Is the Past Infinite?

It is difficult for me to address time as an isolated subject. I must first outline a skeleton of my own cosmological model. The cosmic foam of our universe is the ether foam of a super-universe, and the ether foam of our universe is the cosmic foam of a sub-universe. The median size bubble in our cosmic foam is roughly 10^24 meter across; the median size bubble in our ether foam is roughly 10^-35 m across. Space can be measured in ether-foam bubbles; a cubic meter is roughly 10^105 bubbles, and that number remains constant as our space expands. The expansion of our space stretches the bubble walls of our cosmic foam, causing them to pop. The same thing occurs in the sub-universe, causing its cosmic foam bubble walls to pop.

Space expands by increasing the number of ether-foam bubbles. When a wall separating two bubbles pops, two bubbles become one. That is a reduction in the number of bubbles. For the number of bubbles to increase, bubble walls must un-pop; a new wall must form across the middle of a bubble, dividing it in two. So when a cosmic-foam bubble in the sub-universe pops, one of our ether-foam bubbles must un-pop. In other words, the arrow of time reverses from one universe to the next. The expansion of space aims the arrow of time.

Turning the clock backward, we can imagine our space shrinking. This reduces the scale factor between the cosmic foam and the ether foam. Today, the factor is about 10^59 to 1. Suppose we run the clock back to a time when the scale factor was unity; we reach a time when our cosmic foam existed at the same scale as our ether foam. That could be adopted as a convenient marker for the beginning of time as we know it. Before that, the roles of cosmic foam and ether foam are reversed; the super-universe and sub-universe swap roles.

I am not saying that we CAN turn the clock back that far with any degree of confidence in what might have existed back then. If seems likely that our cosmic foam may have undergone numerous phase shifts since the beginning of time as we know it. Our cosmic foam, today, consists of bubbles surrounded by walls of galaxies. So what was it before galaxies came into existence? Did it have a foamy texture even then? I doubt if we can ever know that. Nevertheless, it does make sense to me to postulate a beginning of time as we know it, and that sets an arbitrary but finite limit to the age of our universe.

Our universe is an insignificant subset of an infinitely greater fractal universe. That greater universe exists outside of time because time runs both forward and backward within it. Time exists within the greater universe, but the greater universe has no beginning or end.

That previous thread is mine. I thought I was still logged in.

past exist only as a numerical order.............Attachment #1: Time_measured_with_Clocks_12.10..pdf

If there was no friction pendulum will never reach an equilibrium and it will continue moving for ever and ever.("continuing into the future" does not seem to be my words)

As far as entropy concerns: it increases as slice ("conceivable universe") is moving from poles to the equator and decreases as slice is moving from the equator to poles. So (b) and may (a) are "feasible" possibilities.

Passage of time is human perception. It is also how we order and make sense of numerous, many dimensional spatial changes that are occurring continuously. The experience comes from within the organism and is in part due to circadian rhythm controlled by the pituitary gland in response to light levels and in part interpretation of the sequence of spatial change experienced by that organism. Ageing is cumulative detrimental changes in the spatial configuration of the material components of the body including damage to the mitochondria the "energy production" units of the cells.

The tick of a clock gives a scale within 3D space that can be used to measure other spatial change. It is a man made construction that works together with human perception to give time a -seemingly- non spatial existence in the world. With a clock we can now see time passing second by second rather than having to rely on the astronomical changes, change in the earth relative to the sun, position of stars or moon in the sky etc. This takes us away from what time is to the universe.

There are changes with in 3D space that we can observe and measure. Distance A to B for example. Taking the 3 dimensions to be the only spatial dimensions that exist A to B is a fixed and unchanging. The position of A and the position of B do not change. The only other change that is occurring according to this viewpoint is change in time. However if distance A to B is measured on the earth's surface, then A to B is changing spatial position because of the earth's rotation about its axis, the orbit of the earth around the sun, procession of the solar system with in the galaxy and so on to bigger scales of change of position with in the material object universe. The cumulative effect of change at many scales, mostly circular motion is a change in -spatial- position. The object has not just moved A to B it has moved A to B' at the new universal position. When the object arrives at B' it is not in the same spatial position as B was, when the object left A. Some say prove it. Well do the celestial bodies of the universe move relative to each other or are they static? If it is assumed that they are moving as our observations show, then the passage of time for an object is a change in its absolute spatial position in the (object) universe. Nothing moves A to B even if we see and measure it doing so. It is only possible for that to be so if the whole universe is static. The 4th dimensional change is a universal spatial change that is generally unobserved and so is unaccounted for, except as passage of time. That is to say relative local spatial change within a particular frame of reference(3D vector) against overall universal change, encompassing changes over many many scales, planetary , solar system , galactic, galaxy cluster, object universal (scalar spatial change).

