Dear Aaron,

I hope that I am still on your spreadsheet.

In my post of may 12 I sent you my idea of "consciouss time travel" but the illustration did not come along, so here it is again.

If you are interested in the whole artiocle I will sent it to your private mail,

mine is

wilhelmus.d@orange.fr

best regards

WilhelmusAttachment #1: figure_4_ENM_HOPPER.jpg

  • [deleted]

  • Post #78

Hi Robert,

I agree, we've been having an interesting discussion. I will get right to replying:

2) Because the laws of physics basically work the same forward as backward, photons traveling forward through time are equivalent to photons traveling backward through time in the other direction. In a sense it's not clear that photons travel through time at all, since the interval along the path of a photon is zero. We can think of the photons that leave a flashlight just as well as photons that arrive at the bulb from the future. So I think we need another explanation for the arrow of time--for, specifically, why we appear to get useful information only from photons arriving from the past--beside the finite speed of light.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This is an excellent point. I have not seen this point being raised in discussions of the arrow of time. However, I can see no reason why photons from the future would be compelled to interact with an eye or a camera, the way they can be conceived as converging upon a flashlight bulb. I think the answer to the why question you suggest comes down to a fundamental difference between emitters and absorbers.

3) Observing the arrival of radiation from a gamma-ray burst 20 years in the future requires information about the burst to travel from the source to us in the present day faster than the speed of light in apparent violation of general relativity. Using the same principle we could send a message faster than the speed of light by transmitting it to the future of a receiver with a foreknowledge machine. Maybe this is possible, but it suggests that foreknowledge machines depend on the existence of exotic spacetimes, which makes me skeptical.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Why would we need such a complicated arrangement, if we merely want to transmit superluminal messages? Superluminal signals that penetrate Faraday cages were pioneered in the late 19th century by everyone-should-know-who, and other methods have been found by other researchers since then. This may come as something of a shock (a pun in this context), but I will be happy to provide you with references. It would be more convenient to do so over email. (Go to my ebook's page on amazon.com and hover over my name to link to my author page, where you will find my email address.)

Now, I don't want to promote the idea that superluminal messages would violate relativity--this is not a simplistic area of science--there is a lot more mathematical leeway than that involved. Even superluminal travel would not necessarily violate relativity, as Miguel Alcubierre's work has so famously shown.

Other researchers have discovered a totally different way to achieve superluminal travel without running afoul of Einstein's powerful dilemma, as I will also share with you. Believe it or not, there is more than one way around the seemingly unbreakable limitation which arises through the interaction of E=mc2 and f=ma.

Again, it has been a pleasure. I look forward to continuing our discussion and expanding it over email. Also, I am nearing the end of your article, so I will be opening a new discussion with you shortly, on your page.

Warmly,

Aaron

Aaron,

Your farsighted, anticipatory, structured and generous nature provides smart and diligent guidelines to communicate in a visionary way. Definitely your essay in a poetic, analytic and challenging way provides an insight that provokes thoughts, and questions. Reading it brought up the question of how is it possible to follow the biophoton in the future? Is it contained in the biophoton a blue print to follow different structures in the layers of time or is it free? I appreciate you shared the possibility of the emergent new questionnaire of how to lead the self to a positive reality.

The suggestion of reading the posts of the persons mentioned in your post are worth reading and help to see constructive interpretations of your work, visions and eagers of helping human kind.I also appreciate you shared a score that is clever, enthusiastic and smart structured grades for qualifying other essays.

Wishing you success in achieving a way to download Cassandra and Everett machines from the intangible world into the tangible one to help human kind. We could share in the future platforms of work, meanwhile Black Sky Thinking wants to invite human kind to develop inner world to make rational cybernetics of cosmos, Cassandra/Everett machines would be an excellent tool to structure a rational platform for a positive reality for human kind. I remain thankful of your post and with the best wishes for you.

Kindly Orenda

Aaron,

Interesting essay, I have a strict belief to theory of relativities so I think we cannot get to access to the future and i do believe we can not calculate what happens in the future exactly as numerics lost some information in the process of quantization. (this maybe the same statement of recent hawking's chaos quantum condition in his black hole denying. anyway, we've got the degree of freedom to imagination and this always makes us building the future indeed so i think your essay would be deserved as rated.

ryoji

Dear Aaron,

I agree with most other comments pointing out that your essay is fun reading, and gives the definite impression that you have worked hard on the topic. But I see the following problems.

