obtuse cones and gram matrices with nonnegative inverse abstract we study some properties of gram matrices with nonnegative inverse which lead to the constructions of obtuse cones these constructions can find applications in optimization problems e g in methods of convex feasibility problems or of convex minimization problems

understanding complex dynamics by means of an associated riemann surface we provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated riemann surface thanks to this geometric description an explicit formula for the period of the orbits can be derived which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem for rational values of this ratio and generic values of the initial data all orbits are periodic and the system is isochronous for irrational values of the ratio there exist periodic and quasi periodic orbits for different initial data moreover the dependence of the period on the initial data shows a rich behavior and initial data can always be found with arbitrarily large periods

the efficiency of ballstep subgradient level methods for convex optimization we study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we establish convergence and efficiency estimates for simple ballstep level controls without requiring that the feasible set be compact our framework may handle accelerations based on cheap projections surrogate constraints and conjugate subgradient techniques

block iterative surrogate projection methods for convex feasibility problems abstract a unified framework is presented for studying the convergence of projection methods for finding a common point of finitely many closed convex sets in r n every iteration approximates each set by a half space given either by an approximate projection of the current iterate or by an aggregate inequality derived from the convex inequalities describing this set the next iterate is found by projecting the current one on a surrogate half space formed by taking a convex combination of the half space inequalities convergence to a solution is established under weak conditions that allow various acceleration techniques and choices of aggregating weights the resulting methods are block iterative and hence lend themselves to parallel implementation we show that the idea of forming cut maps via surrogate inequalities encompasses many classical as well as recently proposed methods for set intersection problems and convex feasibility problems with nondifferentiable inequalities and linear equations and inequalities

parallel projected aggregation methods for solving the convex feasibility problem convergence conditions are established for new sequential and parallel projected aggregation methods pams that find a feasible point of a large system of convex inequalities and linear equations

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

a simultaneous projections method for linear inequalities abstract an iterative method for solving general systems of linear inequalities is considered the method a relaxed generalization of cimmino s scheme for solving linear systems was first suggested by censor and elfving each iterate is obtained as a convex combination of the orthogonal projections of the previous iterate on the half spaces defined by the linear inequalities the algorithm is particularly suitable for implementation on computers with parallel processors we prove convergence from any starting point for both consistent and nonconsistent systems to a feasible point in the first case and to a weighted least squares type solutions in the second

dynamical systems on infinitely sheeted riemann surfaces this paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi valuedness of the solutions as functions of complex time in this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral the associated riemann surface r c is known to be an infinitely sheeted covering of the complex time plane ramified at an infinite set of points whose projection in the plane is dense the main novelty of this paper is that the geometrical structure of these infinitely sheeted riemann surfaces is described in great detail which allows to study global properties of the flow such as asymptotic behaviour of the solutions periodic orbits and their stability or sensitive dependence on initial conditions the results are then compared with a numerical integration of the equations of motion following the recent approach of calogero the real time trajectories of the system are given by paths on r that are projected to a circle on the complex plane due to the branching of r the solutions may have different periods or may not be periodic at all

on the connection between hyperelliptic separability and painleve integrability we consider systems of odes which are associated with some physically significant examples shallow water equilibrium solutions travelling waves of the harry dym equation a lotka volterra system of competing species and the geodesic flow on the triaxial ellipsoid the first three are shown to share the following properties i they are hyperelliptically separable systems hss and after a suitable nonlinear time transformation become algebraically completely integrable aci and ii they are of the weak painleve type and become full painleve after the application of this transformation the geodesic flow on the other hand although it passes the usual painleve test does not possess a full set of free constants and thus one may not conclude whether it has the painleve property or not this system is also hss and becomes aci after the application of a suitable nonlinear time transformation we also combine our geometric analytical investigation with a numerical analysis of the system in the complex plane and show that there is perfect correspondence between the results of the two approaches this correspondence strengthens the reliability of such numerical studies and helps us better understand their implication in cases where such nonlinear transformations to complete integrability are not available

