submitted by spike77wbs to borntoday [link] [comments] |
submitted by spike77wbs to borntoday [link] [comments] |
submitted by spike77wbs to borntoday [link] [comments] |
submitted by spike77wbs to borntoday [link] [comments] |
submitted by spike77wbs to borntoday [link] [comments] |
submitted by abhiswami2004 to memes [link] [comments] |
I wanna know exactly what you know.What's the relevance of this to the Goths? Elvie's hypothesis from TW that human brains are a support mechanism for a quantum-based consciousness field is confirmed true by proto-miller (at least in the Show). I think it is very revealing that the extra 'technobabble' was included in the show (which has the authors as producers) but wasn't in the original Abbadon's gate. The protomolecule are more advanced in their ability to manipulate the quantum fields underlying consciousness, and the Goths are more advanced still than the protomolecule builders.
Oh, so you wanna talk about the non-local quantum hologram, the phase-conjugate adaptive waves resonating in micro-tubules in the brain, which of course requires some closed-timeline curves and Lorentzian manifold, and you catch up, I'll wait.
Those words aren't technobabble, they actually make sense as a vague description of how the protomolecule goop is affecting Holden's brain to produce hallucinations.
If Holden knew enough physics to understand what some of them meant it would scare the wits out of him. I can't even conceive of a technology this advanced; no human being can. Miller's off-the-cuff line was final proof that the Protomolecule builders are geological eras more advanced than humanity.
For starters, a closed timelike curve is a particle that loops backwards into its own past! If Time Travel, at least on tiny scales, is allowable (since FTL is equivalent to time travel and the ring is FTL, this shouldn't be surprising) all conservation laws are immediately broken.
A 'non-local quantum hologram' probably refers to using quantum non-locality to transmit information - faster than light, faster than anything, infinitely fast. This is theoretically impossible - despite what you might have heard, the 'nonlocal' way that some quantum systems seem to behave can't be used to transmit anything faster than light. Clearly whoever built the protomolecule doesn't care what humans think is impossible.
The remark about a Lorentzian Manifold refers to the general mathematical description of how space behaves in general relativity. 'A Lorentzian Manifold' could be any kind of space with any number of dimensions, as long as it follows the laws of general relativity. So bring it all together, a closed timeline curve is produced by some quantum gravitational process to acausally affect the microtubules in the neurons in Holden's brain.
Finally, the reference to 'adaptive waves resonating in micro-tubules' refers to an unproven hypothesis by Roger Penrose that links human consciousness to quantum effects - possibly something to do with how Miller's consciousness merged with the protomolecule goop, and with Holden? For all we know, the Protomolecule builders have advanced in this direction too.
“It’s about the nature of consciousness.” “That may be a wider context than I was looking for, Major.”What Elvi is getting at here is what philosophers of mind call Multiple-realizability. This is the claim (accepted by most philosophers and basically all neuroscientists) that there's nothing special about neurons per se, that experience and subjective, first-person consciousness can be supported by lots of different physical systems as long as they all are doing the same kind of thing. What that kind of thing is, we have no idea.
“Bear with me,” Elvi said. “Unless we’re reaching for religious explanations, which I’m not the person to comment on, consciousness is a property of matter. That’s trivial. We’re made out of matter, we’re conscious. Minds are a thing that brains do. And there’s an energetic component. We know that neurons firing is a sign that a particular kind of conscious experience is happening. So, for instance, if I’m looking at your brain while you imagine something, I can guess reliably whether you’re imagining a song or a picture by seeing if your visual or auditory cortex is lighting up.”
“All right,” Trejo said.
“There’s no reason to believe that a brain is the only structure capable of having that combination of structure and energy. And in fact, there’s a fair amount of evidence that the gate builders had a conscious structure—a brain-like thing—where the material component wasn’t at all the same kind of thing we use. Anecdotally, we’ve found at least one brain-like structure that was a diamond the size of Jupiter.”
“I don’t know what that means,” Trejo said.Here, Elvi is vague but apparently pointing towards Roger Penrose's idea that humanlike intelligence exploits a successor to quantum physics as we understand it today, to make decisions that are literally uncomputable by any normal information-processing system. Though she seems to be focussing more on the idea that it's having experiences that is supported by quantum processes, which isn't Penrose's idea per se.
“Like we don’t have a steel chamber in fusion reactors. We have magnetic bottles. Magnetic fields that perform the same basic function as matter. The older civilization appears to have developed its consciousness in a form that relied more on energetic fields and maybe structures in unobservable matter than the stuff we made a brain from. There’s also some implication that quantum effects have something to do with our being aware. If that’s true for us, it was probably true for them.
“My thesis—the one I was working on before I came here—explored the idea that our brains are kind of a field combat version of consciousness. Not too complex. Not a lot of bells and whistles, but takes a lot of punishment and keeps functioning. Our brain may actually have a kick-starting effect, so when the quantum interactions that underlie having experiences break down, they’re easier to start up again. Does that make sense?”It does make sense, sort of - the idea that quantum interactions underlie consciousness is a big pill to swallow, given what we know about quantum physics as it is now. The idea that these interactions could be detached from the underlying physical structure; it would need new and very strange physics, but this is science fiction, so who am I to judge?
Orch-OR (the theory directly linking consciousness to quantum gravity) is not taken seriously, it isn’t even a hypothesis, it isn’t even falsifiable, and we have far better theories of consciousness based on information theory and neurophysiology that have already produced confirmed and clinically significant results.As far as I know the only line in either books or show that point to the objective wave-function collapse in microtubules a la Penrose explanation is that one line from Miller. Perhaps he was oversimplifying some vastly more complex theory of everything into English. Similarly, Elvi's own explanation doesn't go much further than "there seems to be some kind of deep connection between fundamental physics and consciousness, such that you need something more than information processing, and it might be related to quantum" - she doesn't specifically mention objective wave function collapse.
But, that’s the way the Expanse is heading unfortunately. It is worth noting several things here: that the AdS/CFT correspondence and ER=EPR would be directly related to consciousness if Penrose and Hammeroff were actually correct, because both are describing in a fundamental way how quantum information affects the structure of spacetime. And secondly, even if we assume that consciousness is not quantum in origin directly, both may still be somehow related because consciousness appears to truly be a phenomenon of information processing, and information (albeit quantum information) appears to underlie the structure of space time if the above models are correct.
And finally, it’s worth noting that one of the actual legitimate (but still very, very incomplete) theories of consciousness that we have - Tononi’s IIT - has an interesting informational relationship with quantum mechanics as well. For further info on that, read Tegmark’s Perceptronium paper.
My point is, even if the phenomenon of consciousness is not specifically quantum in origin, it may still be related to the universe on a fundamental level and a true “theory of everything” would likely incorporate that. Most of my colleagues do suspect that consciousness has a deep root in physics, and it is particularly telling that there seems to be a confluence starting between neuroscience, information theory, and physics. So even though I disagree with the premise of how the Expanse interprets that part of the story, I can still get behind the general idea that beings with complete power over space time would also have complete power over consciousness, because it may actually be information (it-from-bit) that defines both the structure of reality and consciousness, albeit at different spatiotemporal scales and orders of magnitudes.
That's why the goths are pissed, the waste heat probably gets piped to their dimension with fuck knows what consequences
Riemann Hypothesis states that the real part of all non-trivial zeros of the Riemann zeta function, or ζ(s) = Σ(k=1 to ∞) 1/k^s = 0, equals one-half. For the non-trivial zero, s, a complex number, we have s = a + bi where Re(s)= a = 1/2.If I had $1 for every "proof" of the Riemann hypothesis I've seen where the writer starts by trying to find s such that 1+1/2s+1/3s+1/4s+⋯=0 I'd probably be lounging on a beach in the Caribbean right now. The problem here is that the Dirichlet series for ζ(s) only converges when the real part of s is greater than 1, and this series is never 0 where it converges. So analyzing only this series will not be helpful.
