the relevance of convex analysis for the study of monotonicity abstract we introduce new representations for maximal monotone operators we relate them to previous representations given by krauss fitzpatrick martinez legaz and thera we show their usefulness for the study of compositions and sums of maximal monotone operators

sinc interpolation of discrete periodic signals the paper introduces a method for the sinc interpolation of discrete periodic signals the convolution of the sinc kernel with the infinite sequence of a periodic function is rewritten as a finite summation the method is equivalent to trigonometrical interpolation by fourier series expansion

a new condition for maximal monotonicity via representative functions abstract in this paper we give a weaker sufficient condition for the maximal monotonicity of the operator s a t a where s x x t y y are two maximal monotone operators a x y is a linear continuous mapping and x y are reflexive banach spaces we prove that our condition is weaker than the generalized interior point conditions given so far in the literature this condition is formulated using the representative functions of the operators involved in particular we rediscover some sufficient conditions given in the past using the so called fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator respectively

a weaker regularity condition for subdifferential calculus and fenchel duality in infinite dimensional spaces abstract in this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping this condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior point regularity conditions given so far in the literature moreover it provides a weak sufficient condition for fenchel duality regarding convex optimization problems in infinite dimensional spaces as an application we discuss the strong conical hull intersection property chip for a finite family of closed convex sets

notes on the interpolation of discrete periodic signals using sinc function related approaches considering the sinc interpolation of discrete periodic signals this correspondence establishes the equivalence of two recently published methods furthermore the interpolation methods are equivalent to a fractional delay filter designed using the frequency sampling technique a numerically stable and computationally attractive form of the equivalent methods is presented

the number of reidemeister moves needed for unknotting there is a positive constant c1 such that for any diagram d representing the unknot there is a sequence of at most 2 c1n reidemeister moves that will convert it to a trivial knot diagram where n is the number of crossings in d a similar result holds for elementary moves on a polygonal knot k embedded in the 1 skeleton of the interior of a compact orientable triangulated p l 3 manifold m there is a positive constant c2 such that for each t 1 if m consists of t tetrahedra and k is unknotted then there is a sequence of at most 2 c 2t elementary moves in m which transforms k to a triangle contained inside one tetrahedron of m we obtain explicit values for c1 and c2

comments on sinc interpolation of discrete periodic signals in a recent paper by t schanze see ibid vol 43 p 1502 3 1995 the convolution of the sinc kernel with the infinite sequence of a periodic function was expressed as a finite summation the expression obtained however is not numerically stable when evaluated at or near integer values of time this correspondence presents a numerically stable formulation that is equivalent to the results reported in the above cited paper and that when sampled is also shown to be equivalent to the inverse discrete fourier transform idft

on extension results for n cyclically monotone operators in reflexive banach spaces abstract in this paper we provide some extension results for n cyclically monotone operators in reflexive banach spaces by making use of the fenchel duality in this way we give a positive answer to a question posed by bauschke and wang 2007 4

an invariant of regular isotopy this paper studies a two variable laurent polynomial invariant of regular isotopy for classical unoriented knots and links this invariant is denoted lk for a link k and it satisfies the axioms 1 regularly isotopic links receive the same polynomial 2 lo0 1 3 l al l a l 4 l small diagrams indicate otherwise identical parts of larger diagrams regular isotopy is the equivalence relation generated by the reidemeister moves of type ii and type iii invariants of ambient isotopy are obtained from l by writhe normalization

realizable ideal d c and c d conversions for band limited periodic signals and characterization of single carrier block transmission for a real band limited periodic signal there are reconstruction formulas that realize the ideal discrete to continuous d c conversion within one period of the signal however the previous results may not be suitable for general complex band limited periodic signals for complex band limited periodic signals in this paper we first propose the reconstruction formulas which lead to realizable forms of ideal d c conversion we then derive the correlation function of the interpolation pulse and this helps us construct a matched filter that can implement the exact sampling c d conversion of a complex band limited periodic signal when there exists noise in the sampling process comparing with the ideal impulse sampling the proposed c d conversion can maintain finite output noise variance the correlation function of the output noise is also given and the oversampling effect of noise is analyzed for a complex interpolation pulse in general we cannot decouple the ideal d c or c d conversion into two independent d c or c d conversions for the real and imaginary parts of the signal we also give the extended forms and decomposition forms of the proposed d c and c d conversions our results are applied to derive the ideal d c and c d conversions in the precoded orthogonal frequency division multiplexing pofdm system the resulted interpolation pulses are used to characterize the single carrier sc block transmission under the pofdm architecture we give characteristics that distinguish the sc block transmission from traditional sc transmission

dualization of generalized equations of maximal monotone type this paper develops a simple duality framework for generalized equations defined by set valued mappings from a linear space to another the original problem is related to two auxiliary problems of the similar form corresponding to lagrangian and dual problems in the theory of convex programming as in convex programming the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically in particular dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations also a given problem can often be solved more easily by way of a dual method the strongest results of this paper concern monotone mappings in this context the duality framework yields several new criteria for maximal monotonicity of composite mappings these results are useful theoretically as well as in numerical solution of generalized equations the duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator splitting methods and other special methods can be applied

fitzpatrick functions and continuous linear monotone operators the notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex lower semicontinuous and proper function and any not necessarily symmetric continuous linear positive operator it was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing fitzpatrick functions in this paper we study fitzpatrick functions of continuous linear monotone operators defined on a hilbert space a novel characterization of skew operators is presented a result by brezis and haraux is reproved using the fitzpatrick function we investigate the fitzpatrick function of the sum of two operators and we show that a known upper bound is actually exact in finite dimensional and more general settings cyclic monotonicity properties are also analyzed and closed forms of the fitzpatrick functions of all orders are provided for all rotators in the euclidean plane

