Asymptotic iteration method for eigenvalue problems

An asymptotic interation method for solving second-order homogeneous linear differential equations of the form y'' = lambda(x) y' + s(x) y is introduced, where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to Schroedinger type problems, including some with highly singular potentials, are presented.Comment: 14 page

Coulomb plus power-law potentials in quantum mechanics

We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for the Coulomb plus power-law potential V(r)=-1/r+ beta sgn(q)r^q, where beta > 0, q > -2 and q \ne 0. We show by envelope theory that the discrete eigenvalues E_{n\ell} of H may be approximated by the semiclassical expression E_{n\ell}(q) \approx min_{r>0}\{1/r^2-1/(mu r)+ sgn(q) beta(nu r)^q}. Values of mu and nu are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, psi(r)= r^{\ell+1}e^{-(xr)^{q}}. We give detailed results for V(r) = -1/r + beta r^q, q = 0.5, 1, 2 for n=1, \ell=0,1,2, along with comparison eigenvalues found by direct numerical methods.Comment: 11 pages, 3 figure

2-Thiabicyclo[3.2.0]hepta-3,6-Dienes. 1. Aluminum Chloride Catalyzed Thermal [2 + 2] Cycloaddition of 2-Butynedinitrile with Alkyl-Substituted Thiophenes

Tetra- and Trialkylthiophenes 4–6 React with 2-Butynedinitrile in the Presence of 1 Equiv of Aluminum Chloride in Dichloromethane at 0 °C Via [2 + 2] Cycloaddition to 2-3.2.0] hepta-3,6-Diene-6,7-Dicarbonitriles 7–10 as the Major Products. Reaction of 4 and 5 Also Gave Small Amounts (4% and 6%) of 1,2-Benzenedicarbonitriles 11 and 12 Which Result from [4 + 2] Cycloaddition and Subsequent Extrusion of Sulfur. 4,5,6,7-Tetrahydro-1,3-Dimethylbenzo[C]thiophene (6) Reacts with 2-Butynedinitrile under the Same Conditions to Give an Ene Adduct (13) in Addition to the [2 + 2] Cycloadduct 10. 2,5-Dimethylthiophene (3) Gave a 2:1 Reaction Product 14, the Formation of Which Can Be Explained Via Friedel-Crafts Alkylation of 3 by the Initially Formed [2 + 2] Cycloadduct. the 2-3.2.0] hepta-3,6-Dienes 7–10 and the Ene Adduct 13 Show Intramolecular Charge-Transfer Absorptions in the UV Spectra. © 1982, American Chemical Society. All Rights Reserved

Semiclassical energy formulas for power-law and log potentials in quantum mechanics

We study a single particle which obeys non-relativistic quantum mechanics in R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2, then q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\ell} may be represented exactly by the semiclassical expression E_{n\ell}(q) = min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) = ln(r). By writing one power as a smooth transformation of another, and using envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are monotone increasing. Recent refinements to the comparison theorem of QM in which comparison potentials can cross over, allow us to prove for n = 1 that Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q} is monotone decreasing. Thus P(q) cannot increase too slowly. This result yields some sharper estimates for power-potential eigenvlaues at the bottom of each angular-momentum subspace.Comment: 20 pages, 5 figure

2-Thiabicyclo[3.2.0]-hepta-3,6-dienes. 2. Thermal and photochemical isomerization of 2-thiabicyclo[3.2.0]hepta-3,6-dienes. An example of the antarafacial-antarafacial Cope rearrangement

Thermal Rearrangement of 2-3.2.0] hepta-3,6-Diene-6,7-Dicarbonitrile8 1–3 to their 4,5-Dicarbonitrile Isomers 4–6 Takes Place at 110–140 °C in Yields of 82–84%. the Reactions Are First Order with Rates Almost Independent of the Polarity of the Solvent. Activation Parameters (ΔH‡ = 120–140 KJ Mol−1 and ΔS‡ ≈ 0 J K−1 Mol−1) Are in Agreement with a Concerted Symmetry-Allowed Antarafacial-Antarafacial Cope Rearrangement. Compounds 1–3 Rearrange Photochemically Via Cleavage of the C-1-S Bond to the 1,7-Dicarbonitrile Isomers 7–9. Isomer 9 Rearranges Further on Prolonged Irradiation to Give 10. a Tentative Mechanism for This Rearrangement is Given. © 1982, American Chemical Society. All Rights Reserved

Ohio Economic Insects and Related Anthropods

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Engineering Students in a Global World: Lehigh University\u27s Global Citizenship Program

As the world grows smaller and companies become increasingly international, engineering students with a global education will be in high demand. Given the constraints of the typical undergraduate engineering curriculum, it can be extremely challenging for engineering students to participate in activities designed to promote global awareness. Lehigh University’s answer to this challenge is the Global Citizenship Program (GCP), a “backpack” program designed to be accessible to any student on campus. The “backpack” metaphor means that all students should be able to assemble their own particular mix of additional courses and experiences in order to develop a global perspective that deepens their core disciplinary training. The GCP at Lehigh provides focus and structure to engineering students, providing opportunities for study abroad and organizing their humanities and social science electives into a coherent package of curricular and co-curricular experiences that maximizes the educational potential of these few non-engineering opportunities

Aerospike Rockets for Increased Space Launch Capability

The US Department of Defense DOD increasingly depends on space assets for everyday operations. Precision navigation communications and intelligence, surveillance, and reconnaissance satellites are highly leveraged space assets. The launch vehicles that place these satellites in orbit are a major limitation of current space systems. If higher-performing launch vehicles were available, many satellites could accommodate additional capabilities, whether in terms of more sensor channels, types of payloads, electrical power, or propellant for orbital maneuvering and station keeping. Space assets are typically designed to conform to a particular launch vehicle s limitations e.g., engineers might design a satellite to be carried by a Delta IV-2 medium launch vehicle. Essentially, this choice of vehicle fixes the maximum mass of the satellite and, thus, its capabilities. If a launcher capable of placing more mass in the desired orbit were available at similar cost, the satellite s design could allow for additional capability. Furthermore, some payloads are too heavy for present-day launch vehicles to place into a particular orbit. A better-performing launcher would enable us to put those payloads into the desired orbits, permitting new missions and capabilities

Solutions for certain classes of Riccati differential equation

We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also investigated.Comment: 10 page