Bounds on isoperimetric values of trees

Let G = (V, E) be a finite, simple and undirected graph. For S subset of V, let delta(S, G) = {(u, v) is an element of E : u is an element of S and v is an element of V - S} be the edge boundary of S. Given an integer i, 1 <= i <= vertical bar V vertical bar, let the edge isoperimetric value of G at i be defined as b(e)(i, G) = min(S subset of V:vertical bar S vertical bar=i)vertical bar delta(S, G)vertical bar. The edge isoperimetric peak of G is defined as b(e)(G) = max(1 <= j <=vertical bar V vertical bar)b(e)(j, G). Let b(v)(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi: 10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees. The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as T-d(2)), c(1)d <= b(e) (T-d(2)) <= d and c(2)d <= b(v)(T-d(2)) <= d where c(1), c(2) are constants. For a complete t-ary tree of depth d (denoted as T-d(t)) and d >= c log t where c is a constant, we show that c(1)root td <= b(e)(T-d(t)) <= td and c(2)d/root t <= b(v) (T-d(t)) <= d where c(1), c(2) are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T = (V, E, r) be a finite, connected and rooted tree - the root being the vertex r. Define a weight function w : V -> N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index eta(T) be defined as the number of distinct weights in the tree, i.e eta(T) vertical bar{w(u) : u is an element of V}vertical bar. For a positive integer k, let l(k) = vertical bar{i is an element of N : 1 <= i <= vertical bar V vertical bar, b(e)(i, G) <= k}vertical bar. We show that l(k) <= 2(2 eta+k k)

Mandelic acid-4-hydroxylase, a new inducible entyme from Image

A soluble fraction of Image catalyzed the hydroxylation of mandelic acid to Image -hydroxymandelic acid. The enzyme had a pH optimum of 5.4 and showed an absolute requirement for Fe2+, tetrahydropteridine, NADPH. Image -Hydroxymandelate, the product of the enzyme reaction was identified by paper chromatography, thin layer chromatography, UV and IR-spectra.

The Structure of Monosodium Phosphoenolpyruvate

CDH406P-.Na +.H20 , M r = 208.0, is monoclinic, Cc, a = 11.423 (2), b = 23.253 (5), c - 6.604 (1) A, fl = 123.63 (1) °, U = 1460.6 A 3, D x =. 1.89 Mg m -a, Z = 8, 2(Mo Ka) = 0.7107 A, p(Mo Ka) = 0.44 mm -~, F(000) = 840. Final R = 0.063 for 1697 reflections.The two crystallographically independent molecules of phosphoenolpyruvate (PEP) (A and B) are almost mirror images of each other, the mirror being the planar enolpyruvate group. The torsion angle C(3)-C(2)- O(1)-P(1) is 122.6 in A and -112.0 ° in B, in contrast to -209.1 ° in PEP.K. The enolic C(2)-O(1) has a partial double-bond character [1.401 (A), 1.386A (B)]. The high-energy P~O bond (1.595 and 1.610A) is comparable to that in PEP.K (1.612 A). Na(1) has six nearest neighbours while Na(2) has only five. The Na + ions are involved in binding only the phosphates of different molecules, in contrast to the K ÷ ion in PEP. K, which binds to both the phosphate and carboxyl ends of the same molecule. The planar carboxyl groups stack on each other at an average distance of 3.2 A instead of forming hydrogen-bonded dimers usually found in carboxylate structures.

The Raman spectrum of cyclohexanol

The Raman spectrum of cyclohexanol has been studied in detail in the liquid state at 30° C. and at about 68° C. and in the solid state at about 13° C. The O-H stretching frequency of cyclohexanol has been found to extend from 3106-3571 cm.-1 in the liquid state at 30° C. and from 3204-3652 cm.-1 at 68° C. The 38 lines recorded in the present investigation have been following frequency shifts: 342, 408, 458, 478, 555, 653, 789, 834, 843, 863, 887, 920, 966, 978, 1024, 1047, 1070, 1139, 1173, 1184, 1210, 1235, 1252, 1301, 1329, 1346, 1362, 1438, 1448, 1464, 2660, 2684, 2710, 2854, 2896, 2925, 2940, 3106 to 3511 (band). Those lines which are italicized are the additional lines observed for the first time. The Raman lines at 966 cm.-1 and 1070 cm.-1 have been assigned to C-OH stretching vibrations of the axial and equatorial isomers. The ratio of the integrated intensity of the 1070 cm.-1 line to the 966 cm.-1 gave the equilibrium constant K as 2·896 at 30° C. and as 2·66 at 68° C. Knowing K, the free energy different Δ F was calculated and it was found to be 0·64 Kcal./mole at 30° C. and 0·66 Kcal./mole at about 68° C. Reasonable assignment has been made for most of the observed Raman lines.

