Dear Edwin,

I would like to now fulfill my promise that I made to discuss your theory further. I do not wish to duplicate points that you have discussed with others, and in light of the extensive discussions you have already had, there is really only one thing that strikes me about it.

It appears to me that much of the opposition you are running into is due to the fact that your idea presents an answer to a question that nobody is currently asking, and that, from a strategic perspective, a more effective approach for you would be to get people to first ask the relevant question, and then present your theory as a solution.

In order for someone to seriously consider your argument, they have to first be willing to question whether the Stern-Gerlach experiment really has been misinterpreted all along for the last 90 years, which is what your assertion amounts to. But that is something that is probably only slightly less controversial than the claim that a single flip of an ordinary coin could result in many outcomes other than heads or tails. If I wanted to defend your position, I would first and foremost gather as much raw data from published SG experiments and perform statistical analyses to show that interpreting its outcomes in binary terms is a mistake. I am quite frankly astonished that you have so far only listed (as far as I can tell) the raw data from a single experiment (and, as far as I can tell, without statistical analysis), and done little more than just to assert that the outcomes of the SG experiment are not binary.

As much as I am sympathetic to non-mainstream viewpoints, I don't think you can blame mainstream physicists if they don't find this convincing. The burden for gathering the evidence is on you, and for such an established result, it is very high.

So my suggestion would be, for the moment, not to focus on promoting your theory, but on planting doubts that the SG experiment has been correctly interpreted. And to do that, you will need massive evidence. Let me be clear that, personally, I am very skeptical that this can be done, but I'd like to see any novel idea in "the marketplace of ideas" have a fair shot. If you don't do this (and under the supposition that there really was a mistaken interpretation), I think you are only depriving yourself of a fair shot.

I hope you found my suggestion useful.

Best wishes,

Armin

    Ed,

    I accept that there is merit in Armin's view: It appears to him that "much of the opposition you are running into is due to the fact that your idea presents an answer to a question that nobody is currently asking."

    But I write to reinforce my own view. "Your opposition in part arises from this fact: you do not answer questions that are asked by me, a fellow local-realist and former colleague."

    And, as I see it, my view is the more serious: On 1 July 2014 (with receipt acknowledged) I sent you a copy of a paper in which I offered two local models of Bell's (1964) that your theory addresses. (It included the same for Aspect's experiment.)

    Now, from the answers I've received to date (and as you know from the brief analyses I've included in this Forum) I find my ideas hiding in your model. And thus "Bell's constraints" (your term) hide there too,* since they are explicitly included in my paper.

    So, surely: In the interests of fair play and the collaborative discussion and development of ideas here, your answers to my questions would help to clarify many issues.

    * partly hiding by means of associated factors.

    NB: Regarding the experiment that you propose, I make absolutely no claim there in that (independent of Bell) QM experiments are at the heart of all my models.

    PS: Regarding your ongoing sensitivity to releasing the high-level Mathematica that your model employs: I make no claim on that.

    With best regards; Gordon Watson: Essay Forum. Essay Only.

    Dear Edwin Eugene,

    As I told you in my Essay page, I have read your interesting Essay. I have a couple of comments:

    1) I did not know Korzybski's work, thanks for pointing out it. I will take infos about.

    2) Although I am not sure that the main claims of the Essay are correct, I find the work interesting and important within the debate determinism/randomness, classical/quantum physics. I find indeed the randomness of quantum mechanics to be neither completely satisfactory nor the final physical answer concerning our understanding of the world. More, the reading enjoyed me. Thus, I will give you an high score.

    I wish you best luck in the Contest.

    Cheers, Ch.

      Dear Eugene,

      As you have mentioned in your essay :FQXi asks why math is so 'unreasonably' effective in fundamental physics.

      The above feature extraction is based on distances obtained from these simplest math operations, and these math operations are easily constructed from physical structures ( atoms, molecules, DNA, proteins, cells, organisms, neural nets, and logic machines ) that can function as gates, implementing AND and NOT logic operations, which can be combined to count to produce integers and to add to produce distance maps and then compare distance maps to get difference maps (gradients) from measurements. The nature of the process of making math maps is thus rooted in the physical universe.

      Math maps imposed on the physical territory form the substance of physics."

