Adjoint Space Research Articles

Differential operators on Hermite Sobolev spaces

In this paper, we compute the Hilbert space adjoint ∂∗ of the derivative operator ∂ on the Hermite Sobolev spaces \(\mathcal {S}_{q}\). We use this calculation to give a different proof of the ‘monotonicity inequality’ for a class of differential operators (L,A) for which the inequality was proved in Infin. Dimens. Anal. Quantum Probab. Relat. Top.2(4) (2009) 515–591. We also prove the monotonicity inequality for (L,A), when these correspond to the Ornstein–Uhlenbeck diffusion.

About Leibniz cohomology and deformations of Lie algebras

We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding whether it has more Leibniz deformations than just the Lie ones. We also give the complete description of the Leibniz (and Lie) universal infinitesimal deformation of the 4-dimensional diamond Lie algebra, used in the WZW model, and study the case of its 5-dimensional analogue by computing Massey products.

Sobolev-Trenogin Principle in Construction of the Boundary Value Problem Adjoint to the Linear Multipoint Problem

The research has been done in two directions. In a linear case, the adjoint boundary problem have been built. We have managed to do it in a classical continuous case without resort to such terms of functional analysis as an adjoint space, adjoint operator, etc. In a non-linear case, we have considered the problem with a small parameter and discussed an issue of applicability of some aspects of a theory of bifurcation of the nonlinear equations' solutions (Trenogin et al., 1991). We have built a boundary problem adjoint to the linear multipoint problem. We have studied unicity of its linear and adjoint differential operators with multipoint boundary conditions and generalized it for the m-point problem.

Small codimension subvarieties in homogeneous spaces

We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in X 2 where X is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of X. Finally we prove transplanting theorems à la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.

Automorphisms of elementary adjoint Chevalley groups of types A l , D l , and E l over local rings with 1/2

It is proved that every automorphism of an elementary adjoint Chevalley group of type A l , D l , or E l over a local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the normalizer of that Chevalley group in GL(V) (V is an adjoint representation space).

Suppressing random dopant-induced fluctuations of threshold voltages in semiconductor devices

This article introduces a robust and computationally efficient technique for the design of fluctuation-resistant structures (fault-tolerant) semiconductor devices. This technique can be applied to the computation of the doping profiles that minimize the intrinsic variations in various parameters induced by random dopant fluctuations. The technique is based on the evaluation of doping sensitivity functions, which are defined as elements of the space adjoint to the space of square integrable functions generated by all possible doping variations. The optimized doping profiles are computed by minimizing the standard deviation of fluctuations of different parameters, and constraints are taken into consideration by using the Lagrange multiplier method. The technique introduced here can be applied to any semiconductor device, such as metal-oxide-semiconductor field-effect transistors (MOSFETs), silicon-on-insulator (SOI) devices, and fin field-effect transistors, and can be used in the framework of any transport model. The technique is applied to the minimization of the random dopant-induced fluctuations of threshold voltages in 25 nm channel length MOSFETs and double-gate fully depleted SOI devices. It is shown that, by carefully designing the doping profiles, random dopant-induced fluctuations can be suppressed between 16% in the case of constrained optimization and 35% in the case of unconstrained optimization for devices with channel lengths smaller than 25 nm. Analytical equations are derived for the optimum doping profiles that minimize the effects of random dopant fluctuations on the threshold voltage in MOSFETs. It is shown that, in both long-channel and short-channel devices, the size of the undoped region should be at least 14 of the width of the depletion region in order to suppress efficiently the random dopant-induced fluctuations.

Bases of rearrangement invariant spaces

We prove that if E is a rearrangement-invariant space, then a boundedly complete basis exists in E, if and only if one of the following conditions holds: 1) E is maximal and E ≠ L1[0, 1]; 2) a certain (any) orthonormal system of functions from L∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L∞, represents a boundedly complete basis in E. As a corollary, we state the following assertion: Any (certain) orthonormal system of functions from L∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L∞, represents a spanning basis in a separable rearrangement-invariant space E, if and only if the adjoint space E* is separable. We prove that in any separable rearrangement-invariant space E the Haar system either forms an unconditional basis, or a strongly conditional one. The Haar system represents a strongly conditional basis in a separable rearrangement-invariant space, if and only if at least one of the Boyd indices of this space is trivial.