Then there is the question of where this dimension of spatial change can possibly exist if all of space is defined by the 3 spatial dimensions.There is only one place for it to be. Running from the exterior to the interior of every mass. Movement along this "scalar" spatial dimension is then able to account for gravity. The object does not move into itself with in a single 3D space but from one 3D space to the next in that orientation. The 4th dimension in the electromagnetic equations of maxwell become the same scalar spatial orientation.

It is the ordering of the change that brings in time. However only that which exists in atemporal now actually materially exists. If one considers that which did exist but no longer exists one has invented the past. Which has no physical existence in atemporal now only in the mind or in the calculation or as an image. If one predicts what will exist then one has created a hypothetical future. Which has no existence in atemporal now only in the mind or the calculation. The experienced present is a patchwork of time delayed data unique to each observer position.It is not the same as the existential reality of the material universe at atemporal now.Past, present and future belong to the image universe. Image and object are not the same thing!!!

Along the 4th dimension of the material object universe there is absolute space that has been passed through, is now occupied, and which will be passed through. Of course the spatial position an object occupied and other objects occupied prior to current atemporal now effects its current change. It is not existential past effecting the present but former objective now (the spatial arrangement and relative positions of all components of the material universe) universe having already influenced current objective now. Progression is from one universal configuration to the next, atemporal now to atemporal now. Atemporal now may seem to be in the future as it exists ahead of delayed transmission and interpretation of any data that can be formed into perception of that which exists.Can things exist before we are aware of them? Certainly. Research has shown that the fear response is detectable in humans before conscious awareness of any threatening stimulus. Immediate preparation of the organism for rapid response gives a survival advantage. The material organism can thus react before the conscious observer function of the organism has awareness of another material configuration presenting a threat. The conscious observer is thus always observing the past, a patchwork image of former material reality.

There is not a material copy of that which has already happened and the future is not already written and in material existence. There is no going back to the past, only back to space that has been passed through. There is no traveling to the future, only to space that has not yet been passed through. Aft space and afore space.It would require billions of universes worth of matter and energy for all of those temporal copy universes to exist. They are fantasy realms imo nothing more. I do not assume my reflection in the mirror to be a real independently existing person, even though I see it clearly.It is an image, a trick of the light.

The image of past events that are seen are just that- images created from the delayed transmission of photons. The material object universe is not the same as the image universe. The material object universe can be described by 4 dimensions, 3 vector spatial one scalar spatial. The image universe can be described by 4 dimensions. 3 vector spatial and one time. The image universe THE UNIVERSE is an image!!! The material object universe that we can not observe is the material object. How can it be said to exist if it can not be observed? There has to be an object to create an image. We could not see the image that we see now if there had not been an object universe. We could not see the 8 minute delayed image of the sun if there was not an object sun existing in atemporal Now. Einstein's work has to do with the -appearance of things- to an observer. Newtons work is to do with the existential matter of the universe. Perhaps too much importance has been given to the appearance of things and erroneous interpretations of the meaning of observation have been made.

At any distance at all from the observer there is time delay in information transfer and the image is non contemporaneous with the observer. The greater the distance the greater the discrepancy. We therefore operate within the image our minds create from a patchwork of time distorted information. That is not the material object reality. That is important.

Considering an eternal material universe one must also consider entropy. By chance, under the influence of forces and organic organization particles come to positions that are more stable than a change in that position and so structure and organization occur. However though matter can come together into structures these too can be broken down. Bodies decay, rocks erode, stars explode.On a universal scale entropy and organization are both ongoing processes.A dynamic equilibrium.I consider it more likely that there is continuous recycling of the universe rather than cataclysmic creation and destruction events of the entire universe simultaneously. The big bang scenario has arisen from the assumption of an expanding universe which is in turn based upon interpretation of red shift data, which may be incorrect. As it takes into account only 3 spatial dimensions of change.

