necessary and sufficient small gain conditions for integral input to state stable systems a lyapunov perspective this paper is concerned with conditions for the stability of interconnected nonlinear systems consisting of integral input to state stable iiss systems with external inputs the treatment of iiss and input to state stable iss systems is unified both necessary conditions and sufficient conditions are investigated using a lyapunov formulation in the presence of model uncertainty this paper proves that for the stability of the interconnected system at least one subsystem is necessarily iss which is a stronger stability property in the set of iiss the necessity of a small gain type property is also demonstrated this paper proposes a common form of smooth lyapunov functions which can establish the iiss and the iss of the interconnection comprising iiss and iss subsystems whenever the small gain type condition is satisfied the result covers situations more general than the earlier study and removes technical conditions assumed in the previous literature global asymptotic stability is discussed as a special case

thermomechanically coupled sensitivity analysis and design optimization of functionally graded materials this paper presents a systematic numerical technique for performing sensitivity analysis of coupled thermomechanical problem of functionally graded materials fgms general formulations are presented based on finite element model by using the direct method and the adjoint method in the modeling of spatial variances of material properties the graded finite element method is employed to conduct the heat transfer analysis and structural analysis and their sensitivity analysis the design variables are the volume fractions of fgms constituents and structural shape parameters the design optimization model is then constructed and solved by the sequential linear programming slp numerical examples are presented to demonstrate the accuracy and the applicability of the present method

the index of an eisenstein ideal and multiplicity one mazur s fundamental work on the eisenstein ideal of prime level has a variety of arithmetic applications in this article we generalize some of his work to square free level more specifically we attempt to compute the index of an eisenstein ideal and the dimension of the mathfrak m torsion of the modular jacobian variety where mathfrak m is an eisenstein maximal ideal in many cases the dimension of the mathfrak m torsion is 2 in other words a multiplicity one theorem holds

brief stability of extremum seeking feedback for general nonlinear dynamic systems while the mainstream methods of adaptive control both linear and nonlinear deal only with regulation to known set points or reference trajectories in many applications the set point should be selected to achieve a maximum of an uncertain reference to output equilibrium map the techniques of the so called extremum control or self optimizing control developed for this problem in the 1950 1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment in this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block we allow the plant to be a general nonlinear dynamic system possibly non affine in control and open loop unstable whose reference to output equilibrium map has a maximum and whose equilibria are locally exponentially stabilizable

cuspidal class number of the tower of modular curves x1 npn we consider a cuspidal class number which is the order of a subgroup of the full cuspidal divisor class group of x1 npn with p nmid n and n 1 by studying the second generalized bernoulli numbers we obtain results similar to ones ferrero and washington in ann math 2 109 2 377 395 1979 washington in invent math 49 87 97 1978 about the relative class numbers of cyclotomic mathbb z p extension of an abelian number field

simultaneous optimization of shape and flow parameters for combined free and forced convection in vertical channels simultaneous optimization of shape and flow parameters is performed for a combined free and forced convection flow through vertical rectangular channels with moving walls the laminar flow is assumed to be fully developed in the axial direction the wall velocity the axial pressure gradient and the channel height in the transverse plane are taken as the optimization parameters the sensitivity expressions of both the objective function and the flow rate constraint of optimization are obtained in terms of the relevant physical variables as well as adjoint variables which satisfy additional p d e s all equations are discretized using the finite element method numerical results are provided for the present constrained optimization problem for various values of the problem parameters which include the moving wall segment size and the rayleigh number the results indicate that with increased rayleigh number the optimal values of the wall velocity and the axial pressure gradient are increased while the optimal value of the channel height is decreased general sensitivity expressions are also presented in the appendix which might be utilized for arbitrary boundary variations along with arbitrary optimization objectives in other investigations 1997 by john wiley sons ltd

shape sensitivity analysis and design optimization of linear thermoelastic solids in this study a numerical scheme is developed for shape sensitivity analysis and design optimization of linear quasistatic thermoelastic solids in this scheme the finite element method is used as the analyzer for analyzing stress temperature shape sensitivity and design velocity field based upon the method of material derivatives both the techniques of the direct differentiation method and the adjoint variable method are applied to derive the shape sensitivity equations the shape optimization formulations discussed here include boundary integrals of displacements and heat fluxes as well as domain integrals of stresses and areas numerical results show that the proposed scheme works well in terms of accuracy

