an anhomomorphic logic for quantum mechanics although various schemes for anhomomorphic logics for quantum mechanics have been considered in the past we shall mainly concentrate on the quadratic or grade 2 scheme in this scheme the grade 2 truth functions are called coevents we discuss properties of coevents projections on the space of coevents and the master observable we show that the set of projections forms an orthomodular poset we introduce the concept of precluding coevents and show that this is stronger than the previously studied concept of preclusive coevents precluding coevents are defined naturally in terms of the master observable a result that exhibits a duality between preclusive and precluding coevents is given some simple examples are presented

a clustering technique for the identification of piecewise affine systems we propose a new technique for the identification of discrete time hybrid systems in the piecewise affine pwa form this problem can be formulated as the reconstruction of a possibly discontinuous pwa map with a multi dimensional domain in order to achieve our goal we provide an algorithm that exploits the combined use of clustering linear identification and pattern recognition techniques this allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures moreover the clustering step used for classifying the datapoints is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

optimal control of a class of linear hybrid systems with saturation we consider a class of first order linear hybrid systems with saturation a system that belongs to this class can operate in several modes or phases in each phase each state variable of the system exhibits a linear growth until a specified upper or lower saturation level is reached and after that the state variable stays at that saturation level until the end of the phase a typical example of such a system is a traffic signal controlled intersection we develop methods to determine optimal switching time sequences for first order linear hybrid systems with saturation that minimize criteria such as average queue length worst case queue length average waiting time and so on first we show how the extended linear complementarity problem elcp which is a mathematical programming problem can be used to describe the set of system trajectories of a first order linear hybrid system with saturation optimization over the solution set of the elcp then yields an optimal switching time sequence although this method yields globally optimal switching time sequences it is not feasible in practice due to its computational complexity therefore we also present some methods to compute suboptimal switching time sequences furthermore we show that if there is no upper saturation then for some objective functions the globally optimal switching time sequence can be computed very efficiently we also discuss some approximations that lead to suboptimal switching time sequences that can be computed very efficiently finally we use these results to design optimal switching time sequences for traffic signal controlled intersections

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

quantum integrals and anhomomorphic logics the basic arena for a probabilistic structure is a set a of events corresponding to a is a dual structure a of coevents we call a an anhomomorphic logic and the coevents are given by truth functions from a to the two element boolean algebra z2 0 1 one of the main goals of a physical theory is to describe physical reality and a coevent a z2 provides such a description in the sense that an event a a actually occurs if and only if a 1 the quantum integral over an event a with respect to a coevent is defined and its properties are treated integrals with respect to various coevents are computed quantum systems are frequently described by a quantum measure which gives the propensity a that an event a occurs for a if a 0 whenever a 0 we say that is preclusive preclusivity is a reality filter because it eliminates coevents that do not describe a possible reality for the system a quantum measure that can be represented as a quantum integral with respect to a coevent i

symmetric multilinear forms and polarization of polynomials we study a generalization of the classical correspondence between homogeneous quadratic polynomials quadratic forms and symmetric alternating bilinear forms to forms in n variables the main tool is combinatorial polarization and the approach is applicable even when n is not invertible in the underlying field

code loops in dimension at most 8 abstract code loops are certain moufang 2 loops constructed from doubly even binary codes that play an important role in the construction of local subgroups of sporadic groups more precisely code loops are central extensions of the group of order 2 by an elementary abelian 2 group v in the variety of loops such that their squaring map commutator map and associator map are related by combinatorial polarization and the associator map is a trilinear alternating form using existing classifications of trilinear alternating forms over the field of 2 elements we enumerate code loops of dimension d dim v 8 equivalently of order 2 d 1 512 up to isomorphism there are 767 code loops of order 128 and 80826 of order 256 and 937791557 of order 512

quasialgebra structure of the octonions abstract we show that the octonions are a twisting of the group algebra of z 2 z 2 z 2 in the quasitensor category of representations of a quasi hopf algebra associated to a group 3 cocycle in particular we show that they are quasialgebras associative up to a 3 cocycle isomorphism we show that one may make general constructions for quasialgebras exactly along the lines of the associative theory including quasilinear algebra representation theory and an automorphism quasi hopf algebra we study the algebraic properties of quasialgebras of the type which includes the octonions further examples include the higher 2n onion cayley algebras and examples associated to hadamard matrices

on extraspecial left conjugacy closed loops abstract a loop q is said to be left conjugacy closed lcc if l x l y l x 1 is a left translation for all x y q we describe all lcc loops q such that q z is an elementary abelian p group where z q is a central subloop of order p we single out those that are right conjugacy closed as well and show their connection to trilinear mappings and quadratic forms isomorphism classes are determined for the case z z q i e for the extraspecial loops

