Rm Int Research Articles

Orbital relaxation and the third-order induction energy in symmetry-adapted perturbation theory

Theoretical investigations of the induction interaction between closed-shell molecules which fully account for the orbital relaxation effects are presented. Explicit expressions for the third-order induction energy in terms of molecular integrals and orbital energies are given and implemented within the sapt2008 program for symmetry-adapted perturbation theory (SAPT) calculations. Numerical investigations for the He–He, He–LiH, Ar–Ar, H2–CO, H2O–H2O, and H2O–NH3 model dimers show that the orbital relaxation increases the third-order induction interaction by 15 to 50% at near-equilibrium geometries, with the largest effect observed for complexes involving highly polar monomers. At large intermonomer separations, the relaxed third-order induction energy perfectly recovers the difference $$\delta E^{\rm HF}_{\rm int}$$ between the Hartree–Fock interaction energy and the sum of the uncorrelated SAPT contributions through second order in the intermolecular interaction operator. At the near-equilibrium geometries, the sum of the relaxed third-order induction and exchange-induction energies reproduces, however, only a small fraction (6 to 15%) of $$\delta E^{\rm HF}_{\rm int}$$ for the nonpolar systems and about 40 to 60% for the polar ones. A comparison of the complete SAPT calculations with the coupled-cluster treatment with single, double, and noniterative triple excitations [CCSD(T)] suggests that the pure SAPT approach with all the available third-order corrections is more accurate for nonpolar systems while for the polar ones the hybrid approach including $$\delta E^{\rm HF}_{\rm int}$$ gives better results.

A visible shoreline theorem for staircase paths

Let C be an orthogonal polygon in the plane, bounded by a simple closed curve, and assume that C is starshaped via staircase paths. Let \(P \subseteq {\mathbb{R}}^2\backslash({\rm int} C)\). If every four points of P see a common boundary point of C via staircase paths in \({\mathbb{R}}^2\backslash({\rm int} C)\), then there is a boundary point b of C such that every point of P sees b (via staircase paths in \({\mathbb{R}}^2\backslash({\rm int} C)\)). The number four is best possible, even if C is orthogonally convex.

The quasiconvex hull for the five-gradient problem

In [8 Chapter 4.3] Kirchheim and Preiss gave an example of a set K consisting of five 2 × 2 symmetric matrices without rank-one connections, for which there exists a Lipschitz mapping u satisfying \({Du \in K}\) . In the present paper we construct the rank-one convex hull of K. As a corollary we obtain that for each \({F \in {\rm int}\,K^{rc}}\) there exists a Lipschitz mapping u satisfying $$Du \in K\quad{\rm and}\quad u(x) = Fx\,{\rm for}\,x\,\in\,{\partial} \Omega \,.$$ Moreover, we show that the rank-one convex hull of K and the quasiconvex hull of K are equal.

High-Temperature Microwave Performance of Submicron AlGaN/GaN HEMTs on SiC

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The effect of temperature on the small-signal radio-frequency (RF) performance of submicron AlGaN/GaN high-electron-mobility transistors on SiC has been studied from room temperature (RT) up to 600 K. A relation between ambient and channel temperatures has been established by means of finite-element simulations. The thermal behavior of the intrinsic parameters <formula formulatype="inline"><tex Notation="TeX">$C_{\rm gs}$</tex></formula>, <formula formulatype="inline"><tex Notation="TeX">$C_{\rm gd}$</tex></formula>, <formula formulatype="inline"><tex Notation="TeX">$g_{m, {\rm int}}$</tex></formula>, and <formula formulatype="inline"><tex Notation="TeX">$g_{\rm ds}$</tex></formula> has been extracted accurately from RF measurements by means of the small-signal equivalent circuit. Main dc parameters <formula formulatype="inline"><tex Notation="TeX">$(I_{D}, g_{m, {\rm ext}})$</tex></formula> show reductions close to 50% between RT and 600 K, mainly due to the decrease in the electron mobility and drift velocity. In the same range, <formula formulatype="inline"><tex Notation="TeX">$f_{T}$</tex></formula> and <formula formulatype="inline"><tex Notation="TeX">$f_{\max}$</tex> </formula> suffer a 60% decrease due to the reduction in <formula formulatype="inline"><tex Notation="TeX">$g_{m, {\rm ext}}$</tex> </formula> and a slight increase of <formula formulatype="inline"><tex Notation="TeX">$C_{\rm gs}$</tex></formula> and <formula formulatype="inline"><tex Notation="TeX">$C_{\rm gd}$</tex></formula>. An anomalous thermal evolution of <formula formulatype="inline"><tex Notation="TeX">$C_{\rm gd}$</tex></formula> at low <formula formulatype="inline"><tex Notation="TeX">$I_{D}$ </tex></formula> has been identified, which is indicative of the presence of traps. </para>

