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Paper Count: 2804
Search results for: marula nut shells activated carbon
2804 Adsorption of Lead (II) and Copper (II) Ions onto Marula Nuts Activated Carbon
Authors: Lucky Malise, Hilary Rutto, Tumisang Seodigeng
Heavy metal contamination in waste water is a very serious issue affecting a lot of industrialized countries due to the health and environmental impact of these heavy metals on human life and the ecosystem. Adsorption using activated carbon is the most promising method for the removal of heavy metals from waste water but commercial activated carbon is expensive which gives rise to the need for alternatively activated carbon derived from cheap precursors, agricultural wastes, or byproducts from other processes. In this study activated bio-carbon derived from the carbonaceous material obtained from the pyrolysis of Marula nut shells was chemically activated and used as an adsorbent for the removal of lead (II) and copper (II) ions from aqueous solution. The surface morphology and chemistry of the adsorbent before and after chemical activation with zinc chloride impregnation were studied using SEM and FTIR analysis respectively and the results obtained indicate that chemical activation with zinc chloride improves the surface morphology of the adsorbent and enhances the intensity of the surface oxygen complexes on the surface of the adsorbent. The effect of process parameters such as adsorbent dosage, pH value of the solution, initial metal concentration, contact time, and temperature on the adsorption of lead (II) and copper (II) ions onto Marula nut activated carbon were investigated, and their optimum operating conditions were also determined. The experimental data was fitted to both the Langmuir and Freundlich isotherm models, and the data fitted best on the Freundlich isotherm model for both metal ions. The adsorption kinetics were also evaluated, and the experimental data fitted the pseudo-first order kinetic model better than the pseudo second-order kinetic model. The adsorption thermodynamics were also studied and the results indicate that the adsorption of lead and copper ions is spontaneous and exothermic in nature, feasible, and also involves a dissociative mechanism in the temperature range of 25-45 °C.
Keywords: adsorption, isotherms, kinetics, marula nut shells activated carbon, thermodynamics
2803 An Efficient Activated Carbon for Copper (II) Adsorption Synthesized from Indian Gooseberry Seed Shells
Authors: Somen Mondal, Subrata Kumar Majumder
Removal of metal pollutants by efficient activated carbon is challenging research in the present-day scenario. In the present study, the characteristic features of an efficient activated carbon (AC) synthesized from Indian gooseberry seed shells for the copper (II) adsorption are reported. A three-step chemical activation method consisting of the impregnation, carbonization and subsequent activation is used to produce the activated carbon. The copper adsorption kinetics and isotherms onto the activated carbon were analyzed. As per present investigation, Indian gooseberry seed shells showed the BET surface area of 1359 m²/g. The maximum adsorptivity of the activated carbon at a pH value of 9.52 was found to be 44.84 mg/g at 30°C. The adsorption process followed the pseudo-second-order kinetic model along with the Langmuir adsorption isotherm. This AC could be used as a favorable and cost-effective copper (II) adsorbent in wastewater treatment to remove the metal contaminants.
Keywords: activated carbon, adsorption isotherm, kinetic model, characterization
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2802 Efficiency of Modified Granular Activated Carbon Coupled with Membrane Bioreactor for Trace Organic Contaminants Removal
Authors: Mousaab Alrhmoun, Magali Casellas, Michel Baudu, Christophe Dagot
The aim of the study is to improve removal of trace organic contaminants dissolved in activated sludge by the process of filtration with membrane bioreactor combined with modified activated carbon, for a maximum removal of organic compounds characterized by low molecular weight. Special treatment was conducted in laboratory on activated carbon. Tow reaction parameters: The pH of aqueous middle and the type of granular activated carbon were very important to improve the removal and to motivate the electrostatic Interactions of organic compounds with modified activated carbon in addition to physical adsorption, ligand exchange or complexation on the surface activated carbon. The results indicate that modified activated carbon has a strong impact in removal 21 of organic contaminants and in percentage of 100% of the process.
Keywords: activated carbon, organic micropolluants, membrane bioreactor, carbon
2801 Synthesis and Characterization of Green Coke-Derived Activated Carbon by KOH Activation
Authors: Richard, Iyan Subiyanto, Chairul Hudaya
Activated carbon has been playing a significant role for many applications, especially in energy storage devices. However, commercially activated carbons generally require complicated processes and high production costs. Therefore, in this study, an activated carbon originating from green coke waste, that is economically affordable will be used as a carbon source. To synthesize activated carbon, KOH as an activator was employed with variation of C:KOH in ratio of 1:2, 1:3, 1:4, and 1:5, respectively, with an activation temperature of 700°C. The characterizations of activated carbon are obtained from Scanning Electron Microscopy, Energy Dispersive X-Ray, Raman Spectroscopy, and Brunauer-Emmett-Teller. The optimal activated carbon sample with specific surface area of 2,024 m²/g with high carbon content ( > 80%) supported by the high porosity carbon image obtained by SEM was prepared at C:KOH ratio of 1:4. The result shows that the synthesized activated carbon would be an ideal choice for energy storage device applications. Therefore, this study is expected to reduce the costs of activated carbon production by expanding the utilization of petroleum waste.
Keywords: activated carbon, energy storage material, green coke, specific surface area
2800 Adsorption of Chromium Ions from Aqueous Solution by Carbon Adsorbent
Authors: S. Heydari, H. Sharififard, M. Nabavinia, H. Kiani, M. Parvizi
Rapid industrialization has led to increased disposal of heavy metals into the environment. Activated carbon adsorption has proven to be an effective process for the removal of trace metal contaminants from aqueous media. This paper was investigated chromium adsorption efficiency by commercial activated carbon. The sorption studied as a function of activated carbon particle size, dose of activated carbon and initial pH of solution. Adsorption tests for the effects of these factors were designed with Taguchi approach. According to the Taguchi parameter design methodology, L9 orthogonal array was used. Analysis of experimental results showed that the most influential factor was initial pH of solution. The optimum conditions for chromium adsorption by activated carbons were found to be as follows: Initial feed pH 6, adsorbent particle size 0.412 mm and activated carbon dose 6 g/l. Under these conditions, nearly %100 of chromium ions was adsorbed by activated carbon after 2 hours.
Keywords: chromium, adsorption, Taguchi method, activated carbon
2799 Viable Use of Natural Extract Solutions from Tuberous and Cereals to Enhance the Synthesis of Activated Carbon-Graphene Composite
Authors: Pamphile Ndagijimana, Xuejiao Liu, Zhiwei Li, Yin Wang
Enhancing the properties of activated carbon is very imperative for various applications. Indeed, the activated carbon has promising physicochemical properties desired for a considerable number of applications. In this regard, we are proposing an enhanced and green technology for increasing the efficiency and performance of the activated carbon to various applications. The technique poses on the use of natural extracts from tuberous and cereals based-solutions. These solutions showed high potentiality to be used in the synthesis of activated carbon-graphene composite with only 3 mL. The extracted liquid from tuberous sourcing was enough to induce precipitation within a fraction of a minute in contrast to that from cereal sourced. Using these extracts, a synthesis of activated carbon-graphene composite was successful. Different characterization techniques such as XRD, SEM, FTIR, BET, and Raman spectroscopy were performed to investigate the composite materials. The results confirmed a conjugation between activated carbon and graphene material.
Keywords: activated carbon, cereals, extract solution, graphene, tuberous
2798 The Adsorption of Zinc Metal in Waste Water Using ZnCl2 Activated Pomegranate Peel
Authors: S. N. Turkmen, A. S. Kipcak, N. Tugrul, E. M. Derun, S. Piskin
Activated carbon is an amorphous carbon chain which has extremely extended surface area. High surface area of activated carbon is due to the porous structure. Activated carbon, using a variety of materials such as coal and cellulosic materials; can be obtained by both physical and chemical methods. The prepared activated carbon can be used for decolorize, deodorize and also can be used for removal of organic and non-organic pollution. In this study, pomegranate peel was subjected to 800W microwave power for 1 to 4 minutes. Also fresh pomegranate peel was used for the reference material. Then ZnCl2 was used for the chemical activation purpose. After the activation process, activated pomegranate peels were used for the adsorption of Zn metal (40 ppm) in the waste water. As a result of the adsorption experiments, removal of heavy metals ranged from 89% to 85%.
Keywords: activated carbon, adsorption, chemical activation, microwave, pomegranate peel
2797 Application of Nitric Acid Modified Cocos nucifera, Pennisetum glaucum and Sorghum bicolor Activated Carbon for Adsorption of H₂S Gas
