Hepatocellular carcinoma (HCC) early diagnosis is challenging. Moreover, for patients with al-pha-fetoprotein (AFP)-negative HCC, this challenge is augmented. MicroRNAs (miRs) profile may serve as potential HCC molecular markers. We aimed to assess plasma homo sabines (hsa)-miR-21-5p, hsa-miR-155-5p, hsa-miR-192-5p, hsa-miR-199a-5p expression levels, as panel of biomarkers for HCC in chronic hepatitis C virus (CHCV) patients with liver cirrhosis (LC), es-pecially AFP-negative HCC cases, a step toward non-protein coding (nc) RNA precision medicine. Subjects and Methods: 79 patients enrolled with CHCV infection with LC, subclassified into LC group without HCC (n=40) and LC with HCC (n=39) in comparison with 15 apparently healthy control subjects. Real-time quantitative PCR was used to measure plasma hsa-miR-21-5p, hsa-miR-155-5p, hsa-miR-192-5p, and hsa-miR-199a-5p. Results: Plasma hsa-miR-21-5p and hsa-miR-155-5p demonstrated significant upregulation, while hsa-miR-199a-5p demonstrated significant downregulation in the HCC group (n=39) when compared to LC group (n=40). Hsa-miR-21-5p expression was positively correlated with serum AFP, insulin, and insulin re-sistance (r=0.5, p<0.001, r=0.334, p=0.01, and r=0.303, p=0.02, respectively). According to the ROC curves, for differentiating HCC from LC, combining AFP with each of hsa-miR-21-5p, hsa-miR-155-5p, and miR199a-5p improved the diagnostic sensitivity to 87%, 82%, and 84%, re-spectively, vs 69% for AFP alone, with an acceptable specificity of 77.5%, 77.5%, and 80%, respec-tively, and AUC= 0.89, 0.85, and 0.90, respectively vs. 0.85 for AFP alone. Hsa-miR-21-5p/hsa-miR-199a-5p and hsa-miR-155-5p/hsa-miR-199a-5p ratios discriminated HCC from LC at AUC=0.76 and 0.71, respectively, with sensitivities=94% and 92%, and specifici-ties=48% and 53%, respectively. Upregulation of plasma hsa-miR-21-5p was considered as an independent risk factor for HCC development [OR=1.198(1.063–1.329), p=0.002]. Conclusion: Combining each of hsa-miR-21-5p, hsa-miR-155-5p, and hsa-miR-199a-5p with AFP made possi-ble to identify HCC development in LC patients’ cohort with higher sensitivity than using AFP alone. Hsa-miR-21-5p/hsa-miR-199a-5p and hsa-miR-155-5p/hsa-miR-199a-5p ratios are poten-tial HCC molecular markers for AFP-negative HCC patients. Hsa-miR-21-5p was linked to insu-lin metabolism in HCC patients’ group as well as being an upregulated independent risk factor for the emergence of HCC from LC in CHCV patients.

The structure and phenomena arising from charge transfer in cold field emission modus in a single closed carbon nanotube under cold field emission conditions were studied. By studying microphotographs in various microscopes - AFM, SEM, TEM it was found inhomogeneities in the form of two types of superlattices. The conductivity of the emitter section of a nanodiode circuit with an emitter made of an individual closed carbon nanotube (CNT) and arrays of closed CNTs was studied. High values of cold field emission were observed, as well as nonlinearity and anomalies of the current-voltage characteristic (I-V characteristics), which manifested themselves in the form of peaks at low and high current values. Peaks in the CVC have distinct areas of negative differential conductivity. It is argued, that the anomalously high currents in the I–V characteristics of cold field emission of electrons from a nanoemitter made of closed CNTs can be associated with a sharp increase in the density of electron states at energies near the van Hove singularity.

Because imbalanced power will cause the loss of the propulsion motor of the railway vehicle, and the increase in temperature will shorten the service life of the electric vehicle. Not only this, but also increase the cost of electricity and maintenance. In the past, the industry only focused on methods to improve power quality such as load capacity, relay setting, and harmonic resolution. Now, the consider of three-phase unbalance rate (TPUR) must be applied. I propose special transformers wiring (STW) to improve the three unbalance rates and provide different transformer wiring methods. According to the IEEE Committee, in the future, power companies will need to install balanced relay stations to improve three-phase unbalance rate. the internal regulations of Taipower must be less than 4.5% (voltage unbalance rate (NPSUR)of 2.5% and motor temperature rise of 12.5%). the derivation of the transformer "three-phase unbalance rate" model is the focus of the railway system. This research is based on the model derivation of different wiring methods to improve the hot problem caused by the three-phase imbalance and improve the service life of the train. And pointed out that Scott, Le-Blanc, Modified-Woodbridge three wiring methods can be applied to future railway system routes to improve the three-phase unbalance rate, in line with the IEEE standard of less than 2%.

