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 Subjects -> CHEMISTRY (Total: 881 journals)     - ANALYTICAL CHEMISTRY (54 journals)    - CHEMISTRY (616 journals)    - CRYSTALLOGRAPHY (21 journals)    - ELECTROCHEMISTRY (28 journals)    - INORGANIC CHEMISTRY (43 journals)    - ORGANIC CHEMISTRY (48 journals)    - PHYSICAL CHEMISTRY (71 journals) CHEMISTRY (616 journals)                  1 2 3 4 | Last

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 Acta Metallurgica Sinica (English Letters)Journal Prestige (SJR): 0.576 Citation Impact (citeScore): 2Number of Followers: 7      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1006-7191 - ISSN (Online) 2194-1289 Published by Springer-Verlag  [2350 journals]
• Young’s Modulus Enhancement and Measurement in CNT/Al Nanocomposites
• Authors: Zi-Yun Yu; Zhan-Qiu Tan; Gen-Lian Fan; Ren-Bang Lin; Ding-Bang Xiong; Qiang Guo; Yi-Shi Su; Zhi-Qiang Li; Di Zhang
Pages: 1121 - 1129
Abstract: Abstract Young’s modulus is a critical parameter for designing lightweight structure, but Al and its alloys only demonstrate a limited value of 70–72 GPa. The introduction of carbon nanotubes (CNTs) is an effective way to make Al and its alloys stiffer. However, little research attention has been paid to Young’s modulus of CNT/Al nanocomposites attributed to the uncertain measurement and unconvincing stiffening effect of CNTs. In this work, improved Young’s modulus of 82.4 ± 0.4 GPa has been achieved in 1.5 wt% CNT/Al nanocomposite fabricated by flake powder metallurgy, which was determined by resonance test and 13.5% higher than 72.6 ± 0.64 GPa of Al matrix. A comparative study and statistical analysis further revealed that Young’s modulus determined by tensile test was relatively imprecise (83.1 ± 4.0 GPa) due to the low-stress microplasticity or interface decohesion during tensile deformation of CNT/Al nanocomposite, while the value (98–100 GPa) was highly overestimated by nanoindentation due to the “pile-up” effect. This work shows an in-depth discussion on studying Young’s modulus of CNT/Al nanocomposites.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0730-8
Issue No: Vol. 31, No. 11 (2018)

• Ni-Based Metallic Glass Composites Containing Cu-Rich Crystalline
Nanospheres
• Authors: Yao-Yao Xi; Jie He; Xiao-Jun Sun; Wang Li; Jiu-Zhou Zhao; Hong-Ri Hao; Ting Xiong
Pages: 1130 - 1136
Abstract: Abstract In this work, a quaternary Ni–Cu–Nb–Ta system has been designed to obtain composite microstructure with spherical crystalline Cu-rich particles embedded in amorphous Ni-rich matrix. The alloy samples were prepared by using single-roller melting-spinning method. The microstructure and thermal properties of the as-quenched alloy samples were characterized by X-ray diffraction, scanning electron microscopy, high-resolution transmission electron microscopy, and differential scanning calorimetry. It shows that the spherical crystalline Cu-rich particles are embedded in the amorphous Ni-rich matrix. The average size of the Cu-rich particles is strongly dependent upon the Cu content. The effect of the alloy composition on the behavior of liquid–liquid phase separation and microstructure evolution was discussed. The phase formation in the Ni-based metallic glass matrix composite was analyzed.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0756-y
Issue No: Vol. 31, No. 11 (2018)

• Crevice Corrosion of Several Supper Stainless Steels in the Simulated
LT-MED Environment
• Authors: Chang-Gang Wang; Xiao-Fang Li; Jie Wei; Xin Wei; Fang Xue; Rong-Yao Ma; Jun-Hua Dong; Wei Ke
Pages: 1137 - 1147
Abstract: Abstract Susceptibility and morphological characteristics of crevice corrosion for SS316, SS904L, SS254sMo and SS2507 in the simulated low-temperature multi-effect distillation environment were investigated by cyclic polarization test, scanning electron microscope and laser microscope. The results show that the crevice corrosion resistance of four kinds of stainless steel is ranked as SS254sMo ≈ SS2507 > SS316 > SS904L. There are “cover” structures over the edge of active crevice corrosion regions of SS904L, SS254sMo and SS2507, but SS316 is an exception. Galvanic corrosion characteristics appeared in the crevice of duplex supper stainless steel SS2507.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0728-2
Issue No: Vol. 31, No. 11 (2018)

• Evolution of the Corrosion Product Film and Its Effect on the
Erosion–Corrosion Behavior of Two Commercial 90Cu–10Ni Tubes in
Seawater
• Authors: Okpo O. Ekerenam; Ai-Li Ma; Yu-Gui Zheng; Si-Yu He; Peter C. Okafor
Pages: 1148 - 1170
Abstract: Abstract The composition and structural evolution of the corrosion product film of two commercial 90Cu–10Ni tubes, namely Tube A and Tube B, after being immersed in natural seawater for 1, 3, and 6 months were characterized by scanning electron microscopy, energy-dispersive X-ray spectroscopy, X-ray diffraction, and X-ray photoelectron spectroscopy, and its effect on the erosion–corrosion behavior of the tubes was determined through a rotating cylinder electrode system using various electrochemical techniques. For the freshly polished samples used as contrast samples, the flow velocity mainly enhanced the cathodic reaction at low flow velocities while both the anodic and the cathodic reactions were remarkably accelerated at higher flow velocities. The corrosion product films formed on the two commercial 90Cu–10Ni tubes after being immersed in seawater for up to 6 months are of a complex three-layer or multilayer structure. The structural evolution of the films is out of sync for the two tubes. A continuous residual substrate layer depleted of Ni was observed in the inner layer of the films on Tube B after 30, 90, and 180 days’ immersion, while it was observed in the film on Tube A only after 180 days’ immersion. The nature of the inner layer plays a crucial role in the erosion–corrosion resistance of the 90Cu–10Ni tubes at higher flow velocity. The film with a compact and continuous inner layer of Cu2O doped with Ni2+ and Ni3+ which bonds firmly with the substrate could survive and even get repaired with the increased flow velocity. The film on Tube B possessing a hollow and discontinuous inner layer composed of the residual substrate was degraded rapidly with increasing rotation speed in spite of its quite good resistance at the stagnant or lower speed conditions.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0745-1
Issue No: Vol. 31, No. 11 (2018)

• Corrosion Behavior of Epoxy-Coated Rebar with Pinhole Defect in Seawater
Concrete
• Authors: Yao-Zong Mao; Ying-Hua Wei; Hong-Tao Zhao; Chen-Xi Lv; Hai-Jiao Cao; Jing Li
Pages: 1171 - 1182
Abstract: Abstract Experiments were carried out to investigate the corrosion behavior of epoxy-coated rebar (ECR) with pinhole defect (diameter in hundreds of microns) immersed in the uncarbonated/carbonated simulated pore solution (SPS) of seawater concrete. Corrosion behavior was analyzed by electrochemical impedance spectroscopy. The composition and morphology of corrosion products were characterized by X-ray diffraction, energy-dispersive spectrometry and scanning electron microscopy. Meanwhile, oxide film produced by preheating before spray coating was investigated by X-ray photoelectron spectroscopy and Mott–Schottky technology. Results indicated that corrosion behavior of ECR with pinhole defect exhibited three stages when immersed in the uncarbonated/carbonated SPS. In the initial stage, steel in defect was passivated when exposed in the uncarbonated SPS and corroded when exposed in the carbonated SPS, due to competitive adsorption between chloride and hydroxyl ions. In the second stage, the oxide film under coating reconstituted (the thickness and defects density decreasing) in the uncarbonated SPS, which was caused by the synergy between high hydroxide and chloride activity, while in the carbonated SPS, crevice corrosion happened under the coating around pinhole, because of the different oxygen concentrations cell at the coating/steel interface. In the third stage, localized corrosion occurred under the coating around the pinhole in the uncarbonated SPS, which was probably induced by ion diffusion at the nano-scale coating/steel interface. The corrosion products adjacent to the defects were re-oxidized from FeCl2·4H2O and Fe2(OH)3Cl to Fe2O3·H2O, and the corrosion area was expanded outward in the carbonated SPS.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0755-z
Issue No: Vol. 31, No. 11 (2018)

• Dissimilar Joining of Pure Copper to Aluminum Alloy via Friction Stir
Welding
• Authors: Farhad Bakhtiari Argesi; Ali Shamsipur; Seyyed Ehsan Mirsalehi
Pages: 1183 - 1196
Abstract: Abstract In this study, the dissimilar friction stir welding (FSW) butt joints between aluminum alloy 5754-H114 and commercially pure copper were investigated. The thickness of welded plates was 4 mm and the aluminum plate was placed on the advancing side. In order to obtain a suitable flow and a better material mixing, a 1-mm offset was considered for the aluminum plate, toward the butt centerline. For investigating the microstructure and mechanical properties of FSWed joints, optical microscopy and mechanical tests (i.e., uniaxial tensile test and microhardness) were used, respectively. Furthermore, the analysis of intermetallic compounds and fracture surface was examined by scanning electron microscopy and X-ray diffraction. The effect of heat generation on the mechanical properties and microstructure of the FSWed joints was investigated. The results showed that there is an optimum amount of heat input. The intermetallic compounds formed in FSWed joints were Al4Cu9 and Al2Cu. The best results were found in joints with 1000 rpm rotational speed and 100 mm/min travel speed. The tensile strength was found as 219 MPa, which reached 84% of the aluminum base strength. Moreover, maximum value of the microhardness of the stir zone (SZ) was attained as about 120 HV, which was greatly depended on the grain size, intermetallic compounds and copper pieces in SZ.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0741-5
Issue No: Vol. 31, No. 11 (2018)

