Max Function Research Articles

Conjugate Lyapunov functions for saturated linear systems

Based on a recent duality theory for linear differential inclusions (LDIs), the condition for stability of an LDI in terms of one Lyapunov function can be easily derived from that in terms of its conjugate function. This paper uses a particular pair of conjugate functions, the convex hull of quadratics and the maximum of quadratics, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a parametertized LDI system with an effective approach which enables optimization on the parameter of the LDI along with the optimization of the Lyapunov functions. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates the effectiveness of this paper's methods.

Predicting the execution times of parallel-independent programs using Pearson distributions

Predicting the execution time of parallel programs involves computing the maximum or minimum of the execution times of the tasks involved in the parallel computation. We present a method to accurately compute the distribution of the largest (Max) and the smallest (Min) execution time of the composite of a number of parallel programming tasks, each having an independent, stochastic, arbitrary workload. The Max function applies to the general case that the composite task completes at the time its longest constituent task terminates. The Min function applies when the completion of its shortest task terminates the whole parallel process, such as in a parallel searching program. Both the Min and Max density function of a constituent task are characterized in terms of a Pearson distribution. Due to its accuracy, the presented method is especially of interest when the performance of time critical parallel applications must be derived. Both prediction methods are tested against three well-known distributions. Furthermore, the Max prediction method is also tested against a number of measured real-life data parallel programs with different degree of parallelism. The results show excellent accuracy of better than 1% with a very few exceptions in extreme situations.

Bit metric generation for Gray coded QAM signals

A simple and general bit metric generation method by using the log likelihood ratio (LLR) for a Gray coded QAM signal over an additive white Gaussian noise channel is proposed. The mathematical max (min) functions of the conventional LLR expression are replaced with basic arithmetic functions. Since the derived LLR expression can be used as a soft bit metric for input to a conventional binary turbo decoder, it is suitable for practical applications with a turbo decoder or other related applications as an alternative to conventional methods.

Recursive approximation of the high dimensional max function

This paper proposes a smoothing method for the general n-dimensional max function, based on a recursive extension of smoothing functions for the two-dimensional max function. A theoretical framework is introduced, and some applications are discussed. Finally, a numerical comparison with a well-known smoothing method is presented.

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR MINMAX FRACTIONAL OPTIMAL CONTROL PROBLEMS CONTAINING ARBITRARY NORMS

Both parametric and parameter-free necessary and sufficient optimality conditions are established for a class of nondiffer-entiable nonconvex optimal control problems with generalized fractional objective functions, linear dynamics, and nonlinear inequality constraints on both the state and control variables. Based on these optimality results, ten Wolfe-type parametric and parameter-free duality models are formulated and weak, strong, and strict converse duality theorems are proved. These duality results contain, as special cases, similar results for minmax fractional optimal control problems involving square roots of positive semi definite quadratic forms, and for optimal control problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here contain various extensions of a number of existing duality formulations for convex control problems, and subsume continuous-time generalizations of a great variety of similar dual problems investigated previously in the area of finite-dimensional nonlinear programming.

Low molecular weight inhibitors of Myc-Max interaction and function.

c-Myc is helix-loop-helix-leucine zipper (HLH-ZIP) oncoprotein that is frequently deregulated in human cancers. In order to bind DNA, regulate target gene expression, and function in a biological context, c-Myc must dimerize with another HLH-ZIP protein, Max. A large number of c-Myc target genes have been identified, and many of the encoded proteins are transforming. Such functional redundancy, however, complicates therapeutic strategies aimed at inhibiting any single target gene product. Given this consideration, we have instead attempted to identify ways by which c-Myc itself could be effectively disabled. We have used a yeast two-hybrid approach to identify low-molecular-weight compounds that inhibit c-Myc-Max association. All of the compounds prevented transactivation by c-Myc-Max heterodimers, inhibited cell cycle progression, and prevented the in vitro growth of fibroblasts in a c-Myc-dependent manner. Several of the compounds also inhibited tumor growth in vivo. These results show that the yeast two-hybrid screen is useful for identifying compounds that can be exploited in mammalian cells. More specifically, they provide a means by which structural analogs, based upon these first-generation Myc-Max inhibitors, can be developed to enhance antitumor efficacy.