      This is because mathematical and physical structures both are creations of Vibration as my Mathematical Structure Hypothesis states. As you have mentioned that Math maps the physical territory and mathematical operators structures,but even mathematical structures are mapped by some laws of invariance.The best example evident is the Riemann Zeta function structure which lies at the boundary of mathematics and physics.It can be deciphered that even certain laws of invariance maps the mathematical structure itself. So, mathematical structures are not only maps but their intrinsic structures are also mapped by certain laws of invariance.Its not mathematics explaining physics or physics explaining mathematics rather their intrinsic laws of invariance,order,symmetry match each other and thats why even Bell's locality-at-distance is also valid for mathematical structures because they are also creations of vibration like physical reality. Bell's locality-at-distance fundamentally exists because we are so addicted to the phenomenon of causation but the truth is that Time,Space & Causations are like the glass through which Absolute is seen.In the Absolute there is neither Time,Space or Causation;which I have described.

      Anyway your essay is great.

      Regards,

      Pankaj Mani

        Dear Edwin,

        Thank you for commenting on my wall and inviting me to read your essay. I just read it, and I have some questions. I apologize if they may seem naive to you.

        1. Is there a typo in your proof of the energy exchange theorem, in the box at page 4?

        2. Is theta the angle made by the particle's magnetic moment with the external magnetic field? This is what I understand from page 4. Or is it the angle between a and b, the directions in which Alice and Bob measure (this is what I understand from page 6)?

        3. Do the two particles share the same theta?

        4. When you apply the energy exchange theorem, what are the two modes M0 and M1? Are they the particle and the magnetic field of the SG device which measures its spin? Or they are the two particles?

        5. Could you show me more precisely where you derive the Bell correlations from your theory?

        6. Is your theory local? If so, where exactly you explain locally the correlations?

        Best wishes,

        Cristi Stoica (link to my essay)

          Dear Neil Bates,

          Thank you for your kind comment. As I noted on your thread I find your views generally compatible with my own, and found your treatment of dimension fascinating. I especially liked your statement

          "All that math knows and can tell us in effect, is about math. When we think it is telling us something about "the world", we are just finding out about the model that we're using."

          In my essay I discuss Bell's over-simplified physics model of Stern-Gerlach based on precession in a constant field, which leads to a null result; 0 not ±1. This contradiction is the basis upon which Bell builds his model, with well-known "logical" consequences. My model, of course does not lead to such a contradiction. Instead, it leads to local realism that produces the same correlation as quantum mechanics. As you indicate, the model did not fall together overnight, but the pieces do fall into place, after, as you say "teasing apart the background physics." It is a complex problem, and, as Jonathan noted above, of a "self-concealing nature", so I do not expect everyone to be convinced right off the bat. It does go against 50 years of gospel. But I am gratified by the number of people who make the effort to understand.

          Thank you sincerely for your kind observations,

          Edwin Eugene Klingman

          Dear Gary Hansen,

          Thanks for reading and commenting. As you note in your first paragraph, we do agree on the overview. Your second paragraph observes that my essay is rather technical and mathematical for a "well-educated but non-specialist audience". I plead guilty to that. I'm sure you are correct that many in the audience must be left wondering about the distinction between tricks and truth.

          Unfortunately, the local audience of other authors, which is the one I tend to address, generally have no problem identifying Bell, as he is almost a saint in the community. But they have been tricked by Bell for 50 years, and it is necessary to become quite technical to reveal the trick. Even then, many find it hard to believe, because it has been ingrained into them that local models cannot produce quantum results. After 50 years this has become a visceral conviction, and can not be successfully addressed with a physics-lite treatment.

          I'm very glad that you enjoyed what you understood. It is unfortunate that everything cannot be understood at first reading [not by me at least] but very fortunate that re-reading complex essays increases their understandability.

          Thanks for plowing through my essay. It's appreciated.

          Edwin Eugene Klingman

          Dear Jayakar,

          Thanks for your very clear statement about use of math to map the physical state of structures. I agree that it is best to recognize mystery at the beginning of the universe, and not in our basic theories via such a mystical ideas as "collapse of the wave function" and "non-local entanglement".

          I will read your essay and respond. I'm very biased in favor of "continuum mechanics", but against "string-matter" [as I understand it] and look forward to seeing how you manage this.

          Thank you for reading and responding.

