In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.

This paper identifies a criterion for choosing the largest set of rejected hypotheses in high-dimensional data analysis where Multiple Hypothesis testing is used in exploratory research to identify significant associations among many variables. The method neither requires predetermined thresholds for level of significance, nor uses presumed thresholds for false discovery rate. The upper limit for number of rejected hypotheses is determined by finding maximum difference between expected true hypotheses and false hypotheses among all possible sets of rejected hypotheses. Methods of choosing a reasonable number of rejected hypotheses and application to non-parametric analysis of ordinal survey data are presented.

This paper investigates the search for an exact analytic solution to a temporal first-order differential equation that represents the number of customers in a non-stationary or time-varying

Variational Auto-Encoders (VAEs) are deep generative models used for unsupervised learning, however their standard version is not topology-aware in practice since the data topology may not be taken into consideration. In this paper, we propose two different approaches with the aim to preserve the topological structure between the input space and the latent representation of a VAE. Firstly, we introduce InvMap-VAE as a way to turn any dimensionality reduction technique, given an embedding it produces, into a generative model within a VAE framework providing an inverse mapping into original space. Secondly, we propose the Witness Simplicial VAE as an extension of the Simplicial Auto-Encoder to the variational setup using a witness complex for computing the simplicial regularization, and we motivate this method theoretically using tools from algebraic topology. The Witness Simplicial VAE is independent of any dimensionality reduction technique and together with its extension, Isolandmarks Witness Simplicial VAE, preserves the persistent Betti numbers of a data set better than a standard VAE.

Liquid biopsies are an integral part of the diagnosis of cancer. Here, we have extended previous validation studies of a new targeted NGS panel to include the detection of copy number amplifications (CNAs), fusions, and exon skipping variants. Detection of these gene classes included specimens from clinical and healthy donors and cell lines (fusions: ROS1, EML4-ALK, NTRK1), (exon skipping: MET exon 14), and (CNAs: HER2, CDK6, EGFR, MYC and MET). The limit of detection (LOD) for fusion/skipping was 42 copies (QC threshold = 3 copies) and was verified using 3 additional fusion/skipping variants. LOD for CNAs was 1.40-fold-change (QC threshold = 1.15-fold change) and was verified with 3 additional CNAs. In precision studies, all fusion/skipping variants were detected in all runs and all days of testing (n=18/18; 100%); average CV for repeatability was 20.5% (range 8.7% - 34.8%), and for reproducibility was 20.8% (range 15.7% - 30.5%). For CNAs, 28/29 (96.6%) copy gains were detected. For CNAs, average CV of 1.85% (range 0% to 5.49%) for repeatability and 6.59% (range 1.65% to 9.22%) for reproducibility. The test panel meets the criteria for being highly sensitive and specific and extends its utility for the detection of clinically relevant variants in cancer.

It has been more than a century since Erlang published his paper in 1909. The classical queueing system for the M/D/c queue has been investigated by many authors. The key is to find the roots of an equation using explicit or numerical methods. We present an alternate form of explicit solution(s) of this/these root(s) using the Lambert-W function in this paper. In addition, the unsaturated probabilities of the number of customers in the system is derived explicitly in terms of this/these root(s). Finally, the mean number of customers in the system and the mean queueing delay are obtained from these probabilities. The exact queueing delay formula for the M/D/c queue is expressed in closed-form in terms of the root(s). Numerical results show that they are in excellent agreement with the numerical results published by Seelen, Tijms and Van Hoorn.

The influence of linear and nonlinear modes of ADC driver on the operational amplifier (Op-Amp), which has different values of maximum output voltage slew rate, on the effective number of bits of ADC is considered. It is shown that the effective number of bits of ADC when op-amps on bipolar transistors with input signal amplitudes exceeding 50-100 mV is determined by the nonlinear modes of its input differential stage.

