On Some Properties of the Norm of the Spectral Geometric Mean

In this paper, we consider the norms related to spectral geometric means and geometric means. When A and B are positive and invertible, we have ||A-1#B|| ≤ ||A-1σsB||. Let H be a Hilbert space and B(H) be the set of all bounded linear operators on H. Let A ∈ B(H). If ||A#X|| = ||AσsX||, ?X ∈ B(H)++, then A is a scalar. When is a C*-algebra and for any , we have that ||logA#B|| = ||logAσsB||, then is commutative.

Geometric Mean Decomposition Based Hybrid Precoding for Millimeter-Wave Massive MIMO

Hybrid precoding can reduce the number of required radio frequency(RF)chains in millimeter-Wave(mmWave) massive MIMO systems. However, existing hybrid precoding based on singular value decomposition(SVD) requires the complicated bit allocation to match the different signal-to-noise-ratios(SNRs) of different sub-channels. In this paper,we propose a geometric mean decomposition(GMD)-based hybrid precoding to avoid the complicated bit allocation. Specifically,we seek a pair of analog and digital precoders sufficiently close to the unconstrained fully digital GMD precoder. To achieve this, we fix the analog precoder to design the digital precoder, and vice versa. The analog precoder is designed based on the orthogonal matching pursuit(OMP) algorithm, while GMD is used to obtain the digital precoder. Simulations show that the proposed GMD-based hybrid precoding achieves better performance than the conventional SVD-based hybrid precoding with only a slight increase in complexity.

On Maps Preserving Unitarily Invariant Norms of the Spectral Geometric Mean

We consider maps on positive definite cones of von Neumann algebras preserving unitarily invariant norms of the spectral geometric means. The main results concern Jordan *-isomorphisms between C*-algebras, and show that they are characterized by the preservation of unitarily invariant norms of those operations.

The Hawking Hubble Temperature as the Minimum Temperature, the Planck Temperature as the Maximum Temperature, and the CMB Temperature as Their Geometric Mean Temperature

Using a rigorous mathematical approach, we demonstrate how the Cosmic Microwave Background (CMB) temperature could simply be a form of geometric mean temperature between the minimum time-dependent Hawking Hubble temperature and the maximum Planck temperature of the expanding universe over the course of cosmic time. This mathematical discovery suggests a re-consideration of Rh=ctcosmological models, including black hole cosmological models, even if it possibly could also be consistent with the Λ-CDM model. Most importantly, this paper contributes to the growing literature in the past year asserting a tightly constrained mathematical relationship between the CMB temperature, the Hubble constant, and other global parameters of the Hubble sphere. Our approach suggests a solid theoretical framework for predicting and understanding the CMB temperature rather than solely observing it.1.

Accurate mirror-based camera pose estimation with explicit geometric meanings

An accurate closed form solution is proposed to estimate camera pose by several mirrored reference object images acquired via a planar mirror under different unknown poses. Compared with state-of-the-art methods, our method is more accurate when there are more than three images and has explicit geometric meanings. This method also properly handles cases in which some of the mirror poses are parallel. The central idea is to minimize an error metric based on all reflections of rotation, which enables the camera rotation to be estimated directly by SVD of sum of mirrored camera rotations. After that, the camera translation is computed by solving a large system of linear equations to minimize object space collinearity error. Both synthesized data and real data experiments show the advantages of our approach.

Trio-Geometric mean-based three-stage Runge–Kutta algorithm to solve initial value problem arising in autonomous systems

In the present worldwide scenario,plenty of problems arising in science and engineering which can be modeled as differential equations and out of these,autonomous system has become a subject of great interest.Several laws of physics in which time is considered as an independent variable are expressed as autonomous systems.In this paper,Runge–Kutta(RK)three-stage geometric mean method is used to solve the initial value problem arises in autonomous systems.The method is discussed in detail,convergence of method is discussed,the accuracy and efficiency of the method are proved by considering a numerical example.The result is compared to some other methods and proposed method is found to be more efficient.The detailed analysis of error estimation confirms that proposed method is more efficient as compared to other methods.

