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# Real Analysis - Science topic

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Questions related to Real Analysis
Question
Of course every Banach space has type 1 and it is known that if 0<p<1 then every quasi-Banach space has type p if and only if it is locally-p-convex. Below is a question and some comments which I think are worthy of discussion. I am not certain of some comments so any corrections or insights would be appreciated. If this discussion gets off the ground, I can provide some extra information and references.
Question: Must a quasi-Banach space of type 1 equal its Mackey topology completion?
1. If there is a non-locally convex space of type 1 with a separating dual then the answer to the above question is negative. However, I don't know of such a space.
2. The Ribe space R is a non-locally convex space whose dual vanishes on a one-dimensional subspace L such that the quotient R/L is the Banach space of absolutely summable scalar sequences. Thus, the completion of R with respect to its Mackey topology is a pseudo-normed space with a 1-dimensional null subspace. Thus, this is a complete pseudo-normed space and is known to have type 1.
3. Consider the Lorentz spaces L1,q for q in [1,inf].
• L1,1 is the Banach space L1 and is therefore type 1 and of course complete in its Mackey topology.
• L1,inf is the non-locally-convex space weak-L1. This space has a complicated, non-trivial dual. I am not sure how to determine if this space is a complete pseudo-normed space in its Mackey topology but it is known that this space does NOT have type 1 but is locally-p-covex for every 0<p<1.
• If 1<q<inf, L1,q has a trivial dual space. Thus, it is complete in its Mackey topology as this is the trivial space {0}. This space is known to have type 1.
As I remember, if we have a topological vector space (X,T) - T the linear topology on X, then a linear topology U on X is Mackey topology iff all linear and continuous functionals on X relatively to T are also continuous relatively to U. In other words, a Mackey topology is the maximal topology on X, preserving the continuous dual of (X,T). And, if (X,T) is locally convex( T is generated by the family of Minkowski functionals on X), then T is right Mackey topology on X.
Please correct me, if I'm wrong!
Now let's consider the Lorentz space L1,q. If 1/q+1/r=1 and L1,r may be identified with the dual of L1,q, I think this fact would help to obtain an answer to your question. But I don't know anything about duals of Lorentz spaces and my assertion is probably not true.
Question
Does there exist a real sequence, say (xn), with the following properties?
(i) (xn) has non-zero terms and converges to zero
(ii) The limit set of abs(xn+1\xn), i.e. the set containing the limits of all convergent subsequences of abs(xn+1\xn), contains at least two points and it is bounded above by a number less than 1.
I concur with Lahbib Oubbi
Question
In Mathematics, some proofs are not convincing since the assumption fails in the Proofs.
If equality is Changed to "approximately equal to" then the Proof becomes more Perfect. But Uniqueness cannot be guaranteed.
In Page number 291, Introduction to Real Analysis Fourth Edition authored by Bartle and Sherbert, the Proof of Uniqueness theorem is explained.
That Proof is not perfect.
Reason : Initially epsilon is assumed as positive and so not equal to zero. Before Conclusion of the Proof, epsilon is considered as zero and written as two limits are equal.
The equality cannot hold since epsilon is not zero. Only the possibility to write is Two limits are approximately equal.
Since Epsilon is arbitrary, never imply epsilon is zero.
I hope the authors and other mathematicians will notice this error and will change in new editions.
The proof is fine.
It is based on the following fact:
If x < epsilon for every positive epsilon, then x ≤ 0.
This is easily proved by contradiction.
Suppose that x < epsilon for every positive epsilon and that x is not less or equal than 0.
Then x>0, hence for epsilon=x/2 it is epsilon>0 and epsilon<x; this contradicts the hypothesis.
Question
A sequential, if-and-only-if criterion of Riemann integrability that follows from the Riemann’s definition of the integral, states that a function on a closed and bounded interval is Riemann integrable if and only if any respective sequence of Riemann sums is Cauchy for any tagged partition of the interval with norm tending to zero .
Although this is not a new result, it is not mentioned in standard textbooks [2-5].
However, if the previous criterion is used as a starting point to define the integral, it provides a number of educational advantages over the traditional definitions of Riemann and Darboux.
The uniqueness, the linearity, and the order properties of the integral all follow naturally and easily from the respective properties of sequence limit .
Moreover, the boundedness of any integrable function is not necessary to be assumed beforehand, as it can be easily proved using that a convergent sequence is bounded .
Further, this approach has the advantage of being “infinitesimal-friendly” (so to speak), as the integral can be realized as an infinite sum of infinitesimal quantities , thus giving the instructor the opportunity to further introduce the concept of infinitesimal.
Appealing mainly to mathematicians, I wonder if a work presenting such an approach would be publishable in educationally-oriented journals of mathematics.

 Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis. John Wiley & Sons, Inc., Fourth Edition, 2011.
 Stephen Abbott, Understanding Analysis. Springer, Second Edition, 2015.
 Michael Spivak, Calculus. Publish or Perish, Inc., Third Edition, 1994.
 Walter Rudin, Principles of Mathematical Analysis. McGraw-Hill, Inc., Third Edition, 1976.
Then send to professor Bartle your manuscripts regarding Riemann integrability, to evaluate the correctness of your statements and proofs! I am convinced that he will not refuse you and will give you the right answer!
Also, send your manuscripts to a peer reviewed mathematical journal, for publication. You will receive the suitable answer! I think you know what this will be!
If you are not a mathematician, why do you think that things that great mathematicians of the 20th century did not discover, you can discover? It is impertinent to combine your preprints with books written by reputed mathematicians such as Bartle, Michael Spivak or Walter Rudin!
Yes, my way to approach our discussion was no more a scientific way, after finding that you don't understand the correct scientific path! I don't understand how certain people end up giving opinions about things they don't master perfectly!
No more words, no more messages! Good luck!
Question
The limit superior/inferior of a sequence in R can be introduced in different ways.
One way to define the previous limits is by considering the sequence of suprema/infima of the successive tails of the initial sequence. The limit superior/inferior is then defined as the infimum/supremum of the sequence of suprema/infima of the successive tails of the initial sequence.
Another way to introduce the limit superior/inferior of a sequence is by considering the set of real numbers with the property that only a finite number of terms of the sequence are greater/less than those real numbers. This is a more general, and more abstract, way to introduce the limit superior/inferior, as the previous set contains the sequence of suprema/infima of the successive tails of the initial sequence.
Finally, a more informative, but also more restrictive, way to introduce the limit superior/inferior is by stating that it is the supremum/infimum of the limit set of the sequence, i.e. it is the supremum/infimum of the subsequential limits of the sequence.
What, do you think, is the best way to introduce the subtle concept of limit superior/inferior?
The concepts of limsup (liminf) that we define for numerical sequences, were generalized by the Polish topologist, Kazimierz Kuratowski in the 1920-ies. Amazing, how topology, set theory, and analysis intertwine.
Question
I want malwares that are used in advanced persistent threat  APT
samples for analysis
Dear Khalid Abdulrazzaq Alminshid,
VirusSign and VirusShare have the best and well-known datasets. It is worth noting that you need to provide them with an official introduction letter from your institute for indicating academic and research purposes.
Regards,
Danial.
Question
How to prove or where to find the integral inequality (3.3) involving the Laplace transforms, as showed in the pictures here? Is the integral inequality (3.3) valid for some general function $h(t)$ which is increasing and non-negative on the right semi-axis? Is the integral inequality (3.3) a special case of some general inequality? I have proved that the special function $h(t)$ has some properties in Lemma 2.2, but I haven't prove the integral inequality (3.3) yet. Wish you help me prove (3.3) for the special function $h(t)$ in Lemma 2.2 in the pictures.
Dear Dr. Feng Qi ,
The further, the more you will be convinced of the correctness of my words. Maybe. I wrote these words for the reason that I have passed a similar path here in August and September this year:
Certain personalities promised to destroy my proof - to show that my proof was wrong. Then they forgot it, apparently? I made a reminder 5 days ago. But, no one reacted and insignificant discussions were continued.
My answer is this: waiting is always the best tactic and strategy. Remember the RF's mathematician G. Perelman. This person generally did not give a damn about the opinion of the professional community of mathematicians and was absolutely indifferent to any action that disturbed him. This is a real Hero!
Best Regards,
Sergey
Question
I'm looking to apply for MS programs but before that I want to clearify which direction should I go as it will help me to write a better Statement of purpose.I'm intrested to do research in area of Analysis i.e Real Analysis, Functional Analysis etc..
I think that a good topic is differential equations (either about discrete or continuous variable, of integer or fractional order).
Question
I am looking at a pde of the form
D^2 u + f u = k u,
here D^2 denotes the laplacian, u and f are complex functions on IR^n. I want as much information as possible about each PDE. For example, how many solutions exist for each value of k and properties of the solutions themselves, such as smoothness, boundedness, stationary points etc. Can you recommend your favourite reference volumes for eigenvalue problems for PDEs? I have a bunch of resources on PDEs in general, so that is not what I am looking for. Instead, I am looking for the subtleties that mainly relates to eigenvalue problems.
Thank you for your time.
I hope that the attached articles are useful in your research.
Best regards
Question
We know that mathematicians study different mathematical spaces such as Hilbert space, Banach space, Sobolev space, etc...
but as engineers, is it necessary for us to understand the definition of these spaces?
Yes, we do have to know about the spaces, at least during our university studies, to enables us to expand our mind into abstact level, that will be very usefull for design activities
Question
A new paper in "Constructive Mathematical Analysis" by Prof. Michele Campiti.
G. C, Rota wrote his thesis under Nelson Dunford that characterized the extensions of OD operators via the concept of boundary values. It appeared in Communications of Pure and Applied Mathematics, in 1958. Might what to check that out relative to this paper.
Question
Is there any transformation from fuzzy to the real or complex domain??
See the following references may are interesting
(1) [IEEE 7th Seminar on Neural Network Applications in Electrical Engineering, 2004. NEUREL 2004. 2004 - Belgrade, Serbia and Montenegro (Sept. 23-25, 2004)] 7th Seminar on Neural Network Applications in Electrical Engineering, 2004. NEUREL 2004. 2004 - From fuzzy to real sets
(2) [IEEE  Second IEEE International Conference on Fuzzy Systems - San Francisco, CA, USA (28 March-1 April 1993)] [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems - From fuzzy logic to fuzzy truth-valued logic for expert systems: a survey
de Mantaras, R.L., Godo, L.
Question
Recent research has shown key important aspects with measurability of theoretical postulates verifying hypotheses parameters, processes, phenomena, and models.
Tensor matrices have necessarily roles to bring about complex nature manifesting spatially and temporarily. Explaining everything that is existing in terms of the fundamental entities have lead to realization of geometric topology space tensor manifold time evolving event gridnetwork.
