A class of completely monotonic functions involving the polygamma functions

Let $\Gamma (x)$ Γ ( x ) denote the classical Euler gamma function. We set $\psi _{n}(x)=(-1)^{n-1}\psi ^{(n)}(x)$ ψ n ( x ) = ( − 1 ) n − 1 ψ ( n ) ( x ) ($n\in \mathbb{N}$ n ∈ N ), where $\psi ^{(n)}(x)$ ψ ( n ) ( x ) denotes the nth derivative of the psi function $\psi (x)=\Gamma '(x)/\Gamma (x)$ ψ ( x ) = Γ ′ ( x ) / Γ ( x ) . For λ, α, $\beta \in \mathbb{R}$ β ∈ R and $m,n\in \mathbb{N}$ m , n ∈ N , we establish necessary and sufficient conditions for the functions $$ L(x;\lambda ,\alpha ,\beta )=\psi _{m+n}(x)-\lambda \psi _{m}(x+ \alpha ) \psi _{n}(x+\beta ) $$ L ( x ; λ , α , β ) = ψ m + n ( x ) − λ ψ m ( x + α ) ψ n ( x + β ) and $-L(x;\lambda ,\alpha ,\beta )$ − L ( x ; λ , α , β ) to be completely monotonic on $(-\min (\alpha ,\beta ,0),\infty )$ ( − min ( α , β , 0 ) , ∞ ) .As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

Random Compressed Coding with Neurons

Classical models of efficient coding in neurons assume simple mean responses--'tuning curves'--such as bell shaped or monotonic functions of a stimulus feature. Real neurons, however, can be more complex: grid cells, for example, exhibit periodic responses which impart the neural population code with high accuracy. But do highly accurate codes require fine tuning of the response properties? We address this question with the use of a benchmark model: a neural network with random synaptic weights which result in output cells with irregular tuning curves. Irregularity enhances the local resolution of the code but gives rise to catastrophic, global errors. For optimal smoothness of the tuning curves, when local and global errors balance out, the neural network compresses information from a high-dimensional representation to a low-dimensional one, and the resulting distributed code achieves exponential accuracy. An analysis of recordings from monkey motor cortex points to such 'compressed efficient coding'. Efficient codes do not require a finely tuned design--they emerge robustly from irregularity or randomness.

Improving correlation accuracy of crashworthiness applications by combining the CORA and MADM methods

With the increasing use of Computer Aided Engineering, it has become vital to be able to evaluate the accuracy of numerical models. This research poses the problem of selection of the most accurate and relevant correlation solution to a set of corridor variations. Specific methods such as CORA, widely accepted in industry, are developed to objectively evaluate the correlation between monotonic functions, while the Minimum Area Discrepancy Method, or MADM, is the only method to address the correlation of non-injective mathematical variations, usually related to force/acceleration versus displacement problems. Often, it is not possible to differentiate objectively various solutions proposed by CORA, which this paper proposes to answer. This research is original, as it proposes a new innovative correlation optimisation framework, which can select the best CORA solution by including MADM as a subsequent process. The paper and the methods are rigorous, having used an industry standard driver airbag computer model, built virtual test corridors and compared the relationship between different CORA and MADM ratings from 100 Latin Hypercube samples. For the same CORA value of ‘1’ (perfect correlation), MADM was capable to objectively differentiate between 13 of them and provide the best correlation possible. The paper has recommended the MADM settings n = 1; m = 2 or n = 3; m = 2 for a congruent relationship with CORA. As MADM is performed subsequently, this new framework can be implemented in already existing industrial processes and provide automotive manufacturers and Original Equipment Manufacturers (OEM) with a new tool to generate more accurate computer models.

On Some Complete Monotonicity of Functions Related to Generalized k − Gamma Function

In this paper, we presented two completely monotonic functions involving the generalized k − gamma function Γ k x and its logarithmic derivative ψ k x , and established some upper and lower bounds for Γ k x in terms of ψ k x .

Monotonic Functions Method and Its Application to Staging of Patients with Prostate Cancer According to Pretreatment Data

Prostate cancer is the second most frequent malignancy (after lung cancer). Preoperative staging of PCa is the basis for the selection of adequate treatment tactics. In particular, an urgent problem is the classification of indolent and aggressive forms of PCa in patients with the initial stages of the tumor process. To solve this problem, we propose to use a new binary classification machine-learning method. The proposed method of monotonic functions uses a model in which the disease’s form is determined by the severity of the patient’s condition. It is assumed that the patient’s condition is the easier, the less the deviation of the indicators from the normal values inherent in healthy people. This assumption means that the severity (form) of the disease can be represented by monotonic functions from the values of the deviation of the patient’s indicators beyond the normal range. The method is used to solve the problem of classifying patients with indolent and aggressive forms of prostate cancer according to pretreatment data. The learning algorithm is nonparametric. At the same time, it allows an explanation of the classification results in the form of a logical function. To do this, you should indicate to the algorithm either the threshold value of the probability of successful classification of patients with an indolent form of PCa, or the threshold value of the probability of misclassification of patients with an aggressive form of PCa disease. The examples of logical rules given in the article show that they are quite simple and can be easily interpreted in terms of preoperative indicators of the form of the disease.

On Landau-Kolmogorov problem on the interval for absolutely monotonic functions

We solve the Landau-Kolmogorov problem and the Kolmogorov problem for three positive numbers on the class of functions which are absolutely monotonic on a finite interval.

On inequalities of Kolmogorov type for classes of functions that are multiply monotonic on a finite interval

We solve the problem about exact constants in additive Kolmogorov-type inequalities on the class of multiply monotonic functions, defined on a finite interval.