The object of this work is to determine the correlation on the Nusselt number on the individual rows of a four-row tubular finned heat exchanger with continuous fins with a staggered tube arrangement using CFD modelling. Correlations for calculating Darcy-Weisbach friction factors on individual tube rows were also determined. Relationships for the Nusselt number and friction factor derived for the entire exchanger based on CFD modelling were compared with those available in the literature determined using experimental data. The maximum relative differences between the Nusselt number for a four-row exchanger determined experimentally and by CFD modelling are in the range from 22% for a Reynolds number based on a tube's outside diameter of 1,000 to 30% for a Reynolds number of 13,000. The maximum relative differences between the friction factor for a four-row exchanger determined experimentally and by CFD modelling are in the range of 50% for a Reynolds number based on a tube outer diameter of 1,000 to 10% for a Reynolds number of 13,000. The CFD modeling performed shows that in the range of Reynolds numbers based on hydraulic diameters from 150 to 1,400, the Nusselt number for the first row in a four-row finned heat exchanger is about 22% to 15% higher than the average Nusselt number for the entire exchanger. In the range of Reynolds number changes based on hydraulic diameter from 2,800 to 6,000, the Nusselt numbers on the first and second rows of tubes are close to each other. Correlations on Nusselt numbers and friction factors derived for individual tube rows can be used in the design of plate-fin and tube heat exchangers used in equipment such as air-source heat pumps, automotive radiators, air-conditioning systems and in air hot-liquid coolers. In particular, the correlations can be used to select the optimum number of tube rows in the exchanger.

Finding out how cells with the same genome change fates in differentiation commitment is a challenge of biology. We used MCF-7 breast cancer cells treated with the ErbB2 ligand heregulin (HRG), which induces differentiation, to address if and how the constitutive pericentromere-associated domains (PADs) may be involved in this process. PAD-specific repressive heterochromatin (H3K9me3) and active euchromatin (H3K4me3) marking, centromere (CENPA) labelling, qPCR, acridine-orange-DNA structural test, and microscopic image analysis were applied. We found a two-step DNA unfolding, at 15-20 min and 60 min after HRG treatment, coinciding with bi-phasic activation of the early response genes (c-FOS family) and two steps of critical phase transition which were revealed in transcriptome studies. In control, the distribution of PAD number and size displays a power-law scaling with a boundary at the nucleolus. PADs’ clustering correlates with centromere numbers. 15 min after HRG treatment, the unravelling of PADs occurs, coinciding with the first step of euchromatin unfolding. The second step is associated with transcription of long-non-coding-RNA from satellite III DNA. We hypothesize that splitting of the PAD clusters under the critical size threshold of the silencing domain abrupts position effect variegation. It allows the first genome transcription avalanche to occur, starting differentiation commitment.

New insights into the causes and mechanisms of congenital heart disease (CHD) have been gained through the use of next-generation sequencing. Examination of the whole exome sequence detects detrimental gene variations modifying single or contiguous nucleotides, that are characterised as pathogenicity based on statistical assessments of families. and correlates with congenital heart disease, elevated expression during heart development, and reduction of harmful protein-coding mutations in the general population. Patients with CHD and extracardiac abnormalities enriched for gene classes meeting these criteria. CHDs and extracardiac defects, supporting a common set of pathways in the organogenesis of CHDs. Single-cell transcriptomics data reveal the expression of genes associated with CHD in specific cell lineages, and emerging evidence suggests that genetic variants disrupt multicellular genes essential for cardiogenesis. Metrics and units are being tracked in whole genome sequence studies.

In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.

Background: Cervical cancer prevention in regions with limited access to screening and HPV vaccination necessitates innovative approaches. This study explored the potential of a test-and-treat strategy using mRNA HPV tests to impact cervical cancer prevention in a high-prevalence HIV population. Methods: A cervical screening study was conducted at three South African hospitals involving 710 under-screened, non-pregnant women (25 to 65 years) without known cervical diseases. Cytology, HPV testing, colposcopy, and biopsies were performed concurrently. Histopathologists determined final histological diagnoses based on biopsy and LLETZ histology. mRNA-HPV-genotyping for 3 (16, 18, 45) to 8 (16, 18, 31, 33, 35, 45, 52, 58) high-risk types was performed on leftover liquid-based cytology material. The preventive potential of the test-and-treat approach was estimated based on published data, reporting the causative HPV types in cervical cancer tissue from South African women. Treatment was provided as needed. Results: HPV positivity rate more than doubled from 3-type (15.2%; 95% CI: 12.6-17.8) to 8-type mRNA (31.5%; 95% CI: 28.8-34.9) combinations, significantly higher among HIV-positive women. CIN3+ prevalence among HIV-positive women (26.4%) was double that of HIV-negative women (12.9%) (p<0.01). The 6-type combination showed the best balance of sensitivity, specificity and treatment group size, and effectiveness to prevent cervical cancer. A 4-type combination (16, 18, 35, 45) could potentially prevent 77.6% (95% CI: 71.2-84.0) of cervical cancer burden by treating 20% and detecting 41.1% of CIN3 cases in the study group. Similarly, a 6-type combination (16, 18, 31, 33, 35, 45), treating 25% and including 62% of CIN3 cases, might prevent 85% of cervical cancer cases (95% CI: 79.6-90.6) among HIV-positive and negative women. Conclusion: Employing mRNA HPV tests within a test-and-treat approach holds huge promise for targeted cervical cancer prevention in under-screened populations. Testing for mRNA of the 6 highest-risk HPV types in this population and treating them all is projected to effectively prevent progression from CIN3 to invasive cervical cancer while reducing overtreatment in resource-constrained settings.