Einstein's General Theory of Relativity, Quantum to Particle Theory of Everything, String Theory among others have measurability in mind a proof of model requirement automatically. For example, Schwartzchild blackhole mathematics helped to identify, observe, and measure singularity blackhole consequently, the recent telescopic photos observing directly, proving validity with General Theory of Relativity tensor predictive capability.
Providing the thumb rules, below certain associative relationships might connect mathematics with physics to measure model......
. typically scalars, scalar matrices are helpful to get statistical measurements that are analyzable observationally experimentally......
. tensors have stochastical vector matrices that aren't amenable to direct measurements. Hence transforming tensors or matrix tensors to scalar matrix systems are key to make measurable operational parametric graphical experimental observational gridnetworks.
My analyses metrix protocol techniques have yielded a rough estimate of overall globally ~80% of objects universally are measurable statistically. This will mean ~20% are uncertainity stochastic probabilities with a few% inherent immeasurable tensor network, aether may be example. I have space time sense 2x2 tensor grid part of a large tensor matrix that if transformed to 5 dimensional like scalar matrix natural manifolds protocol will help eventually in the quantitative grand unified theory of everything. There are more to come after our QFM/EM modeling going on with collaborative platform TEI.
Given below are a few references that are associated, not exhaustive, suggestions welcome. Additions editions expansions!!!!!
(4)
(6) Zurek, Wojciech H. (2003). "Decoherence, einselection, and the quantum origins of the classical". Reviews of Modern Physics. 75 (3): 715. arXiv:quant-ph/0105127. Bibcode:2003RvMP...75..715Z. doi:10.1103/revmodphys.75.715 & Dan Stahlke. "Quantum Decoherence and the Measurement Problem" (PDF). Retrieved 2011-07-23.
I go for simpler models by changing the postulates (modeling). I'm unsure what that 20% is. Not measured because nobody has done it or cannot be measured.
Question
Dear All,
I am hoping that someone of you have the First Edition of this book (pdf)
Introduction to Real Analysis by Bartle and Sherbert
The other editions are already available online. I need the First Edition only.
It would be a great help to me!
Thank you so much in advance.
Sarah
Search by the elements of real analysis by Bartle and Sherbert.
(For its first edition )
Question
Here smooth means infinitely differentiable. Clearly (df_x)*(df_x) has all its eigen values nonnegative and has atleast one +ve eigen value. Can we say that E_f is smooth or continuous. If not true then, are there some known conditions under which it is well behaved (smooth or continuous)?
What can we say if f has constant rank?
If you are interested in the smallest positive (i.e. non-zero) eigenvalue without requiring that the matrix is invertible, then there is no chance to get continuity. Say, it may happen that the eigenvalues of a 2x2 matrix A(x) are a(x) < b(x) that behave as follows: a(x) is positive and tends to a(z) = 0 as x goes to given point z, but b(x) tends to b(z) > 0. Then at point z the eigenvalues are 0 and b(z), so smallest positive eigenvalue is b(z). Then, E_f(x) = a(x) in the small neighborhood of z, so lim_{x \to z} E_f(x) =0, but E_f(z) = b(z), which is not equal to 0.
In other words, when the smallest eigenvalue takes value 0, the smallest POSITIVE eigenvalue jumps from being the really smallest to the second smallest one, which spoils the continuity.
On the other hand, if one speaks about the smallest eigenvalue of non-negative symmetric matrix, permitting the value 0, then it seems to me that one can demonstrate that it depends continuously on matrix.
Question
In the theory of the stability of the differential operators, one could prove the stability results based on spectra of an operator, (all eigenvalues must be negative for example).
one problem with the above method is that not all linear operators are self-adjoint (for examples operators in convection diffusion form) and their corresponding eigenvalue problem can not be solved analytically, hence spectra of the operator can not be calculated analytically. On the other hand there is a definition related to spectra, which is called pseudo-spectra, that somehow evaluates the approximated spectrum , even for non self-adjoint operators.
I want to know is it possible to establish stability results for a differential operator based on pseudo-spectra?
The question you ask is a very broad one. Since differential operators are unbounded - the definition of the on which one defines the operator is important. When one goes to infinite dimensions, there might not even be a point spectrum (eigenvalues) - the spectrum may be only the continuous spectrum and residual spectrum. If the operator has a self joint extension or a normal extension then the problem becomes more tractable but for non-normal operators the problem is quite complex.
At one time a class of operators "spectral operators" were defined, see Dunford and Schwartz, "Linear Operators", vol III, to address the issues on non-normal operators. Proving a non-normal operator - even an ordinary differential operator - was not a trivial task. The goal of this effort was to expand the concept of Jordan Canonical form to first bounded linear operators on a Hilbert/Banach space and then to unbounded linear operators, e.g., differential operators. For compact operators or operators with compact resolvents - that has been solved. For bounded normal and unitary operators, that has been solved (spectral theorem). For unbounded operators with a compact resolvent, that has been pretty much solved (by using the spectral theorem the spectral theorem). For unbounded self adjoint operators, there is also a spectral theorem. However, for non-normal bounded operators and non-self adjoint unbounded operators - there is no general result on a canonical form.
In these problems stability questions be quite problem specific.
Question
Let f be the pdf of a n dimensional N(0,C) distribution i.e up to a multiplicative constant, f(x)=exp(−0.5 xC−1x).
Which vector fields F are so that div(F)=f ?
According to Helmholtz decomposition in 3 dimensions, you can allways write F= -grad (u)+curl (A), where u is a scalar field and A is a vector field. So, operating with the divergence in both sides you get lap (u)=-f, where "lap" stands for Laplacian. Then you use boundary conditions on F to select those for u and A. For instance, in the relevant case of full 3-dimensional space demanding that F and all its derivatives vanish at infinity, one solution to your problem is simply F=-grad (u), where u is the convolution of your Gaussian with 1/|r|^2. Additional solutions can be found by adding to the above solution the curl of any vector field A so that curl (A) vanishes at infinity (i'm not sure whether such a vector field A exists, if we additionally demand it to be smooth).
Question
For example, when we xerox a document, the printer takes the A4 page to the same. Are there other better examples?
One of the most practical areas of Mathematics is the theory of differential equations.  it plays a vital role in Physics and Engineering.  Fixed point theory guarantees that these equations have solutions.  This is one very practical result that should be noted.
Question
Iis there any discrete analogue m of star convexity at 0'  star convexity at 0',?
That is, in the same way that midpoint convexity can be seen to be a discrete analogue of convexity (and entails it under certain regularity assumptions, continuity etc)?
For example, where F:[0,1] \to [0,1],
star convexity at 0 :
"\for all x \in  dom (F)=[0,1],:(\forall t \in [0,1]) :"F(t * x_0 +(1-t) * x )<= t * F(x_0)+(1-t)* F(x)  where x_0 =0 "
Generally with F(0)=0  t it becomes :
For all x \in  dom (F)=[0,1],:(\forall t \in [0,1]) F(tx)<=tF(x)
(it is convexity restricted so that the first argument is some specific minimum value usually x0=0
'
such as '\mid-star convexity at x0=0' for F:[0,1] \to [0,1]  F(0)=0'
mid-star convexity at x0=0 \for all x \in  dom (F)=[0, F(1/2x)<=1/2F(x) which under certain regularity assumptions ensure the star convexity of the function or that its a retract \forall x in dom(F)=[0,1]
F(x)<=x which given F(1)=1 is generally entailed by star convexity
( i call this mid-star convexity with F(0)=0  and x0=0\in dom(F) for F:[0,1]\to [0,1]
strictly monotone increasing bi-jection F:[0,1]\to [0,1]
-where F is absolutely continuous
-F at1/2/3 times continuously differentiable
with F(1)=1 and F(0)=0 F(1/2)=1/2
I presume not as not all star convex functions are continuous to begin with?
Great.
Question
Is there exist at least 9 distinct a bi-j-ective, diffeomorphic& homoeomorphic. analytic  functions F(x,y), dom i of two variables in the x,y, cartesian plane
F(x,y); dom F:[0,1]\times[0,srt(3)/2],\toΔ^2, unit 2 probability simplex