If I understand correctly, foreknowledge machines (FM) either return information (`viewer foreknowledge`) that must be definite and correct, or nothing, depending on where you point them in future spacetime. Cassandra machines are different: they would always show you the future, but they turn out to be impossible (leading to contradiction).

Then, is it correct to say that FMs are Cassandra machines that sometimes does not show anything?

After discussing FMs and related issues, you come to the topic of the contest, and you claim that FMs can be beneficial for `steering our future`. But: When they do not show anything, they can`t be useful. And when they show something, we can be sure that what the viewer sees will definitely happen (since the foreseen future is inviolable). Then, why bother acting? Can`t we comfortably sit and listen to the Beatles vinyl records, while waiting for the foreseen future to happen, whether good or bad?

At p. 7, at the bottom of the second paragraph of Section `Removing the Element of Surprise` you seem to provide a sort of proof that the usage of FMs would tend to eliminate undesirable outcomes. I feel that the crucial passage of the argument is: `However, truly undesirable outcomes that are feasible for a civilization to avoid would strongly invite a VCO, but in such cases there could be no explanation as to why a VCO does not occur.` I could not grasp the logic here. In particular, why would that strongly invite a VCO? Also, words such as `strongly` and `overwhelmingly` suggest a probabilistic argument, while the general context would seem more that of a definite, yes/no type of argument. In any case, given the limited space you had, I could not find clear and convincing argumentation for either.

Thanks for any clarification

Tommaso

    Hi Tommaso,

    Thanks for your interest in my essay, and I'm glad you enjoyed it. Yes, I did work hard on this topic, to produce Understanding Future-Viewing Machines and Time Travel, and while I did have to write for several hours every day to produce my FQXi essay, as only two weeks remained to write it after I finished and published that longer offering, most of the really hard work was already behind me. Now, I wish I had had two more pages of space to play with in order to include everything I would have wanted to include in my FQXi essay, but, in the end, meeting the contest guidelines had to take precedence.

    Now, I will go through your questions point by point:

    "[I]s it correct to say that FMs are Cassandra machines that sometimes does not show anything?"

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    There are a few important points and distinctions here that will clarify. FMs merely act the way Cassandra machines would act, and--if used in an interference viewing scenario--would act like Everett machines, or would fail to operate altogether.

    To answer your question, a Cassandra machine is defined as an imaginary type of future viewer which would always be able to give definite and correct information about future events, in every circumstance. So, substituting this definition into your question would yield:

    "[I]s it correct to say that FMs are... [an imaginary type of future viewer which would always be able to give definite and correct information about future events, in every circumstance] that sometimes does not show anything?"

    With this substitution, it becomes obvious that a future viewer which sometimes shows ambiguous information about the future or sometimes does not show anything about the future cannot be a Cassandra machine in any way whatsoever. So, a FM cannot be described as a Cassandra machine in any way whatsoever, even if it would behave identically to a Cassandra machine in, say, 95% of the circumstances it would encounter.

    "After discussing FMs and related issues, you come to the topic of the contest, and you claim that FMs can be beneficial for `steering our future`. But: When they do not show anything, they can`t be useful. And when they show something, we can be sure that what the viewer sees will definitely happen (since the foreseen future is inviolable). Then, why bother acting? Can`t we comfortably sit and listen to the Beatles vinyl records, while waiting for the foreseen future to happen, whether good or bad?"

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Wouldn't the manner in which we would respond to viewer foreknowledge also be available in viewer foreknowledge? Yes, it is true that we could be sure of every future we might see in confirmed viewer foreknowledge. However, what you are proposing is a situation wherein a society would attempt to contradict a given instance of viewer foreknowledge--but engaging in such an attempt would have led to an interference viewing scenario in the first place, so that society would not have had viewer foreknowledge to contradict. Upon receiving viewer foreknowledge, we could not all just sit back and listen to "Let It Be," if that would contradict what we have seen--such a combination would be physically impossible within the context of the very concept of FMs.

