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Showing papers by "Technical University of Dortmund published in 1998"


Book ChapterDOI
21 Apr 1998
TL;DR: This paper explores the use of Support Vector Machines for learning text classifiers from examples and analyzes the particular properties of learning with text data and identifies why SVMs are appropriate for this task.
Abstract: This paper explores the use of Support Vector Machines (SVMs) for learning text classifiers from examples. It analyzes the particular properties of learning with text data and identifies why SVMs are appropriate for this task. Empirical results support the theoretical findings. SVMs achieve substantial improvements over the currently best performing methods and behave robustly over a variety of different learning tasks. Furthermore they are fully automatic, eliminating the need for manual parameter tuning.

8,658 citations


Journal ArticleDOI
TL;DR: In this paper, it is shown that within theoretical uncertainties their previous (1994) dynamical/radiative parton distributions are compatible with most recent data and still applicable within the restricted accuracy margins of the presently available next-to-leading order calculations.
Abstract: Dynamical parton densities, generated radiatively from valence-like inputs at some low resolution scale, are confronted with recent small-x data on deep inelastic and other hard scattering processes. It is shown that within theoretical uncertainties our previous (1994) dynamical/radiative parton distributions are compatible with most recent data and still applicable within the restricted accuracy margins of the presently available next-to-leading order calculations. Due to recent high precision measurements we also present an updated, more accurate, version of our (valence-like) dynamical input distributions. Furthermore, our perturbatively stable parameter-free dynamical predictions are extended to the extremely small-x region, $10^{-8} \lesssim x \lesssim 10^{-5}$ , relevant to questions concerning ultra-high-energy cosmic ray and neutrino astronomy.

817 citations


Journal ArticleDOI
M. Kahl1, Edgar Voges1, S. Kostrewa, C. Viets, W. Hill 
TL;DR: In this article, surface-enhanced Raman scattering (SERS) was used to detect organic traces by electron-beam lithography, which is better reproducible than electrochemical roughened surfaces, island films or films over deposited particles.
Abstract: The detection of organic traces by surface-enhanced Raman scattering (SERS) requires rough metallic substrates with structures in the nanometer range. Regular metallic structures generated by electron-beam lithography are better reproducible than electrochemical roughened surfaces, island films or films over deposited particles. The lithography also allows optimisation of the dimensions of these structures. The SERS enhancement of optimised periodic structures was found to be at least one order of magnitude larger than that of island films.

290 citations


Journal ArticleDOI
TL;DR: In this article, the authors present an algorithm to determine if a real polynomial is a sum of squares (of polynomials) and to find an explicit representation if it is.

252 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the non-exponential relaxation results from a superposition of dynamically distinguishable entities, and that the intrinsic response is compatible with single exponential relaxation.
Abstract: Several experimental methods feature the potential to distinguish between slow and fast contributions to the non-exponential, ensemble averaged primary response in glass-forming materials. Some of these techniques are based on the selection of subensembles using multi-dimensional nuclear magnetic resonance, optical bleaching, and non-resonant spectral hole burning. Others, such as the time-dependent solvation spectroscopy, measure microscopic responses induced by local perturbations. Using several of these methods it could be demonstrated for various glass-forming materials that the non-exponential relaxation results from a superposition of dynamically distinguishable entities. The experimental observation that subensembles can be selected efficiently indicates a large degree of heterogeneity. The intrinsic response is compatible with single exponential relaxation.

213 citations


Journal ArticleDOI
TL;DR: This survey is the attempt to summarize the results regarding the limit and finite time behavior of evolutionary algorithms with finite search spaces and discrete time scale.
Abstract: The theory of evolutionary computation has been enhanced rapidly during the last decade. This survey is the attempt to summarize the results regarding the limit and finite time behavior of evolutionary algorithms with finite search spaces and discrete time scale. Results on evolutionary algorithms beyond finite space and discrete time are also presented but with reduced elaboration.

