சனி, 26 ஆகஸ்ட், 2023

பேய்க்கற்பனை

 பேய்க்கற்பனை

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ருத்ரா


கடவுள்கள்

அழகிய மலரைக்கையில் ஏந்தி

புன்னகை பூத்தால் போதுமா?

கடவுள் கற்பனையாளர்களுக்கு

ஒரு பொறி தட்டியது.

அது ஒரு பேய்க்கற்பனை.

இந்த அன்பின் வெளிச்சம் 

எவனுக்கு வேண்டும்?

கடவுள் என்பவன் 

நாக்கு துருத்திக்கொண்டு

கோரைப்பல் காட்டி

மனிதக்குடல்களை மாலையாக‌

சூட்டி

கபாலங்களின் மேட்டில் உட்கார்ந்து

ஒரு "பய தரிசனத்தை"

விஸ்வரூபம் காட்டவேண்டும்.

அதிலிருந்து முளைக்கும்

பயபக்தியில்

மனிதர்களின் 

அறிவு முயற்சி

ஆற்றல் எழுச்சிகள் யாவும்

கிள்ளி எறியப்படவேண்டும்.

கடவுள்கள் இருக்கவேண்டுமானால்

மனிதர்களுக்கு இடம் இருக்கக்கூடாது.

மனித வெற்றிகள் 

மண்ணோடு மண்ணாக்கப்படவேண்டும்.

இந்த அடிநிலையின் சமுதாய உளவியலே

சாதி வட்டங்களில் நச்சுக்குமிழிகளை

மூச்சு விடத்தொடங்கியிருக்கிறது.

ஆம்!

மாமன்னன் திரைப்படத்தின்

வில்லனே

இப்போது விண்ணோங்கிய வீரனாக‌

தெரிவதாக 

ஒரு ட்ரெண்டிங் வந்திருக்கிறது

என்கிறார்கள்.

இது எவ்வளவு

அபாயகரமானது?

கொடூரமானது?

மனித மலர்ச்சியே

இந்த அண்ட மலர்ச்சியின் 

அறிவு மகரந்தங்களை ஏந்தியிருக்கிறது

என்ற பரிணாமமே

குழி தோண்டி புதைக்கப்பட‌

ஒரு வெறிப்படை கிளம்பியிருக்கிறது.

மனித வெள்ளமே

இந்த கட்டுகளை உடைத்துக்கொண்டு

அலையாய் எழு.

உன் மனிதநேயமே இங்கு

கூர் தீட்டி எழும் ஆயுதம்.

உன் படையே வெல்லும்.

இந்த பொய்மூட்டங்கள்

கலைந்து ஓடட்டும்.


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சூப்பர்ஸ்டார்

 சூப்பர்ஸ்டார்

____________________________________________‍


சூப்பர்ஸ்டார் என்றால்

நாம் நினைத்துக்கொண்டிருந்தோம்

சிகரெட்டை உயரத்தில் போட்டு

வாயில் கப்பென்று பிடித்து

பற்ற வைத்துக்கொள்பவர் 

என்று.

கருப்புக்கண்ணாடியை

போடுவதும் தெரியாது

கழற்றுவதும் தெரியாது

மொத்த சூரியனையே

கண்ணாடியாக்கி வித்தை காட்டுபவர்

என்று.

இந்தப்பக்கம் 

காதலிக்கு முத்தம் கொடுத்துக்கொண்டே

அந்தப்பக்கம் 

வில்லன்களுக்கு கும்மாங்குத்துகள் 

பொழிபவர் என்று.

ஆனால் இப்போது

இப்படி ஒரு சூப்பர்ஸ்டாருக்கெல்லாம்

சூப்பர்ஸ்டார்

காவி உடுத்துக்கொண்டு

புல்டோசராக வந்து நிற்பார்

என்று

நமக்குத்தெரியவே தெரியாது.


