robust exact differentiation via sliding mode technique abstract the main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises the proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for lipschitz s constant of the derivative

barrier function based adaptive super twisting controller in this article a variable gain super twisting algorithm based on a barrier function is proposed for a class of first order disturbed systems with uncertain control coefficient and whose disturbances derivatives are bounded but the upper bounds of those derivatives are unknown the specific feature of this algorithm is that it can ensure the convergence of the output variable and maintain it in a predefined neighborhood of zero independent from the upper bound of the disturbances derivatives moreover thanks to the structure of the barrier function it forces the gain to decrease together with the output variable and the control signal follows the absolute value of the disturbances

non planar curvature and branching of arteries and non planar type flow in this study magnetic resonance imaging techniques have been used to examine the geometry of arterial curvature and branching in casts and in vivo and to measure the distribution of axial velocity in the associated flow it is found contrary to a widely held view that the geometry is commonly non planar moreover relatively small values of the parameters which render the geometry non planar appear significantly to affect the velocity distribution the findings suggest that non planarity is an important factor influencing arterial flows including wall shear the implications are not restricted to vascular biology pathology and surgery but may extend to the design of general piping systems

adaptation of levant s differentiator based on barrier function abstractan algorithm for adaptation of levant s differentiator ld gains is designed for the case when the upper bound of second derivative of base signal exists but it is unknown the barrier fun

finite element simulation of pulsatile flow through arterial stenosis the problem of blood flow through a stenosis is solved using the incompressible navier stokes equations in a rigid circular tube presenting a partial occlusion calculations are based on a galerkin finite element method the time marching scheme employs a predictor corrector technique using a variable time step results are obtained for steady and physiological pulsatile flows computational experiments analyse the effect of varying the degree of stenosis the stricture length the reynolds number and womersley number the method gives results which agree well with previous computations for steady flows and experimental findings for steady and pulsatile flows

on the shear thinning and viscoelastic effects of blood flow under various flow rates the aim of this paper is to describe and discuss the results of numerical comparative study performed in order to demonstrate and quantify some of the most relevant non newtonian characteristics of blood flow in medium sized blood vessels namely its shear thinning and viscoelastic behavior the models studied in this work are the classical newtonian and oldroyd b models as well as their generalized shear thinning modifications numerical tests are performed on three dimensional geometries namely an idealized axisymmetric stenosis and a realistic stenosed carotid bifurcation reconstructed from medical images the numerical solution of the system of governing equations is obtained by a finite volume method on a structured grid model sensitivity tests are achieved with respect to the characteristic flow rate to evaluate its impact on the observed non newtonian effects

numerical investigation of the non newtonian blood flow in a bifurcation model with a non planar branch the pulsatile flow of non newtonian fluid in a bifurcation model with a non planar daughter branch is investigated numerically by using the carreau yasuda model to take into account the shear thinning behavior of the analog blood fluid the objective of this study is to deal with the influence of the non newtonian property of fluid and of out of plane curvature in the non planar daughter vessel on wall shear stress wss oscillatory shear index osi and flow phenomena during the pulse cycle the non newtonian property in the daughter vessels induces a flattened axial velocity profile due to its shear thinning behavior the non planarity deflects flow from the inner wall of the vessel to the outer wall and changes the distribution of wss along the vessel in particular in systole phase downstream of the bifurcation the velocity profiles are shifted toward the flow divider and low wss and high shear stress temporal oscillations characterized by osi occur on the outer wall region of the daughter vessels close to the bifurcation secondary motions become stronger with the addition of the out of plane curvature induced by the bending of the vessel and the secondary flow patterns swirl along the non planar daughter vessel a significant difference between the non newtonian and the newtonian pulsatile flow is revealed during the pulse cycle however reasonable agreement between the non newtonian and the rescaled newtonian flow is found calculated results for the pulsatile flow support the view that the non planarity of blood vessels and the non newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology

adaptive continuous higher order sliding mode control this paper is concerned with the development of an adaptation structure which can be applied to conventional super twisting and higher order sliding mode schemes the objective is to alter the modulation gains associated with these schemes in such a way that they are as small as possible to mitigate chattering effects but large enough to ensure that sliding can be maintained in the presence of bounded and derivative bounded uncertainties in all the proposed schemes the equivalent control is used to drive the adaptive mechanism the approach is based on a novel dual layer nested adaptive methodology which is quite different to the existing schemes proposed in the sliding mode literature the new adaptive schemes do not require knowledge of the minimum and maximum allowed values of the adaptive gain and in their most general form do not require information about the bound on the disturbances and their derivatives

