Sum Of Fractions Research Articles

From Ancient Egyptian Fractions to Modern Algebra

From Ancient Egyptian Fractions to Modern Algebra

Statistics of ranks, determinants and characteristic polynomials of rational matrices

Abstract We consider the set of m × n {m\times n} matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions.

Comparison between Co(II) and Ni(II) cycling at goethite-water interfaces: Interplay with Fe(II)-catalyzed recrystallization

Cobalt (Co) and nickel (Ni) are critical metals for modern renewable energy technologies as well as essential micronutrients for terrestrial plant health and marine primary production. Both metals are commonly surface-adsorbed onto and/or structurally-incorporated into iron (oxyhydr)oxide minerals, such as goethite (α-FeOOH), that are ubiquitously present in soils and sediments at the Earth’s surface. In sub- and anoxic environments, aqueous ferrous iron (Fe(II)) generated from dissimilatory Fe(III) reduction can induce the recrystallization of goethite and subsequently influences the speciation and mobilization of associated metals, e.g., Co(III) being reduced to Co(II). While it is generally considered that divalent Co and Ni behave similarly at goethite-water interfaces, there lacks a direct comparison of their cycling through goethite in response to Fe(II)-catalyzed recrystallization. Here, under circumneutral anoxic conditions with and without addition of aqueous Fe(II), we performed batch reaction experiments on Co(II)-substituted goethite and Co(II) and Ni(II) sorption experiments on pure goethite. The redox state and coordination environment of Co associated with goethite were determined using high energy resolution fluorescence detected X-ray absorption spectroscopy at the Co K-edge, and the distribution of solid-bound Co and Ni in goethite following sorption was revealed by sequential dissolution. Our results show that substitution of Co(II) for Fe(III) in goethite has less negative feedback on Fe(II)-catalyzed recrystallization than Ni(II), which may be related to their differing structural distortions as evidenced by EXAFS. In the absence of Fe(II), aqueous Co(II) is more favourably adsorbed onto and incorporated deeper into goethite than Ni(II), suggesting that Co(II) is more structurally compatible with goethite. The presence of Fe(II) markedly enhances the structural incorporation of both Co(II) and Ni(II) as a result of goethite recrystallization, which in turn can be inhibited by the accumulation of metals at the mineral surface. Furthermore, structurally-incorporated Co(II) is more difficult to be released back into solution (i.e. sum of aqueous and extract fractions) than Ni(II) during Fe(II)-catalyzed goethite recrystallization. Overall, our work highlights the significant differences between Co(II) and Ni(II) in their cycling at goethite-water interfaces and intricate interplay with Fe(II)-catalyzed recrystallization. This improves our understanding of their mobilization and distribution in anoxic goethite-rich systems and can provide useful insights for their enrichment and extraction from natural and artificial laterites generated from the enhanced weathering of ultramafic mine tailings.

Mechanical properties of Mg-Ce-Y-Zr-Zn-Ca microalloyed magnesium alloys composed of near-atomic solute

A new extruded Mg-0.43Ce-0.17Y-0.17Zn-0.12Zr-0.06Ca (wt%) alloy was prepared by adding five alloying elements in accordance with the principle of near-equivalent atomic weights, with the atomic number percentages of the various alloying elements added close to each other and with the sum of the mass fractions of all the added alloying elements being controlled to be less than 1 wt%. The alloy has a prominent extrusion weave after extrusion and possesses a bimodal organization consisting of fine dynamically recrystallized (DRXed) grains (0.91 μm) and coarse unDRXed grains with strong dislocation storage capacity, achieving a yield strength of 348 MPa and a plasticity of 10.3 %.

The Effect of Tillage Systems and Weed Control Methods on Grain Yield and Gluten Protein Compositional and Content-Related Changes in Hybrid Bread Wheat