          Best wishes,

          Edwin Eugene Klingman

          Dear Christian Corda,

          Thanks for reading my essay and responding. I'm glad that you have discovered Korzybski. His 'map' and 'territory' is always a good distinction to keep in mind.

          I'm even more pleased that you both enjoyed my essay and found it important in the context of the issues currently debated in physics.

          The issue of Bell is far too complex to be understood and decided on the basis of one essay, so I am not surprised that most of the serious physicists who have looked at my essay remain "unsure". That is quite appropriate. What I hoped for was to introduce the idea that, whereas Bell's math and logic have been tested for 50 years, his oversimplified physics has been accepted without question, because it agrees with the 1925 Goudsmit and Uhlenbeck view of 'spin' and with Pauli's simple constant-field eigenvalue equation. As Jonathan notes above, the problem has a "self-concealing nature" that must be seen through before progress can be made.

          And I hoped that, by showing that a local model that takes the inhomogeneous field into account actually yields the quantum mechanical correlation unless the physics information is thrown away by enforcing [unreasonable and unrealistic] constraints, it would catch the attention of serious physicists, who might then be stimulated to wonder how this is so, and thus begin the process of looking beyond Bell's overly-simply physics model. Valid math and logic applied to a faulty model, based on faulty assumptions, produce faulty conclusions, such as "no local model can...".

          Thank you very much for reading and providing very valuable feedback to me.

          I wish you the best also.

          Edwin Eugene Klingman

          Dear Pankaj Mani,

          Thanks for your comments. We seem to agree that the math maps physical structures and that "math maps on physical territory form the substance of physics."

          I will look at your essay on vibration. As I tend to a continuum-based interpretation of reality, vibration certainly plays a significant role in my physics, but I will study your essay and respond on your page. I have some difficulty envisioning Bell's non-locality as purely vibration-based, although for the photon-based experiments I do not rule this out.

          Thanks for your kind words about by essay.

          Best regards,

          Edwin Eugene Klingman

          Dear Cristi Stoica,

          Thank you for reading my essay, which I know is in conflict with your current view. Thus I'm really grateful to you for making the effort. I will try to answer your questions.

          1. Yes, there is a typo in my energy exchange theorem, as I note above on Mar 15, 2015 @ 00:19 GMT. My essay posted on Jan 9 and I tried to submit a corrected version on Jan 10, but the [correct] FQXi policy is to not change essays after they post. It is an obvious typo and has not seemed to throw anyone off, as it is cancelled in the same line.

          2. Also, as discussed in one of the many comments above, the angle θ (with one exception) always refers to the local angle between the spin and the magnetic field in the local Stern-Gerlach apparatus. Only in the figures on page 7 [where θ is shown as the horizontal axis) does θ represent the angle between Alice's setting a and Bob's setting b, which is the angle that appears in the QM correlation, -a.b. I apologize for any confusion. In Stern-Gerlach sources the angle is usually θ = (λ,B) while in Bell sources θ = (a,b).

          3. No, the particles do not show the same θ. The local θ in Alice's device is θ = (λ,a) while the local θ in Bob's device is θ = (-λ,b). Only the local θ has relevance for the local physics that leads to the non-±1 scattered deflection.

          4. The energy modes M0 and M1 are local. M0 is the θ-dependent precession energy associated with configuration -λ.B that is initially not aligned but vanishes when the spin λ aligns with the local field. M1 is the θ-dependent vertical component of the kinetic energy that did not exist when the particle entered on the horizontal axis with only horizontal velocity. Thus the precession energy vanishes and the deflection energy appears locally, and the local conservation follows the Energy-Exchange theorem. And as the deflection is θ-dependent, this dependence can be seen in the measurment data, but is not present in the [incomplete] quantum mechanical formulation, hence is 'hidden'.

          5. The theory [based on energy exchange] calculates a local deflection for Alice denoted by A(λ,a) where both the spin λ and the setting a are randomly generated. Similarly for Bob. These produce scattering or deflections represented by the local θ-dependent position of the particle on the detecting screen. A(λ,a) is read from Alice's screen (as calculated by the theory) and B(-λ,b) is read from Bob's screen, (also calculated by the theory.) It is these two values that are multiplied in pairwise fashion to produce the correlation. But the definition of the expectation value also contains the distribution of values AB, so, as A and B are calculated for 3,000,000 sets of random numbers, the distribution of the results is determined by computer, not from a formula, but from actual data, in a multichannel-analyzer-like analysis. This is used to compute the correlation shown on page 7. The basic formula or definition of expectation value is