Fact III: The sum of the complex conjugate pairs of non-trivial zeros, s = a + bi and s' = c + di where ζ(s) = Σ(k=1 to ∞) 1/k^s = 0 and ζ(s') = Σ(k=1 to ∞) 1/k^s' = 0, of the Riemann zeta function equals one according to the Fundamental Theorem of Arithmetic and the Harmonic Series (H):(Note: Euler and others have proven that there exists an infinite set of primes in H. And that the divergence of H is a key reason for that result.)If s' is the complex conjugate of s=a+bi then why not just write s'=a-bi instead of s'=c+di? Or why not write s=σ+it instead, as this is a fairly standard way to write a non-trivial zero in literature on the topic? Sure, this isn't bad math per say, but it's pretty bad notation. Also, s+s'=1 always only if the Riemann hypothesis is true and this would have nothing to do with the fundamental theorem of arithmetic or the harmonic series! They have already assumed the Riemann hypothesis is true before they've done anything!
Therefore, according to Facts I, II, III, and IV, we have:Then they make some final notes where they try to rewrite the harmonic series using some underexplained ideas about prime gaps and says
k^(1/2) ≤ k^a ≤ k, k^(1/2) ≤ k^c ≤ k, and a + c = 1.
Hence, k^a = k^c = k^(1/2) which implies a = c = 1/2. Riemann Hypothesis is true! Riemann was right!
There are infinitely many more positive integers than there are prime numbers, or prime numbers have a zero density relative to the positive integers, and prime numbers generate the positive even integers efficiently so that gaps between two consecutive prime numbers increase without bound.which is true in the sense of natural density for sure, so why not just say that? Using the phrase "infinitely many more" makes it sound like cardinality. Saying "so that gaps between two consecutive prime numbers increase without bound" makes it look like they're saying all prime gaps become larger as we increase through the sequence of primes, this isn't necessarily true although it's statistically something we should expect. The existence of arbitrarily large prime gaps is true though and isn't hard to prove, but they did not prove it in any of what was written and it's not the same as what they said.
Keywords: π(*):= Odd Prime Counting Function and Fundamental Theorem of Arithmetic (FTA) Goldbach conjecture states every positive even integer is the sum of two prime numbers. (We count one as prime in the sense of additive number theory outside of the FTA.)What? The parenthetical here is so strange. Additive number theorists don’t take 1 to be prime and they have no reason to do so.
Therefore, e ≠ p + q over S, (p,q є S) , implies the following system of equations over S, 1 = e - n1 * q1, 3 = e - n2 * q2, ..., pk = e - nk * qk, according to the Fundamental Theorem of Arithmetic where 1 < qj ≤ (nj * qj)^.5 ≤ nj for 1 ≤ j ≤ k where pj, qj є S and nj is a positive integer. Note: If qj = 1, then nj є S, or nj is an odd prime less than e.and this last sentence is what they try to base their argument on. They attempt argue that for every even number e>2, the probability that an equation of the form p=e-1q doesn't show up goes to 0. Which would mean that it's likely that e=p+q.
In addition, empirical evidence has confirmed the validity of the conjecture for all positive even integers up to at least an order of 10^18. Therefore, we conclude the conjecture is true.If you proved it, why do you need to test it empirically?
Okay.
- x^n+y^n=z^n for n > 2. I begin the proof by assuming there exists an integral (positive integer) solution to equation one for some n > 2. Equation one becomes with some algebraic manipulation, 2. x^n=z^n-y^n = (z^(n/2)+y^(n/2))*(z^(n/2)-y^(n/2)).
Now that I have factored the right side of equation two, Fermat, the great French mathematician and respectable jurist, made I believe the next logical and crucial step.Any evidence that Fermat did what you're about to do?
He factored the left side as well, x^n, with the help of an extra real variable, Ɛ, such that 0 < Ɛ < n . I have the following equation, x^n = x^(n/2+Ɛ/2)* x^(n/2-Ɛ/2) = (z^(n/2)+y^(n/2))*(z^(n/2)-y^(n/2) ). This equation implies x^(n/2+Ɛ/2)= z^(n/2)+y^(n/2) and x^(n/2-Ɛ/2) = z^(n/2)-y^(n/2).Ah yes, if ab=cd then a=c and b=d. Everyone knows that! Eventually, after a few more lines, the author concludes
However, (1/4)^(1/n) is not a rational number, a ratio of two whole numbers, for n > 2. This implies the right side of equation five is not a positive integer. This contradicts my assumption that y is a positive integer. Thus, Fermat’s Last Theorem is true, and Fermat was right!It's so easy now, Fermat's last theorem obviously just reduces to knowing 1/41/n is irrational for n>2. How did nobody see this before?
Proof of the Collatz Conjecture: Suppose there exists a sequence, S’={n0, n1, n2, …} that does not converge to one, or nk ≠ 1 or nsub(k-r) ≠ 2^µ over S’ for all kϵ ℕ where rIt's obvious that hailstone sequences don't converge, so the “does not converge to one” bit is irrelevant. Here the fundamental error is same error as in their attempted proof of the Goldbach conjecture; they think making a probabilistic argument in favor of the conjecture being true is the same thing as proving it. Lots of other basic little details are also wrong, but I'll just look at one:
From a given positive integer, n, we obtain the maximum positive odd integer, n0 > 7, by repeated division of n by 2.What is n here, the starting number? What if n is odd? We'd have to 3n+1 it first, not divide by 2. Even if n is even the first odd number we hit once we finish dividing by 2 is not the maximum odd number in its hailstone sequence, this is easy to see starting with n=22.
