text:
# Is my reasoning about economic growth and environmental sustainability correct?

Below is my attempt to answer question "Can we have economic growth and environmental sustainability?".

Please correct me if I got something wrong in my reasoning.

Let's see what is counted as economic growth and environmental sustainability:

a)Economic growth happens when we produce more goods and services than our Production Possibility Frontier (aka PPF) previously allowed. In its turn, this usually happens when we have more labor and/or capital and/or land than previously. The exception for this are gains from specializing in producing and selling goods in which given country holds comparative advantage. If we just started using our labor/capital/land more efficiently we still would be inside our Production Possibility Frontier, although it would lead to increase in production of goods and services.

b)Environmental sustainability happens when following conditions are met, according to Herman Daly:

1. For renewable resources, the rate of harvest should not exceed the rate of regeneration (sustainable yield);

2. [For pollution] The rates of waste generation from projects should not exceed the assimilative capacity of the environment (sustainable waste disposal); and

3. For nonrenewable resources the depletion of the nonrenewable resources should require comparable development of renewable substitutes for that resource.

• Considering this, we can have economic growth thanks to producing and selling goods in what we have comparative advantage, while reducing production of goods in what have have comparative disadvantage. To put it simply, more free trade, less subsidies of uncompetitive industries - more growth. And it's compatible with environmental sustainability.
• We can increase labor (which is understood as amount of work done) by, say, curing people who are unable to work due to an illness. Or by punishing people who can work, but don't want to work (like USSR did). Or by making people work for longer hours. Or by making sure that workers are really working instead of, say, wasting time playing games and checking their Facebook. Or by training part-time workers to be qualified for a full-time job. Such measures can lead to economic growth and are environmentally sustainable.
• As for land, there does NOT seem to be any environmentally sustainable way to get more of it. If only you don't seriously consider scenario when one country gets territory that previously belonged to another country. But it's a zero-sum game.
• Capital is understood here as all man-made goods which are used for further production of wealth. We can increase our capital in an environmentally sustainable way if we will increase capital thanks to more efficient use of existing labor, land and capital. Again, increased production of capital (and other goods) due to more efficient use of existing labor, land and capital does NOT itself mean economic growth. But after capital will be increased it will be possible to produce outside previously established PPF, meaning that economic growth have occured.

It is on a correct track but I would not say it is 100% correct as you miss the main source of growth which is technological progress. Consider the following Cobb-Douglas production function with all 4 main factors of production, capital ($$K$$), human capital ($$H$$), labor ($$L$$), land ($$M$$) then the standard production function would look something like:

$$F(K,H,L,M) = A K^{\alpha}H^{\beta}L^{\gamma}M^{1 - \alpha - \beta - \gamma}$$

Well you are correct to say that production $$F$$ increases if either more capital is put to work, or more labor hours are employed ($$K$$ or $$L$$ increases). However, there can be limits to how far you can push capital accumulation and still be sustainable. Machines after all need some raw materials. Also, people only have 24 hours a day so there is a limit there. Land is for all practical purposes fixed (at least before we can make an effective use of real estate on moon or mars - and I am saying that just half-jokingly). However, you missed one important factor of production: human capital. That is things like better education which as far as I can see can increase almost without bounds.

Second thing you missed is $$A$$ that is the total factor productivity, which makes any factor more productive and loosely speaking it is the technology avaiable. For example, thanks to our better understanding the computer technology modern computers are more faster than the old ones yet they are produced with the same amount resources (or sometimes even less resources - think of the old bus size punch card computers). This does not apply just to capital, thanks to telecomunication technology you can make better use of your human capital, thanks to better management techniques better use of labor, thanks to agricultural techniques like crop rotation better use of the same land.


text:
0

# How much water does it take to make 1 gallon of gasoline?

Wiki User

2010-02-10 23:01:55

This question can be answered in three ways. It takes 1851 gallons of water to refine a barrel of crude oil. One barrel of crude oil produces 19 gallons of gasoline and 10 gallons of diesel fuel, in this respect it takes 97 gallons of water to produce a gallon of gasoline. If you combine gasoline and diesel, it takes 63 gallons of water to produce a gallon of "fuel." A total of 42 gallons of petroleum products are produced from a barrel of crude oil, in this respect it takes 44 gallons of water to produce each gallon.

Wiki User

2010-02-10 23:01:55
🙏
1
🤨
1
😮
0

format: plain


text:
User francois ziegler - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:01:30Z http://mathoverflow.net/feeds/user/19276 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/134211/the-orbit-g-cdot-x-cap-mathfrakt-for-x-in-mathfrakt-singular/134219#134219 Answer by Francois Ziegler for The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular Francois Ziegler 2013-06-20T04:25:31Z 2013-06-20T04:25:31Z
Yes, it's still true that $G\cdot X\cap\mathfrak t=W\cdot X$.

The inclusion $\supset$ is clear. Conversely, suppose $g\cdot X\in\mathfrak t$. Since $\mathfrak t$ consists exactly of all $T$-fixed points in $\mathfrak g$, it follows that $t\cdot g\cdot X=g\cdot X$ for all $t\in T$. Hence $g^{-1}Tg$ is contained in the stabilizer $G_X$. So $T$ and $g^{-1}Tg$ are two maximal tori in $G_X$. Hence they are conjugate by some $h\in G_X$: $T = h^{-1}g^{-1}Tgh$. So now $gh$ normalizes $T$, and the Weyl group element it represents still sends $X$ to $g\cdot X$. QED.
 http://mathoverflow.net/questions/134142/where-did-sophus-lie-write-the-group-commutator-for-two-one-parameter-groups/134164#134164 Answer by Francois Ziegler for Where did Sophus Lie write the group commutator for two one parameter groups. Francois Ziegler 2013-06-19T16:04:39Z 2013-06-20T03:58:05Z
I believe you're not going to find exactly what you want in Lie, because he never formalized flows (or finite transformations) and their commutation as you do. Maybe the closest would be this, from Über Differentialinvarianten, Math. Ann. 24 (1884) 537-578:

... erhalten wir folgenden Fundamentalsatz, den ich 1872 entdeckt habe:

Satz 3. Enthält eine kontinuierliche Gruppe die beiden infinitesimalen Transformationen: $$Bf=\sum\xi_\varkappa\frac{\partial f}{\partial x_\varkappa} \quad\textit{und:}\quad Cf=\sum\eta_\varkappa\frac{\partial f}{\partial x_\varkappa},$$ so enthält sie ebenfalls die infinitesimale Transformation: $$\sum_i(B\eta_i-C\xi_i)\frac{\partial f}{\partial x_i},$$ deren Symbol bekanntlich auf die beiden äquivalenten Formen: $$B(C(f)) - C(B(f)) = (B, C)$$ gebracht werden kann.

As you can see, his definition of the bracket of vector fields is always as the commutator of the derivations they define on functions (something that goes back to Jacobi). What this Satz states, then, is that the finite transformations (or flow) generated by the infinitesimal commutator $(B,C)$ belong to the group generated by (the flows of) $B$ and $C$. Not surprisingly, Lie's proof is by expanding the flows to second order.

Lie may or may not have stated this Satz elsewhere before 1884, but I doubt he ever wrote a formula for, much less definition of, the bracket as limit of commutators of finite transformations.

Update As to your question of who (esp. first) expressed the bracket as a derivative of commutators of flows: I don't know (my impression is that these things developed slowly in a sort of consensus), but one might argue that the formula $$[V,T]=\frac{d}{ds}\frac{d}{dt}e^{-sV}e^{tT}e^{sV}\Bigr|_{s=t=0}$$
 is on p. 240 of Poincaré, Sur les groupes continus, Trans. Cambridge Philos. Soc. 18 (1900) 220-255.

Further update Trotter's formula that you also mention now is indeed called "Lie-Trotter" by e.g. Chernoff [1968,1974] or Chorin et al. [1978]. The latter write (sic):

... the equation $dx/dt=Ax+Bx$ leads to the 1875 formula of S. Lie [38]: $$\exp\{A+B\} = \lim_{n\to\infty}(\exp\{A/n\}\exp\{B/n\})^n.\tag{*}$$ [38] Lie, S., and Engel, F., Theorie der Transformationsgruppen, 3 Vols., Teubner, Leipzig, 1888.

The problem is that [38] is not from 1875, nor does it contain anything remotely like formula ($*$) (I am ready to bet a lot of money). I may be wrong but until someone finds that elusive 1875 paper, I would tend to date ($*$) from around von Neumann [1929, p. 19].
 http://mathoverflow.net/questions/133661/calculus-book-in-the-spirit-of-the-18th-century/133679#133679 Answer by Francois Ziegler for Calculus book in the spirit of the 18th century Francois Ziegler 2013-06-13T18:20:22Z 2013-06-14T03:07:53Z
I like Analysis by its History, by Hairer and Wanner. Chapter I "Introduction to Analysis of the Infinite" covers precalculus as Euler would. Chapter II "Differential and Integral Calculus" proceeds essentially as you desire (through envelopes, caustics, curvature, and differential equations). Then Chapters III and IV 'redo' everything with Weierstrassian rigor, and move on to several variables.
 http://mathoverflow.net/questions/129310/reference-request-localisation-de-g-modules/129333#129333 Answer by Francois Ziegler for Reference request - localisation de g-modules Francois Ziegler 2013-05-01T17:43:57Z 2013-05-01T17:43:57Z
http://gallica.bnf.fr/ark:/12148/bpt6k6226873r/f29
 http://mathoverflow.net/questions/126868/surjective-submersion-and-fibrations/126875#126875 Answer by Francois Ziegler for surjective submersion and fibrations Francois Ziegler 2013-04-08T15:58:50Z 2013-04-08T15:58:50Z
No. To see this, take a bona fide fibration such as $\operatorname{pr}_2:\mathbf{R}^2\to\mathbf{R}$ and remove a point from the domain.