          < AB > = SUM [ p(AB) (AB) ]

          6. Yes, my theory is local, in that critical settings a and b never appear together, unlike quantum mechanics where a and b do occur in the expectation value. Of course a goes into the calculation of A(λ,a) but it is combined into a product term and cannot be factored out, so it is not present as a in the result. [Just as, if a = 4 and λ = 9, the product term 36 implicitly contains both a and λ but they cannot be explicitly factored out.] Nor is b factorable from Bob's numeric result B(-λ,b). Thus only the [computed or measured] numbers,not containing a and b, are used, and yet, given the physics of energy exchange - based on local conservation - the correlation that results is -a.b. Mine is the only theory that is truly and provably local.

          The above is a very subtle point, and if you still have questions on this point I will be happy to try to answer them.

          Thank you for your best wishes, and especially thanks for taking the time to read and study what you viewed ahead of time as almost certainly a waste of time (kind of like perpetual motion).

          My best wishes for you Cristi,

          Edwin Eugene Klingman

          Dear Armin,

          I regret that the thread of your well formed comment was broken by nonsense.

          Thanks for returning after you've had more time to review my essay. You boil it down to opposition arising from my presenting an answer to a question no one is asking. You you are probably correct in this. The treatment of Joy Christian, for example, has certainly deterred many from asking this question. It has apparently even prevented many of the establishment from reading my essay, and certainly from commenting. In short, it is a taboo question to ask whether Bell was wrong.

          You correctly observed that "in order for someone to seriously consider [my] argument, they have to first be willing to question whether the Stern-Gerlach experiment really has been misinterpreted all along for the last 90 years, which is what [my] assertion amounts to."

          You suggest that I collect all the SG data and statistically analyze it to show that interpreting its outcomes and binary terms is a mistake. You make a good point. But as I have a personal subscription to Phys Rev Letters, I am not in a university environment with access to all different journals, and therefore I frequently run into pay-walls. Moreover, there is abundant evidence on other FQXi threads (JC's, specifically) that people will argue statistics until the cows come home. So while your suggestion is a good one, it seems not best for me with neither access to the data nor much competence in statistics.

          I certainly agree with you that the burden of proof is on me. And as others have reminded me, great claims require great proof. In my opinion, it will be easier to conduct a new SG type experiment to explicitly test for θ-dependence than it will be to gather all data and statistically analyze it, so it is my intent to perform or have performed this specific experiment. Of course, even experiments can be and are ignored, if they go against the grain (see, e.g. Martin Tajmar).

          On the other hand, I think it is incontrovertible that Bell's interpretation of Stern-Gerlach leads to a contradiction. He interprets SG as a constant field through which dipoles precess, which leads to zero deflection, while the entire content of his theory requires ±1 deflection, an obvious contradiction. And it does not take much to see that when the non-constant (gradient) term is added to the Hamiltonian, then Pauli's eigenvalue equation should be affected. These are simply issues of logic that any physicist should be able to follow, and one would think they might be caused to wonder about this aspect of Bell.

          In addition it is easy to show that the local model I derive does reproduce the quantum mechanical correlations [see page 7] against all gospel, and one would think this would arouse curiosity among 'real' physicists, especially when the correlation fails if Bell's constraints are imposed.

          Finally, there is matter of intuition. In this contest at least Phil Gibbs and Ken Wharton have expressed that "intuition" is a thing to be wary of. Most of us are familiar with the theory that says we evolved in a macro-sized world, and therefore our intuition - whatever it is - is simply not suited to the microworld and should not be expected to be so. But my own theory of consciousness does not view consciousness as an artifact, but more as inherent in nature, not quite panpsychism, but close, and in this view intuition is less 'scale-dependent' and more in tune with the true nature of the world, in which case the intuitive rejection of non-locality is not to be dismissed.

          Nevertheless, you have put quite a bit of effort into analyzing the context in which my theory is presented, and have made quite sensible suggestions. For this I thank you sincerely. Yet, as Tom Phipps remarks, the establishment knows how to close ranks in defense of the status quo, and "this means that progress can only occur from inside, and at a snail's pace."