Mandelic Acid 4-Hydroxylase - New Inducible Enzyme From Pseudomonas-Convexa

A soluble fraction of catalyzed the hydroxylation of mandelic acid to -hydroxymandelic acid. The enzyme had a pH optimum of 5.4 and showed an absolute requirement for Fe2+, tetrahydropteridine, NADPH. -Hydroxymandelate, the product of the enzyme reaction was identified by paper chromatography, thin layer chromatography, UV and IR-spectra

The Amenability Studies on Beneficiation of Low Grade Phyllite Deposits of Tungsten From Degana, Rajasthan

The phyllite deposit of Degana, Rajasthan, containing tungsten values in the form of wolframite, (Fe, MnWO sub 4 ) finely dispersed in the quartz groundmass, has been quantitatively analysed to give 0.063% WO sub 3 , 6.66% Fe sub 2 O sub 3 , 14.30% Al sub 2 O sub 3 and 67.4% SiO sub 2 . The major gangue minerals identified are quartz, iron oxides and mica along with minor amounts of graphite, fluorite and sulphides. The amenability of the ore to gravity concentration, magnetic separation and a combination of the processes has been studied. A combination of tabling on --100 mesh ground ore and dry magnetic separation of the tabled concentrate gave a final concentrate containing 1.834% WO sub 3 with an overall recovery of only 4.6%. The complex mineralogy combined with fine dispersion of very low W values have contributed to the low recoveries and grades. Graph, photomicrographs. 10 ref.--AA

Kinematics of pantograph masts

This paper deals with the kinematics of pantograph masts. Pantograph masts have widespread use in space application as deployable structures. They are over constrained mechanisms with degree-of-freedom, evaluated by the Grübler–Kutzback formula, as less than one. In this paper, a numerical algorithm is used to evaluate the degree-of-freedom of pantograph masts by obtaining the null space of a constraint Jacobian matrix. In the process redundant joints in the masts are obtained. A method based on symbolic computation, to obtain the closed-form kinematics equations of triangular and box shaped pantograph masts, is presented. In the process, the various configurations such masts can attain during deployment, are obtained. The closed-form solution also helps in identifying the redundant joints in the masts. The symbolic computations involving the Jacobian matrix also leads to a method to evaluate the global degree-of-freedom for these masts.

Photochromism of alkylisophthalaldehydes in the solid state: structure–reactivity correlations

The crystal and molecular structures of the photochromic compounds 2,5-dimethylisophthalaldehyde (I) and 5-isopropyl-2-methylisophthalaldehyde (II) have been determined by single crystal X-ray analyses. The intramolecular gamma-hydrogen abstraction process involved in the photoenolisation of I and II in the solid state has been rationalised in the light of relevant geometrical parameters.

On limiting cases in the flexure of simply supported regular polygonal plates

Limiting solutions are derived for the flexure of simply supported many-sided regular polygons, as the number of sides is increased indefinitely. It is shown that these solutions are different from those for simply supported circular plates. For axisymmetric loading, circular plate solutions overestimate the deflexions and the moments by significant factors.

Unsteady free convection flow over an infinite vertical porous plate due to the combined effects of thermal and mass diffusion, magnetic field and Hall currents

The unsteady free convection flow over an infinite vertical porous plate, which moves with time-dependent velocity in an ambient fluid, has been studied. The effects of the magnetic field and Hall current are included in the analysis. The buoyancy forces arise due to both the thermal and mass diffusion. The partial differential equations governing the flow have been solved numerically using both the implicit finite difference scheme and the difference-differential method. For the steady case, analytical solutions have also been obtained. The effect of time variation on the skin friction, heat transfer and mass transfer is very significant. Suction increases the skin friction coefficient in the primary flow, and also the Nusselt and Sherwood numbers, but the skin friction coefficient in the secondary flow is reduced. The effect of injection is opposite to that of suction. The buoyancy force, injection and the Hall parameter induce an overshoot in the velocity profiles in the primary flow which changes the velocity gradient from a negative to a positive value, but the magnetic field and suction reduce this velocity overshoot.

Preparation of a ZrO2–Al2O3 nanocomposite by high-pressure sintering of spray-pyrolyzed powders

ZrO2–Al2O3 powders were synthesized by spray pyrolysis. These powders were sintered at 1 GPa in the temperature range of 700–1100 °C. The microstructural evolution and densification are reported in this paper. The application of 1 Gpa pressure lowers the crystallization temperature from ∼850 to <700 °C. Similarly, the transformation temperature under 1 GPa pressure for γ → α–Al2O3 reduces from ∼1100 to 700–800 °C range, and that for t → m ZrO2 reduces from ∼1050 to 700–800 °C range. It was possible to obtain highly dense nanocrystalline ZrO2–Al2O3 composite at temperatures as low as 700 °C. The effect of high pressure on nucleation and transformation of phases is discussed.

Altruistic Wasps?

Polistes dominulus is one of the most common social wasps in Europe and is an invasive species in the United States. Its wide prevalence has made it one of the best-studied social wasps. In most social wasps, the female wasps live in a colony and organize themselves into a behavioral dominance hierarchy such that only the dominant alpha individual (the queen) reproduces while the rest function as apparently altruistic, sterile subordinates (workers), building the nest, foraging for food and pulp, and feeding and caring for the brood. Why should workers invest their time and energy helping to rear the queen's brood, rather than found their own nests and rear their own brood—something they are quite capable of? On page 874 of this issue, Leadbeater et al. (1) show that the subordinates indeed produce their own offspring and this raises interesting questions about the links between altruism, direct reproduction, and the evolution of social behavior.

Analysis of a hybrid noise control system for a duct

This letter proposes the combination of a passive muffler and an active noise control system for the control of very high‐level noise in ducts used with large industrial fans and similar equipment. The analysis of such a hybrid system is presented making use of electroacoustic analogies and the transfer matrix method. It turns out that a passive muffler upstream of the input microphone can indeed lower the acoustic pressure and, hence, the power requirement of the auxiliary source. The parameter that needs to be optimized (or maximized) for this purpose is a certain velocity ratio that can readily be evaluated in a closed form, making it more or less straightforward to synthesize the configuration of an effective passive muffler to go with the active noise control system.