An Integral Representation for the Weighted Geometric Mean and Its Applications

By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.

SHARP BOUNDS FOR TOADER-TYPE MEANS IN TERMS OF TWO-PARAMETER MEANS

In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2) sin^(2))1/2 tdt.

Fecal Enterococci Levels in Selected Tributaries of the Pampanga River Basin, Philippines, and Their Relation to Land Use

This study aims to generate data which can be used as a potential starting point for the updating of the Philippine Water Quality Criteria and the determination of the true impact of land use to the fecal contamination of the Pampanga River Basin (PRB), the largest subwatershed of Manila Bay. Levels of fecal indicator bacteria (FIB) were determined in the selected tributaries of the PRB, representing three land use categories, namely, the forest/woodland (control), agricultural and residential lands. FIB were quantified in order to investigate the potential contribution of the selected areas in the fecal contamination of the PRB. The study was conducted in 2021 covering March, May, June, July, and September to represent the dry (March and May) and wet (June, July, and September) seasons. Counts of FIB, namely thermotolerant coliform, E. coli, and enterococci were qualitatively correlated with the results of the ocular survey and key informant interview based on known fecal contributors and their relevant rainfall data. FIB counts of water bodies in the selected agricultural and residential land use categories had Geometric Mean (GM) counts that are statistically greater than those of bodies of water near the representative forest/woodland (control), and exceeded the acceptable GM limits for all FIB, regardless of the season. Notably, the GM values recorded for the waters near the selected forest/woodland (control) passed the water quality criteria for all measured FIB parameters for both seasons. Furthermore, enterococci levels in the control site were statistically lower during the wet season. These initial findings suggest that agricultural and residential land use categories could be major contributors to the unacceptable water quality of tributaries of the Pampanga River Basin. The prevalence of thermotolerant coliforms and E. coli was noted regardless of rainfall and land use, indicating these FIB may not be adequate as water quality indicators. With their ability to survive and persist in fecally contaminated sediments in water bodies and in nutrient-poor environments, enterococci could be more definitive indicators of fecal contamination and microbiological quality of environmental waters.

Like a Sum Is Generalized into an Integral, a Product May Be Generalized into an Inteduct

It is well known that an integral is nothing but a continuous form of a sum. Is it possible to do the same thing with a product? The answer is yes and done for the first time in this publication. The new operator is called inteduct. As an integral is a proper tool to calculate the arithmetic mean of a function, the inteduct gives the geometric mean of a function. This defines a new branch of mathematics. Most applications may lay way ahead. Only some are discussed here. One is applying the inteduct to probability theory. There it is possible e.g., to determine a function for a life expectation rather than just a numerical value. Another application is to distinguish chaos from randomness within numerically given values. At least for the logistic map there exists a direct connection between Lyapunov exponent and inteduct. To distinguish between chaos and randomness is particularly important in finance. While randomness implies ergodicity, chaos is non-ergodic. And many fundamental financial theories from portfolio theory to market efficiency require ergodicity.

Geometric-Harmonic Mean and Characterizations of Some Mean-Values

The purpose of this paper is to provide a direct proof on the fact that the geometric-harmonic mean of any two positive numbers can be calculated by a first complete elliptical integral, and then to give new characterizations of some mean-values.

The Influence of Different Land Use Manners on Soil Aggregate Characteristics of Consolidation and Returning to Field in Hollow Village of Hilly Area

