Nonlinear Clusters Research Articles

Electronic structure, absorption spectra and oxidation dynamics in polyynes and dicyanopolyynes.

The advent of femtosecond to attosecond experimental tools has made now possible to study such ultrafast carrier dynamics, e.g., the spatial and temporal charge density evolution, after an initial oxidation or reduction in molecules, candidates for atomic wires like polyynes and dicyanopolyynes. Here, we study the electronic structure and hole transfer in symmetric molecules containing carbon, nitrogen and hydrogen, the first members in the series of polyynic carbynes and dicyanopolyynes, using methods based on density functional theory (DFT): constrained DFT (CDFT), time-dependent DFT (TDDFT) and real-time TDDFT (RT-TDDFT), with Löwdin population analysis, comparing many levels of theory and obtaining convergence of the results. For the same purposes, we develop a tight binding (TB) variant using all valence orbitals of all atoms. This TB variant is applied here in linear molecules, but it is also adequate for electronic structure, charge transfer and charge transport of non-linear molecules and clusters of molecules. We calculate the electronic structure, the time-dependent dipole moment and the probabilities of finding the hole at each site, their mean over time values, the mean transfer rates from the oxidation site to other sites and the frequency content (using charge as well as dipole moment oscillations). We take into account zero-point motion. The initial conditions for RT-TDDFT are obtained by CDFT. For TB, we explore different initial conditions: we place the hole at a particular orbital or distribute it among a number of orbitals; it is also possible to include phase differences between orbitals. Finally, we compare with available experimental data.

Sparse kernel k-means clustering

Clustering is an essential technique that groups similar data points to uncover the underlying structure and features of the data. Although traditional clustering methods such as k-means are widely utilized, they have limitations in identifying nonlinear clusters. Thus, alternative techniques, such as kernel k-means and spectral clustering, have been developed to address this issue. However, another challenge arises when irrelevant variables are present in the data; this can be mitigated by employing variable selection methods such as the filter, wrapper, and embedded approaches. In this study, with a particular focus on kernel k-means clustering, we propose an embedded variable selection method using a tensor product space along with a general analysis of variance kernel for nonlinear clustering. Comprehensive experiments involving simulations and real data analysis demonstrated that the proposed method achieves competitive performance compared to existing approaches. Thus, the proposed method may serve as a reliable tool for accurate cluster identification and variable selection to gain insights into complex datasets.

Harmonic balance formulation for nonlinear wave-scattering clusters

Nonlinear oscillators can be used as a scattering mechanism to passively manipulate propagating elastic waves in an energy-dependent manner. However, the interaction of incident elastic waves with clusters of discrete nonlinear oscillators introduces considerable modeling challenges that must be overcome to leverage this passive scattering mechanism. This paper proposes a harmonic balance (HB) approach based on the theory of multiple scattering to solve the nonlinear scattering problem. Specifically, we compute periodic solutions for finite arrays of nonlinear oscillators coupled to a linear elastic continuum to resolve the nonlinear scattering wavefields as well as their stability properties. Additionally, we show that the problem can be recast into an equivalent set of nonlinear oscillators that are coupled through the Green’s function of the elastic continuum. Moreover, the formulation is general in the sense that it is applicable to a general class of linear elastic continua with a known Green’s function. We demonstrate our HB approach with an infinite Euler–Bernoulli beam example to demonstrate the passive tunability of the scattering array’s frequency response with respect to energy and other system parameters, and we confirm the HB beam solutions with direct finite element simulations. Furthermore, we highlight the ubiquity of this approach by studying a nonlinear scattering array of oscillators on a infinite plate and show the sensitivity of the HB solutions to configuration symmetry by comparing the nonlinear solutions of perfectly symmetric to perturbed scattering arrays. The efficacy of the discussed methodology for a broad array of nonlinear scattering problems is thus emphasized.

A simulation study of magnetic nanoparticle clustering in a fluid flow

The assembly of magnetic nanoparticles has been promisingly used in biomedical applications. Several computational studies have addressed the study of cluster formation in the colloidal regime. However, these studies did not mention the formation of clusters in nonequilibrium conditions, such as fluid flow. To fill this gap, we develop computational models of magnetohydrodynamic systems to study their behavior within a flow in terms of cluster shape formation. All simulations were performed by Verlet integration through LAMMPS, a molecular dynamics framework. Through simulations, we discover that the fluid flow promotes formation of non-linear clusters, and linear clusters tend to orient toward the direction of the flow. These phenomena are investigated by using the equilibrium of magnetic torque and fluid torque. The analysis shows that the formation of non-linear clusters are mainly assisted by a bias-orientation of the unstable linear clusters, while the bias-orientation is originated from the torque cancellation in the direction of the flow.