In this article, we estimate the value of Government Expenditure on Education-GEE in the context of Environmental, Social and Governance-ESG dataset of the World Bank. We use data from 193 countries in the period 2011-2020. We use Panel Data with Fixed Effects, Panel Data with Random Effects, Pooled Ordinary Least Squares-OLS, and Weighted Least Squares-WLS. Our results show that the value of GEE is positively associated among others to “Case of Death, by communicable disease and maternal, prenatal and nutrition conditions”, and “Unemployment”, and negatively associated among others to “Hospital Beds” and “Government Effectiveness”. Furthermore, we apply the k-Means algorithm optimized with the Elbow Method and we find the presence of four clusters. Finally, we confront eight machine learning algorithms for the prediction of the future value of GEE. We found that the Polynomial Regression is the best predictive algorithm. The Polynomial Regression predicts an increase in GEE of 7.09% on average for the analysed countries.

In this article, we estimate the role of Political Stability and Absence of Violence and Terrorism-PS in the context of Environmental, Social and Governance-ESG data at world level. We analyse data from 193 countries in the period 2011-2020. We apply Panel Data with Fixed Effects, Panel Data with Random Effects and Pooled Ordinary Least Square-OLS. We found that PS is positively associated, among others, to Population Density and Government Effectiveness, and negatively associated, among others, to Research and Development Expenditure and Maximum 5-day Rainfall. Furthermore, we apply the k-Means algorithm optimized with the application of the Elbow Method and we find the presence of four clusters. Finally, we propose a confrontation among eight different machine-learning algorithms for the prediction of PS and we find that the Polynomial Regression shows the higher performance. The Polynomial Regression predicts an increase in the level of PS of 0.25% on average for the analysed countries.

Although congenital sensorineural hearing loss (SHL) in the bilateral cochleae mainly results from genetic abnormalities, chronic SHL progressing in later life is often influenced by systemic immune disturbances, including autoimmunity, chronic inflammation, and immunosenescence. We have investigated the relationship between the inner ear and systemic immunity and reviewed the possibilities to prevent SHL, including autoimmune SHL and age-related SHL. We also demonstrated two lymphocyte populations, interleukin 1 receptor type II (IL-1R2)-positive T cells (T1R2) and naturally occurring regulatory T cells (nTregs) in CD4⁺ T cells, which increase with aging, suppress host immune function and promote organ degeneration. Alterations in systemic immunity by fewer microbial antigen challenges in the living environment, elimination of immune suppressive lymphocytes, or immune rejuvenation with a reconstituted thymus may contribute not only to renew the cochlear function in SHL, but also to extend the healthy life of functional organs in a vigorous and youthful body, one of humanity’s greatest dreams.

Chen, et al., compared two modified polyurethane-based wound dressings (PUE and PUESi), both intended for use in chronic wounds, with a negative control (gauze) and a positive control (PolyMem) via a series of preliminary in vitro and rodent model tests. Future studies may find that one or both of these two new dressings are beneficial. However, the results of this first study cannot be relied upon. Although they exhibited in-depth knowledge of manufacturing techniques, the study designers seem far less familiar with the science of wound healing and with PolyMem wound dressings. This helps explain the flaws in the methods that lead to many of their tests not being translatable to real-world settings. In addition, the interpretation of some of the test results is questionable.

We tested the possibility of using “adverse events count” (AEC) as a drug-risk indicator and side-effect severity indicator. Data from 3938 adverse event (AE) reports for COVID-19 vaccines (Comirnaty, Moderna, Vaxzevria, and Janssen) and 6869 AE reports for other medicines were collected from the Bulgarian Drug Agency database (01.01.2018–31.03.2022). AEC was modeled with zero-adjusted negative binomial (ZANBI) and zero-adjusted Poisson inverse Gaussian (ZAPIG) regression models, which account for zero absence and overdispersion. The models’ fit was checked with residual diagnostic plots and parametric correspondence to normality. Explanatory variables were: age, sex, sequence number (order of submission), severity of AE, and vaccine type. Average AEC was higher in severe vs. non-severe AE, and in females vs. males; it decreased with age and was lower in Comirnaty than other vaccines. Variability of AEC for COVID-19 data decreased with sequence number in severe AE and increased in non-severe AE. Full ZANBI models had greater sensitivity than parametric ZANBI or ZAPIG models. Results showed correlation between AEC and AE severity, suggesting AEC as a simple and reliable measure of drug-risk and side-effect severity for clinicians and regulators, especially for COVID-19 vaccines.