redundant logarithmic arithmetic a number system that offers advantages in some situations over conventional floating point and sign logarithmic number systems is described redundant logarithmic arithmetic like conventional logarithmic arithmetic relies on table lookups to make the arithmetic unit simpler than an equivalent floating point unit the cost of 32 bit subtraction in a redundant logarithmic number system is lower than previously published logarithmic subtraction methods the total memory requirement for a 29 bit redundant logarithmic unit is 16 k words compared to 22 k words by the best previously published conventional sign logarithm unit assuming similar addition techniques are employed a redundant logarithmic number system can be implemented with online arithmetic which would be impractical for a conventional sign logarithm number system the disadvantages of redundant arithmetic are typical of redundant number systems first the redundancy doubles the storage requirements for data values second the representation can become ill conditioned especially as a result of iterated multiplications third division and square root operations are more difficult to implement in redundant logarithmic arithmetic

non integrable systems with algebraic singularities in complex time dynamical arguments are presented which suggest that there are non integrable systems without clustering of singularities without infinite singularities or singularities with an infinite number of branches in the complex t plane several examples with only algebraic singularities are studied for which strong numerical evidence is presented for non integrability and infinitely sheeted solutions weak painleve potentials are also analysed from this point of view and all integrable cases are found to possess only finitely sheeted solutions

lns with co transformation competes with floating point the logarithmic number system has been proposed as an alternative to floating point arithmetic multiplication division and square root operations are accomplished with inexpensive fixed point methods but addition and subtraction are considerably more challenging recent work has demonstrated that these operations too can be done with similar speed and accuracy to their fp equivalents but the necessary circuitry is complex in particular it is dominated by the need for large rom tables for the storage and interpolation of non linear functions we describe a new co transformation procedure that eliminates much of the rom space and allows the easy synthesis of the remainder in logic and we then evaluate several interpolation methods that might benefit from it synthesised 32 bit implementations are compared with floating point units and show substantial reductions in delay with equivalent accuracy and area

a hybrid number system processor with geometric and complex arithmetic capabilities the architecture design and performance of a hybrid number system processor are described the processor performs multiplication division square root and square in the logarithmic number system lns domain however the input output addition and subtraction are all executed in the 32 b ieee standard floating point number system with the lns multiplier and pipelined architecture the processor is able to perform the geometric and complex arithmetic very effectively the processor is also shown to compare well to an existing 32 b floating point dsp digital signal processor chip for the same level of cmos technology the performance ratios between the hybrid number system and the floating point processor are shown to be 6 4 1 and 8 1 for division and square root respectively for the complex fft fast fourier transform algorithm the ratio is around 2 1

a method of projection onto an acute cone with level control in convex minimization we study a projection method for convex minimization problems in each step we construct a certain polyhedral function that approximates the objective function from below the affine pieces of the approximating function are linearizations of the objective function obtained in previous steps the linearization are determined by a linearly independent system of subgradients which generates an obtuse cone the current approximation of the solution is projected onto a sublevel set of the polyhedral function with level which estimates the optimal value and is updated in each iteration the projection can be easily obtained by solving a system of linear equations and is actually the projection onto a translated acute cone dual to the constructed obtuse cone the construction ensures long steps between succesive approximations of the solution such long steps are important for the good behavior of the method

the efficiency of subgradient projection methods for convex optimization part i general level methods we study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we present several variants and show that they enjoy almost optimal efficiency estimates in another paper we discuss possible implementations of such methods in particular their projection subproblems may be solved inexactly via relaxation methods thus opening the way for parallel implementations they can also exploit accelerations of relaxation methods based on simultaneous projections surrogate constraints and conjugate and projected conditional subgradient techniques

fpga designs with optimized logarithmic arithmetic using a general polynomial approximation approach we present an arithmetic library generator for the logarithmic number system lns the generator produces optimized lns arithmetic libraries that improve significantly over previous lns designs on area and latency we also provide area cost estimation and bit accurate simulation tools that facilitate comparison between lns and floating point designs

modifications and implementation of the ellipsoid algorithm for linear programming we give some modifications of the ellipsoid algorithm for linear programming and describe a numerically stable implementation we are concerned with practical problems where user supplied bounds can usually be provided our implementation allows constraint dropping and updates bounds on the optimal value and should be able to terminate with an indication of infeasibility or with a provably good feasible solution in a moderate number of iterations

the analytic structure of dynamical systems and self similar natural boundaries abstract in this paper we investigate the analytic complex time structure of the movable singularities for several dynamical systems in general it is found that there exists a direct connection between the occurencce of a certain type of multiple valuedness of the singularities and the existence of a class of remarkable self similar natural boundaries for these systems an asymptotic description of the distribution of singularities in the natural boundary is developed this provides a description of the fine scale structure of these natural boundaries that agrees closely with the numerical calculations

a new class of isochronous dynamical systems a new class of isochronous dynamical systems is introduced and briefly discussed these systems feature in their phase space a fully dimensional region part of which can be explicitly identified where all their solutions are completely periodic periodic in all their degrees of freedom with the same period but in other regions of their phase space their evolution might be quite complicated