submitted by ben1996123 to badmathematics [link] [comments]
#! /usbin/python #---------- # This unusual and intriguing algorithm was originally invented # by Michael V. Klibanov, Professor, Department of Mathematics and Statistics, # University of North Carolina at Charlotte. It is published in the following # paper: # M.V. Klibanov, A.V. Kuzhuget and K.V. Golubnichiy, # "An ill-posed problem for the Black-Scholes equation # for a profitable forecast of prices of stock options on real market data", # Inverse Problems, 32 (2016) 015010. #---------- # Script assumes it's called by crontab, at the opening of the market #----- import numpy as np import pause, datetime from bs4 import BeautifulSoup import requests # Quadratic interpolation of the bid and ask option prices, and linear interpolation in between (https://people.math.sc.edu/kellerlv/Quadratic_Interpolation.pdf) def funcQuadraticInterpolationCoef(values): # There is 'scipy.interpolate.interp1d' too y = np.array(values) A = np.array([[1,0,0],[1,-1,1],[1,-2,4]]) return np.linalg.solve(A,y) # https://en.wikipedia.org/wiki/Polynomial_regression def funcUab(t,coef): return coef[2]*t**2 + coef[1]*t + coef[0] def funcF(s, sa, sb, ua, ub): return (s-sb)*(ua-ub)/(sa-sb) + ub # Initialize the volatility and option lists of 3 values optionBid = [0] # dummy value to pop in the loop optionAsk = [0] # dummy value to pop in the loop volatility = [0] # dummy value to pop in the loop # Initalization for the loop Nt = 4 # even number greater than 2: 4, 6, ... Ns = 2 # even number greater than 0: 2, 4, ... twotau = 2 # not a parameter... alpha = 0.01 # not a parameter... dt = twotau / Nt # time grid step dimA = ( (Nt+1)*(Ns+1), (Nt+1)*(Ns+1) ) # Matrix A dimensions dimb = ( (Nt+1)*(Ns+1), 1 ) # Vector b dimensions A = np.zeros( dimA ) # Matrix A b = np.zeros( dimb ) # Vector b portfolio = 1000000 # Money 'available' securityMargin = 0.00083 # EMPIRICAL: needs to be adjusted when taking into account the transaction fees (should rise, see the article p.8) # Wait 10mn after the opening of the market datet = datetime.datetime.now() datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) pause.until(datet) # Record the stock and option values and wait 10mn more def funcRetrieveStockOptionVolatility(): # Stock stock_data_url = "https://finance.yahoo.com/quote/MSFT?p=MSFT" stock_data_html = requests.get(data_url).content stock_content = BeautifulSoup(stock_data_html, "html.parser") stock_bid = content.find("td", {'class': 'Ta(end) Fw(600) Lh(14px)', 'data-test': "BID-value"}) print(stock_bid) stock_ask = content.find("td", {'class': 'Ta(end) Fw(600) Lh(14px)', 'data-test': "ASK-value"}) print(stock_ask) stockOptVol[0] = stock_bid.text.split()[0] stockOptVol[1] = stock_ask.text.split()[0] # Option option_data_url = "https://finance.yahoo.com/quote/MSFT/options?p=MSFT&date=1631836800" option_data_html = requests.get(option_data_url).content option_content = BeautifulSoup(option_data_html, "html.parser") call_option_table = content.find("table", {'class': 'calls W(100%) Pos(r) Bd(0) Pt(0) list-options'}) calls = call_option_table.find_all("tr")[1:] it = 0 for call_option in calls: it+=1 print("it = ", it) if "in-the-money " in str(call_option): itm_calls.append(call_option) print("in the money") itm_put_data = [] for td in BeautifulSoup(str(itm_calls[-1]), "html.parser").find_all("td"): itm_put_data.append(td.text) print(itm_put_data) if itm_put_data[0] == 'MSFT210917C00220000': # One single option stockOptVol[2] = float(itm_put_data[4]) stockOptVol[3] = float(itm_put_data[5]) stockOptVol[4] = float(itm_put_data[-1].strip('%')) else: otm_calls.append(call_option) print("out the money") print("bid = ", option_bid, "\nask = ", option_ask, "\nvol = ",option_vol) return stockOptVol # Record option and volatility stockOptVol = funcRetrieveStockOptionVolatility() optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[4]) # Wait another 10mn to record a second value for the quadratic interpolation datet = datetime.datetime.now() datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) pause.until(datet) stockOptVol = funcRetrieveStockOptionVolatility() optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[4]) tradeAtTimeTau = False tradeAtTimeTwoTau = False # Run the loop until 30mn before closure datet = datetime.datetime.now() datetend = datetime.datetime(datet.year, datet.month, datet.day, datet.hour + 6, datet.minute + 10) while datet <= datetend: datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) optionBid.pop(0) optionAsk.pop(0) optionVol.pop(0) stockOptVol = funcRetrieveStockOptionVolatility() stockBid = stockOptVol[0] stockAsk = stockOptVol[1] optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[5]) # Trade if required if tradeAtTimeTau == True or tradeAtTimeTwoTau == True: # sell if tradeAtTimeTau == True: portfolio += min(optionAsk[2],sellingPriceAtTimeTau) * 140 # sell 140 options bought 10mn ago tradeAtTimeTau = tradeAtTimeTwoTau sellingPriceAtTimeTau = sellingPriceAtTimeTwoTau sellingPriceAtTimeTwoTau = false else: # forecast the option when no trading # Interpolation coefa = funcQuadraticInterpolationCoef(optionAsk) # quadratic interpolation of the option ask price coefb = funcQuadraticInterpolationCoef(optionBid) # quadratic interpolation of the option bid price coefs = funcQuadraticInterpolationCoef(optionVol) # quadratic interpolation of the volatility sigma sa = stockAsk # stock ask price sb = stockBid # stock bid price ds = (sa - sb) / Ns # stock grid step for k in range (0, Ns+1): # fill the matrix and the vector for j in range (0, Nt+1): Atemp = np.zeros( dimA ) btemp = np.zeros( dimb ) print("k = {k}, j = {j}".format(k=k,j=j)) if k == 0: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcUab(j*dt,coefb) elif k == Ns: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcUab(j*dt,coefa) elif j == 0: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcF( k*ds+sb, sa, sb, funcUab(j*dt,coefa), funcUab(j*dt,coefb) ) elif j == Nt: # do nothing pass else: # main case akj = 0.5*(255*13*3)* funcUab(j*dt, coefs)**2 * (k*ds + sb)**2 dts = (twotau-dt)/Nt * (sa-sb-ds)/Ns #---------- #----- Integral of the generator L #---------- #----- time derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] = dts / dt**2 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] = dts / dt**2 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] = - dts / dt**2 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] = - dts / dt**2 # k,j-1 ~ k,j+1 #---------- #----- stock derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k+1,j ~ k+1,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = 4 * akj**2 * dts / ds**4 # k,j ~ k,j Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k+1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k,j ~ k+1,j #----- Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k-1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k+1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k-1,j ~ k+1,j #---------- #----- time and stock derivatives #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+0) ] = akj * dts / (dt*ds**2) # k,j+1 ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = akj * dts / (dt*ds**2) # k+1,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+0) ] = - akj * dts / (dt*ds**2) # k,j-1 ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = - akj * dts / (dt*ds**2) # k+1,j ~ k,j-1 #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+0) ] = -2 * akj * dts / (dt*ds**2) # k,j+1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = -2 * akj * dts / (dt*ds**2) # k,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+0) ] = 2 * akj * dts / (dt*ds**2) # k,j-1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = 2 * akj * dts / (dt*ds**2) # k,j ~ k,j-1 #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+0) ] = akj * dts / (dt*ds**2) # k,j+1 ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = akj * dts / (dt*ds**2) # k-1,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+0) ] = - akj * dts / (dt*ds**2) # k,j-1 ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = - akj * dts / (dt*ds**2) # k-1,j ~ k,j-1 #---------- #---------- #----- Regularisation term - using alpha = 0.01 #---------- #---------- #----- H2 norm: 0 derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += alpha # k,j ~ k,j #----- coef = funcF( k*ds+sb, sa, sb, funcUab(j*dt,coefa), funcUab(j*dt,coefb) ) btemp[ (k+0)*(Nt+1)+(j+0) ] += alpha * 2 * coef #---------- #----- H2 norm: time derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**2 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**2 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] += -alpha / dt**2 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] += -alpha / dt**2 # k,j-1 ~ k,j+1 #----- coef = ( funcF( k*ds+sb, sa, sb, funcUab((j+1)*dt,coefa), funcUab((j+1)*dt,coefb) ) \ - funcF( k*ds+sb, sa, sb, funcUab((j-1)*dt,coefa), funcUab((j-1)*dt,coefb) ) ) / dt btemp[ (k+0)*(Nt+1)+(j+1) ] += alpha * 2 * coef btemp[ (k+0)*(Nt+1)+(j-1) ] += - alpha * 2 * coef #---------- #----- H2 norm: stock derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**2 # k+1,j ~ k+1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += alpha / ds**2 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += -alpha / ds**2 # k+1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += -alpha / ds**2 # k-1,j ~ k+1,j #----- coef = ( funcUab(j*dt,coefa) - funcUab(j*dt,coefb) ) / (sa - sb) btemp[ (k+1)*(Nt+1)+(j+0) ] += alpha * 2 * coef btemp[ (k-1)*(Nt+1)+(j+0) ] += - alpha * 2 * coef #---------- #----- H2 norm: stock and time derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k+1,j+1 ~ k+1,j+1 Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k-1,j+1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k-1,j-1 ~ k-1,j-1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k+1,j-1 ~ k+1,j-1 #---------- Atemp[ (k+1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k+1,j+1 ~ k-1,j+1 Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k+1,j+1 ~ k+1,j-1 Atemp[ (k+1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k+1,j+1 ~ k-1,j-1 #----- Atemp[ (k-1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k-1,j+1 ~ k+1,j+1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k+1,j-1 ~ k+1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k-1,j-1 ~ k+1,j+1 #---------- Atemp[ (k-1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k-1,j+1 ~ k+1,j-1 Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k-1,j+1 ~ k-1,j-1 #----- Atemp[ (k+1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k+1,j-1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k-1,j-1 ~ k-1,j+1 #---------- Atemp[ (k+1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k+1,j-1 ~ k-1,j-1 #----- Atemp[ (k-1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k-1,j-1 ~ k+1,j-1 #---------- coef = ( funcUab((j+1)*dt,coefa) - funcUab((j+1)*dt,coefb) \ - funcUab((j-1)*dt,coefa) + funcUab((j-1)*dt,coefb) ) / (dt * (sa - sb)) btemp[ (k+1)*(Nt+1)+(j+1) ] += alpha * 2 * coef / (ds*dt) btemp[ (k-1)*(Nt+1)+(j+1) ] += - alpha * 2 * coef / (ds*dt) btemp[ (k-1)*(Nt+1)+(j-1) ] += - alpha * 2 * coef / (ds*dt) btemp[ (k+1)*(Nt+1)+(j-1) ] += alpha * 2 * coef / (ds*dt) #---------- #----- H2 norm: stock second derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**4 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += 4 * alpha / dt**4 # k,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**4 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / dt**4 # k,j+1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] += -2 * alpha / dt**4 # k,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**4 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**4 # k,j-1 ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] += -2 * alpha / dt**4 # k,j ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / dt**4 # k,j-1 ~ k,j #---------- #----- H2 norm: time second derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k+1,j ~ k+1,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += 4 * alpha / ds**4 # k,j ~ k,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k+1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k,j ~ k+1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k,j ~ k,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k,j ~ k,j #----- Atemp[ (k+0)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k-1,j ~ k,j #---------- coef = ( funcF( k*ds+sb, sa, sb, funcUab((j+1)*dt,coefa), funcUab((j+1)*dt,coefb) ) \ - 2 * funcF( k*ds+sb, sa, sb, funcUab((j+0)*dt,coefa), funcUab((j+0)*dt,coefb) ) \ + funcF( k*ds+sb, sa, sb, funcUab((j-1)*dt,coefa), funcUab((j-1)*dt,coefb) ) ) / dt**2 btemp[ (k+0)*(Nt+1)+(j+1) ] += alpha * 2 * coef / dt**2 btemp[ (k+0)*(Nt+1)+(j+0) ] += - alpha * 4 * coef / dt**2 btemp[ (k+0)*(Nt+1)+(j-1) ] += alpha * 2 * coef / dt**2 #---------- #---------- #----- Boundary de-computation #---------- if k+1 == Ns: Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = 0 # k+1,j ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] = 0 # k+1,j+1 ~ k+1,j+1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 ~ k+1,j-1 btemp[ (k+1)*(Nt+1)+(j+0) ] = 0 # k+1,j btemp[ (k+1)*(Nt+1)+(j+1) ] = 0 # k+1,j+1 btemp[ (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 if k-1 == 0: Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = 0 # k-1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] = 0 # k-1,j+1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 ~ k-1,j-1 btemp[ (k-1)*(Nt+1)+(j+0) ] = 0 # k-1,j btemp[ (k-1)*(Nt+1)+(j+1) ] = 0 # k-1,j+1 btemp[ (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 if j-1 == 0: Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] = 0 # k,j-1 ~ k,j-1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 ~ k+1,j-1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 ~ k-1,j-1 btemp[ (k+0)*(Nt+1)+(j-1) ] = 0 # k,j-1 btemp[ (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 btemp[ (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 #---------- pass print("-----") print("Atemp = ") print(Atemp) print("-----") print("btemp = ") print(btemp) print("-----") print("-----") A = A + Atemp b = b + btemp print("-----") print("A = ") print(A) print("-----") print("b = ") print(b) print("-----") print("-----") input("Press Enter to continue...") # Conjugate gradient algorithm: https://en.wikipedia.org/wiki/Conjugate_gradient_method x = np.zeros(N).reshape(N,1) r = b - np.matmul(A,x) p = r rsold = np.dot(r.transpose(),r) for i in range(len(b)): Ap = np.matmul(A,p) alpha = rsold / np.matmul(p.transpose(),Ap) x = x + alpha * p r = r - alpha * Ap rsnew = np.dot(r.transpose(),r) if np.sqrt(rsnew) < 1e-16: break p = r + (rsnew / rsold) * p rsold = rsnew print("it = ", i) print("rsold = ", rsold) # Trading strategy sm = (sa + sb)/2 if x[Ns/2*(Nt+1)+Nt/2] >= optionAsk[0] + securityMargin: tradeAtTimeTau = True sellingPriceAtTimeTau = x[Ns/2*(Nt+1)+Nt/2] portfolio -= 140 * optionAsk # buy 140 options if x[Ns/2*(Nt+1)+Nt] >= optionAsk[0] + securityMargin: tradeAtTimeTwoTau = True sellingPriceAtTimeTwoTau = x[Ns/2*(Nt+1)+Nt] portfolio -= 140 * optionAsk # buy 140 options pause.until(datet) # Wait 10mn before the next loop pause.until(datet) datet = datetime.datetime.now() # Time should be around 20mn before closure datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) if tradeAtTimeTau == True: # sell stockOptVol = funcRetrieveStockOptionVolatility() optionAsk.pop(0) optionAsk.append(stockOptVol[3]) portfolio += min(optionAsk[2],sellingPriceAtTimeTau) * 140 # Wait 10mn more to sell the last options pause.until(datet) # it should be around 10mn before closure if tradeAtTimeTwoTau == True: # sell stockOptVol = funcRetrieveStockOptionVolatility() optionAsk.pop(0) optionAsk.append(stockOptVol[3]) portfolio += min(optionAsk[2],sellingPriceAtTimeTwoTau) * 140 # Market closureDon't put money on this as I'm still debugging (I bet you half a bitcoin I have mistaken a few indices in the H_2 norm)... Here is the discretisation formula I used, to copy-paste on latexbase:
\documentclass[12pt]{article} \usepackage{amsmath} \usepackage[latin1]{inputenc} \title{Klibanov algorithm} \author{Discretisation formula} \date{\today} \begin{document} \maketitle Let $$ a_{k,j} = \frac12\sigma(j\delta_\tau)^2\times(255\times13\times3)\times(k\delta_s+s_a)^2, $$ then \begin{alignat*}{3} J_\alpha(u) = & \sum_{k=1}^{N_s} \sum_{j=1}^{N_t} \left| \frac{u_{k,j+1} - u_{k,j-1}}{\delta_\tau} + a_{k,j} \frac{u_{k+1,j} - 2u_{k,j} + u_{k-1,j}}{\delta_s^2}\right|^2\frac{2\tau - \delta_\tau}{N_t}\frac{s_a - s_b - \delta_s}{N_s}\\ & + \alpha \sum_{k=1}^{N_s} \sum_{j=1}^{N_t} \left| u_{k,j} - F_{k,j}\right|^2 \\ & \qquad + \left| \frac{u_{k,j+1} - u_{k,j-1}}{\delta_t} - \frac{F_{k,j+1} - F_{k,j-1}}{\delta_t}\right|^2 \\ & \qquad + \left| \frac{u_{k+1,j} - u_{k-1,j}}{\delta_s} - \frac{u_{a,j} - u_{b,j}}{s_a - s_b}\right|^2 \\ & \qquad + \left| \frac{(u_{k+1,j+1} - u_{k-1,j+1}) - (u_{k+1,j-1} - u_{k-1,j-1})}{\delta_s\delta_t} \right. \\ & \qquad \qquad \left. - \frac{(u_{a,j+1} - u_{b,j+1}) - (u_{a,j-1} - u_{b,j-1})}{(s_a-s_b)\delta_t}\right|^2 \\ & \qquad + \left| \frac{u_{k,j+1} - 2u_{k,j} + u_{k,j-1}}{\delta_\tau^2} - \frac{F_{k,j+1} - 2F_{k,j} + F_{k,j-1}}{\delta_\tau^2} \right|^2 \\ & \qquad + \left| \frac{u_{k+1,j} - 2u_{k,j} + u_{k-1,j}}{\delta_s^2}\right|^2 \end{alignat*} %% \left| \right|^2 with $\tau = 1$ unit of time (for example 10mn). \end{document}Let me know if you see something wrong... And if you want to contribute, feel free