    Among other things, the extra two pages I wish I could have included in my FQXi article would have explained this sort of thing. I'm glad you asked this particular question, Tommaso, as I am very happy for the opportunity to provide clarification on this point.

    (Continued in next post)

    • [deleted]

    • Post #83

    (This post continues the misplaced post below, so that post should be read first.)

    "At p. 7, at the bottom of the second paragraph of Section `Removing the Element of Surprise` you seem to provide a sort of proof that the usage of FMs would tend to eliminate undesirable outcomes. I feel that the crucial passage of the argument is: `However, truly undesirable outcomes that are feasible for a civilization to avoid would strongly invite a VCO, but in such cases there could be no explanation as to why a VCO does not occur.` I could not grasp the logic here. In particular, why would that strongly invite a VCO? Also, words such as `strongly` and `overwhelmingly` suggest a probabilistic argument, while the general context would seem more that of a definite, yes/no type of argument. In any case, given the limited space you had, I could not find clear and convincing argumentation for either."

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    I again commend you, Tommaso, for another excellent set of observations. The argument is essentially intended to be probabilistic, but insofar as it is, it is a probabilistic argument of the strongest variety. In looking over what I wrote, I see that I might have changed one thing to clarify this. The corresponding sentence in my paper could have instead read:

    "This amazing effect would emerge because any outcome that a civilization with foreknowledge machines would wish to avoid, which would also be feasible for that civilization to avoid, [would overwhelmingly tend] ...not [to] emerge as viewer foreknowledge in the first place."

    To see that the meaning of what I wrote and the meaning of the above sentence are nearly indistinguishable in the context of the extreme probabilities involved, let's look at an analogy. Remember the steamroller scene from Austin Powers? (Youtube it before reading further, if you don't know it.) The guy clearly doesn't want to be flattened by the steamroller, as he is yelling "noooo!" Yet, even though he has so much time to avoid the steamroller, he ends up flat.

    That scene provides a good analogy because the man is faced with an outcome that he wishes to avoid, which is also feasible for him to avoid. Now, could what happened to him in the movie happen to anyone in real life? Yes, such an event is logically possible. Accordingly, at most, one can only make a probabilistic argument to conclude that it would overwhelmingly tend not to happen. However, most people would say that such a thing simply could not happen, and leave it at that--after all, the fact that it is so unrealistic and so absurd as to be essentially impossible, explains why it is so funny.

    Now, with that being said, "essentially impossible" and "impossible" are two different things. You were very right to characterize the argument I seemed to be presenting as a probabilistic argument. As such, the words, "could not emerge as viewer foreknowledge in the first place," are incongruent, since these words signal a deductive argument.

    However, there is another way to look at this. A civilization cannot see an outcome in viewer foreknowledge that they will ultimately wish to avoid, and which will be feasible for them to avoid, because the combination of those two things would mean that they would act to avoid such an outcome--but, if they did so act and did avoid the outcome (because it had indeed been feasible for them to do so), this would mean that they must not have received viewer foreknowledge (of the outcome they avoided) in the first place. So, in this context, the statement "any outcome that a civilization with foreknowledge machines would wish to avoid, which would also be feasible for that civilization to avoid, could not emerge as viewer foreknowledge in the first place," is actually true, due to the meaning of the italicized part. I now remember that that is why I included this wording, rather than something like the wording of the above modified sentence (which I also considered).

    Your skills of observation are very keen to identify an issue here. There is indeed an issue here, and I will endeavor very much to sort all of this out.

    After providing all of this clarification, I do not know how much of your question remains, so I will wait until you absorb my two posts and ask further questions before continuing.

    Thank you very much, Tommaso, for writing to me to seek clarification. I look forward to another interaction with you when you can find the time, and I will read and rate your paper, as well as comment about it on your page soon.

    Warmly,

    Aaron

    PS - I'll be rating your essay Aaron (and all the others on my review list) some time between now and May 30. I still hope you'll be able to review mine. All the best, and bye for now, - Mike

    Hi George,

    I have been catching up on all the many questions and essays recently, after completing my whirlwind of a semester. Thank you for your excellent two-part question, I will get right to it:

    "[W]ould a truly rational human, having in hand one of your foreknowledge machines, be inclined to give up the human struggle to make his or her own choices? Would they still be human?"