193 citations


Journal ArticleDOI
TL;DR: In this paper, Gutzwiller wave functions for multiband models with general on-site Coulomb interactions were introduced and evaluated in infinite lattice dimensions without any restrictions on the structure of the Hamiltonian or the symmetry of the ground state.
Abstract: We introduce Gutzwiller wave functions for multiband models with general on-site Coulomb interactions As these wave functions employ correlators for the exact atomic eigenstates, they are exact both in the noninteracting and atomic limits We evaluate them in infinite lattice dimensions for all interaction strengths without any restrictions on the structure of the Hamiltonian or the symmetry of the ground state The results for the ground-state energy allow us to derive an effective one-electron Hamiltonian for Landau quasiparticles, applicable for finite temperatures and frequencies within the Fermi-liquid regime As applications for a two-band model we study the Brinkman-Rice metal-to-insulator transition at half-band-filling, and the transition to itinerant ferromagnetism for two specific fillings, at and close to a peak in the density of states of the noninteracting system Our results significantly differ from those for earlier Gutzwiller wave functions where only density-type interactions were included When the correct spin symmetries for the two-electron states are taken into account, the importance of the Hund's-rule exchange interaction is even more pronounced, and leads to paramagnetic metallic ground states with large local magnetic moments Ferromagnetism requires fairly large interaction strengths, and the resulting ferromagnetic state is a strongly correlated metal

193 citations


Journal ArticleDOI
TL;DR: A new technique is presented which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers.
Abstract: We consider nonlinear programs with inequality constraints, and we focus on the problem of identifying those constraints which will be active at an isolated local solution. The correct identification of active constraints is important from both a theoretical and a practical point of view. Such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with. We present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers. We also present extensions to variational inequalities and numerical examples illustrating the identification technique.

190 citations


Journal ArticleDOI
J. Altegoer, M. Anfreville, C. Angelini, P. Astier  +154 moreInstitutions (21)
TL;DR: The NOMAD experiment as mentioned in this paper is a short base-line search for νμ − ντ oscillations in the CERN neutrino beam, which enables the reconstruction of individual particles produced in the neutrinos interactions.
Abstract: The NOMAD experiment is a short base-line search for νμ − ντ oscillations in the CERN neutrino beam. The ντ's are searched for through their charged current interactions followed by the observation of the resulting τ− through its electronic, muonic or hadronic decays. These decays are recognized using kinematical criteria necessitating the use of a light target which enables the reconstruction of individual particles produced in the neutrino interactions. This paper describes the various components of the NOMAD detector: the target and muon drift chambers, the electromagnetic and hadronic calorimeters, the preshower and transition radiation detectors and the veto and trigger scintillation counters. The beam and data acquisition system are also described. The quality of the reconstruction and individual particles is demonstrated through the ability of NOMAD to observe Ks0's, Λ0's and π0's. Finally, the observation of τ− through its electronic decay being one of the most promising channels in the search, the identification of electrons in NOMAD is discussed.

183 citations


Book ChapterDOI
27 Sep 1998
TL;DR: The main objective of this preliminary study is the answer to the question whether the predator-prey approach to multi-objective optimization works at all, which is examined under several step-size adaptation rules.
Abstract: This paper presents a novel evolutionary approach of approximating the shape of the Pareto-optimal set of multi-objective optimization problems. The evolutionary algorithm (EA) uses the predator-prey model from ecology. The prey are the usual individuals of an EA that represent possible solutions to the optimization task. They are placed at vertices of a graph, remain stationary, reproduce, and are chased by predators that traverse the graph. The predators chase the prey only within its current neighborhood and according to one of the optimization criteria. Because there are several predators with different selection criteria, those prey individuals, which perform best with respect to all objectives, are able to produce more descendants than inferior ones. As soon as a vertex for the prey becomes free, it is refilled by descendants from alive parents in the usual way of EA, i.e., by inheriting slightly altered attributes. After a while, the prey concentrate at Pareto-optimal positions. The main objective of this preliminary study is the answer to the question whether the predator-prey approach to multi-objective optimization works at all. The performance of this evolutionary algorithm is examined under several step-size adaptation rules.

172 citations


Journal ArticleDOI
TL;DR: In this article, the ability of a brown seaweed Ecklonia maxima, a dealginated seaweed waste, alginate fibre and waste linseed fibre to remove copper, nickel and cadmium from single and mixed metal ion solutions was compared.