This paper attempts to develop a Dynamic Mode Decomposition (DMD) based Reduced Order Model (ROMs) which can quickly but accurately predict the forces and moments experienced by a road vehicle such that they be used by an on-board controller to determine the vehicle's trajectory. DMD can linearize a large dataset of high-dimensional measurements by decomposing them into low-dimensional coherent structures and associated time-dynamics. This ROM then also can be applied to predict the future state of the fluid flow. Existing literature on DMD is limited to low Reynolds number applications. This paper presents DMD analyses of the flow around an idealized road vehicle, called the Ahmed body, at a Reynolds number of 2.7E6. The high dimensional dataset used in this paper was collected from a computational fluid dynamics (CFD) simulation performed using Menter's shear stress transport (SST) turbulence model within the context of Improved Delayed Detached Eddy Simulations (IDDES). The DMD algorithm, as available in literature, was found to suffer nonphysical dampening of the medium-to-high frequency modes. Enhancements to the existing algorithm were explored, and a modified DMD approach is presented in this paper which includes: (a) a requirement of higher sampling rate to obtain higher resolution of data, and (b) a custom filtration process to remove spurious modes. The modified DMD algorithm thus developed was applied to the high Reynolds number, separation dominated flow past the idealized ground vehicle. The effectiveness of the modified algorithm was tested by comparing future predictions of force and moment coefficients as predicted by the DMD-based ROM to the reference CFD simulation data, and found to offer significant improvement.

Recent wastewater testing suggests the poliovirus is reemerging, raising questions about the efficacy of the current inactivated polio vaccine and mass vaccination to eradicate poliomyelitis. The present study reassessed reported outcomes from the placebo-control study of the 1954 Salk inactivated poliomyelitis vaccine field trial, which studied a population of 1,829,916 U.S. school children from the first three grades. The placebo-control study of the trial reported that the polio vaccine was 80% to 90% effective in reducing cases of laboratory confirmed paralytic poliomyelitis, 60% to 70% effective against type 1 of the virus, and greater than 90% effective against types 2 and 3 of the poliovirus. However, these outcomes are reported as relative risk reductions, and the absolute risk reductions of the 1954 polio vaccine field trial were never reported to the public. The present paper’s analysis reveals to the public for the first time that the inactivated poliomyelitis vaccine's absolute risk reduction of laboratory confirmed paralytic poliomyelitis in the placebo-control study is 0.01% to 0.02%, an extraordinarily low clinical effect that impugns the Salk polio vaccine’s reported effectiveness. Furthermore, absolute risk reductions for the three poliovirus types are 0.01%, 0.00%, and 0.01%, respectively. Outcome reporting bias in the 1954 inactivated poliomyelitis vaccine field trial casts doubt on eradication of the poliovirus through mass vaccination.

Deterministic-stochastic empirical mode decomposition (EMD) is used to obtain low-frequency (non-diffusive, i.e., background velocity) and high-frequency (diffusive, i.e., eddies) components from a Lagrangian drifter‘s trajectory. Eddy characteristics are determined from the time series of eddy trajectories from individual Lagrangian drifter such as the eddy radial scale, eddy velocity scale, eddy Rossby number, and eddy-background kinetic energy ratio. A long-term dataset of the SOund Fixing And Ranging float time series obtained near the California coast by the Naval Postgraduate School from 1992 to 2004 at depth between 150 and 600 m (http://www.oc.nps.edu/npsRAFOS/) is used as an example to demonstrate the capability of the deterministic-stochastic EMD.