Authors: Z. N. Ali, O. A. Babatunde, S. Garba, H. M. S. Haruna
The potency of modified and unmodified activated carbons prepared from shells of Cocos nucifera (coconut shell), straws of Pennisetum glaucum (millet) and Sorghum bicolor (sorghum) for adsorption of hydrogen sulphide gas were investigated using an adsorption apparatus (stainless steel cylinder) at constant temperature (ambient temperature). The adsorption equilibria states were obtained when the pressure indicated on the pressure gauge remained constant. After modification with nitric acid, results of the scanning electron microscopy of the unmodified and modified activated carbons showed that HNO3 greatly improved the formation of micropores and mesopores on the activated carbon surface. The adsorption of H2S gas was found to be highest in modified Cocos nucifera activated carbon with maximum monolayer coverage of 28.17 mg/g, and the adsorption processes were both physical and chemical with the physical process being predominant. The adsorption data were well fitted into the Langmuir isotherm model with the adsorption capacities of the activated carbons in the order modified Cocos nucifera > modified Pennisetum glaucum > modified Sorghum bicolor > unmodified Cocos nucifera > unmodified Pennisetum glaucum > unmodified Sorghum bicolour.
Keywords: activated carbon adsorption, hydrogen sulphide, nitric acid, modification, stainless steel cylinder
2796 Adsorption of Paracetamol Using Activated Carbon of Dende and Babassu Coconut Mesocarp
Authors: R. C. Ferreira, H. H. C. De Lima, A. A. Cândido, O. M. Couto Junior, P. A. Arroyo, K. Q De Carvalho, G. F. Gauze, M. A. S. D. Barros
Removal of the widespread used drug paracetamol from water was investigated using activated carbon originated from dende coconut mesocarp and babassu coconut mesocarp. Kinetic and equilibrium data were obtained at different values of pH. Babassu activated carbon showed higher efficiency due to its acidity and higher microporosity. Pseudo-second order model was better adjusted to the kinetic results. Equilibrium data may be represented by Langmuir equation. Lower solution pH provided better removal efficiency as the carbonil groups may be attracted by the positively charged carbon surface.
Keywords: adsorption, activated carbon, babassu, dende
2795 The Utilization of Tea Residues for Activated Carbon Preparation
Authors: Jiazhen Zhou, Youcai Zhao
Waste tea is commonly generated in certain areas of China and its utilization has drawn a lot of concern nowadays. In this paper, highly microporous and mesoporous activated carbons were produced from waste tea by physical activation in the presence of water vapor in a tubular furnace. The effect of activation temperature on yield and pore properties of produced activated carbon are studied. The yield decreased with the increase of activation temperature. According to the Nitrogen adsorption isotherms, the micropore and mesopore are both developed in the activated carbon. The specific surface area and the mesopore volume fractions of the activated carbon increased with the raise of activation temperature. The maximum specific surface area attained 756 m²/g produced at activation temperature 900°C. The results showed that the activation temperature had a significant effect on the micro and mesopore volumes as well as the specific surface area.
Keywords: activated carbon, nitrogen adsorption isotherm, physical activation, waste tea
2794 Comparative Evaluation of Kinetic Model of Chromium and Lead Uptake from Aqueous Solution by Activated Balanitesaegyptiaca Seeds
Authors: Mohammed Umar Manko
A series of batch experiments were conducted in order to investigate the feasibility of Balanitesaegyptiaca seeds based activated carbon as compared with industrial activated carbon for the removal of chromium and lead ions from aqueous solution by the adsorption process within 30 to 150 minutes contact time. The activated samples were prepared using zinc chloride and tetraoxophophate(VI) acid. The results obtained showed that the activated carbon of Balanitesaegyptiaca seeds studied had relatively high adsorption capacities for these heavy metal ions compared with industrial Activated Carbon. The percentage removal of Cr (VI) and lead (II) ions by the three activated carbon samples were 64%, 70% and 71%; 60%, 66% and 60% respectively. Adsorption equilibrium was established in 90 minutes for the heavy metal ions. The equilibrium data fitted the pseudo second order out of the pseudo first, pseudo second, Elovich ,Natarajan and Khalaf models tested. The investigation also showed that the adsorbents can effectively remove metal ions from similar wastewater and aqueous media.
Keywords: activated carbon, pseudo second order, chromium, lead, Elovich model
2793 Elimination of Phosphorus by Activated Carbon Prepared from Algerian Dates Stones
Authors: A. Kamarchoua, A. A. Bebaa, A. Douadi
The current work has a goal of the preparation of activated carbon from the stones of dates from southern Algeria (El-Oued province) using a simple pyrolysis proceeded by chemical impregnation in sulphuric acid. For the preparation of the carbon, we choose the diameter of the pellets (0.5-1)mm, activation by acid and water (1:1), carbonization at 450˚C. The prepared carbon has the following characteristics: specific surface 125.86 m2/g, methylene blue number 40, CCE = 0.3meq.g/l, IR and micrographics SEM. The activated carbon thus obtained is used at the water purification in wastewater treatment plant (WWTP) at Kouinine, El- Oued province, to totally eliminate phosphorus. We analyzed the water at the WWTP before the purification procedure. In this study, we have looked at the effect of the following parameters on the adsorption of carbon: the pH, the contact time (Tc) and the agitation speed (Va). The best conditions for phosphorus adsorption are: pH=4 or pH >5, Tc = 60 min and Va = 900 rotations per minute.
Keywords: activated carbon, date stones, pyrolysis, phosphate pollutants
2792 Zinc Adsorption Determination of H2SO4 Activated Pomegranate Peel
Authors: S. N. Turkmen Koc, A. S. Kipcak, M. B. Piskin, E. Moroydor Derun, N. Tugrul
Active carbon can be obtained from agricultural sources. Due to the high surface area, the production of activated carbon from cheap resources is very important. Since the surface area of 1 g activated carbon is approximately between 300 and 2000 m2, it can be used to remove both organic and inorganic impurities. In this study, the adsorption of Zn metal was studied with the product of activated carbon, which is obtained from pomegranate peel by microwave and chemical activation methods. The microwave process of pomegranate peel was carried out under constant microwave power of 800 W and 1 to 4 minutes. After the microwave process, samples were treated with H2SO4 for 3 h. Then prepared product was used in synthetic waste water including 40 ppm Zn metal. As a result, removal of waste Zn in waste water ranged from 91% to 93%.
Keywords: activated carbon, chemical activation, H₂SO₄, microwave, pomegranate peel
2791 Preparation of Activated Carbon from Lignocellulosic Precursor for Dyes Adsorption
Authors: H. Mokaddem, D. Miroud, N. Azouaou, F. Si-Ahmed, Z. Sadaoui
The synthesis and characterization of activated carbon from local lignocellulosic precursor (Algerian alfa) was carried out for the removal of cationic dyes from aqueous solutions. The effect of the production variables such as impregnation chemical agents, impregnation ratio, activation temperature and activation time were investigated. Carbon obtained using the optimum conditions (CaCl2/ 1:1/ 500°C/2H) was characterized by various analytical techniques scanning electron microscopy (SEM), infrared spectroscopic analysis (FTIR) and zero-point-of-charge (pHpzc). Adsorption tests of methylene blue on the optimal activated carbon were conducted. The effects of contact time, amount of adsorbent, initial dye concentration and pH were studied. The adsorption equilibrium examined using Langmuir, Freundlich, Temkin and Redlich–Peterson models reveals that the Langmuir model is most appropriate to describe the adsorption process. The kinetics of MB sorption onto activated carbon follows the pseudo-second order rate expression. The examination of the thermodynamic analysis indicates that the adsorption process is spontaneous (ΔG ° < 0) and endothermic (ΔH ° > 0), the positive value of the standard entropy shows the affinity between the activated carbon and the dye. The present study showed that the produced optimal activated carbon prepared from Algerian alfa is an effective low-cost adsorbent and can be employed as alternative to commercial activated carbon for removal of MB dye from aqueous solution.
Keywords: activated carbon, adsorption, cationic dyes, Algerian alfa
2790 Magnetic Activated Carbon: Preparation, Characterization, and Application for Vanadium Removal
Authors: Hakimeh Sharififard, Mansooreh Soleimani
In this work, the magnetic activated carbon nanocomposite (Fe-CAC) has been synthesized by anchorage iron hydr(oxide) nanoparticles onto commercial activated carbon (CAC) surface and characterized using BET, XRF, SEM techniques. The influence of various removal parameters such as pH, contact time and initial concentration of vanadium on vanadium removal was evaluated using CAC and Fe-CAC in batch method. The sorption isotherms were studied using Langmuir, Freundlich and Dubinin–Radushkevich (D–R) isotherm models. These equilibrium data were well described by the Freundlich model. Results showed that CAC had the vanadium adsorption capacity of 37.87 mg/g, while the Fe-AC was able to adsorb 119.01 mg/g of vanadium. Kinetic data was found to confirm pseudo-second-order kinetic model for both adsorbents.
Keywords: magnetic activated carbon, remove, vanadium, nanocomposite, freundlich
2789 Decoloriation of Rhodamine-B Dye by Pseudomonas putida on Activated Carbon
Authors: U. K. Ghosh, A. Ullhyan