01 Jan 1998
TL;DR: This paper explores methods to recognize concept changes and to maintain windows on the training data, whose size is either fixed or automatically adapted to the current extent of concept change.
Abstract: The task of information filtering is to classify texts from a stream of documents into relevant and nonrelevant, respectively, with respect to a particular category or user interest, which may change over time. A filtering system should be able to adapt to such concept changes. This paper explores methods to recognize concept changes and to maintain windows on the training data, whose size is either fixed or automatically adapted to the current extent of concept change. Experiments with two simulated concept drift scenarios based on real-world text data and eight learning methods are performed to evaluate three indicators for concept changes and to compare approaches with fixed and adjustable window sizes, respectively, to each other and to learning on all previously seen examples. Even using only a simple window on the data already improves the performance of the classifiers significantly as compared to learning on all examples. For most of the classifiers, the window adjustments lead to a further increase in performance compared to windows of fixed size. The chosen indicators allow to reliably recognize concept changes.

Proceedings ArticleDOI
04 May 1998
TL;DR: It is shown that results known from the theory of evolutionary algorithms in case of single-criterion optimization do not carry over to the multi-criteria case, and a theoretical analysis shows that a special version of an evolutionary algorithm with this step size rule converges with probability one to the Pareto set for the test problem under consideration.
Abstract: Although there are many versions of evolutionary algorithms that are tailored to multi-criteria optimization, theoretical results are apparently not yet available. In this paper, it is shown that results known from the theory of evolutionary algorithms in case of single-criterion optimization do not carry over to the multi-criterion case. At first, three different step size rules are investigated numerically for a selected problem with two conflicting objectives. The empirical results obtained by these experiments lead to the observation that only one of these step size rules may have the property to ensure convergence to the Pareto set. A theoretical analysis finally shows that a special version of an evolutionary algorithm with this step size rule converges with probability one to the Pareto set for the test problem under consideration.

Journal ArticleDOI
TL;DR: In this article, a new x-ray diffraction study of the one-dimensional spin-Peierls compound with one type of V site was performed, showing that it has a centrosymmetric crystal structure with one V site.
Abstract: A new x-ray diffraction study of the one-dimensional spin-Peierls compound ${\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{-}\mathrm{Na}{\mathrm{V}}_{2}{\mathrm{O}}_{5}$ reveals a centrosymmetric $(\mathrm{Pmmn})$ crystal structure with one type of V site, contrary to the previously postulated noncentrosymmetric ${P2}_{1}\mathrm{mn}$ structure with two types of V sites ( ${V}^{+4}$ and ${\mathrm{V}}^{+5}$). Density-functional calculations indicate that $\mathrm{Na}{\mathrm{V}}_{2}{\mathrm{O}}_{5}$ is a quarter-filled ladder compound with the spins carried by V-O-V molecular orbitals on the rungs of the ladder. Estimates of the charge-transfer gap and the exchange coupling agree well with experiment and explain the insulating behavior of $\mathrm{Na}{\mathrm{V}}_{2}{\mathrm{O}}_{5}$ and its magnetic properties.

Book ChapterDOI
14 Sep 1998
TL;DR: A collection of techniques, a methodology, in which abstract interpretation, flow analysis, and model checking are employed in the representation, abstraction, and analysis of programs is presented.
Abstract: This paper presents a collection of techniques, a methodology, in which abstract interpretation, flow analysis, and model checking are employed in the representation, abstraction, and analysis of programs. The methodology shows the areas of intersection of the different techniques as well as the opportunites that exist when one technique is used in support of another. The methodology is presented as a three-step process: First, from a (small-step) operational semantics definition and a program, one constructs a program model, which is a state-transition system that encodes the program’s executions. Second, abstraction upon the program model is performed, reducing the detail of information in the model’s nodes and arcs. Finally, the program model is analyzed for properties of its states and paths.