Like a river deposits silt and erodes silt, carving out meanders. The sea deposits sand to build beaches, but also erodes beaches and cliffs.The universe builds stars, planets and lifeforms but also disintegrates stars, planets and lifeforms. Energy is stored in structures and energy is released. Matter is stored in structures and is released to build new structures. Energy is never destroyed. It is eternal if one thinks in terms of time.It is merely atemporal comprising only ongoing spatial change as in material reality everything exists only in space, at an ever changing spatial position. The time dimension is the dimension of the fantasy realms, past , present and future created by light images. The scalar spatial dimension is the dimension of absolute change of the material object universe. Do physical processes occur within the image of the universe or within the material object universe? I say within the object universe.

I don't think it is possible to get something from nothing although some very clever mathematics has been done to show just that. I think it involves lots of assumptions about initial conditions and so allowing rule breaking.

It is far more plausible to me to assume that there was no absolute beginning. We see other objects around us have beginnings and endings,because of the occurrence of spatial changes in the arrangement of the matter comprising those objects.. from life forms, to rocks, buildings, volcanoes even planets and stars. So it is natural to assume that the universe must also have a beginning and end. This is not necessarily so.

A classic car left out unattended in the elements will rust and cease to function and eventually little will be left. However if it is protected and continuously maintained and restored it remains pristine. One car may be dismantled to provide parts for another. One beach may be eroded another formed and built up from the same sand.

If the universe through continuous change builds up and destroys structures recycling energy and matter then this process can continue without end. Far more like the simultaneous erosion and deposition processes we observe on the earth. Rather than the imagined one way street of entropy winding down the energy of the universe into a cold dead future. There is as much reason for it to be in a state of continuous eternal change giving a balance of destruction and organization as there is for it to be absolutely static and dead or non existent.

We see other objects come from somewhere or being made from other things and apply that to the universe, saying where did it come from? or what was it made from? Even who made it? This reasoning does not necessarily apply to the material universe. There may be intermittent local vastly destructive events that lead on to a new creation process whereby dust accumulates into new planets and stars etc but this need not encompass the entire material object universe simultaneously.

Hi William Orem,

I read the abstract of your linked -work-. Would have read more if easily accessible and free. The singularity that provides the whole universe from nothing is such a problematic idea for me that a sensible theory that avoids such a structure seems worthwhile and interesting. My main problem with your outline is that you allow local time reversal. What exactly does this mean to you? Is it just spatial processes running in the opposite direction or is it an exact reversed replay of events?

Many processes can run in either direction. Sand can be deposited sand can be eroded. Groups of organisms come together, groups of organisms disperse. Chemicals can combine and chemicals can be broken apart.This all occurs within what we regard as constant direction of the arrow of time. The processes may change direction but the hands of the clock continue to move in the same direction as does the progression of the sun in the sky.

What constitutes time in those local regions where time reversal occurs?Is it what is happening or is it what it is compared to? What determines time there, the local clock, the outside reference frame clock of an outside observer?, the direction of absolute movement?

If time is reversed rather than just overall processes, gravity also runs backwards and becomes anti gravity as do all forces, pushes pull, pulls push. Atomic spins are reversed, magnetic poles that attracted now repel. How is the reversal of all physics accomplished? Is it just wild speculation or is there some compelling reason to make that assumption?. As you can tell I am very skeptical about this possibility. As an "armchair philosopher" I see no reason to assume that it happens. Though I would be very interested to hear your explanation of why it is good and works.

I think that all change in a material object universe is spatial and spatial change can take place in either direction, without effecting time. Time is an artificial measurement of change and allows perception of the the sequence of change, either using celestial changes or a clock of some kind. The sequence of spatial positions can repeat itself without time being repeated or be reversed without time reversal, such as the swing of a pendulum or any oscillation. At each oscillation the object has a new absolute position in the universe although occupying the same local space reference frame. That is it is within the bounds of an certain 3d space under consideration but due to the movements of the celestial bodies over many scales the cumulative effect is a change of position unaccounted for except as time. Only by reversing the movements of all of the celestial bodies and reversing the effects of all physical processes can time run backwards. It is nearly as preposterous as a singularity. Rule breaking.