• Precipitation Characteristics and Mechanism of Vanadium Carbides in a
V-Microalloyed Medium-Carbon Steel
• Authors: Xiao-Lin Pan; Minoru Umemoto
Pages: 1197 - 1206
Abstract: Abstract The precipitation characteristics and mechanism of vanadium carbides during isothermal transformation at 650 °C in a V-microalloyed medium-carbon steel were investigated through scanning electron microscopy and transmission electron microscopy as well as dilatometry test. Five morphologies of vanadium carbides were found to precipitate at different nucleation sites during the transformation. Two kinds of interphase precipitation form simultaneously in both pro-eutectoid and pearlitic ferrites. The linear arrays of fine interphase precipitates are parallel to the γ/α interface, and the fine needles of interphase precipitates are perpendicular to the γ/α interface. The vanadium carbides of long or short fibers, coarse particles and fine particles form in both pro-eutectoid and pearlitic ferrites, displaying different precipitation distributions and orientation relationships with ferrite. The precipitation mechanisms of vanadium carbide precipitates with different modes were proposed, and the precipitation sequence of various vanadium carbide precipitates was finally ascertained.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0775-8
Issue No: Vol. 31, No. 11 (2018)

• Fe-Based Powders Prepared by Ball-Milling with Considerable Degradation
Efficiency to Methyl Orange Compared with Fe-Based Metallic Glasses
• Authors: Sheng-Hui Xie; Guang-Qiang Peng; Xian-Meng Tu; Hai-Xia Qian; Xie-Rong Zeng
Pages: 1207 - 1214
Abstract: Abstract In this study, the degradation efficiencies of zero-valent iron (ZVI) powders with different structures and components were evaluated for methyl orange (MO). The results show that the structure is an essential factor that affects degradation, and added non-metallic elements help optimize the structure. The amorphous and balled-milled crystalline Fe70Si10B20 has comparative degradation efficiencies to MO with t1/2 values of 6.9 and 7.0 min, respectively. Increasing the boron content can create a favorable structure and promote degradation. The ball-milled crystalline Fe70B30 and Fe43.64B56.36 powders have relatively short t1/2 values of 5.2 and 3.3 min, respectively. The excellent properties are mainly attributed to their heterogeneous structure with boron-doped active sites in ZVI. Composition segregation in the nanoscale range in an amorphous FeSiB alloy and small boron particles in the microscale range embedded in large iron particles prepared by ball-milling, both constitute effective galvanic cells that promote iron electron loss and therefore decompose organic chemicals. These findings may provide a new, highly efficient, low-cost commercial method for azo dye wastewater treatment using ZVI.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0751-3
Issue No: Vol. 31, No. 11 (2018)

• EBSD Study of Microstructural and Textural Changes of Hot-Rolled
Ti–6Al–4V Sheet After Annealing at 800 °C
• Authors: Ji-Ying Xia; Lin-Jiang Chai; Hao Wu; Yan Zhi; Yin-Ning Gou; Wei-Jiu Huang; Ning Guo
Pages: 1215 - 1223
Abstract: Abstract In this paper, electron backscatter diffraction and various other characterization and analysis techniques including X-ray diffraction, electron channeling contrast imaging and energy-dispersive spectrometry were jointly employed to investigate microstructural and textural changes of a hot-rolled Ti–6Al–4V (TC4) sheet after annealing at 800 °C for 5 h. In addition, the hardness variation induced by the annealing treatment is rationalized based on revealed microstructural and textural characteristics. Results show that the TC4 sheet presents a typical dual-phase (α + β) microstructure, with α-Ti as the major phase and short-rod-shaped β-Ti (minority) uniformly distributed throughout the α matrix. Most of α grains correspond to the un-recrystallized structures with a typical rolling texture (c//TD and <11–20>//ND) and dense low angle boundaries (LABs). After the annealing, the stored energy in the as-received specimen is significantly reduced, along with greatly decreased LABs density. Also, the annealing allows recrystallization and grain growth to occur, leading to weakening of the initial texture. Furthermore, the water quenching immediately after the annealing triggers martensitic transformation, which makes the high-temperature β phases be transformed into submicron α plates. The hardness of the annealed specimen is 320.5 HV, lower than that (367.0 HV) of the as-received specimen, which could be attributed to reduced LABs, grain growth and weakened texture. Nevertheless, the hardening effect from the fine martensitic plates could help to suppress a drastic hardness drop.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0768-7
Issue No: Vol. 31, No. 11 (2018)

• Effect of Heat Treatment on the Microstructure and Mechanical Properties
of the Modified 718 Alloy
• Authors: Da-Wei Han; Lian-Xu Yu; Fang Liu; Bin Zhang; Wen-Ru Sun
Pages: 1224 - 1232
Abstract: Abstract M718 alloy with an extra high Mo content of 7.50 wt% which reduced Nb addition and increased Al and Ti additions within the composition specifications of 718 alloy has been designed to increase the service temperature of 718 alloy. And the effect of the heat treatment on the microstructure and mechanical properties of M718 alloy has been investigated in this study. The results showed that Laves phase precipitated on the grain boundaries of M718 alloy instead of δ-Ni3Nb phase in 718 alloy, and γ′′ and γ′ phases precipitated in the matrix of M718 alloy as that in 718 alloy. Increasing the solution temperature from 960 to 1050 °C noticeably reduced the intergranular precipitation of Laves phase. The precipitation of Laves phase was appropriate at 1020 °C for improving the grain boundary cohesion. Increasing the two-stage aging temperatures markedly increased the sizes of γ′′ and γ′ phases. As a result, the strength of M718 alloy increased.
PubDate: 2018-11-01
DOI: 10.1007/s40195-018-0790-9
Issue No: Vol. 31, No. 11 (2018)

• Effects of Passes on Microstructure Evolution and Mechanical Properties of
Mg–Gd–Y–Zn–Zr Alloy During Multidirectional Forging
• Authors: Bing Li; Bu-Gang Teng; De-Gao Luo
Pages: 1009 - 1018
Abstract: Abstract The multidirectional forging (MDF) process was conducted at temperature of 753 K to optimize the mechanical properties of as-homogenized Mg–13Gd–4Y–2Zn–0.6Zr alloy containing long-period stacking ordered phase. The effects of MDF passes on microstructure evolution and mechanical properties were also investigated. The results show that both the volume fraction of dynamic recrystallization (DRX) grains and mechanical properties of the deformed alloy enhanced with MDF passes increasing till seven passes. The average grain size decreased from 76 to 2.24 μm after seven passes, while the average grain size increased to 7.12 μm after nine passes. The microstructure after seven passes demonstrated randomly oriented fine DRX grains and larger basal (0001)<11 $$\bar{2}$$ 0> Schmid factor of 0.31. The superior mechanical properties at room temperature (RT) with ultimate tensile strength (UTS) of 416 MPa and fracture elongation of 4.12% can be obtained after seven passes. The mechanical properties at RT after nine passes are inferior to those after seven passes due to the coarsening of DRX grains, which can be ascribed to the static recovery resulting from the repeated heating at the interval of MDF passes. The elevated temperature mechanical properties of the deformed alloy after seven passes and nine passes were investigated. When test temperature was below 523 K, the elevated temperature tensile yield strength and UTS after seven passes are superior to those after nine passes, while they are inferior to that after nine passes as temperature exceeds 523 K.
PubDate: 2018-10-01
DOI: 10.1007/s40195-018-0769-6
Issue No: Vol. 31, No. 10 (2018)

• CO Catalytic Oxidation of Pt-Loaded Perovskite BaTiO 3 Near
Ferroelectric-Phase Transition Temperature
• Authors: Si-Min Yin; Jia-Jie Duanmu; Yong-Feng Yuan; Shao-Yi Guo; Zhi-Chao Zhu; Zhao-Hui Ren; Gao-Rong Han
Pages: 1031 - 1037
Abstract: Abstract Perovskite BaTiO3 (BTO) nanocrystals with a size of 150–200 nm have successfully been synthesized via a facile hydrothermal method by employing titanate nanowires as synthetic precursor. Tetragonality and spontaneous ferroelectric polarization of BTO nanocrystals have been determined by X-ray diffraction and transmission electron microscopy investigations. BTO nanocrystals loaded with Pt nanoparticles in a size of 2–5 nm have been explored as a catalyst towards CO oxidation to CO2. It is interesting to find that CO catalytic conversion rate over Pt-BTO nanocrystals gradually decreased and further increased near 100 °C when the catalytic temperature keeps increasing, whereas the conversion behavior in oxides is expected to be enhanced upon the catalytic temperature grows. Using differential scanning calorimetry and first-principle calculations, the observed catalytic behavior has been discussed on the basis of the ferroelectric polarization effect and the ferroelectric–paraelectric transition of BTO nanocrystals with a Curie temperature of ~ 110 °C. Below Curie temperature, CO catalytic oxidation could be significantly tailored by ferroelectric polarization of BTO nanocrystals via a promoted dissociation of O2 molecules. The findings suggest that a ferroelectric polarization in perovskite oxides could be an alternative way to modify the CO catalytic oxidation.
PubDate: 2018-10-01
DOI: 10.1007/s40195-018-0757-x
Issue No: Vol. 31, No. 10 (2018)

• Effect of Pre-deformation on Grain Ultrafining by Intercritical
Deformation in Low-Carbon Microalloyed Steels
• Authors: Ba Li; Qing-You Liu; Shu-Jun Jia; Yi Ren; Bing Wang
Pages: 1038 - 1048
Abstract: Abstract In this study, the effect of pre-deformation at recrystallization and non-recrystallization zone on the grain ultrafining by the subsequent intercritical deformation (ID) was investigated on low-carbon microalloyed steel. The results showed that ultrafine grain microstructure with an average size of ~ 1.0 μm was fabricated through pre-deformation in the recrystallization zone followed by ID. When pre-deformed at the non-recrystallization zone prior to ID, the grain size increased to 1.6 μm with a heterogeneous distribution along with the well-developed dynamic recovery of ferrite. The grain ultrafining mechanism was attributed to the combined action of the deformation-induced ferrite transformation and the continuous dynamic recrystallization. In particular, the continuous dynamic recrystallization process during ID occurred on the pro-eutectoid ferrite as a result of the subgrain rotation mechanism and the absorbing dislocations mechanism.
PubDate: 2018-10-01
DOI: 10.1007/s40195-018-0767-8
Issue No: Vol. 31, No. 10 (2018)

