Abstract: We prove a sharp analog of Young’s inequality on SN, and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions to optimizers in a monotonic manner. This strategy also works for the generalization of Young’s inequality on RN to more than three functions, and leads to significant new information about the optimizers and the constants.

Abstract: A wide range of cost functionals that describe the criteria for designing optimal pulses can be reduced to two basic functionals by the introduction of product spaces. We extend previous monotonically convergent algorithms to solve the generalized pulse design equations derived from those basic functionals. The new algorithms are proved to exhibit monotonic convergence. Numerical tests are implemented in four-level model systems employing stationary and/or nonstationary targets in the absence and/or presence of relaxation. Trajectory plots that conveniently present the global nature of the convergence behavior show that slow convergence may often be attributed to "trapping" and that relaxation processes may remove such unfavorable behavior.

Abstract: An e-test for a property P of functions from D = {1,..., d} to the positive integers is a randomized algorithm, which makes queries on the value of an input function at specified locations, and distinguishes with high probability, between the case of the function satisfying P, and the case that it has to be modified in more than ed places to make it satisfy P. We prove that an e-test for a property of integer sequences, such as the property of the sequence being a monotone non-decreasing sequence, that depends (in a strict sense) only on the order relations between the sequence members, cannot perform less queries (in the worst case) than the best e-test which uses only comparisons between the queried values. In addition, we show that an adaptive algorithm for testing that a sequence is monotone non-decreasing performs no better than the best non-adaptive one, with respect to query complexity. From this follows a tight lower bound on tests for this property.

Abstract: We show that the classical monotonicity conditions can be moderated in four theorems of P. Chandra.

Abstract: We develop algorithms for obtaining regularized estimates of emission means in positron emission tomography. The first algorithm iteratively minimizes a penalized maximum-likelihood (PML) objective function. It is based on standard de-coupled surrogate functions for the ML objective function and de-coupled surrogate functions for a certain class of penalty functions. As desired, the PML algorithm guarantees nonnegative estimates and monotonically decreases the PML objective function with increasing iterations. The second algorithm is based on an iteration dependent, de-coupled penalty function that introduces smoothing while preserving edges. For the purpose of making comparisons, the MLEM algorithm and a penalized weighted least-squares algorithm were implemented. In experiments using synthetic data and real phantom data, it was found that, for a fixed level of background noise, the contrast in the images produced by the proposed algorithms was the most accurate.

Abstract: Generalized convex functions preserve many valuable properties of mathematical programming problems with convex functions. Generalized monotone maps allow for an extension of existence results for variational inequality problems with monotone maps. Both models are special realizations of an abstract equilibrium problem with numerous applications, especially in equilibrium analysis (e.g., Blum and Oettli, 1994). We survey existence results for equilibrium problems obtained under generalized convexity and generalized monotonicity. We consider both the scalar and the vector case. Finally existence results for a system of vector equilibrium problems under generalized convexity are surveyed which have applications to a system of vector variational inequality problems. Throughout the survey we demonstrate that the results can be obtained without the rigid assumptions of convexity and monotonicity.

Abstract: In this paper, we investigate majority rule with an infinite number of voters We use an axiomatic approach and attempt to extend May’s Theorem characterizing majority rule to an infinite population The analysis hinges on correctly generalizing the anonymity condition and we consider three different versions We settle on bounded anonymity as the appropriate form for this condition and are able to use the notion of asymptotic density to measure the size of almost all sets of voters With this technique, we define density q-rules and show that these rules are characterized by neutrality, monotonicity, and bounded anonymity on almost all sets Although we are unable to provide a complete characterization applying to all possible sets of voters, we construct an example showing that our result is the best possible Finally, we show that strengthening monotonicity to density positive responsiveness characterizes density majority rule on almost all sets

Abstract: We present a higher order modal fixed point logic (HFL) that extends the modal μ-calculus to allow predicates on states (sets of states) to be specified using recursively defined higher order functions on predicates. The logic HFL includes negation as a first-class construct and uses a simple type system to identify the monotonic functions on which the application of fixed point operators is semantically meaningful. The model checking problem for HFL over finite transition systems remains decidable, but its expressiveness is rich. We construct a property of finite transition systems that is not expressible in the Fixed Point Logic with Chop [1] but which can be expressed in HFL. Over infinite transition systems, HFL can express bisimulation and simulation of push down automata, and any recursively enumerable property of a class of transition systems representing the natural numbers.

Abstract: The dependence of the Gaussian input information rate on the line-of-sight (LOS) matrix in multiple-input multiple-output coherent Rician fading channels is explored. It is proved that the outage probability and the mutual information induced by a multivariate circularly symmetric Gaussian input with any covariance matrix are monotonic in the LOS matrix D, or more precisely, monotonic in D'D in the sense of the Loewner partial order. Conversely, it is also demonstrated that this ordering on the LOS matrices is a necessary condition for the uniform monotonicity over all input covariance matrices. This result is subsequently applied to prove the monotonicity of the isotropic Gaussian input information rate and channel capacity in the singular values of the LOS matrix. Extensions to multiple-access channels are also discussed.

Abstract: A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.