    (I will use the female pronoun.) It would be very unlikely, if not impossible, for a person to be able to see (or eventually be told, by a third party) detailed aspects of her own personal future, for either of these things would cause the future-viewing machine to run into an interference viewing scenario in the first place. However, in general, the future of a civilization's course would not fall into that restricted category.

    So, getting back to your question, in effect, a human being could not see or know her own future choices, but the larger results of everyone's contributions could be known to all. Now, to bring about the larger results that have been seen, a foreknowledge machine operator could see a relevant person's future choices and delegate someone to interact with her in some subtle way, hinting her toward her contribution to the viewed result.

    As you can see, every human being would have to make all of his or her own choices. The differences in the lifestyle of a future-sighted civilization (as opposed to our presently future-blind civilization) consist in the fact that everyone would know the combined end result of everyone's actions, and, most importantly, that that end result would automatically be beneficial to a significant majority of actors (unless avoiding it would literally not have been achievable within the timeframe). This is because, otherwise, the conditions of a VCO would emerge. However, viewer foreknowledge of a given future means that a VCO will not, and so could not, emerge. (For if it did, what had been received could not have been viewer foreknowledge in the first place.) Therefore, viewer foreknowledge automatically leads to an amplification of beneficial outcomes for an entire "future-sighted" community (which in this case is assumed to be a future version of our whole civilization).

    So, to answer the first part of your question, everyone would still make all their own choices, because no individual could know exactly what her own choices will be. Of course, this automatically answers the second part of your question too.

    Wow, George, thanks for your insightful question. I has created an amazing opportunity for everyone to gain greater clarity. I will be reading your article very soon, and then I will post to your page.

    Warm best wishes,

    Aaron

    Aaron,

    Since time grows short, I revisited those reviewed and found that I rated you on May 12th.

    JIm

    Aaron,

    Thanks for the post and the comment. Will contact.

    Jim

    Thanks, Aaron. Excellent clarification in your reply. Now, here's another question - your description of foreknowledge machine and the influence on individual human behaviors seems to echo in a remarkable way the laws of Divine Providence as postulated by Emanuel Swedenborg, here paraphrased: All humans have free will; God wants all humans to achieve their maximum potential for good (in spite of their inherent tendencies to be selfish / evil); At every moment in a person's life, God is providing learning opportunities and feedback to each human seeking to help him/her achieve that maximum potential; Of necessity, God's divine providence must be invisible to the human awareness or they would then perceive their actions or choices as compulsions or rebel against the very thing God wants to achieve.

    How is the foreknowledge machine different?

    Many thanks - George

    All the most useful clarifications, gathered into one place (Part I):

    A response to a comment made by Eckard Blumschein, on his page:

    Thanks for explaining why you were bored with my essay. However, my essay is not about prediction or forecasts in the slightest. The machines I am referring to are not some kind of simulation generators. No data would need to be added, because they do not compute anything.

    Instead, they literally look into the future to see it for what it will be. Now, obviously, we have not invented such machines yet. However, in order for something to be recognized as feasible, it must first be recognized as logically possible. What I have done is to analyze the logic of a future-viewing machine, to isolate the kind of future-viewing machine that is logically possible. The initial step in this process is proving that a naive kind of future-viewing machine is not logically possible. I recommend that you take a look at a recent conversation I had with Robert de Neufville on my page.

    By the way, I agree wholeheartedly with you that an article about some kind of a predictive simulation machine would be tremendously boring.

    _____________________________

    Comment from Robert de Neufville, on this page:

    "I agree that better predictive technology would be valuable. But machines that are occasionally wrong are still extremely useful if their error rates are low. We don't necessarily need theoretically perfect foresight."

    You may know this, but I have to respond in the following way every time someone describes foreknowledge machines as a predictive technology. Foreknowledge machines don't predict anything. One could base predictions on viewer foreknowledge and they would always be right (that is why they would be "predictively useful," to distinguish them from Everett machines which could not help us know anything about which future will come to pass), but foreknowledge machines themselves aren't involved in prediction at all.

    I am now fully convinced that I need to emphasize this distinction much more clearly, since five or six other people have also arrived at the same (understandable) misconception (which I did not sufficiently guard against, as the possibility of confusion here had not occurred to me).