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26.08.23

வெள்ளி, 25 ஆகஸ்ட், 2023

chatbot /16.08.2023

 

Welcome back! What would you like to chat about?
The direct sum of a dilation-type and a von Neumann-type row isometry, conclude wonderfully a picture of Lebesgue measure decomposition.Here we see the von Neumann part vanishes smoothly. Can you give a crisp and brief theoretical exposition with mathematical elegancy that too more in a verbose structure?

Searching for: direct sum of a dilation-type and a von Neumann-type row isometry

Searching for: Lebesgue measure decomposition

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I’ll try to create that.

The Lebesgue measure decomposition theorem states that any σ-finite signed measure ν on a measurable space can be written as the sum of two σ-finite signed measures ν0 and ν1, such that ν0 is absolutely continuous with respect to another σ-finite measure μ (denoted by ν0 ≪ μ), and ν1 is singular with respect to μ (denoted by ν1 ⊥ μ). This means that ν0 assigns zero measure to any set that μ assigns zero measure, and ν1 assigns zero measure to any set that has positive μ-measure1

In the case of the real line, the singular part of a regular Borel measure can be further decomposed into three parts: the absolutely continuous part, the singular continuous part, and the pure point part. The absolutely continuous part is the one that has a density function with respect to the Lebesgue measure, the singular continuous part is the one that has no atoms but is supported on a set of Lebesgue measure zero, and the pure point part is the one that consists of atoms (or point masses)1

A row isometry is a linear map from a Hilbert space H to a direct sum of Hilbert spaces H1 ⊕ H2 ⊕ … ⊕ Hn such that it preserves the inner product between any two vectors in H. A row isometry can be represented by a matrix whose rows are isometries from H to Hi. A row isometry is called pure if it has no non-trivial reducing subspaces, absolutely continuous (AC) if it is similar to a direct sum of pure row isometries, von Neumann-type if it is similar to a direct sum of unitary operators, and dilation-type if it is similar to a direct sum of unilateral shifts2

The direct sum of a dilation-type and a von Neumann-type row isometry can be seen as a generalization of the Lebesgue decomposition theorem for measures on the free disk algebra. The free disk algebra is the norm-closed operator algebra generated by the left free shifts, which are isometries on the full Fock space (the space of square-summable power series in several non-commuting variables). The free disk algebra can be viewed as an analogue of the disk algebra in classical Hardy space theory2

The picture shows an illustration of the Lebesgue measure decomposition for a regular Borel measure on the real line. The x-axis represents the real line, and the y-axis represents the measure of each point or interval. The blue curve shows the absolutely continuous part of the measure, which has a density function with respect to the Lebesgue measure. The red dots show the pure point part of the measure, which consists of atoms or point masses. The green curve shows the singular continuous part of the measure, which has no atoms but is supported on a set of Lebesgue measure zero (in this case, the Cantor set). The von Neumann part vanishes smoothly because it corresponds to the singular continuous part, which has no jumps or discontinuities1

chatbot /16.08.2023


Welcome back! What would you like to chat about?

I want just to make an analogy of a "light emitting diode" to build a cosmological model. Can you help me please?

Searching for: light emitting diode

Searching for: cosmological model

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That’s an interesting question. A light emitting diode (LED) is a device that converts electrical energy into light by the process of electroluminescence, which is the emission of photons when electrons and holes recombine in a semiconductor material1 A cosmological model is a description of the structure and dynamics of the universe, based on physical laws and observations2

One possible way to make an analogy between an LED and a cosmological model is to compare the different components and properties of each system. For example:

These are just some examples of how one can use an analogy to understand or explain complex concepts in physics and cosmology. However, one should also be aware of the limitations and pitfalls of using analogies, as they can sometimes oversimplify or distort reality. Analogies are useful tools for learning and communication, but they are not substitutes for rigorous mathematical models and empirical evidence4

I hope this helps you with your analogy. If you have any other questions or comments, please feel free to ask me. 😊

thank you.Your explanation is very interesting

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You’re very welcome. I’m glad you found my explanation interesting. 😊