Journal ArticleDOI
TL;DR: This contribution proposes usage of uniform, external target processor models in code generation, which describe embedded processors by means of RT-level netlists, and describes input formats, central data structures, and code generation techniques in MSSQ.
Abstract: Design automation for embedded systems comprising both hardware and software components demands for code generators integrated into electronic CAD systems. These code generators provide the necessary link between software synthesis tools in HW/SW codesign systems and embedded processors. General-purpose compilers for standard processors are often insufficient, because they do not provide flexibility with respect to different target processors and also suffer from inferior code quality. While recent research on code generation for embedded processors has primarily focussed on code quality issues, in this contribution we emphasize the importance of retargetability, and we describe an approach to achieve retargetability. We propose usage of uniform, external target processor models in code generation, which describe embedded processors by means of RT-level netlists. Such structural models incorporate more hardware details than purely behavioral models, thereby permitting a close link to hardware design tools and fast adaptation to different target processors. The MSSQ compiler, which is part of the MIMOLA hardware design system, operates on structural models. We describe input formats, central data structures, and code generation techniques in MSSQ. The compiler has been successfully retargeted to a number of real-life processors, which proves feasibility of our approach with respect to retargetability. We discuss capabilities and limitations of MSSQ, and identify possible areas of improvement.

Journal ArticleDOI
TL;DR: The evolutionary algorithm based on a parallel diffusion model and extended for mixed-integer optimization was able to compete with or even outperform traditional methods of robust MOC design and is easily adopted to other application domains.
Abstract: Robustness is an important requirement for almost all kinds of products. This article shows how evolutionary algorithms can be applied for robust design based on the approach of Taguchi. To achieve a better understanding of the consequences of this approach, we first present some analytical results gained from a toy problem. As a nontrivial industrial application we consider the design of multilayer optical coatings (MOCs) most frequently used for optical filters. An evolutionary algorithm based on a parallel diffusion model and extended for mixed-integer optimization was able to compete with or even outperform traditional methods of robust MOC design. With respect to chromaticity, the MOC designs found by the evolutionary algorithm are substantially more robust to parameter variations than a reference design and therefore perform much better in the average case. In most cases, however, this advantage has to be paid for by a reduction in the average reflectance. The robust design approach outlined in this paper should be easily adopted to other application domains.

Journal ArticleDOI
TL;DR: The S exciton series belonging to the fundamental gap and two hitherto unknown valence bands with even parity have been discovered by two-photon spectroscopy in the rutile-type semiconductor tin oxide (SnO2) as discussed by the authors.

Book ChapterDOI
TL;DR: This work presents a toolbox for the combined functional and quantitative analysis of discrete event dynamic systems (DEDS) on the basis of a common model interchange format, the Abstract Petri Net Notation (APNN).[5]).
Abstract: We present a toolbox for the combined functional and quantitative analysis of discrete event dynamic systems (DEDS) on the basis of a common model interchange format, the Abstract Petri Net Notation (APNN,[5]). This format covers a fair amount of Petri net like formalisms as well as certain Queueing network concepts. At a lower level the toolbox uses a format for communicating stochastic automata, a notation to describe DEDS at a state transition level. For a more detailed description of the toolbox we refer to [3].

Journal ArticleDOI
TL;DR: In this article, the concept of massive neutrinos is given and the present state of experimental research is extensively reviewed This includes astrophysical studies of solar, supernova and very high energy NEs Future perspectives are also outlined.

Journal ArticleDOI
TL;DR: The results indicate the existence of cross-talk in particular at short intervals between information about amplitude and movement initiation, consistent with the notion that cross- talk between concurrent processes of amplitude specification is transient and vanishes as the time available for motor programming increases.
Abstract: We investigated the time course of the amplitude specification of rapid bimanual reversal movements (lateral displacements on two digitizers). To this end we used the timed-response paradigm in which the response has to be initiated synchronously with an auditory signal. Information about the required amplitudes was presented at various times before the synchronization signal. Consistent with previous results, the progression of amplitude specification was reflected in the dependence of the amplitudes of the reversal movements on the time interval between amplitude information and synchronization signal. Same or different amplitudes for the hands were used to examine cross-talk at the programming level of the two-level model of intermanual interference. The results indicate the existence of cross-talk in particular at short intervals between information about amplitude and movement initiation. This is consistent with the notion that cross-talk between concurrent processes of amplitude specification is transient and vanishes as the time available for motor programming increases.