The use of simplified tillage systems and weed control methods using reduced herbicide doses in wheat production technology is one of the basic requirements of sustainable agriculture in terms of obtaining high-quality grain of this species. The aim of three-year field studies was to determine the yield and quality characteristics of hybrid wheat grain depending on two tillage systems (CT and RT) and four weed control methods: mechanical (M) and mechanical–chemical, using recommended herbicide doses (MH100) and doses reduced by 25 and 50% (MH75, MH50). A comparable grain yield, number of grains per spike, weight of one thousand wheat grains, and amount of gluten and ω gliadin subunits (GLI) were obtained in the RT and CT systems. The CT system increased protein content (by 15.2 g kg−1) and the increase in the sum of gluten protein fractions was higher for glutenins (GLU) and their LMW and HMW subunits (from 20.9 to 29.8%). The application of the method with the recommended herbicide dose (MH100), compared to M and MH50, resulted in an increase in grain yield by 0.89 and 1.04 t ha−1, respectively, as well as in the sum of GLI (by 8.4 and 12.3%) and GLU (by 13.7 and 25.3%). The application of the herbicide dose reduced by 25% (MH75) compared to the recommended dose (MH100), especially in the RT system, did not cause a significant decrease in protein content and the amount of GLI and GLU, while reducing grain yield (by 0.26 t ha−1) and the amount of gluten (by 3.1%).

A comprehensive kinetic modeling study on NH3/H2, NH3/CO and NH3/CH4 blended fuels

Ammonia (NH3), as an ideal zero-carbon fuel, plays a positive role in mitigating global warming. By cofiring NH3 with highly reactive small molecule fuels such as hydrogen (H2), carbon monoxide (CO) and methane (CH4), the reactivity of NH3 combustion can be significantly enhanced. In this work, a detailed kinetic mechanism for NH3/CH4/H2/CO containing 146 species and 1099 reactions was developed and extensively validated with selected experimental data from the literature on laminar burning velocity (LBV), ignition delay time (IDT), and species concentrations measured in burner-stabilized flames (BSF), plug flow reactors (PFR) and jet-stirred reactors (JSR), covering temperatures of 273–2000 K, pressures of 0.053–40 atm and equivalence ratios of 0.1–2. In addition, the detailed mechanism was simplified to include 53 species and 353 reactions using three simplification methods. Kinetic modeling analysis showed that the chemical and transport effects of H2, CO and CH4 are the main factors enhancing NH3 combustion, which rising with the increasing initial temperature and decreasing ammonia blending ratio. Among the four reactants, H2 is consumed first, followed by CH4 and NH3, with CO reacting last. By modifying the Metghalchi-Kech power law equation, it was found that a clear linear relationship exists between LBV and the sum of the maximum mole fractions of O, H, OH, and NH2 radicals.

All solutions of the Diophantine equation 1/x+1/y+1/z=u/(u+2)

In the history of mathematics, many mathematical researchers have investigated the Diophantine equation in the form , where and are positive integers. Without loss of generality, we may assume that . This Diophantine equation, also known as the Egyptian fraction equation of length 3, is to write the fraction as a sum of three fractions with the numerator being one and the denominators being different positive integers. Examples of research such as, in 2021, Sandor and Atanassov studied and found that the Diophantine equation has forty-four positive integer solutions. In this paper, we will study and find the complete positive integer solutions of the Diophantine equation , by using elementary methods of number theory and computer calculations. In the process, we can see that . Then, we will consider separately the value of a positive integer in nine cases. The first case is impossible. For the second and third cases, we will separate to consider the value of . For the remaining cases, we will separate to consider the value of . The research results showed that all positive integer solutions of the Diophantine equation are eighty-seven positive integer solutions. Moreover, from the steps to find the above positive integer solutions, we expect that it can be used to find the complete positive integer solutions of the Diophantine equation , where is a positive integer with .

First data on polybrominated diphenyl ethers and temporal trend of polychlorinated biphenyls in European brown bear as a bioindicator species

The contamination status on persistent organic pollutants (POPs) in the Croatian terrestrial ecosystem was investigated by analyzing two classes of contaminants in adipose tissue of male European brown bear (Ursus arctos, N = 27) collected in 2021 and 2022. In addition to seven indicator polychlorinated biphenyls (PCBs), for the first time seven polybrominated diphenyl ethers (PBDEs) were analyzed in terrestrial species from this region. Sum of mass fractions detected in adipose tissue ranged from 0.011 to 0.463 ng g−1 lipid weight (lw) for the PBDEs, and from 0.652 to 30.17 ng g−1 lw for PCBs, with two orders of magnitude difference in the median value for these two groups of POPs (PBDEs: 0.022 ng g−1 lw; PCBs: 2.307 ng g−1 lw). PCBs dominated over PBDEs with a share of over 94 % of the total mass fractions of both groups of analyzed POPs. Results of the comparison of the PCB levels in the adipose tissue of brown bears reported here and in previous investigation from this region showed decrease in PCB-28, but increase of PCB-118 and PCB-180, while other congeners seemed to have reached a steady-state level in the environment over a ten-year period. Among the compounds analyzed in this study, BDE-47, PCB-153 and PCB-180 were the dominant congeners. Somewhat higher sum of PCBs was measured in April compared to October, although not statistically significant, indicating possible impact of winter hibernation causing concentration of lipophilic compounds due to body fat loss. Revealed POP levels in brown bears are consistent with the data on the low contamination of the Croatian terrestrial ecosystem in general.

