, our product actually activates that glute 4 transporter. This is roughly anywhere from 40 to 60% of us watching this have that as a deactivated transporter. So as a trainer, I used to tell my clients to get mad at them like, it's your fault. You're doing this. You need to do this better.  And the reality is it probably wasn't any of those things. It most likely was this activator. So we have to keep it activated. So you're now in a path to activate that glute 4 transporter and balance out some of those hormonal things that are going on to ultimately get the results you're looking for and sustain it. And maybe you won't even have to follow a full ketogenic diet to do that. Maybe it's going to be a hybrid down the road with drinking ketone. So I just want to show everybody if you feel sometimes that you're fighting uphill and not getting the results you wanted. I was going to say to keep on that sustained path, what sort of things should she be doing or continue to do or add in to keep getting those results? Now, I mean, obviously it's not all about just getting a high ketone level. What are some other things that she could add in or start doing or changing along the way? Yeah, well, one of the things that people don't realize and I know Tanya has a little bit of a respiratory thing going on right now is where you're encouraging or move your body. But when your body metabolizes and burns body fat, you have to excrete out the aftermath, right? The exhaust of the fat. And there's three ways that your body does that. One through urine, one through sweat and one through your breath. So what's important about that is stay hydrated and physical activity because you breathe more and you're more likely to sweat. So physical activity will be very, very important. Now, that doesn't have to be crossfit, which Liam and I are all for. It doesn't have to be running marathons. You have to just start moving your body if it's starting to start with walking. But you do have to advance that as you get in better shape. You're going to want to challenge yourself a little bit more because otherwise you're not going to get as much breath or as much sweat coming out. So I would say those two water and exercise or activity is really important. We can talk about sleep in there is really important too. And then the last thing is fasting. Fasting has been proven now to be one of those key components to not only short-term success, but long-term success. And what does that look like for you, Tana? Are you someone that's sort of fasting at the moment, would you say? Or are you a breakfast person? Or let's give you some skills and tools that we can maybe help you with that. One of the things I started doing when I joined that 10-day challenge that you guys had on the Kato Beginners like page.  So I get up in the morning, I drink my Kato drink, and then I don't eat anything until 12. And then like I'll eat at 12 if I'm hungry. Sometimes I don't eat until two or three because I'm not hungry. So I just kind of eat when I'm hungry. What you'll notice when you're on the, when you do the Kato or West like is that you're not hungry. Right.   So that's why we like the intermittent fasting. And in long term, we do teach some cycling fasting, meaning that you're going to want to feed every once in a while. But as a whole, intermittent fasting is a great way to start fasting approach to get make, to maximize your experience and results.  Some people do two a day. William and I probably are primarily two a days. Some days I do do three, but most of the time I just do two meals a day and every once in a while I just do one. Yeah, usually I either eat two or just one.  Also, I did pick up your eye, one of the packages of eye fast. So I'm looking forward to doing the eye fast. Yeah, I get mine and soon, Tana, we're going to do it together. So I'm really excited about that as well. Yeah. And I know you've had some really good results lately because I saw a little post here on social media. Tell everyone about the results you've got in their head in just the last while. Like my first goal was because I told Liam that I was at 388 and when you weigh that kind of, when you weigh that, you can't like buy a treadmill. Like they want you to buy like a bow flex or like 1899 because the weight ratings are they're not very many weight rated treadmills out there that aren't like $2,000. So my first goal was to lose enough to buy one that I could afford that was, you know, in a decent price range. So I had to drop about 40 pounds to do that and I did that pretty quickly. And I got my treadmill and now I'm just working on my next goal, which is another, I'm just going to do it. You know, one of my, it's not really pertains to weight loss, but it does kind of pertain to weight loss. My favorite programming teacher used to say, how do you eat elephant one bite at a time? She was talking about computer bites, but it's still a good analogy. I mean, seriously, like if you, and I think Liam and Dustin, we were all talking about this. You can really self-sabotage yourself by making the goals paintable. Right. And then like the negative self-talk. I was talking about that earlier. So you have a bad day and then instead of like just brushing it off and worrying about the next day you beat yourself up. Yeah. And consistently do that.  