CO-DOM(F)=IM(F)=delta_2: {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0\in [0,1]\forall (x1,x2,x3)\in [0,1]
F(0,0)=(1,0,0)
F(1,0)=(0,1,0)
F(1/2, sqrt(3)/2)=(0,0,1)
F(x=1/2,y=sqrt(3)/6))=(1/3,1/3,1/3)_(1/2, sqrt(3)/6)   =
The inverse function being
i=(x,y)=F-1(<p1i,p2i,p3i>_i=(x,y))= <x=[2p2+p3+1]/2,y= p3 *[sqrt(3)/2]]
F(x=1/2,y=sqrt(3)/6))=(1/3,1/3,1/3)_(1/2, sqrt(3)/6) ;
Incidentally it also has to accomodate the 8 element boolean algebra of events on each pt as well  <p1,p2,p3>, pi1+p2+p3=1 pi>=0 omega=1, emptyset =0,
PR(A or B)=PR(A)+PR(B)= p1+p2 >=0
and <p1+p2, p2+p3, p3+p4)
ie F(x,y)= <p1,p2,p3>,\to <1,0, p1, p2, p3, p1+p2, p2+p3, p3+p1}
and as every element of the simplex must be present at very least six times,
It actually must consist of at least six identical simplexes, that where the euclidean
F(x,y, {1.2,3.4.5.6})=Δ^2\cup_i=1-6, these can be the same simplex but with the order, of each pi in each <p1, p2, p3> interchanged
<Omega={A,B,C}, F= {{Omega},{ emptyset}, {A}, {B}, {C}, {A V B}, {AVC}, {B VD)
PR(A)=p1
PR(B)=p2 PR(C)=p3
PR(A or B)=PR(A)+PR(B)= p1+p2 >=0
PR(A or C)=PR(A)+PR(C)=p2+p3
PR(B or C)=PR(B)+PR(C)=p1+p3
where in addition there is a further triangle probability function that is ranked by this chances. The triangle frame function G(x,y,1,6)={{1,0, g1,g2,g3, g1+g2, g2+g3, g1+g3}, 1<=gi, g1+g2, g2+g3, g3+g1>=0, g1+g2 +g3=1;
such that for F(x,y,{i,6]  on the same coordinates, \forall Ei\in(A, B, C,A or B, A or C, B or C)
{{1,0, g1,g2,g3, g1+g2, g2+g3, g1+g3}= [G(x,y,{1,,,6,1), G(x,y,{1,,,6},2), G(x,y,{1,6},3), G(x,y,{1,,,6},4), G(x,y,{1,,,6},5),..... G(x,y,{1,,,6},8),]
; G(x,y,{1,,6},1)=1, G(x,y,{1,,,6},2)=0
1>G(x,y,{1,,6},3)=g1,>0 in the interior
1>G(x,y,{1,,6},4)=g2 >0,
1>G(x,y,{1,,6},5)=g3>0
1>G(x,y,{1,,6},5)=g1+g2>0, >{g1,g2}
1>G(x,y,{1,,6},7)=g2>0
1>G(x,y,{1,,6},8)=g1+g3>0, g1+g3>g3, g1+g3>g1
1>G(x,y,{1,,6},7)=g2+g3>0, g2+g3> (g2, g3)
g1+g2+g3=1
1>G(x,y,{1,,6},5)=g1+g2>0
\forall i in [i in 6} G(0,0,i,)=(0,0,1),
G(0,0,i\in [1,6},3)=G(0,0,i\in [1,6},, 1),=G(0,0,i\in [1,6},5)=G(0,0,i\in [1,6},8}=1
G(0,0,i\in [1,6},2)=G(0,0,i, 4),=G(0,0,i\in [1,6},{7}=,G(0,0,i\in [1,6},{8})=0
\forall i in [i in 6} G(1,0)=(0,1,0)
G(1/2, sqrt(3)/2)=(0,0,1)=
G(x=1/2,y=sqrt(3)/6, {1,,,6}))=(1/3,1/3,1/3)=(1,0,1/3.1/3.1/3. 2/3,2/3.2/3}
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, G(x,y,i,j)=gj>1/3, iff F(x,y,i,j)=pj>1/3,
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, G(x,y,i,j)=gj>1/3, iff F(x,y,i,j)=pj>1/3,
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, G(x,y,i,j)=gj=1/2, iff F(x,y,i,j)=pj=1/2,
\forall {x,(y)}\forall,l {i....6},\forall j \in {1,,,8}, F(x,y,i,j)=pt<|=|>F(x1,y,1i1,t)=pt[\forany {x1,(y1)}\foranyl,l1 {i....6},forany t \in {1,,,8},  iff G(x,y,i,j)=gj|<|=|> G(x1,y1,i1,t)=gt,  ,
\forall {x,(y)}\forall,l {i....6},\forall t \in {1,,,8}, \forall {x1,(y1)}\forall,i1 {i....6},\forall t_1 \in {1,,,8},  such that
F(x,y,i,,t1)+F(x1,y1,i1,t2)=p_t(y,x,i)+p_t1(x1,x1,i1)<|=|> F(x2,y2,i2, t3)+F(x3,y3,i3,t4) =p_t3(x2,x2,i2)+ p_t4(x3,x3,i3)\forany{x2,(y2),i3},(x3,y3,i3}dom(F) such that, forany (t2,t3) \in {1,,,8},  where t2 @,sigma F(x2,y2,i3_, t3 in sigma @,F( x3,y3, t3) iff
G(x,y,i,,t1)+G(x1,y1,i1,t2)=g_t1(y,x,i)+g_t2(x1,x1,i1)<|=|> G(x2,y2,i2,j,t3)+ G(x3,y3,i, t4)=g_t3(y2,x2,i2)+g_t4(x3,x3,i3)
where
, iff F(x,y,i,j)=pj=1/2,
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 0<G(x,y,i,j)=gj<1/3, iff 0<F(x,y,i,j)=pj<1/3,
\forall {x,(y),l {i....6},\forall j \in {1,,,8},2/3 <G(x,y,i,j)=gj>1/3, iff 2/3>F(x,y,i,j)=pj>1/3,
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, G(x,y,i,j)=gj=2/3 iff F(x,y,i,j)=pj=2/3}
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 1>G(x,y,i,j)=gj>2/3 iff 1>F(x,y,i,j)>2/3
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 1>G(x,y,i,j)=gj>0iff 1>F(x,y,i,j)>0
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 1>G(x,y,i,j)=gj= 0 iff F(x,y,i,j)=0
\forall {x,(y),l {i....6},\forall j \in {1,,,8}, 1>G(x,y,i,j)=gj= 1 iff F(x,y,i,j)=1
forall
_(1/2, sqrt(3)/6)   =
{{1,0, p1,p2,p3, p1+p2, p2+p3, p1+p3}
\forall Et\in(A, B, C,A or B, A or C, B or C), \forall Ej\in(A, B, C,A or B, A or C, B or C)
ie for t\in {1,......8), j in {1,,,,,8}
G(,x,y,{i,,,,6},t,) @ x,y,i>G(x,y,{1,,,6},j)  or G(,x,y,{i,,,,6},t,)=G(x,y,{1,,,6,j,) or G(,x,y,{i,,,,6},t,) @ x,y,i<G(x,y,{1,,,6},j)
G(,x,y,{i,,,,6},t,)_t @ x,y,i>G(x,y,{1,,,6},j) iff F((x,y, {i,6},t)>F(x,y,{i,,6},j)= PR(F(x,y,{i,,6})>PR(x,y,{i,,,6}
G(Ei)=G(Ej) iff P(Ei)>PR(Ej) ie g1=g2 iff p1=p2,
g1+g2= g3 iff p3=p1+p2,
G(Ei<G(Ej) iff P(Ei)>PR(Ej)
G(0,0)=(1,0,0)
G(1,0)=(0,1,0)
G(1/2, sqrt(3)/2)=(0,0,1)=
F(x=1/2,y=sqrt(3)/6))=(1/3,1/3,1/3)_(1/2, sqrt(3)/6)   =
where on each vector, it is subject to the same constraints F-1{x,y,i} G(A)+G(B)+G(C)=1, G(A v B)=G(A)+G(B), G(sigma)=1, G(emptyset)=0 etc whenver G-1(x,y, i \in {1,,6})=F-1(x,y,{i,6}
for all of the 8 elements in the sigma algebra of each of the uncountably many vectors in the interior of each of the six simplexes of uncountably many vectors
and all elements F_i in the algebra of said vector in each in simplex,  except omega, 0, G(
Such in addition every element of Δ^1, the unit one probability simplex, set of all two non non negative numbers which sum to one, are present and within the image of the function; described  by triples like (0, p, 1-p) on the edges of the triangle in cartesian coordinates
to, the unit 2 probability simplex
consisting of every triple of three real non-negative numbers, which sum to 1. Is the equilateral triangle, ternary plot representation using cartesian coordinates over a euclidean triangle bi-jective and convex hull. Do terms  p[probability triples go missing.
I have been told that in the iso-celes representations (ie the marshak and machina triangle) that certain triple or convex combinations of three non -negative  values that sum to one are not present.
Simply said, does there exist a bijective, homeomorphic (and analytic) function F(x,y)of two variables x,y, from the x-y plane to to the probability 2- simplex; delta2   where delta2,  the set ofi each and every triple of three non negative numbers which sum to one <p1, p2, p3> 1>p1, p2 p3>=0; p1+p2+p3=1
F(x,y)=<p1, p2, p3>  where F maps each (x,y) in dom(F) subset R^2 to one and only to element of the probability simplex delta2  subset (R>=o)^3; and where the inverse function,  F-1 maps each and every element of delta 2
<p1, p2, p3>;p1+p2+p3  P1. p2. p3. >=0 , that is in the ENTIRE probability simplex, delta 2  uniquely to every element of the dom(F), the prescribed Cartesian plane.
Apparently one generally has to use a euclidean triangle,  with side lengths of one in Cartesian coordinates, often an altitude of one however is used as well according to the book attached attached, last attachment p 169.
(which suggests that certain elements of the simplex will go missing there will be no pt in Dom (F), such F(x,y)=<p1,p2, p3> for some <p1,p2,p3> in S the probability simplex
is in-vertible and has a unique inverse, such that there exists no <p1, p2, p3> in the simplex such that there is no element (xi,yi)of dom(F) such that F(xi,yi)= <p1, p2, p3> in
F(x,y), that is continuous and analytic
map to every vector in the simplex, ie there exists no set of three non negative three numbers p1, p2 p3 where p1+p2 +p3=1 such that
ie for each of the nine F,
Where CODOM(F)=IM(F)=delta_2: {p(i)=<p_1i,p_2i,p_3i>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0}
where p(i( described a triple and i whose  Cartesian index is i= (x,y), ie F(x,y)=p(i)<p_1i,p_2i,p_3i>_i).
and
,
CO-DOM(F)=IM(F)=delta_2: {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0\in [0,1]\forall (x1,x2,x3)\in [0,1]
F(0,0)=(1,0,0)
F(1,0)=(0,1,0)
F(1/2, sqrt(3)/2)=(0,0,1)
(with a continuous inverse)he car-tesian plane, incribed within an equilateral triangle to the delta 2,
coDOM(F)=IM(F)=delta_2: {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0\in [0,1]\forall (x1,x2,x3)\in [0,1]
where, no triple goes missing, and where delta 1, the unit 1 probability simplex subspace, (the set of all 2=real non negative numbers probability doubles which sum to one, described as triples with a single zero entry),
delta _1 subset IM(F)=codom(F)=Dela_2
and each probability value in [0,1] , that is each and every real number in [0,1] occurs infinitely times many for each of the p1-i, p2_2, p3_3 , on some such vector,
1. and one each degenerate double
2, And which contains, as a proper subset, the unit 1 probability simplex, delta 1 (set of all probability doubles)contained within the IM(F)=dom(F)=delta2; in the form of a set of degenerate triples, delta_1*, subset delta 2=IM(F)=codom(F) ( the subset of vectors in the unit 2 (triple) probability simplex  with one and only one, 0, entry),
ie <0.6, 0.4,0>, <0, 0.6, 0.4>
3. where for each pi /evctor (degenerate triple) in the degenerate subset  delta 1 * of delta  2; the map delta1*=delta 1 is the identity (that is no double goes missing). The unit 2 simplex (set of all real non negative triples= im(F) must contain along the edges of the equilateral, every element of delta 1, every set of two number which sum 1).