Abstract: The Lame problem in a 2D domain with a crack under a non-penetration condition is considered as a variational inequality. A primal-dual active set method is proposed as an efficient numerical solution technique and compared to a previously employed iterative method for a penalized formulation. Sufficient conditions for monotonic convergence of a discretized version of the proposed algorithm are given and numerical experiments are presented.

Abstract: A new approach is proposed for optimizing a polynomial fractional function under polynomial constraints, or more generally, a synomial fractional function under synomial constraints. The approach is based on reformulating the problem as the optimization of an increasing function under monotonic constraints.

Abstract: In this article, we study the class of increasing and convex along rays (ICAR) functions over a cone. Apart from studying its basic properties, we study them from the point of view of Abstract Convexity. Further, we study the relation between the ICAR and Lipschitz functions and the properties under which an ICAR function has a Lipschitz behaviour. We also study the class of decreasing and convex along rays functions (DCAR).

Abstract: We show that for a convex function the following, rather modest conditions, are equivalent to monotonicity under local operations and classical communication. The conditions are: 1)invariance under local unitaries, 2) invariance under adding local ancilla in arbitrary state 3) on mixtures of states possessing local orthogonal flags the function is equal to its average. The result holds for multipartite systems. It is intriguing that the obtained conditions are equalities. The only inequality is hidden in the condition of convexity.

Abstract: We give necessary and sui¬ƒcient conditions for the existence of symmetric equilibrium without ties in common values auctions, with multidimensional independent types and no monotonic assumptions. When the conditions are not satisfied, we are still able to prove the existence of pure strategy equilibrium with an exogenous and explicit tie breaking mechanism. As a basis for these results, we obtain a characterization lemma that is valid under a general setting, that includes non-independent types, asymmetrical utilities and any attitude towards risk. Such characterization gives a basis for an intuitive interpretation for the behavior of the bidder: to bid in order to equalize the marginal benefit and the marginal cost of bidding

Abstract: Contrary to widespread perception, there is ever since 1994 a unified, general type independent theory for the existence of solutions for very large classes of nonlinear systems of PDEs. This solution method is based on the Dedekind order completion of suitable spaces of piece-wise smooth functions on the Euclidean domains of definition of the respective PDEs. The method can also deal with associated initial and/or boundary value problems. The solutions obtained can be assimilated with usual measurable functions or even with Hausdorff continuous functions on the respective Euclidean domains. It is important to note that the use of the order completion method does not require any monotonicity condition on the nonlinear systems of PDEs involved. One of the major advantages of the order completion method is that it eliminates the algebra based dichotomy "linear versus nonlinear" PDEs, treating both cases with equal ease. Furthermore, the order completion method does not introduce the dichotomy "monotonous versus non-monotonous" PDEs. None of the known functional analytic methods can exhibit such a performance, since in addition to topology, such methods are significantly based on algebra.

Abstract: In this paper, we investigate a nonlinear Urysohn integral equation on unboundedinterval. We show that under some assumptions that the equation has monotonic solutions belonging to the space of functions being Lebesgue integrable on unbounded interval. The main tool used in our study is the technique associated with measures of weak noncompactness and measures of noncompactness in strong sense.

Abstract: This paper presents a novel approach for constrained state estimation from noisy measurements. The optimal trending algorithms described in this paper assume that the trended system variables have the property of monotonicity. This assumption describes systems with accumulating mechanical damage. The performance variables of such a system can only get worse with time, and their behavior is best described by monotonic regression. Unlike a standard Kalman filter problem, where the process disturbances are assumed to be gaussian, this paper considers a random walk model driven by a one-sided exponentially distributed noise. The main contribution of this paper is in studying recursive implementation of the monotonic regression algorithms. We consider a moving horizon approach where the problem size is fixed even as more measurements become available with time. This enables us to perform efficient online optimization, making embeded implementation of the estimation computationally feasible.

Abstract: In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.

Abstract: A large sample test of proportionality of two cumulative incidence functions is developed for randomly right censored competing risks data without assuming that the K ≥ 2 risks are independent. The test is tailored to detecting monotonicity of the ratio of the two cumulative incidence functions and the test statistic is proved to be asymptotically normally distributed. In addition, it is shown that the proposed test can be readily adapted to test whether the censoring random variable and observable survival time have proportional hazards. The procedures are illustrated through application to two data sets well known in the survival analysis literature.

Abstract: is always non-negative, for any positive integer n and all points x1, . . . , xn in H is said to be positive definite on Hilbert space. In Schoenberg (1938), it was shown that a function is positive definite on Hilbert space if and only if it is completely monotonic, and this characterization is of central importance in the theory of radial basis functions and learning theory. In this paper, we present a short geometric proof of this beautiful fact.

Abstract: This paper investigates the possibility of using a truncated finite impulse response (FIR) model approximation to implement a well-known gradient type repetitive control algorithm. As a result it is in fact shown that the algorithm iteratively solves a model predictive control related cost function. Furthermore, it is shown how accurate the FIR approximation of the original system has to be in order for the algorithm to converge to zero tracking error. Under certain assumptions on the plant model it is shown that the algorithm results in monotonic convergence in the l/sub /spl infin//-norm. The algorithm is applied in real-time to a nonminimum mass-damper-spring system, and experimental results are compared to the theoretical results.