    {Note: In all future work, including the re-writing of my current offerings on this topic, I will not use 'predictively useful.' It's use was clearly a mistake, as I see that it is unnecessarily confusing. Instead, I will only use 'outcome-informative.'}

    Now that that is out of the way, I think you may have been expressing that a future-viewing machine which sometimes delivers a view of the future that is wrong would still be useful. However, when would we trust it and when would we assume that it could be wrong? A predictive technology that is sometimes wrong is fine--that's the nature of prediction--but a viewing technology cannot sometimes be wrong, and be useful. That would be like a telescope which sometimes shows Blackbeard and Captain Ahab walking on the deck of a distant ship, when only Blackbeard is on board. See what I mean?

    All the most useful clarifications, gathered into one place (Part II):

    Question submitted by George Gantz, on this page:

    "[W]ould a truly rational human, having in hand one of your foreknowledge machines, be inclined to give up the human struggle to make his or her own choices? Would they still be human?"

    (I will use the female pronoun.) It would be very unlikely, if not impossible, for a person to be able to see (or eventually be told, by a third party) detailed aspects of her own personal future, for either of these things would cause the future-viewing machine to run into an interference viewing scenario in the first place. However, in general, the future of a civilization's course would not fall into that restricted category.

    So, getting back to your question, in effect, a human being could not see or know her own future choices, but the larger results of everyone's contributions could be known to all. Now, to bring about the larger results that have been seen, a foreknowledge machine operator could see a relevant person's future choices and delegate someone to interact with her in some subtle way, hinting her toward her contribution to the viewed result.

    As you can see, every human being would have to make all of his or her own choices. The differences in the lifestyle of a future-sighted civilization (as opposed to our presently future-blind civilization) consist in the fact that everyone would know the combined end result of everyone's actions, and, most importantly, that that end result would automatically be beneficial to a significant majority of actors (unless avoiding it would literally not have been achievable within the timeframe). This is because, otherwise, the conditions of a VCO would emerge. However, viewer foreknowledge of a given future means that a VCO will not, and so could not, emerge. (For if it did, what had been received could not have been viewer foreknowledge in the first place.) Therefore, viewer foreknowledge automatically leads to an amplification of beneficial outcomes for an entire "future-sighted" community (which in this case is assumed to be a future version of our whole civilization).

    So, to answer the first part of your question, everyone would still make all their own choices, because no individual could know exactly what her own choices will be. Of course, this automatically answers the second part of your question too.

    ______________________________

    In response to Kevin O'Malley, on this page:

    Now, as far as the practicality of my idea is concerned, I wonder how many people would consider the idea of cell phones to be practical in 1879? Only approximately 110 years later, cell phones were starting to become a widespread phoneomenon (yes, that is a real typo, but it was so funny I decided to leave it), and look at where they are today. Practicality is not a good measure of the usefulness of an idea if one allows themselves to think in terms of decades or centuries of progressive surprising developments.

    I have to give a reply I've given many times before. Foreknowledge machines do not forecast or predict anything. They see the future as it will happen, or if they encounter an interference viewing scenario they give only vague information or fail to operate. So, foreknowledge machines cannot be described as prediction machines to any extent, even though a person could be wildly successful by using one claim that they have made predictions. However, a person who uses a foreknowledge machine and then claims to make predictions about what he has seen would be lying: They have not predicted anything, because the foreknowledge machine has not predicted anything. When a person receives viewer foreknowledge and knows that it is viewer foreknowledge within a complete theory of foreknowledge machines and sufficient experience with the machines themselves, they would know they have seen the future for exactly what it will be.

    Here's a parallel: Can you claim to predict something you have just seen in a telescope? That would be an absurd word to use. You can predict that your friend with good eyesight will also see a distant fixed object when she looks in a telescope locked to a tripod, but once having seen it, to express that you "predict" the thing itself is wrong.

    ______________________________

    From Tommaso Bolognesi, on this page:

    "...when... [foreknowledge machines] show something, we can be sure that what the viewer sees will definitely happen...[,] since... [viewer foreknowledge is inviolable]. Then, why bother acting? Can`t we comfortably sit and listen to the Beatles vinyl records, while waiting for the foreseen future to happen, whether good or bad?"

