For any odd positive integer $x$, define $(x_n)_{n\geqslant 0} $ and $(a_n )_{n\geqslant 1} $ by setting $x_{0}=x, \,\, x_n =\cfrac{3x_{n-1} +1}{2^{a_n }}$ such that all $x_n $ are odd. The 3x+1 problem asserts that there is an $x_n =1$ for all $x$. Usually, $(x_n )_{n\geqslant 0} $ is called the trajectory of $x$. In this paper, we concentrate on $(a_n )_{n\geqslant 1} $ and call it the E-sequence of $x$. The idea is that, we consider any infinite sequence $(a_n )_{n\geqslant 1} $ of positive integers and call it an E-sequence. We then define $(a_n )_{n\geqslant 1} $ to be $\Omega-$convergent to $x$ if it is the E-sequence of $x$ and to be $\Omega-$divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the $\Omega-$divergence of all non-periodic E-sequences implies the periodicity of $(x_n )_{n\geqslant 0} $ for all $x_0$. The principal results of this paper are to prove the $\Omega-$divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences $(a_n )_{n\geqslant 1} $ with $\mathop {\overline {\lim } }\limits_{n\to \infty } \cfrac{b_n }{n}>\log _23$ are $\Omega-$divergent by using the Wendel's inequality and the Matthews and Watts's formula $x_n =\cfrac{3^n x_0 }{2^{b_n }}\prod\limits_{k=0}^{n-1} {(1+\cfrac{1}{3x_k })} $, where $b_n =\sum\limits_{k=1}^n {a_k } $. These results present a possible way to prove the periodicity of trajectories of all positive integers in the 3x + 1 problem and we call it the E-sequence approach.

The effect of divergence of chute sidewalls with three different baffle block geometries namely USBR, trihedral and semicircular blocks, as well as the depth and dimensions of the scour hole downstream of the chute were studied using a physical model. For this purpose, 9 models of baffled chutes were designed and constructed with divergence ratios of 1.45, 1.75, and 2.45 and without divergence (with a divergence ratio of 1). Comparing the results on the effect of block geometry at different divergence ratios revealed that the use of blocks proposed in this study instead of standard USBR blocks reduced the mean and maximum scour hole by 50%. For a given block geometry, the mean depth, maximum depth, and length of scour hole were reduced by 75%, 58%, and 50%, respectively.

The advances mobile communications has seen in recent years has rendered the radio spectrum a limited and, hence, an expensive resource. Therefore, technologies that support unlicensed access to spectrum are needed. Therefore, the adoption of novel modulation schemes becomes of utmost importance to obtain better spectral-localization and reduce the OOBE (\textit{Out of Band Emission}) inherent to OFDM (\textit{Orthogonal Frequency Division Multiplexing}) and, consequently, mitigating the interference between secondary (\textit{unlicensed}) and primary users. In this scenario, we assess the gain in the bit error probability using f-OFDM (\textit{filtered-OFDM}) in MIMO systems, both used in the 5G RANGE project.

In many application problems, F-transform algorithms are very efficient. In F-transform techniques, we replace the original signal or image with a finite number of weighted averages. The use of weighted average can be naturally explained, e.g., by the fact that this is what we get anyway when we measure the signal. However, most successful applications of F-transform have an additional not-so-easy-to-explain feature: the partition requirement, that the sum of all the related weighting functions is a constant. In this paper, we show that this seemingly difficult-to-explain requirement can also be naturally explained in signal-measuring terms: namely, this requirement can be derived from the natural desire to have all the signal values at different moments of time estimated with the same accuracy.

The main ideas of F-transform came from representing expert rules. It would be therefore re reasonable to expect that the more accurately the membership functions describe human reasoning, the more efficient will be the corresponding F-transform formulas. We know that an adequate description of our reasoning corresponds to complicated membership functions -- however, somewhat surprisingly, most efficient applications of F-transform use the simplest possible triangular membership functions. There exist some explanations for this phenomenon which are based on local behavior of the signal. In this paper, we supplement this local explanation by a global one: namely, we prove that triangular membership functions are the only one that provide the accurate description of appropriate global characteristics of the signal.