4. And where among-st these degenerate vectors in delta 1* (the doubles inscribed as triples with a single 0), (not the vertices), must contain, for  each, and every of the two convex, combinations, or  positive real numbers in the unit 1  probability simplex (those which sum to one) at each of them, at least three times, such that every real number value p in (0,1), such that :
p1 +1-p =1,  occurs  at least six times, among-st six distinct degenerate, double vectors <p1, p2,. p3}{i\in {1...6}
that for all p in (0,1)and there exists six distinct degenerate triple vectors mapped to six distinct points in the plane
5. in addition  in must contain  the unit 2 simplex  as the sum of the entries in each triple.
ie among-st the triples <p1, p2, p3> in IM (F) it must be that \for reals, r, in (0,1) and for each possible value of p1+p2 assumes that value , infinitinely many times,
p2+p3 on a distinct vector assumes that value r in (0,1) infinitely many times.
p1+p3 must assume that value r in (0,1) infinitely many times, , Corresponding there must not exist some real value in (0,1) such that one of p1, p2 p3 assumes that value e, and moreover, not infintiely times, and but no sum value on some vector  (ie element of F(p1+p2, p2+p3, p3+p1) that also assumes that value and infinitely times. The entire  unit interval of values must for each such rin [0,1]and for each of three distinct sums in Fsimplex must be contained and assumed individuallyt infinitely many
6, Finally for every element of a 2or more distinct vectors such that the two elements  (p1, p2 , p3) sumto one
such that for any given p1\in vector 1 , p2\in vector 2 ,p1+p2=1
for any given p1 in vector 1 p3 in vector 2, p1+p3=1
for any given p1 in vector 1 p1in vector 2, p1+p1=1
such that for any given p2\in vector 1 , p2\in vector 2 ,p2+p2=1
for any given p2 in vector 1 p3 in vector 2, p2+p3=1 ,
for any given p3 in vector 1 p3 in vector 2, p3+p3=1
6, Finally for every 3 elements (,p1 p2 , p3, p1, +p2, p3+p1, p2+ p3)  in 2 or 3or more distinct vectors v1, v2, v3, such that the three elements in  the three distinct vectors,in  sum to  any of these numbers \forall n \in {\forall n\in {1....48}n/28,1, 4/3, 1.25, 1.5, 4/3. 1.75, 1.85, 2}, or such that  or any 2 elements (p1, p2 , p3, p1 +p2 p1+p3, p2+p3) in common vector v1 and another for all possible other elements in (p1, p2 , p3, p1 +p2 p1+p3, p2+p3) in distinct vector v3 sum to those values these must be present
Moreover it must also at least extend to any given 4/5/6/7/8 /9/10distinct  elements of (p1, p2 , p3, p1 +p2 p1+p3, p2+p3) that sum to for all possible combinations of being in up to 10 distinct vectors, 9 distinct vectors two elements in common,,,,,,,,,,,,,,,,, such that for any such one of all such combinations there must be uncountable many versions of each element of p1, p2 , p3, p1 +p2 p1+p3, p2+p3) in each combination individually for each of the above sum values & for each of the above cardinalities of entries,.
More over the entire simplex must be present for each of foralll of the ten possible combinations  or sum  term number values,10
and for all r each different combinations of elements that sum to that those values in (p1, p2 p3, p1+p2.....)
and for each of the distinct number of distinct vectors that could  be present in that sum upl to ten
and, that could sum to each of all those approximately 70 distinct values. that could to those values, and for each of the different number of terms in each sum,. the entire simplex must be present and every such value in [0,1[ must be assumed individually for every term in every
(1)for each of the term length sum (any given 2 that sum to one, any given three which sum to one, any given four that sum one m any given five which sum to one, any three which sum to 2, any given four which sum to 2 any five which sum to 2, any given four which sum to three, any five which sum to 3
(2)for all of the 70 or so , values mentioned d
(3)for all number of distinct vectors of which those terms are in that sum are associated
(4) for all 6 distinct terms types of  in the sum p2 p3, p1+p2.....)
forall n \in {\forall n\in {1....48}n/28,1/2, 2/3, 1,1.125, 4/3, 1.25, 1.5, 5/3. 1.75, 1.877,11/6, 2,2.25, 2.33, 2.5, 2.66, 2.75, 3, 3.25, 3.33 ,3.5,3.666.3.75, 4, 3.333, 4.5, 4.666, 5, 5.5, 6, 6.5, 7 7.333, 7.5, 7,666, 8,  }
IE there vcant be any GAPS
elements  elements such that in  four or five distinct vectors  with no elments in common
, three or four distinct vectors, two elements in a common vector,the other three/2 being in distinct vectors when there are fouir elements
three distinct vectors, with two elements in two  common vectors, or three elements in one vector common vector,  and two and the other two in either one or two common vectors and the other elements or in 2 elements in one common vector and one in a   common vector, 2  distinct vectors with 2  elements common to each of either one or /two of the vectors, 2 vectors with  3 elements in one vectors and 2 in the other
n a distinct vectors,
such p1 vector 1 +p2 in vector 2 +p3 in vector 3=1
such p1 vector 1 +p3 in vector 2 +p2 in vector 3=1
such p1 vector 1 +p2 in vector 2 +p2 in vector 3=1
such p1 vector 1 +p3 in vector 2 +p2 in vector 3=1
such p1 vector 1 +p1 in vector 1 +p1 in vector 2
such p1 vector 1 +p1 in vector 1 +p3 in vector 2
such p1 vector 1 +p1 in vector 1 +p2 in vector 2
such p1 vector 1 +p1 in vector 1 +p2 in vector 3
such p1 vector 1 +p1 in vector 2 +p2 in vector 3
such p1 vector 1 +p1 in vector 3 +p2 in vector 3
such p1 vector 1 +p3 in vector 1 +p2 in vector 3
such p1 vector 1 +p2 in vector 1 +p2 in vector 3
such p1 vector 1 +p3 in vector 1 +p2 in vector 3=1
such p1 vector 1 +p2 in vector 2 +p2 in vector 3=1
such p1 vector 1 +p2 in vector 3 +p3 in vector 3=1
such p1 vector 1 +p1 in vector 2 +p1 in vector 3=1
such p1 vector 1 +p1 in vector 2 +p1 in vector 3=1
such p2 vector 1 +p2 in vector 2 +p2 in vector 3=1
such p3 vector 1 +p3 in vector 2 +p3 in vector 3=1
such p3 vector 1 +p3 in vector 1 +p3 in vector 3=1
such p3 vector 1 +p3 in vector 2 +p3 in vector 2=1
such p3 vector 1 +p3 in vector 3 +p3 in vector 3=1
such p3 vector 1 +p3 in vector 2 +p3 in vector 2=1
such p3 vector 1 +p3 in vector 3 +p3 in vector 3=1
such p2 vector 1 +p1 in vector 2 +p3 in vector 3=1
such p2 vector 1 +p1 in vector 2 +p1 in vector 3=1
such p2 vector 1 +p3 in vector 2 +p2 in vector 3=1
such p2 vector 1 +p2 in vector 2 +p2 in vector 2=1
such p1 vector 1 +p2 in vector 1 +p2 in vector 2=1
such p1 vector 1 +p2 in vector 1 +p3 in vector 3=1
such p1 vector 1 +p2 in vector 1 +p3 in vector 3=1
such p1 vector 1 +p2 in vector 1 +p3 in vector 3=1
uch that for any given p1\in vector 1 , p2\in vector 2 ,p1+p2=1
for any given p1 in vector 1 p3 in vector 2, p1+p3=1
for any given p1 in vector 1 p1in vector 2, p1+p1=1
such that for any given p2\in vector 1 , p2\in vector 2 ,p2+p2=1
for any given p2 in vector 1 p3 in vector 2, p2+p3=1 ,
for any given p3 in vector 1 p3 in vector 2, p3+p3=1
for any given p1 in vector 1 p1+p2 in vector 2 p1+(p1+p2)=1
for any given p1 in vector 1 p1+p3 in vector 2 p1+(p1+p3)=1
for any given p1 in vector 1 p2+p3 in vector 2 p1+(p2+p3)=1
for any given p2 in vector 1 p1+p2 in vector 2; p2+(p1+p2)=1
for any given p2 in vector 1 p1+p3 in vector 2; p2+(p1+p3)=1
for any given p2 in vector 1 p2+p3 in vector 2 p2+(p2+p3)=1
for any given p3 in vector 1 p2+p3 in vector 2 p3+(p2+p3)=1
for any given p3 in vector 1 p2+p3 in vector 2 p3+(p2+p3)=1
for any given p3 in vector 1 p2+p3 in vector 2 p3+(p2+p3)=1
for any given p1+p2 in vector 1 p1+p2 in vector 2  p1+p2)+(p1+p2)=1
for any given p1+p2 in vector 1 p1+p3 in vector 2 p1+p2)+(p1+p3)=1
for any given p1+p2 in vector 1 p2 +p3in vector 2( p1+p2)+(p2+p3)=1
for any given p2+p3 in vector 1 p3 +p1in vector 2 (p2+p3)+(p3+p2)=1
for any given p2+p3 in vector 1 p2 +p3in vector 2
for any given p1+p3 in vector 1 p3+p1in vector 2
for any given p2+p1 in vector 1 p3 +p2in vector 2
for any given p2+p1 in vector 1 p3 in vector 2
such that for any given p2\in vector 1 , p1\in vector 2 ,p1+p2=1
for any given p1 in vector 1 p2 in vector 2, p1+p3=1
for any given p1 in vector 1 p3in vector 2, p1+p1=1
plus each of the
such that for any given p3\in vector 1 , p2\in vector 2 ,p1+p2=1
for any given p3 in vector 1 p1 in vector 2, p1+p3=1
for any given p3 in vector 1 p3n vector 2, p1+p1=1
for all of the 36 or distinct sombination such that p1 in one vector and one of (p1, p2, p3,p1+p2 p2+p3, p3+p1) on a disitnct vector sum to one , for all reals in [0,1]each of these combination must obtain  infinitely many times, each oif these combinations must be surjective/ bijective with regard to the unit 1 probability simplex, and for each such combination listed, each of the two terms must assume individually ,for each  of the uncountably many real values i m [0,1],, uncountably many times.
for each of the way, and each of the 36 values in each of  that one the 36 and obtain infinitely many time, assume each value within in [0,1] infinitely many time,, and
p2\in vector 2 , p3\in vector 2
, p3 p1
, p3+p1=1, p2+p1=1, p3+p2=1,
p2+p3=1
where these are entries on distinct vectors each of these entries must contain infinitely many distinct probability
to either 1, 2, 3, or the
Moreover, it must that both horizontally and vertically, the entire set of element in the sum to 2 non negative simplex
Where CODOM(F)=IM(F)=delta 1, it must be such that the for bijective function G which for each element of Im (F) , G: maps <p1,p2, p3> \in IM (F)=delta 2 ,to G(<p1+p2, p2+p3, p3+p1)>
, ie F(x,y)=<p1,p2, p3) then G[(x,y))=G(F-1(p1,p2, p3)])=<p1+p2, p2+p3, p1+p3>
such that G is also a bi-jective and analytic diffeomorphism onto
delta_2: {p(i)=<p_1i,p_2i,p_3i>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=2, 1>=p_1i,x_p2i,x_3_i>=0}, the set of all three real number that sum to 2.
ie dom (G)=dom (F) and and thus for any<p1, p2 p3,> domain we compute F-1(,p1, p2, p3) to get the cartesian coordinates of that vector and feed them into G, where G computes the probabilities of the disjunctive events
(unit 2 probability simplex)
that delta_2: {p(i)=<p_1i,p_2i,p_3i>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=2, 1>=p_1i,x_p2i,x_3_i>=0}
in these sums in this sense.