• Tribocorrosion Behavior of 304 Stainless Steel in 0.5 mol/L Sulfuric
Acid
• Authors: Ming Liu; De-Li Duan; Sheng-Li Jiang; Ming-Yang Li; Shu Li
Pages: 1049 - 1058
Abstract: Abstract The tribocorrosion behavior of 304 stainless steel was studied through comparing the damage behavior of 304 stainless steel in dilute sulfuric acid to that in distilled water by a reciprocating tribotester. The re-passivation behavior, the surface and sectional morphological features, as well as the change of microhardness of samples were studied, and the tribocorrosion mechanism was also discussed. The experimental results reveal that the damage of stainless steel in dilute sulfuric acid was caused by the steel’s mechanical removal and electrochemical dissolution. The wear mechanism of stainless steel is abrasive wear, which mainly consists of micro-cutting and peeling. The synergetic action between corrosion and wear is notable. The corrosive environment leads to the embrittlement of the surface layer, and the wear destroys the passivation film and causes galvanic corrosion.
PubDate: 2018-10-01
DOI: 10.1007/s40195-018-0773-x
Issue No: Vol. 31, No. 10 (2018)

• Control of Secondary Phases by Solution Treatment in a N-Alloyed High-Mn
Cryogenic Steel
• Authors: Xiao-Jiang Wang; Xin-Jun Sun; Cheng Song; Shuai Tong; Luo-Jin Liu; Huan Chen; Wei Han; Feng Pan
Pages: 1059 - 1072
Abstract: Abstract The secondary phases of the steels have significant effects on the microstructure and mechanical properties, making controlling these secondary phases important. The control of MnS inclusions and AlN precipitates in a N-alloyed high-Mn twin-induced plastic cryogenic steel via solution treatment was investigated with several different techniques including microstructural characterization, 298 K tensile testing, and 77 K impact testing. The solutionizing temperature (ST) increased from 1323 to 1573 K, where the elongated MnS inclusions and large-sized AlN precipitates became spheroidized and dissolved. The aspect ratio of the MnS inclusions decreased as the ST increased and the number density increased. The impact toughness of the steels showed anisotropy and low impact energy values, due to the elongated MnS inclusions and large-sized AIN precipitates. The anisotropy was eliminated by spheroidizing the MnS inclusions. The impact energy was improved by dissolving the large-sized AlN precipitates during the solution treatment. The austenite grain size increased when the dissolution of the AlN precipitate increased, but the effect of the grain size on the yield strength, toughness, and the strength–ductility balance was weak.
PubDate: 2018-10-01
DOI: 10.1007/s40195-018-0759-8
Issue No: Vol. 31, No. 10 (2018)

• Friction and Wear Behavior of AlTiN-Coated Carbide Balls Against SKD11
Hardened Steel at Elevated Temperatures
• Authors: Rui Wang; Hai-Juan Mei; Ren-Suo Li; Quan Zhang; Teng-Fei Zhang; Qi-Min Wang
Pages: 1073 - 1083
Abstract: Abstract In this study, AlTiN coatings were deposited on YT14 cemented carbide balls by arc ion plating technique. The friction and wear behavior of the AlTiN-coated balls against SKD11 hardened steel was investigated by sliding tests using a ball-on-disk tribometer at various temperatures from 25 to 700 °C in air. The results showed that the friction and wear behavior was significantly influenced by the testing temperature. Obvious fluctuations were observed in the friction curves at elevated temperatures, which could be attributed to the formation and rupture of unstable Fe and Cr oxide layers. As the temperature increased from 25 to 500 °C, the wear rate of the coated balls increased from the scale of 10−21–10−20 m3/N m, and then decreased to 10−22 m3/N m as the temperature further increased to 700 °C. It was also found that the friction and wear behavior of the coated balls was directly dependent on the counterpart materials. As the temperature increased, the main wear mechanism of the coated balls changed from mild abrasive wear and adhesive wear to abrasive wear failure at 500 °C, and then transferred to adhesive wear and mild oxidation wear at 700 °C. For SKD11 hardened steel, the primary wear mechanism changed from delamination wear to abrasive wear and then transferred to plastic deformation and fatigue wear, accompanied by adhesive wear and tribo-oxidation wear.
PubDate: 2018-10-01
DOI: 10.1007/s40195-018-0753-1
Issue No: Vol. 31, No. 10 (2018)

• Achieving High Strength and High Electrical Conductivity in a CuCrZr Alloy
Using Equal-Channel Angular Pressing
• Authors: Yun-Xiang Tong; Yu Wang; Zhi-Min Qian; Dian-Tao Zhang; Li Li; Yu-Feng Zheng
Pages: 1084 - 1088
Abstract: Abstract In the present work, a CuCrZr alloy characterized by ultrafine grains and nanoscale particles was prepared by equal-channel angular pressing (ECAP) at 450 °C. A desired combination of a tensile strength (580 MPa) and an electrical conductivity (81% International Annealed Copper Standard) is simultaneously obtained in the as-ECAP-processed CuCrZr alloy without additional aging treatment. The improved properties can be mainly attributed to the ultrafine grains and nanoscale precipitates. This processing may pave a way to develop the CuCrZr alloys having high strength and high electrical conductivity for engineering applications.
PubDate: 2018-10-01
DOI: 10.1007/s40195-018-0766-9
Issue No: Vol. 31, No. 10 (2018)

• Influence of Cryorolling on the Precipitation of Cu–Ni–Si Alloys: An
In Situ X-ray Diffraction Study
• Authors: Wei Wang; Zong-Ning Chen; En-Yu Guo; Hui-Jun Kang; Yi Liu; Cun-Lei Zou; Ren-Geng Li; Guo-Mao Yin; Tong-Min Wang
Pages: 1089 - 1097
Abstract: Abstract The effect of cryorolling on the precipitation process of deformed Cu–Ni–Si alloys was investigated through in situ synchrotron X-ray diffraction technique. The results demonstrate that the precipitation process is significantly accelerated by cryorolling. Cryorolling produces higher dislocation density, which provides more heterogeneous nucleation sites for Ni2Si precipitates, hence promotes precipitation. In the early stage of aging, the enhanced nucleation of precipitates accelerates the depletion of supersaturation, and finer precipitates are obtained. In addition, recrystallization is promoted as a result of high stored energy in the cryorolled Cu–Ni–Si alloys, which facilitates the formation of discontinuous precipitation in the late stage of aging.
PubDate: 2018-10-01
DOI: 10.1007/s40195-018-0781-x
Issue No: Vol. 31, No. 10 (2018)

• Effect of Sintering Temperature and Heating Rate on Crystallite Size,
Densification Behaviour and Mechanical Properties of Al-MWCNT
Nanocomposite Consolidated via Spark Plasma Sintering
• Authors: Lavish Kumar Singh; Alok Bhadauria; Subhodeep Jana; Tapas Laha
Abstract: Abstract Powder mixture of ball-milled aluminium and functionalized multi-walled carbon nanotubes was compacted via spark plasma sintering (SPS) to study effects of sintering temperature and heating rate. An increase in sintering temperature led to an increase in crystallite size and density, whereas an increase in heating rate exerted the opposite effect. The crystallite size and relative density increased by 85.0% and 14.3%, respectively, upon increasing the sintering temperature from 400 to 600 °C, whereas increasing the heating rate from 25 to 100 °C/min led to respective reduction by 30.0% of crystallite size and 1.8% of relative density. The total punch displacement during SPS for the nanocomposite sintered at 600 °C (1.96 mm) was much higher than that of the sample sintered at 400 °C (1.02 mm) confirming positive impact of high sintering temperature on densification behaviour. The maximum improvement in mechanical properties was exhibited by the nanocomposite sintered at 600 °C at a heating rate of 50 °C/min displaying microhardness of 81 ± 3.6 VHN and elastic modulus of 89 ± 5.3 GPa. The nanocomposites consolidated at 400 °C and 100 °C/min, in spite of having relatively smaller crystallite size, exhibited poor mechanical properties indicating the detrimental effect of porosity on the mechanical properties.
PubDate: 2018-08-17
DOI: 10.1007/s40195-018-0795-4

• Effects of W Addition on the Electrochemical Behaviour and Passive Film
Properties of Fe-Based Amorphous Alloys in Acetic Acid Solution
• Authors: Dan-Dan Liang; Xian-Shun Wei; Chun-Tao Chang; Jia-Wei Li; Yong Wang; Xin-Min Wang; Jun Shen
Abstract: Abstract The electrochemical behaviour and passive film properties of Fe–Cr–Mo–W–C–B–Y amorphous alloys in acetic acid solution were investigated. The potentiodynamic polarisation and Nyquist curves demonstrated that W addition significantly enhanced the corrosion resistance. Mott–Schottky plots and angle-resolved X-ray photoelectron spectra indicated that passive films with different W contents exhibited dipolar (p–n) semiconducting characteristics separated by flat-band potentials. The outer and inner oxide layers of the passive films were modified by reducing the acceptor and donor densities. Moreover, W addition favoured the formation of a thicker and more stable passive film to inhibit the dissolution of alloy elements.
PubDate: 2018-08-17
DOI: 10.1007/s40195-018-0791-8

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text:
# Counting strings over $\mathbb{Z}2^d$ with Given Elementary Symmetric Function Evaluations

Abstract : Let $\alpha$ be a string over $\mathbb{Z}_q$, where $q = 2^d$. The $j$-th elementary symmetric function evaluated at $\alpha$ is denoted $e_j(\alpha)$ . We study the cardinalities $S_q(m;\mathcal{T} _1,\mathcal{T} _2,\ldots,\mathcal{T} _t)$ of the set of length $m$ strings for which $e_j(\alpha) = \tau _i$. The $\textit{profile}$ k$(\alpha) = ⟨k_1,k_2,\ldots,k_(q-1) ⟩$ of a string $\alpha$ is the sequence of frequencies with which each letter occurs. The profile of $\alpha$ determines $e_j(\alpha)$ , and hence $S_q$. Let $h_n$ : $\mathbb{Z}_{2^{n+d-1}}^{(q-1)}$ $\mapsto \mathbb{Z}_{2^d} [z]$ mod $z^{2^n}$ be the map that takes k$(\alpha)$ mod $2^{n+d-1}$ to the polynomial $1+ e_1(\alpha) z + e_2(\alpha) z^2 + ⋯+ e_{2^n-1}(\alpha)$ $z^{2^{n-1}}$. We show that $h_n$ is a group homomorphism and establish necessary conditions for membership in the kernel for fixed $d$. The kernel is determined for $d$ = 2,3. The range of $h_n$ is described for $d$ = 2. These results are used to efficiently compute $S_4(m;\mathcal{T} _1,\mathcal{T} _2,\ldots,\mathcal{T} _t)$ for $d$ = 2 and the number of complete factorizations of certain polynomials.
Keywords :
Document type :
Conference papers

Cited literature [4 references]

https://hal.inria.fr/hal-01229695
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### Citation

Charles Robert Miers, Franck Ruskey. Counting strings over $\mathbb{Z}2^d$ with Given Elementary Symmetric Function Evaluations. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.897-908, ⟨10.46298/dmtcs.2352⟩. ⟨hal-01229695⟩

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NAG FL Interfaceg02jff (lmm_init)

1Purpose

g02jff preprocesses a dataset prior to fitting a linear mixed effects regression model via g02jhf.