Not the best translation but interesting. Thanks to u/kokoniqq for the heads up! submitted by AR_MR_XR to AR_MR_XR [link] [comments] 1. Optical display scheme in AR glasses Augmented reality technology, or AR technology, is to provide users with virtual information through images, videos, 3D models and other technologies while displaying real scenes, so as to achieve the ingenious integration of virtual information and the real world. It is the tipping point of the next information technology. According to authoritative predictions, augmented reality glasses will replace mobile phones as the next generation of collaborative computing platforms. Augmented reality technology represented by augmented reality glasses is currently emerging in various industries, especially in the security and industrial fields. Augmented reality technology embodies unparalleled advantages and greatly improves the way of information interaction. At present, the optical display solutions in the more mature augmented reality technology are mainly divided into prism solutions, birdbath solutions, free-form surface solutions, off-axis holographic lens solutions, and lightguide solutions. 1.1 Prism scheme The prism scheme takes Google Glass as an example. As shown in Figure 1, the optical display system is mainly composed of a projector and a prism. The projector projects the image, and then the prism reflects the image directly into the human retina, superimposing it with the real image. Since the system is above the human eye, it is necessary to focus the eye to the upper right to see the image information, and this system has a natural contradiction between the field of view and the volume. The Google Glass system has a small field of view, with only a 15-degree field of view, but the optical lens has a thickness of 10mm, and the brightness is not enough, and the image has a large distortion, so the product was withdrawn by the company shortly after entering the market. Figure 1. Physical image of Google Glass glasses products 1.2 Birdbath solution The optical design in the Birdbath solution is to project light from the display source onto a 45-degree beam splitter. The beam splitter has reflection and transmission values (T), allowing the light to be partially reflected in the percentage of R, while the rest Transmitted in T value. At the same time, T allows users to see physical objects in the real world and digital images generated by the display at the same time. The light reflected from the beam splitter bounces onto the combiner. The synthesizer is generally a concave mirror that redirects light to the eyes. AR headsets using this optical display solution mainly include Lenovo Mirage AR headsets (Figure 2(a)) and ODG R8 and R9 (Figure 2(b)). Among them, ODG has a 50-degree field of view, and its thickness exceeds 20mm. Figure 2. (a) Mirage headset device; (b) ODG R9 headset device 1.3 Free-form surface scheme The free-form surface scheme generally uses a free-form surface mirror with a certain reflection/transmission (T) value. The free-form surface is a complex and unconventional surface shape that is different from a spherical or aspherical surface, which is used to describe the surface shape of the lens. The mathematical expression of is relatively complicated and often does not have rotational symmetry. The light from the display directly hits the concave mirrocombiner and is reflected back into the eyes. The ideal position of the display source is centered and parallel to the mirror surface. Technically speaking, the ideal position is for the display source to cover the user's eyes, so most designs move the display "off-axis" and set it above the forehead. The off-axis display on the concave mirror has distortion, which needs to be corrected on the software/display side. Since free-form surfaces can not only provide more degrees of freedom for the design of optical systems, significantly improve the optical performance of the system, but also bring more flexible structural forms to system design, so it has become a research hotspot in the field of optical design in recent years. Among the most representative companies are Epson of Japan (shown in Figure 3) and the Meta series of Dream Vision Corporation of the United States (shown in Figure 4). Although the AR glasses of Epson of Japan are gambling in terms of color, saturation and image quality, they only have a field of view of 23 degrees and a thickness of 13mm. Although the Meta2 series of AR glasses from American Dreamland Vision has a 90-degree field of view, its thickness exceeds 50mm, and the weight of the optical and mechanical system alone is about 420 grams. Figure 3. AR glasses developed by Epson in Japan. (a) The actual product; (b) The imaging light path. Figure 4. AR helmet developed by American Dreamland Vision. (a) The actual product; (b) The imaging light path. From the above, it can be seen that there is an unavoidable contradiction in the prism scheme, birdbath scheme, and free-form surface scheme, that is, the larger the field of view, the thicker the optical lens and the larger the volume. It is precisely because of this. The irreconcilable contradiction limits its application in smart wear, that is, augmented reality glasses. 1.4 Holographic lens solution The holographic lens solution uses the unique optical characteristics of the holographic lens. The principle is to record a holographic collimating lens (Hd) and a simple linear grating (Hg) on the same holographic dry plate, and the holographic collimating lens will emit the display source. The beam is collimated into a plane wave and diffracted into the substrate for total internal reflection transmission, while the line grating diffracts the beam into the human eye. This system uses holographic optical elements as coupling elements. It has a compact structure and reduces the difficulty of designing and processing holographic optical elements. At the same time, it reduces the dispersion of the holographic lens. It also has the advantages of large FOV and small size, so it is quickly adopted by people. accept. However, due to the relatively small eye movement range, and the holographic lens has complex aberrations and severe dispersion, the imaging effect of the holographic lens is not ideal. The representative manufacturer currently adopting the holographic lens solution is North. As shown in Figure 5, it is the physical map of North's AR glasses products based on the holographic lens solution and the schematic diagram of the imaging optical path. Figure 5. AR glasses based on holographic lens solution developed by North Company. (a) The actual product; (b) The imaging light path. 1.5 Optical waveguide solution The optical waveguide solution has advantages in terms of clarity, viewing angle, volume, etc., so it has become the best optical display solution in augmented reality glasses, and is expected to become the mainstream optical display solution for AR glasses. AR glasses based on waveguide technology are generally composed of three parts: display module, waveguide and coupler. The light emitted by the display module is coupled into the optical waveguide by the in-coupling device, travels forward in the form of total reflection in the waveguide, and when it reaches the out-coupling device, it is coupled out of the optical waveguide and enters the human eye for imaging. Because the optical path is folded by the waveguide, the general system volume is relatively small. According to the principle of the coupler, the optical waveguide technology used in AR glasses based on waveguide technology can be divided into two types: geometric waveguide and diffractive optical waveguide. The geometric waveguide solution generally includes a sawtooth structure waveguide and a polarized film array mirror waveguide (referred to as a polarized array waveguide). Among them, the mainstream polarized arrayed waveguide uses multiple semi-transparent and semi-reflective film layers that are placed in parallel and have a certain split ratio to achieve image output and exit pupil expansion, thereby having a thin, thin, large field of view and eye movement range. And the advantage of uniform color. The diffractive optical waveguide schemes mainly include surface relief grating waveguide scheme and volume holographic grating waveguide scheme. The embossed grating waveguide solution is manufactured using nano-imprint lithography technology. Although it has the advantages of large field of view and large eye movement range, it will also bring challenges to field of view and color uniformity, and related micro-nano processing technology. It is also a huge challenge, and the production cost is high. The volume holographic grating waveguide solution has advantages in color uniformity (no rainbow effect) and realization of a single-chip full-color waveguide, so it has attracted great interest from AR optical module manufacturers. Figure 6 is the basic display principle of the waveguide solution. The coupling area is used to couple the light beam of the micro-projector into the waveguide sheet, so that the light beam satisfies the conditions of total reflection propagation in the waveguide sheet, and the coupling area is used for total reflection. The propagating light beam couples out of the waveguide and reaches the human eye. The coupling area can be mirrors, prisms, relief gratings and volume holographic gratings. The decoupling area can be half mirrors, relief gratings and volume holographic gratings arranged in an array. This article will explain in detail the polarization array waveguide scheme in geometric optical waveguide technology and the surface relief grating waveguide scheme and volume holographic grating waveguide scheme in diffractive optical waveguide technology, and the preparation and processing technology of surface relief grating and volume holographic grating At the same time, it further introduces the