The research aimed to explore the influence of different land use manners on soil aggregate,and provide scientific basis for improving soil stability and production performance of consolidation and returning to field in hollow village of hilly area. After consolidation and returning to field in hollow village of hilly area of Chengcheng County,Shaanxi,5 kinds of land use manners were set for 1-year plantation test,and they were corn( C treatment),wheat( W treatment),vegetable( V treatment),medicinal material( M treatment) and control( no plantation: CK treatment). Soil aggregate distribution,mean mass diameter( WMD),geometric mean diameter( GMD),aggregate failure rate( PAD),unstable aggregate number( ELT) and fractal dimension( D) at 0-40 cm of soil layer were measured and analyzed by dry and wet sieving methods. The results showed that( i) soil aggregate number and size at 0-40 cm of soil layer by each treatment were all significantly better than CK treatment,and > 0. 25 mm of aggregate content by dry sieving method( DR0. 25) and >0. 25 mm of aggregate content by wet sieving method( WR0. 25) at 0-40 cm of soil layer in each treatment showed declining trend with soil layer depth increased;( ii) MWD and GMD sequences of each treatment at 0-40 cm of soil layer by dry and wet sieving methods were both W treatment > C treatment > M treatment > V treatment > CK treatment,and C treatment was conducive to increasing large aggregate content of surface soil,while W treatment was conducive to increasing large aggregate content of lower soil;( iii) the analysis by wet sieving method showed that PAD and ELT at 0-40 cm of soil layer in each treatment both showed similar " Z" shape trend,and each treatment was significantly lower than CK;( iv) D sequence at 0-40 cm of soil layer in each treatment was C treatment < W treatment < M treatment < V treatment < CK treatment. D showed good linear relationship with> 0. 25 mm of aggregate content at 0-40 cm of soil layer by dry and wet sieving methods,and they were respectively R2= 0. 74 and R2= 0. 67. Corn and wheat plantation after consolidation and returning to field in hollow village was conducive to improving large aggregate content at 0-40 cm of soil layer,increasing the stability of soil layer and improving soil structure.

Thinking Probabilistically Revisited

Probability distributions are derived for real-world situations where the environment may be subject to high volatility involving radical revisions in probability judgments.A simple procedure is outlined deriving an initial probability distribution which may then be adjusted to reflect additional or new information.The trade-off between minimal computation and maximum information is examined.

Effect of catalyst diesel particulate filter aging and catalyst loadings on particulate emission characteristics from a diesel vehicle

In this study,the effect of new and used catalyzed diesel particulate filter(CDPF)with different catalyst loadings on the particulate emissions including the particle mass(PM),particle number(PN),particle size distribution(PSD)and geometric mean diameter(GMD)from a diesel vehicle were investigated based on a heavy chassis dynamometer.Results showed that more than 97.9%of the PN and 95.4%of the PM were reduced by the CDPF,and the reduction efficiency was enhanced by the catalyst loading.After using the CDPF,the PSD transformed from bimodal to trimodal with the peak shifting towards smaller particle size,more nucleation mode particles were reduced compared with accumulation mode ones,but the reduction effect on the accumulation mode particles was more significantly influenced by the catalyst loading.Notably,the CDPF increased the accumulation mode particles proportion,producing a larger GMD.For the used CDPF,its reduction effect on the particulate emissions enhanced,especially for the PM in accumulation mode.The PSD returned to bimodal,but the peak at accumulation mode began to be higher than that at nucleation mode,illustrating that more nucleation mode particles was removed.The aging of the CDPF resulted in greater effect on the PN-based PSD than that of PM-based PSD,but the effect of catalyst loading on the PN and PM emission factors was weakened.The used CDPF further increased the GMD,and the effect of catalyst loading on the GMD was strengthened,a higher catalyst loading led to a reduction in the GMD.

Comparisons of Two Ensemble Mean Methods in Measuring the Average Error Growth and the Predictability

In this paper, taking the Lorenz system as an example, we compare the influences of the arithmetic mean and the geometric mean on measuring the global and local average error growth. The results show that the geometric mean error （GME） has a smoother growth than the arithmetic mean error （AME） for the global average error growth, and the GME is directly related to the maximal Lyapunov exponent, but the AME is not, as already noted by Krishnamurthy in 1993. Besides these, the GME is shown to be more appropriate than the AME in measuring the mean error growth in terms of the probability distribution of errors. The physical meanings of the saturation levels of the AME and the GME are also shown to be different. However, there is no obvious difference between the local average error growth with the arithmetic mean and the geometric mean, indicating that the choices of the AME or the GME have no influence on the measure of local average predictability.