I have not read the entirety of that work and therefore may have completely misunderstood. Please forgive me if that is the case.I expect that it is constructed to fit with space-time relativity etc. However that space-time model is based upon time delay ( perhaps I should say spatial sequence progression delay) distortion of the image formed from photon data. It tells us what will be observed (image ) not what is ( material reality). The space-time universe thus formed is not the material universe that exists in purely spatial simultaneous now,(not the present).

Your thoughts, explanations of reasoning etc would be much appreciated.

William Orem,

Oops- just noticed that it was Anthony Aguirre, Steven Gratton's work that was being refereed to you for comment William not your own work. I would still be interested to know your own opinions on the question of actual time reversal outside of mathematical acceptability. What local time reversal actually means to you etc?

Hi Anthony aguirre,

So sorry for that mix up. I would like to address the previous post and the questions raised to you, as you put up the link to your work. Also have you moved on from this ( link in 2007) or is it still a reflection of your current thinking/ research ?

Thinking some more about magnetism it doesn't properly reverse with time reversal. An object that was repelled will be attracted towards but will not stick to the now not quite attracting pole.

Time forwards. Imagining a magnet being dropped onto a repulsing pole of a stronger magnet. It would fall and then be repelled by the magnet. It might be displaced laterally away from the repulsing pole or turn so that attracting poles come together.During the repulsing phase the poles of the magnet do not touch in this imagined case.

Time is reversed. The smaller magnet is now repulsed by the formerly attracting pole if it had turned turned, so turning again or moves towards the formerly repulsing pole from its laterally displaced position. However it is not attracted so sticking to that pole. There is still a barrier to the poles coming together. Instead of the expected complete magnetic attraction it is overcome by anti-gravity and rises into the air. Isn't this a problem because this physics of magnetism doesn't seem to run properly backwards.Is this film running backwards type of time reversal the wrong way to be thinking about it?

There is also the problem of compounds and mixtures and structures that are more energetically stable in their current fixed form than as the ingredients or parts that they were made from. When time is reversed the value of energy would also seem to be reversed so now it is more energetically favorable for formerly stable objects to be undone back to ingredients and parts. Cold objects warm up spontaneously etc.

When time reversal is mentioned by physicists are they referring only to reversal of certain physical processes or all processes, physics, chemistry and biology. If it is reversal of only certain processes then it isn't time reversal its just a reverse reaction or process in forwards time. I understand that time reversal is mathematically allowable but am I failing to comprehend something that makes time reversal plausible and scientifically allowable? Are all physics rules and processes fully reversible? No it would seem.

Does anybody remember the lessons learned from Olber's Paradox? The universe cannot be infinite, and Olber's Paradox explains why. Check it out.

Perhaps the reality of the Paradox is yet another reason behind some scientists inventing parallel universes (actually, somewhat of an oxymoron; the universe is the system of All interacting things; therefore it is and can only be one), since a finite universe points to the idea of a creator whatever that may be...Heaven forbid.

Mike Archer

Hi, Philip:

I don't see how the Wikipedia article you mention resolves the paradox in a way that permits the universe to be infinite, but it does mention at the get-go that:

"In order to explain Olbers' paradox, it is necessary to account for the relatively low brightness of the night sky in relation to the circle of our sun. The universe is only finitely old, and stars have existed only for part of that time."

"The universe is only finitely old,"...

Did you miss this?

Actually, it looks like only one of the possible alternate explanations does not require a finite universe, but even such notables as Paul Davies has clearly set forth the resolution of Olber's Paradox for the observable universe that must be finite, even though he is also "seeking" other "universes" as well.

If there is but one universe as all sane people understand as the system of all interacting things (Since it is all, one cannot add anything to to it. Since you like math, what is the problem with "all plus one?" You'll need some logic as well.), the resolution of Olber's Paradox demonstrates that the universe is indeed finite.

And so to get around this, people start proposing such things as "degrees of reality." Now, who decides how many degrees are involved at any given time? By the bye, who determined there are only degrees? How did he or she know this? Why,...I'll bet it's just another theory without proof. Friday the 13th indeed.

As for double slit experiments, and let's add some aspects of quantum mechanics as well, I have recently joined, so to speak, the camp of Roger Penrose who recognizes the unwarranted philosophy grafted onto the imprecision of such mathematics and experiments that "appear" to demonstrate certain things that, in any event, never seem to impact the classical world in any appreciable way. Why is that? Just imagination? We can't figure it out yet, but we know it's there, kinda like the claim that one cannot define pornography, but he knows it when he sees it. Of course, we can't even see many things claimed to be there by some physicists, and these people laugh at other people who claim belief in an invisible divine being.