And it's super important as well that when things do slow down or they do stop, you know, lean on us to maybe ask the questions of like, so why is that happening? Right now, once again, I'm doing this right. I'm doing that right. I'm doing this right. And I'm doing that right. And what I find all the time is people don't know what they don't know, right? You could be doing something that you think is, you know, tip top spot on the right thing to be doing. But where you're at in your journey, you may be thinking or doing it a little bit too early or a little bit too late or not in the right place. So, you know, Dustin, if we could give Tana maybe some, give her three tips for the next three months of her next journey that are going to help her stay sustainable, stay on the road. And I guess the main thing is just be happy where she's at. Yeah, I think the number one tip you want to have is what is one thing that you're excited to do or see or be in three months? Like what's that one thing? Maybe it's you're looking forward to a trip. Maybe it's you want to fit into a certain piece of clothing. Maybe it's you want to go for a 5K walk. But do you have one thing that you're ultimately looking forward to achieving or doing or seeing in the next few months? So like, not necessarily in the next few months, but I do have a lot of stuff I want to do like that I can't do. Like, so I think I shared with Liam that I had knee replacement.  So in three months, right, let's set a goal. Let's set the let's say let's say that there's a hike you want to do, right? I want you to write that hike down, take some pictures if you have pictures of it, put it up in your house. Because what happens is if there's not something drawn you to moving forward, it's easy to slip back, right? So then what we can do is when you're on your treadmill or you're on your bike or you're moving your body a little bit at home, we can set a goal that moves you towards that destination. I would put that I would put that picture right up on your on your on your workout machines so that you can see what you're working towards. Same thing with clothing.   And then when it fits, you go celebrate do something that celebrates yourself moving in the direction of where you're going. And I want to interject like that one of the videos you guys shared in during the challenge, I think it was on the in one of the emails was about the non scale victories. And that's really important to focus on because like I just posted a picture of me like in a shirt like this one size smaller than I was wearing. So those are those non scale victories that you need to focus on.  So I figured I'm just going to like buy a horrible problem. This keto thing can be expensive. You have to buy all new clothes. I would say find a tribe, you know, we're going to help you do that too and find a tribe to do this with others. What I can tell you is having a goal or a vision or a destination, assessing yourself with with a clear mind and clear vision with your clothing and then having a tribe to go do it with will ultimately lead to your greatest successes. And we're here to coach you be the coaches behind the scenes and help you guide you to that journey. But to me those are the secret weapons for the long term success in the next three months as we do that you get 510 people doing this with you. And you're now accountable to them. You're becoming their coach. It brings more power to you to help you stay accountable. I was going to say I have one friend doing it with me now. So like it does really help if you have other people around you that are that can encourage you when you're having those days. And that's the reason I like the Kato for beginners website too because we all kind of like are on there trying to support each other. So you know when you have that bad day, like you can shrug it off a whole lot easier if you have somebody to talk to.  You can be challenging sometimes, right? Like everybody does it. You know, like everybody does it. We're enemies. That's why Dustin and I always like, you know, try and get people on board like programs, coaching programs, free groups, whatever it is. Because you look at a professional athlete, a professional athlete is the best athlete in the world. But they've got coaches and they've got nutritionists and they've got all the team on their side. I can say to anybody out there right now, name a professional athlete that's done it alone. Yeah, there isn't one. There isn't one, right? But you really do have to watch out for the negative self-talk because you can really self-sabotage yourself. I know I've done it, you know, a lot in my lifetime and that's what I was telling you in the beginning.  And then 10 years later you find yourself in the same place. So, you know, like learn from my mistakes and don't like self-sabotage yourself. Every day is a new day. Shrug it off and go forward. Dustin sees it all the time, right Dustin? What percentage do you reckon the mindset is? I say it's 70%. I sometimes fight between 60 and 70%, but I think the mindset of it is, because people know what to do, they just, something interferes with it. You can really be your own worst enemy, honestly. Usually that's the case. I've been there, so like I said, my motto is like every day is a new day. So, like I just approach every day as like a new day. Like so yesterday was a bad day. Today is going to be a better day. 100%, one bite at a time, right? One bite at a time. And my only other advice would be to track what you're eating because you really don't realize what you're eating until you become accountable for it. So, I track what I eat and eventually you won't have to do it because you'll be aware. But in the beginning it's really important because you don't realize all the stuff you're sticking in your mouth until you have to write it down. Right. I know when I started fasting longer I realized how much little food I would put in my mouth for my kids. I was like, I didn't realize. I didn't even think about that. You know, there's a lot of these little things that I was doing that I didn't realize I was even doing. Even if it was low carb, I was still doing it not thinking about it. Some things we need to think about too that are like low carb things we're picking up are not necessarily the greatest. Right. Like I know I was chewing a lot of gum and one of the people I was talking to was like, ooh, that's one of the bad ones. Like did you look at the sugar alcohol they're using? So be aware of what you're putting in your mouth that's low carb necessarily isn't, you know, make sure you're reading the labels and are aware.  