research and development situation of Goolton Technology in this field. Figure 6. Schematic diagram of the waveguide solution 2. Polarization array waveguide 2.1 Principle of Polarization Array Waveguide The waveguide lens of the polarization array waveguide technology usually adopts a plurality of semi-transmissive and semi-reflective coatings placed in parallel and with a certain split ratio to achieve image output and exit pupil expansion. The semi-transparent and semi-reflective coating has angular selectivity. , And the array is arranged. The schematic diagram of its working principle is shown in Figure 7.After the light emitted by the image source is collimated by the eyepiece system, it is coupled into the waveguide by the reflective surface of the waveguide. The light in each field of view propagates in the waveguide according to the total reflection theorem, and the light enters the semi-transparent On the reverse side, part of it reflects out of the waveguide, and the other part of the transmission continues to propagate. Then this part of the advancing light meets another mirror, and the above-mentioned "reflection-transmission" process is repeated until the last mirror in the mirror array reflects all the remaining light out of the waveguide into the human eye. Since the waveguide can have multiple semi-transparent and semi-reverse surfaces, and each semi-transparent and semi-reverse surface forms an exit pupil, the exit pupil can be expanded when the substrate thickness is very thin to achieve a large field of view and large eye movement range. After multiple reflections, the emitted light can be "adjusted" to be more uniform. Figure 7. Schematic diagram of the working principle of the array optical waveguide The pupil dilation technology of this technology is more complicated in design. Full consideration should be given to stray light, human eye compatibility, and various performance indicators when designing. In addition, uniformity is also an intuitive indicator of the end user experience. How to control the reflection and transmittance of multiple coatings, how to optimize the whole machine, and how to control the coating process can ensure the uniformity of the entire eye movement range. the focus of research. For this reason, Goolton Technology independently developed and designed optical modules based on polarization arrayed waveguide technology, and after repeated attempts to summarize, obtained epoch-making results. 2.2 Goolton Technology-"Seven-fold, dodecahedron" ultra-short-focus AR optical module M3010 Recently, Goolton Technology released a new "seven-fold, dodecahedron" ultra-short-focus AR optical module M3010 (Figure 8), which uses specially selected materials and process combinations to successfully eliminate the inherent noise and streaks of peer products Difficult problems such as perception, ghosting, distortion, dispersion, etc., have broken through the limits of AR display technology at this stage in terms of imaging clarity, maximum brightness, color uniformity, weight, volume, power consumption, light leakage, etc., and all indicators are in In the forefront of the world, it truly integrates all the advantages of optical waveguide modules such as extremely thin, extremely light and extremely high color reproduction, and exerts its performance to the extreme. Figure 9 shows the product specifications of the optical module M3010 based on polarization array waveguide technology recently launched by Goolton Technology. Figure 8. Goolton Technology-(a) Seven-fold optical path; (b) Product display diagram of optical module M3010 based on polarization arrayed waveguide technology Figure 9. Goolton Technology-Product Specifications of Optical Module M3010 Based on Polarization Arrayed Waveguide Technology Goolton Technology’s "seven-fold, dodecahedron" ultra-short-focus optical module M3010 has the following super performance. 1. Small: Based on the anisotropic characteristics of the crystal material, the multiplexing of optical devices is realized, and the light that originally propagated in one direction in the optical waveguide is folded into 7 segments, which reduces the volume of the projector unit by 85%; 2. : Weight is about 33 grams; 3. Transparency: The light transmittance of the waveguide lens exceeds that of ordinary architectural glass windows, which can reach more than 85%; 4. Thin: Refraction distortion is less than 2mm; 5. Color: ultra-high contrast, resolution, color reproduction The M3010 comes standard with LCOS as the image source, and the resolution can reach 1920*1080. It provides optical resolution close to the limit resolution of the human eye, completely eliminates the sense of screen boundary, and the image quality is clear and delicate, and the image contrast Sharp, no graininess. The maximum brightness can reach 5000nit, and the color gamut coverage is more than 100% RGB, reaching the level of professional monitors. 6. Zero light leakage: Thanks to the light splitting film array waveguide sheet and optical structure exclusively developed by Goolton Technology, the M3010 module will not leak light when working, and will not expose the content displayed on the screen to the outside world, regardless of the concealment requirements The extremely high military helmet is also the AR glasses for consumer entertainment, this feature is very important; 7.Super vision: M3010 adopts top-down structure, the horizontal field of view is completely unobstructed, and the entire field of view is fully visible. While improving the user experience, it also solves the problem of security risks caused by users wearing glasses to block the line of sight; 8. Low power consumption: battery life can reach about 10 hours; 9. Mass production reaches 10K pieces per year, and the mass production yield is stable. The cost has reached the world-class level; 10. Ultra-strict environmental testing standards: In the face of extreme high and low temperature environments, as well as high humidity and continuous salt spray, Goolton Technology’s "seven-fold, dodecahedron" ultra-short-focus AR optics The module M3010 can work stably with reliability far exceeding the industry average; 11. Fully accept customization: AR glasses have rich landing scenes, based on the powerful functions of M3010, technology companies from all walks of life can work in their familiar Customize a wide range of intelligent AR products in the field. We firmly believe that Goolton Technology’s "seven-fold, dodecahedron" ultra-short-throw optical module M3010 will definitely be able to kick off the next generation of display technology revolution, and provide better and more advanced technology for companies that also hold the spirit of craftsmanship. Powerful AR products and high-quality services. 3. Diffractive optical waveguide 3.1 Surface relief grating waveguide The relief grating waveguide solution is to use relief grating (SRG) instead of traditional catadioptric optical device (ROE) as the coupling in, coupling out and exit pupil expansion device in the waveguide solution. The schematic diagram of its working principle is shown in Figure 10. Figure 10. Schematic diagram of the principle of diffractive optical waveguide and surface relief grating Commonly used relief gratings are mainly one-dimensional gratings, including tilted gratings, trapezoidal gratings, blazed gratings and rectangular grating structures. Figure 11(a) shows the scanning electron microscope (SEM) image of the tilted grating. The two-dimensional grating is mainly a hexagonal cylindrical grating structure commonly used in waveguides. Figure 11(b) shows the SEM image of the two-dimensional cylindrical grating structure. The feature sizes of the above grating structures are all nanometers. Currently, the most representative products of the embossed grating waveguide solution are Microsoft's HoloLens series 12(a) and WaveOptics' embossed grating waveguide series 12(b). Figure 11. (a) Tilted grating structure diagram; (b) Two-dimensional cylindrical grating structure diagram Figure 12. (a) HoloLens2 from Microsoft; (b) Embossed grating waveguide from WaveOptics 3.2 Volume holographic grating waveguide The volume holographic grating waveguide solution uses a volume holographic grating as the coupling in and out of the waveguide. A volume holographic grating is an optical element with a periodic structure. It is generally exposed through a double-beam holographic exposure, directly on the micron-level thickness of the photopolymer The internal interference of the film forms interference fringes with light and dark distribution, which causes the periodic change of the refractive index inside the material. This period is generally a nano-scale grating structure, which is an order of magnitude with the wavelength of visible light, so the light can be effectively modulated, and the incident light can be diffracted to change the direction of light transmission. Combining the volume holographic grating and the waveguide film, the diffraction efficiency of the volume holographic grating can be adjusted by designing the relevant parameters of the