Isn't that interesting? If one cannot prove the existence of a divine being, but still believes in one because of observations of the ways the world works, and perhaps some sophisticated philosophical reasoning etc., that person is said to be supremely ignorant, no matter how rational he may be.

However, if one cannot prove the existence of a mathematical universe or hypothetical particle and on and on, but still believes in such things because they fit into highly speculative formulae, why that person is said to be brilliant. And in greater support of the fellow's brilliance, we must marvel at the claim that such things can never be observed. Where are your clothes, Emperor?

Ah, yes. The residents of many glass houses don't even recognize their own faith which is less rational than the faith in a divine being may be. To be sure, a finite universe points to a divine being, so for the atheist/marginal agnostic, this must be opposed by any means no matter how irrational they may be. In a roundabout way, such wasted would-be science is so silly that it makes the philosopher's God appear much more reasonable. An unintended consequence of being so radically opposed to a Divine Being (not to be confused with a fractal universe), another Tower of Babel is being constructed by various physicists who speak in "mathematical" tongues that fewer and fewer can understand because at bottom it is more gibberish.

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Now, what good is inferring any supposed reality like other universes that cannot be observed or tested? Can you appreciate the honesty of someone like Lee Smolin who laments the fact that more and more mathematical-type theories are being proposed and asked to be accepted even though they can never be scientifically tested? Indeed, no longer science, but mere fantasy and ideology wrapped in some mathematical garb.

The arrow of time? Talk about a non-existent problem. Go back to ancient philosophers for the best understanding of time as the measure of motion. Hone in on this concept: The measure of motion. Stay focused on this and do not attempt to graft anything else onto it. This takes away all problems and fantasies regarding going backwards or sidewise. And if you possess a most sophisticated and rational understanding of relativity, you will realize that it is a most absolutist theory of reality despite the misnomer that allows people to falsely rationalize. But here's the key: the event is real no matter who or what perceives it and when. The event is real and absolute.

Enough for now, and take heart if the above is a bit too much common sense for you to handle. Why,... in an alternate universe, you are quite a rational fellow,...only you'll never be able to interact with him, so you may be doomed to be locked in your foam forever. Ha, Ha!

Good luck!

Mike Archer

Would there even be a paradox if Mr. Olbers and those before him had the ability to see in microwaves? The paradox states roughly: "Why is the night sky dark? The answer is that it isn't. No matter where you look, you see a very uniform distribution of microwaves. I realize the current accepted explanation for the CMB is from the SLS from the Big Bang, but what prevents it from being evidence of a eternal past with an infinite number of nearly uniform sources redshifted into the microwave spectrum via accelerated expansion?

Greetings, UFO Pilot:

Nice flight above California recently.

Now, instead of talking about irreleveant microwaves, try to think about the resolution of Olber's Paradox by the realization that the universe cannot be infinite in time since thermal dynamic equilibrium would have already been reached and we'd either be on fire or in ice. Olber's Paradox remains a problem if the universe is accepted as infinite in time and space.

Pleased to enlighten you to basic reality.

Mike Archer

"Cosmic foam bubbles? Can they be measured, observed, investigated empirically in any way? What is the foam and where does it come from?"

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The Sloan Digital Sky Survey has mapped over half a million galaxies in 3D. Of course, there are more gaps than mapped regions. Our view is blocked by the disk of our own galaxy, and we can't see what lies beyond other galaxies. The task of mapping all the visible galaxies may take decades.

I haven't actually seen the map in stereo vision, but those who have describe it as a giant bubble bath with walls of galaxies surrounding the voids. A void is the interior of a bubble. They believe dark matter completes the fabric of the bubble walls. This is what I am calling the cosmic foam.

We are just beginning to get statistical measurements, and it appears that the median bubble size in the cosmic foam is roughly 10^24 meters across. That being the case, a median-size bubble occupies roughly one millionth of the volume of the visible universe. (I am saying that, by definition, half the volume of a region is contained in bubbles larger than the median size.)

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"This looks like another one of those "math creating reality" claims instead of math reflecting testable reality."