ECT-B-splines defined by generalized divided differences

ECT-spline curves are generated from different local ECT-systems via connection matrices. If they are nonsingular, lower triangular and totally positive there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized either to form a nonnegative partition of unity or to have integral one. In this paper such ECT-B-splines are defined by generalized divided differences. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines [Mühlbach and Tang, Calculation of ECT-B-splines and of ECT-spline curves recursively, in preparation]. There is an ECT-spline space naturally adjoint to every ECT-spline space. We also construct B-splines via generalized divided differences for this space and study relations between the two adjoint spaces.

A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically thecascade algorithm in wavelet theory. Let\(\mathcal{H}\) be a Hilbert space, and let π be a representation ofL∞(\(\mathbb{T}\)) on\(\mathcal{H}\). LetR be a positive operator inL∞(\(\mathbb{T}\)) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on\(\mathcal{H}\) (bounded, but noncontractive) such that $$\pi (f){\rm M} = M\pi (f(z^2 ))andM*\pi (f)M = \pi (R*f),f \in L^\infty (\mathbb{T}),$$ where the * refers to Hilbert space adjoint. We give a complete orthogonal expansion of\(\mathcal{H}\) which reduces π such thatM acts as a shift on one part, and the residual part is\(\mathcal{H}\)(∞) = ∩ n [M n \(\mathcal{H}\)], where [M n \(\mathcal{H}\)] is the closure of the range ofM n . The shift part is present, we show, if and only if ker (M*)≠{0}. We apply the operator-theoretic results to the refinement operator (or cascade algorithm) from wavelet theory. Using the representation π, we show that, for this wavelet operatorM, the components in the decomposition are unitarily, and canonically, equivalent to spacesL2(E n ) ⊂L2(ℝ), whereEn ⊂ ℝ, n=1,2,3,..., ∞, are measurable subsets which form a tiling of ℝ; i.e., the union is ℝ up to zero measure, and pairwise intersections of differentE n 's have measure zero. We prove two results on the convergence of the cascale algorithm, and identify singular vectors for the starting point of the algorithm.

Augmented Lagrangian Variational FormulationFor A Plastic-hardening-damage Coupled Model

A model for elastic-plastic hardening materials with coupled damage based on internal variables is described. Each kinematic variable is partitioned in the sum of a reversible and an irreversible part. The first enters as argument of the free energy, the other enters as argument of a dissipation functional D. that generates a generalised elastic domain K for the thermodynamic variables as the level set at 1 of the dissipation function. Following a general procedure for multivocal potentials, a family of variational principles can be obtained, whose stationary points represent the solution of the structural problem. 1. Aims and Limitations of the work It is presented a new phenomenological model for a class of elasticplastic-damaging materials, based on an internal variable description of the modifications of the microstructure. No attempt is made to give any physical interpretation for the internal variables. A consistent thermodynamic development of the basic equations is presented, based on the introduction of two functional ruling the reversible and the irreversible phenomena. Mild regularity hypotheses are made on these functional, that guarantee the mathematical treatment of the equations with the tools of convex analysis. The most significant point of the model is the consistent derivation of a (convex) elastic region from a dissipation functional of the kinematic variables (internal and external). In this way it is obtained a single elastic domain in the extended space of stresses and thermodynamic forces dual of the hardening and damage internal variables -, for which Transactions on Engineering Sciences vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533 98 Damage and Fracture Mechanics normality rule holds naturally. According to the choice of the dissipation functional, and therefore of the generalised elastic domain, coupling between plasticity and damage can be easily modelled. Various kinds of behaviour can be recovered, as the decrease of the yield strength or the degradation of the hardening modulus. The different damages are clearly distinguished in the model according to whether they influence the reversible or the irreversible part of the constitutive equations. Principal aim of the model is to use it in developing variational principles amenable for simple numerical treatment. It is shown that it is possible to obtain a series of energy functional as straightforward generalisation of those valid in the field of plasticity with internal variables. Particular attention is devoted to mixed principles, that the authors have shown in previous works to be particularly useful in the numerical analysis (Cuomo ). The paper will present the main features of the model, while the consequences of particular choices of the form of the governing potentials will not be discussed. The identification of the model parameters is not faced in the present work 2. State variables and ambient spaces Kinematic internal variables are introduced for describing both material hardening and damage, defined in suitable vector spaces. Conjugate thermodynamic variables are defined in adjoint spaces. Each kinematic variables is partitioned as the sum of a reversible and an irreversible part. The first enter as argument in the Helmoltz free energy F, that rules the generalised elastic (reversible) behaviour. The irreversible parts enter as arguments in a dissipation functional D, that rules the dissipating phenomena. Schematically it is: £ = £^,+£ E%> deformation space a = «^+cx^ e9 internal strain space co = (0 v (4) Linearity assumptions have been introduced, so small deformation only are considered. The operator C accounts for the possibility that internal microdeformations result into overall deformations, Owen; v is an external damage potential. In most of the classic works on damage and hardening it is assumed that d, C? reduce to the null operator. From the virtual power equality the equilibrium conditions follow in the usual way, Cc+C|'x=/ Ci'C = f (5) The second equation is non classical; in it t are external source of damage work. This contribution has been examined by Fremond 4. Constitutive equations 4.1 Free energy Many different forms of the free energy have been assumed in the literature. Following the pioneering works of Kachanov it is often introduced a damaging effect on the stored elastic energy of the type Transactions on Engineering Sciences vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533