Meigniez on p. 3778 lists a number of sufficient conditions that a submersion $f$ be a fibration. The best known, already noted by Damian and Donu, is that $f$ be proper as in Ehresmann's Theorem (proved e.g. in Bröcker and Jänich, (8.12)).
 http://mathoverflow.net/questions/123926/reference-request-lascouxs-formulas-for-chern-classes-of-tensor-products-and-sy/124027#124027 Answer by Francois Ziegler for Reference request: Lascoux's formulas for Chern classes of tensor products and symmetric powers Francois Ziegler 2013-03-08T22:03:11Z 2013-03-08T22:03:11Z
You say you don't have access to Lascoux's paper, but it is actually available online at the BNF:

http://gallica.bnf.fr/ark:/12148/bpt6k62341359/f397.image

 http://mathoverflow.net/questions/122793/spectacular-applications-of-functional-analysis-in-resolutions-of-apparently-unre/122808#122808 Answer by Francois Ziegler for spectacular applications of functional analysis in resolutions of apparently unrelated problems Francois Ziegler 2013-02-24T16:22:09Z 2013-02-24T16:22:09Z
One should probably mention Gelfand's proof of Wiener's theorem that, if a nowhere zero periodic $f$ has absolutely convergent Fourier series, then so does $1/f$.

There is also Kantorovich's famous note "On a problem of Monge":

Monge, in his memoirs of 1781, considering the problem of the most rational ways of transporting earth from an embankment to an excavation, proposed the following problem: divide two equal volumes into infinitesimal particles and associate them one to another so that the sum of the path lengths multiplied by the volumes of the particles be minimum possible.

In connection with this problem, Monge created the geometrical theory of congruences. As to the problem itself, he conjectured, but did not proved rigorously, that the paths of the mass translocation form a family of normals to a certain family of surfaces.

The same problem was studied later by Dupin, but a rigorous proof of the Monge theorem was given only a century later, in 1884, in a 200-page memoirs by Appell. (...)

Meanwhile, this assertion follows immediately from the abstract theorem mentioned above. (...)

 http://mathoverflow.net/questions/122477/stabilizers-for-nilpotent-adjoint-orbits-of-semisimple-groups/122520#122520 Answer by Francois Ziegler for Stabilizers for Nilpotent Adjoint Orbits of Semisimple Groups Francois Ziegler 2013-02-21T10:33:16Z 2013-02-21T10:33:16Z
The component groups $\pi_0(C_G(X))$ are known by work of Alekseevskiĭ, MR557505 = MR2140712, which contains additional references and can be read on Google books. Another good place to look is the book by Collingwood & McGovern, Nilpotent orbits in semisimple Lie algebras.

(Out of curiosity, what other topological invariants are you after?)
 http://mathoverflow.net/questions/119402/why-all-irreducible-representations-of-compact-groups-are-finite-dimensional-e/119423#119423 Answer by Francois Ziegler for Why all irreducible representations of compact groups are finite-dimensional ? [EDIT: Subtleties: AC,etc] Francois Ziegler 2013-01-20T20:16:22Z 2013-01-22T01:38:29Z
(Addressing only the title question.) There is a short proof avoiding Peter-Weyl and the theory of compact operators. It is due to Nachbin and is reproduced in Hewitt-Ross, Abstract Harmonic Analysis 1, p. 344. A slight simplification of it runs as follows: Pick a unit vector $v$ in your representation space $V$. Schur's lemma gives $$\int_G gv(gv,\cdot)dg = \lambda1 \tag 1$$ where $\lambda = \int_G|(v,gv)|^2dg>0$ (sandwiching (1) with $(v,\cdot v)$). Now let $W\subset V$ be finite-dimensional, and write $E=E^2$ for the orthogonal projection $V\to W$. We get $$\int_GEgv(gv,E\cdot)dg = \lambda E, \tag 2$$ whence, taking traces in (2), $\lambda\dim(W)=\int_G||Egv||^2dg\leqslant\operatorname{vol}(G)$. Thus, the dimension of any finite-dimensional subspace is bounded, as was to be shown.
 http://mathoverflow.net/questions/109395/is-there-a-geometric-intuition-underlying-the-notion-of-normal-varieties/117329#117329 Answer by Francois Ziegler for Is there a "geometric" intuition underlying the notion of normal varieties? Francois Ziegler 2012-12-27T15:03:40Z 2012-12-27T22:19:12Z
Regarding the question "Who was the person who invented this notion?", a paper of H. T. Muhly provides interesting background (as well as a geometric interpretation) for projectively normal:

In the terminology of the Italian School an algebraic variety is called "normal" if its system of hyperplane sections is complete. O. Zariski applies the term "normal" to an algebraic variety whose associated ring of homogeneous coordinates is integrally closed. The two concepts are not equivalent. Zariski refers to a variety which satisfies the former condition as "normal in the geometric sense" and to one which satisfies the latter condition as "normal in the arithmetic sense".

(...)

The object of this note is to characterize geometrically those algebraic varieties which are normal in the arithmetic sense. To this end we propose the following theorem: A necessary and sufficient condition that the $r$-dimensional algebraic variety $V_r$ be normal in its ambient projective space $P_n$ is that for every integer $m$ the linear system cut out on $V_r$ by the hypersurfaces of order $m$ in $P_n$ be complete.

 http://mathoverflow.net/questions/117051/the-origin-of-sets/117063#117063 Answer by Francois Ziegler for The origin of sets? Francois Ziegler 2012-12-23T02:26:24Z 2012-12-27T13:42:34Z
Euler in Lettres à une princesse d'Allemagne sur divers sujets de physique et de philosophie, 17-24 feb 1761, writes about objects he calls spaces (my emphasis):

As a general notion encompasses an infinity of individual objects, one regards it as a space within which all these individuals are enclosed: thus, for the notion of man, one makes a space (fig. 39) in which one conceives that all men are comprised. For the notion of mortal, one also makes a space (fig. 40), where one conceives that everything mortal is comprised. Then, when I say that all men are mortal, that comes down to the former figure being contained in the latter.

(...)

These round figures or rather these spaces (for it doesn't matter what shape we give them) are very well-suited to facilitating our reflections (...)

etc., and illustrates this with what we would call ensemblist diagrams (fig. 39 to 89), famously reproduced on Swiss banknotes. The applications he gives here are to everyday logic, so perhaps less mathematical than intended by the question. (I don't know if he ever wrote again on the subject.)
 http://mathoverflow.net/questions/115839/what-is-the-structure-of-the-space-of-solutions-of-a-non-linear-ode/115854#115854 Answer by Francois Ziegler for What is the structure of the space of solutions of a non linear ODE? Francois Ziegler 2012-12-08T22:50:47Z 2012-12-09T19:56:26Z
As Robert Bryant observed, something like the solubility of $F$ for $u^{(n)}$ needs to be assumed. Then by the trick (due I believe to D'Alembert) of setting $x=(t,u,\dots,u^{(n-1)})$ and dt/ds=1, the equation can always be rewritten $$\frac{dx}{ds}=f(x).$$ This is what most geometers would call the "standard ODE", wherein $f$ is a smooth vector field on the manifold where $x$ evolves.

In this setting, the space of (maximal connected) solution curves is indeed always a (not necessarily Hausdorff) manifold. I don't know who first wrote this, but according to this recent obituary a contender would be J.-M. Souriau in his 1970 book "Structure des systèmes dynamiques". Translation of the relevant passage:

The manifold of motions of a system

Jean-Marie Souriau has observed that the set of maximal solutions of a differential system on a differentiable manifold possesses itself a natural structure of differentiable manifold, not always separated: thus one can speak of the system's manifold of motions. This very simple property is rarely mentioned in the usual courses on differential calculus.

 http://mathoverflow.net/questions/112657/maximal-subgroups-of-finite-simple-groups/113998#113998 Answer by Francois Ziegler for maximal subgroups of finite simple groups Francois Ziegler 2012-11-20T23:20:28Z 2012-11-29T19:39:24Z
The recent book of Malle and Testerman, Linear algebraic groups and finite groups of Lie type, has several chapters on the subject. From the MR review:

some important recent developments are treated here for the first time. For instance, the authors describe the classification of the maximal subgroups of simple algebraic groups, and this is used in their subsequent analysis of the subgroup structure of finite groups of Lie type.

(etc. The review goes on to describe the Aschbacher/Liebeck-Seitz results mentioned in other replies.)

