This study aimed to compare procedural learning skills between Spanish-speaking preschool children (ages 4 years to 4 years, 11 months) with Developmental Language Disorder (DLD) and their typically developing (TD) peers. Using the Serial Reaction Time task, participants (30 DLD and 30 TD children) respond to visual stimuli in a sequenced manner over four blocks, followed by a random order block. The task assessed reaction time (RT) and accuracy. Results showed a significant interaction between group and block for reaction time (RT) and accuracy, with children with DLD exhibiting longer RTs and accuracy deficits across blocks. In contrast, the TD group showed higher RT efficiency and accuracy in the sequential blocks and as expected, decreased performance in the random block according to the experimental manipulation. Overall, the results of this investigation suggest that there was no implicit learning in the DLD group, as indicated by the SRT task paradigms of procedural memory. These findings align with the Procedural Deficit Hypothesis (PDH), suggesting that linguistic deficits in the TD population may derive from a general dysfunction of the procedural memory system domain in the Spanish context.

Prime numbers are the atoms of mathematics and mathematics is needed to make sense of the real world. Finding the Prime number structure and eventually being able to crack their code is the ultimate goal in what is called Number Theory. From the evolution of species to cryptography, Nature finds help in Prime numbers. One of the most important advances in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In that paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x) and the roots (zeros) of the zeta function defined by: Where ζ(z) is a function of a complex variable z that analytically continues the Dirichlet series. Riemann also formulated a conjecture about the location of the zeros of RZF, which fall into two classes: the "trivial zeros" -2, -4, -6, etc., and those whose real part lies between 0 and 1. Riemann's conjecture Riemann hypothesis [RH] was formulated as this: [RH] The real part of every nontrivial zero z* of the RZF is 1/2. Proving the RH is, as of today, one of the most important problems in mathematics. In this paper we will provide a proof of the RH. The proof of the RH will be built following these five parts: PART 1: Description of the RZF PART 2: The C-transformation PART 3: Application of the C-transformation to in Re(z) 0 to obtain ζ(z)=X(z)-Y(z) PART 4: Analysis of the values of z such that X(z)=Y(z), and |X(z)|=|Y(z)|, that equates to ζ(z)=0 Proof that |X(z)|=|Y(z)| only if Re(z)=1/2 Conclude that ζ(z)=0 only if Re(z)=1/2 for Re(z) 0 PART 5: We will also prove that all non-trivial zeros of ζ(z) in the critical line of the form are not distributed randomly. There is a relationship between the values of those zeros and the Harmonic function that leads to an algebraic relationship between any two zeros.

This work uses a new approach to investigate the Riemann hypothesis, drawing conclusions about its trueness. It is based on the more general method presented in a very recent publication, by the same author, showing in detail how to approach that very well-known and interesting problem. This is achieved by means of the theory of dynamical systems (Poincaré index associated to equilibria of 2-dimensional systems) and the study of the zeros of the Dirichlet eta function, defined by a Dirichlet series. By using the well-known fact that the zeros of the eta function include all zeros of the Riemann zeta function in the (open) critical strip, excluded the critical line, ((0,1/2) ∪ (1/2,1)) × (−∞,∞), the development proceeds using only the eta function. In addition, the open and simply connected region (1/2,1) × (0,∞) is used along the text, taking into account the symmetries of zeros of the functions under analysis in the critical strip. The basic line of proof is to find the mathematical expression for the Poincaré index of the vector field associated to the eta function, assuming the existence of a zero of the eta function outside the critical line (in (1/2,1) × (0,∞)), and investigating the resulting unfoldings. Eventually, an inconsistency occurs and the proof ends by contradiction.

Aim: This study aimed to carrying out a prospective survey on socio-demographics and lifestyle factors among a population of Caribbean immigrants living in the US. Materials and Methods: The data were processed and analysed using the SPSS software and Excel. Crosstabulations were done. The Chi-square test was use to evaluate different hypotheses in this study. Statistical significance was defined as p<0.05.Results: Gender was found statistically significant difference with the country of birth of the Caribbean immigrants (p=0.038), and in the cleanness of their neighbourhoods (p=0.045). There were differences in occupations between males and females (p=0.001). Males were less unemployed than females (p=0.011). Gender also showed statistically significant difference in how easy the immigrants balanced their work and personal life (p=0.044). Age groups depicted differences in the physical health of the immigrants (p=0.001). The use of alcohol and tobacco was not an important risk factor among participants (p=0.529).Conclusions: These facts suggest that socio-demographics among a population of Caribbean immigrants were significantly different among genders. However the use of tobacco and alcohol showed not significant differences among the immigrants.