          I am not quite as old as Tom, but I am not well suited to a 'snail's pace' at my age. Better to present the logic, the history, the analysis, the model, the results, and the interpretation that contradicts Bell and then focus on an experiment that will prove [or not] my theory.

          Thank you very much for your well thought out and friendly, supportive, suggestion.

          Edwin Eugene Klingman

          • [deleted]

          • Post #251

          Gordon

          Last year, you sent me several papers for review and comment, and at that time it appeared you may have shown the mathematical basis for my physics-based Bell argument, and you have been given full credit for your relevant contributions in my QSLR paper [reference 2]. However after several months of examining that path, I became convinced that your math did not solve the Bell problem, and yours is, in fact, a non-local approach as you depend upon bringing both Alice's and Bob's remote settings into your local calculation, as does quantum mechanics. As I am only interested in a local model, this disqualifies your approach as far as I'm concerned. As to this paper, I have properly credited all the contributions and sources.

          After developing my computer simulation, I realized that it is Bell's insistence on suppressing the physics by imposing the ±1 constraints that is at the root of Bell's error. As you continue to apply these constraints, your model does not resemble my model in any way, nor does it address the problem. You have a formal, non-physical, approach that yields a non-local calculation of Bell's theorem.

          As witnessed on other threads, when the code from models is introduced, all physics discussion goes out the window and the topic focuses on coding. My objective is to provide enough insight into the physics that others, skilled in both physics and computing, can independently generate the same results by following the logic I lay out, without being influenced by whatever code I have used. Bell's theorem must be discussed at the level of physics. It is not a math problem, per se, nor does your mathematical approach, devoid of physics, solve Bell's problem.

          Edwin Eugene Klingman

          Dear Edwin,

          Sorry for taking so long to come back but this isn't a facile topic. I had to read your paper a few times, revisit both Bell and SG and then go through (almost all) the comments, since I had some questions and I suspected I can find the answers there (and I did). I can say it was an interesting and exciting read so I felt motivated to put some effort into understanding it.

          I think your writing style is both enjoyable and clear, though I'm sure that you would have been more comfortable if you had a couple of extra pages, option that was taken away by the contest rules. I think that if Scientific American or any other magazine would want to write an article for the public, they could because everything you present is a problem of logic. Surely some readers have problems with the physics because it's difficult to imagine moving scenarios, but a short animated clip can very easily show how and why precession and deflection in magnetic fields influence where a particle lands in an experiment. When I started I wasn't very familiar with the topic, but right now I have at least a feeling of understanding or intuition, so you shouldn't be worried that your paper would be unclear for most readers; I know you mentioned this was a concern for you.

          I will shortly outline my key take-aways to check my understanding. You are noticing that Bell starts with quantized spin so we revisit the experiment which established that. We start with a formulation of the movement of an uncharged particle through a magnetic field and show that there exists energy exchange between the particle's precession (magnetic moment) and the deflection on the field gradient, and thus the trace of the real spin is preserved (I mean real as in 3D coordinate system of real numbers) through the position of the particle on the SG screen. So in a set of two anticorrelated particles, one shouldn't expect to find two values of spin, but any two opposite values of spin, a more general result meaning that spin is not necessarily quantized. The usual entanglement correlation is due to conservation of the original angular momentum of the particles from source to screen. The SG quantization depends on the length (and strength?) of the field generated by the device.

          What hasn't been asked before, and therefore I can ask now without asking you to revisit the same topic time and again, is about the experiment you mention close to the end. You said that theta-dependent scattering should be testable; are you referring to the Alice and Bob setup in page 6 (a spin correlated SG pair) or do you have something else in mind? If you do have something else in mind, do you develop it in another paper? I'd like to try to read it. To rephrase, what is the experiment you'd like to do if you would receive funding? Also since my understanding is that the same correlations can be obtained with a classical model, do you expect to find any difference between an experiment you might set up and a Bell experiment? I know that in your setup you can calculate the results in advance but what I mean is, will the shape on the screen be different or should the result look the same? Should the reasoning allow you to make different predictions from Bell or SG? Again, if you develop a setup anywhere else, I'd be interested to try to understand it.