Also Robert A. Wilson's The finite simple groups addresses the question, for the classical groups in a devoted section (3.10), for the others by systematically giving references to original papers classifying the maximal subgroups.
 http://mathoverflow.net/questions/114626/fundamental-motivation-for-several-complex-variables/114635#114635 Answer by Francois Ziegler for Fundamental motivation for several complex variables Francois Ziegler 2012-11-27T06:54:59Z 2012-11-29T14:03:50Z
To say that $F$ is holomorphic is to say in a sense that it depends on only "half" of the variables $(\partial F/\partial\bar z_j=0)$. This is analogous to the wave functions of quantum mechanics depending only on $q$, i.e. only on "half" the phase space variables $(p,q)$. The analogy is made precise by the theory of polarizations in geometric quantization and the orbit method.

For instance, a reductive Lie group has some of its irreducible unitary representations attached to hyperbolic coadjoint orbits (consisting of matrices with real eigenvalues), others to elliptic orbits (consisting of matrices with imaginary eigenvalues). While the former admit real polarizations which lead to representations in $L^2$ sections over a space half the dimension (i.e., real analysis), the latter admit complex polarizations which lead to representations in holomorphic sections over the orbit (or Dolbeault cohomology thereof; i.e., complex analysis).

A classic example of the elliptic story is the Borel-Weil(-Bott) realization of all irreducible representations of compact Lie groups. This involves holomorphic functions on the complexified group in an essential way.
 http://mathoverflow.net/questions/112125/weakest-assumption-for-pointwise-convergence-of-fourier-series/112138#112138 Answer by Francois Ziegler for Weakest assumption for pointwise convergence of Fourier series Francois Ziegler 2012-11-12T02:07:21Z 2012-11-25T02:50:36Z
The function must be integrable in a certain sense defined by Denjoy and others. Here is an interesting survey paper on the subject:

One of the problems in the theory of trigonometric series $$\frac12a_0+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)\tag{1.1}$$ is that of suitably defining a trigonometric integral with the property that, if the series (1.1) converges everywhere to a function $f(x)$, then $f(x)$ is necessarily integrable and the coefficients, $a_n$ and $b_n$, given in the usual Fourier form. It is well known that a series may converge everywhere to a function which is not Lebesgue summable nor even Denjoy integrable (...) The problem has been solved by Denjoy [4; 5], Verblunsky [14], Marcinkiewicz and Zygmund [10], Burkill [1; 2], and James [8]. (...) The solutions are described, mainly in the order in which they were published, in §§2-7 below.

 http://mathoverflow.net/questions/112281/polarisation-in-a-nighbourhood-of-a-lagrangian-submanifold/112296#112296 Answer by Francois Ziegler for Polarisation in a nighbourhood of a Lagrangian submanifold Francois Ziegler 2012-11-13T16:14:15Z 2012-11-17T21:43:11Z
Arnol'd (p. 3314) puts it that way:

Weinstein's Theorem. Some neighborhood of any Lagrangian submanifold in any symplectic manifold is symplectomorphic to some neighborhood of this Lagrangian submanifold in any other symplectic manifold, for instance in its own cotangent bundle space.

(The resulting neighborhood then has the obvious transverse polarization by fibers of the cotangent bundle.) Unless I am mistaken, Weinstein proves this in Symplectic manifolds and their lagrangian submanifolds, Theorem 6.1 and Corollary 6.2 (which he points out goes back to Souriau).
 http://mathoverflow.net/questions/102560/reference-request-for-the-list-of-maximal-subgroups-of-su3-1/111532#111532 Answer by Francois Ziegler for Reference request for the list of maximal subgroups of SU(3,1) Francois Ziegler 2012-11-05T07:44:02Z 2012-11-06T13:03:41Z
Mohamed Selim Taufik, On maximal subalgebras in classical real Lie algebras: "This paper is concerned with the classification of irreducible maximal subalgebras of the classical real Lie algebras su(p,q), sv(p,q) and si(p,q). We use the results of E. B. Dynkin, who classified the maximal subalgebras of the classical Lie algebras in the complex case. (...)"

Boris P. Komrakov Maximal subalgebras of real Lie algebras and a problem of Sophus Lie: "A classification of the maximal proper subalgebras of the simple real finite-dimensional Lie algebras is presented without proof. Contributions by A. A. Morozov, E. B. Dynkin, M. Berger and M. S. Taufik are mentioned."
 http://mathoverflow.net/questions/109222/jacobi-method-on-first-order-partial-differential-equations/109268#109268 Answer by Francois Ziegler for Jacobi method on first order partial differential equations Francois Ziegler 2012-10-10T01:52:41Z 2012-10-10T01:52:41Z
Chapter VII of É. Goursat's book, Leçons sur l'intégration des équations aux dérivées partielles du premier ordre, exposes the method and ends with 14 examples of applying it (pp. 168-169).
 http://mathoverflow.net/questions/109164/synthetic-approach-to-hyperbolic-geometry/109172#109172 Answer by Francois Ziegler for Synthetic approach to hyperbolic geometry? Francois Ziegler 2012-10-08T19:36:04Z 2012-10-09T17:40:38Z
I'd say your best bet is with works from the early 20th century, when this sort of thing was in fashion:

Julian Lowell Coolidge, The elements of non-Euclidean geometry (1909)

Horatio Scott Carslaw, The elements of non-Euclidean plane geometry and trigonometry (1916)

Duncan M'Laren Young Sommerville, The elements of non-Euclidean geometry (1919)

 http://mathoverflow.net/questions/107795/examples-of-non-kahler-compact-symplectic-manifolds/107814#107814 Answer by Francois Ziegler for Examples of non-Kahler compact symplectic manifolds. Francois Ziegler 2012-09-22T02:45:28Z 2012-09-22T02:45:28Z
I would recommend the Tralle-Oprea book, Symplectic manifolds with no Kähler structure.
 http://mathoverflow.net/questions/106068/structure-of-the-unitary-representation-l2n-m-when-n-is-a-nilpotent-lie-gr/106073#106073 Answer by Francois Ziegler for Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group Francois Ziegler 2012-08-31T21:43:37Z 2012-09-01T15:27:02Z
The representation you're looking at is $\mathrm{Ind}_M^N1$ and as such, its decomposition into irreducibles is very well understood using Kirillov's orbit method. (Essentially, the irreducibles that enter correspond to the coadjoint orbits in the image of the moment map $T^*(N/M)\to\mathfrak n^*$.)

I'd say the basic paper on the subject is this one by Corwin, Greenleaf, and Grélaud. It has references to the earlier work by Kirillov himself, and you'll find more in mathscinet's forward references to reviews citing it.
 http://mathoverflow.net/questions/102597/history-question-autobiography-of-mathematicians/105491#105491 Answer by Francois Ziegler for History Question: AUTObiography of Mathematicians Francois Ziegler 2012-08-26T01:08:58Z 2012-08-26T01:08:58Z
Analyse des travaux scientifiques de Henri Poincaré faite par lui-même, Acta Math. 38 (1921), 3-135.

Apparently written in 1901 at the request of Mittag-Leffler, this is not quite an autobiography but more in the style of the "Notices sur les travaux scientifiques" that many French scientists wrote, often as candidates to the Academy of Sciences. A 19th century example is Darboux (1884).
 http://mathoverflow.net/questions/102656/symplectic-formulation-of-statistical-physics/102674#102674 Answer by Francois Ziegler for Symplectic formulation of statistical physics Francois Ziegler 2012-07-19T14:40:29Z 2012-07-22T07:09:36Z
You want to read Chapter IV "Statistical Mechanics" in Structure of Dynamical Systems (1970 French original available here) by J.-M. Souriau, one of the pioneers of symplectic mechanics.

Given a symplectic manifold $X$ on which a Lie group $G$ acts with moment map $\Psi$, Souriau defines the Gibbs states as the probability measures on $X$ that maximize entropy for a given mean value of $\Psi$. He shows (thm 16.219) that they have the form
 $$\text{const}\times e^{-\langle\Psi(\cdot),\beta\rangle}\lambda, \qquad (\lambda=\text{Liouville measure})$$ for some $\beta\in\mathfrak g$ which generalizes the "inverse temperature" when $G=\mathbf R=$ {time translations}. The rest of the Chapter is devoted to the study of these states; in particular when $G=$ SO(3), $\beta$ can be interpreted as a rotation vector, and the fact that planets revolve in a common plane as the equality of their equilibrium $\beta$s. See also Définition covariante des équilibres thermodynamiques, Suppl. Nuovo Cimento 1 (1966), 203–216.

Later Souriau developed a general-relativistic viewpoint on dissipative processes which explains why they preserve the mean value of $\Psi(x)$. Thus, quoting from this summary to whet your appetite: "the first principle of thermodynamics [loses] its primitive status and [becomes] a necessary consequence of the invariance of the symplectic structure in gravitational gauge transformations." For more details, see Thermodynamique et géométrie, Lecture Notes in Math. 676 (1978), 369–397 or scanned preprint.
 http://mathoverflow.net/questions/102763/the-use-of-parentheses-to-mean-i-wont-tell-you-this-again/102768#102768 Answer by Francois Ziegler for the use of parentheses to mean "I won't tell you this again" Francois Ziegler 2012-07-20T19:57:13Z 2012-07-20T20:05:57Z
Re: "Does this use of parentheses have a name?",

preterition |ˌpretəˈri sh ən|

noun (...) the rhetorical technique of making summary mention of something by professing to omit it.