    Wouldn't the manner in which we would respond to viewer foreknowledge also be available in viewer foreknowledge? Yes, it is true that we could be sure of every future we might see in confirmed viewer foreknowledge. However, what you are proposing is a situation wherein a society would attempt to contradict a given instance of viewer foreknowledge--but engaging in such an attempt would have led to an interference viewing scenario in the first place, so that society would not have had viewer foreknowledge to contradict. Upon receiving viewer foreknowledge, we could not all just sit back and listen to "Let It Be," if that would contradict what we have seen--such a combination would be physically impossible within the context of the very concept of... [foreknowledge machines].

    Dear Mr. Aaron M. Feeney

    Irrespective of existence of CTC, the physical analysis of such things is very useful for development of physics.

    But you forget

    1. references of Hadley , which tries to connect quantum mechanics and general relativity with the principle of CTCs.

    2. One form of double slit experiment exists, where only after take off of electrons it is decided, if both slits will be opened or only one. Unfortunately, I do not remember details of this thought experiment and link for it. But this experiment is connected with CTCs.

    3. It is also an option that Libet experiment, in psychology, is connected with CTCs or with foreknowledge. This is also a link which would give total survey of possible existence of CTCs of your paper.

    In this contests, Luca Zimmermann gives essay about nature of time. Maybe this one is also important for you.

    Physics of time is also connected with consciousness, with QM and with special relativity. In this paper, I try to prove that time does not flow if matter does not exist.

    It would be useful to analyse consequences of time-machines, thus if it is possible that such device could cause some contradictions, for instance, that some thing arise without that any one do anything.

    My name Janko is west Slavic name, in English it means John.

    About the evaluations I also think, that nobody deserve 1, so this mark is unfair. I proposed my system for evaluation . But, FQXi is maybe the only way that someone read our essay so this is also one benefit of this. One the newest idea is that everyone gives also the quiz of its essay. If someone does no pass this quiz, he cannot evaluate it.

    My essay

    Best regards

    Janko Kokosar

    dear Aaron,

    You promised me a reaction on my post of may 19.

    and also a substantive remark (and maybe rating) of my essay (topic 1991)

    best regards

    Wilhelmus

      Hi Wilhelmus,

      Thanks for the reminder. Sorry for the delay. Yours will be the first one I do today. (Over these next four days, I will complete all the others too.)

      Warmly,

      Aaron

      Hi Aaron,

      I want to first extend my gratitude for your thorough response. It's beyond doubt that you put substantial thought into your explanation, and I'm genuinely appreciative to have encountered some novel concepts (which I am admittedly still trying to parse, although I realize that others may have, in their comments, posed similar questions as I have, and you may have already responded to those). The concept of an interference-free foreknowledge machine (a term which I think is well-worded) is something of a confusing notion to me, but one that I'm certainly willing to consider. I wonder how it'd be possible for such a machine to deliver information from a future to a past (subjectively, our present) without interfering with the progress of time, thereby creating a different set of future information. I apologize if my question seems rudimentary - I realize that there may be something I overlooked, but I sincerely appreciate the consideration you put into your essay, and I will definitely be doing some more thinking about it in the future. (Then again, perhaps my future self already has, and all I have to do is try to get in touch with him for his insights)

      =)

      Wishing you the best regards in the competition and in all else.

        Hi Alex,

        Your confusion, as you describe it, is well-founded. Within the definitions given in my writings, there is no such conceptual thing as an, "interference-free foreknowledge machine." That combination of words is contradictory. It is absolutely essential to the concept of foreknowledge machines that they must encounter interference in certain situations, and that they can only deliver viewer foreknowledge if they do not encounter interference. Acting to bring about a given future that has been seen, because it is desirable, is the most important example of a case wherein there would be no interference. For instance, whenever a foreknowledge machine operator who works for an airline would see that a given flight will land safely, she would approve it for takeoff.

        I do discuss the naive conception of interference-free future-viewing machines (future-viewing machines, not foreknowledge machines), and I term them Cassandra machines. However, it is easy to show that such machines are logically impossible.

        I believe these comments will clear everything up for you. Thank you for visiting. Please feel free to post back with your thoughts, or any additional questions.

        Warmly,

        Aaron