2Specification

Fortran Interface
 Subroutine g02jff ( hlmm, n, y, wt, dat, nff, nrf, nvpr,
 Integer, Intent (In) :: nrndm, n, lddat, sddat, lrcomm, licomm Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: fnlsv, nff, rnlsv, nrf, nvpr, icomm(licomm) Real (Kind=nag_wp), Intent (In) :: y(n), wt(*), dat(lddat,sddat) Real (Kind=nag_wp), Intent (Out) :: rcomm(lrcomm) Character (1), Intent (In) :: weight Type (c_ptr), Intent (In) :: hddesc, hfixed, hrndm(nrndm) Type (c_ptr), Intent (Inout) :: hlmm
#include <nag.h>
 void g02jff_ (void **hlmm, void **hddesc, void **hfixed, const Integer *nrndm, void ***hrndm[], const char *weight, const Integer *n, const double y[], const double wt[], const double dat[], const Integer *lddat, const Integer *sddat, Integer *fnlsv, Integer *nff, Integer *rnlsv, Integer *nrf, Integer *nvpr, double rcomm[], const Integer *lrcomm, Integer icomm[], const Integer *licomm, Integer *ifail, const Charlen length_weight)
The routine may be called by the names g02jff or nagf_correg_lmm_init.

3Description

g02jff must be called prior to fitting a linear mixed effects regression model via g02jhf.
The model is of the form:
 $y=Xβ+Zν+ε$
 where $y$ is a vector of $n$ observations on the dependent variable, $X$ is an $n$ by $p$ design matrix of fixed independent variables, $\beta$ is a vector of $p$ unknown fixed effects, $Z$ is an $n$ by $q$ design matrix of random independent variables, $\nu$ is a vector of length $q$ of unknown random effects, $\epsilon$ is a vector of length $n$ of unknown random errors.
Both $\nu$ and $\epsilon$ are assumed to have a Gaussian distribution with expectation zero and variance/covariance matrix defined by
 $Var ν ε = G 0 0 R$
where $R={\sigma }_{R}^{2}I$, $I$ is the $n×n$ identity matrix and $G$ is a diagonal matrix. It is assumed that the random variables, $Z$, can be subdivided into $g\le q$ groups with each group being identically distributed with expectation zero and variance ${\sigma }_{i}^{2}$. The diagonal elements of matrix $G$ therefore take one of the values $\left\{{\sigma }_{i}^{2}:i=1,2,\dots ,g\right\}$, depending on which group the associated random variable belongs to.
The model therefore contains three sets of unknowns: the fixed effects $\beta$, the random effects $\nu$ and a vector of $g+1$ variance components $\gamma$, where $\gamma =\left\{{\sigma }_{1}^{2},{\sigma }_{2}^{2},\dots ,{\sigma }_{g-1}^{2},{\sigma }_{g}^{2},{\sigma }_{R}^{2}\right\}$.
Case weights can be incorporated into the model by replacing $X$ and $Z$ with ${W}_{c}^{1/2}X$ and ${W}_{c}^{1/2}Z$ respectively where ${W}_{c}$ is a diagonal weight matrix.
The design matrices, $X$ and $Z$, are constructed from an $n×{m}_{d}$ data matrix, $D$, a description of the fixed independent variables, ${\mathcal{M}}_{f}$, and a description of the random independent variables, ${\mathcal{M}}_{r}$. See Section 11 for further details.

4References

Rao C R (1972) Estimation of variance and covariance components in a linear model J. Am. Stat. Assoc. 67 112–115
Wolfinger R, Tobias R and Sall J (1994) Computing Gaussian likelihoods and their derivatives for general linear mixed models SIAM Sci. Statist. Comput. 15 1294–1310

5Arguments

1: $\mathbf{hlmm}$Type (c_ptr) Input/Output
On entry: must be set to c_null_ptr or, alternatively, an existing G22 handle may be supplied in which case g02jff will destroy the supplied G22 handle as if g22zaf had been called.
On exit: holds a G22 handle to the internal data structure containing a description of the model. You must not change the G22 handle other than through the routines in Chapters G02 or G22.
2: $\mathbf{hddesc}$Type (c_ptr) Input
On entry: a G22 handle to the internal data structure containing a description of the data matrix, $D$, as returned in hddesc by g22ybf.
3: $\mathbf{hfixed}$Type (c_ptr) Input
On entry: a G22 handle to the internal data structure containing a description of the fixed part of the model ${\mathcal{M}}_{f}$ as returned in hform by g22yaf.
If hfixed is c_null_ptr then the model is assumed to not have a fixed part.
4: $\mathbf{nrndm}$Integer Input
On entry: the number of elements used to describe the random part of the model.
Constraint: ${\mathbf{nrndm}}\ge 0$.
5: $\mathbf{hrndm}\left({\mathbf{nrndm}}\right)$Type (c_ptr) Input
On entry: a series of G22 handles to internal data structures containing a description of the random part of the model ${\mathcal{M}}_{r}$ as returned in hform by g22yaf.
6: $\mathbf{weight}$Character(1) Input
On entry: indicates if weights are to be used.
${\mathbf{weight}}=\text{'U'}$
No weights are used.
${\mathbf{weight}}=\text{'W'}$
Case weights are used and must be supplied in array wt.
Constraint: ${\mathbf{weight}}=\text{'U'}$ or $\text{'W'}$.
7: $\mathbf{n}$Integer Input
On entry: $n$, the number of observations in the dataset, $D$.
Constraint: $1\le {\mathbf{n}}\le {n}_{d}$, where ${n}_{d}$ is the value supplied in nobs when hddesc was created.
8: $\mathbf{y}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: $y$, the vector of observations on the dependent variable.
Constraint: ${\mathbf{y}}\left(i\right)\ne 0.0$ for at least one $i=1,2,\dots ,n$.
9: $\mathbf{wt}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array wt must be at least ${\mathbf{n}}$ if ${\mathbf{weight}}=\text{'W'}$.
On entry: if ${\mathbf{weight}}=\text{'W'}$, wt must contain the diagonal elements of the weight matrix ${W}_{c}$.
If ${\mathbf{wt}}\left(i\right)=0.0$, the $i$th observation is not included in the model and the effective number of observations is the number of observations with nonzero weights.
If ${\mathbf{weight}}=\text{'U'}$, wt is not referenced and the effective number of observations is $n$.
Constraint: if ${\mathbf{weight}}=\text{'W'}$, ${\mathbf{wt}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$.
10: $\mathbf{dat}\left({\mathbf{lddat}},{\mathbf{sddat}}\right)$Real (Kind=nag_wp) array Input
On entry: the data matrix, $D$. By default, ${D}_{ij}$, the $\mathit{i}$th value for the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,{m}_{d}$, should be supplied in ${\mathbf{dat}}\left(i,j\right)$.
If the optional parameter Storage Order, described in g22ybf, is set to $\mathrm{VAROBS}$, ${D}_{ij}$ should be supplied in ${\mathbf{dat}}\left(j,i\right)$.
If either ${y}_{i}$, ${w}_{i}$ or ${D}_{ij}$, for a variable $j$ used in the model, is NaN (Not A Number) then that value is treated as missing and the whole observation is excluded from the analysis.
11: $\mathbf{lddat}$Integer Input
On entry: the first dimension of the array dat as declared in the (sub)program from which g02jff is called.
Constraints:
• if the optional parameter Storage Order, described in g22ybf, is set to $\mathrm{VAROBS}$, ${\mathbf{lddat}}\ge {m}_{d}$;
• otherwise ${\mathbf{lddat}}\ge n$.
12: $\mathbf{sddat}$Integer Input
On entry: the second dimension of the array dat as declared in the (sub)program from which g02jff is called.
Constraints:
• if the optional parameter Storage Order, described in g22ybf, is set to $\mathrm{VAROBS}$, ${\mathbf{sddat}}\ge n$;
• otherwise ${\mathbf{sddat}}\ge {m}_{d}$.
13: $\mathbf{fnlsv}$Integer Output
On exit: the number of levels for the overall subject variable in ${\mathcal{M}}_{f}$. If there is no overall subject variable, ${\mathbf{fnlsv}}=1$.
14: $\mathbf{nff}$Integer Output
On exit: the number of fixed effects estimated in each of the fnlsv subject blocks. The number of columns, $p$, in the design matrix $X$ is given by $p={\mathbf{nff}}×{\mathbf{fnlsv}}$.
15: $\mathbf{rnlsv}$Integer Output
On exit: the number of levels for the overall subject variable in ${\mathcal{M}}_{r}$. If there is no overall subject variable, ${\mathbf{rnlsv}}=1$.
16: $\mathbf{nrf}$Integer Output
On exit: the number of random effects estimated in each of the rnlsv subject blocks. The number of columns, $q$, in the design matrix $Z$ is given by $q={\mathbf{nrf}}×{\mathbf{rnlsv}}$.
17: $\mathbf{nvpr}$Integer Output
On exit: $g$, the number of variance components being estimated (excluding the overall variance, ${\sigma }_{R}^{2}$). This is defined by the number of terms in the random part of the model, ${\mathcal{M}}_{r}$ (see Section 11 for details).
18: $\mathbf{rcomm}\left({\mathbf{lrcomm}}\right)$Real (Kind=nag_wp) array Communication Array
On exit: a communication array as required by the routines g02jgf or g02jhf.
19: $\mathbf{lrcomm}$Integer Input
On entry: the dimension of the array rcomm as declared in the (sub)program from which g02jff is called.
20: $\mathbf{icomm}\left({\mathbf{licomm}}\right)$Integer array Communication Array
On exit: a communication array as required by the routines g02jgf or g02jhf.
If licomm or lrcomm are too small and ${\mathbf{licomm}}\ge 2$, then ${\mathbf{ifail}}={\mathbf{192}}$ and ${\mathbf{icomm}}\left(1\right)$ holds the minimum required value for licomm and ${\mathbf{icomm}}\left(2\right)$ holds the minimum required value for lrcomm.
21: $\mathbf{licomm}$Integer Input
On entry: the dimension of the array icomm as declared in the (sub)program from which g02jff is called.
22: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=11$
On entry, hlmm is not c_null_ptr or a recognised G22 handle.
${\mathbf{ifail}}=21$
hddesc has not been initialized or is corrupt.
${\mathbf{ifail}}=22$
hddesc is not a G22 handle as generated by g22ybf.
${\mathbf{ifail}}=31$
hfixed has not been initialized or is corrupt.
${\mathbf{ifail}}=32$
hfixed is not a G22 handle as generated by g22yaf.
${\mathbf{ifail}}=33$
A variable name used when creating hfixed is not present in hddesc.
Variable name: $〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=34$
The fixed part of the model contains categorical variables, but no intercept or main effects terms have been requested.
${\mathbf{ifail}}=41$
On entry, ${\mathbf{nrndm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrndm}}\ge 0$.
${\mathbf{ifail}}=51$
$i=〈\mathit{\text{value}}〉$.
${\mathbf{hrndm}}\left(i\right)$ has not been initialized or is corrupt.
${\mathbf{ifail}}=52$
$i=〈\mathit{\text{value}}〉$.
${\mathbf{hrndm}}\left(i\right)$ is not a G22 handle as generated by g22yaf.
${\mathbf{ifail}}=53$
No model has been specified.
${\mathbf{ifail}}=54$
A variable name used when creating hrndm is not present in hddesc.
Variable name: $〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=61$
On entry, weight had an illegal value.
Constraint: ${\mathbf{weight}}=\text{'U'}$ or $\text{'W'}$.
${\mathbf{ifail}}=71$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=72$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${n}_{d}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\le {n}_{d}$, where ${n}_{d}$ is the value supplied in nobs when hddesc was created.
${\mathbf{ifail}}=73$
On entry, no observations due to zero weights or missing values.
${\mathbf{ifail}}=91$
On entry, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{wt}}\left(i\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{wt}}\left(i\right)\ge 0.0$.
${\mathbf{ifail}}=101$
On entry, column $j$ of the data matrix, $D$, is not consistent with information supplied in hddesc, $j=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=102$
Column $j$ of the data matrix, $D$, required rounding more than expected when being treated as a categorical variable, $j=〈\mathit{\text{value}}〉$.
All output is returned using the rounded value(s).
${\mathbf{ifail}}=111$
On entry, $n=〈\mathit{\text{value}}〉$ and ${\mathbf{lddat}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lddat}}\ge n$.
${\mathbf{ifail}}=112$
On entry, ${m}_{d}=〈\mathit{\text{value}}〉$ and ${\mathbf{lddat}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lddat}}\ge {m}_{d}$.
${\mathbf{ifail}}=121$
On entry, ${m}_{d}=〈\mathit{\text{value}}〉$ and ${\mathbf{sddat}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sddat}}\ge {m}_{d}$.
${\mathbf{ifail}}=122$
On entry, $n=〈\mathit{\text{value}}〉$ and ${\mathbf{sddat}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sddat}}\ge n$.
${\mathbf{ifail}}=191$
On entry, ${\mathbf{licomm}}=〈\mathit{\text{value}}〉$ and ${\mathbf{lrcomm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{licomm}}\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{lrcomm}}\ge 〈\mathit{\text{value}}〉$. The minimum array sizes for licomm and lrcomm are held in the first two elements of icomm repectively.
${\mathbf{ifail}}=192$
On entry, ${\mathbf{licomm}}=〈\mathit{\text{value}}〉$ and ${\mathbf{lrcomm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{licomm}}\ge 〈\mathit{\text{value}}〉$ and ${\mathbf{lrcomm}}\ge 〈\mathit{\text{value}}〉$. icomm is not large enough to hold the minimum array sizes.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