dynamics of coronary artery curvature obtained from biplane cineangiograms abstract a system was developed to track the centerlines axes of human coronary arteries during the cardiac cycle from biplane cineangiograms the system employs a time interpolation technique on one image plane and a smoothing technique on the other to create simultaneous image pairs the image pairs are then used to reconstruct the three dimensional axes of the vessels during the cardiac cycle once the vessel axes are reconstructed acceleration velocity and geometric parameters such as curvature can be obtained in the present work a sequence of human angiograms is used to obtain the axial and temporal variation of the curvature of a portion of the left anterior descending coronary artery distal to the third diagonal branch during two cardiac cycles for the case studied the change in curvature during the cycle ranged from 0 25 to 1 8 cm 1 depending upon the position of the site

on the potentialities of 3d 1d coupled models in hemodynamics simulations this work comprises a step towards the quantitative and qualitative analysis of coupled local and global hemodynamics phenomena in the arterial system the aim of this work is to present some numerical examples to put in evidence the importance of the use of 3d 1d coupled models in hemodynamics problems when carrying out simulations of rather complex situations accordingly some cases for which classical modeling cannot be applied are identified and solved the results obtained here allow us to assess some interrelations between local pointwise quantities defined at the level of the 3d model and global mean quantities defined at the level of the 1d model

a study of wall shear stress in 12 aneurysms with respect to different viscosity models and flow conditions recent computational fluid dynamics cfd studies relate abnormal blood flow to rupture of cerebral aneurysms however it is still debated how to model blood flow with sufficient accuracy common assumptions made include newtonian behaviour of blood traction free outlet boundary conditions and inlet boundary conditions based on available literature these assumptions are often required since the available patient specific data is usually restricted to the geometry of the aneurysm and the surrounding vasculature however the consequences of these assumptions have so far been inadequately addressed this study investigates the effects of 4 different viscosity models 2 different inflow conditions and 2 different outflow conditions in 12 middle cerebral artery aneurysms the differences are quantified in terms of 3 different wall shear stress wss metrics involving maximal wss average wss and proportion of aneurysm sac area with low wss the results were compared with common geometrical metrics such as volume aspect ratio size ratio and parent vessel diameter and classifications in terms of sex and aneurysm type the results demonstrate strong correlations between the different viscosity models and boundary conditions the correlation between the different wss metrics range from weak to medium no strong correlations were found between the different wss metrics and the geometrical metrics or classifications

outflow boundary conditions for three dimensional finite element modeling of blood flow and pressure in arteries flow and pressure waves emanate from the heart and travel through the major arteries where they are damped dispersed and reflected due to changes in vessel caliber tissue properties and branch points as a consequence solutions to the governing equations of blood flow in the large arteries are highly dependent on the outflow boundary conditions imposed to represent the vascular bed downstream of the modeled domain the most common outflow boundary conditions for three dimensional simulations of blood flow are prescribed constant pressure or traction and prescribed velocity profiles however in many simulations the flow distribution and pressure field in the modeled domain are unknown and cannot be prescribed at the outflow boundaries an alternative approach is to couple the solution at the outflow boundaries of the modeled domain with lumped parameter or one dimensional models of the downstream domain we previously described a new approach to prescribe outflow boundary conditions for simulations of blood flow based on the dirichlet to neumann and variational multiscale methods this approach termed the coupled multidomain method was successfully applied to solve the non linear one dimensional equations of blood flow with a variety of models of the downstream domain this paper describes the extension of this method to three dimensional finite element modeling of blood flow and pressure in the major arteries outflow boundary conditions are derived for any downstream domain where an explicit relationship of pressure as a function of flow rate or velocities can be obtained at the coupling interface we developed this method in the context of a stabilized semi discrete finite element method flow rate and pressure distributions are shown for different boundary conditions to illustrate the dramatic influence of alternative boundary conditions on these quantities