Although Shannon mutual information has been widely used, its effective calculation is often difficult for many practical problems, including those in neural population coding. Asymptotic formulas based on Fisher information sometimes provide accurate approximations to the mutual information but this approach is restricted to continuous variables because the calculation of Fisher information requires derivatives with respect to the encoded variables. In this paper, we consider information-theoretic bounds and approximations of the mutual information based on Kullback--Leibler divergence and Rényi divergence. We propose several information metrics to approximate Shannon mutual information in the context of neural population coding. While our asymptotic formulas all work for discrete variables, one of them has consistent performance and high accuracy regardless of whether the encoded variables are discrete or continuous. We performed numerical simulations and confirmed that our approximation formulas were highly accurate for approximating the mutual information between the stimuli and the responses of a large neural population. These approximation formulas may potentially bring convenience to the applications of information theory to many practical and theoretical problems.

This paper considers estimating the ratio of two distributions with different parameters and common supports. We consider a Bayesian approach based on the Log--Huber loss function which is resistant to outliers and useful to find robust M-estimators. We propose two different types of Bayesian density ratio estimators and compare their performance in terms of Bayesian risk function with themselves as well as the usual plug-in density ratio estimators. Some applications such as classification and divergence function estimation are addressed.

Here, we have discovered a X-ray excited long afterglow phosphor β-NaYF₄: Tb³⁺. After the irradiation of X-ray, the green emission can persist for more than 240 h. After 36 h, the afterglow intensity arrived at 0.69 mcd•m^-2, which can clearly be observed by naked eyes. Even after 84 h, the afterglow emission brightness still reached 0.087 mcd•m^-2. Also, combined with the results of thermoluminescence and photoluminescence, the super long afterglow emission of β-NaYF₄: Tb³⁺ can be ascribed to the tunneling model associated with F centers. More importantly, the super long green afterglow emission of β-NaYF₄: Tb³⁺ has been successfully used as in vivo light source to activate g-C₃N₄ for photodynamic therapy（PDT）and bacteria destruction. Furthermore, super long persistent luminescence of β-NaYF₄: Tb³⁺ could be repeatedly charged by X-ray for many circulations, which indicates that the phosphors have high photo stability under repeated cycles of alternating X-ray irradiation.

Background: Identification of cancer biomarkers that are differentially expressed (DE) under two biological conditions is an important task in many microarray studies. There exist several methods in the literature in this regards and most of these methods designed especially for unpaired samples, which does not satisfy the requirements of paired samples where the gene expressions are taken from the same patients before and after treatment. Furthermore, the traditional biomarker identification methods based on either p-values or fold change (FC) values. However, sometimes, p-value based results do not comply with FC based results due to the smaller variance of gene expressions. There are some methods that combine both p-values and FC values to solve this problem. But, these methods also show weak performance for small-sample case in presence of outlying expressions. To overcome this problem, in this paper an attempt is made to develop a hybrid robust SAM-FC approach by combining rank of FC values and rank of p-values based on SAM statistic using minimum β-divergence method, which is designed for paired samples. This method introduces a weight function known as β-weight function. This weight function produces larger weights corresponding to usual/normal expressions and smaller weights for unusual/outlying expressions. The β-weight function plays the significant role on the performance of the proposed method. Results: The proposed method uses β-weight function as a measure of outlier detection by setting β=0.2. We unify both classical and robust estimates using β-weight function such that maximum likelihood estimators (MLEs) are used in absence of outliers and minimum β-divergence estimators are used in presence of outliers to obtain reasonable p-values and FC values in the proposed method. We examined the performance of proposed method in a comparison of some popular methods (t-test, SAM, LIMMA, Wilcoxon, WAD, RP and FCROS) using both simulated and real gene expression profiles for both small-and large-sample cases. From the simulation and a real spike in data analysis results we observed that the proposed method outperforms other methods for small-sample case in presence of outliers and it keeps almost equal performance with other robust methods (Wilcoxon, RP and FCROS) otherwise. From a head-and-neck cancer (HNC) dataset the proposed method identified 2 genes (CYP3A4, NOVA1) that are significantly enriched in linoleic acid metabolism, drug metabolism, steroid hormone biosynthesis and metabolic pathways. The survival analysis through Kaplan-Meier curve revealed that combined effect of these 2 genes has prognostic capability and they might be promising biomarker of HNC. Moreover, we retrieved the 12 candidate drugs based on gene interaction from glad4u and drug bank databases. Conclusion The identified drugs showed statistical significance and critical role of the proteins indicate that these proteins might be therapeutic target in cancer. Thus, elucidating the associations between the drugs identified in the present study require further investigations.