ordinary representations and modular forms this paper establishes criteria for a representation gal q q gl2 qp to be modular in the sense that there is a newform f such that f fontaine and mazur have conjectured that is modular provided it is unramified outside of a finite set of primes irreducible has odd determinant and the restriction to a decomposition group dp at p satisfies certain conditions the following conjecture is a special case of fm95 ref 1 conjecture 3c

asymptotic models for optimizing the contour of multiply connected cross section of an elastic bar in torsion an asymptotic solution is obtained for the problem of maximizing the torsional rigidity of elastic multiply connected cylindrical bars for a given area of cross section the shapes of the inner contours of the multiply connected cross section are specified while the outer contour is determined as a result of the shape optimization we apply the method of matched asymptotic expansions to construct a first order asymptotic model the conditions for unique solvability of the asymptotic model have been established under some restrictions imposed on the location of the inner contours and their polarization matrices the economy achieved by optimization is estimated

design sensitivity analysis for nonlinear thermal systems abstract the lagrange multiplier method and convolution theory are used to formulate design sensitivities for nonlinear transient thermal systems variations of a general design functional are expressed in explicit form with respect to variations in structural shape and the conventional design quantities i e material properties applied loads prescribed boundary conditions and initial conditions the design functional depends on these explicit quantities and the implicit repsonse fields temperature temperature gradient heat flux vector and reaction surface flux a finite element implementation of the methodology illustrates the formulation

extremum seeking based adaptive control for electromagnetic actuators in this paper we present a learning based adaptive method to solve the problem of robust trajectory tracking for electromagnetic actuators we merge a nonlinear backstepping controller that ensures bounded input bounded states stability with a multi variable extremum seeking model free learning algorithm the learning algorithm is used to estimate online the uncertain parameters of the model in this sense we propose a learning based adaptive controller we present a proof of stability of this learning based nonlinear controller when considering uncertainties with linear parametrisation the efficiency of this approach is shown on a numerical example

variational approach by means of adjoint systems to structural optimization and sensitivity analysis ii structure shape variation abstract for a linear elastic structure the first variation of an arbitrary stress strain and displacement functionals corresponding to variation of shape of external boundaries or interfaces is derived by using the solutions for primary and adjoint systems the application to optimal design is next presented and the relevant optimally conditions are derived from general expressions the path independent integrals used in fracture mechanics are rederived as a particular case of general expressions

comparison between the variational and implicit differentiation approaches to shape design sensitivities the most commonly used approach to the shape design sensitivity problem results from the implicit differentiation of the discretized equilibrium equations the most general implementation of this technique requires that finite differences be used to differentiate the element stiffness matrices proper choice of the step size is necessary to obtain high levels of accuracy and to avoid round off errors furthermore since it is necessary to operate on the element matrices this method is difficult to implement into a general purpose finite element program a more recent shape design sensitivity formulation based upon variational calculus avoids having to differentiate the discretized equations and results in an analytical expression for the derivative the approach is based upon the total derivative of the variational state equation and uses an adjoint variable technique for design sensitivity analysis only structural response data on the boundary of the structure is necessary thereby making implementation into a general purpose program less difficult this paper attempts to compare the two different techniques and point out the similarities two test problems are shown to demonstrate the accuracy of the variational approach

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

deformations of certain reducible galois representations ii we examine the deformation theory of two dimensional mod p reducible indecomposable galois representations removing or relaxing many of the hypotheses of part i we are able to prescribe local conditions on our deformations by allowing ramification at a set of primes congruent to 1 mod p not 1 mod p 2 and satisfying some other splitting conditions the new hypotheses admit irreducible ordinary characteristic zero lifts of many mod p representations which by skinner wiles allows us to conclude the modularity of such representations

optimal arrangement of holes in a two dimensional heat conductor by a special boundary integral method the position size and surface temperature of circular holes inside a two dimensional heat conductor are optimized to produce a minimum variation in surface temperature over a portion of the outer boundary this problem whic arises in thermal desing of moulds and dies resembles those encountered in structural shape optimization because the internal geometry of the heat conductor depends on the design variables in this paper some of the traditional difficulties associated with shape optimization are overcome by analysing steady heat conduction with a special boundary integral method developed for two dimensional regions with circular hole this approach eliminates the need to regenerate a finite element mesh over the interior of the region each time the geometry is changed during the design process it also increases the efficiency of the analysis by reducing the number of unknowns in the numerical discretization of the region since the objective function depends only on the boundary temperatures there is no need to determine temperatures in the interior the analysis method is applied to two problems arising in optimal thermal design of compression moulds these examples show that the number of holes choson for the design strongly affects their resulting optimal arrangement as well as the ultimate uniformity of the cavity surface temperature