AI is extraordinarily useful technology but it is becoming increasingly powerful with rapidly growing capabilities to disrupt and harm human society. We call on international and national organizations and individuals to join forces in banning the development of superintelligent AI and introducing regulations to prevent and mitigate AI-caused harm.

In this article, I estimate the value of “Individual Using Internet”-IUI in the context of Environmental, Social and Governance-ESG database of the World Bank. I use data from 193 countries for the period 2011-2020. I found that among others the value of IUI is positively associated to “Methane Emissions” and “People Using Safely Managed Sanitation Services” and negatively associated among others to “Fossil Fuel Energy Consumption” and “Renewable Energy Consumption”. I apply the k-Means algorithm for the clusterization optimized with the Elbow Method and we find the presence of three clusters. Finally, I confront eight machine-learning algorithms to predict the future value of IUI. I found that the best predictive algorithm is Linear Regression and that the value of IUI is expected to decrease on average of 0.30% for the analysed countries.

We analyze the question of GDP Growth-GDPG rate in the context of Environmental, Social and Governance-ESG framework. We use World Bank data for 193 countries in the period 2011-2020 using different econometric techniques i.e., Panel Data with Fixed Effects, Panel Data with Random Effects, Pooled Ordinary Least Squares-OLS. We found that GDPG rate is positively associated, among others, to “Government Effectiveness” and “Prevalence of Undernourishment” and negatively associated among others to “Unemployment” and “Research and Development Expenditure”. Furthermore, we have applied the k-Means algorithm optimized with the Elbow method and we found the presence of four clusters in the sense of GDPG rate. Finally, we confront eight machine learning algorithms to predict the value of GDPG rate and we found that the Polynomial Regression is the best predictor. The Polynomial Regression predicts an increase of GDPG rate equal to 2.88% on average for the analysed countries.

We estimate the determinants of Emissions-COE in the context of Environmental, Social and Governance-ESG model at world level. We use data of the World Bank for 193 countries in the period 2011-2020. We found that the level of COE is positively associated, among others to “Methane Emissions”, “Research and Development Expenditures”, and negatively associated among others to “Renewable Energy Consumption” and “Mean Drought Index”. Furthermore, we have applied a cluster analysis with the k-Means algorithm optimized with the Elbow Method and we find the presence of four cluster. Finally, we apply eight machine-learning algorithms for the prediction of the future value of COE and we find that the Artificial Neural Network-ANN algorithm is the best predictor. The ANN predicts a reduction in the level of COE equal to 5.69% on average for the analysed countries.

We estimate the value of Research and Development Expenditures as a percentage of GDP-RDE in the context of Environmental, Social and Governance-ESG model. We use the ESG World Bank database. We analyze data from193 countries in the period 2011-2020. We apply a set of econometric techniques i.e. Pooled Ordinary Least Squares-OLS, Panel Data with Random Effects, Panel Data with Fixed Effects, Weighted Least Squares-WLS. We found that the level of RDE is positively associated, among others, to “Nitrous Oxide Emissions” and “Scientific and Technical Journal Articles”, and negatively associated, among others to “Heat Index 35”, “Maximum 5-day Rainfall”. Furthermore, we perform a cluster analysis with the application of the k-Means algorithm optimized with the Elbow Method. The results show the presence of four clusters. Finally, we confront eight different machine-learning algorithms to predict the future value of RDE. We find that Linear Regression is the best predictive algorithms. RDE is expected to growth on average of 0.07% for the analysed countries.