\textsc{Background} Often, in biology, we are faced with the problem of exploring relevant unknown biological hypotheses in the form of myriads of combination of factors that might be affecting the pathway under certain conditions. Currently, a major problem in biology is to cherry pick the combinations based on expert advice, literature survey or guesses for investigation. The search and wet lab testing of these combinations costs a lot in terms of time, investment and energy. In a recent development of the PORCN-WNT inhibitor ETC-1922159 for colorectal cancer, a list of down-regulated genes were recorded in a time buffer after the administration of the drug. The regulation of the genes were recorded individually but for a majority, it is still not known which higher ($\geq 2$) order combinations might be playing a greater role in the pathway. \textsc{Results} The pipeline provides a prioritised list of important $2^{nd}$ order combinations of a range of family of genes involved in the Wnt pathway. More specifically, it reveals the various unexplored FZD-WNT combinations that have been untested till now in the pathway. In relation to ETC-1922159 affected combinations, the down-regulation of LGR-RNF family after the drug treatment is evident in these rankings as it takes bottom priorities for LGR5-RNF43 combination. The LGR6-RNF43 takes higher ranking than LGR5-RNF43, indicating that it might not be playing a greater role as LGR5 during the Wnt enhancing signals. These rankings confirm the efficacy of the proposed search engine design. \textsc{Conclusion} A pipeline has been developed to prioritise an $n^{th}$ order combination of factors that affect a signaling pathway. It takes into account the sensitivity indices computed from variance based (SOBOL) and density-kernel based (HSIC) methods to estimate the influence of each factor or combination of factors. These are then fed as feature vectors into a powerful support vector ranking algorithm that produces a ranked list of the interactions/combinations.

To extend the calculation power of tensor analysis, we introduce four new definition of tensor calculations. Some useful tensor identities have been proved. We demonstrate the application of the tensor identities in continuum mechanics: momentum conservation law and deformation superposition.

Phylostratigraphic analysis is a way to look anew on phylogenetic data in the evolutionary aspect. It allows counting the evolutionary age based on the analysis of genes, their orthologs and finding the last common ancestor. We performed phylostratigraphic analysis of Arabidopsis thaliana genes associated with several types of abiotic stresses (heat, cold, water-related, light, osmotic, salt, and oxidative) determined by the Gene Ontology annotation. Comparison of the distributions of ages of genes associated with stresses of different type has shown the heat stress to involve older genes while the light stress – younger genes. At the same time, all types of stress are characterized by a significantly higher proportion of old genes (common to all eukaryotes) compared to the whole set of A.thaliana genes. This can be explained by the involvement of basic molecular processes in plant cells into the stress response. Reconstruction and graphical analysis of the gene network of the heat stress educed several clusters associated with different response functions. Some of these clusters contain only ancient genes. The results obtained show that the phylostratigraphic analysis reveals the fundamental features of the organization of gene networks and their evolution.

This paper discusses the computer-aided (CAD) classification between Hepatocellular Carcinoma (HCC), i.e., the most common type of liver cancer, and Liver Abscess, based on ultrasound image texture features and Support Vector Machine (SVM) classifier. Among 79 cases of liver diseases, with 44 cases of HCC and 35 cases of liver abscess, this research extracts 96 features of Gray-Level Co-occurrence Matrix (GLCM) and Gray-Level Run-Length Matrix (GLRLM) from the region of interests (ROIs) in ultrasound images. Three feature selection models, i) Sequential Forward Selection, ii) Sequential Backward Selection, and iii) F-score, are adopted to determine the identification of these liver diseases. Finally, the developed system can classify HCC and liver abscess by SVM with the accuracy of 88.875%. The proposed methods can provide diagnostic assistance while distinguishing two kinds of liver diseases by using a CAD system.

A new series of 1,3,4-oxadiazoles derivatives was synthesized, characterized and evaluated for their in vitro and in vivo anti-thrombotic activity. Compounds (3a-3i) exhibited significant clot lysis with respect to negative control and reference drug streptokinase (30,000 IU) while enhanced clotting time (CT) values were observed (130-342 sec) for these tested compounds than the standard drug heparin (110 sec.). High affinity towards 1NFY with greater docking score was observed for the compounds (3a, 3i, 3e, 3d and 3h) than the control ligand RPR200095. In addition, very good inhibitory potential against factor Xa (F-Xa) was observed with higher docking scores (5612-6270) with ACE values (–189.68 to –352.28 kcal/mol) than the control ligand RPR200095 (Docking score 5192; ACE –197.81 kcal/mol. In vitro, in vivo and in silico results proposed that these newly synthesized compounds can be used as anti-coagulant agents.