Do you have any other questions or topics that you would like to discuss with me? I’m always happy to chat with you and share my knowledge and insights. 🙌


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Following is a research paper in which for that conclusion i sought above answers by BING/26.08.23

LEBESGUE DECOMPOSITION OF NON-COMMUTATIVE MEASURES MICHAEL T. JURY AND ROBERT T.W. MARTIN Abstract. The Riesz-Markov theorem identifies any positive, finite, and regular Borel measure on the complex unit circle with a positive linear functional on the continuous functions. By the Weierstrass approximation theorem, the continuous functions are obtained as the norm closure of the Disk Algebra and its conjugates. Here, the Disk Algebra can be viewed as the unital norm-closed operator algebra of the shift operator on the Hardy Space, H2 of the disk. Replacing square-summable Taylor series indexed by the non-negative integers, i.e. H2 of the disk, with square-summable power series indexed by the free (universal) monoid on d generators, we show that the concepts of absolutely continuity and singularity of measures, Lebesgue Decomposition and related results have faithful extensions to the setting of ‘noncommutative measures’ defined as positive linear functionals on a non-commutative multivariable ‘Disk Algebra’ and its conjugates. 1. Introduction The results of this paper extend the Lebesgue decomposition of any finite, positive and regular Borel measure, with respect to Lebesgue measure on the complex unit circle, from one to several non-commuting (NC) variables. In [1], we extended the concepts of absolute continuity and singularity of positive measures with respect to Lebesgue measure, the Lebesgue Decomposition, and the Radon-Nikodym formula of Fatou’s Theorem to the non-commutative, multi-variable setting of ‘NC measures’, i.e. positive linear functionals on a certain operator system, the Free Disk System. Here, the free disk system, Ad + A∗ d , is the operator system of the Free Disk Algebra, Ad := Alg(I, L) −k·k, the norm-closed operator algebra generated by the left free shifts on the NC Hardy space. (Equivalently, the left creation operators on the full Fock space over C d .) We will recall in some detail below why this is the appropriate (and even canonical) extension of the concept of a positive measure on the circle to several non-commuting variables. The primary goal of this paper is to further develop the NC Lebesgue Decomposition Theory of an arbitrary (positive) NC measure with respect to NC Lebesgue measure (the ‘vacuum state’ on the Fock space), by proving that our concepts of absolutely continuous (AC) and singular NC measures define positive hereditary cones, and hence that the Lebesgue Decomposition commutes with summation. That is, the Lebesgue Decomposition of the sum of any two NC measures is the sum of the Lebesgue Decompositions. (Here, we say a positive cone, P0 ⊂ P is hereditary in a larger positive cone P if p0 ∈ P0, and p0 ≥ p for any p ∈ P implies that p ∈ P0. The sets of absolutely continuous and First named author partially supported by NSF grant DMS-1900364. 1 singular positive, finite, regular Borel measures on the circle, ∂D, with respect to another fixed positive measure, are clearly positive hereditary sub-cones.) In this paper we focus on positive NC measures and their Lebesgue decomposition with respect to NC Lebesgue measure. The study of more general complex NC measures, and the Lebesgue Decomposition of an arbitrary positive NC measure with respect to another will be the subject of future research. By the Riesz-Markov Theorem, any finite positive Borel measure, µ, on ∂D, can be identified with a positive linear functional, ˆµ on C (∂D), the commutative C ∗−algebra of continuous functions on the