In this article, we analyse the determinants of Regulatory Quality-RQ for 193 countries in the period 2011-2020. We use a database from ESG-Environment Social Governance of the World Bank. We apply OLS, Panel Data with Fixed Effects and Panel Data with Random Effects. We found that the variables that have the most positive impact on RQ, among others, are “GHG Net Emission”, “Mean Drought Index”, and “Heat Index”. We also found that the variables that have the most negative impact on RQ are among others “Renewable Energy Consumption”, “Voice and Accountability” and “Rule of Law”. Furthermore, we have applied the k-Means algorithm optimized with the Elbow Method and we find the presence of five clusters. In adjunct, we confront eight machine learning algorithms to predict the value of RQ and we found that the best predictor is Polynomial Regression. The predictive level of RQ for the analysed countries is expected to diminish of -1,29%. In the end, we present a network analysis with the Euclidean distance and we found the presence of a structure of seven networks using augmented data.

In this article we estimate the level of Government Effectiveness-GE in 193 countries in the period 2011-2020 using data of the ESG World Bank Database. Different econometric techniques are used i.e. Panel Data with Random Effects, Panel Data with Fixed Effects, and Pooled OLS. Results show that GE is positively related among others to “Control of Corruption”, “Political Stability and Absence of Violence/Terrorism”, and negatively associated with “Percentage Annual GDP Growth”. We perform a cluster analysis with the k-Means algorithm optimized with the Elbow Method and we find the presence of four clusters. Finally, we confront eight machine learning algorithms for the prediction of GE. Results show that the Polynomial Regression is the best predictive algorithm. The value of GE is expected to growth on average by 15.97%.

Goals: To use visual logistics for interpreting COVID-19 molecular and rapid antigen test (RAgT) performance, determine prevalence boundaries where risk exceeds expectations, and evaluate benefits of recursive testing along home, community, and emergency spatial care paths. Methods: Mathematica/open access software helped graph relationships, compare performance patterns, and perform recursive computations. Results: Tiered sensitivity/specificity comprise: T1) 90%/95%; T2) 95%/97.5%; and T3) 100%/≥99%, respectively. In emergency medicine, median RAgT performance peaks at 13.2% prevalence, then falls below T1, generating risky prevalence boundaries. RAgTs in pediatric ERs/EDs parallel this pattern with asymptomatic worse than symptomatic performance. In communities, RAgTs display large uncertainty with median prevalence boundary of 14.8% for 1/20 missed diagnoses, and at prevalence >33.3-36.9% risk 10% false omissions for symptomatic subjects. Recursive testing improves home RAgT performance. Home molecular tests elevate performance above T1, but lack adequate validation. Conclusions: Widespread RAgT availability encourages self-testing. Asymptomatic RAgT and PCR-based saliva testing present the highest chance of missed diagnoses. Home testing twice, once just before mingling, and molecular-based self-testing help avoid false omissions. Community and ER/ED RAgTs can identify contagiousness in low prevalence (<22%). Real-world trials of performance, cost-effectiveness, and public health impact could identify home molecular diagnostics as the optimal diagnostic portal.

A method for calibrating models of agricultural production and resource use presented by Howitt [1] for policy analysis is proposed to leverage multidisciplinary agricultural research at the National Center for Alluvial Aquifer Research (NCAAR). An illustrative example for Sunflower County, MS is presented to show how plot-level research can be extended to draw systemic region or basin wide implications. A hypothetical improvement in yields for dryland soybean varieties is incorporated to the model and shown to have a positive impact on aquifer outcomes and producer profits. The example illustrates that a change in one practice-crop combination can have system-wide impacts as evidenced by the change in acreages for all crops and practices.

Immune checkpoint receptors play a crucial role in steering tumor progression, prompting a focused investigation into the role of TIM3 in breast cancer. Analysis of TCGA BRCA data unveiled diverse correlations between TIM-3 expression and immune cell infiltration across subtypes. In our in-house cohort studied through IHC, significant changes in TIM3 expression with tumor grades were observed. Higher TIM3 expression correlated with increased disease-free survival in pan breast cancer, marking the first revelation of TIM3's positive role in pan BC prognosis. The study indicates that alluring TIM3 response, rather than suppression, may enhance breast cancer prognosis.

In this article we estimate the level of Control of Corruption for 193 countries in the period 2011-2020 using data from the ESG World Bank Database. Various econometric techniques are applied i.e.: Panel Data with Random Effects, Panel Data with Fixed Effects, Pooled OLS, WLS. Results show that “Control of Corruption” is positively associated, among others, to “Government Effectiveness” and “Political Stability and Absence of Violence/Terrorism”, while it is negatively associated among others to “Agriculture, Forestry, and Fishing Value Added as Percentage of GDP” and “GHG Net Emissions/Removals by LUCF”. A cluster analysis implemented with the k-Means algorithm optimized with the Elbow Method shows four clusters. A confrontation among eight Machine Learning algorithms is proposed for the prediction of Control of Corruption. Polynomial Regression is the best predictor for the training data. The level of Control of Corruption is expected to growth by 10.36% on average.