Dynamic orthogonal range queries in OLAP

We study the problem of pre-computing auxillary information to support on-line range queries for the sum and max functions on a datacube. For a d-dimensional datacube with size n in each dimension, we propose a dynamic range max data structure with O( α d ( s, n) L d ) query time, O( α d ( s, n) L 2 d n d/ L ) update time and O( s d ) storage where L and s are parameters chosen by the user before the construction of the data structure such that L∈{1,…,⌈ log n⌉} and s is a non-negative multiple of n; and α( s, n) is the functional inverse of Ackermann's function. Moreover, the data structure can be initialized in time linear to its size. There are three major techniques employed in designing the data structure, namely, a technique for trading query and update times, a technique for trading query time and storage and a technique for extending one-dimensional data structures to d-dimensional ones. Our techniques are also applicable to range queries over any semigroup and group operation, such as min, sum and count.

Duality models for some nonclassical problems in the calculus ofvariations

Parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonconvex variational problems with generalized fractional objective functions and nonlinear inequality constraints containing arbitrary norms. Based on these optimality criteria, ten parametric and parameter‐free dual problems are constructed and appropriate duality theorems are proved. These optimality and duality results contain, as special cases, similar results for minmax fractional variational problems involving square roots of positive semidefinite quadratic forms as well as for variational problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here subsume various existing duality formulations for variational problems and include variational generalizations of a great variety of cognate dual problems investigated previously in the area of finite‐dimensional nonlinear programming by an assortment of ad hoc methods.

Metal speciation and pH effect on Pb, Cu, Zn and Cd biosorption onto Sphaerotilus natans: Langmuir-type empirical model

Biosorption data of lead, copper, zinc and cadmium onto Sphaerotilus natans at different equilibrium pH (3–5 units) were here reported and analysed. Experimental results outlined the positive effect of pH increase on pollutant uptake and also the biomass affinity series (Pb>Cu>Zn>Cd) reflecting the hydrolytic properties of metals. An original empirical model was proposed to represent the effect of pH on heavy metal biosorption inserting q max vs. pH empirical functions into the classical Langmuir isotherm.

Separation of joint plan equilibrium payoffs from the min–max functions

This paper concerns infinitely repeated and undiscounted two-person non-zero-sum games of incomplete information on one side. Following an improvement on the proof of equilibrium existence, it establishes sufficient conditions for the existence of an equilibrium with payoffs superior to what the players would receive from observable deviation. To do this, we show that a no-where empty convex valued upper hemi-continuous correspondence from a compact metric space to a compact and convex subset of a Euclidean space can be approximated by Hausdorff-continuous convex valued correspondences that contain the original correspondence. This paper can be considered to be a first step toward a theory of equilibrium selection for these games. Examples are presented that show both the difficulty of and the desirability for stronger results than those presented here.

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR A CLASS OF NONSMOOTH CONTINUOUS-TIME GENERALIZED FRACTIONAL PROGRAMMING PROBLEMS

Both parametric and parameter-free stationary-point-type and saddle-point-type necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time generalized fractional programming problems with Volterra-type integral inequality and nonnegativity constraints. These optimality criteria are then utilized for constructing ten parametric and parameter-free Wolfe-type and Lagrangian-type dual problems and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be obtained for two important special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. All the results developed here are also applicable to continuous-time programming problems with fractional, discrete max, and conventional objective functions, which are special cases of the main problem studied in this paper.

Dynamical properties of min-max networks.

In this paper we study the dynamical behavior of a class of neural networks where the local transition rules are max or min functions. We prove that sequential updates define dynamics which reach the equilibrium in O(n2) steps, where n is the size of the network. For synchronous updates the equilibrium is reached in O(n) steps. It is shown that the number of fixed points of the sequential update is at most n. Moreover, given a set of p < or = n vectors, we show how to build a network of size n such that all these vectors are fixed points.