volume holographic grating (such as material refractive index n, refractive index modulation factor and thickness, etc.). The schematic diagram of the working principle of the volume holographic grating waveguide technology is shown in Figure 13.The image generated by the microdisplay becomes parallel light after passing through the collimating system. The diffraction effect changes the propagation direction of parallel light. When the light in the waveguide meets the condition of total reflection, it is confined to travel in the waveguide direction without loss. When the parallel light propagates to the holographic grating at the out-coupling end, the condition of total reflection is destroyed, and the light is diffracted again and becomes parallel light that exits the waveguide and enters the human eye for imaging. When the coupled holographic grating and the modulated out holographic grating have the same periodic structure and mirror symmetry, the dispersion can be effectively eliminated. Figure 13. Schematic diagram of the working principle of volume holographic grating waveguide technology The representative manufacturers that adopted volume holographic grating waveguide solutions in the early days were Sony and Digilens. With the maturity of this technology, the number of companies participating in the optical research of holographic grating diffraction waveguides is increasing, mainly including TruLife and WaveOptics in the United Kingdom, and Akonia in the United States. Wait. Sony has produced a high-brightness single-green volume holographic grating waveguide. As shown in Figure 14, the structure uses a double-sided volume holographic grating as the coupling end, achieving a transmittance of 85% and a display brightness of 1000cd/m2. However, due to the small thickness of the volume holographic grating, the efficiency of the system is low. In addition, it can only be used for monochromatic display and has been discontinued. Digilens launched a double-layer full-color volume holographic grating waveguide, as shown in Figure 15. This structure realizes color by using multiple monochromatic gratings, which can effectively reduce the crosstalk of system colors, but the efficiency of the system is not high, and because of its The double-layer waveguide structure makes the system more difficult to manufacture. Figure 14. Sony's double-sided volume grating structure holographic waveguide. (a) The actual product; (b) The imaging light path. Figure 15. Digilens full-color volume holographic grating waveguide. (a) The actual product; (b) The imaging light path. Goolton Technology uses the holographic material exposure method to combine the RGB three colors into a diffractive waveguide, and uses the principle of coherent recording and diffraction to transmit the image to the human eye for display. There are three aspects: simulation design, materials, and process preparation. Simulation design requires self-written complex calculation models; materials mainly refer to the photosensitive materials in HOE. For holographic optical waveguides, low shrinkage ratio, high efficiency and high uniformity before and after manufacturing are required; in terms of process, more holographic technology is required Manufacturing light path and exposure experience, it is very related to the materials used. Figure 16 shows the corresponding display effect of the single-layer full-color volume holographic grating waveguide developed by Goolton Technology, with a field of view of 30°. Figure 16. The display effect of the single-layer full-color volume holographic grating waveguide developed by Goolton Technology 4. Micro-nano manufacturing of diffractive optical waveguides 4.1 Micro-nano fabrication of surface relief grating waveguide Surface relief gratings can be divided into one-dimensional and two-dimensional gratings from the dimension, and can be divided into straight gratings, blazed gratings and inclined gratings in structure. Since the augmented reality optical waveguide is used in the visible light band, in order to achieve greater diffraction efficiency and field of view, its characteristic size is generally hundreds of nanometers or even tens of nanometers, and its performance has a small tolerance for errors. Processing and preparation posed great challenges. The current preparation of diffractive optical waveguides is basically based on semiconductor preparation processes (such as photolithography and etching processes). However, because these methods are limited by their complicated and expensive equipment, the production cost is very high, and they are not suitable for mass production of optical modules. Shown in Figure 17 is the process flow chart of surface relief grating template preparation or small batch preparation, including its scanning electron micrograph. For straight gratings, the process is relatively mature. First, a resist layer is spin-coated on the substrate, and the grating is patterned by interference exposure or electron beam exposure, and then reactive ion etching (RIE) or inductively coupled plasma etching is used ( ICP) transfer the pattern to the substrate and remove the resist layer to complete the preparation of the straight grating. Due to the uniformity problem, the oblique grating optical waveguide represented by HoloLens cannot be directly prepared by the reactive etching scheme, so the preparation process is more complicated, and focused ion beam etching (FIBE) and ion beam etching are required. Beam etching, IBE), reactive ion beam etching (reactive ion beam etching, RIBE) technology. Considering efficiency and uniformity comprehensively, RIBE is a more suitable solution. First, a hard mask (such as Cr) layer is plated on the substrate by physical or chemical methods, and then a resist layer is spin-coated. Also use interference exposure or electron beam exposure for patterning, and then transfer the resist pattern to the Cr layer through a chlorine dry etching process. After the etching process, the remaining resist layer is stripped by oxygen plasma method. Next, the fluorine-based RIBE process is used to incident the substrate with an ionized argon ion beam at an oblique angle. After the reactive ion beam etching, the Cr mask is removed by a standard wet etching process to obtain an oblique grating with excellent uniformity. Figure 17. Surface relief grating template or small batch preparation process The above-mentioned semiconductor-based manufacturing process is expensive and not suitable for mass production and processing of grating waveguides. Therefore, the replication process of diffractive optical waveguides was developed to achieve mass production, and this large-scale manufacturing process relies on optical resins with high refractive index. At present, Magic Leap and WaveOptics have carried out verification of related processes. The replication process includes hot embossing, UV-nano imprint lithography and micro contact printing (also known as soft lithography). Among them, ultraviolet nanoimprint lithography is a common method in mass production of surface relief grating waveguides. The specific process flow is shown in Figure 18. The process can be divided into two stages: nanoimprint work mold preparation stage and mass production stage. First, the pattern is processed on the silicon wafer to be used as a template through the above-mentioned template preparation process, and UV resin is spin-coated on a larger silicon wafer through nanoimprint technology and more templates are printed on it. The printed structure is then exposed to ultraviolet light to fix the resin. Finally, the multi-pattern imprinting mold is mass-produced by repeating the above process. In the mass production process, multi-pattern molds are used to produce surface-relief grating waveguides, then functional coatings are used to cover the waveguides, and laser cutting technology is used to separate them, and finally the waveguides of different structures are stacked to realize the preparation of optical modules. Figure 18. Mass production process flow of surface relief grating replication 4.2 Micro-nano manufacturing of volume holographic grating waveguide The key element of the volume holographic waveguide is the volume holographic grating. The preparation of the volume holographic grating makes use of the characteristics of holographic technology. Two plane light waves with a certain angle excited by laser interfere with each other, and the interference pattern is exposed and attached to the substrate. It is obtained by forming interference fringes on the photosensitive material, and the material properties change according to the intensity distribution of light. Finally, a material with periodic changes in refractive index is obtained. The materials for preparing volume holographic waveguides include silver halide, dichromate gelatin, photosensitive polymers, holographic polymer dispersed liquid crystals, and other more exotic materials. Holographic technology is a method that uses the principle of optical coherence to record and obtain the amplitude and phase information of an object