Activated carbon prepared from mustard stalk was applied to decolorize Rhodamine-B dye bearing synthetic wastewater by simple adsorption and simultaneous adsorption and biodegradation (SAB) using Pseudomonas putida MTCC 1194. Results showed that percentage of Rhodamine-B dye removal was 82% for adsorption and 99.3% for SAB at pH 6.5, adsorbent dose 10 g/L and temperature 32ºC.
Keywords: activated carbon, mustard stalk, Rhodamine-B, adsorption, SAB, Pseudomonas putida
2788 Adsorption Studies of Lead from Aqueos Solutions on Cocount Shell Activated Carbon
Authors: G. E. Sharaf El-Deen, S. E. A. Sharaf El-Deen
Activated carbon was prepared from coconut shell (ACS); a discarded agricultural waste was used to produce bioadsorbent through easy and environmental friendly processes. This activated carbon based biosorbent was evaluated for adsorptive removal of lead from water. The characterisation results showed this biosorbent had very high specific surface area and functional groups. The adsorption equilibrium data was well described by Langmuir, whilst kinetics data by pseudo-first order, pseudo-second order and Intraparticle diffusion models. The adsorption process could be described by the pseudo-second order kinetic.
Keywords: coconut shell, activated carbon, adsorption isotherm and kinetics, lead removal
2787 Activated Carbons Prepared from Date Pits for Hydrogen Storage
Authors: M. Belhachemi, M. Monteiro de Castro, M. Casco, A. Sepúlveda-Escribano, F. Rodríguez-Reinoso
In this study, activated carbons were prepared from Algerian date pits using thermal activation with CO2 or steam. The prepared activated carbons were doped by vanadium oxide in order to increase the H2 adsorption capacity. The adsorbents were characterized by N2 and CO2 adsorption at 77 K and 273K, respectively. The hydrogen adsorption experiments were carried at 298K in the 0–100 bar pressure range using a volumetric equipment. The results show that the H2 adsorption capacity is influenced by the size and volume of micropores in the activated carbon adsorbent. Furthermore, vanadium doping of activated carbons has a slight positive effect on H2 storage.
Keywords: hydrogen storage, activated carbon, vanadium doping, adsorption
2786 Comparative Study of Fenton and Activated Carbon Treatment for Dyeing Waste Water
Authors: Prem Mohan, Namrata Jariwala
In recent years 10000 dyes are approximately used by dying industry which makes dyeing wastewater more complex in nature. It is very difficult to treat dyeing wastewater by conventional methods. Here an attempt has been made to treat dyeing wastewater by the conventional and advanced method for removal of COD. Fenton process is the advanced method and activated carbon treatment is the conventional method. Experiments have been done on synthetic wastewater prepared from three different dyes; acidic, disperse and reactive. Experiments have also been conducted on real effluent obtained from industry. The optimum dose of catalyst and hydrogen peroxide in Fenton process and optimum activated carbon dose for each of these wastewaters were obtained. In Fenton treatment, COD removal was obtained up to 95% whereas 70% removal was obtained with activated carbon treatment.
Keywords: activated carbon, advanced oxidation process, dyeing waste water, fenton oxidation process
2785 Evaluation of the Adsorption Adaptability of Activated Carbon Using Dispersion Force
Authors: Masao Fujisawa, Hirohito Ikeda, Tomonori Ohata, Miho Yukawa, Hatsumi Aki, Takayoshi Kimura
We attempted to predict adsorption coefficients by utilizing dispersion energies. We performed liquid-phase free energy calculations based on gas-phase geometries of organic compounds using the DFT and studied the relationship between the adsorption of organic compounds by activated carbon and dispersion energies of the organic compounds. A linear correlation between absorption coefficients and dispersion energies was observed.
Keywords: activated carbon, adsorption, prediction, dispersion energy
2784 Production of Clean Reusable Distillery Waste Water Using Activated Carbon Prepared from Waste Orange Peels
Authors: Joseph Govha, Sharon Mudutu
The research details the treatment of distillery waste water by making use of activated carbon prepared from orange peels as an adsorbent. Adsorption was carried out at different conditions to determine the optimum conditions that work best for the removal of color in distillery waste water using orange peel activated carbon. Adsorption was carried out at different conditions by varying contact time, adsorbent dosage, pH, testing for color intensity and Biological Oxygen Demand. A maximum percentage color removal of 88% was obtained at pH 7 at an adsorbent dosage of 1g/20ml. Maximum adsorption capacity was obtained from the Langmuir isotherm at R2=0.98.
Keywords: distillery, waste water, orange peel, activated carbon, adsorption
2783 Fabrication of Activated Carbon from Palm Trunksfor Removal of Harmful Dyes
Authors: Eman Alzahrani
Date palm trees are abundant and cheap natural resources in Saudi Arabia. In this study, an activated carbon was prepared from palm trunks by chemical processes. The chemical activation was performed by impregnation of the raw materials after grinding with H3PO4 solution (63%), followed by placing of the sample solution on a muffle furnace at 400ºC for 30 min, and then at 800ºC for 10 min. The morphology of the fabricated material was checked using scanning electron microscopy that showed the rough surfaces on the carbon samples. The use of fabricated activated carbon for removal of eosin dye from aqueous solutions at different contact time, initial dye concentration, pH and adsorbent doses was investigated. The experimental results show that the adsorption process attains equilibrium within 20 min. The adsorption isotherm equilibrium was studied by means of the Langmuir and Freundlich isotherms, and it was found that the data fit the Langmuir isotherm equation with maximum monolayer adsorption capacity of 126.58 mg g-1. The results indicated that the home made activated carbon prepared from palm trunks has the ability to remove eosin dye from aqueous solution and it will be a promising adsorbent for the removal of harmful dyes from waste water.
Keywords: activated carbon, date palm trunks, H3PO4 activation, adsorption, dye removal, eosin dye, isotherm
2782 Removal of Nitrate and Phosphates from Waste Water Using Activated Bio-Carbon Produced from Agricultural Waste
Authors: Kgomotso Matobole, Natania De Wet, Tefo Mbambo, Hilary Rutto, Tumisang Seodigeng
Nitrogen and phosphorus are nutrients which are required in the ecosystem, however, at high levels, these nutrients contribute to the process of eutrophication in the receiving water bodies, which threatens aquatic organisms. Hence it is vital that they are removed before the water is discharged. This phenomenon increases the cost related to wastewater treatment. This raises the need for the development of processes that are cheaper. Activated biocarbon was used in batch and filtration system to remove nitrates and phosphates. The batch system has higher nutrients removal capabilities than the filtration system. For phosphate removal, 93 % removal is achieved at the adsorbent of 300 g while for nitrates, 84 % removal is achieved when 200 g of activated carbon is loaded.
Keywords: waste water treatment, phosphates, nitrates, activated carbon, agricultural waste
2781 Reduction of Nitrogen Monoxide with Carbon Monoxide from Gas Streams by 10% wt. Cu-Ce-Fe-Co/Activated Carbon
Authors: K. L. Pan, M. B. Chang
Nitrogen oxides (NOₓ) is regarded as one of the most important air pollutants. It not only causes adverse environmental effects but also harms human lungs and respiratory system. As a post-combustion treatment, selective catalytic reduction (SCR) possess the highest NO removal efficiency ( ≥ 85%), which is considered as the most effective technique for removing NO from gas streams. However, injection of reducing agent such as NH₃ is requested, and it is costly and may cause secondary pollution. Reduction of NO with carbon monoxide (CO) as reducing agent has been previously investigated. In this process, the key step involves the NO adsorption and dissociation. Also, the high performance mainly relies on the amounts of oxygen vacancy on catalyst surface and redox ability of catalyst, because oxygen vacancy can activate the N-O bond to promote its dissociation. Additionally, perfect redox ability can promote the adsorption of NO and oxidation of CO. Typically, noble metals such as iridium (Ir), platinum (Pt), and palladium (Pd) are used as catalyst for the reduction of NO with CO; however, high cost has limited their applications. Recently, transition metal oxides have been investigated for the reduction of NO with CO, especially CuₓOy, CoₓOy, Fe₂O₃, and MnOₓ are considered as effective catalysts. However, deactivation is inevitable as oxygen (O₂) exists in the gas streams because active sites (oxygen vacancies) of catalyst are occupied by O₂. In this study, Cu-Ce-Fe-Co is prepared and supported on activated carbon by impregnation method to form 10% wt. Cu-Ce-Fe-Co/activated carbon catalyst. Generally, addition of activated carbon on catalyst can bring several advantages: (1) NO can be effectively adsorbed by interaction between catalyst and activated carbon, resulting in the improvement of NO removal, (2) direct NO decomposition may be achieved over carbon associated with catalyst, and (3) reduction of NO could be enhanced by a reducing agent over carbon-supported catalyst. Therefore, 10% wt. Cu-Ce-Fe-Co/activated carbon may have better performance for reduction of NO with CO. Experimental results indicate that NO conversion achieved with 10% wt. Cu-Ce-Fe-Co/activated carbon reaches 83% at 150°C with 300 ppm NO and 10,000 ppm CO. As temperature is further increased to 200°C, 100% NO conversion could be achieved, implying that 10% wt. Cu-Ce-Fe-Co/activated carbon prepared has good activity for the reduction of NO with CO. In order to investigate the effect of O₂ on reduction of NO with CO, 1-5% O₂ are introduced into the system. The results indicate that NO conversions still maintain at ≥ 90% with 1-5% O₂ conditions at 200°C. It is worth noting that effect of O₂ on reduction of NO with CO could be significantly improved as carbon is used as support. It is inferred that carbon support can react with O₂ to produce CO₂ as O₂ exists in the gas streams. Overall, 10% wt. Cu-Ce-Fe-Co/activated carbon is demonstrated with good potential for reduction of NO with CO, and possible mechanisms will be elucidated in this paper.