In the recent past about 40 botulinum neurotoxin (BoNT) subtypes belonging to serotypes A, B, E, and F pathogenic to humans were identified among hundreds of independent isolates. BoNTs are the etiological factors of botulism and represent potential bioweapons, but are also recognized pharmaceuticals for efficient counteraction of hyperactive nerve terminals in a variety of human diseases. The detailed biochemical characterization of subtypes as the basis for development of suitable countermeasures and possible novel therapeutic applications is lagging behind the increase in new subtypes. Here we report the primary structure of a ninth subtype of BoNT/F. Its amino acid sequence diverges by at least 8.4% at the holotoxin and 13.4% on the enzymatic domain level from all other known BoNT/F subtypes. We found that BoNT/F9 shares the scissile Q⁵⁸/K⁵⁹ bond in its substrate vesicle associated membrane protein 2 with the prototype BoNT/F1. Comparative biochemical analyses of four BoNT/F enzymatic domains showed that the catalytic efficiencies decrease in the order F1 > F7 > F9 > F6 and vary by up to factor eight. K_M values increase in the order F1 > F9 > F6 ≈ F7, whereas k_cat decreases in the order F7 > F1 > F9 > F6. Comparative substrate scanning mutagenesis studies revealed a unique pattern of crucial substrate residues for each subtype. Based upon structural coordinates of F1 bound to an inhibitor polypeptide the mutational analyses suggest different substrate interactions in the substrate binding channel of each subtype.

Nowadays, we live in a world in which people are facing with a lot of data that should be stored or displayed. One of the key methods to control and manage this data refers to grouping and classifying them in clusters. Today, clustering has a critical role in information retrieval methods for organizing large collections inside a few significant clusters. One of the main motivations for the use of clustering is to determine and reveal the hidden and inherent structure of a set of data. Ensemble clustering algorithms combine multiple clustering algorithms to finally reach an overall clustering system. Ensemble clustering methods by lack of information fusing utilize several primary partitions of data to find better ways. Since various clustering algorithms look at the different data points, they can produce various partitions from such data. It is possible to create a partition with high performance by combining the partitions obtained from different algorithms, even if the clusters to be very dense from each other. Most studies in this area have examined all the initial clusters. In this study, a new method is used in which the most sustainable clusters are utilized instead of all primary produced clusters. Consensus function based on co-association matrixes used to select more stable clusters. The most stable clusters selection method is done by cluster stability criterion based on F-measure. Optimization functions are used to optimize the obtained final clusters. The genetic algorithm is the optimizer used in this article to find the ultimate clusters participated in a consensus. Experimental results on several datasets show that the output of proposed method is various clusters with high stability.

The Inverted F Antenna (IFA) with the parasitic element on a finite conducting plane is proposed in the range frequency of 0.1 to 5.5 GHz and its characteristics are analyzed numerically. In this research, the parasitic element and the main IFA are investigated to obtain the resonance frequency for multiband operation purposes. The parasitic element is placed beneath adjusted to the main IFA to derive more frequency bands. The distance between the parasitic element and main horizontal element extremely affects the performance of the IFA. It is found that when the parasitic element is located closer to the conducting plane, this element is coupled by the current on the conducting plane. Consequently, the return loss bandwidth becomes narrower. Therefore, the gain of the proposed IFA becomes a bit higher. The antenna gain is about 8.21 dB at band #3 (λ_1.747), 7.43 dB at band #5 (λ_2.967) and 8.82 dB at band #6 (λ_4.023). This occurs when the calculation condition is antenna height h1 = 23 mm, h2 = 21 mm, horizontal antenna elements L = 173.2 mm, L1 = 140.9 mm and Lp = 152 mm, shorted element Ls = 30.7 mm, the distance between parasitic element and shorted element pyl = 5 mm. While the size of conducting plane is considered pxp+pxm by pyp+pym as 57.5+57.5 mm by 200+50 mm. In the numerical analysis, the electromagnetic simulator WIPL-D based on Method of Moment is used. The results show that the proposed IFA has UHF and SHF channel receiver which are suitable for advanced wireless service (band #3), mobile phones, Bluetooth, maritime service, radiolocation service (band #5) and radars, mobile phones, commercial wireless LAN (band #6).