Nonlinear multiple scattering of flexural waves in elastic beams: Frequency conversion and non-reciprocal effects

Multiple scattering from a cluster of nonlinear point attachments on a beam is formulated and analyzed. The nonlinear equation of motion is solved using a perturbation strategy. The analytical solution is implemented for two common mass–spring–damper systems and for either nonlinear stiffness or nonlinear damping of power-law dependence. The nonlinearity generates harmonic waves at frequencies that are multiples of the incident-wave frequency, and also shifts the linear response of the fundamental frequency. In addition to asymmetric reflection, which is also achievable using linear scatterers with damping, the cluster of nonlinear scatterers can produce asymmetric transmission, thus breaking reciprocity of the system. Analytical results for a single scatterer are presented and validated using Finite Element Method simulations with perfect agreement. A two-scatterer nonlinear cluster acts as a filter that is tunable by the incident amplitude and a three-scatterer cluster that acts as a non-reciprocal frequency converter are considered, demonstrating the potential of the proposed framework for the design of nonlinear scatterer clusters with more complicated scattering properties.

Molecular Tailoring Approach for Estimating Individual Intermolecular Interaction Energies in Benzene Clusters.

There is no general method available for the estimation of individual intermolecular interaction energies in weakly bound molecular clusters, and such studies are limited only to the dimer. Recently, we proposed a molecular tailoring approach-based method for the estimation of individual O-H···O hydrogen bond energies in water clusters. In the present work, we extend the applicability of this method for estimating the individual intermolecular interaction energies in benzene clusters, which are expected to be small. The basis set superposition error (BSSE)-corrected individual intermolecular interaction energies in linear (LN) benzene clusters, LN-(Bz)n n = 3-7, were calculated to be in the range from -1.75 to -2.33 kcal/mol with the cooperativity contribution falling between 0.05 and 0.20 kcal/mol, calculated at the MP2.5/aug-cc-pVDZ level of theory. In the case of non-linear (NLN) benzene clusters, NLN-(Bz)n n = 3-5, the BSSE-corrected individual intermolecular interaction energies exhibit a wider range from -1.16 to -2.55 kcal/mol with cooperativity contribution in the range from 0.02 to -0.61 kcal/mol. The accuracy of these estimated values was validated by adding the sum of interaction energies to the sum of monomer energies. These estimated molecular energies of clusters were compared with their actual calculated values. The small difference (<0.3 kcal/mol) in these two values suggests that our estimated individual intermolecular interaction energies in benzene clusters are quite reliable.

An 8-yr Meteotsunami Climatology across Northwest Europe: 2010–17

AbstractMeteotsunamis are shallow-water waves that, despite often being small (~0.3 m), can cause damage, injuries, and fatalities due to relatively strong currents (&gt;1 m s−1). Previous case studies, modeling, and localized climatologies have indicated that dangerous meteotsunamis can occur across northwest Europe. Using 71 tide gauges across northwest Europe between 2010 and 2017, a regional climatology was made to understand the typical sizes, times, and atmospheric systems that generate meteotsunamis. A total of 349 meteotsunamis (54.0 meteotsunamis per year) were identified with 0.27–0.40-m median wave heights. The largest waves (~1 m high) were measured in France and the Republic of Ireland. Most meteotsunamis were identified in winter (43%–59%), and the fewest identified meteotsunamis occurred in either spring or summer (0%–15%). There was a weak diurnal signal, with most meteotsunami identifications between 1200 and 1859 UTC (30%) and the fewest between 0000 and 0659 UTC (23%). Radar-derived precipitation was used to identify and classify the morphologies of mesoscale precipitating weather systems occurring within 6 h of each meteotsunami. Most mesoscale atmospheric systems were quasi-linear systems (46%) or open-cellular convection (33%), with some nonlinear clusters (17%) and a few isolated cells (4%). These systems occurred under westerly geostrophic flow, with Proudman resonance possible in 43 out of 45 selected meteotsunamis. Because most meteotsunamis occur on cold winter days, with precipitation, and in large tides, wintertime meteotsunamis may be missed by eyewitnesses, helping to explain why previous observationally based case studies of meteotsunamis are documented predominantly in summer.

