Restricted Paths Research Articles

Bijective Lattice Path Proof of the Equality of the Dual Jacobi-Trudy Determinants

We give a bijective lattice path proof of the equality of the dual Jacobi-Trudy determinant formulas for Schur polynomials. Related ideas have appeared in [1, pp.304-306] and [2, p.24]. We remark that the same bijection works for the case of flagged skew Schur polynomials [2, 8] and that a determinant for $q$-counting restricted lattice paths [7] follows from the bijection.

Higher dimensional restricted lattice paths with diagonal steps

In this paper, restricted minimal lattice paths with horizontal, vertical, and diagonal steps, in two and higher dimensions are discussed. The Delannoy numbers, the numbers of unrestricted minimal lattice paths with diagonal steps, and some of their properties are introduced. The recurrence on the Delannoy numbers is extended to higher dimensions. The relation in two dimensions between the restricted minimal lattice paths and the Delannoy numbers is shown through the use of Andre's reflection principle. This relation is generalized from two dimensions to higher dimensions and is found to be in the form of a determinant. The relation between unrestricted and restricted weighted minimal lattice paths in two dimensions is shown by the extension of Andre's reflection principle.

Combinatorics of representations of $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))$$ atq=0

Theq=0 combinatorics for\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\) is studied in connection with solvable lattice models. Crystal bases of highest weight representations of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\) are labelled by paths which were introduced as labels of corner transfer matrix eigenvectors atq=0. It is shown that the crystal graphs for finite tensor products ofl-th symmetric tensor representations of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\) approximate the crystal graphs of levell representations of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\). The identification is made between restricted paths for the RSOS models and highest weight vectors in the crystal graphs of tensor modules for\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\).

Superconductivity in the filamentary conductor TaSe 3

Superconductivity in a quasi-one-dimensional conductor TaSe 3 has been systematically studied. Resistivity below T c (2.0 K) is strongly non-linear for current densities higher than 0.1 Amm −2, reflecting severely restricted current paths in the crystals. The diamagnetic susceptibility χ does not exhibit any indication of an abrupt change at T c. Furthermore, χ shows no significant variation of diamagnetism, with magnetic fields parallel and perpendicular to the crystalline b-axis over the low magnetic field range of 7.2 mOe to 4.0 Oe. The diamagnetism is quite weak and the value of χ is much less than −3.0 × 10 −3 emu cm −3 even at 0.40 K. The superconducting thin filaments in the whisker crystal would have an effective radius much smaller than the London penetration depth λ and this penetration reduces the values of χ. These filaments are individually fairly well isolated above 0.40 K.

Means for quieting electric power transformers

An invention, now under development, comprises means that, without affecting the insulating or cooling capacity of the transformer design, create regions of low acoustic impedance at appropriate locations to facilitate the accommodation of oscillatory movement of the oil proximate to the source of any such movement. This significantly reduces the pressure fluctuations developed in conventional transformer designs where any movement of the core or windings must be accommodated by a combined movement of the tank walls and oscillatory flow of oil back and forth over relatively long and restricted paths from the expanding to the contracting regions. The physical concept and experimental designs will be discussed.

A determinant for q-counting n-dimensional lattice paths

A bijective proof of Gessel and Viennot is extended to a proof of an n-dimensional q-analogue of Kreweras's determinant formula for counting restricted lattice paths. In the proof the determinant has direct combinatorial significance.

Water Pathways in Wheat Leaves. IV. The Interpretation of Images of a Fluorescent Apoplastic Tracer

Sections of wheat leaves fed with the fluorescent apoplastic tracer sulforhodamine G (SR) through the xylem were prepared by freeze-substitution and resin embedding. The distribution of fluorescence intensity (FI) of the tracer was measured by microspectrofluorometry at a resolution of 0.4 �m. SR was found to move within cell walls in restricted paths less than 200 nm wide. The name 'nanopaths' is suggested for these. The highest FI was found around the mestome-sheath / parenchyma-sheath border on the xylem side, and was shown to be due, not to binding of the tracer to wall components, but to the generation of a very high concentration of SR there by the separation of water from the solute. This separation cannot be evaporative but must be osmotic, and is presented as evidence of a major symplastic water movement starting at the parenchyma sheath cell membrane. The main resistance to water loss from the veins is at the mestome sheath and appears to be controlled by the suberised lamellae.

Q-Counting n-dimensional lattice paths

n-dimensional lattice paths are enumerated by generating functions which are Gaussian multinomial coefficients in the case of unrestricted paths. Convolutions for path counts are studied which yield a q-Vandermonde convolution and a determinant of Gaussian multinomial coefficients as the generating function for certain restricted paths.

A quasiclassical trajectory study of the H2+F2 reactions

A theoretical investigation of the reaction dynamics of the reactions H2(v=0,2,4, or 6) + F2(v=0 or 6) → H+HF+F, 2HF, F2H+H, H2F+F, F2+2H, and H2+2F has been accomplished using Monte Carlo averaging over quasiclassical trajectories on a reasonable semiempirical potential-energy surface. The details of the dynamical mechanism of the reaction processes were investigated by examining geometrically restricted reaction paths and selected restricted initial states. Reaction cross sections have been computed and estimates of rate coefficients have been made. The computed thermal rates are many orders of magnitude lower than thermal collision frequencies. Reaction thresholds are lowered significantly for vibrationally excited hydrogen.

Distribution of crossings in restricted paths

Distribution of crossings in restricted paths

Suitability of the flash method for measuring the thermal diffusivity of cracked specimens

The accuracy of thermal diffusivity measurements by the flash method on specimens containing cracks has been assessed from a theoretical treatment of heat flow through a specimen containing a model crack. It was found that the thermal diffusivity as measured by the flash method is probably within 5% of the value derived from a steady-state heat-flow method. Measurements on Armco iron specimens containing restricted heat-flow paths confirmed this conclusion.

On Lattice Paths with Several Diagonal Steps

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.

ORGANIC MENTAL SYNDROME WITH PHENOMENA OF EXTINCTION AND ALLESTHESIA

CLINICAL neurologists are rarely inclined to study their patients from a psychologic point of view with the same intensity as they study them from the neurophysiologic standpoint. The clinician readily accepts the observations made by the physiologist on synaptic or neuronal function. He seems to understand these findings because his neurologic orientation has been in this direction. Current teaching stresses that nerve impulses are transmitted in sensory and motor nerves over definite, restricted paths, and in the central nervous system from one point to another through intercellular connections. There is little emphasis on theories which postulate collective action within the nervous system. Such theories have been proposed by Jackson, 1 Goldstein, 2 Lashley, 3 Kluver 4 and others. 5 After reviewing his experimental work, Lashley stated that behavior seems to be determined by masses of excitation, by the form or relations or proportions of excitation within general fields of activity

On the Poisons of Plant-Lice*

Although applied entomology has made enormous progress during recent years both in its scope and importance, we cannot deny that it has hitherto moved only in certain restricted paths and is therefore in danger of becoming one-sided, for the subject is taken up at present almost exclusively with a study of lifehistories, with parasites and the empirically ascertained methods of combatting injurious species.† Thus it has come about that up to the present time we know very little concerning the secretion of poisons by the sucking plant-parasites and the effeCts of these poisons on the plants. Most attention has been paid to this question by those who have investigated the origin of plant galls. Hence it may be of interest to record some investigations which I have made on the poisons of plant-lice.