Intake, in vivo digestibility and protein utilization of wethers fed timothy or tall fescue when harvested at different dates in the first regrowth cycle

The objective of this study was to evaluate the effects of harvest date in the first regrowth cycle and grass species on intake, in vivo digestibility and its relation to protein utilization in wethers fed timothy or tall fescue silages. Timothy and tall fescue were harvested at regular (RTI and RTF, respectively) or late date (LTI and LTF, respectively) in the first regrowth, creating four experimental silages. Eight wethers were used in a duplicated 4 × 4 Latin square. Experimental periods lasted for 4 wk and wethers were fed ad libitum during the first 3 wk, with intake recorded during the third week. During the fourth week, wethers were fed 80% of ad libitum, and feces and urine were collected during the last 4 d. Wethers receiving RTI showed the greatest DM intake when expressed as kg/d or as percentage of body weight (BW) (P≤0.05). The intake of neutral detergent fiber (aNDFom) was affected by forage species only, where animals fed timothy silages had greater aNDFom intake than animals fed tall fescue silages (P<0.001). Intakes of CP and sum of the protein fractions A, B1 and B2 (AB1B2) were affected by the interaction between harvest date and forage species, where wethers fed RTI showed the greatest intakes of CP (P=0.001) and AB1B2 (P=0.02). Harvesting the forages at late date decreased the in vivo digestibility in wethers but only for timothy, where animals fed LTI silage showed the lowest DM (P<0.001), organic matter (OM) (P<0.001), aNDFom (P=0.02) and acid detergent fiber (P=0.004) digestibility, and a tendency for lower CP digestibility (P=0.07) compared with the other silages. Wethers fed RTI silage showed greater intake of nitrogen (N) (P=0.001) and digestible OM (P=0.003), greater allantoin (P=0.03) and hippuric acid (P=0.05) excretions, greater microbial N flow (P=0.03), and a tendency for greater excretion of fecal N (P=0.09) compared with the other silage-fed animals. In conclusion, delayed harvest decreased in vivo digestibility only in timothy, but even with lower in vivo digestibility wethers fed timothy silages showed a greater intake than wethers fed tall fescue silages, likely due to lower concentration of hydroxycinnamic acids observed in timothy compared with tall fescue as published previously.

Unit fractions with shifted prime denominators

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm { prime}\}$ . The same conclusion holds for the set $\{p-h : p\textrm { prime}\}$ for any $h\in \mathbb {Z}\backslash \{0\}$ , provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

A threshold for the best two-term underapproximation by Egyptian fractions

Let G be the greedy algorithm that, for each θ∈(0,1], produces an infinite sequence of positive integers (an)n=1∞ satisfying ∑n=1∞1/an=θ. For natural numbers p<q, let Υ(p,q) denote the smallest positive integer j such that p divides q+j. Continuing Nathanson’s study of two-term underapproximations, we show that whenever Υ(p,q)⩽3, G gives the (unique) best two-term underapproximation of p/q; i.e., if 1/x1+1/x2<p/q for some x1,x2∈N, then 1/x1+1/x2⩽1/a1+1/a2. However, the same conclusion fails for every Υ(p,q)⩾4. Next, we study stepwise underapproximation by G. Let em=θ−∑n=1m1/an be the mth error term. We compare 1/am to a superior underapproximation of em−1, denoted by N/bm (N∈N⩾2), and characterize when 1/am=N/bm. One characterization is am+1⩾Nam2−am+1. Hence, for rational θ, we only have 1/am=N/bm for finitely many m. However, there are irrational numbers such that 1/am=N/bm for all m. Along the way, various auxiliary results are encountered.