Keywords: nitrogen oxides (NOₓ), carbon monoxide (CO), reduction of NO with CO, carbon material, catalysis
2780 Industrial Wastewater Treatment Improvements Using Activated Carbon
Authors: Mamdouh Y. Saleh, Gaber El Enany, Medhat H. Elzahar, Moustafa H. Omran
The discharge limits of industrial waste water effluents are subjected to regulations which are getting more restricted with time. A former research occurred in Port Said city studied the efficiency of treating industrial wastewater using the first stage (A-stage) of the multiple-stage plant (AB-system).From the results of this former research, the effluent treated wastewater has high rates of total dissolved solids (TDS) and chemical oxygen demand (COD). The purpose of this paper is to improve the treatment process in removing TDS and COD. Thus, a pilot plant was constructed at wastewater pump station in the industrial area in the south of Port Said. Experimental work was divided into several groups adding activated carbon with different dosages to waste water, and for each group waste water was filtered after being mixed with activated carbon. pH and TSS as variables were also studied. At the end of this paper, a comparison was made between the efficiency of using activated carbon and the efficiency of using limestone in the same circumstances.
Keywords: adsorption, COD removal, filtration, TDS removal
2779 Thermal Regeneration of CO2 Spent Palm Shell-Polyetheretherketone Activated Carbon Sorbents
Authors: Usman D. Hamza, Noor S. Nasri, Mohammed Jibril, Husna M. Zain
Activated carbons (M4P0, M4P2, and M5P2) used in this research were produced from palm shell and polyetherether ketone (PEEK) via carbonization, impregnation, and microwave activation. The adsorption/desorption process was carried out using static volumetric adsorption. Regeneration is important in the overall economy of the process and waste minimization. This work focuses on the thermal regeneration of the CO2 exhausted microwave activated carbons. The regeneration strategy adopted was thermal with nitrogen purge desorption with N2 feed flow rate of 20 ml/min for 1 h at atmospheric pressure followed by drying at 1500C. Seven successive adsorption/regeneration processes were carried out on the material. It was found that after seven adsorption regeneration cycles; the regeneration efficiency (RE) for CO2 activated carbon from palm shell only (M4P0) was more than 90% while that of hybrid palm shell-PEEK (M4P2, M5P2) was above 95%. The cyclic adsorption and regeneration shows the stability of the adsorbent materials.
Keywords: activated carbon, palm shell-PEEK, regeneration, thermal
2778 Produced Gas Conversion of Microwave Carbon Receptor Reforming
Authors: Young Nam Chun, Mun Sup Lim
Carbon dioxide and methane, the major components of biomass pyrolysis/gasification gas and biogas, top the list of substances that cause climate change, but they are also among the most important renewable energy sources in modern society. The purpose of this study is to convert carbon dioxide and methane into high-quality energy using char and commercial activated carbon obtained from biomass pyrolysis as a microwave receptor. The methane reforming process produces hydrogen and carbon. This carbon is deposited in the pores of the microwave receptor and lowers catalytic activity, thereby reducing the methane conversion rate. The deposited carbon was removed by carbon gasification due to the supply of carbon dioxide, which solved the problem of microwave receptor inactivity. In particular, the conversion rate remained stable at over 90% when the ratio of carbon dioxide to methane was 1:1. When the reforming results of carbon dioxide and methane were compared after fabricating nickel and iron catalysts using commercial activated carbon as a carrier, the conversion rate was higher in the iron catalyst than in the nickel catalyst and when no catalyst was used.
Keywords: microwave, gas reforming, greenhouse gas, microwave receptor, catalyst
2777 Kinetics, Equilibrium and Thermodynamics of the Adsorption of Triphenyltin onto NanoSiO₂/Fly Ash/Activated Carbon Composite
Authors: Olushola S. Ayanda, Olalekan S. Fatoki, Folahan A. Adekola, Bhekumusa J. Ximba, Cecilia O. Akintayo
In the present study, the kinetics, equilibrium and thermodynamics of the adsorption of triphenyltin (TPT) from TPT-contaminated water onto nanoSiO2/fly ash/activated carbon composite was investigated in batch adsorption system. Equilibrium adsorption data were analyzed using Langmuir, Freundlich, Temkin and Dubinin–Radushkevich (D-R) isotherm models. Pseudo first- and second-order, Elovich and fractional power models were applied to test the kinetic data and in order to understand the mechanism of adsorption, thermodynamic parameters such as ΔG°, ΔSo and ΔH° were also calculated. The results showed a very good compliance with pseudo second-order equation while the Freundlich and D-R models fit the experiment data. Approximately 99.999 % TPT was removed from the initial concentration of 100 mg/L TPT at 80oC, contact time of 60 min, pH 8 and a stirring speed of 200 rpm. Thus, nanoSiO2/fly ash/activated carbon composite could be used as effective adsorbent for the removal of TPT from contaminated water and wastewater.
Keywords: isotherm, kinetics, nanoSiO₂/fly ash/activated carbon composite, tributyltin
2776 Flotation Recovery of Gold-Loaded Fine Activated Carbon Using Emulsified Diesel and Kerosene as Collectors
Authors: Emmanuel Jr. Ballad, Herman Mendoza
The recovery of fine activated carbon with adsorbed gold in the cyanidation tailings of a small-scale gold plant was investigated due to the high amount of gold present. In the study, collectors that were used are kerosene and diesel. Emulsification of the oils was done to improve its collecting property, thus also the recovery. It was found out that the best hydrophile lypophile balance (HLB) of emulsified diesel and kerosene oil is 13 and 12 respectively. The amount of surfactants (SPAN 20 and TWEEN 20) for the best stability of the emulsified oils was found to be 10% in both kerosene and diesel. Optical microscopy showed that the oil dispersion in the water forms spherical droplets like features. The higher the stability, the smaller the droplets and their number were increasing. The smaller droplets indicate better dispersion of oil in the water. Consequently, it will have a greater chance of oil and activated carbon particle interaction during flotation. Due to the interaction of dispersed oil phase with carbon, the hydrophobicity of the carbon will be improved and will be attached to the bubble. Thus, flotation recovery will be increased. Results showed that the recovery of the fine activated carbon using emulsified diesel or kerosene is three times more effective than using pure diesel or kerosene.
Keywords: emulsified oils, flotation, hydrophile lyophile balance, non-ionic surfactants
2775 Effect of Ramp Rate on the Preparation of Activated Carbon from Saudi Date Tree Fronds (Agro Waste) by Physical Activation Method
Authors: Muhammad Shoaib, Hassan M Al-Swaidan
Saudi Arabia is the major date producer in the world. In order to maximize the production from date tree, pruning of the date trees is required annually. Large amount of this agriculture waste material (palm tree fronds) is available in Saudi Arabia and considered as an ideal source as a precursor for production of activated carbon (AC). The single step procedure for the preparation of micro porous activated carbon (AC) from Saudi date tree fronds using mixture of gases (N2 and CO2) is carried out at carbonization/activation temperature at 850°C and at different ramp rates of 10, 20 and 30 degree per minute. Alloy 330 horizontal reactor is used for tube furnace. Flow rate of nitrogen and carbon dioxide gases are kept at 150 ml/min and 50 ml/min respectively during the preparation. Characterization results reveal that the BET surface area, pore volume, and average pore diameter of the resulting activated carbon generally decreases with the increase in ramp rate. The activated carbon prepared at a ramp rate of 10 degrees/minute attains larger surface area and can offer higher potential to produce activated carbon of greater adsorption capacity from agriculture wastes such as date fronds. The BET surface areas of the activated carbons prepared at a ramp rate of 10, 20 and 30 degree/minute after 30 minutes activation time are 1094, 1020 and 515 m2/g, respectively. Scanning electron microscopy (SEM) for surface morphology, and FTIR for functional groups was carried out that also verified the same trend. Moreover, by increasing the ramp rate from 10 and 20 degrees/min the yield remains same, i.e. 18%, whereas at a ramp rate of 30 degrees/min the yield increases from 18 to 20%. Thus, it is feasible to produce high-quality micro porous activated carbon from date frond agro waste using N2 carbonization followed by physical activation with CO2 and N2 mixture. This micro porous activated carbon can be used as adsorbent of heavy metals from wastewater, NOx SOx emission adsorption from ambient air and electricity generation plants, purification of gases, sewage treatment and many other applications.
Keywords: activated carbon, date tree fronds, agricultural waste, applied chemistry