Like we don't need to be on this live. You can do it yourself. I do want to answer one question. It's more fun to work with other people. It's nice to have that community sense and like the people like to help you along the way. Well, I want to just answer one question the lady dropped in there and I thought it was really good question. So I'm going to pop it up here. I might do a live on it a little bit later. She said because you're not hungry, do you need to worry about the calories you take in if you're not getting enough in? And I thought there's a really good question because you want to think about two things. The more that you're restricting your calories, the more that you're fasting, the more important the quality of your food is.  Well, I exercise a lot so I can eat crappy. No, the truth is, if you exercise a lot, your demand for high quality food goes up because you're putting higher stress on your body. So if you're a person right now that's doing some intermittent fasting or doing OMAD or doing one meal a day, I have some different fasting cycles that I like. The quality of your nutrients are really important because you need to still get a very good balanced diet even though you're eating less. Second thing is that I don't encourage people to go low calorie for a long period of time. So I like people to cycle their calories, meaning that once a week, maybe twice a week. At the beginning, you can go a little further with low calorie, but long term you're going to want to cycle your calories, not carbs necessarily calories. So I have a couple of days where you eat more than you would otherwise.  What would your recommendations be on calories? What's a good range, I guess? That's a great question. That's going to vary based off of size, activity level, male, female, muscle mass. So you're looking at the average, they say women, 1,500 men at 2,000. I rarely ever find that to be actually that accurate, but just keep that in mind is I would look at it this way. There's a really good calculator out there if you just type in calorie counter and it gives you a pretty good judgment based off your activity levels. But don't ever hold that as true because if you've been on diets your whole life, you may not actually fit the mold. You might have slowed your metabolism down. And what happens when you do that and you start eating their amount of calories, you start gaining weight because your metabolism is slower. So what I always tell people, that's why I like starting out lower and start moving towards fat burning because psychologically, if you don't start losing, you quit and move on. So I'd rather start lower and slowly work your way back up. And then once you find like a maintenance and you're not losing any more body fat, then we kind of know that that's probably a pretty safe place in your metabolism based off your current body type, your activity levels and your muscle mass.  If you lose 100 pounds and you're the same muscle mass, you're not actually burnt. You're not using as many calories to the day. If you're not, if you're doing the same activity, does that make sense? Because your body does have to carry 100 pounds everywhere. So your calories have to be adjusted off of that. Most likely your calories will actually go down after you lost 100 pounds just naturally because you're carrying less weight. There's a lot of factors in there, so I don't like to overthink that. Typically, if you're following, especially if you're drinking ketones and you're following our plans, you're going to kind of solve the problem in itself because you're not going to be super hungry. When you are hungry, feed the warrior because your body's telling you it needs energy. And if you learn how to listen to that, it gets really good. Even on ketones, there's days that you're going to be more hungry with ketones that tells you your body's looking to be fed, not necessarily binge eat on pizza, but eat higher quality foods at a higher amount. I don't know if you saw that comment down there. Doctors call so, is it? He's done some research of his own on the keto products and I'll say they're really amazing and unique. I must applaud you guys for your work. I wish I had made it. I wish I made those products. I helped launch it. I got that one, but yeah, it's been a great journey and to have bioidentical peer therapy to ketones with the most natural products out there. I mean, I think it's going to be the biggest breakthrough of our, maybe of our lifetime, if not the century. And I hope not, at least to help them all in this space, but right now it's just pretty amazing what it does. I just want to jump in real quick because I've tried a bunch of different keto products. This is the first keto product that actually lives up to what it actually says it's going to do. Like you take the keto pills, they say, oh, you're going to go into ketosis, not. So this is the first product that I've actually taken that actually said it was going to boost my ketones and actually did. So it's pretty amazing, honestly. I think that's amazing. And I think it's a great Z-Gway because we've already been going for 25 minutes.  Because the next three months are just going to be amazing for you. I can't wait to work with you along the way. Help you more in your journey and just get after that three month goal of let's get your hiking. Let's get your bike riding. Let's take one bite at a time and then get after those results. And I'm super excited for you, super pumped for you. And grateful to have you on board, Dustin, running tonight's call and just spreading the word. That's what I'm going to do. Thank you all. All right, guys. Get you later. All right.


Can you do one for my dead sister? Of course I can. Here we got some purples and pinks just for you and the family. I hope this helps sound. It's not much, but I hope you like it and my blessings coming your way. My prayers. Thank you.