light wave. It uses the principle of interference recording and diffraction reproduction to record the interference fringes generated by the interference of the object light wave with amplitude and phase information and the reference light wave into a hologram in the form of intensity distribution, thereby recording all the amplitude and phase information of the object light wave in On photosensitive materials. Holography is an active coherent imaging technology. The holographic recording optical path (as shown in Figure 19(a)) mainly completes two functions. One is to complete the coherent illumination of the measured object, which is formed by the transmission or reflection of the object. The object light wave; the second is to use the reference light wave to interfere with the object light wave to form a hologram. Image Among them, T0 represents the zero-order diffracted light, which corresponds to the transmitted light wave of the reference light wave; T+1 represents the +1-order diffracted light, which carries the information of the original object light wave; T-1 is the -1 order diffracted light, which carries the object light wave Conjugation information. In optical holography, the +1-order diffracted light can form a virtual image of the object, which can be directly observed with eyes, while the -1st-order diffracted light can form a real image of the object, which can be received by the screen. Figure 19. Schematic diagram of the recording and reproduction process of optical holography The diffraction order of the ideal holographic grating is only 0 and ±1 orders. The holographic optical waveguide display uses the 0th order light to be continuously totally reflected in the optical waveguide, while the -1 order light continuously emits from the waveguide surface. The geometrical schematic diagram of grating diffraction is shown in Figure 20. Figure 20. Schematic diagram of holographic grating diffraction geometry Image From the above three equations, it can be concluded that for a specific wavelength, waveguide medium and light incident angle, the grating period that meets the total reflection condition should meet certain conditions. According to its structure, holographic gratings can be divided into transmission type and reflection type holographic gratings. The fundamental difference between the two is that the recording method is different, that is, the propagation direction of the two recording lights is different, which causes the different orientation of the interference fringe surface in the recording material. When the transmission type holographic grating records, the object light and the reference light are incident from the same side of the recording medium, while when the reflection type holographic grating records, the object light and the reference light are incident from both sides of the recording medium. Holographic gratings can be divided into surface holographic gratings and volume holographic gratings according to the relative thickness relationship between the thickness of the recording medium and the interference fringe spacing. The evaluation criteria of surface holographic grating and volume holographic grating are characterized by Q value. When Q≥10, it is volume holographic grating, otherwise it is surface holographic grating. Image The microstructure of volume holographic grating is inside the volume grating, so its diffraction is mainly the volume effect of the material. When the incident light satisfies the Bragg condition, the volume holographic grating will have extremely high diffraction efficiency, and if it deviates from the Bragg condition, the diffraction efficiency will drop rapidly. This characteristic makes the volume holographic grating have obvious angle and wavelength selectivity. When used as a coupling device, a volume holographic grating can couple light with a specific wavelength and angle in the waveguide out of the waveguide without blocking the view of the real scene from the outside, so it is an ideal coupling device. The above-mentioned preparation process of volume holographic grating is only suitable for small batch verification, and for mass production, it is necessary to develop a more economical solution. Companies represented by Sony and DigiLens have developed the processing process of volume holographic waveguide. The roll-to-roll process for preparing volume holographic waveguides is shown in Figure 21. First, the double-beam interference exposure method is used to form volume holographic waveguides in the photosensitive polymer film attached to the roll; the second step is to form high-quality cycloolefin polymer plastic waveguides by injection molding. In order to obtain a qualified image, the warpage of the waveguide must be less than 5um, and the thickness change of the effective area should be less than 1um. Then the transfer process of the holographic optical element is carried out to accurately align and paste the holographic waveguide film with the plastic waveguide; then the plastic holographic waveguide is cut; finally in the color matching process, the red and blue plastic waveguides and the green plastic waveguide are aligned and used with UV The resin encapsulates and fixes it. The plastic substrate should remain flat before and after each processing is a challenge faced by both the stamping and color matching processes. Figure 21. Preparation process of roll-to-roll holographic waveguide |
Mathematical Programming Solvers This section provides an overview of open source as well as commercial optimizers. Which type of mathematical programming problem can be solved by a certain package or function can be seen from the abbreviations in square brackets. Conjugate[z] (90 formulas) Primary definition (1 formula) Specific values (31 formulas) General characteristics (5 formulas) Transformations (29 formulas) Complex characteristics (12 formulas) Differentiation (2 formulas) Representations through equivalent functions (9 formulas) Zeros (1 formula) History (0 formulas) The conjugate matrix of a matrix is the matrix obtained by replacing each element with its complex conjugate, (Arfken 1985, p. 210).. The complex conjugate is implemented in the Wolfram Language as Conjugate[z].. Note that there are several notations in common use for the complex conjugate. Applied physics and engineering texts tend to prefer , while most modern math and theoretical physics Examples of Use. The conjugate can be very useful because.. when we multiply something by its conjugate we get squares like this:. How does that help? It can help us move a square root from the bottom of a fraction (the denominator) to the top, or vice versa.Read Rationalizing the Denominator to find out more: To rationalize the denominator using conjugate in math, there are certain steps to be followed.. Let us understand this by taking one example. Example . Rationalize the denominator \(\frac{1}{{5 - \sqrt 2 }}\) Solution. Step 1: Find out the conjugate of the number which is to be rationalized. The mathematical expressions calculator is a powerful algebraic calculation tool, it is able to analyze the type of expression to calculate and use the appropriate calculator to determine the result. For some calculations, in addition to the result, the different calculation steps are returned. Conjugate Concept. The term conjugate means a pair of things joined together. These two things are exactly the same except for one pair of features that are actually opposite of each other. If you Mathematical function, suitable for both symbolic and numerical manipulation. can be entered as co , conj , or \[Conjugate] . Conjugate automatically threads over lists. Mathematica » The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha » Explore anything with the first computational knowledge engine. Illustrated definition of Conjugate: In Algebra, the conjugate is where you change the sign ( to minus, or minus to ) in the middle of...
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Conjugate of Matrix and it's Properties. To ask your doubts on this topic and much more, click here: http://www.techtud.com/video-lecture/lecture-conjugate-m... Calculating a Limit by Multiplying by a Conjugate - Duration: 4:09. patrickJMT 335,377 views. 4:09. Algebra 1 11.8a - Adding and Subtracting Radicals - Duration: 7:52. ... https://sites.google.com/site/otjinenemath/These videos are intended to be used for anyone who wants, or needs to learn mathematics. These lessons will star... This channel is dedicated to promoting Maths and help people generate interest in Maths by showing a more fun side of Mathematics, as well as the elegance of... Mathematics: About Fourier transform and complex conjugateHelpful? Please support me on Patreon: https://www.patreon.com/roelvandepaarWith thanks & praise t... If you like my videos then please like, comment and subscribe to my channel Senior Series Link - 1. Determinant - https://youtube.com/playlist?list=PLxlX9Whq... Lecture: Mathematical Reasoning About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators ...
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