text:
Who was Michael Benedum?
Michael Benedum and his wife, Sarah, were natives of West Virginia. During his lifetime in the oil and gas business, Mr. Benedum amassed a fortune, ranking him in his day among the 100 wealthiest Americans. The Benedum's established the Foundation in 1944. Mrs. Benedum died in 1951; Mr. Benedum in 1959.
Does the Foundation make grants to individuals?
No, grants are made only to 501(c)(3) organizations and to qualified government units and agencies.
Does the Foundation fund scholarships?
No, the Foundation does not fund scholarship.
How large is the Foundation?
The Foundation's assets totaled over $367 million at the end of 2014.
Who serves on the Foundation board?
The board includes nine members: one family member, four members from West Virginia and four from Southwestern Pennsylvania. Click here for a list of board members.
Does the Foundation support organizations outside West Virginia or Southwestern Pennsylvania?
The Foundation's geographic focus is on these two regions and generally no grants are made to organizations out of the regions.
Does the Foundation accept grant applications electronically?
Yes. See the How To Apply page for instructions.
Does the Foundation fund conferences, publications and media projects?
Grants for conferences, publications and media projects are considered only if they are an integral part of a Foundation-supported program.
How can we contact the Foundation?
Claude Worthington Benedum Foundation
1400 Benedum Trees Building