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

optimal aerodynamic design of airfoils in unsteady viscous flows abstract a continuous adjoint formulation is used to determine optimal airfoil shapes in unsteady viscous flows at re 1 10 4 the reynolds number is based on the free stream speed and the chord length of the airfoil a finite element method based on streamline upwind petrov galerkin supg and pressure stabilized petrov galerkin pspg stabilizations is used to solve both the flow and adjoint equations the airfoil is parametrized via a non uniform rational b splines nurbs curve three different objective functions are used to obtain optimal shapes maximize lift minimize drag and minimize ratio of drag to lift the objective functions are formulated on the basis of time averaged aerodynamic coefficients the three objective functions result in diverse airfoil geometries the resulting airfoils are thin with the largest thickness to chord ratio being only 5 4 the shapes obtained are further investigated for their aerodynamic performance maximization of time averaged lift leads to an airfoil that produces more than six times more lift compared to the naca 0012 airfoil the excess lift is a consequence of the large peak and extended region of high suction on the upper surface and high pressure on the lower surface minimization of drag results in an airfoil with a sharp leading edge the flow remains attached for close to 70 of the chord length minimization of the ratio of drag to lift results in an airfoil with a shallow dimple on the upper surface it leads to a fairly large value of the time averaged ratio of lift to drag 17 8 the high value is mostly achieved by a 447 increase in lift and 16 reduction in drag compared to a naca 0012 airfoil imposition of volume constraint for the cases studied is found to result in airfoils that have lower aerodynamic performance

variational approach by means of adjoint systems to structural optimization and sensitivity analysis i variation of material parameters within fixed domain abstract for a linear elastic structure the first variation of an arbitrary stress strain and displacement functionals corresponding to variation of material parameters within specified domain is derived by using the solution for primary and adjoint systems this variation is of fundamental importance in sensitivity analysis optimal design and identification problems simple examples of optimal stiffness design and identification of stiffness parameters in beams are presented

shape design sensitivity analysis via material derivative adjoint variable technique for 3 d and 2 d curved boundary elements a general approach to shape design sensitivity analysis of three and two dimensional elastic solid objects is developed using the material derivative adjoint variable technique and boundary element method the formulation of the problem is general and first order sensitivities in the form of boundary integrals for the effect of boundary shape variations are derived for an arbitrary performance functional second order quadrilateral surface elements for 3 d problems and quadratic boundary elements for 2 d problems are employed in the solution of primary and adjoint systems and discretization of the boundary integral expressions for sensitivities the accuracy of sensitivity information is studied for selected global performance functionals and also for boundary state fields at discrete points numerical results are presented to demonstrate the accuracy and efficiency of this approach

numerical methods for second order shape sensitivity analysis with applications to heat conduction problems in this paper two finite element based schemes for second order shape sensitivity analysis are presented in the first formulation the av dd method the first order shape sensitivity equation is derived and expressed in terms of state and adjoint variables the resultant equation is then directly differentiated to obtain the second order shape sensitivity equation in the second formulation the dd av method the functional of concern is differentiated twice to yield the second order sensitivity equation in which the second order shape derivatives can be eliminated by introducing a proper adjoint equation a thermal fin problem and a thermal insulation layer problem have been studied to validate the proposed schemes it is shown that both methods yield identical results though the dd av method is computationally more efficient

an application of trefftz method to the sensitivity analysis of two dimensional potential problem this paper presents a new formulation for the sensitivity analysis with the help of the trefftz boundary type solution procedure direct differentiation of the expressions of the field variables leads to the expressions of the sensitivities since the field variables are approximated by linear combination of regular t complete functions the analytical expressions of the sensitivities are also regular sensitivity analysis schemes with respect to variations of dimensions of the object and boundary conditions are explained in detail two dimensional potential problems in the bounded region are considered as a numerical example in order to confirm its validity

optimisation of the torsional rigidity of axisymmetric hollow shafts abstract the torsional rigidity of an elastic axisymmetric hollow shaft of prescribed length and internal cross section is maximised by appeal to the calculus of variations this optimisation is with respect to variation of the outer curved surface of the shaft subject to the isoperimetric constraint that its volume remains fixed a regular perturbation solution for the case of small shaft thickness is presented