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

on a generalization of jacobi s elliptic functions and the double sine gordon kink chain a generalization of jacobi s elliptic functions is introduced as inversions of hyperelliptic integrals we discuss the special properties of these functions present addition theorems and give a list of indefinite integrals as a physical application we show that periodic kink solutions kink chains of the double sine gordon model can be described in a canonical form in terms of generalized jacobi functions

low power logarithmic number system addition subtraction and their impact on digital filters this paper presents techniques for low power addition subtraction in the logarithmic number system lns and quantifies their impact on digital filter vlsi implementation the impact of partitioning the look up tables required for lns addition subtraction on complexity performance and power dissipation of the corresponding circuits is quantified two design parameters are exploited to minimize complexity namely the lns base and the organization of the lns word a roundoff noise model is used to demonstrate the impact of base and word length on the signal to noise ratio of the output of finite impulse response fir filters in addition techniques for the low power implementation of an lns multiply accumulate mac units are investigated furthermore it is shown that the proposed techniques can be extended to cotransformation based circuits that employ interpolators the results are demonstrated by evaluating the power dissipation complexity and performance of several fir filter configurations comprising one two or four mac units simulations of placed and routed vlsi lns based digital filters using a 90 nm 1 0 v cmos standard cell library reveal that significant power dissipation savings are possible by using optimized lns circuits at no performance penalty when compared to linear fixed point two s complement equivalents

order and the ubiquitous occurrence of chaos abstract for a large class of ode s which includes the van der pol equation we determine analytically the asymptotic location of the singularities in the complex t plane by integrating these ode s numerically we show that if the singularities are dense which is the generic case the solution is chaotic in the sense of sensitive dependence on initial conditions in the exceptional case that the singularities are not dense the solution exhibits order taxis chaos is ubiquitous even for first order ode s in complex t

pipelined computation of very large word length lns addition subtraction with polynomial hardware cost a novel pipelined method is proposed to compute the addition subtraction in very large word length logarithmic number system lns arithmetic digit parallel additive normalization and digit on line multiplicative normalization methods are adopted to compute the exponential and logarithmic functions respectively in lns addition subtraction these two methods can both be implemented in a pipelined and regular architecture the size of the required lookup tables is now proportional to a third order polynomial function instead of an exponential function of the word length the total size of the tables in a 32 bit lns unit is estimated to be less than 53 5 kbits and the total size of the table s in a 64 bit lns unit will be less than 471 kbits furthermore the hardware cost of the other circuits in the proposed lns unit is only proportional to the square of the word length this study also develops a simple leading zero bits prediction technique that can significantly enhance the precision in lns addition subtraction computation when compared to floating point arithmetic design our approach still suffers from large hardware cost and pipeline latency however the proposed approach has provided a theoretical advancement in the development of very large word length lns arithmetic

applying features of ieee 754 to sign logarithm arithmetic various features found in standard floating point arithmetic ieee 754 are examined in light of their appropriateness for sign logarithm arithmetic the emphasis is on a 32 b word size comparable to ieee 754 single precision although other word sizes are possible a multilayer sign logarithm format is considered the lowest layer similar to previous implementations would provide only normalized representations but would not provide representations for zero denormalized values infinities and nans the highest layer would provide most of the features found in ieee 754 including zeros denormalized values infinities and nans novel algorithms for implementing logarithmic denormalized arithmetic are presented simulation results show that the error characteristics of the proposed logarithmic denormalized arithmetic algorithms are similar to those of the denormalized floating point arithmetic in ieee 754

the transition from regular to irregular motions explained as travel on riemann surfaces we introduce and discuss a simple hamiltonian dynamical system interpretable as a three body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on riemann surfaces the interest of this phenomenology illustrating the onset in a deterministic context of irregular motions is underlined by its generality suggesting its eventual relevance to understand natural phenomena and experimental investigations here only some of our main findings are reported without detailing their proofs a more complete presentation will be published elsewhere

