This one, by Oleg Golberg, appeared in the issue of Mathematical Reflections (MR) 2007. I don’t have a solution yet, but I think I should be able to solve it sooner or later. If you find a solution, you should send it to MR by Jan 19. Here is the problem anyway.

For all integers , prove that

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March 7, 2019 at 9:14 pm

tusudjuindeed this inequality is very important and must used by scholars observed prof dr mircea orasanu and prof drd horia orasanu and followed that this can be extended for important functions or operators as LAGRANGIAN OPERATORS or CONSTRAINTS OPTIMIZATIONS As a simple example of such a generalized momentum, we consider the angular momentum of a particle in a central potential. If we use polar coordinates r, to describe the motion of a single particle in the plane, then the Lagrangian has the form L = T–V = m(2 + r22)/2–V(r), and the angular momentum of the system is represented by L/.

If the Lagrangian (17) is not an explicit function of time, then a derivation formally equivalent to that in Sec. IVB (with time as the single variable) shows that the function ( 1L/i)–L, sometimes called12 the energy function h, is a constant of the motion of the system, which in the simple cases we cover13 can be interpreted as the total energy E of the system.

If the Lagrangian (17) depends explicitly on time, then this derivation yields

2 . FORMULATION

We will calculate the flux out of the box to find an expression for divergence.

We continue by simplifying these integrals and then grouping N terms and M terms together.

We see that the flux equals the area of the box times an expession of M and N. This expression is defined as the divergence. The notation used to show divergence is

Also note that divF=flux/area.

In terms of fluid flow, the divergence can be interpreted as the flux over an infinitely small loop.

We neglect to show the entire process for circulation, but through a similar method to the one shown above, it can be demonstrated that:

Once again, our result is the area of the box times a (somewhat different) expression of M and N. This expression is defined as the curl.

Note that the curl is equivalent to the circulation/area.

Also, curl is a vector quantity, but in 2 dimensions, this can be neglected since the direction of the vector will always be perpendicular to the 2 dimensions. A more complete definition will be given later in the course.

Green’s Theorem

For circulation:

For flux:

R is the region enclosed by C.

Green’s Theorem converts the line integral into an area integral, which is often easier to solve.

Examples will be given next lecture.

The Hankel transform (of order zero) is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel and also called the Fourier-Bessel transform. It is defined as

(1)

(2)

Let

(3)

(4)

so that

May 4, 2019 at 10:19 pm

prof drd horia orasanuin thus are observable many situations appear important aspects that are studied fundamentals problems stated by prof dr mircea orasanu and prof drd horia orasanu that are followed with of adelic generalized functions, as linear continuous functionals on the space of Schwartz-Bruhat functions, are considered. The importance of adelic generalized functions in adelic quantum mechanics is demonstrated. In particular, adelic product formula for Gauss integrals is derived, and the connection between the functional relation for the Riemann zeta function and quantum states of the harmonic oscillator is stated.”

________________________________________

P.G. Rooney, “Another proof of the functional equation for the Riemann zeta function”, Journal of Mathematical Analysis and Applications 185 (1994) 223-228

[abstract:] “A new proof of the functional equation for the Riemann zeta function is given, based on the theory of Mellin multiplier transformations.”

________________________________________

A. Ossicini, “An alternative form of the functional equation for Riemann’s zeta function”, Atti Semin. Mat. Fis. Univ. Modena e Reggio Emilia 56 (2008–2009) 95–111

[abstract:] “In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann’s Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work “Remarques sur un beau rapport entre les series des puissances tant directes que reciproques” in Memoires de l’Academie des Sciences de Berlin 17, (1768), permit to define a special function, named $A(s)$, which is fully symmetric and is similar to Riemann’s “XI” function. To be complete we find several integral representations of the $A(s)$ function and as a direct consequence of the second integral representation we obtain also an analytic continuation of the same function using an identity of Ramanujan.”

treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for $\zeta(s)$. Riemann provides two different proofs of this. We present here, after showing the first one, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper

May 10, 2019 at 11:05 am

prof drd horia orasanuthese subject must extended observed prof dr mircea orasanu and prof drd horia orasanu and followed that these aspects must posters

May 16, 2019 at 11:26 pm

prof dr mircea orasanuin these cases are considered more situations as observed for prof dr mircea orasanu and prof drd mircea orasanu and followed in special for HANKEL transform and consequences expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind Jν(kr). The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r axis. The necessary coefficient Fν of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval. ) support for such a position and his associated critique of arguments put by those in the mathematics education community he labelled as ‘multiculturalists’ is both misleading and flawed in several respects

May 18, 2019 at 11:06 am

prof dr mircea orasanufor many situations can e applied other theorems and post observed prof dr mircea orasanu and prof drd horia orasanu with followed unlimited consequences and thus the future of teacher education at the present time, I believe that it is relevant to consider the wider social and political context in which schools and institutions of teacher education are placed at this time. In particular I wish to draw attention to what Prime Minister Tony Blair had to say in his speech to the 1998 Labour Party Conference, where he argued that:

The centre-left may have lost in the battle of ideas in the 1980s, but we are winning now. And we have won a bigger battle today: the battle of values. The challenge we face has to be met by us together: one nation, one community.

When a young

June 12, 2019 at 3:33 am

prof dr mircea orasanuin many situations appear solutions as observed pr and followed for developments of dr mircea orasanu mircea and prof drd horia orasanu and followed for developments some Camille Jordan’s father, Esprit-Alexandre Jordan (1800-1888), was an engineer who had been educated at the École Polytechnique. Camille’s mother, Joséphine Puvis de Chavannes, was the sister of the famous painter Pierre Puvis de Chavannes who was the foremost French mural painter of the second half of the 19th century. Camille’s father’s family were also quite well known; a grand-uncle also called Ennemond-Camille Jordan (1771-1821) achieved a high political position while a cousin Alexis Jordan (1814-1897) was a famous botanist