the importance of diagonal dominance in the iterative solution of equations generated from the boundary element method the unsymmetric matrix equations generated from the boundary element method bem can be solved iteratively with convergence to the correct solution guaranteed if the boundary element system of equations can be first transformed into an equivalent diagonally dominant system a transformation is presented which selectively annihilates terms in the coefficient matrix of the system ax b until an equivalent row diagonally dominant system if available is obtained the new row diagonally dominant system is well suited for use with jacobi and gauss seidel point iterative equation solvers the diagonal dominizing transformation presented in this work is not suitable for large systems of equations but is useful as a research tool for studying the importance of diagonal dominance in the iterative solution of equations generated from the bem a simple laplacian problem is used to examine the structure of the bem equations and to introduce the diagonal dominizing transformation the importance of diagonal dominance is shown by comparing iterative convergence of positive definite symmetric positive definite and diagonally dominant systems of bem equations obtained from a plane strain elasticity problem

tangent operators sensitivity expressions and optimal design of non linear elastica in contact with applications to beams the tangent operator and design sensitivity expressions for non linear elastica subject to frictionless unilateral contact are derived and computed via the finite element method the sensitivity computations are then combined with a numerical programming package to create an optimal design environment to exemplify the optimal design environment a beam s contact surface is contoured to minimize bending stress this work combines the research on the sensitivity analysis of non linear elastic bodies subject to constraints finite strain non linear elastic beam analysis and contact analyses the analysis is valid for any smooth contact surface and specialized for the case in which the surface is represented by a cubic spline the direct differentiation method is utilized to perform the sensitivity analysis in an example problem the sensitivity analysis is verified by finite difference computations and then combined with a numerical optimization program to design the stop profile of a valve

an adjoint method for shape optimization in unsteady viscous flows a new method for shape optimization for unsteady viscous flows is presented it is based on the continuous adjoint approach using a time accurate method and is capable of handling both inverse and direct objective functions the objective function is minimized or maximized subject to the satisfaction of flow equations the shape of the body is parametrized via a non uniform rational b splines nurbs curve and is updated by using the gradients obtained from solving the flow and adjoint equations a finite element method based on streamline upwind petrov galerkin supg and pressure stabilized petrov galerkin pspg stabilization techniques is used to solve both the flow and adjoint equations the method has been implemented and tested for the design of airfoils based on enhancing its time averaged aerodynamic coefficients interesting shapes are obtained especially when the objective is to produce high performance airfoils the effect of the extent of the window of time integration of flow and adjoint equations on the design process is studied it is found that when the window of time integration is insufficient the gradients are most likely to be erroneous

variational principle for shape design sensitivity analysis in this paper the adjoint method of design sensitivity analysis is stated as a general variational principle that is applicable to a wide range of problems the principle gives a very simple and straightforward method of obtaining the design sensitivity expression for a functional dependent on the state fields the sensitivity expression involves certain adjoint fields and explicit design variations of the functional and the governing equations for the primary state fields the principle is stated in terms of an augmented functional that is defined by adding to the response functional whose sensitivity is desired the equilibrium equation of the primary problem then explicit design variation of the augmented functional gives the total design variation of the response functional stationarity of the augmented functional with respect to the state fields defines the adjoint problem for the response functional the principle is proved and applied to several classes of linear and nonlinear problems such as field problems structural problems and dynamic response problems it is shown that the design sensitivity expressions derived in the literature using long procedures are obtained quite routinely by use of the principle

strong modularity of reducible galois representations in this paper we call strongly modular those reducible semi simple odd mod l galois representations for which the conclusion of the strongest form of serre s original modularity conjecture holds under the assumption that the serre weight k satisfies l k 1 we give a precise characterization of strongly modular representations hence generalizing a classical theorem of ribet pertaining to the case of conductor 1 when the representation rho is not strongly modular we give a necessary and sufficient condition on the primes p not dividing nl for which it arises in level np where n denotes the conductor of rho this generalizes a result of mazur on the case n k 1 2