STABLE RANK OF THE C*-ALGEBRAS OF NILPOTENT LIE GROUPS

The stable rank of the C*-algebras of simply connected nilpotent Lie groups is determined by the dimension of the fixed point subspaces of the real adjoint spaces of the Lie algebras under the coadjoint actions, which is a generalization of the result obtained by A. J-L. Sheu. This equality is no longer affirmative in the case of non-nilpotent Lie groups.

An identification method of nonlinear systems by volterra series

AbstractNumerical models whose input‐output relations are expressed by Volterra series, are used to identify nonlinear systems from observed set of inputs and outputs according to given evaluation criterion. Parameters of systems to be identified are Volterra kernels of various orders. Since these kernels are multivariable functions, considerable calculation and memory capacities are required for identification. This paper presents a method without Volterra series which are multivariable functions for identification and output estimation. Our identification method follows the report by Hsieh; the evaluation function is expressed using inner products in Hilbert space, employing operators from fixed input and the Hilbert space to which outputs belong. Next, the Volterra kernel of each order is regarded as a mapping from adjoint space, and the optimization is done for one variable function in adjoint space. The computing method is as follows: the problem in Hilbert space is replaced by a problem in a finite‐dimensional vector space by discretizing the time axis, the orthogonal vector system is obtained, and the optimization by one‐dimensional time sequence is done. Afterward, it is shown that the output estimation can be done for other inputs without obtaining Volterra kernels which are multivariable functions, by using the adjoint solution obtained as described in the foregoing, and constructing its computer algorithm.

Adjoint Versus Forward Solution of Anisotropic Transport Problems

The sensitivity of the accuracy of the forward space and adjoint space calculation of a detector response in a source driven system to the order of the angular quadrature used is investigated. It i...

Partial widths and resonance normalization.

Several methods of obtaining properly normalized resonance wave functions and partial widths in multichannel resonance problems are investigated and explicit relationships between these methods are demonstrated. In particular, the normalizations studied include those based on the definition of a resonance state in terms of the poles of the total Green's function or T matrix, methods based on a study of adjoint resonance states in an adjoint space, normalizations which define formal partial widths which sum precisely to the total width, and the simple branching-ratio method. All methods are applied to a model problem consisting of two open channels and one closed channel.

THE STATIC AXIALLY SYMMETRIC SELF-DUAL YANG-MILLS FIELDS AND SURFACES OF NEGATIVE CURVATURE

An explicit geometric picture about the complete integrability of the static axially symmetric self-dual Yang-Mills equation and the gravitational Ernst equation is presented. The corresponding soliton surfaces in adjoint space (3-dimensional Minkowski space) has negative variable curvature. The Riccati equation is also given, so that the integrability of the Bäcklund transformation gets the confirmation.