Performance-Enhanced DMT System With Joint Precoding and Probabilistic Constellation Shaping

The promising solution for short-reach optical communication, discrete multitone (DMT) system suffer from high peak-to-average power ratio (PAPR). It is easy for the DMT signal to enter the nonlinear region of the components in the systems, which distorts the signal and limits the modulation efficiency. It is well-known that the precoding technique can suppress the PAPR of DMT without loss of spectral efficiency. We found that, in the I/Q plane, the nonlinear distortion clusters in the constellation points with higher power due to the joint precoding and inverse fast Fourier transform. The principle of this phenomenon has been studied. To cope with the specially distributed nonlinear distortions induced by precoding, we adopted the probabilistic constellation shaping (PCS) technique to reduce the probability of constellation points with high power, resulting in the overall performance improvement. Joint precoding and PCS significantly improve the resistance to nonlinear distortions for DMT systems. Simulations and experiments of 16-QAM and 64-QAM DMT transmission systems were conducted to evaluate the performance. The experimental results show that at the NGMI threshold of 0.92, up to 4 dB enhancement of the receiver sensitivity has been achieved by the proposed scheme when 4-GBd 64-QAM signals are transmitted.

Entertainment media in the context of hybrid war in the post-Soviet countries: the case of Ukraine

The article provides a complex analysis of how entertainment media can serve to undermine a country’s resilience and security amidst hybrid war using the case of Ukraine as an example. The paper documents that before the launch of Russia’s aggression against Ukraine in 2014, Russian media products had been heavily present in Ukrainian media space, including the entertainment segment. In 2017, Ukraine restricted access to some Russian media products and social media in its territory in an effort to counter disinformation and the use of user data by Russian security services via their access to the social media based in the Russian jurisdiction. Despite the measures taken by the state to address security challenges, build resilience and fight disinformation in the media, the influence of Russian entertainment media in shaping public opinion remains significant. In this paper, the authors analyze segments of the media space where Russian entertainment products are present in Ukraine, the tools used by Russia to enforce its narratives through media content, and the ways Ukraine has responded to these. This paper aims at demonstrating the role of entertainment mass media in the resilience of countries and how it is used in the context of hybrid war. Also, it looks at the efforts to counter this influence. The research shows that Russian entertainment media and content act as a soft power or cultural affinity element alongside misinformation or manipulations via news or information content. By using historical references, demonstrating civilizational and moral superiority, showing Slavic brotherhood, Russia and russocentric forces use entertainment media to shape and manipulate public opinion. As content consumption switches from linear media, such as television, to non-linear clusters of conventional and digital outlets, the room for the distribution of manipulative messages and narratives expands. Among other things, this undermines the resilience of countries and endangers their national security, especially in the hybrid war context. Much is being done to counter this impact. Ukraine’s restrictive measures against some Russian media, social networks and content have been effective in that they have decreased the consumption and the trust for Russian media amongst Ukrainian audiences. Offering alternative content, produced domestically and internationally, has contributed to diversification of the content, moving the audiences from the Russocentric cultural product to a more diverse one.

Analysis on the Features of NOCT and NMOT Tests With Photovoltaic Module

The operating temperature of the photovoltaic (PV) module plays a major role among the parameters affecting the energy yield of photovoltaic (PV) power generation systems. This paper compared the requirements of Nominal Operating Cell Temperature (NOCT), which is used to estimate the PV module operating temperature according to IEC 61215:2005, with those of the recently revised IEC 61215: 2016 Nominal Module Operating Temperature (NMOT). Through this, each feature of the parameters used to calculate the NOCT and NMOT could be determined, and it was assumed that the NMOT is lower than NOCT. And, among the parameters, the wind speed was found to be a significant variable affecting the temperature, and non-linear clusters and skewed data are improved if some data in the morning were deleted considering the period of module activation. Moreover, it is found that the method summing up the daily data for the calculation can reduce the correlation and deviation of NMOT.