Integer-valued Euler–Jackson’s finite differences

Let D be a Dedekind domain and q be a nonzero element of D which is not a root of unity. We are interested here in the (D)-algebra \({{{\rm Int}^{[k]}_J}(S,D)}\) formed by the polynomials whose values on the subset \({S=\{q^n\mid n\in \mathbb{N}\}}\) belong to D as well as the values of their k first Euler–Jackson’s finite differences. Mainly, we describe the characteristic ideals of \({{{\rm Int}^{[k]}_J}(S,D)}\) and, in the case where D = V is a discrete valuation domain, we give bases of the V-module \({{{\rm Int}^{[k]}_J}(S,V)}\) . Moreover, the description of the maximal ideals of the D-algebra \({{{\rm Int}^{[k]}_J}(S,D)}\) allows us to characterize the subsets S such that \({{\rm Int}_J^{[\infty]}(S,D)}\) is equal to the D-algebra \({{\rm Int}^{(\infty)}(S,D)}\) formed by the polynomials whose values on S belong to D as well as the values of all their derivatives.

ON THE SOLUTION EXISTENCE OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH DISCONTINUOUS MULTIFUNCTIONS

In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set $K$ in a normed space $X$, a subset $D$ in a Hausdorff topological vector space $Y$, and a closed convex cone $C$ in $R^n$. Let $\Gamma: K\to 2^K$, $\Phi : K\rightarrow 2^{D}$ be two multifunctions and $f : K\times D\times K\to R^n$ be a single-valued mapping. Find a point $(\hat x, \hat y)\in K\times D$ such that \begin{gather} (\hat x, \hat y)\in \Gamma(\hat x)\times\Phi(\hat x),\,\, {\rm and}\,\, \{f(\hat x, \hat y, z): z\in\Gamma(\hat x)\}\cap (-{\rm Int }C)=\emptyset. \notag \end{gather} We prove some existence theorems for the problem in which $\Phi$ can be discontinuous and $K$ can be unbounded.

ON THE SOLUTION EXISTENCE OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH DISCONTINUOUS MULTIFUNCTIONS

In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set $K$ in a normed space $X$, a subset $D$ in a Hausdorff topological vector space $Y$, and a closed convex cone $C$ in $R^n$. Let $\Gamma: K\to 2^K$, $\Phi : K\rightarrow 2^{D}$ be two multifunctions and $f : K\times D\times K\to R^n$ be a single-valued mapping. Find a point $(\hat x, \hat y)\in K\times D$ such that \begin{gather} (\hat x, \hat y)\in \Gamma(\hat x)\times\Phi(\hat x),\,\, {\rm and}\,\, \{f(\hat x, \hat y, z): z\in\Gamma(\hat x)\}\cap (-{\rm Int }C)=\emptyset. \notag \end{gather} We prove some existence theorems for the problem in which $\Phi$ can be discontinuous and $K$ can be unbounded.

Projective re-normalization for improving the behavior of a homogeneous conic linear system