          Warm regards,

          Alma

            I realized that I forgot to ask you something. I tried to search online for new revisions of the SG experiment but for me it proved impossible to find relevant information. Upper on the page in the comments, there's mentioning of a serial SG where particles are prepared in one spin position, then go through a second field and still end up in both the upper and the lower half plane. Do you know if that experiment has ever been performed and what's the result? I know you did a lot of research on the topic, so I tried to find answers to my questions in the references you used in the other two papers but couldn't; there's just too much information. If you encountered a paper that acts as a hub and points out the most known SG type experiments, can you please tell me which one is it?

            I know you already have lots of comments and I'm sorry to burden you further but you made me curious and you're very nice and answer everyone.

            Alma

            Alma,

            If I may interject, it is frustrating finding reference to SG experimentation and I've noticed that there is often more about deflection of electrons than there is about neutral atoms with a magnetic moment. I think in searching, one must be aware that the focus is on the typical shaping of the magnets themselves which produces an inhomogeneous field intensity, whereas uniform magnets with flat surfaces facing each other produce a homogeneous intensity at least throughout the region bounded by the surface area of the faces.

            But electron streams behave differently than do neutrons (or neutral atoms) which possess a magnetic moment, and the Quantum Mechanical standard model treats electrons as point particles because the electrical charge does not exhibit a pole. The 'negative' charge is uniformly spherical so it doesn't present a differentiated directional attitude at any time in crossing the field, it will be deflected the same amount relative only to magnetic field intensity in accord with Faraday's right hand rule. Like in a cathode ray tube. Also, science lacks a general definition of 'charge', positive and negative are merely operational definitions and though the inverse square law holds true in experimental measurement, there is no theoretical basis that limits the intensity of charge and so mathematically it goes to a singularity of infinite intensity. So it gets treated as a point particle.

            "Spin" is a property of the electron point particle which has no correlation to a classical physical rotation. It is a measurement function that can be used to establish an ad hoc directional attitude in the otherwise homogeneous spherical negative charge field of the elusive electron. IF (!) there is a physical rotation experienced by an electron giving rise to a magnetic dipole moment, that magnetic moment is overwhelmed by interaction of the charge field and the directional field of the magnets. And IF (!) it exists and persists it aligns as if it were the same as the axis of the ad hoc measurement schemata, and consequently has no 'wobble' which would precess.

            So electrons are not what were used in the original Stern-Gerlach experiments that John Bell referred to. S-G used neutral silver atoms which possess a magnetic dipole moment which precesses. Good Luck finding reference to S-G type laboratory studies of that sort. SG magnet arrays seem to be used primarily for electron traps. Cheers jrc

            JRC,

            Thanks for your comment. I am not sure why you mention electrons. I explicitly mentioned uncharged particles so, worst case scenarios, I was thinking neutrinos or you haven't read my comment :)

            All of my 2 questions to Edwin are genuine and I am not making any assumptions. I asked what kind of experiment would he make should he receive funding and if he can point me out some direction for further reading. I'm sure he would gladly answer the first and if he won't answer the second I guess I'll just remain curious, tough luck. I did spend a lot of time trying to understand his work (because it's very interesting) and reply to him and I am sure he does rather appreciate it. Thanks for wishing me good luck in finding SG type experiments. Cheers Alma

            Dear Edwin,

            Your thorough and well-presented technical argument is something I need to spend more time with. But the point of the essay is very interesting and probably worthy of a good deal of discussion. When I began studying mathematics I was most intrigued by, what my teachers often called, counter-intuitive results. I enjoyed the fact that a formalism of our own making could produce these kinds of surprises. There was some non-obvious thing about the relationship between mathematical thought and intuition. This relationship is even more interesting when one recognizes the crucial role mathematical intuition plays in developing mathematics. And so while I very much enjoy the way you use mathematics to bring a physical idea more in line with our intuition, I think I will be slow to accept that the counter-intuitive ideas are incorrect.

            I do appreciate your comment on my essay and hope we find opportunities to continue to communicate,

            Best,

            Joselle

              Dear Edwin,

              Thank you for the answers. Although you provided detailed answers to my questions, I still don't get it. I think I need more details. Could you please show me the formula by which Alice and Bob calculate A(λ,a), and what are the inputs? Then how to calculate from these the expectation value, so we can see if we get the same correlations as QM? Perhaps if you have a concrete example, that would be great. Sorry for not being able to find these myself, I also looked in your references [2,3], but I missed them.

              Best regards,

              Cristi

              Dear Alma,

              As this comment will probably be hidden, I will answer you in a new thread below.

              Edwin Eugene Klingman