Small zeros of quadratic forms with linear conditions

Given a quadratic form and M linear forms in N+1 variables with coefficients in a number field K, suppose that there exists a point in K^N+1 at which the quadratic form vanishes and all the linear forms do not. Then we show that there exists a point like this of relatively small height. This generalizes a result of D.W. Masser.     

ON THE INTERPRETATION OF ABSORPTION SPECTRA OF LEAVES–II. THE NON-ABSORBED RAY OF THE SIEVE EFFECT and THE MEAN OPTICAL PATHLENGTH IN THE REMAINDER OF THE LEAF

Abstract— The sieve effect and scattering within leaves are analysed by the use of a simple model. By plotting the leaf transmittance (corrected for light not entering the leaf) vs the transmittance of an equivalent amount of homogeneous plastid pigments, an intercept is found where the latter is zero. This minimum transmittance represents the fraction of the leaf area devoted to the ray of the sieve effect which strikes no chloroplasts. It varied between 7% and 0.2% in non-senescent leaves. When this was subtracted from the leaf spectrum, the peak absorbance was greater than that of the homogeneous leaf pigments in all cases. The ratio of the leaf absorbance to that of the homogeneous pigments, at the same wavelength, is the apparent optical pathlength, which increases with decreasing absorbance. By plotting this ratio vs the absorbance of the equivalent homogeneous pigment, an intercept is found where the latter is zero. This intercept is interpreted as an estimate of the true mean scattering pathlength. Leaves with high chlorophyll contents had low pathlengths (mean and SD = 2.30 ± 0.25); with moderate and low contents, the values were higher (2.75 ± 0.28, 3.95 ± 0.77). Another application of the model gave values between 3 and 4 for the true scattering pathlength.     

Bounds for solid angles of lattices of rank three

We find sharp absolute constants C₁ and C₂ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C₁,C₂]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in RN with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R³. Such spherical configurations come up in connection with the kissing number problem.     

Integral Points of Small Height Outside of a Hypersurface

Let F be a non-zero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.     

Optics of a Bifacial Leaf: 3. Implications for Photosynthetic Performance

On the basis of a preceding study of light and absorption gradients in a bifacial leaf, spatial patterns of photosynthetic rate and photosynthetic saturation have been analyzed for different cases of natural illumination and different distributions of photosynthetic capacity along the depth coordinate. Two quality criteria of the spatially extended leaf photosynthetic performance arose from this analysis. One of them characterizes the range of the acceptable incident quantum fluxes between the lowest economically acceptable one and the highest still not photodamaging one. The second criterion gives the limit for the fractional part of the leaf photosynthetic capacity that can be utilized before the first photoinhibition effect at some depth occurs. It is shown how the range of the acceptable fluxes can be affected by the interplay between light and capacity gradients.     

ESTIMATION OF THE OPTICAL PARAMETERS AND LIGHT GRADIENTS IN LEAVES: MULTI-FLUX versus TWO-FLUX TREATMENT

Abstract
Parameters of light propagation in plant leaves-absorption and scattering coefficients, asymmetry of scattering-have been estimated on the basis of measured transmission and remission as well as internal fluxes. This estimation has been carried through by solving the inverse problem of the 4-flux radiative transfer-a theory considering forward and backward diffuse as well as directed components of the overall radiation in a multiply scattering sample. Using the gained parameters, light flux gradients in a two-layered model leaf have been calculated at different wavelengths. Monte Carlo simulation of absorption spectra performed with the parameters obtained with this treatment is in a good agreement with experimental spectra, thus substantiating the theory. Parallel calculations with the two-flux (Kubelka-Munk) theory provide an estimation of the accuracy and applicability of this more simple treatment. Calculations have been performed for two different plants.     

Frobenius Problem and the Covering Radius of a Lattice

Let and let be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as are non-negative integers. The condition that implies that such a number exists. The general problem of determining the Frobenius number given N and is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.     

ON THE INTERPRETATION OF ABSORPTION SPECTRA OF LEAVES–I. INTRODUCTION and THE CORRECTION OF LEAF SPECTRA FOR SURFACE REFLECTION

Abstract— Reflectance and transmittance spectra of leaves and their sum can be corrected to relate only to the light actually entering the leaf, if the reflectance of the epidermal surface is known. The latter is found if the leaf reflectances at several wavelengths near the transmittance minimum in the red are plotted vs the transmittances of a homogeneous suspension of the native pigment-proteins at the same chlorophyll content per unit area and at the same wavelengths. With non-senescent leaves, the relation is linear and the extrapolation of the pigment transmittance to zero gives the value for the surface reflection. Surface reflectance data (both adaxial and abaxial) are given for the leaves of a number of trees and a few herbs, plus examples of the raw and corrected spectra. With normal, glaucous leaves, the adaxial reflectance averaged 4.5% of the incident light ( n = 23, range = 3.7 −5.9, standard deviation = 0.4). The reflectances of the abaxial surfaces ranged between 7 and 13% since additional near-surface reflection occurred at the inside of the epidermis and in the spongy mesophyll. Reflectance and transmittance data demonstrated strong absorption in the epidermis below 480 nm.     

On well-rounded sublattices of the hexagonal lattice

We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid.     

IMAGE ANALYZER USED FOR DATA ACQUISITION IN PHOTOTROPISM STUDIES

Abstract— Equipment (video system and goniometric device) developed for the automatic acquisition of data, which can generally be used for testing movement responses of small organisms is presented. This relatively cheap setup, especially designed for kinetic testing of phycomyces phototropism, could also be used with very slight modifications to study phototropism in higher plants and even for other purposes in the study of cell and organism movements. It permits continuous automatic observation of bending angle as a function of time. The goniometric device permits a complete geometrical localization of the object to be analyzed, even time-dependent if needed. The video-system hardware consists of a CCD-video camera (with a sensitivity maximum between 700 and 800 nm), light source (equipped with an IR-filter), video monitor, video card (for image digitalization), video recorder, computer and computer interface. The last assures (through a parallel port of the computer) complete control of the setup; time coordination between computer, lamp, camera and video recorder is thus ensured. Both real time processing and analysis of previously recorded data can be performed. The system permits automatic observation of experiments over long time periods and provides a large amount of data assuring good statistics, which can be analysed quickly.