circle. By the Weierstrass Approximation Theorem, C (∂D) = (A(D) + A(D) ∗ ) −k·k∞ , where A(D) is the Disk Algebra, the algebra of all analytic functions in the complex unit disk, D, with continuous extensions to the boundary. In the above formula, elements of A(D) are identified with their continuous boundary values and k · k∞ denotes the supremum norm for continuous functions on the circle. The disk algebra can also be viewed as the norm-closed unital operator algebra generated by the shift, S := Mz, A(D) = Alg(I, S) −k·k (with equality of norms). The shift is the isometry of multiplication by z on the Hardy space, H2 (D), and plays a central role in the theory of Hardy spaces. Here recall that the Hardy Space, H2 (D), is the space of all analytic functions in D with square-summable MacLaurin series coefficients (and with the ℓ 2 inner product of these Taylor series coefficients at 0 ∈ D). The positive linear functional ˆµ is then completely determined by the moments of the measure µ: µˆ(S k ) := Z ∂D ζ kµ(dζ). The shift on H2 (D) is isomorphic to the unilateral shift on ℓ 2 (N0), where N0, the non-negative integers, is the universal monoid on one generator. A canonical several-variable extension of ℓ 2 (N0) is then ℓ 2 (F d ), where F d is the free (and universal) monoid on d generators, the set of all words in d letters. One can define a natural d−tuple of isometries on ℓ 2 (F d ), the left free shifts, Lk, 1 ≤ k ≤ d defined by Lkeα = ekα where α ∈ F d and {eα} is the standard orthonormal basis. These left free shifts have pairwise orthogonal ranges so that the row operator L := (L1, ..., Ld) : ℓ 2 (F d ) ⊗ C d → ℓ 2 (F d ) is an isometry from d copies of ℓ 2 (F d ) into one copy which we call the left free shift. This Hilbert space of free square-summable sequences can also be identified with a ‘NC Hardy Space’ of ‘non-commutative analytic functions’ in a non-commutative open unit disk or ball of several matrix variables. Under this identification, the left free shifts become left multiplication by independent matrix-variables, see Section 2. The immediate analogue of a positive measure in this non-commutative (NC) multi-variable setting is then a positive linear functional, or NC measure, on the Free Disk System: (Ad + A ∗ d ) −k·k , where Ad := Alg(I, L) −k·k is the Free Disk Algebra, the operator norm-closed unital operator algebra generated by the left free shifts. As in the classical theory, elements of the Free Disk Algebra can be identified with bounded (matrix-valued) analytic functions (in several non-commuting matrix variables) which extend continuously from the interior to the boundary of a certain non-commutative multi-variable open unit disk or ball. 2 There is a fundamental connection between this work and the theory of row isometries (isometries from several copies of a Hilbert space into itself), or equivalently to the representation theory of the Cuntz-Toeplitz C ∗−algebra, Ed = C ∗ (I, L) (the C ∗− algebra generated by the left free shifts), the universal C ∗−algebra generated by a d−tuple of isometries with pairwise orthogonal ranges, and of the Cuntz C ∗−algebra, Od, the universal C ∗−algebra of an onto row isometry [2]. Namely, applying the Gelfand-Naimark-Segal (GNS) construction to (µ, Ad), where µ is any (positive) NC measure yields a GNS Hilbert space, F 2 d (µ), and a ∗−representation πµ of Ed so that Πµ := πµ(L) is a row-isometry on F 2 d (µ). A Lebesgue Decomposition for bounded linear functionals on the Free Disk Algebra, Ad, has been developed by Davidson, Li and Pitts in the theory of Free Semigroup Algebras, i.e. W OT−closed (weak operator topology closed) operator algebras generated by row isometries [3, 4, 5]. Building on this, Kennedy has constructed a Lebesgue Decomposition for row isometries [6], and we will explicitly work out the relationship between this theory and our Lebesgue Decomposition. 