ORIGIN late 16th cent.: from late Latin praeteritio(n-), from praeterire ‘pass, go by.’
 http://mathoverflow.net/questions/98997/determinants-and-polynomials-in-matrices/98999#98999 Answer by Francois Ziegler for determinants and polynomials in matrices Francois Ziegler 2012-06-07T01:35:45Z 2012-06-09T23:29:02Z
This is going to sound like massive overkill, but it is "very well known" that the only 1-dimensional polynomial representations of $GL(V)$ (which is what you're looking at) are the nonnegative powers of $\mathrm{det}$.

Reference (I assume from the mention of statistics that you are OK working with base field $\mathbf{R}$ or $\mathbf{C}$): e.g. Procesi on p.278 of Lie Groups lists all irreducible rational representations as all $$S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z},$$ where $\lambda$ runs over a certain set of partitions or Young tableaux; and on p.270 he gives a dimension formula for $S_\lambda(V)$ which is $>1$ unless $S_\lambda(V)$ is trivial.
 http://mathoverflow.net/questions/98956/papers-whose-title-defines-a-new-terminology/98973#98973 Answer by Francois Ziegler for Papers whose title defines a new terminology Francois Ziegler 2012-06-06T18:40:12Z 2012-06-06T18:40:12Z
Diener, Francine; Diener, Marc Chasse au canard. I. Les canards. (French) [Duck hunt. I. The ducks] Collect. Math. 32 (1981), no. 1, 37–74.

Benoît, Éric Chasse au canard. II. Tunnels—entonnoirs—peignes. (French) [Duck hunt. II. Tunnels—funnels—combs] Collect. Math. 32 (1981), no. 2, 77–97.

Callot, Jean-Louis Chasse au canard. III. Les canards ont la vie brève. (French) [Duck hunt. III. Ducks have a short life] Collect. Math. 32 (1981), no. 2, 99–114.

Benoît, Éric; Callot, Jean-Louis Chasse au canard. IV. Annexe numérique. (French) [Duck hunt. IV. Numerical appendix] Collect. Math. 32 (1981), no. 2, 115–119.
 http://mathoverflow.net/questions/98956/papers-whose-title-defines-a-new-terminology/98972#98972 Answer by Francois Ziegler for Papers whose title defines a new terminology Francois Ziegler 2012-06-06T18:21:42Z 2012-06-06T18:21:42Z
Jean-Pierre Serre: Gèbres, Enseign. Math. (2) 39 (1993), 33–85.
 http://mathoverflow.net/questions/98579/deformation-of-lagrangian-manifolds/98582#98582 Answer by Francois Ziegler for Deformation of Lagrangian manifolds Francois Ziegler 2012-06-01T16:18:20Z 2012-06-06T01:05:06Z
You already know that the pair $(M,L)$ of a symplectic manifold and a Lagrangian submanifold is locally isomorphic to $(T^*L, L)$. This is the beginning of Corollary 6.2 of Weinstein, who continues: "and the lagrangian submanifolds of $M$ "near" $L$ are in 1-1 correspondence with "small" closed forms on $L$."

The correspondence in question (explained on the previous page of Weintein's paper) is that "a submanifold of $T^*L$ transversal to the fibres is locally the graph of a 1-form $\sigma:L\to T^*L$. The graph of $\sigma$ is isotropic if and only if... $\sigma$ is a closed 1-form."

In short, the map you want attaches to a closed 1-form (on $L$!) its graph in $M\simeq T^*L$.

Update: This construction identifies a neighborhood of $f_0:L\hookrightarrow M$ in the space of embeddings (Whitney C$^1$ topologized), with a neighborhood of zero in the space of closed 1-forms on $L$. See Thm II.3.8 in Michèle Audin's notes (available here). She concludes that $Z^1(L)$ "can be considered as a neighbourhood of $f_0$ in the “manifold” of deformations of $f_0$, or as its tangent space at $f_0$."
 http://mathoverflow.net/questions/98296/steinmetz-laplace-and-fourier-transforms/98813#98813 Answer by Francois Ziegler for Steinmetz, Laplace and Fourier Transforms Francois Ziegler 2012-06-04T21:10:58Z 2012-06-04T21:10:58Z
(Too long for a comment.) I think the italian Wiki page is wrong. It says the transform "was conceived by the author in 1893 [probably this text] and exposed in his treatise Theory and Calculation of Alternating Current Phenomena four years later."

However, as you can check there is not a trace in these texts (nor in the others authored by Steinmetz and available on archive.org) of what Wikipedia calls "la transformata di Steinmetz" -- i.e. the Fourier isomorphism $L^2(S^1)\to\ell^2(\mathbf{Z})$ -- which by the way, was written explicitly much before 1893.

These texts are famous for introducing complex numbers (in particular the notation $j=\sqrt{-1}$) into electrical engineering, but my impression is that naming the transform in Steinmetz's honor happened much later, perhaps in Italy, with little regard to what he himself actually did (or as the case may be, never did) with it.
 http://mathoverflow.net/questions/98534/when-do-commuting-hamiltonian-flows-have-commuting-generators/98541#98541 Answer by Francois Ziegler for When do commuting Hamiltonian flows have commuting generators? Francois Ziegler 2012-06-01T05:05:30Z 2012-06-04T17:01:43Z
This is not so much an answer as a suggestion to change the question. When $(P,\Omega)$ is prequantizable, i.e. there exists over $P$ a hermitian line bundle with connection $(L,\nabla)$ having curvature $\Omega$ (see e.g. Kostant 1970), then your hamiltonian vector fields $X^g, X^h$ and their flows $\varphi^g, \varphi^h$ lift canonically to $\nabla$-preserving vector fields $\xi^g, \xi^h$ and flows $\psi^g, \psi^h$ on $L$. These commute if and only if $[g,h]=0$.

That, I believe, is the correct "classical analogue" of the quantum facts you allude to.

Conversely, the true quantum analogue of looking at $\varphi^g, \varphi^h$ is looking at the action of $e^{ibG}, e^{iaH}$ not on Hilbert space $\mathcal{H}$ but on projectivized Hilbert space $\mathbb{P}\mathcal{H}$. There they commute iff (disregarding the usual domain questions) $[G,H]$ is a constant multiple of the identity.

In fact the analogy is good enough that $\mathbb{P}\mathcal{H}$ is a (usually infinite-dimensional) symplectic manifold, to which the first paragraph above applies, with $g, h$ the expectation values of $G, H$ and $L^\times \to P$ the tautological projection $\mathcal{H}\setminus\lbrace0\rbrace\to \mathbb{P}\mathcal{H}$. Moreover $\xi^h$ and $\psi^h(a)$ are just $H$ and $e^{iaH}$ (acting on $\mathcal{H}\setminus\lbrace0\rbrace$) -- so we've come full circle.

[P.S.: Regarding functions whose Poisson brackets are constant, you might be interested in this paper of Roels and Weinstein.]

Update regarding your extra question ("Isn't $e^{ibG}e^{iaH}=e^{iaH}e^{ibG}\Leftrightarrow[G,H]=0$ also true on $\mathbb{P}\mathcal{H}$?"): This is a statement about transformations of $\mathcal{H}$, not $\mathbb{P}\mathcal{H}$. Write $\underline{e^{iaH}}$ for the diffeo of $\mathbb{P}\mathcal{H}$ induced by $e^{iaH}\in\mathrm{U}(\mathcal{H})$, and likewise $\underline{iH}$ for the vector field on $\mathbb{P}\mathcal{H}$ induced by $iH\in\mathrm{End}(\mathcal{H})$. Then (exercise!) $e^{iaH}\mapsto\underline{e^{iaH}}$ is a group morphism with kernel the multiples of the identity, and likewise $iH\mapsto\underline{iH}$ is a Lie algebra morphism with kernel the multiples of the identity. Therefore we have

\begin{array}{cccl} \underline{e^{ibG}}.\underline{e^{iaH}}=\underline{e^{iaH}}.\underline{e^{ibG}} & \Leftrightarrow & [\underline{iG},\underline{iH}]=0 &\text{(actions on }\mathbb{P}\mathcal{H})\\ \Updownarrow & & \Updownarrow\\ e^{ibG}e^{iaH}e^{-ibG}e^{-iaH}\in\mathbb{C}\cdot\mathbf{1} & \Leftrightarrow & [iG,iH]\in\mathbb{C}\cdot\mathbf{1} & \text{(actions on }\mathcal{H}) \end{array}