On February 24, 2022 Russia started the full-scale invasion of Ukraine, which created an unprecedented number of refugees. To estimate the influence of this humanitarian disaster on the COVID-19 pandemic dynamics, the averaged daily numbers of cases and reproduction numbers in Ukraine, the UK, Poland, Germany, the Republic of Moldova, and in the whole world were calculated for the period February-April 2022. The registered numbers of cases were compared with ones calculated with the use of the generalized SIR-model and corresponding parameter identification procedure for the previous epidemic waves in Ukraine, Poland, Germany, and the world. Since before February 24, 2022 the estimation of the number of infectious persons per capita in Ukraine 3.6 times exceeded the global figure, the increase of the number the new cases and the pandemic duration was expected. In March 2022 the increase of the averaged number of new cases in the UK, Germany, and worldwide was visible. A simple formula to estimate the effective reproduction number based on the smoothed accumulated numbers of cases is proposed. The results of calculations agree with the figures presented by John Hopkins University and demonstrate a short-term growth of the reproduction number in the UK, Poland, Germany, Moldova, and worldwide in March 2022.

Extensive genome structure variation, such as copy number variations (CNVs) and presence/absence variations, is the basis for the remarkable maize genetic diversity; however, the effect of CNVs on maize herbivory defense remains largely underexplored. Here, we report that the naturally occurring duplication of the maize 9-lipoxygenase gene, ZmLOX5, leads to increased resistance of maize to herbivory by fall armyworms (FAW). Previously, we showed that ZmLOX5-derived oxylipins are required for defense against chewing insect herbivores and identified several inbred lines, including Yu796, that contained duplicated CNVs of ZmLOX5, referred to as Yu796-2×LOX5. To test whether introgression of the Yu796-2×LOX5 locus into the herbivore-susceptible B73 background that contains a single ZmLOX5 gene is a feasible approach to increase resistance, we generated a series of near-isogenic lines that contained either 2, 1 or 0 copies of the Yu796-2×LOX5 locus in the B73 background by six back crosses (BC6). Droplet digital PCR (ddPCR) confirmed the successful introgression of the Yu796-2×LOX5 locus in B73. The resulting B73-2×LOX5 inbred line displayed increased resistance against FAW associated with increased expression of ZmLOX5, increased wound-induced production of its primary oxylipin product, the α-ketol, 9-hydroxy-10-oxo-12(Z),15(Z)-octadecadienoic acid (9,10-KODA), and the downstream defense hormones regulated by this molecule, 12-oxo-phytodienoic acid (12-OPDA) and abscisic acid (ABA). Surprisingly, wound-induced JA-Ile production was not increased in B73-2×LOX5, resulting from the increased JA catabolism. Furthermore, B73-2×LOX5 displayed enhanced drought tolerance likely due to increased ABA and 12-OPDA content. Taken together, this study revealed that the duplicated CNV of ZmLOX5 quantitively contributes to maize antiherbivore defense and presents proof-of-concept evidence that the introgression of naturally occurring duplicated CNVs of a defensive gene into the productive, but susceptible, crop varieties is a feasible breeding approach for enhancing plant resistance to herbivory and tolerance to abiotic stress.

Dirac's Large Number Hypothesis (LNH), proposed in 1937, has captivated the scientific community with its exploration of profound correlations between cosmic and atomic scales. This hypothesis delves into the intricate interplay between the infinitesimally small and the vastly large, offering potential insights into the fundamental nature of the universe. As we continue to uncover the mysteries of the cosmos, the LNH oscillates on the precipice of scientific acceptance—neither fully validated nor entirely dismissed. In this review paper, we embark on a comprehensive journey through Dirac's LNH, shedding light on its theoretical underpinnings, its implications for universe models, and its resonance with the Anthropic Principle. We delve into the possibilities of variable gravitational constants and continuous mass creation, inviting further exploration into the intricacies of our cosmic symphony. By embracing the ongoing quest for understanding, we endeavor to unravel the profound harmonies that connect the infinitesimal with the infinite, contributing to the symphony of knowledge in theoretical physics and cosmology.

