Some Roots of an Equation Involving Bessel Functions

Bogert, B. P.

doi:10.1002/sapm1951301102

Cited by 12 publications

(5 citation statements)

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“…(A5) For our purposes, a was estimated from a curve fitted to discrete values of a expressed as a function of vjr ( Youngs & Gardner, 1963;Bogert, 1951), where r^ is the radius of the cylinder of tissue (assumed to be equivalent to half the distanee between bundles);…”

Section: Discussionmentioning

confidence: 99%

Water storage and osmotic pressure influences on the water relations of a dicotyledonous desert succulent

Schulte¹,

Smith²,

Nobel³

1989

Plant Cell & Environment

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Water storage and nocturnal increases in osmotic pressure affect the water relations of the desert succulent Ferocactus acanthodes, which was studied using an electrical circuit analog based on the anatomy and morphology of a representative individual. Transpiration rates and osmotic pressures over a 24‐h period were used as input variables. The model predicted water potential, turgor pressure and water flow for various tissues. Plant capacitances, storage resistances and nocturnal increases in osmotic pressure were varied to determine their role in the water relations of this dicotyledonous succulent. Water coming from storage tissues contributed about one‐third of the water transpired at night: the majority of this water came from the nonphotosynthetic, water storage parenchyma of the stem. Time lags of 4 h were predicted between maximum transpiration and maximum water uptake from the soil. Varying the capacitance of the plant caused proportional changes in osmotically driven water movement but changes in storage resistance had only minor effects. Turgor pressure in the chlorenchyma depended on osmotic pressure, but was fairly insensitive to doubling or halving of the capacitance or storage resistance of the plant. Water uptake from the soil was only slightly affected by osmotic pressure changes in the chlorenchyma. For this stem succulent, the movement of water from the chlorenchyma to the xylem and the internal redistribution of water among stem tissues were dominated by nocturnal changes in chlorenchyma osmotic pressure, not by transpiration.

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Section: Discussionmentioning

confidence: 99%

Water storage and osmotic pressure influences on the water relations of a dicotyledonous desert succulent

Schulte¹,

Smith²,

Nobel³

1989

Plant Cell & Environment

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“…For a narrow annular region with Neumann boundary condition on the inner radius and Dirichlet boundary condition on the outer, the eigenvalues in (1.1) are given by (1.6) where x is a root of the mixed cross-product Bessel equation (A.6) with v = N. (The opposite boundary configuration can be obtained by appropriate redefinitions of x and y [5], [20].) As indicated in the expression (A.7), the roots tend to infinity as y -» 1 (cf.…”

Section: Narrow Annular Regionmentioning

confidence: 99%

“…As indicated in the expression (A.7), the roots tend to infinity as y -» 1 (cf. [5]). That this must be so can be deduced immediately from the fact that the left side of (A.6) in the case y = 1 is just the Wronskian ( [2], page 500)…”

Section: Narrow Annular Regionmentioning

confidence: 99%

Eigenvalues of the Laplacian with Neumann boundary conditions

Gottlieb

1985

J. Aust. Math. Soc. Series B, Appl. Math

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Various geometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with appropriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases of Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two-and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poisson summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.

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“…(5) have been tabulated by Bogert (1951) as a function of 2(= b/a) for n = 1 and various values of 2, including 2 = 2(1) 10 and 2 = 20. Additional values were computed here and the extended Table 4.…”

Section: F(t)/f(oe)= 1 -~ Ae -~ (6) N=lmentioning

confidence: 99%

Calcium fluxes in internally dialyzed giant barnacle muscle fibers

Russell

Blaustein

1975

J. Membrain Biol.

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Calcium-45 fluxes have been examined in isolated giant barnacle muscle fibers subjected to internal solute control by means of "internal dialysis". The 45Ca efflux was dependent upon the concentrations of both total and ionized internal Ca (Ca2+ buffered with EGTA). With a total Ca concentration of 2.0 mM and a 1:2 Ca/EGTA ratio (nominal [Ca2+]i =0.13 muM), the Ca efflux averaged 1.2 pmoles/cm2 sec. Under identical conditions, the mean Ca influx was only 0.36 pmoles/cm2 sec. The Ca efflux may not be attributed to leak of the CaEGTA complex, since a 2.5-fold increase in the EGTA concentration (nominal [Ca2+]i=0.032 muM) reduced the 45Ca efflux by one-third. Furthermore, when EDTA was used to buffer the internal Ca concentration (in the absence of internal Mg), the steady efflux of 14C-EDTA was only about 10% of the 45Ca efflux (in parallel experiments). The timecourse of the 45Ca fluxes also appeared anomalous in the 45Ca influx reached a steady level much more rapidly than 45Ca efflux in fibers of comparable diameters. If the muscle fibers are treated as right circular cylinders, these data imply that the apparent diffusion coefficient for inwardly-moving Ca is much larger than for outwardly-moving Ca. In contrast to Ca efflux, the outward diffusion of 22Na, 14C-EDTA and 3H2O appears to be limited primarily by the permeability of the dialysis tube wall. Some, but not all, of the anomalous behavior of the Ca fluxes can be reconciled if the deep, branched infoldings of the barnacle muscle surface membrane are taken into account.

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Some Roots of an Equation Involving Bessel Functions

Cited by 12 publications

References 0 publications

Water storage and osmotic pressure influences on the water relations of a dicotyledonous desert succulent

Water storage and osmotic pressure influences on the water relations of a dicotyledonous desert succulent

Eigenvalues of the Laplacian with Neumann boundary conditions

Calcium fluxes in internally dialyzed giant barnacle muscle fibers

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