Whether and how P2X7 receptor knockout (KO) modulates central post-stroke pain (CPSP) induced by lesions of the ventrobasal complex (VBC) of the thalamus regarding behavioral, molecular, and electrical recording parameters remains unclear. Following the experimental protocols of a rat hemorrhage CPSP model, the VBC of the thalamus of wild-type and P2X7 receptor KO mice was injected with 10 mU/0.2 μl type IV collagenase to produce an animal model of stroke-like thalamic hemorrhage. Behavioral data showed that the CPSP group induced thermal and mechanical pain, while the P2X7 receptor KO group showed reduced thermal and mechanical pain behaviors. Molecular assessments revealed that the CPSP group had lower expression of NeuN and KCC2 and higher expression of GFAP, IBA1, and BDNF. The P2X7 KO group showed lower expression of GFAP, IBA1, and BDNF and higher expression of KCC2 than the CPSP group. The expression of NKCC1, GABAa receptor, and TrkB did not differ significantly between the control, CPSP, and P2X7 receptor KO groups. Muscimol application increased multiunit numbers and [Cl-] influx in the membrane in the CPSP group, while P2X7 receptor KO reduced multiunit activity and [Cl-] outflux. P2X4 receptor expression was significantly decreased in the 100 kb but not the 50 kb site in the P2X7 receptor KO group. Altogether, the P2X7 hypothesis of CPSP was proposed, wherein P2X7 receptor KO changed the CPSP pain behavior, numbers of astrocytes and microglia, CSD amplitude of the anterior cingulate cortex and the medial dorsal thalamus, BDNF and KCC2 expression, [Cl-], and P2X4 activation in 100 kb with P2X7 receptors. The present findings have implications for the clinical treatment of CPSP symptoms.

In this paper we investigate some properties of square-free numbers. In particular, we study a function that counts the number of square-free numbers no larger than the given number n. We show that this function has asymptotics consistent with the predictions of the Riemann hypothesis and use this to obtain new and better estimates of the density of square-free numbers. Finally, we present a well-known generalization of the concept of a square-free number to monoids and research on square-free factorizations.

The core motivation for this study emanates from Voronin S.'s seminal work on the universality of the zeta function. This universality positions the zeta function as a pivotal element in the conceptual framework of global artificial intelligence, suggesting its potential as a foundational component. A critical barrier to harnessing this promising universality has been the absence of efficient algorithms capable of actualizing this concept.

: Aim: The aim in this study was to conduct a prospective survey regarding the sociodemographics and lifestyle risk factors for cancer of a population of Caribbean immigrants living in the U.S. Materials and Methods: The data were processed and analysed using SPSS software 27 and Excel. Crosstabulations were performed. The chi-square test was used to evaluate different hypotheses. Statistical significance was defined as p &lt; 0.05. Results: Statistically significant differences in the country of birth of the Caribbean immigrants were found for sex (p = 0.038) and the cleanness of their neighbourhoods (p = 0.045). We found differences in occupations between men and women (p = 0.001). Men were less unemployed than women (p = 0.011). Sex also showed statistically significant differences in how easily the immigrants balanced their work and personal life (p = 0.044). Age groups depicted differences in the physical health of the immigrants (p = 0.001). The use of alcohol and tobacco was not an important risk factor among participants (p = 0.529). Conclusions: These findings suggest that the sociodemographics risk factors for cancer among a population of Caribbean immigrants were significantly different between the sexes. However, the use of tobacco and alcohol showed insignificant differences among the immigrants.

The physiological mechanism of membrane potential generation relies on the passage of mobile ions across the plasma membrane, which is selectively permeable to mobile ions. The existence of a membrane and the transport of ions across it are essential for the generation of membrane potential. However, a long-forgotten theory, Association-Induction hypothesis, asserts that the heterogeneous distribution of charges in space, due to ion adsorption, is the true cause of membrane potential generation. Using an artificial living cell model consisting of a viscous aqueous solution, the mechanism of membrane potential generation is restudied in this work. The results indicate that membrane potential can be generated even in the absence of a plasma membrane and that ion adsorption leading to a heterogeneous spatial charge distribution is an essential factor for membrane potential generation. Consequently, the authors invites people to reconsider the Association-Induction hypothesis as a genuine mechanism for generating membrane potential.

One of the most important unresolved mysteries in mathematics is the Riemann Hypothesis, which suggests a fundamental connection between the non-trivial zeros of the Riemann zeta function and the distribution of prime numbers. Here, we explore the fascinating union of trigonometric functions, prime numbers, and the Riemann zeta function through an examination of the zeros in the statement ln(sec(π·n log(n))). We demonstrate a strong mathematical connection between these components, providing information on the mysterious properties of prime numbers and their complex relationships to basic mathematical operations. Our thorough investigation adds to the current discussion of the Riemann Hypothesis by offering possible solutions and deepening our comprehension of the intricate relationship between number theory and analytic functions.