In this paper we study the homogeneous conic system \(F: Ax = 0, x \in C\setminus \{0\}\) . We choose a point \(\bar s \in {\rm int}C^*\) that serves as a normalizer and consider computational properties of the normalized system \(F_{\bar s} : Ax = 0, \bar s^T x = 1, x \in C\) . We show that the computational complexity of solving F via an interior-point method depends only on the complexity value \(\vartheta\) of the barrier for C and on the symmetry of the origin in the image set \(H_{\bar s} := \{Ax {\bar s}^Tx = 1, x \in C\}\) , where the symmetry of 0 in \(H_{\bar s}\) is$$ {\rm sym}(0, H_{\bar s}) := {\rm max}\{\alpha : y \in H_{\bar s} \Rightarrow -\alpha y \in H_{\bar s} \} .$$We show that a solution of F can be computed in \(O(\sqrt{\vartheta}{\rm ln}(\vartheta/{\rm sym}(0, H_{\bar s}))\) interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region \(F_{\bar s}\) and the image set \(H_{\bar s}\) and prove the existence of a normalizer \({\bar s}\) such that \({\rm sym}(0,H_{\bar s}) \ge 1/m\) provided that F has an interior solution. We develop a methodology for constructing a normalizer \({\bar s}\) such that \({\rm sym}(0, H_{\bar s}) \ge 1/m\) with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomial-time, the normalizer will yield a conic system that is solvable in \(O(\sqrt{\vartheta}{\rm ln}(m\vartheta))\) iterations, which is strongly-polynomial-time. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1,000 × 5,000.

Uniform Fat Segment and Cusp Properties for Compactness in Shape Optimization

A general compactness theorem for shape/geometric analysis and optimization is given for a family of subsets verifying the uniform fat segment property in a bounded open holdall with or without constraints on the De Giorgi [11] or the γ-density perimeter of Bucur and Zolesio [3]. The uniform fat segment property is shown to be equivalent to the uniform cusp property introduced in [9] with a continuous non-negative cusp function. This equivalence remains true for cusp functions that are only continuous at the origin. The equivalence of sets verifying a segment property with their C0-graph representation is further sharpened for sets with a compact boundary. Our C0-graph characterization is shown to be equivalent to both the uniform cusp property and the uniform segment property. It is used to formulate sufficient conditions on the local graphs of a family of subsets of a bounded open holdall to get compactness. A first condition assumes that the local graphs are bounded above by a cusp function; a second condition which requires that the local graphs be equicontinuous turns out to be equivalent to the first one. The respective solutions of the Laplacian with homogeneous Dirichlet or Neumann boundary condition are shown to be continuous with respect to domains in that family. In the Dirichlet case for 1 < p < ∞, we prove the (1,p)-stability of compact sets in the sense of Herdberg [14] under the weaker almost everywhere assumption rather than quasi everywhere. It is also shown that for the family of measurable crack free sets \(\Omega\) in a bounded open holdall \(D\colon \ v\in \{w\in W^{1,p}_0(D)\colon \ w= 0 \mbox{ almost everywhere on } D\backslash \Omega\}\) implies \(v|_{{\rm int} {\Omega}}\in W^{1,p}_0({\rm int} {\Omega})\).

An Optimal Alternative Theorem and Applications to Mathematical Programming

Given a closed convex cone P with nonempty interior in a locally convex vector space, and a set $$A \subset Y $$ , we provide various equivalences to the implication $$A \cap (-{\rm int}\,P) = \emptyset \Longrightarrow {\rm co}(A)\cap (-{\rm int}\, P) = \emptyset, $$ among them, to the pointedness of cone(A + int P). This allows us to establish an optimal alternative theorem, suitable for vector optimization problems. In addition, we present an optimal alternative theorem which characterizes two-dimensional spaces in the sense that it is valid if, and only if, the space is at most two-dimensional. Applications to characterizing weakly efficient solutions through scalarization; the zero (Lagrangian) duality gap; the Fritz---John optimality conditions for a class of nonconvex nonsmooth minimization problems, are also presented.

Bianchi type-I cosmology with scalar and spinor fields

We consider a system of interacting spinor and scalar fields in a gravitational field given by a Bianchi type-I cosmological model filled with perfect fluid. The interacting term in the Lagrangian is chosen in the form of derivative coupling, i.e., ${\cal L}_{\rm int} = \frac{\lambda}{2} \vf_{,\alpha}\vf^{,\alpha} F$, with $F$ being a function of the invariants $I$ an $J$ constructed from bilinear spinor forms $S$ and $P$. We consider the cases when $F$ is the power or trigonometric functions of its arguments. Self-consistent solutions to the spinor, scalar and BI gravitational field equations are obtained. The problems of initial singularity and asymptotically isotropization process of the initially anisotropic space-time are studied. It is also shown that the introduction of the Cosmological constant ($\Lambda$-term) in the Lagrangian generates oscillations of the BI model, which is not the case in absence of $\Lambda$ term. Unlike the case when spinor field nonlinearity is induced by self-action, in the case in question, wehere nonlinearity is induced by the scalar field, there exist regular solutions even without broken dominant energy condition.