8Parallelism and Performance

g02jff makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

10Example

This example fits a random effects model with three random submodels and two fixed effects to a simulated dataset with $90$ observations and $12$ variables. The model is fit using maximum likelihood (ML). Standard labels for the parameter estimates and variance components are obtained from g22ydf. See Section 10 in g02jhf for an example of how to construct custom labels.

10.1Program Text

Program Text (g02jffe.f90)

10.2Program Data

Program Data (g02jffe.d)

10.3Program Results

Program Results (g02jffe.r)

11Algorithmic Details

11.1Fixed Effects Design Matrix, $X$

The fixed effects design matrix, $X$, is constructed from the data matrix $D$ and ${\mathcal{M}}_{f}$, as encoded in hfixed. Details of the construction are described in Section 3 in g22yaf and Section 3 in g22ycf.
It is possible to store the cross-product matrix, ${X}^{\mathrm{T}}X$ in a block diagonal form if ${\mathcal{M}}_{f}$ contains an overall subject effect, ${S}_{f}$. In this context ${S}_{f}$ is defined as a main effect or interaction term that is contained in all other terms. For example, if ${\mathcal{M}}_{f}$ simplifies to ${V}_{1}.{V}_{4}+{V}_{1}.{V}_{2}.{V}_{4}+{V}_{1}.{V}_{2}.{V}_{3}.{V}_{4}$, then ${S}_{f}={V}_{1}.{V}_{4}$. If it is advantageous to do so, g02jff will make use of this block diagonal structure and fnlsv will be set to the number of levels in ${S}_{f}$, otherwise ${\mathbf{fnlsv}}=1$.

11.2Random Effects Design Matrix, $Z$

The random effects design matrix, $Z$, is constructed from the data matrix $D$ and ${\mathcal{M}}_{r}$ which is made up of nrndm submodels, ${\mathcal{M}}_{ri}$, where ${\mathcal{M}}_{ri}$ is encoded in ${\mathbf{hrndm}}\left(i\right)$. Each submodel is made up of two parts, the random effects and a subject term. The random effects are specified as described in Section 3 in g22yaf and the subject term is specified via the g22yaf optional parameter Subject. The design matrix $Z$ is constructed as described in Section 3 in g22ycf using a model constructed from the nrndm submodels. As an example, if there were $3$ submodels:
• `-1+V07+V08+V09 / SUBJECT = V13`
• `-1+V05+V06     / SUBJECT = V11.V12`
• `V03+V04        / SUBJECT = V10.V11.V12`
then $Z$ would be constructed as if g22ycf was called using the model
• `(V07+V08+V09).V13 + (V05+V06).V11.V12 + (V10.V11.V12 + (V03+V04).V10.V11.V12)`
It should be noted that unless specified otherwise (by the inclusion of -1) a submodel will contain an intercept. This results in a term corresponding to the subject term being included in the combined model (V10.V11.V12 in this instance).
The above model expands out further to:
• `V07.V13 + V08.V13 + V09.V13 + V05.V11.V12 + V06.V11.V12 + V10.V11.V12 + V03.V10.V11.V12 + V04.V10.V11.V12`
Each term in the expanded model corresponds to a variance component, so in this case, $g=8$.
When constructing $Z$ all contrast information specified when the submodels are constructed in calls to g22yaf is ignored and dummy variables are used throughout.
It is possible to store the cross-product matrix, ${Z}^{\mathrm{T}}Z$ in a block diagonal form if ${\mathcal{M}}_{r}$ contains an overall subject effect, ${S}_{r}$. In this context ${S}_{r}$ is defined as a main effect or interaction term that is contained in all other subject terms. For example, if the random effects model is constructed from $3$ submodels with subject terms ${V}_{1}.{V}_{4}$, ${V}_{1}.{V}_{2}.{V}_{4}$ and ${V}_{1}.{V}_{2}.{V}_{3}.{V}_{4}$, then ${S}_{r}={V}_{1}.{V}_{4}$ and rnlsv will be set to the number of levels in ${S}_{r}$, otherwise ${\mathbf{rnlsv}}=1$.

12Optional Parameters

As well as the optional parameters common to all G22 handles described in g22zmf and g22znf, a number of additional optional parameters can be specified for a G22 handle holding the description of a linear mixed model, as returned by g02jff in hlmm.
Each writeable optional parameter has an associated default value; to set any of them to a non-default value, use g22zmf. The value of any optional parameter can be queried using g22znf.
Most of the optional parameters described in this section are related to the behaviour g02jhf when fitting the model. These descriptions should therefore be read in conjunction with the documentation for that routine.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 12.1.