Moreover, it also be the case, that there must exist
, p2+p3, p1+p3
see (4) below
i=(x,y), and i(2)=(x2,y2); . i(3)=(x3, y3), i(4)=(x4, y4), i(5)=(x5, y5), i(6)=(x6,y6);
(x6,y6)\neq (x5, y5)\neq (x4, y4)\neq(x3,y3)\neq(x2,y2)\neq (x,y)
where
F(x2, y2)= p(i(2)=<p_1(i2)=p, p_2(i2)=0, p_3_(i2)=1-p>{i2}; p1+p3=1; 0>(p3, p1)<1,    p2=0, p1=p
F(x3, y3)= p(i(3)=<p_1(i3)=p, p_2(i3)=1-p, p_3_(i3)=0>^{i3}; 0>(p1, p2) <1,  p1+p2=1, p3=0, p1=p
F(x4, y4)= p(i(4)=<p_1(i4)=0, p_2(i4)=p, p_3_(i4)=1-p>{i4); 0>(p3, p2) <1,  p3+p2=1, p1=0, p2=p
F(x5, y5)= p(i(5)=<p_1(i5)=1-p, p_2(i5)=p, p_3_(i5)=0>^{i5)  0>(p1, p2) <1,  p1+p2=1, p3=0, p2=p
F(x6, y6)= p(i(6)=<p_1(i6)=1-p, p_2(i6)=0, p_3_(i6)=p>{i6) 0>(p1, p3) <1,  p1+p3=1, p2=0, p3=p
F(x, y)= p(i)=<p_1(i)=0, p_2(i)=1-p, p_3_(i)=p>{i);
0>(p3, p2) <1;  p3+p2=1, p1=0, p3=p
where for all i\in {i,i(1)...i(5)} and p_t1(m)+p_t2(m) +pt3(m)1 etc
,p1(i)+p2(i) +p3(i)=1
p1(i2)+p2(i2) +p3(i2)=1,
such that p_j_i(2)\in p(i(2); p_j_i(2)\=p
\oplus p2 oplus p3 =0, and

where  ONE and only of p1, p2, p3 =0, where  that precise values occurs at least twice in the first entry of two distinct vectors,
<0.6, 0.4, 0>
<0.6, 0, 0.4>, at least twice in the second entry, p2, here p2=0.6
<0, 0.6, 0.4)
<0.4, 0.6, 0)
and in the third entry p3, p3=0.6; at least twice
<0.4,6, 0.6)
<0, 0.4, 6)
. In other words \forall (p1)\in (0,1) and for all p2=in \in (0,1), amd for all  p3 \in (0,1), where p2+p3=1, there exists two distinct vectors (if only in name) such that <p1=0, p2, p3=1-p2), and < p1=0, p2=1-p3, p3>
there exists
\forall p\in [0,1] ;  0<p1, p2<1; <p, p2=1-p>
possible degenerate triple combinations, for each degenerate convex combination in for all real positive p ;  0<p<1 p, 1-p
<0.6, 0.4, 0> <0.4, 0.6, 0> <0.6, 0.4, 0>, <0, 0.6, 0.4)
[delta_1* ]subset delta2=IM(F)=codom(F)= {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0, & \exists in p(i), one and only one(p2_i, p1_i, p3_i)=0; where the other two entries  \neq 0, s.y1< p1 p2>0}
for all convex combinations in delta 1 , all possible probability doubles,  tuples element of [0,1]^2, of non negative real numbers which sum to 1
where map G(delta1*)=delta1 is the identity
or some subset of I^2\subset R^2 to the unit probability simplex, (the triangle simplex of all triples of non-negative numbers <=1,  which sum to one.?
Are such functions convex, that those which use absolute bary-centric coordinates over the probability simplex, when defined over an equilateral triangle with unit length in the Cartesian plane.
seehttps://en.wikipedia.org/wiki/Affine_space#Affine_coordinates
I presume that such function are hardly homogeneous in that infinitely many possitive triples will not be present?
F:I^2, to {<x_1,x_2,x_3>_m; x1+x2+x3=1, (x_1,x_2,x_3)\in [0,1]\forall (x1,x2,x3)\in [0,1]}
from the set of all or triples {<x_1,x_2,x_3>_m; x1+x2+x3=1, (x_1,x_2,x_3)\in [0,1]\forall (x1,x2,x3)\in [0,1]} to a unique index m,\in I^2, a real interval in the cartesian plane?
dom(F):[0,1]\times[0,srt(3)/2]
IM(F)=F: {p(i)=<p_1,p_2,p_3>_i; |\forall p(i)\in [0,1]^3, s.t: p1_i+p2_i+p3_i=1, 1>=p_1i,x_p2i,x_3_i>=0\in [0,1]\forall (x1,x2,x3)\in [0,1]}
where
F(0,0)=(1,0,0)
F(1,0)=(0,1,0)
F(1/2, sqrt(3)/2)=(0,0,1)
F(x=1/2,y=sqrt(3)/6))=(1/3,1/3,1/3)_(1/2, sqrt(3)/6)   =
ie x=2p2+p3]/2,= [2 times 1/3 +1/3]/=1/2
y=srt(3)/2-sqrt(3)/2p1-sqrt(3)/2 times p2= sqrt(3)/2*(p3)= sqrt(3)/2*(1/3)=sqrt(3)/6
i=(x,y)=F-1(<p1i,p2i,p3i>_i=(x,y))= <x=[2p2+p3+1]/2,y= p3 *[sqrt(3)/2]]
, ill have to check the properties there a lot of other roles that it has to fulfill then just this.
Where in addition, every no value of x1+x2, x2+x3, x3+x1 can be missing these must assume each value in [0,1], prefererably infinitely many if positive and <1, and cannot assume a value, that is not assumed by one of the x1,x2, x3, somewhere in the structure,
Preferably this must property contain the unit 1 simplex, as a function x1, x2+x3, where every convex combination which sum to one, of two values must be assumed,  by x1, 1-x, on distinct vectors <x1,x2,x3>m
x2, 1-x2=x1+x3, <x1,x2,x3>m_1 ,m_1\neq m
x3,1-x3 =x2+x3=<x1,x2,x3_m2, m2\neq m1\neq m,
There also cannot be any mismatched between element of the domain on distinct vectors, ie diagonal or vertical sums, where any two of them sum to one, any three of that sum to 1, or 2,
<0.6, 0.25, 0,15>
<0.4, 0.32, 0.28>
<0.26, 0.4, 0.34>
<0.3   0.4 , 0.3>
<0.26, 0.38, 0.36>   <0.35, 0.35, 0.3>, <0.26,0.4, 28>
I presume if its convex it would contain the doubly stochastic matrices or the permutation matrics
,
<0.7, 0.25, 0,15>
<0.8, 0.32, 0.28>
<0.5, 0.32, 0.28> must be a vectors <0.3=1-0.7x1, 0.5=1-x2=0,5, x1=1-0.8=0.2)
and conversely for <x1,x2,x3> there must be triad of three distinct vectors, such that one elements =1-x1, , another =1-x2, and another =1-x3
<x1,x2,x3, >
<y1,y2,y3>, where one of y1,y2,y3, = 1-x1, one of z1,z2,z3, =1-x2,
<z1,z2,z3>
there must be distinct vectors such that <x1,x2,x3>, x1=25+0.32,
as well for x2, x3, x1+x2, x3+x2, x1+x3,
and which sum to 0.15+0.28,  and common vectors, where all of the six events,  or rather 12 events ,   whose collective sum <= 1, lie on a common vector as atomic events <x1,x2,x3> or disjunctive events <x1+x2,x2+x3, x3+x1>
ie a vectors <0.4, 0.25, 0.35>, and one where <x1=0.4, x2, x3> where x1+x2=0.6
< 0.4, 15, 0.45>, <0.6, 0.28,0.12>
<x1,x2,x3> where x1+x2=0.4
, any 'two sums';,  three, sum  of two elements, to one, or any three sum sum to one
, or three elements of distinct vectors which sum to 2,
and the set of three non positive numbers which sum to 2, as s
Where m denote a Cartesian pair of points in the x,y, plane which uniquely denotes a specific vector, built over an equilateral triangle  in Cartesian coordinates,
and with unit length in side,in Cartesian coordinates
-where this is distance in Cartesian coordinates (x,y) in euclidean norm  of each Cartesian coordinate  probability vectors vertex = F-1(1,0,0),F-1(0,1,0),F-1(0,0,1)=1, where the respective euclidean  norm in probability coordinates is  clearly sqrt (2)  sqrt sqrt(1-0)^2+ (1-0)^2+(0-0)^2)in
sqrt (2) in probability coordinates, in 2- norm,1, in 1 norm,
distance from the triangle center/circum-centre and vertexes, (the Cartesian coordinates of the mid spaced probability vector (1/3, 1,3, 1.3)whose distance from each vertex as a euclidean norm in probability =2/3=equal to the 1-norm distance between value in the triple and its relevant vertex)=2/3, where the overall 1-norm difference in probability between the cent-roid and each vertex =4/3,
=  1/sqrt (3), in Cartesian coordinates in euclidean norm,
all three altitudes=medians (distance from each vertex in Cartesian coordinates, to center of each opposing side of the triangle)-in 2-norm again=
=,sqrt(3)/2
the probability vertices whose Im(F)=(1,0,0),(0,1,0),(0,0,1) whose untoward cartesian coordinates are given above.
and the three, apothem (2-norm distance from the cir-cum center, the Cartesian coordinates (1/2,sqrt(3)/6) of the centroids/probability vector, (1/3,1.3, 1.3),)  and the Cartesian coordinates of the  mid point of each side of the equilateral triangle,
= 1/(2 * sqrt(3))
where the Centro id (1/3,1/3,1/3 )is the vector in n simplex, entries are just the average , n-pt average of the unit (the only vector with three values precisely the same, and the sums precisely the same 2,3,2.3,2,3)
mid probability vector all entries 1/n here 1/3, whose Cartesian coordinate are the circumcentre of the triangle, the point were all three medians cross, (ie the Cartesian point equidistant from each vertex.
that being the circumcentre , (1/2, sqrt(3)/6)  =in 2-norm of the  (the pt whose probability coordinates are just the average for an n simplex of a unit vector (1/n, 1/n,1.n)
denoting the distance between the cen are
F-1(1/n, 1/n, 1/n) here F-1(1/3, 1.3, 1.3)
probability/bary-centric coordinates, with side lengths, between the vertics of sqrt (2)
length  sqrt(2) in probability coodinates (euclidean norm ) ie sqrt ([1-0]^2+[0-1)^2+ [0,-0,]^2)=sqrt(2), and have side lengths =1 in cartesian coordinates,  with distance from the centre 1/sqrt (3),  andall median/altitudes /angular bisectors/perpendicular bisectors=sqrt(3)/2, area=sqrt(3)/4 and apothem=1/2sqrt(3)
F(x,y)=<x-1,x_2,x_3>m=(x,y)
Dear Professor Mateljevic,
When you say that the unit disk is a real analytic bijection to R2, and has a bijective analytic function rto R^2, does it have one ; is it b-ijective to the simplex; are all pts in (p1, p2, p3)  in IM(F) of the unit disk function.  Do any go missing. I presumed that there were certain issues, with the surface area of the sphere
F: dom (unit disk) to . I was wondering if a quadratic function of the northern surface of the unit sphere, would be bijective as defined by F:[0, 2pi times [0, pi]in R^2\to < [p1=sinpi* cosphi]^2, p2=[sin pi * sin phi]^2, p3=cos^2phi> in R^3where  p1+p2+p3=1, and should be non-negative due to the squares.