There are unsolved problems related to inflation, gravity, dark matter, dark energy, missing antimatter, and the birth of the universe. Some of them can be better answered by assuming the existence of aether and hypoatoms. Both were created during the inflation in the very early universe. While aether forms vacuum, hypoatoms, composed of both matter and antimatter and believed to be neutrinos, form all observable matter. In vacuum, aether exists between the particle-antiparticle dark matter form and the dark energy form in a dynamic equilibrium: A + A-bar = gamma + gamma. The same reaction stabilizes hypoatoms and generates a 3-dimensional sink flow of aether that causes gravity. Based on the hypoatom structure, the singularity does not exist inside a black hole; the core of the black hole is a hypoatom star or neutrino star. By gaining enough mass, ca. 3 X 10²² M_sun, to exceed neutrino degeneracy pressure, the black hole collapses or annihilates into the singularity, thus turning itself into a white hole or a Big Bang. The universe is anisotropic. Its center, or where the Big Bang happened, is at 0.66^+0.03_-0.01 times the radius of the observable universe at Galactic coordinates (l, b) = (286⁺¹⁰_-10, -43⁺⁷_-6). If we look from the Local Group to the center of the universe, the universe is rotating clockwise.

The concept of gray-zone lymphoma (GZL) has been progressively refined since its in-troduction in the literature in 1998. For several years, it was applied to a rather broad spectrum of conditions, posing the problem of the differential diagnosis between any type of Hodgkin lymphoma (HL) and diffuse large B-cell lymphoma, with special reference to primary mediasti-nal forms (PMBL). Officially recognized as a provisional entity in the 4th and revised 4th editions of the WHO Classification of Tumour of Haematopoietic and Lymphoid Tissues with the term “B-cell lymphoma unclassifiable with features intermediate between diffuse large B-cell lym-phoma and classical Hodgkin lymphoma”, it was limited to tumours, which showed either morphologic features reminiscent of classical HL (CHL) but carrying a complete B-cell pheno-type or conversely provided with a PMBL morphology but revealing CHL phenotypic charac-teristics. The definition of GZL has been further revised in the recently published International Lymphoma Classification and 5th Edition of the WHO Classification of Haemato-lymphoid Tu-mours, which have limited it to mediastinal neoplasms based on emerging molecular evidences. The aim of this review is to critically discuss the issue of GZL, also in the light of the suboptimal response to current therapies.

Hidradenitis suppurativa and Langerhans cell histiocytosis are rare disorders that can reveal similar clinical manifestations, making differential diagnosis difficult. A definitive diagnosis requires biopsy and histological confirmation in suspicious cases. The complex pathophysiology of these diseases has still not been understood, therefore the treatment is still a challenge for physicians. It is seen with predilection in the intertriginous areas of the body, predominantly in the axillary region, rarely requiring a surgical procedure. However perineal, perianal, and gluteal HS usually necessitates some form of surgical treatment. Langerhans cell histiocytosis (histiocytosis X) is a very rare systemic disease manifested by clonal cell proliferation of histiocytic-dendritic nature, with pseudonodular manifestation that deeply infiltrates the hypodermis and ulcerates the epidermis on the surface. Langerhans cells are dendritic, antigen-presenting cells localized in the suprabasal regions of the squamous epithelium of the skin and mucous membranes, which help regulate the immune system. In this report, we present our experience with extensive gluteal, perineal, and perianal hidradenitis suppurativa and Langerhans cell histiocytosis cases, including our treatment methods and outcomes.