Finite linear chains of potassium: A dft calculation of their interatomic distances and average electrical polarizabilities

Interesting physical behaviors can be found in one-dimensional clusters, but they have been studied much less than the three-dimensional ones. Using the model chemistry APFD/6-311 + G(3df) included in the GAUSSIAN 16 software package, we investigated the possible existence of some very small finite linear chains of potassium (Kn,n=2−13) beyond K3 and K4, whose existence has already been reported in the literature. The harmonic vibrational frequencies were always positive for all the optimized structures, guaranteeing that all of them corresponded to local minimums of the potential energy surface. All those systems belonged to the point group D∞h, although as a consequence of the finite size effects, the atoms were not evenly spaced. The odd chains had their atoms spaced more evenly than the even chains. The calculated values of the average electrical polarizabilities per atom α‾(Kn) showed odd-even oscillations as a function of the number of atoms, their minimums (maximums) corresponded to the even (odd) chains, and were always larger than those of the non-linear clusters. In particular, the α‾(K11) and α‾(K13) values were, respectively, 10 and 20 times larger than those for their three-dimensional counterparts.© 2014 Holy Spirit University of Kaslik. Hosting by Elsevier B.V. All rights reserved. Peer review under responsibility of Holy Spirit University of Kaslik.

Longitudinal automatic carrier landing system guidance law using model predictive control with an additional landing risk term

This paper used a model predictive control with an additional term to develop a modified longitudinal guidance law to reduce landing risk in an automatic carrier landing system. The landing risk model was established by using a longitudinal trajectory and touchdown point predictive principle. A traditional MPC was then involved in designing a modified automatic carrier landing system guidance law for the proposed model. The nonlinear landing mathematic model of an F/A-18 carrier-based aircraft was initially established. Considering the processed procedure in the model predictive control algorithm, the corresponding linear landing model was derived on the basis of the equilibrium states of the F/A-18. Second, landing trajectory in the longitudinal plane was analysed so that the predictive principle of the trajectory trend was reasonably addressed. Depending on the experimental sample data of a pilot model, some linear imitating envelopes are transformed from the corresponding nonlinear trajectory clusters. Furthermore, a touchdown point prediction model was further established based on the predicted trajectory and touchdown point. Third, the traditional model predictive control was introduced to integrate the landing risk term in the performance cost function to develop a novel modified algorithm that not only guides the aircraft to automatically approach and land on the carrier, but also eliminates landing risk during the final carrier approach. Linear matrix inequalities were imported to substitute algebraic inequalities derived from this new algorithm to increase calculating speed. A simulation mission was conducted on a semi-physical platform and compared with the traditional model predictive control without the additional term. The theoretical results validated the correctness and robustness of the modified algorithm and its capability to eliminate landing risk during terminal carrier approach.

Synchronization-based scalable subspace clustering of high-dimensional data

How to address the challenges of the "curse of dimensionality" and "scalability" in clustering simultaneously? In this paper, we propose arbitrarily oriented synchronized clusters (ORSC), a novel effective and efficient method for subspace clustering inspired by synchronization. Synchronization is a basic phenomenon prevalent in nature, capable of controlling even highly complex processes such as opinion formation in a group. Control of complex processes is achieved by simple operations based on interactions between objects. Relying on the weighted interaction model and iterative dynamic clustering, our approach ORSC (a) naturally detects correlation clusters in arbitrarily oriented subspaces, including arbitrarily shaped nonlinear correlation clusters. Our approach is (b) robust against noise and outliers. In contrast to previous methods, ORSC is (c) easy to parameterize, since there is no need to specify the subspace dimensionality or other difficult parameters. Instead, all interesting subspaces are detected in a fully automatic way. Finally, (d) ORSC outperforms most comparison methods in terms of runtime efficiency and is highly scalable to large and high-dimensional data sets. Extensive experiments have demonstrated the effectiveness and efficiency of our approach.

Histological evidence for reversible cardiomyocyte changes and serum cardiac troponin T elevation after exercise in rats.