Whilst the triangular functions appear to be the other way around and maximize surface area
Question
Given a1, a2, ..., an positive real numbers, and defined
p= a\prod { i = 1 to n, i <> k} [(ai)- (ak)2]
how to prove that \sum \frac{1}{pi} is positive?
I had an idea of proof, but not sure it would work....
I have the idea written in the attached .png file.
EDIT: See the .png file here.
Thanks Viera answer I've got that the proof for even  n  is quite sufficient. Thank you Viera for keeping calm while giving interesting answers to interesting questions:)
Best regards, Joachim
PS. Meanwhile I have realized, that replacing   ak2  by  ck  and and assuming increasing ordering (without losses, as Viera has noticed) , we are getting for the sum S of the question the following expression with the use of divided difference of order  n-1:
(n-1)! (-1)n-1 S = (n-1)! [ c1, c2, . . . , cn :  f(c) ],
which equals the value   f(n-1)(b)  of the derivative  of  f  of  order  n-1  at some point  b  from [c1 ,  cn ],  where  in this case   f(c) = 1/\sqrt{c}.
For the calculus of divided differences and their reprepresentation see e.g.
Having this and the sign changes of the derivative of the inverse square root one gets positive value of   S, for any choice of   positive   ck -s .
Further conclusion is the  this holds for every negative power of  c  put in place of  f(c),  and also for every Laplace transform  of a positive measure on  R+  since then the derivatives of order  n  have  sign equal   (-1)n  (cf. the William Feller Bible on probability about the completely monotone functions)  JoD
Question
The quadratic term of a Taylor expansion of a multivariate scalar valued function can be expressed in terms of the Hessian. Is there a similar form for vector valued functions, in the sense that all partial derivatives can be arranged in a matrix or tensor?
It has been quite long since the original question was posed, but if anybody is still interested, I think the answer is in the first formula of Section A.1.2 in this paper
Question
Consider the Lyapunov equation given by A'P+PA+I=0, where I is the identity matrix, A is Hurwitz, and P is a positive definite and symmetric n by n matrix. How can we find an upper bound to the Frobenius norm of P, i.e., ||P||_F, using the eigenvalues of A ?
I was able to find a relationship with eigenvalues of P(see attached picture) but I need to find out relation with eigenvalues of A.
Any idea how should I proceed?
In your opinion what is Frobenius Norm?
Question
Is the series in the picture convergent? If it is convergent, what is the sum of the series?
Dear Joachim,
you've right. Just ignore the first part of my answer. May be the double series calculation will lead to some result.
Rgrds,
Tibor
Question
(1)  How  can we   find    the partial sum    of   n1000  instantly ?
(2)  is  there  is  a  simple method to find  partial sum  of the  sequence  f(n)  ?
(3)  Any general method   to compute partial   sum of sequence   ?
(4)  What is the  value  of   Method , if we have   good  approximation  for all differentable  sequence   ?
@ Juan Weisz
That  is more than that   .  It  is   a little secret   have little secret     of  analytic continuity   and discreetness  of  integers
Question
When performing CSP using the toolbox ( from the link http://sccn.ucsd.edu/wiki/Plugin_list_process ) , there are two places where eigen value decomposition is occurring . In the first one , the eigen values have been sorted in descending order but in the second eigen decomposition , it is not . However , the paper " Optimal Spatial Filtering of Single Trial EEG During Imagined Hand Movement " describes in CSP section that the descending order is assumed during the decompositions throughout the section . Should eigen values in both the decompositions ( as per the CSP script ) be arranged in descending order only ? Any help is greatly appreciated . Thanks in advance
Thank you
Question
I has made proposition and prove that (f o f ')(x)= (f 'o f )(x) if f is additive real function and (f o f) (x)=f (x) but I still difficult get example non-trivial. Could you help me? example trivial is an identity function.
I apologize for the inconvenience this
By "additive function" do you mean function on (-\infty, + \infty) that satisfies equation f(x+y) = f(x) + f(y)? Then all such continuous functions are of the form f(x) = cx.
Since in your question you are speaking about the derivative, then you must deal only with continuous functions. So, f(x) = cx and you cannot get other examples for the identity (f o f ')(x)= (f 'o f )(x) with additive functions.
There are only two functions of the form f(x) = cx that satisfy the second identity
(f o f) (x)=f (x), namely f(x) = x and f(x) = -x.
Question
We call local homeomorphism of R a function f:U--->R with: U open neighborhood of 0 \in R, f(0)=0, f injective continuous and open.
We identify two local homeomorphisms of R if and only if they coincide in a suitable neighborhood of 0.
We denote by LH the set of all equivalence classes of local homeomorphisms of R.
LH becomes a group when endowed with the operation of composition of functions.
I am interested in results describing the structure of the group LH.
It seems to me "germs of homeomorphisms at 0"
Question
The string detained between two points can sustain harmonic motion.  At the string midpoint, the potential and kinetic energy are inverse, so that when the string is in the mid-line, the kinetic energy is 1 and the potential energy is 0. When the string is at the boundary the string stops for an instant, so the kinetic energy goes to zero while the potential energy is 1.
A node is defined as a point on a string where there is no movement possible, so that both the kinetic and potential energy are zero.  The fixed-point theorem says if pitch is a real function defined on [0, 1], then there must be a fixed point on the interval.
Since the kinetic and potential energy is just the result of basic trig functions sin and cos, it seems clear to me then that wave reflection cannot occur at the string endpoints.  The endpoints are fixed points which are in effect fulcrums with a fixed-point position so that length L = 1 is a bound variable.  The fulcrum allows the fundamental in the monochord to drive the string on the other side of a node, but the condition for wave reflection does not exist at the node.
When a sin wave crosses zero there is no requirement that the point is fixed, so the boundary cannot simply be added to the wave function arbitrarily without changing the nature of Fourier analysis.
If the waves reflect at endpoints, do they also reflect at nodes that are not enpoints? Of course not!  But then, in the 1/3 mode, what makes the middle wave where there are 4 nodes and 3 waves?  Is the middle wave the reflection of 2 traveling waves between the two non-endpoint nodes?
The boundary condition for traveling wave reflection is 1, 0 , 1 which is clearly a false statement.
Significantly, the frequency and the wave length are bound by the string and not free variables subject to real analysis as continuous variables.  Nodes and waves cannot add at the same point.
If physicists really think there are two traveling waves on a string moving in opposite directions that make a standing wave, but no one can see or demonstrate these waves, then maybe they have action-at-a-distance wrong, too.  After all, the basic error is assuming frequency is continuous, and then using functions with free variables adjoined with arbitrary integers.
But in the typical reflection the kinetic energy goes to zero at the wall while the potential energy is at a maximum so there is a force that explains the reflection,  If a wave travels down a funnel it does not reflect.  So I say these imaginary traveling ways are nonsense.  Where is the equation of motion if the wave is standing?  Where is the stroboscopic image of these two waves that some how reflect to make one.  There is a curve lifting function between the string at rest and the string in wave form that is point-for-point so the only force vectors are orthogonal to the string axis.
Statements about the string without quantifiers are provable.  Can you prove there is a mode lower than the fundamental with wavelength 2L? Of course not.
Question
Greeting and salutation!
I have a initial value differential equation with a unknown parameter, how can i solve it With Matlab ODE solver or other software????
With Best Regards
Hamed
Dear Prof Abram
attached figure is attached for clarification; don't hesitate if more explain is necessary
Thank you so much for your time gift
Hamed
Question
that means, I have only research keywords for 1500 research paper but I dont have the text of them or even the abstracts , so Can I use this co-word analysis or co-occurrence technique just for keywords?
Hello,
Yes, you can, but the research will be very limited.
Take in consideration that the key works could have synonyms in the text itself…. So, you will be missing out on them.
Having 1500 papers….that gives a better perspective of the key words…. You could correlate them with the words/terms that you used to find the papers [well, the keywords of the papers]. Another step is to find synonyms of those key words as well….in order to give you more substance.
But in your research, you have to specify that you analyzed only the key words.
Regards,
Alex
Question
We know that Heine-Borel property and related theorems for topological vector spaces. Do we have similar notions for abelian topological groups? For example when a topological group has this property or not? Can you introduce me a reference about that?
In vector space one can define the norm and it will be a metric space. It is impossible to consider this property in topological groups (in nonmetrizable case).
Question
I have a doubt regarding completeness of C[0, infinity) with respect to sup norm. Is the space Banach Space? Is it a Banach lattice?
Actually the sup norm is not defined on C[0,\infty) since this space contains unbounded functions. If you consider the space C^b[0,\infty) of bounded continuous functions (or the space suggested yb Luiz) , then the sup norm is well defined, and this space is complete (basically since the uniform limit of continuous functions is continuous).
Question
For example, Circle is not a function. Can any one explain with this inverse and implicit function theorem?
1. The  unit  circle   x^2 + y^2=1   is not a function,  but locally  is  a graph of a function .Set  f(x,y)=x^2 + y^2-1.  Then the partial derivatives  are   f'_x=2x    and   f'-y=2y.Hence  if    (x_0,y_0) is a point on the unit circle    and for example  y_0     is not 0,  by the implicit function theorem  locally  there is  a function y=g(x)   which is solution of the equation    f(x,y)=0.Denote   by   g_0(x) the   square root of 1-x^2, for x between -1 and 1. Actually if y_0  >0 , y=g0(x)  is solution    and    if y_0 <0 , y=-g0(x) is solution.  In a similar way we can discuss the case if   x0  is not 0.
Question
This paper provides a simplified exposition (no real analysis) of the economic theory presented in the second part of my 1999 book, Axiomatic Theory of Economics.  It makes no mention of the first part of my book about the foundations of economics.  In this question we will discuss my three-term system of formal logic, specifically with comparison to the attached paper by Steve Faulkner, which was posted in reply to another question that I recently asked.
In Section 4.3 of my 1999 book I write:
“The great crack in the foundation of mainstream logic where first-sense and third-sense truth are confused has been resolved.  Whenever mainstream logic speaks of affirmation they refer to phenomena having been observed that conform to a definition (truth in the first sense) and whenever they speak of negation they refer to the impossibility of phenomena conforming to a definition (falsity in the third sense).  The three senses of truth must be strictly separated…”
The three sense of truth are defined in an earlier section of my book, but suffice it to say that I was not bothered so much by the paradoxes that Gödel addressed but by the fact that, if p is impossible, the statement “some p are q” is false while the statement “all p are q” is true.  This is absurd.  If I told you that all red-headed Eskimos can foresee the future, a logician would have to admit that, within his science, this statement is true.  But everybody else would denounce me as a lunatic:  Eskimos do not have red hair and nobody – regardless of the color of their hair – can foresee the future.  The logical truth value of my statement will not inspire anybody to travel to Alaska to find Sibyl the Eskimo with her flaming red hair.