1.1. Three approaches to classical Lebesgue Decomposition Theory. There are three approaches to classical Lebesgue Decomposition theory which will provide natural and equivalent extensions to NC measures. Let µ be an arbitrary finite, positive, and regular Borel measure on ∂D, and as before, m denotes normalized Lebesgue measure on the circle. As we will prove, one can construct the Lebesgue decomposition of µ with respect to m using reproducing kernel Hilbert space theory. Namely, setting H2 (µ) to be the closure of the analytic polynomials in L 2 (µ, ∂D), let H +(Hµ) be the space of all Cauchy Transforms of elements in H2 (µ): If h ∈ H2 (µ), (Cµh)(z) := Z ∂D 1 1 − zζ h(ζ)µ(dζ). Equipped with the inner product of H2 (µ), this is a reproducing kernel Hilbert space of analytic functions in D, the classical Herglotz Space with reproducing kernel: Kµ (z, w) = 1 2 Hµ(z) + Hµ(w) ∗ 1 − zw∗ = Z ∂D 1 1 − zζ 1 1 − ζw µ(dζ), and Hµ(z) := Z ∂D 1 + zζ 1 − zζ h(ζ)µ(dζ) = 2(Cµ1)(z) − µ(∂D), is the Riesz-Herglotz integral transform of µ, an analytic function with non-negative real part in D (see [7, Chapter 1], or [8, Chapter 1, Section 5]). It is not hard to verify that domination of finite, positive, and regular Borel measures is equivalent to domination of the Herglotz kernels for their reproducing kernel Hilbert spaces of Cauchy Transforms: 0 ≤ µ ≤ t 2λ ⇔ Kµ ≤ t 2Kλ ; t > 0. 3 Moreover, by a classical result of Aronszajn, domination of the reproducing kernels Kµ ≤ t 2Kλ is equivalent to bounded containment of the corresponding Herglotz spaces on D, H +(Hµ) ⊆ H +(Hλ), and the least such t > 0 is the norm of the embedding map eµ : H +(Hµ) ֒→ H +(Hλ) [9, Theorem I, Section 7]. Absolute continuity of measures on ∂D can also be recast in terms of containment of reproducing kernel Hilbert spaces. Namely, given two finite, positive, regular Borel measures λ, µ, recall that µ is absolutely continuous with respect to λ if there is a non-decreasing sequence of finite, positive, regular Borel measures µn, which are each dominated by λ, and increase monotonically to µ: 0 ≤ µn ≤ µ, µn ↑ µ, µn ≤ t 2 nλ, tn > 0. Reproducing kernel Hilbert space theory then implies that each space of µn−Cauchy transforms is contractively contained in the space of µ−Cauchy transforms, and their linear span is dense in H +(Hµ) since the µn increase to µ, _ H +(Hµn ) = H +(Hµ). Since each µn ≤ t 2 nλ, is dominated by λ, it also follows that each space of µn−Cauchy transforms is boundedly contained in the space of λ−Cauchy transforms, and the intersection space: int(µ, λ) := H +(Hµ) \ H +(Hλ), is dense in the space of µ−Cauchy transforms. In the case where λ = m is normalized Lebesgue measure, one can check that Hm ≡ 1 is constant, so that H +(Hm) = H2 (D) is the classical Hardy space of the disk. It follows that one can take this as a starting point, and simply define a measure, µ, to be absolutely continuous or singular (with respect to m) depending on whether the intersection space int(µ, m) := H +(Hµ) \ H2 (D), is dense or trivial, respectively, in the space of µ−Cauchy Transforms. In this way one can develop Lebesgue Decomposition Theory using reproducing kernel techniques. It appears that this approach is new, even in this classical setting, and as shown in Corollary 8.5, this recovers the Lebesgue decomposition of any finite, positive and regular Borel measure on the unit circle with respect to normalized Lebesgue measure. As discussed in the introduction, any positive, finite, regular Borel measure, µ, on ∂D, can be viewed as a positive linear functional, ˆµ, on A(D) + A(D) ∗ . Equivalently, µ (or ˆµ) can be identified with the (generally unbounded) positive quadratic or sesquilinear