and my claim is that these, not $[G,H]=0$, are the quantum analogs of $\varphi^g(b)\varphi^h(a)=\varphi^h(a)\varphi^g(b)$.
 http://mathoverflow.net/questions/134013/square-root-of-a-certain-matrix/134020#134020 Comment by Francois Ziegler Francois Ziegler 2013-06-18T17:56:12Z 2013-06-18T17:56:12Z @unknown: I don't know. But what I sketched in my comment to your question above gives you the solution anyway. The point is that whenever the minimal polynomial factors without multiplicities, as in your case: $(R-1)(R-\lambda_+)(R-\lambda_-)=0$, then one has explicit formulas (which I have now added) for the eigenprojectors and hence a square root. http://mathoverflow.net/questions/134013/square-root-of-a-certain-matrix Comment by Francois Ziegler Francois Ziegler 2013-06-18T17:51:01Z 2013-06-18T17:51:01Z Here the eigenprojectors are, explicitly: $$E_1=\frac{R-\lambda_+}{1-\lambda_+}\frac{R-\lambda_-}{1-\lambda_-}$$ $$E_+=\frac{R-1}{\lambda_+-1}\frac{R-\lambda_-}{\lambda_+-\lambda_-}$$ $$E_-=\frac{R-1}{\lambda_--1}\frac{R-\lambda_+}{\lambda_--\lambda_+}$$ http://mathoverflow.net/questions/134013/square-root-of-a-certain-matrix/134020#134020 Comment by Francois Ziegler Francois Ziegler 2013-06-18T14:11:11Z 2013-06-18T14:11:11Z I think the OP wants a square root of $I+P(r,r,r-1)$, not $I+P(r,r,0)$. http://mathoverflow.net/questions/134013/square-root-of-a-certain-matrix Comment by Francois Ziegler Francois Ziegler 2013-06-18T14:04:17Z 2013-06-18T14:04:17Z Unless I'm mistaken, your matrix has minimal polynomial $(R-1)(R^2-(nr+1)R+r)=0$. From there it is easy to get its spectral form $E_1+\lambda_+E_++\lambda_-E_-$ where $\lambda_\pm=\frac12\left(nr+1\pm\sqrt{(nr+1)^2-4r}\right)$, and from there the square root $E_1+\sqrt{\lambda_+}E_++\sqrt{\lambda_-}E_-$. http://mathoverflow.net/questions/133789/standard-natural-numbers-do-not-form-a-set Comment by Francois Ziegler Francois Ziegler 2013-06-15T04:22:17Z 2013-06-15T04:22:17Z The downvotes and closure of this question seem based on misunderstanding. Indeed as Joel David points out, it is literally true that the naive "set" of standard integers is not a set in the formal sense. See e.g. page 5 of "Analyse Non Standard et Représentation du Réel" here: hal.inria.fr/inria-00163365 . (Whence the title of a famous book by the same author, "Et pourtant... ils ne remplissent pas N!") http://mathoverflow.net/questions/132105/how-can-i-transfer-this-basic-result-from-elementary-analysis-on-euclidean-spaces Comment by Francois Ziegler Francois Ziegler 2013-05-28T15:52:27Z 2013-05-28T15:52:27Z Apart from the title, this seems an exact duplicate of mathoverflow.net/questions/131911/… http://mathoverflow.net/questions/129044/defining-equations-of-a-flag-variety Comment by Francois Ziegler Francois Ziegler 2013-04-29T01:53:59Z 2013-04-29T01:53:59Z Also, Weyman's "Cohomology of Vector Bundles and Syzygies", §3.1, seems to do what you want. books.google.com/… http://mathoverflow.net/questions/119402/why-all-irreducible-representations-of-compact-groups-are-finite-dimensional-e/119423#119423 Comment by Francois Ziegler Francois Ziegler 2013-01-21T03:54:39Z 2013-01-21T03:54:39Z @Aakumadula: You are quite right about Schur's Lemma. (A detailed reference for it would be Hewitt-Ross, p. 324.) So I'd call this proof short and simple, but not quite "elementary". http://mathoverflow.net/questions/109395/is-there-a-geometric-intuition-underlying-the-notion-of-normal-varieties/117329#117329 Comment by Francois Ziegler Francois Ziegler 2012-12-27T22:22:57Z 2012-12-27T22:22:57Z Matthew -- Thanks; you are quite right and I have edited accordingly. (To me the main interest here was the statement that normality has origins earlier than Zariski.) http://mathoverflow.net/questions/115868/existence-of-l2-member-satisfying-positivity-conditions Comment by Francois Ziegler Francois Ziegler 2012-12-09T02:34:31Z 2012-12-09T02:34:31Z Not likely, by "dimension counting". Given $f$, the second integral defines a linear form on part of $f^\perp$, which is still going to have a huge kernel. http://mathoverflow.net/questions/115868/existence-of-l2-member-satisfying-positivity-conditions Comment by Francois Ziegler Francois Ziegler 2012-12-09T02:23:00Z 2012-12-09T02:23:00Z For $g\in L_2$, $\int g(x)\,dx$ need not exist... http://mathoverflow.net/questions/115866/homotopy-pi-4su2z-2 Comment by Francois Ziegler Francois Ziegler 2012-12-09T01:58:15Z 2012-12-09T01:58:15Z <google.com/search?q=Π4(S3)> http://mathoverflow.net/questions/112281/polarisation-in-a-nighbourhood-of-a-lagrangian-submanifold/112296#112296 Comment by Francois Ziegler Francois Ziegler 2012-11-14T20:28:33Z 2012-11-14T20:28:33Z No, there is no reason that the symplectomorphism send your $\alpha$ to the standard 1-form. http://mathoverflow.net/questions/112281/polarisation-in-a-nighbourhood-of-a-lagrangian-submanifold/112296#112296 Comment by Francois Ziegler Francois Ziegler 2012-11-13T19:08:18Z 2012-11-13T19:08:18Z I don't know. By "preserve" do you mean "map $\alpha$ to the standard cotangent bundle 1-form"? Why should it? I will admit that I couldn't tell what you mean by a polarization "whose one form is $\alpha$". That may also be why I don't see why you'd expect uniqueness of the transverse polarization. http://mathoverflow.net/questions/112125/weakest-assumption-for-pointwise-convergence-of-fourier-series Comment by Francois Ziegler Francois Ziegler 2012-11-12T00:50:10Z 2012-11-12T00:50:10Z @Yemon, what's wrong about the ambient class being all functions $S^1\to\mathbf{C}$? Therein the poster wants to characterize the set of all pointwise limits of pointwise convergent Fourier series.


text:
## Mass Theory – Prove that \$ mathcal B ( mathbb R) ^ infty subsetneq mathcal B ( mathbb R ^ infty) \$.

I want to prove that $$mathcal B ( mathbb R) ^ infty subsetneq mathcal B ( mathbb R ^ infty)$$, from where $$mathbb R ^ infty = prod_ {i = 1} ^ infty mathbb R$$. $$mathcal B (A)$$ is the Borel $$sigma –$$Algebra of $$A$$ and $$mathcal B ( mathbb R) ^ infty$$ is the product $$sigma –$$Algebra on $$mathbb R ^ infty$$,

I do not really know how to go about it. I know that $$mathcal B ( mathbb R) ^ infty = sigma ( mathcal C)$$ from where $$mathcal$$ is the amount of cylinder, d. H. the amount of form $$C_ {i_1, …, i_n} (B_1, …, B_n) = {x in mathbb R ^ infty mid x_ {i_1} in B_1, …, x_ {i_n } in B_n }$$ from where $$B_i$$ Borel starts $$mathbb R$$,

Now I know that $$mathcal B ( mathbb R ^ infty)$$ is the $$sigma –$$Algebra generated by open sets of $$mathbb R ^ infty$$, A basis is given by $$left { prod_ {i = 1} ^ infty U_i mid U_i text {open mathbb R and} U_i neq mathbb R text {for a finite number of i } right }.$$

For the recording it is clear whether $$left { prod_ {i = 1} ^ nB_ {j_i} mid prod_ {i = 1} U_i in mathcal B ( mathbb R ^ infty) text {where} U_ {j_i} = B_ {j_i} text {and} U_k = mathbb R text {if k notin {j_1, …, j_n } } right },$$
is a $$sigma –$$Algebra, the open set of included $$mathbb R ^ n$$and thus are cylinders in $$mathcal B ( mathbb R ^ infty)$$,

How can I prove that the inclusion is strict?


text:
Home  >>  CBSE XII  >>  Math  >>  Matrices

# Show that if A and B are square matrices such that AB=BA,then$(A+B)^2=A^2+2AB+B^2$

## 1 Answer

Toolbox:
• $(A+B)^2=A^2+2AB+B^2$
Step1:
To prove
$(A+B)^2=A^2+2AB+B^2$
LHS:-
$(A+B)^2=(A+B)(A+B)$
$\;\;\;\;\;\qquad=A(A+B)+B(A+B)$
$\;\;\;\;\;\qquad=AA+AB+BA+B.B$
$\;\;\;\;\;\qquad=A^2+AB+BA+B^2$
Step2
Given:
AB=BA.
Hence replace AB=BA.
$\Rightarrow A^2+AB+AB+B^2$
$\Rightarrow A^2+2AB+B^2$
Hence proved.
$(A+B)^2=A^2+2AB+B^2$
answered Mar 23, 2013

1 answer

1 answer

1 answer

1 answer

1 answer

1 answer

1 answer


text:
• 10
• 10
• 12
• 12
• 14
• entries
122
121
• views
68723

# O_O

132 views

I've, like, neglected this thing. Not to mention GDNet+ expired on me. I've moved on from my Arena Battle project.. in retrospect, it was getting kind of convoluted. I'm now working on something totally new for me... Project 1 in the C++ Workshop!

No, seriously, it's new! You have to color the output, so I Googled around and found this. Encapsulating it in a class (cConsole), and setting a few functions for easy manipulation, I've created an easy interface with the console screen. I can color output with output(), and check specific keypresses with getKey().