12.1Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
• a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
• the default value.
Keywords and character values are case and white space insensitive.
 Gamma Lower Bound $r$ Default
A lower bound for the elements of ${\gamma }^{*}$, where ${\gamma }^{*}=\gamma /{\sigma }_{R}^{2}$.
 Gamma Upper Bound $r$ Default $\text{}={10}^{20}$
An upper bound for the elements of ${\gamma }^{*}$, where ${\gamma }^{*}=\gamma /{\sigma }_{R}^{2}$.
 Initial Distance $r$ Default $\text{}=100000.0$
The initial distance from the solution.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, g02jhf passes Initial Distance to the solver as stepmx.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, this option is ignored.
 Initial Value Strategy $i$ Default $\text{}=\text{special}$
Controls how g02jhf will choose the initial values for the variance components, $\gamma$, if not supplied.
${\mathbf{Initial Value Strategy}}=0$
The MIVQUE0 estimates of the variance components based on the likelihood specified by Likelihood are used.
${\mathbf{Initial Value Strategy}}=1$
The MIVQUE0 estimates based on the maximum likelihood are used, irrespective of the value of Likelihood.
See Rao (1972) for a description of the minimum variance quadratic unbiased estimators (MIVQUE0).
By default, for small problems, ${\mathbf{Initial Value Strategy}}=0$ and for large problems ${\mathbf{Initial Value Strategy}}=1$.
Constraint: ${\mathbf{Initial Value Strategy}}=0$ or $1$.
 Likelihood $a$ Default $\text{}=\mathrm{REML}$
Likelihood defines whether g02jhf will use the restricted maximum likelihood (REML) or the maximum likelihood (ML) when fitting the model.
Constraint: ${\mathbf{Likelihood}}=\mathrm{REML}$ or $\mathrm{ML}$.
 Linear Minimization Accuracy $r$ Default $\text{}=0.9$
The accuracy of the linear minimizations.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, g02jhf passes Linear Minimization Accuracy to the solver as eta.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, this option is ignored.
 Line Search Tolerance $r$ Default $\text{}=0.9$
The line search tolerance.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, this option is ignored.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, g02jhf passes Line Search Tolerance to the solver as Line Search Tolerance.
 List
 NoList Default
Optional parameter List enables printing of each optional parameter specification as it is supplied. NoList suppresses this printing.
 Major Iteration Limit $i$ Default $\text{}=\text{special}$
The number of major iterations.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, g02jhf passes Major Iteration Limit to the solver as maxcal. In this case, the default value used is $1000$.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, g02jhf passes Major Iteration Limit to the solver as Major Iteration Limit. In this case, the default value used is $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3×g\right)$, where $g$ is the number of variance components being estimated (excluding the overall variance, ${\sigma }_{R}^{2}$).
 Major Print Level $i$ Default $\text{}=\text{special}$
The frequency that monitoring information is output to Unit Number.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, g02jhf passes Major Print Level to the solver as iprint. In this case, the default value used is $-1$ and hence no monitoring information will be output.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, g02jhf passes Major Print Level to the solver as Major Print Level. In this case, the default value used is $0$ and hence no monitoring information will be output.
 Maximum Number of Threads $i$ Default $\text{}=\text{special}$
Controls the maximum number of threads used by g02jhf in a multithreaded library. By default, the maximum number of available threads are used.
In a library that is not multithreaded, this option has no effect.
Constraint: ${\mathbf{Maximum Number of Threads}}\ge 0$.
 Minor Iteration Limit $i$ Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3×g\right)$
The number of minor iterations.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, this option is ignored.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, g02jhf passes Minor Iteration Limit to the solver as Minor Iteration Limit. In this case, the default value used is $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3×g\right)$, where $g$ is the number of variance components being estimated (excluding the overall variance, ${\sigma }_{R}^{2}$).
 Minor Print Level $i$ Default $\text{}=0$
The frequency that additional monitoring information is output to Unit Number.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, this option is ignored.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, g02jhf passes Minor Print Level to the solver as Minor Print Level. The default value of $0$ means that no additional monitoring information will be output.
 Optimality Tolerance $r$ Default
The optimality tolerance.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, this option is ignored.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, g02jhf passes Optimality Tolerance to the solver as Optimality Tolerance.
 Parallelisation Strategy $i$ Default $\text{}=\text{special}$
If ${\mathbf{Maximum Number of Threads}}>0$ then Parallelisation Strategy controls how g02jhf is parallelised in a multithreaded library.
${\mathbf{Parallelisation Strategy}}=1$
g02jhf will attempt to parallelise operations involving $Z$, even if ${\mathbf{rnlsv}}=1$.
${\mathbf{Parallelisation Strategy}}=2$
g02jhf will only attempt to parallelise operations involving $Z$, if ${\mathbf{rnlsv}}>1$.
By default, ${\mathbf{Parallelisation Strategy}}=1$, however, for some models / datasets, this may be slower than using ${\mathbf{Parallelisation Strategy}}=2$ when ${\mathbf{rnlsv}}=1$.
In a library that is not multithreaded, this option has no effect.
Constraint: ${\mathbf{Parallelisation Strategy}}=1$ or $2$.
 Solution Accuracy $r$ Default $\text{}=0.0$
The accuracy to which the solution is required.
• When ${\mathbf{Solver}}=\mathrm{E04LB}$, g02jhf passes Solution Accuracy to the solver as xtol.
• When ${\mathbf{Solver}}=\mathrm{E04UC}$, this option is ignored.
 Solver $a$ Default $\text{}=\text{special}$
Controls which solver g02jhf will use when fitting the model. By default, ${\mathbf{Solver}}=\mathrm{E04LB}$ is used for small problems and ${\mathbf{Solver}}=\mathrm{E04UC}$, otherwise.
If ${\mathbf{Solver}}=\mathrm{E04LB}$, then the solver used is the one implemented in e04lbf and if ${\mathbf{Solver}}=\mathrm{E04UC}$, then the solver used is the one implemented in e04ucf/e04uca.
Constraint: ${\mathbf{Solver}}=\mathrm{E04LB}$ or $\mathrm{E04UC}$.
 Sweep Tolerance $r$ Default $\text{}=\text{special}$
The sweep tolerance used by g02jhf when performing the sweep operation Wolfinger et al. (1994). The default value used is ${\mathbf{Sweep Tolerance}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\epsilon ,\epsilon ×\left(\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}{\left({Z}^{\mathrm{T}}\right)}_{ii}\right)\right)$, where .
 Unit Number $i$ Default $=\text{advisory message unit number}$
The monitoring unit number to which g02jhf will send any monitoring information.


text:
# River landscapes and optimal channel networks

### Abstract

Type
Publication
Proceedings of the National Academy of Sciences

# Abstract

We study tree structures termed optimal channel networks (OCNs) that minimize the total gravitational energy loss in the system, an exact property of steady-state landscape configurations that prove dynamically accessible and strikingly similar to natural forms. Here, we show that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent. We also study the natural river trees in an arbitrary graph in terms of forbidden substructures, which we call k-path obstacles, and OCNs on a d-dimensional lattice, improving earlier results by determining the minimum energy up to a constant factor for every $d \geq 2$. Results extend our capabilities in environmental statistical mechanics.


text:
Ensemble Kalman filter updates based on regularized sparse inverse Cholesky factors
Abstract The ensemble Kalman filter (EnKF) is a popular technique for data assimilation in high-dimensional nonlinear state-space models. The EnKF represents distributions of interest by an ensemble, which is a form of dimension reduction that enables straightforward forecasting even for complicated and expensive evolution operators. However, the EnKF update step involves estimation of the forecast covariance matrix based on the (often small) ensemble, which requires regularization. Many existing regularization techniques rely on spatial localization, which may ignore long-range dependence. Instead, our proposed approach assumes a sparse Cholesky factor of the inverse covariance matrix, and the nonzero Cholesky entries are further regularized. The resulting method is highly flexible and computationally scalable. In our numerical experiments, our approach was more accurate and less sensitive to misspecification of tuning parameters than tapering-based localization.
Authors:
;
Award ID(s):
Publication Date:
NSF-PAR ID:
10277276
Journal Name:
Monthly Weather Review
ISSN:
0027-0644
2. Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda _i\}$ and eigenvectors $\{\boldsymbol{u}_i\}$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda }_i\}$ and eigenvectors $\{\tilde{\boldsymbol{u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties—particularly,more »


text:
# 4.S: Fractions (Summary)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

### Key Terms

 complex fraction A fraction in which the numerator or the denominator contains a fraction. equivalent fractions Two or more fractions that have the same value. fraction A fraction is written $$\frac{a}{b}$$. in a fraction, a is the numerator and b is the denominator. A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included. least common denominator (LCD) The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators. mixed number A mixed number consists of a whole number a and a fraction $$\frac{b}{c}$$ where c ≠ 0. It is written as $$a \frac{b}{c}$$, where c ≠ 0. proper and improper fractions The fraction $$\frac{a}{b}$$ is proper if a < b and improper if a > b. reciprocal The reciprocal of the fraction $$\frac{a}{b}$$ is $$\frac{b}{a}$$ where a ≠ 0 and b ≠ 0. simplified fraction A fraction is considered simplified if there are no common factors in the numerator and denominator.

### Key Concepts

#### 4.1 - Visualize Fractions

• Property of One
• Any number, except zero, divided by itself is one. $$\frac{a}{a}$$ = 1, where a ≠ 0.
• Mixed Numbers
• A mixed number consists of a whole number a and a fraction $$\frac{b}{c}$$ where c ≠ 0.
• It is written as follows: $$a \frac{b}{c} \quad c \neq 0$$
• Proper and Improper Fractions
• The fraction ab is a proper fraction if a < b and an improper fraction if a ≥ b .
• Convert an improper fraction to a mixed number.
1. Divide the denominator into the numerator.
2. Identify the quotient, remainder, and divisor.
3. Write the mixed number as $$quotient \frac{remainder}{divisor}$$.
• Convert a mixed number to an improper fraction.
1. Multiply the whole number by the denominator.
2. Add the numerator to the product found in Step 1.
3. Write the final sum over the original denominator.
• Equivalent Fractions Property: If a, b, and c are numbers where b ≠ 0, c ≠ 0, then $$\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$$.

#### 4.2 - Multiply and Divide Fractions

• Equivalent Fractions Property
• If a, b, c are numbers where b ≠ 0, c ≠ 0, then $$\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$$ and $$\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$$.
• Simplify a fraction.
1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
2. Simplify, using the equivalent fractions property, by removing common factors.
3. Multiply any remaining factors.
• Fraction Multiplication
• If a, b, c, and d are numbers where b ≠ 0 and d ≠ 0, then $$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$.
• Reciprocal
• A number and its reciprocal have a product of 1. $$\frac{a}{b} \cdot \frac{b}{a}$$ = 1.
• #### Table 4.98

Opposite Absolute Value Reciprocal
has opposite sign is never negative has same sign, fraction inverts
• Fraction Division
• If a, b, c, and d are numbers where b ≠ 0, c ≠ 0, and d ≠ 0, then $$\frac{a}{b} + \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$.
• To divide fractions, multiply the first fraction by the reciprocal of the second.

#### 4.3 - Multiply and Divide Mixed Numbers and Complex Fractions

• Multiply or divide mixed numbers.
1. Convert the mixed numbers to improper fractions.
2. Follow the rules for fraction multiplication or division.
3. Simplify if possible.
• Simplify a complex fraction.
1. Rewrite the complex fraction as a division problem.
2. Follow the rules for dividing fractions.
3. Simplify if possible.
• For any positive numbers a and b, $$\frac{-a}{b} = \frac{a}{-b} = - \frac{a}{b}$$.
• Simplify an expression with a fraction bar.
1. Simplify the numerator.
2. Simplify the denominator.
3. Simplify the fraction.

#### 4.4 - Add and Subtract Fractions with Common Denominators

• If a, b, and c are numbers where c ≠ 0, then $$\frac{a}{c} + \frac{b}{c} = \frac{a + c}{c}$$.
• To add fractions, add the numerators and place the sum over the common denominator.
• Fraction Subtraction
• If a, b, and c are numbers where c ≠ 0, then $$\frac{a}{c} - \frac{b}{c} = \frac{a - c}{c}$$.
• To subtract fractions, subtract the numerators and place the difference over the common denominator.

