A Unified Constructive Approach to the Topological Degree in Rn

The paper gives an approach to the topological degree in Rⁿ which takes into account numerical requirements and permits derivation of the known degree computation formulas in a simple way. The new approach subsumes several earlier approaches and represents a general principle of construction of degree computation formulas. The basic idea consists of computing the degree of a continuous function relative to a bounded open subset Ω of Rⁿ by means of an auxiliary function which is defined on a polyhedron approximating Ω and maps into a known fixed convex polyhedron containing the origin of Rⁿ. It is further shown that the topological degree of a continuous function relative to an n-dimensional polyhedron P can be computed alone by means of a subset of the boundary of P .     

Unsymmetrically Substituted Benzonaphthoporphyrazines: A New Class of Cationic Photosensitizers for the Photodynamic Therapy of Cancer*

Abstract— Unsymmetrical zinc(II) complexes of benzonaphthoporphyrazines 5 a-12a bearing between one and eight pyri-dyloxy substituents are synthesized by statistical tetra-merization of 6-(l,l-dimethylethyl)-2,3-naphthalenedi-carbonitrile (1) with 4-(3-pyridyloxy)- or 4,5-bis-(3-pyri-dyloxy)-1,2-benzenedicarbonitrile (2, 3). Methylation of 5a-12a leads to the cationic pyridyloxybenzonaphtho-porphyrazines 5b- 12b having between one to eight positive charges. The Q-band transition in the visible spectra exhibits a bathochromic shift from 680 to 760 nm dependent upon the number of annelated naphthalene rings. The singlet oxygen quantum yields of the benzonaphthoporphyrazines determined by the dye-sensitized photoox-idation of 1,3-diphenylisobenzofurane is surprisingly high (in the range of zinc phthalocyanine). The photoox-idative stabilities of the photosensitizers described quantitatively by first-order kinetics decrease with the number of annelated naphtho groups. A linear correlation between the logarithm of the decomposition rate constant and the position of the highest occupied molecular orbital (HOMO)-level of the photosensitizers is found. Destabi-lization of the HOMO leads to a decrease of the pho-tostability. Due to their adjustable long wavelength absorption, their intramolecular polarity axis and their different hydrophilic/hydrophobic character, these novel compounds may be suitable photosensitizers for the photodynamic therapy of cancer.     

Computation of stationary points via a homotopy method

A homotopy method is presented that locates both minimizers and saddle points of energy functions in an efficient manner. In contrast to other methods, it makes possible the exploration of large parts of potential energy surfaces. Along a homotopy path stationary points of odd and even order occur alternately. A path tracing procedure requiring only gradients and at most one evaluation of the Hessian matrix is given. Test results on a model potential and three MINDO/3 potentials are reported. Received: 6 May 1996 / Accepted: 2 April 1998 / Published online: 23 June 1998     

Potential energy surface exploration with equilibrial paths. Part II: Application

The results of an exploration of an ab initio potential energy surface of the C₂H₄O system with equilibrial paths are presented. In particular, four minimizers associated with the product of an elimination reaction and 26 saddle points, among these 14 saddle points of first order, have been located. Furthermore, twelve valley–ridge inflection points and seven double bifurcation points have been detected.     

Author index

Authors Index

Author index     

Metal chelates of porphyrin derivatives as sensitizers in photooxidation processes of sulfur compounds and in photodynamic therapy of cancer

A number of porphyrin derivatives based on hematoporphyrin, 5,10,15,20-tetrasubstituted porphyrins, phthalocyanines, and naphthalocyanines were prepared either as low-molecular compounds or bonded with methoxypoly(ethylene glycol) or attached to silica of low surface area. The low-molecular weight and the polymer-bonded porphyrins exhibit comparable triplet lifetimes and activities in the photosensitized formation of singlet oxygen. For photoninduced processes, the monomeric state of sensitizers is fundamentally important. The porphyrins have been investigated as sensitizers for photooxidation of thiolates and sulfides, which occursvia singlet oxygen, and, therefore, is much more efficient than the corresponding catalytic dark oxidation. Polymer-bonded porphyrins and long-wavelength absorbing naphthalocyanines incorporated in liposomes exhibitin vivo high accumulation in tumor tissues. Under irradiation, singlet oxygen is produced, and efficient phototherapeutic effects are observed, which may be used for photodynamic cancer therapy.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2071–2082, December, 1994.D. Wöhrle is very thankful to G. Schulz-Ekloff (Bremen), M. Shopova (Sofia), I. Okura (Tokyo), B. Roeder (Berlin), G. Graschew (Heidelberg), and their coworkers for fruitful cooperation. The research was supported by the German Research Society (Deutsche Forschungsgemeinschaft, 436 BUL 113/51/0, Wo 237/ 12-1), the German Cancer Service (Deutsche Krebshilfe, W 15/89/Sch12), Bremen University (FSP 2/90), and the Federal Department of Scientific Research and Tech nology (Bundesministerium far Forschung und Technotogie, Bonn, 13N6290).     

Potential energy surface exploration with equilibrial paths. Part I: Theory

The equilibrial path concept is further developed. Special attention is spent the symmetry conservation along equilibrial paths and symmetry-breaking. Symmetry-breaking can occur only at singular points. The simple singular points of an equilibrial path are valley–ridge inflection points. In contrast to the intrinsic reaction paths and the gradient extremal paths, the equilibrial paths enable to describe the branching of reaction channels.     

EQUIPATH-an equilibrial path tracing routine for the use with the program package GAUSSIAN94

The present article deals with the numerical details important for a successful implementation of the equilibrial path tracing procedure EQUIPATH. The results of an application to the RHF/6-31G(d) potential energy surface of butane are reported and discussed. Sixteen saddle points and several valley-ridge inflection points have been located. For the fragmentation C(4)H(10)--> C(2)H(6)+C(2)H(4) two different reaction paths have been found. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1801-1816, 2001     

Dissipative quantum Church-Turing theorem

We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system. An immediate consequence is that dissipative quantum computing is no more powerful than the unitary circuit model. Our result can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. Formally, we introduce a Trotter decomposition for Liouvillian dynamics and give explicit error bounds. This constitutes a practical tool for numerical simulations, e.g., using matrix-product operators. We also demonstrate that most quantum states cannot be prepared efficiently.