Oh, getKey().. I just love it. I'll code-tag it here, and explain it, although all you C++ gurus out there should be able to understand it just fine. I just like explaining. :D

char cConsole::getKey(){    while (true)    {        GetNumberOfConsoleInputEvents(rHnd, &numEvents);        if (numEvents == 0) continue;        INPUT_RECORD *eventBuffer = new INPUT_RECORD[numEvents];        ReadConsoleInput(rHnd, eventBuffer, numEvents, &numEventsRead);        int i = 0;        for (; i < numEventsRead;)        {            if (eventBuffer.EventType == KEY_EVENT)            {                int j = 0;                while (j <= keys.length())                {                    if (eventBuffer.Event.KeyEvent.wVirtualKeyCode == keys[j])                    {                        delete eventBuffer;                        break;                    }                    j++;                }                if (j == keys.length() + 1)                {                    i++;                    continue;                }                else return keys[j];            }            else            {                i++;                continue;            }        }    }}

To start with, I have a public string variable in cConsole (don't yell!) that the user can set to, say, "123". This function cycles through any events that occur until a key event occurs. Then it cycles through the 'keys' string I mentioned, checking the key event against each character in 'keys', until it finds a match or cycles the whole string. If the iteration variable 'j' is 1 more than the string length, then it didn't find a match, and it continues. Otherwise, it returns the char that was matched, so the calling function can deal with it.

I love it. :D

Very nicely done! Nice clean code, simple to use, and very helpful.

Btw, I'm stealing it when I do my text-based RPG!


text:
Within the Kirby configuration file site/config/config.php there are additional options which can be turned on if desired. Currently there is just one option.

## MathJax

If you wish to display mathematical formulae in your KDoc website, the MathJax option is for you. MathJax is described as "A JavaScript display engine for mathematics that works in all browsers."

Turning on MathJax is as simple as modifying the default value of "false" to "true" in the Kirby config file mentioned earlier. The default settings are as follows.

c::set('kdoc.mathjax', false);
c::set('kdoc.mathjax.url', 'https://cdn.mathjax.org/mathjax/latest/MathJax.js');
c::set('kdoc.mathjax.config', 'TeX-AMS_CHTML');

See further information about these settings in the comments of the config file itself.An example of it's usage is, if you type:

$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$

The output will look like:

$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$

Further information about how to use MathJax can be found at the MathJax website.


text:
Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals
Search results

Search: MSC category 26D15 ( Inequalities for sums, series and integrals )

 Expand all        Collapse all Results 1 - 7 of 7

1. CMB Online first

Atıcı, Ferhan M.; Yaldız, Hatice
 Convex Functions on Discrete Time Domains In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f \geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept, we state and prove discrete Hermite-Hadamard inequality using the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$). Second, we state and prove the discrete fractional Hermite-Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale. Keywords:discrete calculus, discrete fractional calculus, convex functions, discrete Hermite-Hadamard inequalityCategories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15

2. CMB 2014 (vol 58 pp. 188)

Wirths, Karl Joachim
 Telescoping Estimates for Smooth Series We derive telescoping majorants and minorants for some classes of series and give applications of these results. Keywords:telescoping series, Stietjes constant, Hardy's formula, Stirling's formulaCategories:26D15, 40A25, 97I30

3. CMB 2011 (vol 55 pp. 355)

Nhan, Nguyen Du Vi; Duc, Dinh Thanh
 Convolution Inequalities in $l_p$ Weighted Spaces Various weighted $l_p$-norm inequalities in convolutions are derived by a simple and general principle whose $l_2$ version was obtained by using the theory of reproducing kernels. Applications to the Riemann zeta function and a difference equation are also considered. Keywords:inequalities for sums, convolutionCategories:26D15, 44A35

4. CMB 2011 (vol 54 pp. 630)

Fiorenza, Alberto; Gupta, Babita; Jain, Pankaj
 Mixed Norm Type Hardy Inequalities Higher dimensional mixed norm type inequalities involving certain integral operators are characterized in terms of the corresponding lower dimensional inequalities. Keywords:Hardy inequality, reverse Hardy inequality, mixed norm, Hardy-Steklov operatorCategories:26D10, 26D15

5. CMB 2010 (vol 53 pp. 327)

Luor, Dah-Chin
 Multidimensional Exponential Inequalities with Weights We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$\Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p},$$ where 0<$p$, $q$<$\infty$, $\Phi$ is a logarithmically convex function, and $T_k$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin--Cochran-Lee type inequalities. Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regionsCategories:26D15, 26D10

6. CMB 2005 (vol 48 pp. 333)

Alzer, Horst
 Monotonicity Properties of the Hurwitz Zeta Function Let $$\zeta(s,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^s} \quad{(s>1,\, x>0)}$$ be the Hurwitz zeta function and let $$Q(x)=Q(x;\alpha,\beta;a,b)=\frac{(\zeta(\alpha,x))^a}{(\zeta(\beta,x))^b},$$ where $\alpha, \beta>1$ and $a,b>0$ are real numbers. We prove: (i) The function $Q$ is decreasing on $(0,\infty)$ if{}f $\alpha a-\beta b\geq \max(a-b,0)$. (ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a-\beta b\leq \min(a-b,0)$. An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s-1)\zeta(s,x)]^{1/(s-1)}$ is decreasing on $(1,\infty)$. This settles a conjecture of Bastien and Rogalski. Categories:11M35, 26D15

7. CMB 1999 (vol 42 pp. 478)

Pruss, Alexander R.
 A Remark On the Moser-Aubin Inequality For Axially Symmetric Functions On the Sphere Let $\scr S_r$ be the collection of all axially symmetric functions $f$ in the Sobolev space $H^1(\Sph^2)$ such that $\int_{\Sph^2} x_ie^{2f(\mathbf{x})} \, d\omega(\mathbf{x})$ vanishes for $i=1,2,3$. We prove that $$\inf_{f\in \scr S_r} \frac12 \int_{\Sph^2} |\nabla f|^2 \, d\omega + 2\int_{\Sph^2} f \, d\omega- \log \int_{\Sph^2} e^{2f} \, d\omega > -\oo,$$ and that this infimum is attained. This complements recent work of Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang concerning the Moser-Aubin inequality. Keywords:Moser inequality, borderline Sobolev inequalities, axially symmetric functionsCategories:26D15, 58G30
 top of page | contact us | privacy | site map |

© Canadian Mathematical Society, 2015 : https://cms.math.ca/


text:
10,671 Pages

The googolplexitritrex is equal to \( 10\uparrow\uparrow\uparrow\) googolplexiduduex using arrow notation.[1] The term was coined by Wikia user Username5243.

• In Bird's array notation, it would be equal to \([10,[10,[10,[10,[10,100]],2],2],3]\). (Trivially, this is \(\{10,\{10,\{10,\{10,\{10,100\}\},2\},2\},3\}\) in BEAF.)
• In the FGH, this is roughly equal to \(f_4(f_3^2(f_2(f_2(324))))\).

## Sources

Community content is available under CC-BY-SA unless otherwise noted.


text:
# Probability of general Brownian (or non) bridge to be higher than given parameter?

Consider general Brownian bridge W(0)=0; W(T) = a. (Here "general" means: $W(T)\ne 0$).

What is the probability W(t) >= b, for all $t \in [0, T]$ ?

Is there close simple formula in terms of a, b , T ?

EDIT Is the answer equal to $1-exp(-2b(a+b)/T)$ or $exp(-2b(a-b)/T)$ ? (See comment-edit below).

Can the formula be the same for any martingale type stochastic process (e.g. random walk), not only brownian motion, or it somehow depends on the details of stochastic process ? If there is such dependence how the questioned probability will change if we consider the distribition W(t) to have more and more "heavy tails" ?

(I'm sure that questions are well-known for experts, but it is somehow difficult to google the asnwer, so let me ask here).

Remark: If we consider somewhat informally related question - probability for W first hit a, before hitting (-b), (i.e. W(some T)=a and W(t < T) > b) there is simple formula P = b/(a+b), which holds true for any martingale stochastic process. The questions are somewhat different, but still resemble each other and so the simplicity of the answer P=b/(a+b) makes me hope that the answer to my question might be simple and closed form.

EDIT As MattF suggested at a comment we might look at "On the maximum of the generalized Brownian bridge" Theorem 2.1. And the answer seems to be almost there - given by the formula presented in edit above. But I am not 100% sure, since they consider more general case and we need to set u=t which causes division by zero in their formula. Hopefully it is not a big problem - the formula is continuous and just simplifies to $exp( -2\beta(\beta-\eta)/ u )$ in a limit u=t, in their notations.

Is it correct ?

To pass from my notations - I am interested in "min greater" not "max greater", that is why I would need to change sign for their $\eta$ and chnage to "1-their answer" and $\eta -> -a$ $\beta -> b$".

However the answer in my notations $1-exp(-2b(b+a)/T)$ does not seems to be reasonable, because if a=b, the answer should be equal to "1", which seems to contradict the formula.

It seems $exp(-2b(a-b)/T)$, for a>=b would be more reasonable since at least we get "1" for a=b, but it seems not to correspond to the formula from the paper or I made some stupid mistake passing from one notations to the other.

• By googling I find: "the maximum of the generalized Brownian bridge", theorem 2.1. researchgate.net/publication/… – Matt F. May 5 '17 at 20:54
• @MattF. Thank you very very much ! I hope that the paper gives the answer, however I am not completely sure - they consider more general case than I need and we need to set u=t to get my case, which however causes division by zero in their formula - it very might be that it is not a problem we just need to write $\infty$ and their formula simplifies at lot. However I am not 100% sure. – Alexander Chervov May 6 '17 at 12:23

For Brownian bridge, this is more or less a standard exercise. It is simpler to do it first for Brownian motion and then move to Brownian bridge. There are two steps. I assume $b<0$, otherwise the answer is $0$. Let $c=a-b>0$.