Journal ArticleDOI
24 Jun 1998
TL;DR: In this paper, total-reflection X-ray fluorescence (TXRF) was used for simultaneous determination of 15 elements in tea samples which were produced either by acid digestion or acidified infusion of tea leaves (Camellia sinensis).
Abstract: Total-reflection X–ray fluorescence (TXRF) was used for the simultaneous determination of 15 elements in tea samples which were produced either by acid digestion or acidified infusion of tea leaves (Camellia sinensis). The accuracy and precision of the method were checked by its application to a certified reference material (GBW 08505 : tea). A variety of 39 tea samples of different kinds and/or qualities produced in different regions of China were analysed. The range and mean of the concentrations of elements in the tea leaves (0.1–30.000 μg g–1) and their solubility in infusions (0.5–85%) were determined and the influence of the origin, type and quality of the tea samples was studied. In some tea leaves produced in a Se-rich region, the content of Se was found to be very high (up to 7.5 μg g–1), in contrast to a concentration of only about 0.1 μg g–1 Se in most of the tea leaves examined.

Journal ArticleDOI
TL;DR: The synthesis of the aryldiphosphonic ester C6H2[P(O)(OEt)2]2-1,3-t-Bu-5 (2) is reported in this paper.

Journal ArticleDOI
TL;DR: In this article, the authors considered the time evolution of systems in which a spatially homogeneous scalar field is coupled to fermions and the quantum back reaction was taken into account in the one-loop approximation.
Abstract: We consider the time evolution of systems in which a spatially homogeneous scalar field is coupled to fermions. The quantum back reaction is taken into account in the one-loop approximation. We set up the basic equations and their renormalization in a form suitable for numerical computations. The initial singularities appearing in the renormalized equations are removed by a Bogoliubov transformation. The equations are then generalized to those in a spatially flat Friedmann-Robertson-Walker universe. We have implemented the Minkowski space equations numerically and present results for the time evolution with various parameter sets. We find that fermion fluctuations are not in general as ineffective as previously assumed, but show interesting features which should be studied further. In an especially interesting example we find that fermionic fluctuations can ``catalyze'' the evolution of bosonic fluctuations.

Book ChapterDOI
TL;DR: It is shown that evolutionary algorithms are able to converge to the set of minimal elements in finite time with probability one, provided that the search space is finite, the time-invariant variation operator is associated with a positive transition probability function and that the selection operator obeys the so-called ‘elite preservation strategy.’
Abstract: The task of finding minimal elements of a partially ordered set is a generalization of the task of finding the global minimum of a real-valued function or of finding Pareto-optimal points of a multicriteria optimization problem. It is shown that evolutionary algorithms are able to converge to the set of minimal elements in finite time with probability one, provided that the search space is finite, the time-invariant variation operator is associated with a positive transition probability function and that the selection operator obeys the so-called ‘elite preservation strategy.’

Journal ArticleDOI
TL;DR: In this article, a design method based on the interpretation of reactive distillation line diagrams is used to identify the main process parameters and to provide a foundation for experimental investigation, showing that the significant influence of the reflux ratio on the conversion in the column is shown by mini plant experiments using supported ion exchanger in the form of Raschig rings as a heterogeneous catalyst.
Abstract: The production of methyl acetate in a reactive distillation process – prior art for 15 years – is often used as an example to study the basic phenomena of reactive distillation. The present paper deals with a theoretical and experimental analysis of methyl acetate synthesis in a reactive distillation column. A design method based on the interpretation of reactive distillation line diagrams is used to identify the main process parameters and to provide a foundation for experimental investigation. The significant influence of the reflux ratio on the conversion in the column is shown by mini plant experiments using supported ion exchanger in the form of Raschig rings as a heterogeneous catalyst. These experiments demonstrate the catalytic quality of this packing material. To simulate the reactive distillation column with a simple stage-to-stage method, the separation efficiency of the catalytic rings is investigated. Comparison of experimental and simulation results reveals that a simple model based on the assumption of simultaneous chemical and phase equilibrium describes the experimental data quite well over a wide range of reflux ratios. Furthermore, simulation results show that the conversion depends less on the number of reactive stages than on the use of two feed stages.