Effect of catalyzed diesel particulate filter and its catalyst loading on emission characteristics of a non-road diesel engine

In this study, the effects of a diesel oxidation catalyst(DOC) coupled with a catalyzed diesel particulate filter(CDPF) with different catalyst loadings on the power, fuel consumption,gaseous and particulate emissions from a non-road diesel engine were investigated. Results showed that the after-treatment had a negligible effect on the power and fuel consumption.The reduction effect of the DOC on the CO and hydrocarbon(HC) increased with the engine load. Further reductions occurred coupling with the CDPF. Increasing the catalyst loading resulted in a more significant reduction in the HC emissions than CO emissions. The DOC could increase the NO_(2)proportion to 37.9%, and more NO_(2)was produced when coupled with the CDPF below 250℃;above 250℃, more NO_(2)was consumed. The after-treatment could reduce more than 99% of the particle number(PN) and 98% of the particle mass(PM).Further reductions in the PN and PM occurred with a higher CDPF catalyst loading. The DOC had a better reduction effect on the nucleation particles than the accumulation ones, but the trend reversed with the CDPF. The DOC shifted the particle size distribution(PSD) to larger particles with an accumulation particle proportion increasing from 13% to 20%, and the geometric mean diameter(GMD) increased from 18.2 to 26.0 nm. The trend reversed with the CDPF and the accumulation particle proportion declined to less than 10%. A lower catalyst loading on the CDPF led to a higher proportion of nucleation particles and a smaller GMD.

Soil Aggregate Stability and Iron and Aluminium Oxide Contents Under Different Fertiliser Treatments in a Long-Term Solar Greenhouse Experiment

Soil in greenhouses is likely to suffer a gradual decline in aggregate stability. Determination of the effects of different fertiliser practices on soil aggregate stability is important for taking advantage of solar greenhouses. Soil aggregate stability and iron （Fe） and aluminium （A1） oxide contents were investigated in a 26-year long-term fertilisation experiment in greenhouse in Shenyang, China, under eight fertiliser treatments： manure （M）, fertiliser N （FN）, fertiliser N with manure （MN）, fertiliser P （FP）, fertiliser P with manure （MP）, fertiliser NP （FNP）, fertiliser NP with manure （MNP）, and control without any fertiliser （CK）. A wet sieving method was used to determine aggregate size distribution and water-stable aggregates （WSA）, mean weight diameter and geometric mean diameter as the indices of soil aggregate stability. Different fertiliser treatments had a statistically significant influence on aggregate stability and Fe and A1 oxide contents. Long-term application of inorganic fertilisers had no obvious effects on the mass proportion of aggregates. By contrast, manure application significantly increased the mass proportion of macroaggregates at the expense of microaggregates. All treatments, with the exception of FNP, significantly increased the stability of macroaggregates but decreased that of microaggregates when compared with CK. Aggregation under MP and MN was better than that under M and MNP; however, no significant differences were found among inorganic fertiliser treatments （i.e., FN, FP, and FNP）. A positive relation was found between pyrophosphate-extractable Fe and WSA （r=0.269）, but no significant relations were observed between other Fe and Al oxides and aggregate stability.

Immunological effects of a 10-μg dose of domestic hepatitis B vaccine in adults

Objective:To evaluate the immunological effects of three types of domestic 10-μg/dose hepatitis B vaccines in adults compared with a foreign vaccine, and to provide scientific evidence in support of adult hepatitis B vaccination. Methods:Adults from five counties (Deqing, Changxing, Nanxun, Wuxing, Anji) in Huzhou City, Shaoxing County and Tongxiang County, Zhejiang Province, China were selected. Blood samples were taken to assess serum HBsAg, anti-HBs, and anti-HBc using a chemiluminescence immunoassay. Adults, aged 16 to 49 years and who were anti-HBs negative at baseline, received hepatitis B immunizations at 0, 1, and 6 months. Anti-HBs levels were assessed one month after the third and final vaccination. Results:A total of 1 872 adults were immunized and the average positive rate was 89.5%. Four types of hepatitis B vaccine were used, including three from Chinese companies (Shenzhen Kangtai, Dalian High-Tech, and North China Pharmaceutical) and one from a UK company (GlaxoS-mithKline). Their seroconversion rates were 81.67%, 95.05%, 89.64%, and 86.81%, respectively. There was a significant difference between the anti-HBs positive conversion rates of the four types (P<0.005) but the seroconversion rates among the different vaccines were not significantly different (χ 2 =2.123, P=0.145). The average anti-HBs geometric mean titers (GMTs) of non-immune adults immunized with each of the four vaccines were 177.28, 473.23, 246.13, and 332.20 mIU/ml, respectively. There were no sta- tistically significant differences in the GMTs between the three types of domestic vaccine and the foreign vaccine (t= 1.575, P=0.116). Conclusions:Domestic recombinant hepatitis B vaccines can achieve immunization effects comparable to those of a foreign vaccine.