1) By the reflection principle, $$P(W_t\geq b \; \forall t\in [0,T], W_T\in [a,a+dx])=P^{-b}(W_t\geq 0 \;\forall t\in [0,T], W_T\in [c,c+dx])= \frac{1}{\sqrt{2\pi T}} \left(e^{-a^2/2T}-e^{-(a-2b)^2/2T}\right)dx:=F(a,b,T)dx.$$

2) We also have $$P(W_T\in [a,a+dx])=\frac{1}{\sqrt{2\pi T}}e^{-a^2/2T}dx:= G(a,T)dx.$$

So the answer to your question is $F(a,b,T)/G(a,T)$.

For more general random walks, it is hopeless to get explicit formulae (since you do not have a reflection principle, unless the random walk is e.g. simple random walk or a variant of that), and even the analogue of $G$ is not explicit. But asymptotics are available, see F. Caravenna, A local limit theorem for random walks conditioned to stay positive, Probab. Theory Related Fields, 133:508--530, 2005.

• Thank you very much ! The answer seems to be correct - it is correct in limiting cases and also seems same (up to notations) as the paper quoted. But I am not expert in stoch.proc. so I cannot fully understand your argument with reflection principle, if you can give more details it would be very helpful. What is $P^b$ and why it equals to what you write ? – Alexander Chervov May 14 '17 at 9:06
• A reffrence to Reflection principle which close to your discussion and written for simplyminded persons would be also very helpful . Thanks in advance! – Alexander Chervov May 14 '17 at 9:39
• $P^b$ is the law of the process starting at $b$. The equality is just shift-invariance of increments of Brownian motion. As for reflection principle, just google it... en.wikipedia.org/wiki/Reflection_principle_(Wiener_process) – ofer zeitouni May 18 '17 at 19:59
• Wiki does not seem to give similar reflection principle that you use – Alexander Chervov May 18 '17 at 20:54
• Of course it does. The probability in question is to start at $-b$ and end at $c$ w/out touching 0. The reflection principle (and a union-exclusion) tells you that this is the same as hitting $c$ w/out the constraint, minus the probability of hitting $-c$. This is what I wrote. If you prefer and are more analytcally inclined, solve the heat equation with Dirichlet boundary conditions at $0$. – ofer zeitouni May 18 '17 at 23:30

Consider the standard Wiener process $W = (W_t, 0 \leq t \leq 1)$ and let $m = \inf_{0\leq t\leq 1} W_t$, $M = \sup_{0\leq t\leq 1} W_t$. If $a\leq 0 \leq b$ and $a\leq a' < b' \leq b$, then

$$\mathbb{P}(a<m\leq M < b; a' < W_1 < b') = \sum_{k=-\infty}^\infty \left(\Phi(2k(b-a)+b')-\Phi(2k(b-a)+a')\right)\\ - \sum_{k=-\infty}^\infty \left(\Phi(2k(b-a)+2b-a')-\Phi(2k(b-a)+2b-b')\right)$$

This can be found as for example Theorem 9.10 in Billingsley's Convergence of Probability Measures. A similar result for the standard Brownian bridge can be found using the previous result, and is Theorem 9.39 in the same text. It says that:

$$\mathbb{P}\left(a < \inf_t W_t^\circ\leq \sup_t W_t^\circ \leq b\right) = \sum_{k=-\infty}^\infty \left(e^{-2k^2(b-a)^2}-e^{-2(b+k(b-a))^2}\right)$$

Of course, these results hold considering $t\in[0,1]$ and $W^\circ(1)=0$, but these should not be too hard to transfer to the case you are thinking of.

The derivation of these distribution functions relies on analysing the simple random walk, and then using Donsker's theorem to take a limit of this case.

• Thank you for the answer ! However it seems a little bit different from what I ask. You consider W simultaneosly greater m and smaller M, but in my question I do no impose second condition. Any way thank you very much for your answer ! – Alexander Chervov May 14 '17 at 9:08
• I believe your desired distribution can be had simply by taking the limit as $a\to -\infty$. Similarly, if you desire that $W_1=\alpha$, the right distribution should be the limit of $P(a< m \leq M < b; \alpha < W_1 < \alpha + \epsilon)$ as $\epsilon\to 0$ and $a\to -\infty$. At least, that is how one gets distributions for the Brownian bridge from the above. (i.e. the case $\alpha = 0$.) – Vilhelm Agdur May 14 '17 at 12:39


text:
# Publishing house / Banach Center Publications / All volumes

## On the Rankin–Selberg convolution of degree 2 functions from the extended Selberg class

### Volume 118 / 2019

Banach Center Publications 118 (2019), 25-35 MSC: Primary 11M41, 11F30. DOI: 10.4064/bc118-2

#### Abstract

Let $F(s)$ be a function of degree $2$ from the extended Selberg class. Assuming certain bounds for the shifted convolution sums associated with $F(s)$, we prove that the Rankin–Selberg convolution $F\otimes \overline {F}(s)$ has holomorphic continuation to the half-plane $\sigma \gt \theta$ apart from a simple pole at $s=1$, where $1/2 \lt \theta \lt 1$ depends on the above mentioned bounds.

#### Authors

• Jerzy KaczorowskiFaculty of Mathematics and Computer Science
Umultowska 87
61-614 Poznań, Poland
and
Institute of Mathematics of the Polish Academy of Sciences
00-656 Warsaw, Poland
e-mail
• Alberto PerelliDipartimento di Matematica
Università di Genova
via Dodecaneso 35
16146 Genova, Italy
e-mail

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.


text:
On the way to achieve minimalism and essentialism

Category: Life

Why I still haven’t found a life partner

Yesterday I downloaded an audio book from Audible.com – which is called The Mathematics of Love, written by Hannah Fry who seems to be a pretty charming mathematician, who also from time to time making some YouTube Numberphile videos such as this one.

Never mind… The point is in her book she mentioned a method I used previously on a beer conversation with my lovely colleagues, a method can help you calculate how many potential life partners are out there waiting for you to be find. (I guess every single man or woman with a little of brain function would have probably thought about the same question at least once when he or she was half way intoxicated and there happened to be a young couple across the bar kissing as if they were the main actors of the show that evening…)

Anyway, the method is, actually very simple, just using the number of total potential partners, multiply by a few factors each from a certain perspective can help you estimate how many percentage of them are actually can be your life partner. It said to be the same way scientists try estimate the number of highly evolved civilizations that might exist in our galaxy (Drake equation), mentioned by Hannah in her book when referring a paper called Why I don’t have a girlfriend – by Peter Backus.

$G = N \cdot f_W \cdot f_A \cdot f_S \cdot f_U \cdot f_{B1} \cdot f_{B2} \cdot f_{P1} \cdot f_{P2} \cdot f_f$

I remember I did the calculation, once, for the fun of it, together with my colleagues, and the result was quite shocking. So I am going to do it here again:

Suppose I live in Denmark, and the whole population is around N=6,000,000, and we assume:

• $f_W$ = the fraction of people in DK who are women = 0.5;
• $f_A$ = the fraction of the women in DK who are age-appropriate = 10 years out of 70 years (suppose the age is evenly distributed) = 1/7;
• $f_S$ = the fraction of the women with the right age are single = 0.5;
• $f_U$ = the fraction of the women who have a university education = 0.26 (no offense but I have the same reason with the guy mentioned in the paper Why I don’t have a girlfriend;
• $f_{B1}$ = the fraction of the women with the right age with the right education who I find physically attractive = 2 out of 10 = 0.2;
• $f_{B2}$ = the fraction of the women with the right age with the right education who find I physically attractive = 2 out of 10 = 0.2; (Let’s say I am average looking, if you don’t agree, comments below and I will reduce this value);
• $f_{P1}$ = the fraction of the women with the right age with the right education who we both find attractive and who think my personality is acceptable also enjoy being with me, or you may call it have a word to say between = 0.5;
• $f_{P2}$ = chance that I enjoy being with her, same above = 0.5;
• $f_f$ = sadly because I am a foreigner who don’t speak Danish yet so maybe half of the women in the end decided not to date me = 0.5.

Thus, we can easily calculation G, which is the possible number of ladies in Denmark could become a life partner of mine:

$G = 6e10^6 \cdot 0.5 \cdot {1 \over 7} \cdot 0.5 \cdot 0.26 \cdot 0.2 \cdot 0.2 \cdot 0.5 \cdot 0.5 \cdot 0.5 = 279$

My goodness, only 279 women are out there in DK could be my life partner, based on a positive estimation.

However since I live in a town called Odense, suppose if I don’t go out of the town and just try to meet some local people, with the population of 190,000 in Odense, the number G becomes 8.8. It seems not that bad, but if you look at the numbers again, you will see, statistically, out of every thousand girls at the right age, there’s only a half girl could potentially become my life partner. What a lonely planet ?!

The next question to think about is, how to identify those 279 ladies in Denmark, I mean at least efficiently using my spare time. At the moment, I have no idea 🙂

What are the simplest things one can do to make oneself happier?

When I was in a dark period I instituted a simple rule that changed my life.