inversion of a general hyperelliptic integral and particle motion in hořava lifshitz black hole space times the description of many dynamical problems such as the particle motion in higher dimensional spherically and axially symmetric space times is reduced to the inversion of hyperelliptic integrals of all three kinds the result of the inversion is defined locally using the algebro geometric techniques of the standard jacobi inversion problem and the foregoing restriction to the divisor for a representation of the hyperelliptic functions the klein weierstras multi variable function is introduced it is shown that all parameters needed for the calculations such as period matrices and abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and constants the cases of genus two three and four are considered in detail the method is exemplified by the particle motion associated with genus one elliptic and genus three hyperelliptic curves applications are for instance solutions to the geodesic equations in the space times of static spherically symmetric ho ava lifshitz black holes

monotone gram matrices and deepest surrogate inequalities in accelerated relaxation methods for convex feasibility problems abstract the relaxation method for linear inequalities iterates by projecting the current point onto the most violated constraint accelerated methods project onto the intersection of several halfspaces or onto a surrogate halfspace corresponding to a nonnegative combination of constraints we extend todd s conditions for finding best surrogate inequalities via the solution of systems of linear equations our techniques may be used for accelerating various methods for convex feasibility and optimization problems

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

sign logarithm arithmetic for fft implementation sign logarithm arithmetic is applicable to a variety of numerical applications where wide dynamic range and small wordsize are required in this paper the basic sign logarithm arithmetic operations required for signal processing i e addition subtraction and multiplication are reviewed the computational errors are analyzed for fft realization and simulation results are presented which serve to verify the analysis it is shown that the sign logarithm approach provides improved arithmetic quantization error performance for a given word size over fft s implemented with conventional fixed or floating point arithmetic and that the sign logarithm implementation is faster and less complex than conventional approaches

newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ2 νn 1 n z ergodicity isochrony and fractals we study the complexification of the one dimensional newtonian particle in a monomial potential we discuss two classes of motions on the associated riemann surface the rectilinear and the cyclic motions corresponding to two different classes of real and autonomous newtonian dynamics in the plane the rectilinear motion has been studied in a number of papers while the cyclic motion is much less understood for small data the cyclic time trajectories lead to isochronous dynamics for bigger data the situation is quite complicated computer experiments show that for sufficiently small degree of the monomial the motion is generically isochronous with integer period which depends in a quite sensitive way on the initial data if the degree of the monomial is sufficiently high computer experiments show essentially chaotic behavior we suggest a possible theoretical explanation of these different behaviors we also introduce a two parameter family of two dimensional mappings describing the motion of the center of the circle as a convenient representation of the cyclic dynamics we call such a mapping the center map computer experiments for the center map show a typical multifractal behavior with periodicity islands therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity

nonnegative moore penrose inverses of gram operators this paper is concerned with necessary and sufficient conditions for the nonnegativity of moore penrose inverses of gram operators between real hilbert spaces these conditions include statements on acuteness or obtuseness of certain closed convex cones the main result generalizes a well known result for inverses in the finite dimensional case over the nonnegative orthant to moore penrose inverses in possibly infinite dimensional hilbert spaces over any general closed convex cone

the efficiency of subgradient projection methods for convex optimization part ii implementations and extensions in the first part of this paper we studied subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we presented several variants and showed that they enjoy almost optimal efficiency estimates in this part of the paper we discuss possible implementations of such methods in particular their projection subproblems may be solved inexactly via relaxation methods thus opening the way for parallel implementations we discuss accelerations of relaxation methods based on simultaneous projections surrogate constraints and conjugate and projected conditional subgradient techniques

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

integrated circuit logarithmic arithmetic units this correspondence examines integrated circuit logarithmic arithmetic units which include adders subtracters multipliers and dividers the design of these arithmetic units is reviewed and an example arithmetic unit which performs multiplication followed by addition is designed in detail the design results are used to develop a size and speed comparison of integrated circuit logarithmic and fixed point arithmetic units this comparison is exercised through a video signal processing example it is concluded from this comparison that logarithmic arithmetic units are smaller than and as fast as fixed point arithmetic units with comparable capabilities in digital signal processing applications characterized by large dynamic range and moderate computational accuracy requirements further this comparison quantitatively illustrates the interaction of digital signal processing and integrated circuit issues in the design of special purpose digital signal processors