some majorization inequalities in euclidean jordan algebras abstract in this paper we prove various eigenvalue and trace inequalities of objects in the setting of simple euclidean jordan algebras via majorization techniques

the complexity of fault detection problems for combinational logic circuits in this correspondence we analyze the computational complexity of fault detection problems for combinational circuits and propose an approach to design for testability although major fault detection problems have been known to be in general np complete they were proven for rather complex circuits in this correspondence we show that these are still np complete even for monotone circuits and thus for unate circuits we show that for k level k 3 monotone unate circuits these problems are still np complete but that these are solvable in polynomial time for 2 level monotone unate circuits a class of circuits for which these fault detection problems are solvable in polynomial time is presented ripple carry adders decoder circuits linear circuits etc belong to this class a design approach is also presented in which an arbitrary given circuit is changed to such an easily testable circuit by inserting a few additional test points

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

a minimal universal test set for self test of exor sum of products circuits a testable exor sum of products esop circuit realization and a simple universal test set which detects all single stuck at faults in the internal lines and the primary inputs outputs of the realization are given since esop is the most general form of and exor representations our realization and test set are more versatile than those described by other researchers for the restricted grm fprm and pprm forms of and exor circuits our circuit realization requires only two extra inputs for controllability and one extra output for observability the cardinality of our test set for an n input circuit is n 6 for built in self test bist applications we show that our test set can be generated internally as easily as a pseudorandom pattern and that it provides 100 percent single stuck at fault coverage in addition our test set requires a much shorter test cycle than a comparable pseudoexhaustive or pseudorandom test set

lowner s operator and spectral functions in euclidean jordan algebras we study analyticity differentiability and semismoothness of lowner s operator and spectral functions under the framework of euclidean jordan algebras in particular we show that many optimization related classical results in the symmetric matrix space can be generalized within this framework for example the metric projection operator over any symmetric cone defined in a euclidean jordan algebra is shown to be strongly semismooth the research also raises several open questions whose answers would be of strong interest for optimization research

positive and doubly stochastic maps and majorization in euclidean jordan algebras abstract a positive map between euclidean jordan algebras is a symmetric cone order preserving linear map we show that the norm of such a map is attained at the unit element thus obtaining an analog of the operator matrix theoretic russo dye theorem a doubly stochastic map between euclidean jordan algebras is a positive unital and trace preserving map we relate such maps to jordan algebra automorphisms and majorization

derivation of reduced test vectors for bit parallel multipliers over gf 2 m this paper presents an algebraic testing method for detecting stuck at faults in the polynomial basis pb bit parallel bp multiplier circuits over gf 2m the proposed technique derives the test vectors from the expressions of the inner product ip variables without any requirement of the atpg tool this low complexity testing method requires 2m 1 test vectors for detecting single stuck at faults in the and part and multiple stuck at faults in the exor part of the multiplier circuits the test vectors are independent of the multiplier s structure as proposed in t a gulliver et al 1991 but are dependent on m for the multiplier circuits the test set is found to be smaller in size than the atpg generated test set the test set provides 100 percent single stuck at fault coverage

commutation principle for variational problems on euclidean jordan algebras this paper establishes a commutation result for variational problems involving spectral sets and spectral functions the discussion takes places in the context of a general euclidean jordan algebra

commutation principles in euclidean jordan algebras and normal decomposition systems the commutation principle of ramirez seeger and sossa siam j optim 23 2013 pp 687 694 proved in the setting of euclidean jordan algebras says that when the sum of a frechet differentiable function theta and a spectral function f is minimized maximized over a spectral set omega any local minimizer respectively maximizer a operator commutes with the frechet derivative theta a in this paper we extend this result to sets and functions which are just invariant under algebra automorphisms we also consider a similar principle in the setting of normal decomposition systems

convex analysis on the hermitian matrices there is growing interest in optimization problems with real symmetric matrices as variables generally the matrix functions involved are spectral they depend only on the eigenvalues of the matrix it is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues a new approach to this characterization is given via a simple fenchel conjugacy formula we then apply this formula to derive expressions for subdifferentials and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables analogous results hold for hermitian matrices