redpajama_set_name: RedPajamaCommonCrawl


text:
package org.apache.beam.runners.dataflow.worker;

import com.google.api.services.dataflow.model.CounterUpdate;
import edu.umd.cs.findbugs.annotations.SuppressFBWarnings;
import java.util.Map;
import java.util.Map.Entry;
import javax.annotation.Nonnull;
import org.apache.beam.runners.core.metrics.DistributionData;
import org.apache.beam.runners.core.metrics.GaugeCell;
import org.apache.beam.runners.core.metrics.MetricsMap;
import org.apache.beam.sdk.metrics.Counter;
import org.apache.beam.sdk.metrics.Distribution;
import org.apache.beam.sdk.metrics.Gauge;
import org.apache.beam.sdk.metrics.MetricKey;
import org.apache.beam.sdk.metrics.MetricName;
import org.apache.beam.sdk.metrics.MetricsContainer;
import org.apache.beam.vendor.guava.v26_0_jre.com.google.common.base.Function;
import org.apache.beam.vendor.guava.v26_0_jre.com.google.common.base.Predicates;
import org.apache.beam.vendor.guava.v26_0_jre.com.google.common.collect.FluentIterable;
import org.checkerframework.checker.nullness.qual.Nullable;

/**
 * For Dataflow Streaming, we want to efficiently support many threads report metric updates, and a
 * single total delta being reported periodically as physical counters.
 */
@SuppressWarnings({
  "nullness" // TODO(https://issues.apache.org/jira/browse/BEAM-10402)
})
public class StreamingStepMetricsContainer implements MetricsContainer {

  private final String stepName;

  private MetricsMap<MetricName, DeltaCounterCell> counters =
      new MetricsMap<>(DeltaCounterCell::new);

  private MetricsMap<MetricName, GaugeCell> gauges = new MetricsMap<>(GaugeCell::new);

  private MetricsMap<MetricName, DeltaDistributionCell> distributions =
      new MetricsMap<>(DeltaDistributionCell::new);

  private StreamingStepMetricsContainer(String stepName) {
    this.stepName = stepName;
  }

  public static MetricsContainerRegistry<StreamingStepMetricsContainer> createRegistry() {
    return new MetricsContainerRegistry<StreamingStepMetricsContainer>() {
      @Override
      protected StreamingStepMetricsContainer createContainer(String stepName) {
        return new StreamingStepMetricsContainer(stepName);
      }
    };
  }

  @Override
  public Counter getCounter(MetricName metricName) {
    return counters.get(metricName);
  }

  @Override
  public Distribution getDistribution(MetricName metricName) {
    return distributions.get(metricName);
  }

  @Override
  public Gauge getGauge(MetricName metricName) {
    return gauges.get(metricName);
  }

  public Iterable<CounterUpdate> extractUpdates() {
    return counterUpdates().append(distributionUpdates());
  }

  private FluentIterable<CounterUpdate> counterUpdates() {
    return FluentIterable.from(counters.entries())
        .transform(
            new Function<Entry<MetricName, DeltaCounterCell>, CounterUpdate>() {