“A new system of formal logic will now be introduced.  The three terms of this system of logic are P for possible, I for impossible and M for maybe (similar to Zen Buddhism’s mu.)  Following are eleven logical relations concerning the definitions p and q.  These statements are followed by a truth table which shows, in each of the four situations with which one could be presented when observing phenomena’s conformance to p and q, whether the statement affirms its possibility, its impossibility, or says nothing about that situation.”
While I do not have space here to print the entire list of eleven logical relations, I will print the truth table for “p is possible unless q is possible” to give a taste of what I am doing:
Do phenomena conform to definition p?                T   T   F   F
Do phenomena conform to definition q?                T   F   T   F
p is possible unless q is possible                            I   P   M  M
I then use an example from Willard Quine’s Methods of Logic (p. 196) to illustrate how my method works:
Premises:
1)  The guard searched all who entered the building except those who were accompanied by members of the firm.
2)  Some of Fiorecchio’s men entered the building unaccompanied by anyone else [unaccompanied by non-Fioreccio men].
3)  The guard searched none of Fioreccio’s men.
Using my system, by filling in a truth table with P (possible), I (impossible) and M (maybe), we can quickly determine if the statement, “Some of Fioreccio’s men are members of the firm” is proven.  There is no room to print this here, but it is a sixteen-column truth table with four rows of P, I or M for each of the three premises and the relation, “people who work for Fioreccio.”  Below this is another row labeled “result.”
“Now, filling in an I wherever we see one, a P wherever we see one that is not dominated by an I, and an M only where no statement is made either way, we get the result.”
This is in contrast to Dr. Quine’s method (p. 199), which only proves or disproves one statement at a time.  I write:
“From this result [the three-term truth table] one can test the truth of any conclusion one is interested in…  If we were interested in knowing whether the statement ‘All of Fiorecchio’s men entered the building unaccompanied by non-Fiorecchio men’ is implied by the premises, we would need [elaborate what is needed that we do not know] so the conclusion is not proven; it is a maybe.  This is a more insightful ‘maybe’ than we had before analysis, however, as we now know where our investigation must lead.”
REFERENCES
Quine, W.V.  1982.  Methods of Logic.  Cambridge, MA:  Harvard University Press
Victor, I am both a mathematician and a philosopher, and I work a lot with Logic, which is my preferred part.
I have just studied Godel's proof of God and you can see the result of that in the book On Religion, which is found on my profile here.
I think that the aim of Logic is not confusing people, but making things easier and nicer instead.
Priest, like you, loved formalizing anything.
The problem with that is that you end up working in the abstraction, but you then want to come back to reality in the end and, whilst the first move could be acceptable, if things were ever done properly, the second move is usually impossible to to accept.
Basically, I have written a lot about this by now: We can fit a reduced universe into a large one, but there are lots of things that escape logic in human language/discourse and the reverse move is not usually OK.
The sentences you wrote are easily dealt with without us changing them into symbols and, therefore, the need to do that does not exist. We are not simplifying if doing that either, we are actually complicating and probably missing the point, so that that is also not advisable.
What Prof. Faulkner says is true and it actually occurred to me as I read your writing, but in a much more informal way.
If you read more of my work, you will understand why this concern you seem to be having is actually equivocated for Mathematics. Creating new systems of logic, as Priest does, is fun, but may also be completely useless activity.
Apart from learning the concepts involved in the modelling of the systems and applying those, there is nothing to it.
If you want them to be useful, you must make sure you find a situation for which we feel the necessity of coding things. In second place, as you have probably noticed, you should also check if the system already exists.
Remember that we must improve things with them, not confuse people even more or lead them to mistake in reasoning.
Priest has a book with almost all non-classical systems and you could actually use his book, since it works like an encyclopedia, to find out if a system already exists or not.
I think he finished this book in 2000, after he went through it with me.
Question
Dear RG friends:
In two weeks time, I am all set to conduct a technical session on Analysis.
I plan to deliver a long lecture on "Fixed Point Theorems". Of course, Banach fixed point theorem is useful to establish the local existence and uniqueness of solutions of ODEs, and contraction mapping ideas are also useful to develop some simple numerical methods for solving nonlinear equations. Are there any other interesting science / engineering applications?
Kindly let me know! Thank you for the kind help.
With best wishes,
Sundar
As applications :   logic programming and fuzzy logic programming
Nash equilibrium and game theory.
Question
What is the limsup and liminf of 1/n+(-1)^n of an alternating Sequence?
In my view Lim Sup is 1 and Lim inf is -1 for the given sequence.
The simplest answer is the answer of Prof. Ismat Beg, but you may writ it as,
Let yk=infn>k xn, zk=supn>kxn then
limsup xn=the infimum of zk, k rangers over natural number and
liminf xn=the supremum of yk, k rangers over natural number and
This formula is more easy for students.
Question
In 1999 I published a book, Axiomatic Theory of Economics
Since then I have found that economists who have not read even the simplified exposition will invoke the name Kurt Gödel when dismissing my theory.
I know who Gödel is, but I do not see what the foundations of mathematics have to do with me.  I rely only on widely accepted calculus and real analysis results that should be familiar to any practicing engineer.  The antipathy I get from economists has nothing to do with number theory – most of them would be hard pressed to even define a prime – it is all about me stating my assumptions clearly before proving my theorems.
So my question is:
How should I respond to people who invoke Gödel’s name when dismissing my work?
I am reminded of Van Helsing holding up a cross to Dracula, except for economists it is Gödel’s Incompleteness Theorems that ward off the evil logician.
Have other people at Research Gate faced similar criticism?  How did you respond?
FYI  I am NOT a follower of Gerard Debreu.  I have my own theory.  Something else that I have noticed about economists is that they are incapable of recognizing that it is possible to have more than one axiomatic theory that purports to describe the same phenomena.  I have found it impossible to disabuse economists of the belief that Debreu (who was parroting Bourbaki) fully defines the axiomatic method.
Economists claim that the practice of deductive logic rises or falls with the fortunes of this one man, regardless of what axioms the practitioner is using.  I reply that, since Debreu lost all of his followers in 1974 when his theory went down in flames, accusing me (who was eight years old at the time) of having ever been a follower is actually a straw man attack.
I agree with many of the above comments to the effect that Gödel is just a red herring in this context. A supplementary consideration might be to invoke a notion I would assume economists are familiar with, namely "satisficing". An axiomatic system need not attempt to capture everything, not even many things that would be easily capturable if one cared enough to capture them. One may simply want a system that is good enough for the purposes at hand. And such a pragmatic approach can itself be justified on economic grounds.
Question
I need these measurement for the diversity analysis of population of vectors
The (vector) norm not the suitable measure in this problem.
Question
Consider the cubic equation:
$t^3 = 3pt + 2 q \qquad (1)$.
We introduce two variables $u$ and $v$ linked by the condition
$u+v=t\,$
and substitute this in the depressed cubic (1), giving
$u^3+v^3+(3uv-3p)(u+v)-2q=0 \qquad (2)\,.$
At this point Cardano imposed a second condition for the variables $u$ and $v$:
$3uv-3p=0\,$, ie $uv=p$.
As the first parenthesis vanishes in (2), we get $u^3+v^3=2q$ and $u^3v^3=p^3$. Thus $u^3$ and $v^3$ are the two roots of the equation
$z^2 -2 qz + p^3 = 0\,.$
$(z-q)^2 =q^2 - p^3=D$. Hence $z= q + \sqrt{q^2 - p^3}$. Denote by $a_1$ and $a_2$ two roots of $D$ and set $z_k=q +a_k$.
Solutions of equations $w^3=1$ are $1,e^{2\pi i/3},e^{-2\pi i/3}$.
Denote by $\underline{A}=\{u_1,u_2,u_3 \}$ and $\underline{B}=\{v_1,v_2,v_3\}$ solutions of equations $u^3=z_1$ and $v^3=z_2.$
Check that $u_k v_j \in \{p , p e^{2\pi i/3},p e^{-2\pi i/3} \}$. Set $p_1=p , p_2= p e^{2\pi i/3}$, and $p_3=p e^{-2\pi i/3}$.
In discussion with my students (Svetlik,Knezevic,Stankovic,Avalic), we prove the following:
Proposition 1.
The set $\underline{X}= \{u_k + v_j:1\leq k,j \leq 3 \}$ has $9$ elements if $q + \sqrt{q^2 - p^3} \neq 0$ and six elements if $D=0$ and $q\neq 0$.
If $u_0\in A$, $v_0\in B$, then $\underline{X}= \{\omega^k u_0 + \omega^l v_0:k,l=0,1,2\}$ and
the points of $\underline{X}$ forms three (respectively two) equilateral triangles if $D\neq 0$(respectively $D=0$ and $q\neq 0$).
Presently, we did not find origin of this result in the literature.
What are possible generalizations of this results?
Dear Miodrag:
Please allow me to add just a very simple remark (which does not contribute to your question). There is a more intuitive way to Cardano's formula: Why should someone put $u+v=t$ and assume $uv=p$? Here is the idea: We want to solve the cubic equation
$$t^3 = 3pt + 2q \eqno(1)$$
which is difficult. But other cubic equations are apparently easy:
$$(t-u)^3 = v^3 \eqno(2)$$
is solved by $(*)$ $t - u = v$ or $t = u+v$. Now the idea is to transform the easy equation (2) into the difficult'' form (1). In fact,
\begin{eqnarray}
(t-u)^3 &=& t^3 - 3ut^2 + 3u^2t - u^3 \cr
&=& t^3 - 3u(t-u)t -u^3 \cr
&=& t^3 - 3uvt - u^3 \nonumber
\end{eqnarray}
using $(*)$. Thus (2) is of the form (1) with
$$p = uv,\ \ \ 2q = v^3 + u^3.$$
Solving these equations for $u^3$, $v^3$ gives Cardano's formula.
Best regards
Jost
Question
I would really appreciate if anyone can give me a formal citation for this. I know this is a standard problem for which I also have a proof. However this is requried as a part of a more complicated problem and it would be nicer for the writing if I can just simply give a citation.
I would recommend looking at the following paper by Clarke (1975): www.ams.org/tran/1975-205-00/.../S0002-9947-1975-0367131-6.pdf . This paper gives some very general results on functions of the form f(x) = max {g (x,u) : u \in U} (such as when they are continuous or differentiable). It has 1099 citations, according to Google. And I believe it is a well-known paper in optimization circles.
Question
If K and L are convex bodies whose radial functions have the same distribution function, i.e. equi-measurable, what can be said? In particular, does this mean that the two radial functions(of the sphere) are equal up to composition with some measure-preserving transformation? If so, does the convexity of the two bodies imply any regularity of this measure-preserving transformation of the sphere? If possible, how much regularity must the bodies possess to force this transformation to be a rotation or orthonormal linear transformation of the sphere?