Racemization and hydrolysis of (S)‐naproxen 2,2,2‐trifluoroethyl ester in non‐polar solvents by strong neutral bases: implication for ion‐pair kinetic basicity and hydrolysis

AbstractBy using strong neutral bases as catalyst, a detailed investigation of the racemization of (S)‐naproxen 2,2,2‐trifluoroethyl ester was conducted in the non‐polar solvents isooctane, cyclohexane and n‐hexane. The second‐order interconversion constant $k^*_{\rm int}$ as representing the ion‐pair kinetic basicity in isooctane was first estimated and correlated with the equilibrium ion‐pair basicity pKip in tetrahydrofuran, giving slopes of 0.768 and 0.689 for non‐phosphazene and phosphazene bases, respectively, in the Brønsted correlations. The result was further compared with that for (S)‐naproxen 2,2,2‐trifluoroethyl thioester, showing about a 1–2 orders of magnitude enhancement of $k^*_{\rm int}$ for the corresponding thio‐containing analogue. A smaller influence of non‐polar solvents (i.e. isooctane, n‐hexane and cyclohexane) on $k^*_{\rm int}$ was found. Kinetic analysis of the racemization and hydrolysis of (S)‐naproxen 2,2,2‐trifluoroethyl ester in isooctane and n‐hexane containing 7‐methyl‐1,5,7‐triazabicyclo[4.4.0]dec‐5‐ene and water suggests nucleophilic hydrolysis by the base, where the breakdown of tetrahedral intermediates IR1 and IS1 is the rate‐limiting step and the hydrolysis constant khy is in proportion to the product of base and ion‐pair concentrations. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Studies on zero point of charge and intrinsic ionization constant of Zn?Mg?Al hydrotalcite-like compounds

The zero point of charge (ZPC) and the intrinsic ionization constant (\( K_{\rm a}^{{\rm int}} \)) of Zn−Mg−Al hydrotalcite-like compounds (HTlc) with the general formula [(Zn,Mg)1− x Al x (OH)2] x + [(OH,Cl) x ] x − were determined by potentiometric titration (PT). The variation of the ZPC and \( K_{\rm a}^{{\rm int}} \) with x was investigated. For the colloidal particles possessing permanent charges, the ZPC determined by the PT method is the zero point of net charge (ZPNC). The ZPNC ( pH ZPNC) values were 9.63 {[Zn0.27Mg0.36Al0.37 (OH)2]Cl0.13 (OH)0.24 }, 9.68 {[Zn0.13 Mg0.58Al0.29 (OH)2]Cl0.12 (OH)0.17 }, 9.67 {[Zn0.17Mg0.54Al0.29 (OH)2]Cl0.16 (OH)0.12 },10.16 {[Zn0.08Mg0.67Al0.25 (OH)2]Cl0.17 (OH)0.08}, 10.33 {[Zn0.16 Mg0.60 Al0.24 (OH)2] Cl0.16 (OH)0.08} and 10.60 {[Zn 0.19Mg0.60 Al0.21 (OH)2] Cl0.15 (OH)0.06}; the intrinsic ionization constants \( {\rm p}K_{{\rm a2}}^{{\rm int}} \) of the same HTlc samples were 10.31, 10.44, 10.44, 11.02, 11.19 and 11.54. With decreasing x, ZPNC and \( K_{{\rm a2}}^{{\rm int}} \) of HTlc increased, and the acidity of the HTlc samples decreased.