optimisation of homogeneous thermal insulation layers abstract the rate of heat loss from a body surrounded by a layer of homogeneous insulation is minimised by use of the calculus of variations this optimisation is with respect to variation of the outer surface of the insulation layer subject to the isoperimetric constraint that its volume remains fixed both dirichlet and mixed boundary conditions for the temperature field on the outer surface are considered regular perturbation solutions in the case of small layer thickness are presented

computational method for optimization of structural shapes in the direct method of shape optimum design mesh regridding is a necessary step in avoiding the distortion of finite element grids the effect of internal nodal movements on the shape design sensitivity associated with the regridding process has been carefully examined in this paper it shows that to maintain computational accuracy the effects of internal nodal movement should be considered in shape design sensitivity analysis to take the internal nodal movement into account the numerical accuracy of the speed method is investigated a new approach which constructs the velocity field over the element domain is presented although it takes more computational time the new approach does improve the accuracy of shape design sensitivity

accuracy of the domain material derivative approach to shape design sensitivities numerical accuracy for the boundary and domain methods of the material derivative approach to shape design sensitivities is investigated through the use of mesh refinement the results show that the domain method is generally more accurate than the boundary method using the finite element technique it is also shown that the domain method is equivalent under certain assumptions to the direct differentiation approach that evaluates sensitivities by directdifferentiating the implicit discretized equilibrium equation not only theoretically but also numerically

sensitivity analysis for non linear constrained elastostatic systems adjoint and direct differentiation methods are used to formulate design sensitivities for non linear constrained elastostatic systems variations of a general response functional are expressed in explicit form with respect to all design field variations including shape the response functional is expressed over the undeformed configuration however the displacement gradient is included so that field quantities that are defined over the deformed configuration such as cauchy stress may be modelled the finite element method is used to evaluate the sensitivities in an example problem

a non gradient approach to global extremum seeking an adaptation of the shubert algorithm the main purpose of this paper is to adapt the so called shubert algorithm for extremum seeking control of general dynamic plants this algorithm is a good representative of the sampling optimization methods that achieve global extremum seeking on compact sets in the presence of local extrema the algorithm applies to lipschitz mappings the model of the system is assumed unknown but the knowledge of its lipschitz constant is assumed the controller depends on a design parameter the waiting time and tuning guidelines that relate the design parameter and the region of convergence and accuracy of the algorithm are presented the analysis shows that semi global practical convergence in the initial states to the global extremum can be achieved in presence of local extrema if compact sets of inputs are considered numerical simulations for global optimization in the presence of local extrema are provided to demonstrate the proposed approach

modularity of residual galois extensions and the eisenstein ideal for a totally real field f a finite extension f of fp and a galois character gf f unramified away from a finite set of places p p consider the bloch kato selmer group h h1 f 1 in an earlier paper of the authors it was proved that the number d of isomorphism classes of non semisimple reducible residual representations giving rise to lines in h which are modular by some f also unramified outside satisfies d n dimfh this was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible selmer group we show here that if in addition the relevant local eisenstein ideal j is non principal then d n when f q we prove the desired bounds on the congruence module and the selmer group we also formulate a congruence condition implying the non principality of j that can be checked in practice allowing us to furnish an example where d n

on deformation rings of residually reducible galois representations and r t theorems we introduce a new method of proof for r t theorems in the residually reducible case we study the crystalline universal deformation ring r and its ideal of reducibility i of a mod p galois representation 0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non isomorphic constituents 1 and 2 under some assumptions on selmer groups associated with 1 and 2 we show that r i is cyclic and often finite using ideas and results of but somewhat different assumptions from bellaiche and chenevier we prove that i is principal for essentially self dual representations and deduce statements about the structure of r using a new commutative algebra criterion we show that given enough information on the hecke side one gets an r t theorem we then apply the technique to modularity problems for 2 dimensional representations over an imaginary quadratic field and a 4 dimensional representation over q

comparison of shape design sensitivity analysis formulations via material derivative adjoint variable and implicit differentiation techniques for 3 d and 2 d curved boundary element this paper reviews and compares two different approaches to continuum structural shape design sensitivity analysis the material derivative adjoint variable method and the implicit differentiation method using 3 d and 2 d bem employing identical bea models curved element technology numerical integration and equation solving techniques the similarities and differences of both methods are highlighted significant theoretical and computational aspects associated with both methods are presented and accuracy and computational efficiency characteristics are demonstrated advantages and disadvantages of both techniques are identified the characterization provided can aid in the prudent selection between these sensitivity analysis techniques based on the characteristics of the proposed application