on the importance of blood rheology for bulk flow in hemodynamic models of the carotid bifurcation abstract here we present a study on the impact of assumptions on image based hemodynamic simulations of healthy carotid bifurcations in particular we evaluate to which extent assumptions on blood rheology influence bulk flow features driven by the fact that few studies have provided adequate insights into the influence of assumptions to confidently model the 4d hemodynamics within the bifurcation the final goal is to complement integrate and extend with a quantitative characterization of the bulk flow the description currently adopted to classify altered hemodynamics which is based on wall shear stress wss hemodynamic simulations of two image based carotid bifurcation geometries were carried out assuming a reference newtonian viscosity two non newtonian rheology models and newtonian viscosities based on characteristic shear rates wss based and lagrangian based metrics for helical flow quantification and for vorticity dynamics quantification were calculated our findings suggest that the assumption of newtonian rheology 1 could be reasonable for bulk flow metrics differences from non newtonian behavior are lower than 10 2 influences at different levels the wss based indicators depending on the bifurcation model even if in our study it is lower than the major source of uncertainty as recognized by the literature i e uncertainty on geometry reconstruction

the birkhoff theorem for unitary matrices of prime power dimension abstract the unitary birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity if the dimension n of the unitary matrix equals a power of a prime p i e if n p w then the birkhoff decomposition does not need all n possible permutation matrices as the epicirculant permutation matrices suffice this group of permutation matrices is isomorphic to the general affine group ga w p of order only p w p w 1 p w p p w p w 1 p w

dynamic curvature strongly affects wall shear rates in a coronary artery bifurcation model this study was motivated by the need for a better understanding of coronary artery blood flow patterns and their possible role in atherosclerosis formation of particular interest in this study was the effects of the dynamic deformation due to myocardial contraction on wall shear rate patterns in the coronary arteries a better understanding of these effects on wall shear rate in a bifurcation geometry and an evaluation of the importance of these effects was desired a three dimensional computer model of a bifurcation lying on the surface of a sphere with time varying radius of curvature was employed to simulate the motion and deformation of the arteries the results indicated low mean shear rates along the myocardial wall and very high shear rate variations over 100 of the static mean shear rate along the outer wall the results obtained using a quasi static analysis were found to underestimate the dynamic wall shear rate variation along the myocardial and outer walls it was concluded that dynamic geometry effects are important in determining sites of low mean and oscillating wall shear that have been associated with atherogenesis in curved bifurcating arteries

the influence of out of plane geometry on pulsatile flow within a distal end to side anastomosis we present an experimental and computational investigation of time varying flow in an idealized fully occluded 45 distal end to side anastomosis two geometric configurations are assessed one where the centerlines of host and bypass vessels lie within a plane and one where the bypass vessel is deformed out of the plane of symmetry respectively termed planar and non planar flow experiments were conducted by magnetic resonance imaging in rigid wall models and computations were performed using a high order spectral hp algorithm results indicate a significant change in the spatial distribution of wall shear stress and a reduction of the time averaged peak wall shear stress magnitude by 10 in the non planar model as compared to the planar configuration in the planar geometry the stagnation point follows a straight line path along the host artery bed with a path length of 0 8 diameters by contrast in the non planar case the stagnation point oscillates about a center that is located off the symmetry plane intersection with the host artery bed wall and follows a parabolic path with a 0 7 diameter longitudinal and 0 5 diameter transverse excursion a definition of the oscillatory shear index osi is introduced that varies between 0 and 0 5 and that accounts for a continuous range of wall shear stress vector angles in both models regions of elevated oscillatory shear were spatially associated with regions of separated or oscillating stagnation point flow the mean oscillatory shear magnitude considering sites where osi 0 1 in the non planar geometry was reduced by 22 as compared to the planar configuration these changes in the dynamic behavior of the stagnation point and the oscillatory shear distribution introduced by out of plane graft curvature may influence the localization of vessel wall sites exposed to physiologically unfavorable flow conditions

sinkhorn normal form for unitary matrices abstract sinkhorn proved that every entry wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices in this note we prove a recently conjectured analogue for unitary matrices every unitary can be decomposed into two diagonal unitaries and one whose row and column sums are equal to one the proof is non constructive and based on a reformulation in terms of symplectic topology as a corollary we obtain a decomposition of unitary matrices into an interlaced product of unitary diagonal matrices and discrete fourier transformations this provides a new decomposition of linear optics arrays into phase shifters and canonical multiports described by fourier transformations