Essential role for Max in early embryonic growth and development

Loss of Max function in the mouse resulted in generalized developmental arrest of both embryonic and extraembryonic tissues at early postimplantation (approximately E5.5-6.5), coincident with loss or dilution of maternal Max stores in the expanding embryo in vivo and in blastocyst outgrowths in vitro. Developmentally arrested embryos were reduced in size and exhibited widespread cytological degeneration and feeble BrdU incorporation. Max and, by extension, the Myc superfamily, serve essential roles in early mammalian development and a maternal reservoir of Max exists in sufficient amount to sustain Myc superfamily function through preimplantation stages of development.

Are cortical models really bound by the "binding problem"?

Are cortical models really bound by the "binding problem"?

Fuzzy relational calculus approach to multidimensional pattern classification

Our aim is to design a pattern classifier using fuzzy relational calculus (FRC) which was initially proposed by W. Pedrycz. In the course of doing this we introduce a new interpretation of multidimensional fuzzy implication (MFI) to represent our knowledge about the training data set. The new interpretation is basically a set of one-dimensional fuzzy implications. The consequences of all one-dimensional fuzzy implications are finally collected through one intersection operator ‘ ∩’. Subsequently, we consider the notion of a fuzzy pattern vector, which is formed by the cylindrical extension of the antecedent part of each one-dimensional fuzzy implication. Thus, we get a set of fuzzy pattern vectors for the new interpretation of MFI and represent the population of training patterns in the pattern space. We also introduce a new approach to the computation of the derivative of the fuzzy max and min functions using the concept of a generalized function. During the construction of the classifier, based on FRC, we use fuzzy linguistic statements (or fuzzy membership function to represent the linguistic statement) to represent the ranges of features (e.g. feature F i is small/medium/big, etc. ∀ i) for a population of patterns. Note that the construction of the classifier essential depends on the estimate of a fuzzy relation R i between the antecedent part and consequent part of each one-dimensional fuzzy implication. Thus, a set of fuzzy relations is formed from the new interpretation of MFI. This set of fuzzy relations is termed as a core of the classifier. Once the classifier is constructed the non-fuzzy features of a pattern can be classified. At the time of classification of the test patterns, we use the concept of fuzzy singleton to fuzzify the non-fuzzy feature values of the test patterns. The performance of the proposed scheme is tested on synthetic data. Finally, we use the proposed scheme for the vowel classification problem of Indian languages.

Combined Entropic Regularization and Path-Following Method for Solving Finite Convex Min-max Problems Subject to Infinitely Many Linear Constraints

In this paper, we study the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints. The difficulty of such problems arises from the kinks of the max function and it is often suggested that, by imposing certain regularization functions, nondifferentiability will be overcome. We find that the entropic regularization introduced by Li and Fang is closely related to recently developed path-following interior-point methods. Based on their results, we create an interior trajectory in the feasible domain and propose a path-following algorithm with a convergence proof. Our intention here is to show a nice combination of minmax problems, semi-infinite programming, and interior-point methods. Hopefully, this will lead to new applications.

Pharmacokinetic–pharmacodynamic modeling of the coexistence of stimulatory and sedative components for midazolam

Midazolam increased the shorter-response rate and decreased the reinforcement rate of a contingency-controlled timing behavior—a differential-reinforcement-of-low-rate 45-s schedule. The responding rate changes observed were immediately interpretable as functions of midazolam concentration during a 3-h session—a period for investigating the onset, peak, and disappearance of midazolam effect—in rats. That the midazolam pharmacokinetic–pharmacodynamic model was a direct application of our alprazolam pharmacokinetic–pharmacodynamic model implies that both drugs exhibit similar pharmacological effects. The two peaks of the shorter-response rate increases produced by midazolam were modeled as a stimulation-sedation model that consisted of two opposing effect-link sigmoidal E max functions. The stimulation-sedation model suggested that midazolam possesses both stimulatory and sedative effects in a continuous but sequential fashion, and hypothesizes the coexistence of stimulation and sedation components for midazolam; this model may help delineate possible mechanisms for rebound side effects and of tolerance in humans. The reinforcement rate was, then, an index for evaluating the deficit in timing performance.