Rule: When I arrive home from work, the very first thing I tell my wife is the best thing that happened that day.

No exceptions. No complaining. Just the best thing that day, even if it was just a good cup of coffee. This had the effect of starting our evening off on a positive note and it changed our relationship.

What are the simplest things one can do to make oneself happier?

How do I learn to code? What language should I start with?

Answer by Roshan Choxi on Quora:

It’s been my sole focus to answer this question for the last two years, and I think a lot of the resources mentioned here are great but I’ve noticed there are three strategies that successful students consistently use better than anyone else regardless of what resources they use:

1. Focus on habits, not goals
2. Learning alone is painful
3. Build things

Note: some of this is borrowed from my answer to another Quora question: How can I prepare for Bloc?

1. Focus on habits, not goals

It seems counterintuitive that you shouldn’t focus on goals, but hear me out — it’s all about leverage. Anyone who works with me knows that I dweebishly reference the R’as Al Ghul scene in Batman Begins pretty much 3-4 times a day:

Our investors at Bloc are getting tired of board meetings starting with Batman clips.

R’as tells Bruce:

If you focus on building the habit of programming for 20-30 hours a week, you will reach your goal of being a web developer. If you focus on the goal of being a web developer in X months, you get nothing from that but stress and insecurity about how far along you are. Focus on the habit, not the goal. Rub your chest, your arms will take care of themselves.

So here’s what you should do right now: put 15 minutes a day on your calendar to spend time programming. Don’t do more than 15, just focus on doing 15 minutes a day. If you can do it successfully with no excuses for a week, try bumping it up to 20 minutes a day. Don’t try to overextend yourself by doing an hour a day right off the bat, this is going to be a 10,000 hour marathon so we’re focusing on developing the habit right now. The number of minutes you put in isn’t as important as you showing up each day.

2. Learning alone is painful

When I was learning web development, the two biggest social components to my learning were having a mentor and belonging to a community.

Having a mentor

I worked at a small startup called merge.fm while in college. I learned more in the summer I spent working with one of their cofounders than I did in the entire previous year at my university. There’s just something about working alongside an expert who knows more than you that really accelerates your learning, you’re able to pick up on how they think and unveil what you don’t know you don’t know. There’s a reason why mentorship used to be the de facto standard of learning a new trade, it’s very effective.

Belonging to a community

For me, the two communities I belonged to were the Illini Entrepreneurship Network (a student organization at my university) and HackerNews (a large hacker/startup oriented online community).

I didn’t learn what objects and classes were from HackerNews, but I learned a different category of things. I learned that nobody likes Javascript. I learned that Rubyists are the hipsters of programming. I learned that Bret Taylor, Rich Hickey, and John Carmack are programming gods, and that software companies that are truly serious about coffee have kitchens that look like meth labs.

In short, I learned how to talk shop. That turns out to be important when you’re working with other developers, but it’s also the thing that makes you feel like a developer.

3. Build things

In the first year of learning web development, I built:

• A Digg Clone (from a Sitepoint book on Rails, I believe it’s out of date now though)
• An E-Commerce App (from Agile Web Development with Rails 4)
• A GeekSquad-esque App (personal project)
• A Realtime, Online Classroom (personal project)
• A Foreign Language Flashcard App (class project)

I think building real projects is important for many reasons, but the most important one to me is because it’s fun. That’s something that is tragically lost in classical education, but I think it’s important enough to be on this list. Look for resources that show you how to build things, http://ruby.railstutorial.org/ is a good one.

4. Be a cockroach

I secretly added a 4th item for those of you who’ve stuck around to read this far down the page.

Paul Graham once told the founders of Airbnb:

“You guys won’t die, you’re like cockroaches.”

You’ll probably want to quit learning how to code at some point. Like anything worthwhile, it’s difficult and will make you feel stupid at times. This is why #1 on this list is so important — stop worrying so much about whether you’re making progress or how much longer it’ll be until you feel like you’ve “made it.” All you have to do is focus on showing up, for 10-30 hours a week. Be as mindless as a cockroach about everything else, and don’t “die.”

I made the statement years ago which is often quoted that 80 percent of life is showing up. People used to always say to me that they wanted to write a play, they wanted to write a movie, they wanted to write a novel, and the couple of people that did it were 80 percent of the way to having something happen. All the other people struck out without ever getting that pack. They couldn’t do it, that’s why they don’t accomplish a thing, they don’t do the thing, so once you do it, if you actually write your film script, or write your novel, you are more than half way towards something good happening. So that I would say was say my biggest life lesson that has worked. All others have failed me.

Woody Allen

How do I learn to code? What language should I start with?

In other word, keep it simple and do fewer things (or one thing) at a time

I just saw an updated post from my previous colleague – a system architect in a software company. In the post he shared with me some blog pages on the topic of project management and planning. Even it is widely used or considered in software industry, but the issues and challenges they met there are more or less the same with our  issues and challenges seen in project management, in the industry as an automobile sub-suppler.

Here are some of the blog pages (also some of the “laws” that linked in the page are very enlightening.)
Cumulative Flow Diagram
The Project Management Principles That (Almost) Nobody Teaches

One of the key ideas is called Little’s Law:
“.. In order to meet development or service deadline, we must reduce the work-in-process, or finish it before starting a new jobs.”

Note: “work-in-progress” means the number of requests (work units) that are being processed, i.e. they have entered the system, but have not got out yet.


text:
# Parametric Equations and Planar Curves

by Vincent P. Schielack (Texas A&M University) and Kirby C. Smith (Texas A&M University)

College Mathematics Journal
September, 1994

Subject classification(s): Algebra and Number Theory | Linear Algebra
Applicable Course(s): 3.8 Linear/Matrix Algebra

The author shows that the system characterized by

$x(t) = a_1 t^2 + b_1 t + c_1$
$y(t) = a_2 t^2 + b_2 t + c_2$
$z(t) = a_3 t^2 + b_3 t + c_3$

must lie in a plane. He does this with an illustrative example represented by
$x(t) = A [t^2, t, 1]^T$
where $A$ is a constant $3 \times 3$ matrix.

A pdf copy of the article can be viewed by clicking below. Since the copy is a faithful reproduction of the actual journal pages, the article may not begin at the top of the first page.


text:
### Home > CCG > Chapter Ch7 > Lesson 7.1.1 > Problem7-11

7-11.

Solve each system of equations below, if possible. If it is not possible, explain what the lack of an algebraic solution tells you about the graphs of the equations. Write each solution in the form $(x,y)$. Show all work.

1. \begin{aligned}[t] &y = -2x - 1 \\ &y = \frac{1}{2}x - 16 \end{aligned}

By the Equal Values Method,

$-2x-1=\frac{1}{2}x-16$

$x=6$

Substitute $6$ for $x$ in one of the original equations.
Solve for $y$, which will give you the intersection of the two lines.

1. \begin{aligned}[t] &y = x^2 + 1 \\ &y = -x^2 \end{aligned}

By the Equal Values Method,

$x^2+1=-x^2$

$2x^2+1=0$

$x=\frac{0\pm\sqrt{0^2=(4)(2)(1)}}{2\cdot2}\;\to\;x=\frac{0\pm\sqrt{-8}}{4}$


text:
# Mutual Information of the summation of a Chi-square random variable and a Gaussian variable

As the title, $$X$$ is an random variable subject to $$N(0,1)$$, $$N$$ is an random variable subject to $$N(0,\sigma^2)$$, and $$X$$ and $$N$$ are independent. I want to calculate the mutual information $$I(X,Y)$$ between $$X$$ and $$Y=X^2+N$$. The problem can be reduced to calculate the differential entropy $$h(Y)$$ of $$Y$$. However, $$h(Y)=-\int_{-\infty}^{+\infty}p_Y(y)log(p_Y(y))\mathrm{d}y$$, and the $$p_Y(y)$$ can not get the explicit form. How can I calculate $$h(Y)$$ or $$I(X,Y)$$ can be calculated in other ways?

It is not quite true that $$p_Y(y)$$ has no explicit form, it is given by $$p_Y(y)=\int_0^\infty \frac{e^{-(w-y)^2/2 \sigma^2-w/2}}{2 \pi \sigma \sqrt{w}}\,dw=f_1(y)\theta(2y-\sigma^2)+f_2(y)\theta(\sigma^2-2y),$$ with $$\theta(y)$$ the unit step function and $$f_1(y)=\frac{1}{4\sigma\sqrt 2}e^{\frac{\sigma^4-4 \sigma^2 y-4 y^2}{16 \sigma^2}} \sqrt{2y-\sigma^2} \left[I_{1/4}\left(\frac{\left(\sigma^2-2 y\right)^2}{16 \sigma^2}\right)+I_{-1/4}\left(\frac{\left(\sigma^2-2 y\right)^2}{16 \sigma^2}\right)\right],$$ $$f_2(y)=\frac{1}{4\pi\sigma}e^{\frac{\sigma^4-4 \sigma^2 y-4 y^2}{16 \sigma^2}} \sqrt{\sigma^2-2 y}\; K_{1/4}\left(\frac{\left(\sigma^2-2 y\right)^2}{16 \sigma^2}\right).$$ The differential entropy $$h(Y)$$ can be readily evaluated numerically, here is a plot of $$h(Y)$$ versus $$\sigma$$.