strong convergence rate in averaging principle for stochastic fitzhugh nagumo system with two time scales abstract this article deals with averaging principle for stochastic fitzhugh nagumo system with different time scales under suitable conditions the existence of an averaging equation eliminating the fast variable for this coupled system is proved and as a consequence the system can be reduced to a single stochastic ordinary equation with a modified coefficient moreover the rate of convergence for the slow component towards the solution of the averaging equation is of order 1 2

easily testable realizations for generalized reed muller expressions this paper presents a design method of easily testable and exor networks it is an improvement of reddy 1972 and saluja reddy s 1975 methods and has the following features the network uses generalized reed muller expressions grms instead of positive polarity reed muller expressions pprms the average number of products for grms is less than half of that for pprms and is less than that of sum of products expressions sops the network consists of a literal part an and part an exor part and a check part the exor part can be a tree instead of a cascade thus the network is faster the test detects multiple stuck at faults under the assumption that the faults occur at most one part either the literal part the and part the exor part or the check part

convex analysis of spectrally defined matrix functions the purpose of this work is to carry out a systematic study of a special class of convex functions defined over the space s n of symmetric matrices of order n ti n the functions under consideration phi s n ra r cup infty are spectrally defined in the sense that the value phi a depends only on the spectrum lam 1 a ldots lam n a of the matrix a in s n fenchel legendre conjugation first and second order subdifferentiability asymptotic behavior and other concepts of convex analysis are the main ingredients of our exposition

spectral sets and functions on euclidean jordan algebras abstract spectral sets functions in euclidean jordan algebras are generalizations of permutation invariant sets respectively functions in r n in this article we study properties of such sets and functions and show how they are related to algebra automorphisms and majorization we show that spectral sets functions are indeed invariant under automorphisms but the converse may not hold unless the algebra is r n or simple we study schur convex spectral functions and provide some applications we also discuss the transfer principle and a related metaformula

on the impossible class of faulty functions in logic networks under short circuit faults the important problem of recognizing a priori the class of boolean functions which are never obtainable from a given combinational network under short circuit faults is almost unexplored primarily due to lack of understanding of the functional and structural factors that influence the fault behavior in the network in view of this a new concept of impossible class of faulty functions icff is introduced in this correspondence several intriguing properties of icff are uncovered namely the undetectability of input bridging faults the impossibility of the transformation of a fault free function fo to a subset or superset of fo and to other functions belonging to the same p and n equivalence classes of fo etc the closure amongst the fan out free and unate functions under bridging faults is investigated the impact of icff on the testability of the network is also discussed

on closedness and test complexity of logic circuits the concept of closedness of a set of logic functions under stuck type faults is introduced all sets of logic functions closed under stuck type faults are classified for the sets of logic functions closed under stuck type faults the test complexity and the universal test sets are considered it is shown that for each class of linear functions or functions and and functions both the minimum numbers of multiple fault detection tests and multiple fault location tests are exactly n 1 where n is the number of inputs of the circuits and that there exists universal test sets with n 1 tests to detect and locate all multiple faults in such circuits

on the complexity of estimating the size of a test set most np completeness results for test generation problems involve a reduction to the redundancy problem which explicitly encodes the satisfiability problem in this correspondence we investigate the complexity of a more modest problem that of estimating the size of a test set under the constraint that the circuit is irredundant we show that even this constrained problem is np hard in the strong sense

permutation invariant proper polyhedral cones and their lyapunov rank abstract the lyapunov rank of a proper cone k in a finite dimensional real hilbert space is defined as the dimension of the space of all lyapunov like transformations on k or equivalently the dimension of the lie algebra of the automorphism group of k this rank measures the number of linearly independent bilinear relations needed to express a complementarity system on k that arises for example from a linear program or a complementarity problem on the cone motivated by the problem of describing spectral proper cones where the complementarity system can be expressed as a square system that is where the lyapunov rank is greater than equal to the dimension of the ambient space we consider proper polyhedral cones in r n that are permutation invariant for such cones we show that the lyapunov rank is either 1 in which case the cone is irreducible or n in which case the cone is isomorphic to r n in the latter case we show that the corresponding spectral cone is isomorphic to a symmetric cone