reconstruction of l splines of polynomial growth from their local weighted average samples in this paper we study the reconstruction of cardinal l spline functions from their weighted local average samples y n f h n n z where the weight function h t has support in 1 2 1 2 it is shown that there are infinitely many l spline functions which are solutions to the problem for the given data yn and given d n find a cardinal l spline f t c d 1 r satisfying y n f h n n z further it is shown that for d 1 2 and for every nonnegative h supported in 1 2 1 2 there is a unique solution to this problem if both the samples and the l splines are of polynomial growth moreover for d 2 it is shown that for every sample of polynomial growth the above problem has a unique solution of polynomial growth when the weight function h supported in 1 2 1 2 is positive definite

a series of algebras generalizing the octonions and hurwitz radon identity we study non associative twisted group algebras over mathbb z 2 n with cubic twisting functions we construct a series of algebras that extend the classical algebra of octonions in the same way as the clifford algebras extend the algebra of quaternions we study their properties give several equivalent definitions and prove their uniqueness within some natural assumptions we then prove a simplicity criterion

efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form for hybrid systems described by switched linear difference equations linear threshold conditions automata and propositional logic conditions described in mixed logical dynamical form this note describes two algorithms for transforming such systems into an equivalent piecewise affine form where equivalent means that for the same initial conditions and input sequences the trajectories of the system are identical the proposed techniques exploit ideas from mixed integer programming and multiparametric programming

simple graded commutative algebras we study the notion of graded commutative algebra for an arbitrary abelian group the main examples are the clifford algebras already treated in h albuquerque and s majid 2002 2 we prove that the clifford algebras are the only simple finite dimensional associative graded commutative algebras over r or c our approach also leads to non associative graded commutative algebras extending the clifford algebras

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

quantum dynamics without the wavefunction when suitably generalized and interpreted the path integral offers an alternative to the more familiar quantal formalism based on state vectors self adjoint operators and external observers mathematically one generalizes the path integral as propagator to a quantal measure on the space of all conceivable worlds and this generalized measure expresses the dynamics or law of motion of the theory much as wiener measure expresses the dynamics of brownian motion within such histories based schemes new and more realistic possibilities open up for resolving the philosophical problems of the state vector formalism in particular one can dispense with the need for external agents by locating the predictive content of in its sets of measure zero such sets are to be precluded but unrestricted application of this rule engenders contradictions one possible response would remove the contradictions by circumscribing the application of the preclusion concept another response more in the tradition of quantum logic would accommodate the contradictions by dualizing to a space of co events and effectively identifying reality with an element of this dual space

piecewise affine state feedback for piecewise affine slab systems using convex optimization abstract this paper shows that lyapunov based state feedback controller synthesis for piecewise affine pwa slab systems can be cast as an optimization problem subject to a set of linear matrix inequalities lmis analytically parameterized by a vector furthermore it is shown that continuity of the control inputs at the switchings can be guaranteed by adding equality constraints to the problem without affecting its parameterization structure finally it is shown that piecewise affine state feedback controller synthesis for piecewise affine slab systems to maximize the decay rate of a quadratic control lyapunov function can be cast as a set of quasi concave optimization problems analytically parameterized by a vector before casting the synthesis in the format presented in this paper lyapunov based piecewise affine state feedback controller synthesis could only be formulated as a bi convex optimization program which is very expensive to solve globally thus the fundamental importance of the contributions of the paper relies on the fact that for the first time the piecewise affine state feedback synthesis problem has been formulated as a convex problem with a parameterized set of lmis that can be relaxed to a finite set of lmis and solved efficiently to a point near the global optimum using available software furthermore it is shown for the first time that in some situations the global can be exactly found by solving only one concave problem

moufang loops with a unique nonidentity commutator associator square in 9 griess considers a class of loops called code loops which have applicability to a construction of the monster and its nonassociative algebra see 7 these loops turn out to be moufang loops which have a unique nonidentity commutator a unique nonidentity associator and a unique nonidentity square moufang loops with one or more of these properties also play an important role in the authors work on loops which have alternative loop rings 2 5 81 for example a nonassociative loop which has an alternative loop ring over a ring of characteristic different from two must be a moufang loop with a unique nonidentity commutator and associator which coincide in this paper we study moufang loops with the properties in question in section 3 we concentrate on loops which have unique nonidentity commutators and or associators and investigate some of their properties in section 4 we consider moufang loops with a unique nonidentity square and show that these are exactly the code loops of griess