              @SuppressFBWarnings(
                  value = "NP_METHOD_PARAMETER_TIGHTENS_ANNOTATION",
                  justification = "https://github.com/google/guava/issues/920")
              @Override
              public @Nullable CounterUpdate apply(
                  @Nonnull Map.Entry<MetricName, DeltaCounterCell> entry) {
                long value = entry.getValue().getSumAndReset();
                if (value == 0) {
                  return null;
                }

                return MetricsToCounterUpdateConverter.fromCounter(
                    MetricKey.create(stepName, entry.getKey()), false, value);
              }
            })
        .filter(Predicates.notNull());
  }

  private FluentIterable<CounterUpdate> distributionUpdates() {
    return FluentIterable.from(distributions.entries())
        .transform(
            new Function<Entry<MetricName, DeltaDistributionCell>, CounterUpdate>() {
              @SuppressFBWarnings(
                  value = "NP_METHOD_PARAMETER_TIGHTENS_ANNOTATION",
                  justification = "https://github.com/google/guava/issues/920")
              @Override
              public @Nullable CounterUpdate apply(
                  @Nonnull Map.Entry<MetricName, DeltaDistributionCell> entry) {
                DistributionData value = entry.getValue().getAndReset();
                if (value.count() == 0) {
                  return null;
                }

                return MetricsToCounterUpdateConverter.fromDistribution(
                    MetricKey.create(stepName, entry.getKey()), false, value);
              }
            })
        .filter(Predicates.notNull());
  }

  /**
   * Returns {@link CounterUpdate} protos representing the changes to all metrics that have changed
   * since the last time it was invoked.
   */
  public static Iterable<CounterUpdate> extractMetricUpdates(
      MetricsContainerRegistry<StreamingStepMetricsContainer> metricsContainerRegistry) {
    return metricsContainerRegistry
        .getContainers()
        .transformAndConcat(StreamingStepMetricsContainer::extractUpdates);
  }
}

redpajama_set_name: RedPajamaGithub


MMTA's Popular Styles program offers an opportunity for students to learn and perform both solo and combo-style arrangements of jazz, rock, musical theater, and folk tunes that might fall outside the traditional piano repertoire. The professional contemporary musicians on the Popular Styles development team continue to produce and revise materials to provide students with this valuable part of their music experience. The program also places an emphasis on listening to performances and recordings of popular music, and learning by ear.
The MMTA Popular Styles exam program is available in four levels, roughly corresponding to Piano Exam levels 2-5. There is no MMTA program prerequisite for Popular Styles, but students should be finished with level 1 of a piano method or equivalent.
Exam workbooks for levels 1-4 were first published in June 2013 in an electronic-only format; a year later, the first print versions were made available under the moniker "2014 Edition". In June 2015, the workbooks were updated with corrections, clearer formatting, and expanded guidelines, and entitled "2014 Edition, Revised". For this revised edition, no substantive changes were made, so, either the "2014" or "2014, Revised" editions may be used for exams.
There is no syllabus per se for the Popular Styles program. Instead, an exam workbook is offered for each level, serving two purposes: 1) students and teachers use it together to prepare for the exam; and 2) the student brings the workbook to the exam itself, which is open-book.
For their own use, MMTA members may order print editions of the exam workbooks, or download them from the Materials - Popular Styles Exams page for free. For each student, though, MMTA members are asked to order the print editions whether the student takes the exam or not.
Thinking about registering students for an event?
Current MMTA members may log in, then register students online. Non-members may download registration forms using the link(s) provided on Popular Styles Event pages, then submit by mail with a non-member fee.
Popular Styles Exams focus on the development of improvisation, aural learning, and the uniquely American contribution to the world of music. The recommended prerequisite is completion of the first level of any established piano method.
Students use workbooks, lead sheets, backing tracks, and other resources to help them prepare for exams.