the influence of the non newtonian properties of blood on the flow in large arteries unsteady flow in a 90 curved tube a numerical and experimental investigation of unsteady entry ow in a 903 curved tube is presented to study the impact of the non newtonian properties of blood on the velocity distribution the time dependent ow rate for the newtonian and the nonnewtonian blood analog uid were identical for the numerical computation a carreau yasuda model was employed to accommodate the shear thinning behavior of the xanthan gum solution the viscoelastic properties were not taken into account the experimental results indicate that signi cant di erences between the newtonian and non newtonian uid are present the numerical results for both the newtonian and the non newtonian uid agree well with the experimental results since viscoelasticity was not included in the numerical code shear thinning behavior of the blood analog uid seems to be the dominant non newtonian property even under unsteady ow conditions finally a comparison between the non newtonian uid model and a newtonian uid at a rescaled reynolds number is presented the rescaled reynolds number based on a characteristic rather than the high shear rate viscosity of the xanthan gum solution was about three times as low as the original reynolds number comparison reveals that the character of ow of the non newtonian uid is simulated quite well by using the appropriate reynolds number 1999 elsevier science ltd all rights reserved

an efficient approach to study the pulsatile blood flow in femoral and coronary arteries by differential quadrature method abstract in this paper flow analysis for a non newtonian third grade blood in coronary and femoral arteries is simulated numerically blood is considered as the third grade non newtonian fluid under periodic body acceleration motion and pulsatile pressure gradient differential quadrature method dqm and crank nicholson method cnm are used to solve the partial differential equation pde governing equation by which a good agreement between them was observed in the results the influences of some physical parameters such as amplitude lead angle and body acceleration frequency on non dimensional velocity and profiles are considered for instance the results show that increasing the amplitude a g and reducing the lead angle of body acceleration make higher velocity profiles in the center line of both arteries

a unified variational approach for coupling 3d 1d models and its blood flow applications in this work we propose an extended variational formulation in order to handle the problem of the flow of an incompressible fluid in compliant vessels with discontinuous fields this approach makes possible to tackle in a consistent variational manner problems like the coupling between models of different dimensionality the main ideas of the work are presented for a simple heat transfer problem and then are used in the context of modelling blood flow in large vessels a computational implementation of the proposed formulation is also shown and discussed by presenting examples of practical applications

helical flow in carotid bifurcation as surrogate marker of exposure to disturbed shear abstract the main purpose of the study is to verify if helical flow might be a surrogate marker of the exposure to disturbed shear in normal carotid bifurcations based on hemodynamic simulations data of 50 carotid bifurcations we defined appropriate helicity based bulk flow descriptors and multivariate linear regression analysis was performed to assess the potential for a combination of helicity based indicators in describing the exposure to disturbed shear to select the optimal combination of helicity based descriptors in the regression model the akaike information criterion was applied the identified statistical model is composed of two bulk flow descriptors proposed here to quantify helicity intensity and the balance between counter rotating helical flow patterns in the flow field the model revealed the existence of a significant relationship with adjusted squared pearson s correlation coefficient in the range 0 4 0 7 p

numerical treatment of defective boundary conditions for the navier stokes equations we present a formulation for accommodating defective boundary conditions for the incompressible navier stokes equations where only averaged values are prescribed on measurable portions of the boundary in particular we consider the case where the flow rate is imposed on several domain sections this methodology has an interesting application in the numerical simulation of flow in blood vessels when only a reduced set of boundary data are generally available for the upstream and downstream sections

pulsatile flow of non newtonian fluids through arterial stenoses the problem of blood flow through stenoses is solved using the incompressible generalized newtonian model the herschel bulkley bingham and power law fluids are incorporated the geometry corresponds to a rigid circular tube with a partial occlusion calculations are performed by a galerkin finite element method for the pulsatile case a predictor corrector time marching scheme is used with an adaptive time step results are obtained for steady and pulsatile physiological flows computations show that the memory effects taken into account in the model affect deeply the flow compared with newtonian reference case the disturbances are stronger by their vorticity intensity and persist after the geometrical obstacle this is especially true for severe stenoses