Background: Nodal peripheral T-cell lymphomas (nPTCLs) encompass a heterogeneous group of mature and aggressive lymphoid malignancies, including peripheral T-cell lymphoma, not otherwise specified (PTCL/NOS), angioimmunoblastic T-cell lymphoma (AITL) and anaplastic large cell lymphoma (ALCL) ALK-positive and ALK-negative. Their differential diagnosis and prognosis are an issue in the clinical practice. Accurate biomarkers to refine the different subtypes of nPTCLs and to stratify their prognosis are essential to improve their treatment approach. The aim of this study was to test the prognostic impact of GATA3 gene expression, and its capability to discriminate the different subtypes of nPTCLs. Patients and Methods: From 2000 to 2017, 80 patients with nPTCLs were eligible for GATA3 gene expression analysis that was assessed retrospectively by quantitative real time PCR (qRT-PCR) of neoplastic biopsies in Formalin-Fixed Paraffin-Embedded samples (FFPE). Results: Median age was 49 years old (IqR 34-59), 43/80 (53.7%) were male. Median follow-up was 1.72 years. Of them, 36.3% were classified as PTCL/NOS, 31.2% ALK-negative ALCL, 21.2% ALK-positive ALCL and 11.3% AITL. The majority of cases had advanced stage (III/IV). Two-year estimated overall survival (OS) and progression-free survival (PFS) were 52.2% and 39.5%, respectively. The median GATA3 gene expression level was 0.49% (range 0 – 7.07) in all cohort, it was 0.11% for ALK-positive ALCL, 0.46% for ALK-negative ALCL, 0.86% for PTCL/NOS and 0.67% for AITL. The difference of GATA3 gene expression among distinct variants of nPTCLs was statistically significant (p < 0.001). GATA3 gene expression levels ≥ 0.71% discriminate PTCL/NOS from ALK-negative ALCL and AITL with sensitivity of 62% and specificity of 80.3%. GATA3 gene expression levels ≥ median was associated with poor 2-year OS for PTCL/NOS (46.7% x 21.4%, p=0.04) and for ALK-negative ALCL (85.7% x 54.5%, p=0.04). Conclusion: Despite the relative small and heterogeneous group of patients with nPTCLs, GATA3 gene overexpression may be an important biomarker associated with poor prognosis in PTCL/NOS and ALK-negative ALCL. Moreover it may also discriminate different subtypes of nPTCLs. Further studies with larger series of patients should confirm our findings.

Abstract: A prevalence boundary (PB) marks the point in prevalence where the false omission rate, RFO=FN/(TN+FN), exceeds the tolerance limit for missed diagnoses. Objectives were to mathematically analyze rapid antigen test (RAgT) performance, determine why PBs are breeched, and evaluate the merits of testing three times over five days, now required by the US Food and Drug Administration for asymptomatic persons. Equations were derived to compare test performance patterns, calculate PBs, and perform recursive computations. An independent July 2023 FDA-university-commercial evaluation of RAgTs provided performance data used in theoretical calculations. Tiered sensitivity/specificity comprise: Tier-1) 90%, 95%; Tier-2) 95%, 97.5%; and Tier-3) 100%, ≥99%, respectively. Repeating a T2 test improves the PB from 44.6% to 95.2% (RFO 5%). In the FDA-university-commercial evaluation, RAgTs generated sensitivity of 34.4%, which improved to 55.3% when repeated, then 68.5% with the third test. With RFO=5%, PBs were 7.37/10.46/14.22%, respectively. PB analysis suggests RAgTs should achieve clinically proven sensitivity of 91.0-91.4%. When prevalence exceeds PBs, missed diagnoses can perpetuate virus transmission. Repeating low-sensitivity RAgTs delays diagnosis. In homes, high-risk settings, and hotspots, PB breaches may prolong contagion, defeat mitigation, facilitate new variants, and transform outbreaks into endemic disease. Molecular diagnostics can help avoid these potential vicious cycles

Matiyasevich's theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non-negative integers. Smorynski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smorynski's theorem easily follows from Matiyasevich's theorem, (2 ) Hilbert's Tenth Problem for Q has a negative solution if and only if the set of all Diophantine equations with a finite number of rational solutions is not recursively enumerable.

Backward electromagnetic waves are extraordinary waves with contra-directed phase velocity and energy flux. Unusual properties of the coherent nonlinear optical coupling of the phase-matched ordinary and backward electromagnetic waves with contra-directed energy fluxes are described which enable greatly-enhanced frequency and propagation direction conversion, parametrical amplification, as well as control of shape of the light pulses. Extraordinary transient processes that emerge in such metamaterials in pulsed regimes are described. The results of the numerical simulation of particular plasmonic metamaterials with hyperbolic dispersion are presented, which prove the possibility to match phases of such coupled guided ordinary and backward electromagnetic waves. Particular properties of the outlined processes in the proposed metamaterial are demonstrated through numerical simulations. Potential applications include ultra-miniature amplifiers, frequency changing reflectors, modulators, pulse shapers, and remotely actuated sensors.