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.

Minimization of errors in discrete wavelet filtering of signals during ultrasonic measurements and testing

Error minimizing methods for discrete wavelet filtering of ultrasonic meter signals are considered. For this purpose, special model signals containing various measuring pulses are generated. The psi function of the Daubechies 28 wavelet is used to generate the pulses. Noise is added to the generated pulses. A comparative analysis of the two filtering algorithms is performed. The first algorithm is to limit the amount of detail of the wavelet decomposition coefficients in relation to signal interference. The minimum value of the root mean square error of wavelet decomposition signal deviation which is restored at each level from the initial signal without noise is determined. The second algorithm uses a separate threshold for each level of wavelet decomposition to limit the magnitude of the detail coefficients that are proportional to the standard deviation. Like in the first algorithm, the task is to determine the level of wavelet decomposition at which the minimum standard error is achieved. A feature of both algorithms is an expanded base of discrete wavelets ‒ families of Biorthogonal, Coiflet, Daubechies, Discrete Meyer, Haar, Reverse Biorthogonal, Symlets (106 in total) and threshold functions garotte, garrote, greater, hard, less, soft (6 in total). The model function uses random variables in both algorithms, so the averaging base is used to obtain stable results. Given features of algorithm construction allowed to reveal efficiency of ultrasonic signal filtering on the first algorithm presented in the form of oscilloscopic images. The use of a separate threshold for limiting the number of detail coefficients for each level of discrete wavelet decomposition using the given wavelet base and threshold functions has reduced the filtering error.

An Elementary Proof to eγ log log n > Ψ(n)/n for any n > 30

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

An Elementary Proof to the Riemann Hypothesis

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

An Elementary Proof to the Riemann Hypothesis

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

Ferulic Acid and Salicylic Acid Foliar Treatments Reduce Short-Term Salt Stress in Chinese Cabbage by Increasing Phenolic Compounds Accumulation and Photosynthetic Performance

Salinity stress is one of the most damaging abiotic stresses to plants, causing disturbances in physiological, biochemical, and metabolic processes. The exogenous application of natural metabolites is a useful strategy to reduce the adverse effects of stress on crops. We investigated the effect of foliar application of salicylic acid (SA) and ferulic acid (FA) (10–100 μM) on short-term salt-stressed (150 mM NaCl, 72 h) Chinese cabbage plants. Subsequently, proline level, photosynthetic performance, phenolic metabolites with special focus on selected phenolic acids (sinapic acid (SiA), FA, SA), flavonoids (quercetin (QUE), kaempferol (KAE)), and antioxidant activity were investigated in salt-stressed and phenolic acid-treated plants compared with the corresponding controls. Salt stress caused a significant increase in SA and proline contents, a decrease in phenolic compounds, antioxidant activity, and photosynthetic performance, especially due to the impairment of PSI function. SA and FA treatments, with a concentration of 10 μM, had attenuated effects on salt-stressed plants, causing a decrease in proline and SA level, and indicating that the plants suffered less metabolic disturbance. Polyphenolic compounds, especially FA, SiA, KAE, and QUE, were increased in FA and SA treatments in salt-stressed plants. Consequently, antioxidant activities were increased, and photosynthetic performances were improved. FA resulted in a better ameliorative effect on salt stress compared to SA.

Identification of a Novel Cis-Acting Regulator of HIV-1 Genome Packaging

Human immunodeficiency virus type 1 (HIV-1) uptakes homo-dimerized viral RNA genome into its own particle. A cis-acting viral RNA segment responsible for this event, termed packaging signal (psi), is located at the 5′-end of the viral genome. Although the psi segment exhibits nucleotide variation in nature, its effects on the psi function largely remain unknown. Here we show that a psi sequence from an HIV-1 regional variant, subtype D, has a lower packaging ability compared with that from another regional variant, HIV-1 subtype B, despite maintaining similar genome dimerization activities. A series of molecular genetic investigations narrowed down the responsible element of the selective attenuation to the two sequential nucleotides at positions 226 and 227 in the psi segment. Molecular dynamics simulations predicted that the dinucleotide substitution alters structural dynamics, fold, and hydrogen-bond networks primarily of the psi-SL2 element that contains the binding interface of viral nucleocapsid protein for the genome packaging. In contrast, such structural changes were minimal within the SL1 element involved in genome dimerization. These results suggest that the psi 226/227 dinucleotide pair functions as a cis-acting regulator to control the psi structure to selectively tune the efficiency of packaging, but not dimerization of highly variable HIV-1 genomes.

Existence theorems for $ \Psi $-fractional hybrid systems with periodic boundary conditions

<abstract><p>This research paper deals with two novel varieties of boundary value problems for nonlinear hybrid fractional differential equations involving generalized fractional derivatives known as the $ \Psi $-Caputo fractional operators. Such operators are generated by iterating a local integral of a function with respect to another increasing positive function $ \Psi $. The existence results to the proposed systems are obtained by using Dhage's fixed point theorem. Two pertinent examples are provided to confirm the feasibility of the obtained results. Our presented results generate many special cases with respect to different values of a $ \Psi $ function.</p></abstract>