Effect of invasive weed biochar amendment on soil enzymatic activity and respiration of coal mine spoil:a laboratory experiment study

Mining and excavation activities cause massive degradation of land,leading to complete loss of soil resources,vegetation,and biodiversity.Mine spoils support invasive weeds(predominantly Lantana)which can strive in these harsh conditions,causing allelopathy during plantation stage of reclamation.It is hypothesised that biochar produced from invasive weeds will enhance enzymatic activity,CO_(2)flux and overall fertility of coal mine spoil.A 6-month incubation study was conducted on the effect of biochar amendment(2 and 3%,w/w)on mine spoil enzymatic activities(dehydrogenase,invertase,amylase and cellulase),respiration and coal mine spoil fertility.The study showed that biochar significantly improved dehydrogenase(83%)and cellulase activity(78%)at 3%amendment.Geometric mean of enzymatic activities increased from 1.87 in control to 4.51 at 2%and 3.25 at 3%biochar amendment.Mine spoil physio-chemical properties such as soil organic carbon(65%),cation exchange capacity(54%),bulk density(25%)and water holding capacity(19%),were improved significantly com-pared to the unamended mine spoil.Biochar amendment reduced mine spoil CO_(2)flux at 2%(2.85μmol CO_(2)m^(−2)s^(−1))and 3%(2.60μmol CO_(2)m^(−2)s^(−1))compared to control(4.92μmol CO_(2)m^(−2)s^(−1)).The cost of biochar production and application(2%,w/w)in pit plantation during reclamation is estimated to be 844 USD t ha−1(plantation density:1600 trees ha−1).On the basis of present study,biochar preparation from invasive weeds can be used for sustainable reclamation of coal mine spoil.

Distribution and stability of water-stable aggregates as affected by long-term cattle manure application to saline-sodic soil in the black soil region of northeastern China

Saline-sodic soil has a poor structure,low nutrient content,and excessive sodium in the western Heilongjiang Province,resulting in low crop productivity.Experimental treatments were established by applying manure to the soil for 5 years,12 years,and 16 years and soil without manure application was used as a control treatment(CK).The results indicate that the application of manure significantly increased soil macroaggregates,the mean weight diameter(MWD)and the geometric mean diameter(GMD)compared to those for the CK treatment.The soil organic matter(SOM)concentration increased from 17.8 to 47.9 g/kg,the soil pH decreased from 10.18 to 7.89,and the electrical conductivity(EC)decreased from 4.92 to 0.19 dS/m.The soil exchangeable Na^(+)was decreased and exchangeable Ca^(2+)was increased in the treatments with manure application compared with the CK treatment.And a decrease in the CaCO_(3)content was observed in the treatment with manure.Water-stable aggregates(WSAs)of greater than 2.0 mm were the dominant factor driving the changes in the MWD,and WSAs of 1.0-2.0 mm were the dominant factor driving the changes in the GMD.The correlation matrix showed that the SOM and soil exchangeable Ca^(2+)concentration was positively correlated with the stability of the WSAs,while the pH,EC,and soil exchangeable Na^(+)were negatively and significantly correlated.We conclude that the long-term application of manure to saline-sodic soil can increase the proportion of soil macroaggregates and thus increase the stability of WSAs,as a result of the formation of soil macroaggregates mainly caused by the increase in the organic colloidal matter and soil exchangeable Ca^(2+),and by the decrease in soil exchangeable Na^(+).