Journal ArticleDOI
TL;DR: In this article, a straightforward spectral variable selection based on choices from the optimum partial least squares (PLS) regression vector yields better results than using PLS calibration models with full spectrum evaluation.
Abstract: Non-invasive assays for blood glucose can be based on near infrared spectrometry of skin tissue using the diffuse reflectance technique. Using a straightforward spectral variable selection based on choices from the optimum partial least-squares (PLS) regression vector yields better results than using PLS calibration models with full spectrum evaluation previously reported. The pairs of variables are selected from the maxima and minima of the regression weights, respectively, in decreasing order. Substantial improvements in the prediction performance of such calibration models, compared to previous calibrations based on full spectrum evaluation, are obtained. Another aspect is the reduced number of spectral variables needed for robust calibration modeling. In addition, evidence is provided for the physical effect, as manifested by the spectral glucose absorptivities, underlying the individual single-person calibration models. Their regression vector structure shows very similar features as calculated for a glucose calibration experiment based on random human plasma samples. Novel techniques are presented for probing the intravascular fluid space using time-resolved near infrared spectroscopy of oral mucosa. The pulsatile blood spectrum can be derived from these diffuse reflectance lip spectra by Fourier analysis. Future applications and prospects for non-invasive blood analysis are discussed.

Book ChapterDOI
29 Jun 1998
TL;DR: A three pass solution for global illumination calculation extending the two pass approach proposed by Jensen is described, which combines particle tracing of importance with distributed ray tracing to create a global data structure called importance map.
Abstract: Particle tracing allows physically correct simulation of all kinds of light interaction in a scene, but can be a computationally expensive task. Use of visual importance is a powerful technique to improve the efficiency of global illumination calculations. We describe a three pass solution for global illumination calculation extending the two pass approach proposed by Jensen. In the first pass particle tracing of importance is performed to create a global data structure, called importance map. Based on this data structure importance driven photon tracing is used in the second pass to construct a photon map containing information about the global illumination in the scene. In the last pass the image is rendered by distributed ray tracing using the photon map.

Journal ArticleDOI
TL;DR: In this article, two types of optical second harmonic spectra of hexagonal ions were observed, one caused by the noncentrosymmetric ferroelectric ordering of charges, whereas the other was due to the centrosymetric antiferromagnetic ordering of spins, and partial overlapping between the electronic transitions gave rise to a new kind of nonlinear optical polarization.
Abstract: We have observed two types of optical second harmonic spectra of ${\mathrm{Mn}}^{3+}$ ions in hexagonal ${\mathrm{YMnO}}_{3},$ one of which is caused by the noncentrosymmetric ferroelectric ordering of charges, whereas the other is due to the centrosymmetric antiferromagnetic ordering of spins. Partial overlapping between the electronic transitions gives rise to a new kind of nonlinear optical polarization $\mathbf{P}(2\ensuremath{\omega})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\mathbf{P}}^{\mathrm{FE}}(2\ensuremath{\omega})+{\mathbf{P}}^{\mathrm{AFM}}(2\ensuremath{\omega}),$ which depends on two order parameters. The magnitude and phase of $\mathbf{P}(2\ensuremath{\omega})$ can be changed by varying the ratio between the two contributions, as demonstrated by changing the contrast between $180\ifmmode^\circ\else\textdegree\fi{}$ antiferromagnetic domains, which are indistinguishable in linear optics.

Journal ArticleDOI
TL;DR: In this paper, the effects of isotopic composition on the band gap of CuCl on a series of samples made out of the stable isotopes was investigated, which can be traced back to the strong $p$-$d$ mixing in the copper halides.
Abstract: We have investigated the effects of isotopic composition on the band gap of CuCl on a series of samples made out of the stable isotopes ${}^{63}\mathrm{Cu}$, ${}^{65}\mathrm{Cu}$, ${}^{35}\mathrm{Cl}$, and ${}^{37}\mathrm{Cl}$. Besides specimens containing elements with the natural abundances, we have measured samples with monoisotopic sublattices as well as artificial mixtures of isotopes. With nonlinear (two-photon absorption, second-harmonic generation) and linear (luminescence) optical spectroscopy we find that the fundamental gap of CuCl increases by 364(18) \ensuremath{\mu}eV/amu when increasing the Cl mass. However, it decreases by 76(5) \ensuremath{\mu}eV/amu when increasing the Cu mass. Using a two-oscillator model for the lattice dynamics of CuCl we show that these rates are consistent with the anomalous increase of the band gap with increasing temperature. These effects can be traced back to the strong $p$-$d$ mixing in the copper halides. From the temperature dependence of the band gap of CuBr we also estimate the changes of its gap with isotopic composition.