Pure math is always ahead of the game because we pave the roads and identify the dead ends for applied sciences. Sometimes we pave the roads so far ahead of technology that our  results get lost or forgotten. I'm sure there are answers to many applied questions hidden away in decades old pure math publications but the applied scientists don't have the vocabulary to even do a useful search for this information. There needs to be more communication from both sides.
Question
we say Chebyshev wavelet family or basis and why ??
Thank you for answering Mr Hammad Khalil,
But in the legendre one the weight function w(x)=1 but in chebyshev one  we have four kinds of chebyshev wavelets and four weight functions wich are =~1 , my question is :when we construct the discret wavelet we will translate and dilate the wavelet mother, to form the chebyshev wavelet ,and we will translate and dilate the weight function too ,the result that we get is many spaces L2(wn) , where is the orthonormal basis here  and in which space !!!!!
Question
Let ||x||_1\leq ||y||_1. Is it possible to say ||x||_p\leq||y||_p  for 1<p? In another way, the inequality in l1 is protect in lp or not?
The answer to your question is in the attached file "Answer25.02.2015.pdf".
Question
I want to teach real analysis course this semester. I want to teach more about functional analysis and less about real analysis. I need a brief and new book. Could you please guide me?
Thanks Jean for your guide.
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This is motivated by a calculation using Fourier transforms of filters / mollifiers. Since it is such a classical question, I suspect there is a classical answer out there and would appreciate help satisfying my curiosity.
Dear Bill:
If I understand it correctly, you are expanding a function f in Taylor series at a=0, which is natural, as the function is even. As such, all odd-order derivatives of f at 0 must be zero. Thus, we deal only with f^(2k)(0) and this sequence should alternate. That's it. It seems not to exist any reasonable classification of functions with this property.
All the best,
Roman
Question
In the figure lines are on the real axis.
What I am interested in to prove that the left limit of the red interval is to the left of the left blue one and correspondingly for the right end.
I was thinking that we can prove it by the definition of the left limit and right limit but I stuck with in it.
Any suggestions to prove it mathematically will be highly appreciable.
The question is incorrect.
Generally it is not valid.
If you have a concrete function, we can discuss its limit.
Question
Some nowhere differentiable functions are fractional differentiable and comply with the Hölder condition.
Dear Prof Munoz-Vazquez,
I think that the answer is yes. It was proven in
1. S. Samko, A. Kilbas and 0. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon & Breach Sci. Publishers, London- New York, 1993 (See page 239, Lemma 13.1, and page 242, Corollary of Lemma 13.2)
or
2. Ross, Bertram; Samko, Stefan G.; Love, E. Russel. Functions that have no first order derivative might have fractional derivatives of all orders less than one. Real Anal. Exchange 20 (1994/95), no. 1, 140--157 (see Lemma 2' p. 146)
that Hölder continuous functions has some Riemann-Liouville fractional derivatives.
See the attachement.
Question
I need a counter example or proof to confirm this statement: The product of two divergent series may not be divergent.
The product here is Cauchy Product but not pairwise product (in that case is easy to say harmonic series).
DEar Yin Zhao, I think I have a counter example: consider power series
1/f=(1-z)(1+z) and 1/g=(1-z)(1+z^2). Now each of these power series is divergent in z=1 but their ratio is convergent. Note that the Cauchy product of the Taylor series of f and g sums to the product f(z)g(z). Of course some familiarity with power series might help...
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By space of measurable functions, I mean L_0(m) where m is a non-atomic sigma-finite measure space.
In my opinion, you can start with the book by Martin Vaeth "Ideal Spaces" LNM 1664. But, of course, you should think about classes of measurable functions...
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I always wonder about correctness of this fundamental result which is used in most of the proofs in mathematics and how we can construct results without it.
For followers of this thread, the following article may be of interest:
J.S. Russell, Indefinite divisibility:
See page 7, where the author writes:
The reason for the apparent failure of Zorn’s Lemma is that the lemma (like the comprehension scheme) involves an ambiguous quantifier (twice over, in stating the existence of a chain, and in stating that it is maximal). While we can truly say there is_1 a chain that is maximal_1 (i.e. no_1 chain properly contains it), we may yet go on to use a more inclusive quantifier “there is_2 ”, with respect to which that chain is not maximal_2 after all—for there is_2 a chain that properly contains it. This is just what we should expect in the presence of indefinite extensibility.
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I think I may have this but have not seen anything in the Literature.
Dear Geoff DIESTEL
Since you are interested in the "weak-L^{p}" space (introduced by Marcinkiewicz, I suppose), which coincides with the Lorentz space L_{p,\infty} if 1<p<\infty, then it is the dual of another Lorentz space, namely L^{p^{\prime}, 1}. I am not sure if it was known before the general theory of interpolation of Banach spaces by Jacques-Louis LIONS and Jaak PEETRE, and I have described their theory (a little simplified after their "joint work" by Jaak PEETRE's work) in my (second) book
An Introduction to Sobolev Spaces and Interpolation Spaces
Springer 2007, series: Lecture Notes of the Unione Matematica Italiana, Vol. 3
It means that there is a weak \star topology on L_{p,\infty} for which every closed bounded convex set is compact. However, the space is not reflexive, although L^{q,1} is also a dual for 1<q<\infty.
The weak topology on L_{p,\infty} is then different, and it is probably for this topology that you ask the question, and I suppose that your notation D_f(t) means \int_{0}^{t} f^{*}(s) ds, where f^{*} is the non-increasing rearrangement (introduced by Hardy & Littlewood, I suppose).
My guess is that the condition you are looking at does not work, but maybe the answer is in the article which Yuri BRUDNYI mentioned, which I have not taken the time to look at.
Best regards
Luc TARTAR
Question
I want to prove that if f is an increasing function on [a,b], then the function g(x)=f(x)+f(a+b-x) is decreasing on [a,(a+b)/2] and increasing on [(a+b)/2,b].
Since f is an increasing function on [a,b] then of course the function f(a+b-x) is also an increasing function on [a,b]. Then how is it possible that the sum of two increasing functions is decreasing on the subinterval of the domain?
Question
If yes, is the notion of Bernstein polynomial the same in both intervals.
There have been lots of good answers already, (especially by @Aurelian Gheondea) but here is an example why things go completely wrong on an open interval. W.l.o.g. assume a = 0, b = 1. Consider a continuous function f that blows up at 0, e.g. f(x) = 1/x. Any polynomial P on (0,1) extends to [0,1] and so is bounded. Thus
|| f - P ||_\infty = sup_{x \in (0,1)} |f - P| = \infty
You can approximate f on every increasing sequence of subintervals [\epsilon, 1- \epsilon] as well as you like but that just does not get you closer to approximating such a function f in the sup norm. This is why for most approximation problems of continuous functions on topological spaces people use the topology of uniform approximation on compacta (or a cofinal collection of compacta, i.e. a collection of compacta such that for each compactum you can find a larger compactum in the cofinal collection, eg. on the unit interval the collection of all closed intervals or the collection of intervals of the form [\epsilon, 1-\epsilon] , as in the answer of @Miguel Carrion Alvarez)
Question
I am looking for a list of classical spaces which are known to compactly embed into the space of measurable functions and any which are known NOT to compactly embed into L_0. L_0 is just the space of measurable functions over the interval (0,1) equipped with Lebesgue measure.
I believe you consider L^0 equipped with the metric of convergence in measure. Then the first answer to your question is that NO L^p space is compactly embedded into L^0. Indeed, consider a sequence f_n, constructed as follows:
the segment (0,1) is split into 2^n intervals of equal length, f_n=1 on odd intervals and f_n=-1 on even intervals (flipping sign function). Obviously, |f_n| =1 at every point so
|| f_n ||_{L^p} = 1 for all n and p.
On the other hand, \int_0^1 f_n ds =0 and it is easy to verify that
\int_0^1 f_n s^m ds < c 2^{-n} for all m =1,2,3,...
Since polynomials are dense in continuous functions, we conclude that
the measures f_n ds converge to zero. So an L^0-convergent sub-sequence f_k of f_n converges to 0 if exists.
However, | { |f_n| =1} | = 1, so no sub-sequence of f_n converges to 0 in L^0.
I believe, neatly approximating f_n by continuous functions one can show that C(0,1) does not compactly embedded into L^0 either. Probably, even the harmonics f_n(s) = sin( \pi n (s-1/2 ) ) would do.
On the other hand, L^p(0,1) and C(0,1) are continuously embedded into L^0(0,1). Hence spaces compactly embedded in L^p(0,1) and C(0,1) - such as Sobolev and Hoelder spaces and functions of bounded variation - are compactly embedded into L^0.
It seems all about classic functional spaces.
Question
Assume the underlying measure is a probability measure. I think I've heard this is true but I could be wrong.
Dear Geoff Diestel
I am not used to play with weird measured spaces, but on the part of the set where the measure has no atoms, one may use a theorem of Liapounov (not the famous one who worked on ODE) for which Zvi Artstein has given a quite simple proof in the mid 70s (Look for the Extreme Points), and construct a sequence of characteristic functions converging weakly * in L^{\infty} to 1/2 (hence a sequence like the Rademacher one mentioned by Anton Schep).
Luc TARTAR, mathematician
Question
(The inequality is meant to hold for every sequence x_n converging to x.)
Please note that I did not mean >= or liminf, i.e. upper or lower semicontinuous.
I apologize for answering a question with a question, but why do you suspect this type of function should have a name?
Upper/Lower semicontinuous functions are useful for optimization problems, and producing solutions to elliptic equations. Essentially, if f(x) <= liminf f(x_n), we can "chase" minimizing sequences and arrive at local minima (or likewise for local maxima). Thus, we have tools to identify "special" points and have certainty that they satisfy the local min/max properties we desire.
On the other hand, f(x) <= limsup f(x_n) simply says that x is not "special", in some local sense. This doesn't seem to be a useful property for a function without further restrictions. By which I mean, I am unaware of any applications which utilize such a property to derive any profound result.
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I ask because this seems to allow one to replace compactness with weak-sequential-compactness(and weak-compactness for Banach spaces of measurable functions) in some arguments. For example, if X is such a space of measurable functions then any weakly-sequentially-compact subset will have the property that convergence in measure is equivalent to convergence in X.
Maybe Proposition 1 of the paper "Theorems on representation of functions by series"
by K.S. Kazarian and D. Waterman can be helpful for some cases? You can download the paper from my page in RG.
Question
Factorial analysis.
0^0 and 0! are equal to 1 because of the empty product rule. The IEEE standard says that 0^0 should evaluate 1. So most computer software will evaluate it that way. There are no situations where evaluating to 0 is helpful.
There are many formulas that assume 0^0=1, e.g. notation for polynomials and power series, cardinal number arithmetic, the binomial theorem, etc., these all assume 0^0=1. For example, substituting x=0 into x^0+x^1+x^2+... = 1/(1-x) shows 0^0 = 1.
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