In this research a neccessary and sufficient condition for the proof of the Binary Goldbach conjecture is established. It is established that the square of all natural numbers greater or equal to 2 have an additive partition equal to the sum of the square of a natural number greater or equal to zero and a Goldbach partition semiprime. All Goldbach partition semiprimes are odd except 4. This finding is in itself proof all composite even numbers have at least Goldbach partition. The result of the proof of the Binary Goldbach conjecture is used to prove the Andrica and Legendre conjectures. The Riemann hypothesis is examined and sources of non trivial zeroes outside the critical strip are discussed. An example example of a non-trivial zero outside the critical strip is given

Although the dopamine hypothesis of schizophrenia explains the effects of all the available antipsychotics in clinical use, there is an increasing need for developing new drugs for the treatment of positive, negative, and cognitive symptoms of chronic psychoses. Xanomeline-trospium (KarXT) is a drug combination that is based on the essential role played by acetylcholine in the regulation of cognitive processes and the interactions between this neurotransmitter and other signaling pathways in the central nervous system, with a potential role in the onset of schizophrenia, Alzheimer’s disease, and substance use disorders. A systematic literature review that included four electronic databases (PubMed, Cochrane, Clarivate/Web of Science, and Google Scholar) and the US National Library of Medicine database for clinical trials detected 21 sources referring to 14 studies focused on KarXT, out of which only four have published results. Based on the results of these trials, the short-term efficacy and tolerability of xanomeline-trospium are good, but more data is needed before this drug combination may be recommended for clinical use. However, on a theoretical level, the exploration of KarXT is useful for increasing the interest of researchers in finding new, non-dopaminergic, antipsychotics that could be used either as monotherapy or as add-on drugs.

Theoretical physics addresses the fundamental principles of nature and the mysteries of the universe. In the last century, the theory of relativity challenged people's understanding of time and space, while quantum mechanics revealed various forms of energy. However, the concept of gravity remains elusive. All matter in the universe is composed of energy that can be converted into other forms, yet it is always conserved. However, achieving a state of equilibrium, represented by the equal sign, can be challenging to achieve on both macro and quantum scales. A phenomenon that can transition from one state to infinity is called collision and is essential for understanding natural phenomena and physical processes throughout the universe. In the study of general relativity cannot be three-dimensional problems to gain mathematical inspiration, understand the application of new dimensions. Here, we elucidate the nature of gravity, which inspires the solution of the Riemann hypothesis. Our results show that the conduction mode of dimensional energy is defined by Euler's formula.

In this article, it is demonstrated that if the zeta function does not have a sequence of zeros whose real part converges to 1, then it cannot have any zeros in the critical strip, showing that the Riemann Hypothesis is false.

In this article, we present solutions for one of the oldest mathematical problems, the Collatz Conjecture, and provide a proof for the Riemann Hypothesis, utilizing a new number theory based on newly discovered number properties, which will also be presented in this paper. This is made possible through the Sophy-Peter mathematical framework, built upon this new number theory. The Collatz Conjecture will be disproven, but as an alternative, the Oscillating Theorem will be introduced, with its correctness proven within this article. Furthermore, we present the general version of the Riemann zeta function, developed based on the new number theory. The correctness of this function is verified by comparing it with existing results of the zeta function. Using this approach and the Sophy-Peter framework, we have successfully proven the Riemann Hypothesis, long considered a millennium problem. Moreover, since the general zeta function is proven, this implies the correctness of the new number theory and the Sophy-Peter framework.

This paper presents a statistical approach to identify the underlying roughness characteristics in synthetic aperture radar (SAR) intensity data. The physical modeling of this kind of data allows the use of the Gamma distribution in the presence of fully-developed speckle, i.e., when there are infinitely many independent backscatterers per resolution cell, and none dominates the return. Such areas are often called ``homogeneous'' or ``textureless'' regions. The GI0 distribution is also a widely accepted law for heterogeneous and extremely heterogeneous regions, i.e., areas where the fully-developed speckle hypotheses do not hold. We propose three test statistics to distinguish between homogeneous and inhomogeneous regions, i.e., between gamma and GI0 distributed data, both with a known number of looks. The first test statistic uses a bootstrapped non-parametric estimator of Shannon entropy, providing a robust assessment in uncertain distributional assumptions. The second test uses the classical coefficient of variation (CV). The third test uses an alternative form of estimating the CV based on the ratio of the mean absolute deviation from the median to the median. We apply our test statistic to create maps of \(p\)-values for the homogeneity hypothesis. Finally, we show that our proposal, the entropy-based test, outperforms existing methods, such as the classical CV and its alternative variant, in identifying heterogeneity when applied to both simulated and actual data.