Formation of Fe- and Mg-rich smectite and zeolite under alkaline conditions is concerned as secondary minerals after alkaline alteration of bentonite in a repository of radioactive wastes. It might be crucial for safety assessment whether smectite will be formed or not as secondary minerals after alkaline alteration of bentonite. In present paper, Fe- and Mg-rich smectite which are currently interacting with hyperalkaline groundwater was found at Narra in Palawan, Philippines. Mineralogical and geochemical investigation was conducted to understand formation process of smectite and factors determined secondary mineral species. Our study revealed a certain amount of smectite may have been produced under hyperalkaline conditions, altered from amorphous or poorly crystalline material such as M-S-H and F-S-H. Key factor which decides smectite or zeolite as secondary minerals after alkaline alteration of bentonite might be whether nuclei of M-S-H and/or F-S-H will be formed or not. This might be decided by the presence of dissolved Mg2+ and Fe2+ in the system. Our suggested formation process of smectite under alkaline conditions is analogue with generally-accepted model of smectite formation that might have been occurred on early Mars.

Radar transmit signal design is a critical factor for the radar performance. In this paper, we investigate the problem of radar signal waveform design under the small signal power conditions for detecting a doubly spread target, whose impulse response can be modeled as a random process, in a colored noise environment. The doubly spread target spans multiple range bins (range-spread) and its impulse response is time-varying due to fluctuation (hence also Doppler-spread), such that the target impulse response is both time-selective and frequency-selective. Instead of adopting the conventional assumption that the target is wide-sense stationary uncorrelated scattering,we assume that the target impulse response is both wide-sense stationary in range and in time to account for the possible correlation between the impulse responses corresponding to close range intervals. The locally most powerful detector, which is asymptotically optimal for small signal cases, is then derived for detecting such targets. The signal waveform is optimized to maximizing the detection performance of the detector or equivalently maximizing the Kullback-Leibler divergence. Numerical simulations validate the effectiveness of the proposed waveform design for the small signal power conditions and performance of optimum waveform design are shown in comparison to the frequency modulated waveform.

In this paper a robust version of the Wald test statistic for composite likelihood is considered by using the composite minimum density power divergence estimator instead of the composite maximum likelihood estimator. This new family of test statistics will be called Wald-type test statistics. The problem of testing a simple and a composite null hypothesis is considered and the robustness is studied on the basis of a simulation study. Previously, the composite minimum density power divergence estimator is introduced and its asymptotic properties are studied.

Watermelon fusarium wilt caused by Fusarium oxysporum f.sp. niveum (FON) is a destructive soil-borne disease throughout the world leading to serious economic losses and limit watermelon production. Cuminic acid, extracted from the seed of Cuminum cyminum L., belongs to benzoic acid analogues. In this study, the median effective concentration (EC₅₀) values for cuminic acid in inhibiting mycelial growth of FON was 22.53μg/mL. After treatment with cuminic acid, mycelial morphology was seriously influenced; cell membrane permeability and glycerol content were increased markedly, but pigment and mycotoxin (mainly fusaric acid) were significantly decreased. Synthesis genes of bikaverin and fusaric acid both were down regulated compared with the control confirmed by quantitative RT-PCR. In greenhouse experiments, cuminic acid at all concentrations displayed significant bioactivities against FON. Importantly, significant enhancement of activities of SOD, POD, CAT and decrease of MDA content after cuminic acid treatment in watermelon leaves were observed in vivo. These indicated that cuminic acid not only showed high antifungal activity, but also could enhance the self-defense system of the host plant. Above all, cuminic acid showed the potential as a biofungicide to control FON.

Analysis of 49 strains of F. langsethiae originating from northern Europe (Russia, Finland, Sweden, UK, Norway, and Latvia) revealed the presence of spontaneous auxotrophic mutants that reflect natural intraspecific diversity. Our investigations detected that 49.0% of F. langsethiae strains were auxotrophic mutants for biotin, and 8.2% of the strains required thiamine as a growth factor. They failed to grow on vitamin-free media. For both prototrophic and auxotrophic strains, no growth defect was observed in rich organic media. Without essential vitamins, a significant reduction in the growth of the auxotrophic strains results in a decrease of the formation of T-2 toxin and diacetoxyscirpenol. In addition, all analysed F. langsethiae strains were distinguished into two subgroups based on PCR product sizes. According to our results, 26 and 23 strains of F. langsethiae belong to subgroups I and II respectively. We determined that the deletion in the IGS region of the rDNA of F. langsethiae belonging to subgroup II is linked with temperature sensitivity and causes a decrease in strain growth at 30 °C. Four thiamine auxotrophic strains were found in subgroup I, while 21 biotin auxotrophic strains were detected in subgroups II. To the best of our knowledge, the spontaneous mutations in F. langsethiae observed in the present work have not been previously reported.