multidimensional modelling for the carotid artery blood flow abstract in this work a multidimensional 3d 1d fem model of the whole arterial tree is implemented it comprises a 3d compliant model of the carotid bifurcation coupled with a 1d model for the remaining part of the arterial tree with this approach difficulties arising from the treatment of boundary conditions for the 3d model are naturally handled the navier stokes equations are used as the governing equations for the blood flow while an elastic compliant model is implemented for the arterial wall also the a l e formulation is considered within the 3d blood regions taking into account the domain deformations produced by the moving artery wall this 3d model is complemented with a 1d model of the entire arterial tree in order to appropriately set inflow and outflow boundary conditions for the former the reduced 1d model solves the momentum and continuity equations in compliant tubes so as to reproduce the propagation of the pressure pulse in the arterial network also a volumetric flow rate is imposed at the inlet to model the systolic work of the heart the peripheral arteriole beds are simulated with the well known lumped windkessel model a standard geometry of the carotid bifurcation is discretized with p1 bubble p1 tetrahedral elements the obtained results properly reproduce the general flow patterns reported in the literature very good agreement between the outcomes of a pure 1d model and those of the combined multidimensional model was obtained it is worth noting that this kind of model may provide useful information to gain understanding in the genesis and development of arterial diseases

control of nonlinear systems with partial state constraints using a barrier lyapunov function this article addresses the problem of control design for strict feedback systems with constraints on the states to prevent the states from violating the constraints we employ a barrier lyapunov function blf which grows to infinity whenever its arguments approaches some finite limits based on blf based backstepping we show that asymptotic output tracking is achieved without violation of any constraint provided that the initial states and control parameters are feasible we also establish sufficient conditions to ensure feasibility which can be checked offline without precise knowledge of the initial states the feasibility conditions are relaxed when handling the partial state constraint problem as compared to the full state constraint problem in the presence of parametric uncertainties blf based adaptive backstepping is useful in preventing the states from transgressing the constrained region during the transient stages of online parameter adaptation to relax the feasibility conditions asymme

assessment of blood volume flow in slightly curved arteries from a single velocity profile abstract non invasive estimation of arterial blood volume flow bvf has become a central issue in assessment of cardiovascular risk poiseuille and womersley approaches are commonly used to assess the bvf from centerline velocity but both methods neglect the influence of curvature based on the assumption that the velocity in curved tubes as function of the circumferential position for a given radial position can be approximated by a cosine the bvf can also be estimated by averaging velocities at opposite radial positions referred to as the cosine model ctm this study investigates the accuracy of bvf estimation in slightly curved arteries for bvf waveforms obtained in the brachial artery of 6 volunteers computational fluid dynamics simulations were used to compute the influence of curvature on velocity profiles the bvf was then estimated from the simulation results with the ctm and methods based on poiseuille womersley and using the center stream velocity and the velocity waveform at the position where the maximum velocity is observed and compared to the prescribed bvf the simulations show that the influence of curvature is strongest when the flow decelerates for poiseuille and womersley the time average bvf was underestimated by maximally 10 4 and 7 8 for a radius of curvature of 50 and 100 mm respectively the estimation error is lower for the ctm and equals 4 2 and 1 2 for a radius of curvature of 50 and 100 mm respectively from this study we can conclude that the velocity waveform at the position of the maximum rather than the center stream velocity waveform combined with the womersley method should be chosen the ctm improves current estimation techniques if in vivo velocity distributions are available

the birkhoff theorem for unitary matrices of arbitrary dimensions abstract it was shown recently that birkhoff s theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime we prove a generalization of the birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension

birkhoff s polytope and unistochastic matrices n 3 and n 4 the set of bistochastic or doubly stochastic n n matrices is a convex set called birkhoff s polytope which we describe in some detail our problem is to characterize the set of unistochastic matrices as a subset of birkhoff s polytope for n 3 we present fairly complete results for n 4 partial results are obtained an interesting difference between the two cases is that there is a ball of unistochastic matrices around the van der waerden matrix for n 3 while this is not the case for n 4

adaptive continuous twisting algorithm abstractin this paper an adaptive continuous twisting algorithm acta is presented for double integrator acta produces a continuous control signal ensuring finite time convergence of the states to zero moreover the control signal generated by acta compensates the lipschitz perturbation in finite time i e its value converges to the opposite value of the perturbation acta also keeps its convergence properties even in the case that the upper bound of the derivative of the perturbation exists but it is unknown

