Series representations of the remainders in the expansions for certain trigonometric functions and some related inequalities, II

We examine Wilker and Huygens-type inequalities involving trigonometric functions making use of results derived in Part I. The Papenfuss–Bach inequality representing upper and lower bounds for the function x sec2 x − tan x for 0 ≤ x < π/2 is also investigated. An open problem posed by Sun and Zhu concerning this last inequality is established

Correct Approximation of IEEE 754 Floating-Point Arithmetic for Program Verification

Verification of programs using floating-point arithmetic is challenging on several accounts. One of the difficulties of reasoning about such programs is due to the peculiarities of floating-point arithmetic: rounding errors, infinities, non-numeric objects (NaNs), signed zeroes, denormal numbers, different rounding modes, etc. One possibility to reason about floating-point arithmetic is to model a program computation path by means of a set of ternary constraints of the form z = x op y and use constraint propagation techniques to infer new information on the variables' possible values. In this setting, we define and prove the correctness of algorithms to precisely bound the value of one of the variables x, y or z, starting from the bounds known for the other two. We do this for each of the operations and for each rounding mode defined by the IEEE 754 binary floating-point standard, even in the case the rounding mode in effect is only partially known. This is the first time that such so-called filtering algorithms are defined and their correctness is formally proved. This is an important slab for paving the way to formal verification of programs that use floating-point arithmetics.Comment: 64 pages, 19 figures, 2 table

Approximating Mills ratio

Consider the Mills ratio f(x) =1 − Φ(x)/φ(x), x ≥ 0, where φ is the density function of the standard Gaussian law and Φ its cumulative distribution. We introduce a general procedure to approximate f on the whole [0, ∞) which allows to prove interesting properties where f is involved. As applications we present a new proof that 1/f is strictly convex, and we give new sharp bounds of f involving rational functions, functions with square roots or exponential terms. Also Chernoff type bounds for the Gaussian Q-function are studied

The neural circuitry of fear conditioning : a theoretical account

In the last decades, fear conditioning has been established as one of the most successful paradigms for studying the neural substrates of emotional learning. Experimental research has revealed a complex circuitry of brain regions—most prominently the amygdala—underlying the acquisition, extinction and generalization of conditioned fear. As the wealth of experimental data grows, theoretical models that help interpret results and generate new hypotheses play an increasingly important role. In this thesis, two computational models of the neural substrates of fear conditioning are presented. The first model is a biologically realistic spiking neural network model of the central amygdala, the main output structure of the amygdala. Based on a recent experimental study that demonstrated the importance of tonic extrasynaptic inhibition for fear generalization, the effects of changes in neuronal membrane conductance on input processing are analyzed in the model. Consistent with experimental results, it is shown that subpopulation-specific changes in tonic inhibitory conductance increase the responsiveness of the network to phasic inputs, presumably causing the increase in fear generalization. On the basis of this result, the model is analyzed from a functional perspective. It is argued that tonic inhibition in the central amygdala acts as a controller by which network sensitivity is flexibly adjusted to relevant features of the environment, such as predictability of threat, and concrete predictions that follow from this proposition as well as possible adjustment mechanisms are discussed. In addition, a systems level model is presented that is based on a recent high-level approach to conditioning and proposes a specific physiological implementation in the basolateral amygdala, prefrontal cortex and the intercalated cell clusters of the amygdala. It is a central hypothesis of the model that the interaction between fear and extinction neurons in the basal amygdala, which has been described experimentally, is a neural substrate of the switching between socalled latent states, which allow the animal to organize its experience and infer structure in the environment. Important behavioral phenomena are reproduced in the model and the effect of de-activation of model structures is shown to be in good agreement with results from lesion studies. Finally, predictions and questions that follow from the main hypothesis are considered. Taken together, the two models provide a coherent theoretical account of the neural basis of acquisition and extinction of conditioned fear, as well as the control of fear generalization. Importantly, this account combines different levels of analysis. By virtue of this combination, the scope of predictions that can be derived is expanded and the models become more amenable to experimental testing