Egyptian fractions of bounded length

Egyptian fractions of bounded length

Size-fractioned zooplankton biomass in the Barents Sea: macroecological patterns across biogeography, climate, and varying ecosystem state (1989-2020)

The zooplankton community in the Barents Sea was monitored during autumn cruises from 1989-2020, with biomass determined as the depth-integrated dry weight (g m-2) of 3 size fractions (&gt;2 mm, 1-2 mm, &lt;1 mm). A large data set of 4543 sampling stations encompassing a subdivision of the Barents Sea was spatially averaged into 15 polygons based on topography. The resulting data set is used to describe relationships between the biomass of size fractions as expressions of the size structure of the zooplankton communities. Each of the 3 size fractions was positively related to the total zooplankton biomass (sum of fractions). The medium size fraction, which contained on average 48% of the total biomass, showed a strong linear correlation with total biomass (R2 ≈ 0.8). The medium fraction contained the older copepodite stages of Calanus species (C. finmarchicus and C. glacialis), which are dominant species interpreted to drive the overall changes in zooplankton biomass. The biomass of the small size fraction decreased with decreasing total biomass, but less so than the medium fraction. Thus, the small/medium biomass ratio increased as the total biomass decreased. This trend was most pronounced for shallow and central polygons, which coincide with the core feeding area of the large Barents Sea stock of capelin Mallotus villosus. The change in the small/medium biomass ratio is interpreted to reflect a shift from Calanus to smaller zooplankton in response to predation by capelin and other planktivorous predators, and possibly also by warming due to climate change.

Hemp protein: obtaining and functional and technological properties

The aim of this work was to determine the structural and functional characteristics of the two main hemp seed proteins, water-soluble albumin and salt-soluble globulin and alkali-soluble glutelin. The extract of 0.1 M CaCl2 protein meal of hemp seeds was subjected to sequential extraction to obtain three fractions: albumin in the aqueous phase and globulin (edestin) and glutelin in the salt phase at pH-9.0. Data on the amino acid composition showed the presence of an increased content of aromatic and hydrophobic residues in the globulin fraction. Gel electrophoresis showed that the albumin fraction had fewer disulfide bonds and therefore a more open (flexible) structure. The analysis of the content of essential amino acids in protein isolate and UV concentrate and its individual fractions was carried out for a general understanding of the ways of their application. The balance of amino acid content in the UV concentrate is close to the “ideal protein” and in the future it is advisable to use it to isolate bioactive peptides. Thus, the results obtained indicate that the processes of mechanical and chemical action on hemp flour provide raw materials with a high protein content, containing all essential amino acids and characterized by a predominant content of the sum of water and salt-soluble fractions. In connection with the identified changes in the properties of the protein obtained by different methods for release from antinutrients in the future, it is planned to pay attention to this issue separately.

Flexural behavior of high-strength steel bar reinforced UHPC beams with considering restrained shrinkage

Despite the excellent mechanical properties of ultra-high performance concrete (UHPC), UHPC beams with a conventional reinforcement ratio consistently demonstrate limited ductility. Therefore, this study aims to enhance the ductility of reinforced UHPC beams by regulating the tensile interaction between steel fibers and reinforcement. Six UHPC beams with varying fiber volume fractions and reinforcement ratios were tested under flexure. The effect of restrained shrinkage on flexural behavior was emphatically investigated. A cracked section analysis was also implemented to predict cracking behaviors of UHPC beams and was verified by experimental results. The test results revealed an increase in the ductility of UHPC beams with an increasing reinforcement ratio, while a decrease was observed with an increasing fiber volume fraction. The contribution of UHPC tensile capacity decreased significantly with the increase in the reinforcement ratio because of the restrained shrinkage. The stiffness and crack control capacity of reinforced UHPC beams reaches a plateau when the sum of reinforcement ratio and fiber volume fraction exceeds 3.1%. Increasing the fiber content is more effective than increasing the reinforcement ratio in limiting crack development, whereas the converse is true for improving the flexural capacity.

New coefficients of the weighted-sum-of-gray-gases model for gas radiation characteristics of hydrogen/natural gas blends combustion

To achieve carbon neutrality, hydrogen/natural gas blends are gaining momentum as a clean low-carbon fuel for the future. In order to accurately carry out the CFD numerical simulation of hydrogen/natural gas blends combustion, it is imperative to establish a precise model for gas radiation characteristics. The existing weighted-sum-of-gray-gases model (WSGGM) is not suitable for high hydrogen content fuels, such as hydrogen/natural gas blends. Hence, new coefficients of WSGGM is formulated specifically for the combustion of hydrogen/natural gas blends, drawing upon the groundwork established by the statistical narrow-band model (SNB model). This model is specifically designed to be applicable within the ambient pressures, encompassing the temperature regime of 400–2500 K, a path-length range of 0.001–60 m, and a hydrogen blending ratio of 20%–70%. The radiative heat transfer characteristics of the sum of the molar fraction of H2O and CO2 at 0.9, 0.6, and 0.3 were studied according to application conditions. The WSGGM is implemented to compute the radiative heat transfer in non-isothermal and non-homogeneous gas mixtures. When compared to the benchmark model, the deviation of the wall heat flux does not surpass 5.20%. The results demonstrate the excellent adaptability of the WSGG coefficients in simulating the radiative heat transfer during hydrogen/natural gas blends combustion.