EFFECT OF BLOOD VOLUME EXPANSION ON ??VO2MAX, ENDURANCE PERFORMANCE AND CARDIAC FUNCTION IN ENDURANCE ATHLETES 806

Recent work in our laboratory suggests that the greater portion of the enhancement in cardiac function seen in endurance-trained athletes is simply a passive response to their higher blood volume (BV). We speculated that an additional increase in plasma volume (PVexp), (500 ml of 6% dextran) would lead to further enhancements in their endurance performance, ˙VO2 max and cardiac function. Therefore, nine highly trained endurance cyclists(˙VO2 max ≥ 65 ml·kg-1·min-1) were studied employing a double blind, cross-over design; i) before PVexp(Ctrl 1), ii) following sham PVexp (Sham), iii) following restoration to normocythemia (Ctrl 2), iv) after PVexp, andv) upon re-establishment of control haematologic levels (Ctrl 3). PVexp resulted in a 547 ± 61 ml increase in BV (p < 0.05). The consequent hemodilution resulted in a reduction of 6% in both the oxygen-carrying capacity and oxygen content of blood. There were no significant differences in endurance performance or ˙VO2 max among conditions. Following PVexp, left ventricular ejection time was significantly increased but there were no significant changes in left ventricular ejection rate, diastolic filling time or diastolic filling rate. As well, PVexp had no significant effect on arterial blood pressure or total peripheral resistance. Maximal cardiac output (Qmax) and maximal stroke volume (SVmax) were significantly higher following PVexp (3.2% and 3.4% respectively), which was sufficient to offset the hemodilution effects of PVexp and allow˙VO2 max and endurance performance to remain unchanged. These results suggest that in endurance-trained athletes who already have a high BV, an additional increase in PV will not result in a sufficient increase in SVmax and Qmax to overcome the hemodilution effects of acute Pvexp, resulting in no further enhancements in endurance performance, ˙VO2 max or cardiac function. Thus, endurance athletes may be at an optimal BV which is at the limits of their diastolic reserve capacity.

Histone Deacetylases Associated with the mSin3 Corepressor Mediate Mad Transcriptional Repression

Transcriptional repression by Mad–Max heterodimers requires interaction of Mad with the corepressors mSin3A/B. Sin3p, the S. cerevisiae homolog of mSin3, functions in the same pathway as Rpd3p, a protein related to two recently identified mammalian histone deacetylases, HDAC1 and HDAC2. Here, we demonstrate that mSin3A and HDAC1/2 are associated in vivo. HDAC2 binding requires a conserved region of mSin3A capable of mediating transcriptional repression. In addition, Mad1 forms a complex with mSin3 and HDAC2 that contains histone deacetylase activity. Trichostatin A, an inhibitor of histone deacetylases, abolishes Mad repression. We propose that Mad–Max functions by recruiting the mSin3–HDAC corepressor complex that deacetylates nucleosomal histones, producing alterations in chromatin structure that block transcription.

Optimal and sub‐optimal maximum a posteriori algorithms suitable for turbo decoding

AbstractFor estimating the states or outputs of a Markov process, the symbol‐by‐symbol maximum a posteriori (MAP) algorithm is optimal. However, this algorithm, even in its recursive form, poses technical difficulties because of numerical representation problems, the necessity of non‐linear functions and a high number of additions and multiplications. MAP like algorithms operating in the logarithmic domain presented in the past solve the numerical problem and reduce the computational complexity, but are suboptimal especially at low SNR (a common example is the Max‐Log‐MAP because of its use of the max function). A further simplification yields the soft‐output Viterbi algorithm (SOVA). In this paper, we present a Log‐MAP algorithm that avoids the approximations in the Max‐Log‐MAP algorithm and hence is equivalent to the true MAP, but without its major disadvantages. We compare the (Log‐)MAP, Max‐Log‐MAP and SOVA from a theoretical point of view to illuminate their commonalities and differences. As a practical example, we consider Turbo decoding, and we also demonstrate the practical suitability of the Log‐MAP by including quantization effects. The SOVA is, at 10−4, approximately 0.7 dB inferior to the (Log‐)MAP, the Max‐Log‐MAP lying roughly in between. The channel capacities of the three algorithms ‐when employed in a Turbo decoder‐ are evaluated numerically.