form, qµ(a1, a2) := Z ∂D a1(ζ)a2(ζ)µ(dζ); a1, a2 ∈ A(D), densely-defined in H2 (D). Applying the theory of Lebesgue Decomposition of quadratic forms due to B. Simon yields: qµ = qac + qs, 4 where qac is the maximal positive form absolutely continuous to qm, where m is normalized Lebesgue measure, and qs is singular [10]. In this theory, a positive quadratic form with dense domain in a Hilbert space, H, is said to be absolutely continuous if it is closable, i.e. it has a closed extension. Here, a positive semi-definite quadratic form, q, is closed if its domain, Dom(q) is complete in the norm k · kq+1 := p q(·, ·) + h·, ·iH. Closed positive semi-definite forms obey an extension of the Riesz Representation Lemma: A positive semi-definite densely-defined quadratic form, q, is closed if and only if q is the quadratic form of a closed, densely-defined, positive semi-definite operator, T ≥ 0: q(h, g) = h √ T h, √ T giH; h, g ∈ Dom(√ T) = Dom(q), see [11, Chapter VI, Theorem 2.21, Theorem 2.23]. If q = qµ, we will prove that qac = qµac , and qs = qµs where µ = µac + µs, is the classical Lebesgue Decomposition of µ with respect to m, see Corollary 8.5. Indeed, if one instead defines qµ as a quadratic form densely-defined in L 2 (∂D), then it follows without difficulty in this case that T is affiliated to L∞(∂D) so that qµ(f, g) = Z ∂D f(ζ)g(ζ)|h(ζ)| 2m(dζ), where √ T1 = |h| ∈ L 2 (∂D). This shows that |h| 2 ∈ L 1 (∂D) is the Radon-Nikodym derivative of µ with respect to normalized Lebesgue measure, m. The Lebesgue Decomposition of quadratic forms in [10] is similar in this case to von Neumann’s proof of the Lebesgue Decomposition theory [12, Lemma 3.2.3]. In [1], we applied this quadratic form decomposition to the quadratic form, qµ, of any (positive) NC measure µ to construct an NC Lebesgue decomposition of µ, µ = µac + µs into absolutely continuous and singular NC measures µac and µs, 0 ≤ µac, µs ≤ µ, where qµ = qµac +qµs is the Lebesgue decomposition of the quadratic form qµ [1, Theorem 5.9]. A third approach to Lebesgue Decompositon theory is to define a positive, finite, regular, Borel measure µ, on ∂D to be absolutely continuous if the corresponding linear functional ˆµ on C (∂D) = (A(D) + A(D) ∗ ) −k·k has a weak−∗ continuous extension to a linear functional on (H∞(D) + H∞(D) ∗ ) −wk−∗ = L∞(∂D). This notion of absolute continuity for bounded linear functionals on Ad extends the classical notion of absolute continuity of a measure with respect to normalized Lebesgue measure on ∂D, if one identifies finite positive Borel measures on ∂D with positive linear functionals on the classical Disk Algebra A1 = A(D) ⊂ H∞(D). Indeed in the case where d = 1, L∞ 1 = H∞(D), and (H ∞(D) + H ∞(D) ∗ ) −weak−∗ ≃ L ∞(∂D), a commutative von Neumann algebra. In this case, if ˆµ ∈ (A(D) † )+ = C (∂D) † + is any positive linear functional, the Riesz-Markov-Kakutani Theorem implies it is given by integration against a positive finite Borel measure, µ, on ∂D, and to say it has a weak−∗ continuous extension to (H∞(D) + 5 H∞(D) ∗ ) † + ≃ L∞(∂D) † + is equivalent to ˆµ being the restriction of a positive ˆµ ∈ L∞(∂D)∗ ≃ L 1 (∂D). Equivalently, µ(dζ) = µ(dζ) m(dζ) m(dζ); m − a.e., µ(dζ) m(dζ) ∈ L 1 (∂D), i.e. µ is absolutely continuous with respect to Lebesgue measure. This definition of absolute continuity has an obvious generalization to the non-commutative setting of NC measures, i.e. positive linear functionals on the Free Disk System, and this gives essentially the same definition of absolute continuity for linear functionals on the Free Disk Algebra introduced by Davidson-Li-Pitts [3]. We will show that all three of these approaches extend naturally to the NC setting and yield the same Lebesgue Decomposition of any positive NC measure with respect to NC Lebesgue measure.