bending and twisting of an in vivo coronary artery at a bifurcation abstract dynamic changes in the geometric shape and dimensions of a left coronary artery tree were extracted from the computer tomographically reconstructed three dimensional images of an in situ beating heart of an anesthetized dog wireframe models of the left coronary artery tree at 16 different instants of a cardiac cycle were constructed for the study of its flexing motion for quantifying the local bending and twisting of the left coronary artery tree the anatomic landmarks of the bifurcation points are selected as focussed locations at these points the space curves of the tree at different cardiac instants were first derived in parametric forms curvature and torsion expressions are next obtained in terms of the derivatives with respect to the parameter this analysis revealed that during the initial contraction of the heart wall a 2 reduction per millisecond in the radius of curvature occurred near the bifurcation point where the left circumflex coronary artery descends toward the apex of the heart when the left ventricular chamber reached a maximum value the radius of curvature was found to decrease at a rate of 2 3 ms 1 at the end of diastole an increase in the radius of curvature at a rate of 5 7 ms 1 was observed the twisting rates per unit length of artery near the bifurcation point of the selected artery were found to range from 0 62 to 0 63 ms 1

hybrid element based approximation for the navier stokes equations in pipe like domains abstract motivated by its applicability to model blood flow in the cardiovascular system in this work we propose a numerical approximation of the navier stokes equations maintaining comparable accuracy with traditional finite element methods while performing a substantial reduction in the problem size the method is envisaged for domains in which flow phenomena exhibits a dominant axial direction the strategy consists of combining a finite element approximation for the description of fields velocity components and pressure in the primary axial direction while taking a polynomial spectral approximation for the description of fields across the secondary transversal direction in essence this method can be understood as an extension of the so called hierarchical modeling himod approach special attention is given to domains featuring sudden constrictions expansions and bifurcations for which a modified spectral approximation is considered the capabilities and potentialities of the proposed approach to simulate fluid flow phenomena in pipe like domains are presented through several examples

a multidimensional phase locked loop for blind multiuser detection this paper concerns the problem of blind multiuser detection a special case of the blind source separation problem in which the source signals have finite alphabets specifically we address the problem of identifying and resolving the n spl times n unitary matrix ambiguity u that results from whitening the receiver observations where n is the number of sources we propose the multidimensional phase locked loop mpll as a generalization of a scalar decision directed pll to vector valued signals the mpll adapts an estimate of u according to the recursion u spl circ sub k 1 u spl circ sub k r sub k where r sub k is an n dimensional householder like rotation depending on only the kth receiver observation the o n sup 2 complexity of an efficient implementation of the algorithm is extremely low nevertheless simulation results demonstrate good convergence properties and superior steady state performance when compared with prior techniques the algorithm is also able to accommodate large alphabets and shaped alphabets

the nonlinear dynamics of elastic tubes conveying a fluid abstract the kirchhoff equations for elastic tubes are modified to include the effect of fluid flow using the techniques of linear and nonlinear analysis specially developed for the kirchhoff equations the effect of the fluid flow on the basic twist to writhe instability is investigated the results suggest an intriguing modification of the bifurcation threshold due to the flow beyond threshold the buckled tube acquires a slight curvature which modifies the flow rate and results in a correction to nonlinearity of the amplitude equation governing the deformation dynamics

the decomposition of an arbitrary 2w 2w unitary matrix into signed permutation matrices abstract birkhoff s theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices a similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices unitary matrices of dimension equal to a power of 2 say 2 w deserve special attention as they represent quantum qubit circuits we investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix it turns out to be a matrix group isomorphic to the extraspecial group e 2 2 w 1 of order 2 2 w 1 an associated projective group of order 2 2 w equally suffices

the birkhoff theorem for unitary matrices of prime dimension abstract the birkhoff s theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners we prove that any unitary matrix with equal line sums can also be written as a sum of permutation matrices with sum of weights equal 1 furthermore when the matrix dimension is prime we prove that the unitary matrix lies inside a convex complex birkhoff polytope

barrier lyapunov functions for the control of output constrained nonlinear systems in this paper we present control designs for single input single output siso nonlinear systems in strict feedback form with an output constraint to prevent constraint violation we employ a barrier lyapunov function which grows to infinity when its arguments approach some limits by ensuring boundedness of the barrier lyapunov function in the closed loop we ensure that those limits are not transgressed besides the nominal case where full knowledge of the plant is available we also tackle scenarios wherein parametric uncertainties are present asymptotic tracking is achieved without violation of the constraint and all closed loop signals remain bounded under a mild condition on the initial output furthermore we explore the use of an asymmetric barrier lyapunov function as a generalized approach that relaxes the requirements on the initial conditions we also compare our control with one that is based on a quadratic lyapunov function and we show that our control requires less restrictive initial conditions a numerical example is provided to illustrate the performance of the proposed control