This study characterized cardiac troponin T (cTnT) appearance and associated histological evidence of reversible or irreversible changes in myocardial ultrastructure, determined via electron microscopy, in rats undertaking isoproterenol (ISO) infusion or an endurance exercise challenge. Male rats were randomized into ISO and exercise groups. In ISO trials rats were killed 5 h (ISO‐5H) and 24 h (ISO‐REC19H) after a single ISO or saline injection (SAL‐5H; SAL‐REC19H). In the exercise trials rats were killed before, as a control (EXE‐CON), immediately after (EXE‐END5H) and 19 h after (EXE‐REC19H) a 5‐h bout of swimming with 5% body weight attached to their tail. Serum cTnT was quantified by electrochemiluminescence, and myocardial samples in ISO‐REC19H, EXE‐REC19H and SAL‐REC19H were harvested for assessment of specific mitochondrial injury scores using electron‐microscopy. cTnT was undetectable in all control animals (SAL‐5H/SAL‐REC19H and EXE‐CON). cTnT increased in all animals after ISO and exercise but the response was significantly higher (P < 0.05) at ISO‐5H (median [range]: 2.60 [1.76–6.18] μg · L−1) than at EXE‐END5H (median [range]: 0.05 [0.02–0.14] μg · L−1). cTnT returned to baseline at EXE‐REC19H, but had not completely recovered at ISO‐REC19H (median [range]: 0.17 [0.09–1.22] μg · L−1). Mitochondrial “injury scores” were significantly higher (P < 0.05) in ISO‐REC19H compared to EXE‐REC19H and SAL‐REC19H, with no difference between EXE‐REC19H and SAL‐REC19H. Mitochondria from EXE‐REC19H appeared aggregated in nonlinear clusters in a small number of scans. These findings suggest that acute exercise‐induced appearance of cTnT in this animal model is only associated with reversible changes in cardiomyocyte structure.

Density-based geodesic distance for identifying the noisy and nonlinear clusters

Clustering analysis can facilitate the extraction of implicit patterns in a dataset and elicit its natural groupings without requiring prior classification information. For superior clustering analysis results, a number of distance measures have been proposed. Recently, geodesic distance has been widely applied to clustering algorithms for nonlinear groupings. However, geodesic distance is sensitive to noise and hence, geodesic distance-based clustering may fail to discover nonlinear clusters in the region of the noise. In this study, we propose a density-based geodesic distance that can identify clusters in nonlinear and noisy situations. Experiments on various simulation and benchmark datasets are conducted to examine the properties of the proposed geodesic distance and to compare its performance with that of existing distance measures. The experimental results confirm that a clustering algorithm with the proposed distance measure demonstrated superior performance compared to the competitors; this was especially true when the cluster structures in the data were inherently noisy and nonlinearly patterned.

Electron counting approach to the structure and bonding of sandwiched low dimensional palladium clusters

One dimensional transition metal clusters are of theoretical and practical interest due to their special electronic and magnetic properties. One class of such clusters consists of a linear Pd moiety sandwiched between two olefin chains which act as a π-sheath. In this work, we provide a simple model, based on electron counting and basic background knowledge on coordination chemistry, to elucidate the structure and bonding properties of this kind of clusters. We also apply what we have learned from these simple examples to some other low-dimensional non-linear Pd clusters.

A systematic investigation of cooperativity between two types of hydrogen bonding in the nonlinear clusters of an aromatic molecule: Pyrazole

Crystalline pyrazole consists of nonlinear chains of an aromatic molecule. It includes two independent molecules which in turn causes two different types of hydrogen bonds (HBs). These two types of HBs with slight differences in their NH⋯N geometries can be considered as interesting ones in the recent studies of cooperativity between different HBs. These HBs are investigated in several pyrazole clusters by electronic structure calculations. Parameters such as structure, binding energy, charge transfer, chemical shielding and electric field gradient (EFG) parameters calculated at the second order Moller–Plesset perturbation (MP2) and density functional (DF) levels of theory. Both the basis set superposition error (BSSE) and zero point vibrational energy (ZPVE) corrections on the cooperativity enhancement were considered. Changes of different properties of clusters against crystal size were investigated by proposed diagrams fitted to a logarithmic function which renders their extrema in the crystal limit. In each cluster, pyrazole molecules for which their parameters are more affected by cooperativity enhancement were explored employing these fitted diagrams. Most calculated energetic and spectroscopic parameters were in good linear correlations with both the structural parameters and charge transfer along HB (qn(N2)→σ∗(N1–H1)). These correlations in the cases of nuclear magnetic resonance (NMR) and nuclear quadrupolar resonance (NQR) parameters, were explained in the terms of natural charges of bonding (σ(N1H1)) and antibonding (σ*(N1H1)) orbitals. Organizing calculated data for mental clusters with similar molecules and HB types produced better regression values in all linear correlations. According to the experimental CQ of N(2) in solid state and zero charge transfer in the gas phase, the value of charge transfer in the crystalline pyrazole and gas phase value of CQ of N(2) were assessed, respectively. Diagrams of the structural parameters against either crystal size or HB cooperativity propose that experimental structure in the case of both hydrogen and heavy nuclei positions is in doubt and should be revisited.