#### 4.5 - Add and Subtract Fractions with Different Denominators

• Find the least common denominator (LCD) of two fractions.
1. Factor each denominator into its primes.
2. List the primes, matching primes in columns when possible.
3. Bring down the columns.
4. Multiply the factors. The product is the LCM of the denominators.
5. The LCM of the denominators is the LCD of the fractions.
• Equivalent Fractions Property
• If a, b, and c are whole numbers where b ≠ 0, c ≠ 0 then $$\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$$ and $$\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$$.
• Convert two fractions to equivalent fractions with their LCD as the common denominator.
1. Find the LCD.
2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
4. Simplify the numerator and denominator.
• Add or subtract fractions with different denominators.
1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. Write the result in simplified form.
• Summary of Fraction Operations
• Fraction multiplication: Multiply the numerators and multiply the denominators. $$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$.
• Fraction division: Multiply the first fraction by the reciprocal of the second. $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$.
• Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD. $$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$.
• Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD. $$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$$.
• Simplify complex fractions.
1. Simplify the numerator.
2. Simplify the denominator.
3. Divide the numerator by the denominator.
4. Simplify if possible.

#### 4.6 - Add and Subtract Mixed Numbers

• Add mixed numbers with a common denominator.
3. Simplify, if possible.
• Subtract mixed numbers with common denominators
1. Rewrite the problem in vertical form.
2. Compare the two fractions. If the top fraction is larger than the bottom fraction, go to Step 3. If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction.
3. Subtract the fractions.
4. Subtract the whole numbers.
5. Simplify, if possible.
• Subtract mixed numbers with common denominators as improper fractions.
1. Rewrite the mixed numbers as improper fractions.
2. Subtract the numerators.
3. Write the answer as a mixed number, simplifying the fraction part, if possible.

#### 4.7 - Solve Equations with Fractions

• Determine whether a number is a solution to an equation.
1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.
• Addition, Subtraction, and Division Properties of Equality: For any numbers a, b, and c,
• if a = b, then a + c = b + c. Addition Property of Equality
• if a = b, then a - c = b - c. Subtraction Property of Equality
• if a = b, then $$\frac{a}{c} = \frac{b}{c}$$, c ≠ 0. Division Property of Equality
• The Multiplication Property of Equality
• For any numbers ab and c, a = b, then ac = bc.
• If you multiply both sides of an equation by the same quantity, you still have equality.


text:
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text: ColloquiumThursday November 7th, 2019 Time: 4:00 PM - 5:00 PM Title: New techniques for studying the topology and geometry of almost complex manifolds. Speaker: Scott Wilson, Queen's College Location: Math Tower P-131 Abstract: There is a fascinating interplay in the topology and geometry of manifolds among the various tensors one can consider such as non-degenerate bilinear forms and almost complex structures. Each of these are informed by a certain flatness or integrability condition, and have been studied to varying degrees of success. In this talk, I'll focus on the lattermost category, describing a new sequence of cohomology groups that can be defined in the context of almost complex manifolds, which generalizes a well-studied object in the theory of complex manifolds. This will be defined by very elementary means, and some interesting examples will be computed, which show there is a rich theory to be explored involving both topology and geometry. As I'll explain, in dimension 6 the groups can be used to prohibit the existence of an interesting class of compatible metrics, the so-called nearly Kahler metrics, which adds to this fascinating interplay.


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# M26-25

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16 Sep 2014, 01:25
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If $$x$$ and $$y$$ are negative numbers, what is the value of $$\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}$$?

A. $$1+y$$
B. $$1-y$$
C. $$-1-y$$
D. $$y-1$$
E. $$x-y$$

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16 Sep 2014, 01:25
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Official Solution:

If $$x$$ and $$y$$ are negative numbers, what is the value of $$\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}$$?

A. $$1+y$$
B. $$1-y$$
C. $$-1-y$$
D. $$y-1$$
E. $$x-y$$

Note that $$\sqrt{a^2}=|a|$$. Next, since $$x \lt 0$$ and $$y \lt 0$$ then $$|x|=-x$$ and $$|y|=-y$$.

So, $$\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}=\frac{|x|}{x}-\sqrt{(-y)*(-y)}=\frac{-x}{x}-\sqrt{y^2}=-1-|y|=-1+y$$

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25 Oct 2014, 05:44
1
How does abs(x) = -x, and abs(y) = -y when x<0 and y<0? I am do not understand this concept
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25 Oct 2014, 05:47
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delm2010 wrote:
How does abs(x) = -x, and abs(y) = -y when x<0 and y<0? I am do not understand this concept

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25 Oct 2014, 23:33
Hi Bunuel,
I have a query.... wouldn't \sqrt{x^2}/ x would be equal to +1?

I say so since it's already given in the question that x is a -ve number, so by \sqrt{x^2} we know that we need to consider only the -ve Root.

But in the denominator the variable x would be a -ve number. Hence after the resolution of \sqrt{x^2} one part of the question would be |x|/x, where |x| would give us -x(since the -ve root needs to be considered) and the denominator in isolation is again a -ve number... so the fraction becomes \frac{-x}{-x} = 1.

Kindly clarify
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26 Oct 2014, 06:08
avaneeshvyas wrote:
Hi Bunuel,
I have a query.... wouldn't \sqrt{x^2}/ x would be equal to +1?

I say so since it's already given in the question that x is a -ve number, so by \sqrt{x^2} we know that we need to consider only the -ve Root.

But in the denominator the variable x would be a -ve number. Hence after the resolution of \sqrt{x^2} one part of the question would be |x|/x, where |x| would give us -x(since the -ve root needs to be considered) and the denominator in isolation is again a -ve number... so the fraction becomes \frac{-x}{-x} = 1.

Kindly clarify

Next, the point is that the square root function cannot give negative result ($$\sqrt{some \ expression}\geq{0}$$) and $$\sqrt{x^2}=|x|$$, which means that $$\sqrt{x^2}/ x=\frac{|x|}{x}=\frac{-x}{x}=-1$$.

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27 Nov 2014, 13:37
[quote="Bunuel"]Official Solution:

-1-|y|=-1+y

I have not understood the above portion. Could you please tell me how to get this?

I have done it by taking numbers.

Let x=-1 and y=-1 Here i have taken integer value thinking that I will get an answer which can be in option also.

Now putting the value of x and y I am getting the value of question is -2.

Only option D gives me the value. Is my approach correct?
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28 Nov 2014, 05:38
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2
Raihanuddin wrote:
Bunuel wrote:
Official Solution:

-1-|y|=-1+y

I have not understood the above portion. Could you please tell me how to get this?

I have done it by taking numbers.

Let x=-1 and y=-1 Here i have taken integer value thinking that I will get an answer which can be in option also.

Now putting the value of x and y I am getting the value of question is -2.

Only option D gives me the value. Is my approach correct?

-1 - |y| = -1 + y because we are given that y is negative. When y is negative |y| = -y. Hence -1 - |y| = -1 - (-y) = -1 + y.
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16 Oct 2015, 07:57
There is a step that is off in this question. If you substitute negative y from the get go -(-y) = +y * (-y) leaves you with a negative inside the square root and this cannot happen. Should be written as square root of y*lyl
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18 Oct 2015, 13:39
danjbon wrote:
There is a step that is off in this question. If you substitute negative y from the get go -(-y) = +y * (-y) leaves you with a negative inside the square root and this cannot happen. Should be written as square root of y*lyl

That's wrong. $$\sqrt{-y*|y|}=\sqrt{-(negative)*positive}=\sqrt{positive*positive}$$
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31 Aug 2016, 07:25
having a hard time understanding this basic rule: $$\sqrt{x^2}=|x|$$

Assuming it's an integer, shouldn't this be the positive integer of x? why |x|
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31 Aug 2016, 07:48
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xnthic wrote:
having a hard time understanding this basic rule: $$\sqrt{x^2}=|x|$$

Assuming it's an integer, shouldn't this be the positive integer of x? why |x|

MUST KNOW: $$\sqrt{x^2}=|x|$$:

The point here is that since square root function cannot give negative result then $$\sqrt{some \ expression}\geq{0}$$.

So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?

Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.

So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.

What function does exactly the same thing? The absolute value function: $$|x|=x$$, if $$x\geq{0}$$ and $$|x|=-x$$, if $$x<0$$. That is why $$\sqrt{x^2}=|x|$$.

For example if x = -5, then $$\sqrt{x^2}=\sqrt{25}=5=|-5|=|x|$$

Theory on Abolute Values: math-absolute-value-modulus-86462.html
Absolute value tips: absolute-value-tips-and-hints-175002.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

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01 Sep 2016, 00:12
Bunuel wrote:
If $$x$$ and $$y$$ are negative numbers, what is the value of $$\frac{\sqrt{x^2}}{x}-\sqrt{-y*|y|}$$?

A. $$1+y$$
B. $$1-y$$
C. $$-1-y$$
D. $$y-1$$
E. $$x-y$$

Alternate approach is to testing some easy values:

As per question stem, x & y are negative number..

Let x= -1 & y=-1

Substituting the in equation

$$\frac{\sqrt{1}}{-1}-\sqrt{-(-1)*|-1|}$$ = -2

y-1=-1-1=-2 only answer that matches what we get above

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07 Dec 2016, 11:46
1
Hi,

Having trouble wrapping my head around why $$\sqrt{x^2}$$ is equal to |x|

I looked at the post on this page, but did not see any explanation.

For example, how is the equation $$\sqrt{4}$$ = {-2, 2} any different from $$\sqrt{2^2}$$ = ? Is it because the solution of $$\sqrt{x^2}$$ must be equal to the solution of $$(\sqrt{x})^2$$? if so, why?

Thanks in advance for any help!

(with regard to bunnel's explanation to xnthic, throughout the explanation bunnel writes "$$\sqrt{25}$$ = 5" as though this is given, but my understanding is that $$\sqrt{25}$$ could be either positive or negative 5, as either positive or negative 5 squared result in 25. If I am incorrect, then that is most likely the root of my confusion)
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08 Dec 2016, 13:53
blitheclyde wrote:
Hi,

Having trouble wrapping my head around why $$\sqrt{x^2}$$ is equal to |x|

I looked at the post on this page, but did not see any explanation.