Nonexistence of invariant semigroups in affine symmetric spaces

Let \(\left( G,L,\tau \right) \) be an affine symmetric space with \(G\) a simple Lie group, \(\tau \) an involutive automorphism of \(G\) and \(L\) an open subgroup of the \(\tau \)-fixed point group \(G^{\tau }\). It is proved here that the existence of a proper semigroup \(S\subset G\) with \({\rm int}S\neq \emptyset \) and \(L\subset S\) implies that \(\left( G,L,\tau \right) \) is of Hermitian type, as conjectured by Hilgert and Neeb [4]. When \(S\) exists, it turns out that it leaves invariant an open \(L\)-orbit in a minimal flag manifold of \(G\). A byproduct of our approach is an alternate proof of the maximality of the compression semigroup of an open orbit (see Hilgert and Neeb [31]).

A Local (Perturbative) Construction of Observables in Gauge Theories: The Example of QED

Interacting fields can be constructed as formal power series in the framework of causal perturbation theory. The local field algebra $\tilde {\cal F}({\cal O})$ is obtained without performing the adiabatic limit; the (usually bad) infrared behavior plays no role. To construct the observables in gauge theories we use the Kugo-Ojima formalism; we define the BRST-transformation $\tilde s$ as a graded derivation on the algebra of interacting fields and use the implementation of $\tilde s$ by the Kugo-Ojima operator $Q_{\rm int}$. Since our treatment is local, the operator $Q_{\rm int}$ differs from the corresponding operator $Q$ of the free theory. We prove that the Hilbert space structure present in the free case is stable under perturbations. All assumptions are shown to be satisfied in QED.

Standard Uncertainty of Angular Positions and Statistical Quality of Step-Scan Intensity Data

In step-scan diffraction measurements, the diffraction angle 2θ is an observation with standard uncertainty u(2θ). By the law of uncertainty propagation, u(2θ), typically 0.001 < u(2θ) < 0.004°, affects the standard uncertainty u_{\rm total}(y) of the intensity y at each step 2\theta_i, depending on the local slope y'_i = \left. {{\rm d}y}/{{\rm d}(2\theta)}\right|_{2\theta_{i}}, by u_{\rm total}^2(y_i) = u^2_{\rm Poisson}+[y'_iu(2\theta)]^2, where u_{\rm Poisson} = ({y_i})^{1/2} is the conventional Poisson statistics. For the intensity y at 2θ of steepest slope, u_{\rm total}(y) is given by u_{\rm total}^2(y) = u^2_{\rm Poisson}(1+v^2), where v = 2u(2\theta){y_0}^{1/2}/h is the ratio of y'_iu(2\theta) and u_{\rm Poisson}, y_0 is the peak intensity and h the full width at half-maximum of the profile. The error of the intensities at individual steps modifies also the standard uncertainty of the integrated intensity: u_{\rm total}^2(\rm Int) = u^2_{\rm Poisson}({\rm Int}) (1+ {v^2}/{2}). As v depends on y_0^{1/2}/h, it is evident that the importance of the correction increases with increasing count rates and decreasing line width. In most practical cases, y'_iu(2\theta) contributes a multiple of Poisson statistics to the standard uncertainty of intensity. It will be shown that with a realistic weighting scheme the \chi^2 as well as the Durbin–Watson test become more meaningful.

Context-free graph grammars and concatenation of graphs

An operation of concatenation is defined for graphs. This allows strings to be viewed as expressions denoting graphs, and string languages to be interpreted as graph languages. For a class \(K\) of string languages, \({\rm Int}(K)\) is the class of all graph languages that are interpretations of languages from \(K\). For the classes REG and LIN of regular and linear context-free languages, respectively, \({\rm Int}({\rm REG}) = {\rm Int}({\rm LIN})\). \({\rm Int}({\rm REG})\) is the smallest class of graph languages containing all singletons and closed under union, concatenation and star (of graph languages). \({\rm Int}({\rm REG})\) equals the class of graph languages generated by linear HR (= Hyperedge Replacement) grammars, and \({\rm Int}(K)\) is generated by the corresponding \(K\)-controlled grammars. Two characterizations are given of the largest class \(K'\) such that \({\rm Int}(K') = {\rm Int}(K)\). For the class CF of context-free languages, \({\rm Int}({\rm CF})\) lies properly inbetween \({\rm Int}({\rm REG})\) and the class of graph languages generated by HR grammars. The concatenation operation on graphs combines nicely with the sum operation on graphs. The class of context-free (or equational) graph languages, with respect to these two operations, is the class of graph languages generated by HR grammars.