A Five-Year Radar-Based Climatology of Tropopause Folds and Deep Convection over Wales, United Kingdom

AbstractA five-year (2006–10) radar-based climatology of tropopause folds and convective storms was constructed for Wales, United Kingdom, to determine how deep, moist convection is modulated by tropopause folds. Based on the continuous, high-resolution data from a very high frequency (VHF) wind-profiling radar located at Capel Dewi, Wales, 183 tropopause folds were identified. Tropopause folds were most frequent in January with a secondary maximum in July. Based on data from the U.K. weather radar network, a climatology of 685 convective storms was developed. The occurrence of convective storms was relatively high year-round except for an abrupt minimum in February–April. Multicellular lines (43.5%) were the most common morphology with a maximum in October, followed by isolated cells (33.1%) with a maximum in May–September, and nonlinear clusters (23.4%) with a maximum in November–January. Convective storms were associated with 104 (56.8%) of the tropopause folds identified in this study, with the association strongest in December. Of the 55 tropopause folds observed on the eastern side of an upper-level trough, 37 (67.3%) were associated with convective storms, most commonly in the form of multicellular lines. Of the 128 tropopause folds observed on the western side of an upper-level trough, 42 (32.8%) were associated with convective storms, most commonly isolated cells. These results suggest that more organized storms tend to form in environments favorable for synoptic-scale ascent.

Evaluation of the Discriminating Capacity of Rock and Attribute Properties Using Statistical Methods

Statistical rock physics analysis of petrophysical logs for reservoir characterization has been carried out in a Niger delta field. The objective of the study is to determine the discriminating capacity of these rocks and attribute properties in 2-D cross plot space. Classifications were done in terms of Pearson's coefficient of correlation (R 2 ) of the cross plotted property pairs. Low R 2 value gives the most non-linear data clusters with good separation and highest discriminating capacity, while high R 2 value gives linear and poorly separated data clusters and least discriminating capacity. Results indicates that the discriminating capacity of rock and attribute properties were in the order of ρ, PI, λρ, Poisson ratio, VpVs ratio, Zp and Vp for fluid, while ρ, µρ, Zs, Zp, and Vs are for lithology. In all cases of cross plot pairs, ρ appears to have the highest discriminating capacity, while Vs vs VpVs ratio cross plot the least. ρ exhibits dual property because of its heightened sensitivity to lithology and pore fill. The relative value of R 2 and the degree of discrimination into distinct lithology units and fluid types for the investigated properties depends on the quality of well log data used in the analysis and the characteristic property of the reservoir parameters.

Investigative Clinical Study on Prostate Cancer Part IV: Exploring Functional Relationships of Total Testosterone Predicting Free Testosterone and Total Prostate-Specific Antigen in Operated Prostate Cancer Patients

Objectives: To explore, in operated prostate cancer patients, functional relationships of total testosterone (tt) predicting free testosterone (ft) and total PSA. Patients and Methods: 128 operated prostate cancer patients were simultaneously investigated for tt, ft and PSA before surgery. Patients were not receiving 5α-reductase inihibitors, LH-releasing hormone analogues and testosterone replacement treatment. Scatter plots including ft and PSA versus tt were computed in order to assess the functional relationship of the variables. Linear regression analysis of tt predicting ft and PSA was computed. Results: tt was a significant predictor of the response variable (ft) and different subsets of the patient population were assessed according to the ft to tt ratio. PSA was related to tt according to a nonlinear law. tt was a significant predictor of PSA according to an inversely nonlinear law and different significant clusters of the patient population were assessed according to the different constant of proportionality computed from experimental data. Conclusions: In our prostate cancer population, ft was significantly predicted by tt according to a linear law, and the ft/tt ratio was a significant parameter for assessing the different clusters. Also, tt was a significant variable predicting PSA by a nonlinear law and different clusters of the patient population were assessed by the different constants of proportionality. As a theory, we explain the nonlinear relation of tt in predicting PSA as follows: (a) the number of androgen-independent prostate cancer cells increases as tumor volume and PSA serum levels rise, (b) the prevalence of androgen-independent cells producing a substance which inhibits serum LH, and (c) as a result lower levels of serum tt are detected.