Riemann

This result, called the Riemann Integral, can be applied to unsteady isentropic compression waves as well as to expansion waves. By defining a Riemann function ... 

Note that in arriving at (2.58) it was assumed that there was no change in entropy along the rarefaction. This assumption is equivalent to stating that the rarefaction must be propagating into a uniform state. The Riemann Invarient has been defined in terms of the Riemann function... 

By plotting Hugoniot curves in the pressure-particle velocity plane (P-u diagrams), a number of interactions between surfaces, shocks, and rarefactions were solved graphically. Also, the equation for entropy on the Hugoniot was expanded in terms of specific volume to show that the Hugoniot and isentrope for a material is the same in the limit of small strains. Finally, the Riemann function was derived and used to define the Riemann Invarient. 

Riemann Invarient A constant defined by (2.61), which is independent of position on a rarefaction wave that propagates into a uniform state. 

J.K. Dukowicz, A General, Non-Iterative Riemann Solver for Godunov s Method, J. Comput. Phys. 61 (1985). 

P. Colella and H.M. Glaz, Efficient Solution Algorithms for the Riemann Problem for Real Gasses, J. Comput. Phys. 59 (1985). 

Z) is a triangle. The integral in (4,75) is, in Riemann s definition, an improper integral. It can be approximated by a finite sum because the integrand is monotone, which implies, furthermore, that is bounded. I skip the details because considerations of this type are standard in the discussion of improper Riemannian integrals, Equations (4,73) and (4.75), in the notation of (4.56), imply... 

W Riemann and H F Walton, Ion Exchange in Analytical Chemistry, Pergamon Press, Oxford, 1970... 

Voderholzer U, Riemann D, Hornyak M, et al A double-blind, randomized and placebo-controlled study on the polysomnographic withdrawal effects of zopiclone, zolpidem and triazolam in healthy subjects. Eur Arch Psychiatry Clin Neurosci 251 117-123, 2001... 

Oryza sativum X66125 D84400 OS378734 243fdnayysnllsnKgllhsdq262 249fdmyyqnllnqKgllssdq268 255fdlgyfKnvaKRRglfhsdg280 (Riemann et al., 1992) (Ito et al., 2000) (Chittoor et al., 1997)... 

The important point is that v E,k) and A0( ,fe) are multivalued, with a logarithmic branch point at fe = s = 0, while the residual functions m(e, k, zo) and w(e, k) are single valued. The result, as discussed in more detail later, is that the value of the quantum number v depends on the chosen location of the branch cut and on which Riemann sheet is taken, bearing in mind that the branch of arctan( /fe) must be taken according to the appropriate quadrant of the complex k, e) plane. Thus cj) = arctan( /fe) + ti/2 increases smoothly from zero to In around a counterclockwise circle in the k, e) plane, starting at = 0 and < 0. 

The Bohr quantization condition, with quantum number v, corresponds to choosing the lowest Riemann sheet, which requires an additional phase correction of —2ti on crossing the branch cut, which is taken along the positive -axis. Thus the phase term cj) rises to ti at = 0+ and > 0, drops abruptly to —71 on crossing the positive fe = 0 axis, and returns to zero on the negative... 

It is, however, more revealing in the context of monodromy to allow/(s, ) to pass from one Riemann sheet to the next, at the branch cut, a procedure that leads to the construction in Fig. 4, due to Sadovskii and Zhilinskii [2], by which a unit cell of the quantum lattice, with sides defined here by unit changes in k and v, is transported from one cell to the next on a path around the critical point at the center of the lattice. Note, in particular, that the lattice is locally regular in any region of the [k, s) plane that excludes the critical point and that any vector in the unit cell such as the base vector, marked by arrows, rotates as the cell is transported around the cycle. Consequently, the transported dashed cell differs from that of the original quantized lattice. 

Moments of this conditional distribution can be written as standard Riemann Integrals of the pdf fx(z (N)) or as Stieltjes integrals of the cdf Fx(zf(N)) For example, the conditional expectation is written ... 

Miklautz, H., Riemann, J., and Vidic, H. J., The molecular weight distribution of heparin determined with a HPLC-LALLS technique, /. Liq. Chromatogr., 9, 2073, 1986. 

If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series it can in fact be made divergent or oscillatory (Riemanns theorem). 

Himathongham S, Bahari S, Riemann H and Cliver D (1999), Survival of Escherichia coli 0157 H/ and Salmonella typhimurium in cow manure and cow manure slurry , FEMS Microbiology Letters, 178, 251-257. 

Becker, W., Bergmann, A., Biscotti, G., Konig, K., Riemann, I., Kelbauskas, L. and Biskup, C. (2004a). High-speed FLIM data acquisition by time-correlated single photon counting. Proc. SPIE 5223, 1-9. 

Ohm GS (1827) Die Galvanische Kette, Mathematisch Bearbeitet. Riemann, Berlin... 

The interchain interaction of dipole moments collinear to the chain axes decays exponentially with the interchain distance (see Eq. (2.2.11)) and has no contribution descending by the power law y2 in Eq. (3.3.6). Summation over lattice sites and interchain distances involves the Riemann zeta-function... 

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