Current prenatal genetic evaluation showed a significantly increase in non-invasive screening and the reduction of invasive diagnostic procedures. To evaluate the diagnostic efficacy on detecting common aneuploidies, structural chromosomal rearrangements and pathogenic copy number variants (pCNV), we performed a retrospective analysis on a case series initially analyzed by aneuvysion fluorescence in situ hybridization (FISH) and karyotyping then followed by array comparative genomic hybridization (aCGH). Of the 386 cases retrieved from the past decade, common aneuploidies were detected in 137 cases (35.5%), other chromosomal structural rearrangements were detected in four cases (1%), and pCNV were detected in five cases (1.3%). The relative frequencies for common aneuploidies suggested a under detection of sex chromosome aneuploidies. Approximately 9.5% of cases with common aneuploidies showed a mosaic pattern. Inconsistent results between FISH and karyotyping were noted in cases with pseudo-mosaicism introduced by culture artifact or variable cellular proliferation from cells with mosaic karyotypic complements under in vitro cell culture. Based on findings from this case series, cell-based FISH and karyotyping should be performed to detect common aneuploidies, structural chromosomal abnormalities, and mosaic pattern. DNA-based aCGH and reflex FISH should be performed to detect and confirm genomic imbalances and pCNV. Practice points to ensure the diagnostic accuracy and efficacy were summarized.

During the conceptual and preliminary design stage of a ship, designers need to ensure that the selected principal dimensions and parameters are good enough to deliver a stable ship (statically and dynamically) besides deadweight and speed. To support this, the initial intact stability of the proposed ship is required to be calculated, and in doing so, the lightship weight and its detailed breakdown are necessary to be known. After hull steel weight, machinery weight, mainly, marine propulsion and power generation machinery, play a vital role in the lightship weight estimate of a ship due to its robustness. The correct estimation of respective weights improves the accuracy of calculating a ship's initial stability typically to be designed and built. Hence, it would be advantageous for the designer to convince the ship owner. A total of 3006 marine propulsion (main marine diesel) engines and 348 power generation (auxiliary marine diesel) engines/generators of various power output (generators output for auxiliary engines), engine RPM and cylinder number of different engine makers are collected. These are analyzed and presented in both tabular and graphical forms to demonstrate the possible relationship between marine propulsion engine weight and power generation engine weight, and their respective power output, RPM, cylinder number, power-RPM ratio and power-RPM ratio per cylinder.In this article, the authors have attempted to investigate the behavior of marine propulsion engine weight and power generation engine/generator weight regarding engine power output, generator power output, engine RPM and cylinder number (independent variables). Further attempts have been made to identify those independent variables that influence the weight of the marine propulsion engine and power generation engine/generator (dependent variables), and their interrelationships. A mathematical model has thus been developed and proposed, as a guiding tool, for the designer to estimate the weight of main and auxiliary engines more accurately during the conceptual and preliminary design stage.

The aim of the studies was to test the hydraulic pressure losses along the length, local pressure losses, absolute and relative roughness values of the pipeline walls made of modern polymer material. The article presents the results of the experimental studies of hydraulic resistance of pipelines from modern polymer materials, the method of theoretical determination of resistance coefficients in the VALTEC ISO 21003 metal polymer pipeline, the classification and method of determining the values of local hydraulic resistance coefficients of T-Junctions are given, the dependence for determining the Darcy λ hydraulic friction coefficient associated with the value of the equivalent hydraulic roughness coefficient of the inner surface of pipes and the Reynolds number. The experimental data of the hydraulic coefficient of friction Darcy λ was compared with the known theoretical relationships of A.D. Altschul, Colbrooke for smooth pipes, Colbrooke and White, an alternative to the Colbrooke-White, Offengenden equation for low-shear pipes and pipes from polymer materials with Reynolds numbers Re>10^4. Dependence of hydraulic coefficient of friction on relative equivalent roughness of pipeline walls and Reynolds number, specific energy loss per one linear meter on total flow rate in system of hydraulically short pipeline is obtained. The empirical dependence of values of coefficients of local resistance ζ=f(Q2/Q3) from a ratio of the divided expenses of Q2/Q3 of the equal-pass T-Junction α=90 the pipeline to a branch up, on pass in the direct T-Junction and division of streams at various internal pipeline diameters of VALTEC.

Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (“An approach to first-principles electronic structure calculation by symbolic-numeric computation” by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multivariate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one canbuild the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.

This paper offers a novel geometric and visual approach to the renowned Four Color Map Theorem and K5 non-planarity problem while unveiling their profound connection to the "kissing number" problem in 2D. We represent planar graphs through circles and tangents, simplifying complex structures and shedding light on these classic problems. Our proofs by contradiction, rooted in the kissing number concept, reveal that both the Four Color Map Theorem and K5 non-planarity are fundamentally linked, as they pivot around the concept of coloring. This study bridges the realms of geometry and graph theory, providing fresh insights and emphasizing the significance of the kissing number problem in various fields.

Spinors are used for the computation of probability in quantum mechanics. They are treated as elements of a complex vector space in contrast to a real vector space. Spinor theory is abstract mathematics with an ambiguous interpretation. The overall phase of spinors does not affect the computation of probability in quantum mechanics. If two spinors have an overall phase of imaginary number i, then they can be treated like vectors in a real vector space. The square of the magnitude of the number, which is the sum of dot product and the term having basis vector i of cross product of two such vectors, is equal to the probability computed using the corresponding spinors. Therefore, the geometry of such spinors can be easily depicted in a three-dimensional space like vectors in a real vector space. Spinors are not isotropic vectors in Hilbert space. The sum of dot product and cross product of two complex numbers is equivalent to the quotient of the division of one complex number by another. Similarly, the sum of dot product and cross product of two vectors is the quotient of the division of one vector by another. Using the rules of division of vectors, we can find the rules of multiplication of vectors. The rules of multiplication and division of basis vector i match the rules of multiplication and division of imaginary number i. Therefore, basis vector i is nothing but imaginary number i. Multiplication of dot and cross product of two vectors to the second vector will give us the components of the first vector that are parallel and orthogonal to the second vector. Vectors are also made up of complex numbers like spinors. The reason for finding dot product of complex numbers in the process of computing the probability is to ignore the overall phase in contrast to the phase difference. This is misconstrued as complex vector space in quantum mechanics. 3-D number which is nothing but a spinor with new notation is an extension of vector algebra. It can have a real number as a term in addition to the terms i, j and k. In all other respects, it is a vector. 3-D numbers are part of the number system like real numbers and complex numbers. The real numbers are one dimensional numbers, the complex numbers are two-dimensional numbers and the 3-D numbers as well as vectors and spinors are three-dimensional numbers. As polarisation and spin are one and the same, we can straight away apply 3-D numbers to polarisation of light. 3-D numbers will greatly simplify the study of areas of science where three-dimensional space is involved.

Let $f\left(1\right)=1$ , and let $f(n+1)=2^{2^{f\left(n\right)}}$ for every positive integer n. We consider the following hypothesis: if a system S ⊆ {x_i · x_j = x_k : i, j, k ∈ {1, . . . , n}} ∪ {x_i + 1 = x_k : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x₁, . . . , x_n, then each such solution (x₁, . . . , x_n) satisfies x₁, . . . , x_n ≤ f (2n). We prove: (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set $\mathfrak{M}$ ⊆ has a finite-fold Diophantine representation, then $\mathfrak{M}$ is computable.

SARS-CoV-2 Omicron variant spike RBD epitopes in complex with spike (S) -protein D614G mutations now predominate globally and increase infectivity by assembling more functional S proteins into the virion. On average, homeopathic medicine contains less than one molecule per dose on average. Single variable SphericalHarmonicY, LaguerreL, WhittakerM, SphericalBesselJ, LegendreP, LegendreQ, LaguerreL, ChebyshevT, and Hypergeometric1F1 Functions pFqarise in connection with the power series solution of the Schrodinger equation or in the summation of perturbation expansions in quantum mechanics. A quantum interpretation of the homeopathic method is presented here through a novel algebraic topology approach to supersymmetry and symmetry breaking in quantum fields and quantum gravity theory with the aim of developing a wide range of physical-based drug design applications in a Small Molecule Quantized Water Memory Network suggesting that the Hidden Black Hole Paradox of Information Loss might be solved under suitable conditions.