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

orthogonal designs and a cubic binary function orthogonal designs are fundamental mathematical notions used in the construction of space time block codes for wireless transmissions designs have two important parameters the rate and the decoding delay the main problem of the theory is to construct designs maximizing the rate and minimizing the decoding delay all known constructions of cods are inductive or algorithmic in this paper we present an explicit construction of optimal cods we do not apply recurrent procedures and do calculate the matrix elements directly our formula is based on a cubic function in two binary n vectors in our previous work comm math phys 2010 and j pure and appl algebra 2011 we used this function to define a series of non associative algebras generalizing the classical algebra of octonions and to obtain sum of squares identities of hurwitz radon type

gradings on octonions abstract all the different ways of grading any cayley dickson or octonion algebra as well as the associated central simple malcev algebras will be obtained

every continuous nonlinear control system can be obtained by parametric convex programming in this short note we define parametric convex programming pcp in a slightly different manner than it is usually done by extending convexity not only to variables but also to the parameters and we show that the widely applied model predictive control mpc technique is a particular case of pcp the main result of the note is an answer to the inverse question of pcp which feedback laws can be generated by pcp by employing results of convex analysis we provide a constructive proof yet not computational that allows us to conclude that every continuous feedback law can be obtained by pcp

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

consistent histories and the interpretation of quantum mechanics the usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times called consistent histories through a criterion which ensures that within limits which are explicitly defined within the formalism classical rules for probabilities are satisfied the interpretive scheme which results is applicable to closed isolated quantum systems is explicitly independent of the sense of time i e past and future can be interchanged has no need for wave function collapse makes no reference to processes of measurement though it can be used to analyze such processes and can be applied to sequences of microscopic or macroscopic events or both as long as the mathematical condition of consistency is satisfied when applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus

mpc for continuous piecewise affine systems abstract a large class of hybrid systems can be described by a max min plus scaling mmps model i e using the operations maximization minimization addition and scalar multiplication first we show that continuous piecewise affine systems are equivalent to mmps systems next we consider model predictive control mpc for these systems in general this leads to nonlinear nonconvex optimization problems we present a new mpc method for mmps systems that is based on canonical forms for mmps functions in case the mpc constraints are linear constraints in the inputs only this results in a sequence of linear optimization problems such that the mpc control can often be computed in a much more efficient way than by just applying nonlinear optimization as was done in previous work

the shannon sampling series and the reconstruction of signals in terms of linear quadratic and cubic splines the sine kernel function of the sampling series is replaced by spline functions having compact support all built up from the b splines m n the resulting generalized sampling series reduces to finite sums so that no truncation error occurs moreover the approximation error generally decreases more rapidly than for the classical series when w tends to infinity 1 w being the distance between the sampling points for the kernel function varphi t 5m 4 t 4m 5 t with support tfrac 5 2 tfrac 5 2 e g it decreases with order o w 4 provided the signal has a fourth order derivative for the classical series the order is o w 4 log w seven characteristic examples are treated in detail

a new decomposition method for multiuser dc programming and its applications we propose a novel decomposition framework for the distributed optimization of difference convex dc type nonseparable sum utility functions subject to coupling convex constraints a major contribution of the paper is to develop for the first time a class of inexact best response like algorithms with provable convergence where a suitably convexified version of the original dc program is iteratively solved the main feature of the proposed successive convex approximation method is its decomposability structure across the users which leads naturally to distributed algorithms in the primal and or dual domain the proposed framework is applicable to a variety of multiuser dc problems in different areas ranging from signal processing to communications and networking as a case study in the second part of the paper we focus on two examples namely i a novel resource allocation problem in the emerging area of cooperative physical layer security and ii and the renowned sum rate maximization of mimo cognitive radio networks our contribution in this context is to devise a class of easy to implement distributed algorithms with provable convergence to stationary solution of such problems numerical results show that the proposed distributed schemes reach performance close to and sometimes better than that of centralized methods

on the zeros of the generalized euler frobenius laurent polynomial and reconstruction of cardinal splines of polynomial growth from local average samples abstract let h t be a nonnegative measurable function supported in 1 2 1 2 and m d t 1 2 1 2 1 2 1 2 1 2 1 2 t d 1 times be the central b spline of degree d we show that the roots of the generalized euler frobenius laurent polynomial defined by e h d z n z h m d n z n are simple negative and all are different from 1 as a consequence of this result we show that for every sequence y n n z r z of samples having polynomial growth and nonnegative measurable function h supported in 1 2 1 2 there is a unique spline f of degree d with polynomial growth satisfying f h n y n n z the presented work answers affirmatively the open problem posed in perez villalon and portal 2012 9

inverse parametric optimization with an application to hybrid system control we present a number of results on inverse parametric optimization and its application to hybrid system control we show that any function that can be written as the difference of two convex functions can also be written as a linear mapping of the solution to a convex parametric optimization problem we exploit these results in application to the control of systems with piecewise affine dynamics and show that it is possible to model such systems as optimizing processes optimal control problems for such systems can be remodeled as bilevel optimization problems and solved with existing techniques