For example, how is the equation $$\sqrt{4}$$ = {-2, 2} any different from $$\sqrt{2^2}$$ = ? Is it because the solution of $$\sqrt{x^2}$$ must be equal to the solution of $$(\sqrt{x})^2$$? if so, why?

Thanks in advance for any help!

(with regard to bunnel's explanation to xnthic, throughout the explanation bunnel writes "$$\sqrt{25}$$ = 5" as though this is given, but my understanding is that $$\sqrt{25}$$ could be either positive or negative 5, as either positive or negative 5 squared result in 25. If I am incorrect, then that is most likely the root of my confusion)

$$\sqrt{4}=2$$ ONLY, not 2 or -2. The square root function cannot give negative result. Check here: m26-184464.html#p1729559
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05 Dec 2017, 23:09
Bunuel wrote:
blitheclyde wrote:
Hi,

Having trouble wrapping my head around why $$\sqrt{x^2}$$ is equal to |x|

I looked at the post on this page, but did not see any explanation.

For example, how is the equation $$\sqrt{4}$$ = {-2, 2} any different from $$\sqrt{2^2}$$ = ? Is it because the solution of $$\sqrt{x^2}$$ must be equal to the solution of $$(\sqrt{x})^2$$? if so, why?

Thanks in advance for any help!

(with regard to bunnel's explanation to xnthic, throughout the explanation bunnel writes "$$\sqrt{25}$$ = 5" as though this is given, but my understanding is that $$\sqrt{25}$$ could be either positive or negative 5, as either positive or negative 5 squared result in 25. If I am incorrect, then that is most likely the root of my confusion)

$$\sqrt{4}=2$$ ONLY, not 2 or -2. The square root function cannot give negative result. Check here: http://gmatclub.com/forum/m26-184464.html#p1729559

Hi Bunuel, certain application of the sqrt function seem to be contradictory. If sqrt function can't give negative numbers, why do we substitute both -ve and +ve values of $$\sqrt{expression}$$ while solving algebraic equations.
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05 Dec 2017, 23:22
abhishek911 wrote:
Bunuel wrote:
blitheclyde wrote:
Hi,

Having trouble wrapping my head around why $$\sqrt{x^2}$$ is equal to |x|

I looked at the post on this page, but did not see any explanation.

For example, how is the equation $$\sqrt{4}$$ = {-2, 2} any different from $$\sqrt{2^2}$$ = ? Is it because the solution of $$\sqrt{x^2}$$ must be equal to the solution of $$(\sqrt{x})^2$$? if so, why?

Thanks in advance for any help!

(with regard to bunnel's explanation to xnthic, throughout the explanation bunnel writes "$$\sqrt{25}$$ = 5" as though this is given, but my understanding is that $$\sqrt{25}$$ could be either positive or negative 5, as either positive or negative 5 squared result in 25. If I am incorrect, then that is most likely the root of my confusion)

$$\sqrt{4}=2$$ ONLY, not 2 or -2. The square root function cannot give negative result. Check here: http://gmatclub.com/forum/m26-184464.html#p1729559

Hi Bunuel, certain application of the sqrt function seem to be contradictory. If sqrt function can't give negative numbers, why do we substitute both -ve and +ve values of $$\sqrt{expression}$$ while solving algebraic equations.

When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the positive root. That is:

$$\sqrt{9} = 3$$, NOT +3 or -3;
$$\sqrt[4]{16} = 2$$, NOT +2 or -2;

Notice that in contrast, the equation $$x^2 = 9$$ has TWO solutions, +3 and -3. Because $$x^2 = 9$$ means that $$x =-\sqrt{9}=-3$$ or $$x=\sqrt{9}=3$$.
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11 Jun 2018, 02:11
Hi,

Since x is negative, \sqrt{x^2}/-x - \sqrt{-y*-y}
= $$-x/-x$$ - \sqrt{y^2}
= 1-(-y)
= 1+y

Can someone please explain what wrong am I doing here?
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11 Jun 2018, 05:44
Hi,

Since x is negative, \sqrt{x^2}/-x - \sqrt{-y*-y}
= $$-x/-x$$ - \sqrt{y^2}
= 1-(-y)
= 1+y

Can someone please explain what wrong am I doing here?

Why are you changing x to -x there? The stem just says that x is negative, so x stands for some negative number, changing x to -x does not make sense.
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11 Jul 2018, 08:26
Hi,

In one of the other questions, the explanation is provided as follows

If p=−1
then (|p|!)^p=(|−1|!)^−1=1^−1=1

Here |-1| = 1

But in this problem |y| = -y since y is negative

Can you please explain the discrepancy or the misunderstanding in my thought
Re: M26-25   [#permalink] 11 Jul 2018, 08:26

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# M26-25

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text:
# zbMATH — the first resource for mathematics

The lattice of R-subalgebras of a bounded distributive lattice. (English) Zbl 0542.06004
An R-subalgebra of a bounded, distributive lattice L is a sublattice that includes 0 and 1, and is closed under existing relative complements. The collection $$S_ R(L)$$ of all R-subalgebras of L is a lattice under inclusion. It is shown that $$S_ R(L)$$ is dually isomorphic to the lattice of all congruence relations (that is, kernels of morphisms) of the Priestley dual of L. This duality is then used to characterize those L such that $$S_ R(L)$$ is semimodular, modular, distributive, and Boolean. For example, it is shown that $$S_ R(L)$$ is distributive iff $$S_ R(L)$$ is Boolean iff L is a chain or a four element Boolean algebra.
Reviewer: R.S.Pierce

##### MSC:
 06D05 Structure and representation theory of distributive lattices 06D15 Pseudocomplemented lattices
Full Text:


text:
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DAYS

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HOUR

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MINS

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SEC

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Construct Binary Tree from Parent Array
Medium Accuracy: 45.96% Submissions: 25785 Points: 4

Given an array of size N that can be used to represents a tree. The array indexes are values in tree nodes and array values give the parent node of that particular index (or node). The value of the root node index would always be -1 as there is no parent for root. Construct the standard linked representation of Binary Tree from this array representation.

Note: If two elements have the same parent, the one that appears first in the array will be the left child and the other is the right child.

Example 1:

Input:
N = 7
parent[] = {-1,0,0,1,1,3,5}
Output: 0 1 2 3 4 5 6
Explanation: the tree generated
will have a structure like
0
/   \
1     2
/ \
3   4
/
5
/
6

Example 2:

Input:
N = 3
parent[] = {2, 0, -1}
Output: 2 0 1
Explanation: the tree generated will
have a sturcture like
2
/
0
/
1

You don't need to read input or print anything. The task is to complete the function createTree() which takes 2 arguments parent[] and N and returns the root node of the constructed tree.

Expected Time Complexity: O(N)
Expected Auxiliary Space: O(N)

Constraints:
1 ≤ N ≤ 103

We are replacing the old Disqus forum with the new Discussions section given below.


text:
## Course Outline

• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Models with a Quantitative Explanatory Variable
• segmentPART III: EVALUATING MODELS
• segmentChapter 9 - The Logic of Inference
• segmentChapter 10 - Model Comparison with F
• segmentChapter 11 - Parameter Estimation and Confidence Intervals
• segmentPART IV: MULTIVARIATE MODELS
• segmentChapter 12 - Introduction to Multivariate Models
• segmentChapter 13 - Multivariate Model Comparisons
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources

### list College / Advanced Statistics and Data Science (ABCD)

Book
• College / Advanced Statistics and Data Science (ABCD)
• College / Statistics and Data Science (ABC)
• High School / Advanced Statistics and Data Science I (ABC)
• High School / Statistics and Data Science I (AB)
• High School / Statistics and Data Science II (XCD)

## 8.6 Assessing Model Fit with Sum of Squares

Finally, let’s examine the fit of our regression model by running the supernova() function on our model. And at the same time, let’s compare the table we get from the regression model (Height_model) with the one we produced before for the Height2Group_model.

supernova(Height2Group_model)
supernova(Height_model)

Height2Group Model

Analysis of Variance Table (Type III SS)
Model: Thumb ~ Height2Group

SS  df      MS      F    PRE     p
----- --------------- | --------- --- ------- ------ ------ -----
Model (error reduced) |   830.880   1 830.880 11.656 0.0699 .0008
Error (from model)    | 11049.331 155  71.286
----- --------------- | --------- --- ------- ------ ------ -----
Total (empty model)   | 11880.211 156  76.155

Height Model

Analysis of Variance Table (Type III SS)
Model: Thumb ~ Height

SS  df       MS      F    PRE     p
----- --------------- | --------- --- -------- ------ ------ -----
Model (error reduced) |  1816.862   1 1816.862 27.984 0.1529 .0000
Error (from model)    | 10063.349 155   64.925
----- --------------- | --------- --- -------- ------ ------ -----
Total (empty model)   | 11880.211 156   76.155

Remember, the total sum of squares is the sum of squared deviations (or more generally, residuals) from the empty model. Total sum of squares is all about the outcome variable, and isn’t affected by the explanatory variable or variables. And when we compare statistical models, as we are doing here, we always are modeling the same outcome variable.

### Partitioning Sums of Squares

Video Transcript

If you want to try out the app Dr. Ji uses in this video you can click this link to the sum of squares applet. Copy/paste the data below into the little “sample data” box to reproduce Ji’s examples. (Here’s the full link in case that one doesn’t work: http://www.rossmanchance.com/applets/RegShuffle.htm)

 Height2 Group Thumb 0 56 0 60 1 61 0 63 1 64 1 68

 Height Thumb 62 56 66 60 67 61 63 63 68 64 71 68

For any model with an explanatory variable (what we have been calling “complex models”), the SS Total can be partitioned into the SS Error and the SS Model. The SS Model is the amount by which the error is reduced under the complex model (e.g., the Height model) compared with the empty model.

As we developed previously for the group models, SS Model is easily calculated by subtracting SS Error from SS Total. This is the same, regardless of whether you are fitting a group model or a regression model. Error from the model is defined in the former case as residuals from the group means, and in the latter, residuals from the regression line.

It also is possible to calculate the SS Model in the regression model directly, in much the same way we did for the group model. Recall that for the group model, SS Model was the sum of the squared deviations of each person’s predicted score (their group mean) from the Grand Mean. In the regression model, SS Model is calculated in exactly the same way, except that each person’s predicted score is defined as a point on the regression line. The Grand Mean is the same in both cases.