Rings that are Egyptian with respect to a multiplicative set

The notion of an Egyptian integral domain D (where every fraction can be written as a sum of unit fractions with denominators from D) is extended here to the notion that a ring R is W-Egyptian, with W a multiplicative set in R. The new notion allows denominators just from W. It is shown that several results about Egyptian domains can be extended to the W-Egyptian context, though some cannot, as shown in counterexamples. In particular, being a sum of unit fractions from W is not equivalent to being a distinct sum of unit fractions from W, so we need to add the notion of the strictly W-Egyptian ring. Connections are made with Jacobson radicals, power series, products of rings, factor rings, modular arithmetic, and monoid algebras.

Approximation by Egyptian fractions and the weak greedy algorithm

Let 0<θ⩽1. A sequence of positive integers (bn)n=1∞ is called a weak greedy approximation of θ if ∑n=1∞1/bn=θ. We introduce the weak greedy approximation algorithm (WGAA), which, for each θ, produces two sequences of positive integers (an) and (bn) such that(a) ∑n=1∞1/bn=θ;(b) 1/an+1<θ−∑i=1n1/bi<1/(an+1−1) for all n⩾1;(c) there exists t⩾1 such that bn/an⩽t infinitely often.We then investigate when a given weak greedy approximation (bn) can be produced by the WGAA. Furthermore, we show that for any non-decreasing (an) with a1⩾2 and an→∞, there exist θ and (bn) such that (a) and (b) are satisfied; whether (c) is also satisfied depends on the sequence (an). Finally, we address the uniqueness of θ and (bn) and apply our framework to specific sequences.

Beyond "greening" and "browning": Trends in grassland ground cover fractions across Eurasia that account for spatial and temporal autocorrelation.

Grassland ecosystems cover up to 40% of the global land area and provide many ecosystem services directly supporting the livelihoods of over 1 billion people. Monitoring long-term changes in grasslands is crucial for food security, biodiversity conservation, achieving Land Degradation Neutrality goals, and modeling the global carbon budget. Although long-term grassland monitoring using remote sensing is extensive, it is typically based on a single vegetation index and does not account for temporal and spatial autocorrelation, which means that some trends are falsely identified while others are missed. Our goal was to analyze trends in grasslands in Eurasia, the largest continuous grassland ecosystems on Earth. To do so, we calculated Cumulative Endmember Fractions (annual sums of monthly ground cover fractions) derived from MODIS 2002-2020 time series, and applied a new statistical approach PARTS that explicitly accounts for temporal and spatial autocorrelation in trends. We examined trends in green vegetation, non-photosynthetic vegetation, and soil ground cover fractions considering their independent change trajectories and relations among fractions over time. We derived temporally uncorrelated pixel-based trend maps and statistically tested whether observed trends could be explained by elevation, land cover, SPEI3, climate, country, and their combinations, all while accounting for spatial autocorrelation. We found no statistical evidence for a decrease in vegetation cover in grasslands in Eurasia. Instead, there was a significant map-level increase in non-photosynthetic vegetation across the region and local increases in green vegetation with a concomitant decrease in soil fraction. Independent environmental variables affected trends significantly, but effects varied by region. Overall, our analyses show in a statistically robust manner that Eurasian grasslands have changed considerably over the past two decades. Our approach enhances remote sensing-based monitoring of trends in grasslands so that underlying processes can be discerned.

Removing the Froissart Doublets in a Rational Interpolation Based on Loewner Matrix

In order to implement wide band frequency sweeping, the S-parameters can be fitted with an adaptive rational interpolation based on Loewner matrix. However, the errors in the sampling data may lead to Froissart doublets, which look like spikes in the curve. In this paper, a novel technique is proposed to remove these doublets. At first, the rational expression is converted into the sum of partial fractions by solving two generalized eigenvalue problems. After that, the partial fraction term with the smallest imaginary part of the pole and relatively large absolute value is considered to generate the doublets. Removing this term results in a smooth rational polynomial, which is validated by the example of a passive circuit simulated by finite element method(FEM).