Spontaneous ring transformation of a 5-azido-1,2,3-thiadiazole

The reaction of 4-carbethoxy-5-chloro-1,2,3-thiadiazole ($\text{1}$) with sodium azide results in the formation of ethyl α-thiatriazolyldiazoacetate ($\text{3}$) instead of the corresponding azide ($\text{2}$). Two plausible mechanisms for this new rearrangement are formulated.     

Prüfer and Morita Hulls of FCP Extensions

We examine in this paper some properties of Morita and Prüfer hulls of FCP and FIP extensions of rings. Dichotomy phenomena appear in the case of a quasi-local base ring. In the general case, we define relative supports that allow us to introduce the concept of direct factorization of an extension and to characterize these hulls.     

Oscillatory Perturbations of Linear Problems at Resonance

This paper is concerned with a study of bounded perturbations of resonant linear problems. It follows from our results that for certain types of bounded domains Ω ? Rⁿ, n ≥ 2, the Dirichlet problem $\matrix{\Delta u+\lambda_{1}u+g(u)=h(x),\ \ \ x\in\Omega\cr \quad\quad\quad\quad\quad\quad u=0,\ \ \ x\in\partial\Omega,}$ has infinitely many positive solutions, in case λ₁ is the principal eigenvalue of ?Δ subject to trivial Dirichlet boundary conditions, g is a nontrivial periodic nonlinearity of zero mean and ∫_03A9h(x)?(x)dx = 0, where ? is an eigenfunction corresponding to λ₁.     

Multicriteria Semi-Obnoxious Network Location Problems (MSNLP) with Sum and Center Objectives

Locating a facility is often modeled as either the maxisum or the minisum problem, reflecting whether the facility is undesirable (obnoxious) or desirable. But many facilities are both desirable and undesirable at the same time, e.g., an airport. This can be modeled as a multicriteria network location problem, where some of the sum-objectives are maximized (push effect) and some of the sum-objectives are minimized (pull effect).We present a polynomial time algorithm for this model along with some basic theoretical results, and generalize the results also to incorporate maximin and minimax objectives. In fact, the method works for any piecewise linear objective functions. Finally, we present some computational results.     

On the existence of soliton solutions to quasilinear Schrödinger equations

Variational techniques are applied to prove the existence of standing wave solutions for quasilinear Schr?dinger equations containing strongly singular nonlinearities which include derivatives of the second order. Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Direct methods of the calculus of variations and minimax methods like the Mountain Pass Theorem are used. The difficulties introduced by the nonconvex functional are substantially different from the semilinear case. Received: 21 March 2000 / Accepted: 23 February 2001 / Published online: 25 June 2001     

The Liouville-Bratu-Gelfand Problem for Radial Operators

We determine precise existence and multiplicity results for radial solutions of the Liouville-Bratu-Gelfand problem associated with a class of quasilinear radial operators, which includes perturbations of k-Hessian and p-Laplace operators.     

Tensor method for optimal control problems constrained by fractional three-dimensional elliptic operator with variable coefficients

We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two- (2D) and three-dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D $n\times n\times n$ spacial grids. Using the diagonalization of the arising matrix-valued functions in the eigenbasis of the one-dimensional Sturm–Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right-hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low-rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an $\epsilon$-threshold. This method reduces the numerical cost for solving the control problem to $O(n\log n)$ (plus the quadratic term $O(n^2)$ with a small weight), which outperforms traditional approaches with $O(n^3\log n)$ complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by $O(n\log n)$. This essentially outperforms the traditional methods operating with fully populated $n^3\times n^3$ matrices and vectors in ${?}^{n^3}$. Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.     

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