Points of smoothness of convex bodies in a separable Banach space

exists for each y ~ X. This condition is equivalent to the condition that a unique support functional f(x) = llfll = 1 (f ~ X*) exists at x. By the Mazur theorem, the points of smoothness of the sphere of each separable Banach space form a dense G6-subset of S (see [i]). Moreover, each separable Banach space admits an equivalent norm, in which each point of the new unit sphere is a point of smoothness [2, p. 25]. A point x ~ S will be called a fixed point of smoothness of S if it is a point of smoothness of the unit sphere SX** of the second adjoint space X** under the canonical embedding of X in X**. In addition, the norm is also said to be very smooth at the point x ~ S (see [3] and [2, p. 30]). It is clear that if a space is reflexive, then each point of smoothness of its unit sphere is fixed. If the norm is Frech~t-differentiable at a point x ~ S (i.e., the limit (i) exists uniformly with respect to all y ~ X), then x is a fixed point of smoothness (see [3] and [2, p. 30]). Each Banach space with separable adjoint admits an equivalent Frechet-differentiable norm [4], i.e., each point of the new unit sphere is a fixed point of smoothness in this norm. It may be asked as to howmany fixed points of smoothness convex bounded centrally symmetric bodies (unit balls in equivalent norms) of a separable Banach space have? A verbatim repetition of Mazur's arguments (see [i]) in relation to the space X** shows that if the space X has separable second adjoint, then the fixed points of smoothness of its unit sphere form a dense G6-subset of S. However the following theorem is valid.

Self-Dual Yang-Mills Fields in Static Axially Symmetric Case and Related Topics

In this article we will present an explicit geometric picture about the complete integrability of the static axially symmetric SDYM equation and the gravitational Ernst equation, interpret the correspondence between their Bäcklund transformation formulae and the transformations from one focal surface of Weingarten congruence to the other, and give the matrix Riccati equation so that the integrability of the B.T. will be proved. It is shown that for the axially symmetric SDYM equation and gravitational Ernst equation the adjoint space of the group (SL(2r)) is a 3-dimensional Minkowski space, and the corresponding soliton surfaces have negative variable curvature. After introducing the generator R we can explain the B.T. as the rotation around the common tangent between two surfaces of solitons. Using Riccati equation we will confirm in this paper the integrability of B.T. and prove that the B.T. is strong, i.e., the new and old solutions satisfy equations of motion separately. Some related topics are also discussed.

On the approximate solution of time-optimal control problems

In this paper we are concerned with the approximate solution of time-optimal control problems in a nonreflexive Banach SpaceE by sequences of similar problems in Banach spacesEn which are assumed to approximateE in a fairly general sense. The problems under consideration are such that the solution operator of the associated evolution equation is a strongly continuous holomorphic contraction semigroup and the class of controls is taken from the dual of the Phillips adjoint space with respect to the infinitesimal generator of that semigroup. The main object is to establish convergence of optimal controls, transition times and corresponding trajectories of the approximating control problems which can be done by means of some results from the theory of approximation of semigroups of operators. Finally, these abstract convergence results will be applied to time-optimal control problems arising from heat transfer and diffusion processes.

Design Sensitivity Analysis of Planar Mechanism and Machine Dynamics

A method of formulating and automatically integrating the equations of motion of quite general constrained dynamic systems is presented. Design sensitivity analysis is carried out using a state space adjoint variable method that has been employed extensively in optimal control and structural design optimization. Both dynamic analysis and design sensitivity analysis formulations are automated and numerical solution of state and adjoint differential equations are carried out using a stiff numerical integration method that treats mixed systems of differential and algebraic equations. A computer code that implements the method is applied to two numerical examples. The first example concerns a relatively simple slider-crank mechanism. The second example treats a more complex agricultural trip plow that undergoes intermittent motion.

Note on Multiple Objective Dynamic Programming

A technique for finding MINSUM and MINMAX solutions to multi-criteria decision problems, called Multi Objective Dynamic Programming, capable of handling a wide range of linear, nonlinear, deterministic and stochastic multi-criteria decision problems, is presented and illustrated. Multiple objectives are considered by defining an adjoint state space and solving a (N + 1) terminal optimisation problem. The method efficiently generates both individual (criterion) optima and multiple criteria solutions in a single pass. Sensitivity analysis on weights over the various objectives is easily performed.