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I am no mathematician. Numbers are a weakness of my model---not a strength. I use very rough approximations to provide a starting point from which others may someday refine the model. There is no insubstantial mathematical space in my model. Everything consists of waves in a hard, massive medium; and that medium consists of particles which are made of waves in a finer medium, and so on ad infinitum. There are no finite empty spaces to be bridged by insubstantial mathematical forces.

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"How do you know the sub-universe is running backwards if all of your tests run forward, which they must?"

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We cannot subject the sub-universe to tests; if we could it would be part of our universe. We can only infer that it must run backwards because of the way the model explains the expansion of space. If a cubic meter is 10^105 median-size bubbles, and that number is constant, then the expansion of space means the number of bubbles in a region of space must be increasing. When a foam fizzes, bubbles are popping, which decreases the number of bubbles. A bubble wall pops, and two bubbles merge into one. For the number of bubbles to increase, they must be un-popping. New bubble walls must be appearing, dividing one bubble into two. The second law of thermodynamics prevents bubbles from un-popping in forward time. (Note: I am assuming the number of bubbles per cubic meter is constant. More generally, it makes sense that the number of bubbles increases as space expands. In keeping with Occam's razor, I choose constant as the simplest explanation until such time as it leads to a contradiction.)

What makes the bubbles pop is the expansion of space in the sub-universe. The cosmic-foam bubbles of the sub-universe are stretched to their breaking point by expansion of sub-universe space. So that is a forward time cause of popping bubbles, from a sub-universe perspective. From our perspective, those same bubbles are un-popping. The cause of the popping is expansion of sub-universe space, which occurred before the popping from a sub-universe perspective. From our perspective, the cause is after the bubble un-pops. So the effect precedes the cause from our perspective.

Also, a popping bubble generates pressure waves which radiate outward. From our perspective, those pressure waves converge to a point where they seem to cause a new bubble wall to appear. The pressure waves are dark energy, and they are converted to new space.

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"Don't forget common sense just because it's not fashionable in the non-scientific quasi-physics world."

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I'm sure Einstein's heard many similar admonitions from his contemporaries. "Common sense" is a euphemism for thinking well inside the box. Paint by the numbers; don't cross the lines. If we never think outside the box, we'll be condemned to add new patches to the same old flawed concepts for ever.

Oops! That post, above belongs on a different discussion. Too bad I can't delete it.

"We cannot subject the sub-universe to tests."

Therefore, not a scientific inquiry. Just gibberish disguised in pseudo-scientific terms.

Also, I knew Albert Einstein. Albert Einstein was a friend of mine.

Philip: You are no Albert Einstein.

It appears you also think he is still alive by your present tense claim involving the great scientist who thought creatively inside the box known as the universe. Nice cliche on your part, though, regarding your faulty understanding of common sense. It supports your nonsense claims that just can't be tested or observed, yet you go on and on about certain actions causing this or that to pop, but then they reverse themselves and run backwards and effects precede causes.

Lots of stuff going on that can't be observed or tested. A superb example of mythological fantasy "pseudo-physics."

Thanks for the ride, though. I actually enjoyed reading your dazzling illusions.

KII

Hi,

Once Nature has never made anything unique in gender(There no just ONE galaxy, ONE sun, ONE planet or sand grain, but large number of them)and can just mass-produce anything; and once we humans have been wrong every time we thought we had found "the entire"; very great chances are that we are also wrong thinking our local(backyard) universe is - this time - the entire stuff.

Great chances are too that we can never discover too much beyond our backyard-universe, due to our littleness.

Personally, I'm convinced that the Cosmos is structured with infinite 'classes of dimensions'(like our) infinitely and fractaly nested.

In other words, totally infinite, as much in time as in space; despite the horror the term "infinite" cause in large amount of scientists.

Cheers,

Wilton

Yes the past would haveto be infinite or else we wouldn't have an infinite future which we must since 1. we cannot prove its finite and 2. we are not philosophically predestined for anything other than to exist in the moment

Sorry, but your speculation is of the "not even wrong" type. A physical theory, to be taken seriously, need some equations, which have to be sufficient to explain the things we observe around us, and in a quite mathematical, quantitative way.

By the way, don't use popular introductions into string theory or other theories as an example what you have to do in fundamental physics. There are professional papers behind such popular introductions, full of mathematics, and if you want to develop a serious theory, you have to present similar professional papers. That's hard, but such is life.

By the way, using UPPERCASE in such postings is considered to be bad style ("crying").