precoded and vector ofdm robust to channel spectral nulls and with reduced cyclic prefix length in single transmit antenna systems the performance of orthogonal frequency division multiplexing ofdm systems may be degraded when intersymbol interference isi channels have spectral nulls also when isi channels have many taps the data rate overhead due to the insertion of the cyclic prefix is high we first propose a precoded ofdm system that may improve the performance of the ofdm systems for spectral null channels we also propose size k spl times x1 vector ofdm vofdm systems that reduce the cyclic prefix length by k times compared to the conventional ofdm systems the precoding scheme is simply to insert one or more zeros between each two sets of k consecutive information symbols although it can be generalized to a general form this precoding scheme may be able to remove the spectral nulls of an isi channel without knowing the isi channel when no zero is inserted between each two sets of k consecutive information symbols and only each k consecutive symbols are blocked together we obtain vofdm systems both theoretical performance analysis and simulation results are presented finally we compare the combination of the vofdm systems and the unitary matrix modulation with the conventional ofdm systems and the phase shift keying modulation where both differential noncoherent and coherent modulations and demodulations are considered

fitzpatrick functions cyclic monotonicity and rockafellar s antiderivative several deeper results on maximal monotone operators have recently found simpler proofs using fitzpatrick functions in this paper we study a sequence of fitzpatrick functions associated with a monotone operator the first term of this sequence coincides with the original fitzpatrick function and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties it is shown that for any maximal cyclically monotone operator the pointwise supremum of the sequence of fitzpatrick functions is closely related to rockafellar s antiderivative several examples are explicitly computed for the purpose of illustration in contrast to rockafellar s result a maximal 3 cyclically monotone operator need not be maximal monotone a simplified proof of asplund s observation that the rotation in the euclidean plane by n is n cyclically monotone but not n 1 cyclically monotone is provided the fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail we conclude with a new proof of moreau s result concerning the convexity of the set of proximal mappings

tait s conjectures and odd crossing number amphicheiral knots we give a brief historical overview of the tait conjectures made 120 years ago in the course of his pioneering work in tabulating the simplest knots and solved a century later using the jones polynomial we announce the solution again based on a substantial study of the jones polynomial of one possibly his last remaining problem of tait with the construction of amphicheiral knots of almost all odd crossing numbers an application to the non triviality problem for the jones polynomial is also outlined 1 the first knot tables knot theory originated in the late 19th century at that time w thomson lord kelvin p g tait and j maxwell propagated the vortex atom theory in an attempt to explain the structure of the universe they believed that a supersubstance ether makes up all of matter and atoms are knotted tubes of ether knotting is hereby understood as tying a piece of rope and then identifying both its ends so that the tying cannot be any more undone thus in the realm of constructing a periodic table of elements tait began the catalogization of the simplest knots he depicted knots as we still do today by diagrams consisting of a smooth plane curve with transverse self intersections or crossings at each crossing one of the two strands passes over the other the term simplest refers to the number of crossings of the diagram we can define the crossing number of a knot as the minimal crossing number of all its diagrams and say that tait sought the list of knots with given small crossing number the list meant to present each knot by exactly one diagram this entails that knots from different diagrams should be inequivalent in the sense that one cannot turn a closed piece of rope knotted one way into one knotted the other way without cutting the rope the simplest knots are shown in figure 1 the leftmost one of crossing number 0 is the trivial knot or unknot it has some special importance much like the unit element in a group tait completed the list up to 7 crossings little kirkman later conway co and others took over and continued his work in the modern computer age tables have reached the knots of 17 crossings with millions of entries even though tait s vortex atom theory has long been dismissed an account on knot tabulation is received by the editors may 30 2007 2000 mathematics subject classification primary 57m25 secondary 01a55 01a60

maximal monotonicity for the precomposition with a linear operator we give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator a circ t circ a when a is a linear continuous mapping between two reflexive banach spaces and t is a maximal monotone operator as a special case we get the weakest constraint qualification that guarantees the maximal monotonicity of the sum of two maximal monotone operators on a reflexive banach space then we give a weak constraint qualification assuring the bre zis haraux type approximation of the range of the subdifferential of the precomposition to a of a proper convex lower semicontinuous function in nonreflexive banach spaces extending and correcting in a special case an older result due to riahi

a comparison of constraint qualifications in infinite dimensional convex programming in this paper the relationships between various constraint qualifications for infinite dimensional convex programs are investigated using robinson s refinement of the duality result of rockafellar it is demonstrated that the constraint qualification proposed by rockafellar provides a systematic mechanism for comparing many constraint qualifications as well as establishing new results in different topological environments

ber minimized ofdm systems with channel independent precoders we consider the minimization of uncoded bit error rate ber for the orthogonal frequency division multiplexing ofdm system with an orthogonal precoder we analyze the ber performance of precoded ofdm systems with zero forcing and minimum mean squared error mmse receivers in the case of mmse receivers we show that for quadrature phase shift keying qpsk there exists a class of optimal precoders that are channel independent examples of this class include the discrete fourier transform dft matrix and the hadamard matrix when the precoder is the dft matrix the resulting optimal transceiver becomes the single carrier system with cyclic prefix sc cp system we also show that the worst solution corresponds to the conventional ofdm system the conventional ofdm system has the largest ber in the case of zero forcing receivers the design of optimal transceiver depends on the signal to noise ratio snr for higher snr solutions of optimal precoders are the same as those of mmse receivers