optimal control of convective fitzhughnagumo equation we investigate smooth and sparse optimal control problems for convective fitzhughnagumo equation with traveling wave solutions in moving excitable media the cost function includes distributed spacetime and terminal observations or targets the state and adjoint equations are discretized in space by symmetric interior point galerkin sipg method and by backward euler method in time several numerical results are presented for the control of the traveling waves we also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing tikhonov regularization parameter further we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced hessian

riccati based feedback control of the monodomain equations with the fitzhugh nagumo model feedback control for the monodomain equations is studied the dynamics of interest are governed by a coupled pde ode reaction diffusion system with nonmonotone nonlinearity of fitzhugh nagumo type a localized distributed control is used to locally stabilize the nonlinear system this is achieved by a riccati based feedback law determined by the linearized system it is shown that the riccati equation corresponding to the pde variable suffices for exponential stabilization of the linearized pde ode system the theoretical findings are underlined by several numerical examples

extended commutation principles for normal decomposition systems abstract in this paper we study some optimization problems on lewis normal decomposition systems 1996 2000 we extend gowda and jeong s commutation principle 2017 in two directions firstly we replace the sum operation by an arbitrary binary operation monotone in each variable secondly we employ g increasing functions in place of g invariant convex functions we give applications to some classes of matrices we also interpret the obtained results for simple euclidean jordan algebras

convex graph invariants the structural properties of graphs are usually characterized in terms of invariants which are functions of graphs that do not depend on the labeling of the nodes in this paper we study convex graph invariants which are graph invariants that are convex functions of the adjacency matrix of a graph some examples include functions of a graph such as the maximum degree the maxcut value and its semidefinite relaxation and spectral invariants such as the sum of the k largest eigenvalues such functions can be used to construct convex sets that impose various structural constraints on graphs and thus provide a unified framework for solving a number of interesting graph problems via convex optimization we give a representation of all convex graph invariants in terms of certain elementary invariants and we describe methods to compute or approximate convex graph invariants tractably we discuss the interesting subclass of spectral invariants and also compare convex and nonconvex invariants finally we

on the connectedness of spectral sets and irreducibility of spectral cones in euclidean jordan algebras abstract let v be a euclidean jordan algebra of rank n a set e in v is said to be a spectral set if there exists a permutation invariant set q in r n such that e 1 q where v r n is the eigenvalue map that takes x v to x the vector of eigenvalues of x written in the decreasing order if the above q is also a convex cone we say that e is a spectral cone this paper deals with connectedness and arcwise connectedness properties of spectral sets by relying on the result that in a simple euclidean jordan algebra every eigenvalue orbit x y y x is arcwise connected we show that if a permutation invariant set q is connected arcwise connected then 1 q is connected respectively arcwise connected a related result is that in a simple euclidean jordan algebra every pointed spectral cone is irreducible

spectral cones in euclidean jordan algebras abstract a spectral cone in a euclidean jordan algebra v of rank n is of the form k 1 q where q is a permutation invariant convex cone in r n and v r n is the eigenvalue map which takes x to x the vector of eigenvalues of x with entries written in the decreasing order in this paper we describe some properties of spectral cones we show for example that spectral cones are invariant under automorphisms of v that the dual of a spectral cone is a spectral cone when v is simple or carries the canonical inner product and characterize the pointedness solidness of a spectral cone we also show that for any spectral cone k in v dim k 0 1 m 1 m where dim k denotes the dimension of k and m is the dimension of v

error estimates for the semidiscrete galerkin approximations of the fitzhugh nagumo equations l 2 error estimates are obtained for the semidiscrete approximation of the fitzhugh nagumo equations with nonsmooth initial data the equations consist of a nonlinear parabolic equation coupled with an ordinary differential equation smoothing due to the parabolic equation is used to show increased accuracy for the approximation for positive time