Own this house for under $500/mo – Danville Kentucky House for sale https://www.youtube.com/watch?v=GNWpns74zGE Here we are today in one of my favorite towns. I said that last time we shot a house over on Regency, sold in like 10 days so maybe it's good luck, maybe people agree with me that this is just one of the greatest small towns going. Danville, Kentucky, it's a college town, and what a location. People send me a lot of mail, email and say, "Hey, one of these days you're going to send me a Danville Kentucky House for sale that I can afford." This is a house you can afford. Payments on this house are going to be like 450 to 500 dollars a month. You can't rent it for that, I guarantee you that. You can rent this house 750, 800 dollars a month. You go one, two, three houses down and that's the main drag, that is a busy road, 25,000 daily traffic count. What does that mean? Well it means there's a lot of stuff on there so you just buzz on up there to Walmart and all the Lowe's and all the shopping and restaurants and everything. Buzz down this way, more restaurants, downtown, the courthouse, regional medical center, all the stuff. Great spot, I'll put the location here and do a lifestyle map so you can see exactly what you've got. This has got a very cool yard. The property line is somewhere around here, it goes to those bushes. You see there's little stick there in that bush, see it right there? And back to that fence, and that little shed's included so it's got this here and you've got room for actually four cars, two there and two here. You can go in the back door, you can go in the front door. I'm going to show you everything. It has got all redone hardwood floors and you're going to love that. Come on in to this Danville Kentucky House for sale. Alright, so here we go. Let's go on in. I'm going to show you everything. You've got a really neat door here and I think this is like fiberglass or something like that. Might be real wood, I don't think so. I think it's a fiberglass door which means no maintenance. I've got a fiberglass door at my house and love, love, love it because you don't have to do anything to it. Anyway, it's beautiful and it's nice in here. It feels so good, nice and cool and dry, central air. Love it. Check out these floors. This is fantastic. So these are the old floors original with the house. They pull up the carpet and you never know what you're going to get. Here you've got one little spot here, ain't no big deal and the rest of it's just gorgeous. In fact, it's prettier than the new stuff because it's all mellowed out and everything. So I don't know if you call it vintage or whatever but this is beautiful. All redone just recently. They got the satin finish on it and it looks fantastic. You've got beautiful moulding here and then you've got some kind of speaker system thing. I don't know whether they had like a Bose deal or something, you know like a surround-y sound-y thing or whatever and you got all that. Anyway, they did the floors right, done professionally. They even did in the closets so it not cheap out. That's what it looks like inside. Isn't it nice? Let's run in here, we'll look at the kitchen which is compact but it's nice, easy to clean so if you're not needing a huge kitchen, this is going to be right up your alley because it's so easy to take care of. You've got a nice pantry there, full of paint stuff and then check this out. Okay, so here's the deal. In some places in the country, they call it like a 1 1/4 bath, a 1 1/2 bath, a 1 3/4 bath, so that would be a tub, a shower, a toilet and a sink. Here we call it—it's either a full bath or a half bath. Well this ain't a half bath so legally I don't know what you call this but there it is, the toilet. If you need an extra toilet, you've got a thing right here, you could put in a sink if you wanted to and then run your washing machine into the sink or whatever, right? Everybody always used to do that. But anyway, I just thought that was kind of funny. It's nice to have that extra toilet, what the heck. So who knows, the guy who lived here might have been a plumber or something. I don't know but you've got this freakishly huge triple window here which is really nice. I love that. You could put your Christmas tree there in the middle or something and turn all the lights on and drive the neighbours crazy. Here you've got a nice bedroom and no, my camera's not messed up, this is like a mocha colour here and pink here so I'm assuming it was a little one's bedroom. And then you've got– this is what I consider the master bedroom because it's the biggest by far. It's really nice. And this one's really nice too. This one's a little bit smaller than the other one but it's got a really super-duper closet. And of course I don't get along with these doors sometimes so that will be fixed. That one's off the track, we'll fix that. It's no big deal. I'll have Jeremy come out here with his screwdriver. You know, that's pretty funny if you know Jeremy. It's got one of these all house fans. I was up in the attic, it's got a pull-down attic thing and there is some plywood on the floor. I can't remember how much is on the floor. It's not all– you have to put more plywood up there. But you could have tons of storage. You could have like hundreds of square feet of storage, you just need to put some more plywood up there. It would be very simple, you walk it right in the door and then plenty big, right up here and you just nail it down with a nail gun or screw it, doink doink doink doink, and you're all set. Tons of insulation, if I remember correctly, and then you just throw the switch which is right here, and the louvres come up and it's almost like you don't need air conditioning but it's nice to have both, right? And then we've got the bathroom and this is all a closet there with the deal here, and a freakishly large, huge, thing of Germ-X. I have no idea where you would get one like that, maybe Sam's Club perhaps, Costco or some other industrial, military thing. Anyway, you've got a nice shower surround, pretty cloth– shower curtain dealio here, new tilt-in windows, pedestal sink. It looks sharp. Place is nice, okay? Look at that. At the price, it's unbelievable. It's totally clean, totally move in ready, and it's right around the corner from all the stuff so if you get the wicked munchies and you need Taco Bell, it's literally like half a mile away, maybe a quarter mile. Maybe you could walk. Alright, let's go on out the back and there's a little step there. I know, I just found it. So it's got a Goodman system. We love Goodman. If you've ever had any kind of HVAC done in your house, Goodman's a—like an offshoot of one of the big ones and it is now become a big one. We've never had any problems with any Goodman, so kind of love them. I have Train in my house, I've had it serviced a few times. Goodmans we've put in lots of other houses, never had a problem. Seriously. So you've got this great patio back here, concrete patio. You could– you could put like a cover over this or something and have outer space, right? How fun is that? And then this is not the property line. The property line is in the bushes. See that pipe right there? That's the property line, and it comes this way, and then I guess it jiggy jogs over. The shed is included. People like sheds, I know that, that's for sure. And it's all fenced in so the little ones aren't going to get away, or maybe you've got a little Chihuahua or something like that, a doggie, and it's not going to get out because you've got a beautiful fence here. That opens up, you're in. There's a little walk gate over there, you can go on out. That way you can bring your groceries in the back, right in here to the kitchen and the patio. And there's the pantry, you can just drop your potato chips and your lasagna and all that other stuff in there. I have no idea why I said potato chips and lasagna, maybe because I'm hungry. I know I don't look hungry but I am. So here you go, 410 Jean Drive a Danville Kentucky House for sale and it's in the college town. The price, crazy, $450, 500 a month for your mortgage. We can hook you up with the right mortgage broker. We've got great guys who do a lot of our business, you can get whoever you want. You can call us, we'll give you the guys we use and have good luck with, whatever you want. But here you go, and you can get a Royal Housing loan in this area, right now, which is nothing down, which helps a lot, people like that too. Call Jeremy, his phone number's right there. We are going to love to show you this property. And remember the lifestyle map, you are close to everything. That's what people want. And we got jobs here in this county and the next county over, tons and tons of jobs. As always, thanks for watching and don't forget to subscribe. You'll be alerted to properties like this before the general public. Ciao. There's the pantry, you just drop your– your– yeah, drop your stuff in there. And you just nail it down with a nail gun or screw it, doink doink doink– screw it, doink doink doink doink. So we've got this [yells] oh boy. Okay. And a freakishly large thing of Germ-X. I have no idea where you would get one like that, maybe Sam's Club perhaps, Costco, or some other industrial military thing. So if you get the wicked munchies and you need Taco Bell, maybe you could walk. Of course if you've got the munchies at 2 o'clock in the morning, who knows if you want to walk or not. Who knows if you can walk. Let's go on out the back door here and see what we've got. Try that one more time. This time I won't break my neck. Nail it down with a nail gun or screw it, doink doink doink doink. Thanks for watching this video about a Danville Kentucky House for sale You might want to get you in our Exclusive Mailing List, so we can send you your PERFECT homes and land – and farms the second they are for sale (not when they hit the MLS). go to "Your Perfect Property" at Best Real Estate Agent in Kentucky – it only takes a few seconds to fill out your needs and wants and we will start sending you properties that meet YOUR criteria. You'll be 1st to know about properties – It's like a sneak peek – before they hit the market – sent right to your inbox.


text:
\section{Introduction}

Four decades ago, Jan Mycielski posed this question about a regular tetrahedron {\sf T}:\vspace{.3em}

{\sf For ${\cal G}$ the isometry group on 3-space generated by the edge rotations $r_{\sf E}$ of {\sf T}, where Size$(r_{\sf E})$ is the supplement of the dihedral angle of {\sf T}, what can be said
about the orbit, $P{\cal G}:=\{Pf:f\in{\cal G}\}$, of a point $P$ affixed to {\sf T}? }\vspace{.3em}

Mycielski's response, to our recent answer to his question, led to our study of two manifestations, ${\cal W}$ and ${\cal S}$, of the rotation group generated by a set of rotations. We establish sufficient conditions
for the orbits $P{\cal W}$ and $P{\cal S}$ of a point $P$ to be dense in $\Re^d$ for $d\in\{2,3\}$. We have not studied the case where $d\ge4$.

\section{Technicalities}

$\Re:=(-\infty,\infty)$ denotes the set of all real numbers,  ${\Bbb Z}$ is the set of all integers, and ${\Bbb N}:={\mathbb Z}^+$ is the set of all positive integers. When $n\in{\Bbb N}$ then
$[n] := \{1,2,\ldots,n\}$. Finally, ${\mathbb Q}$ is the set of all rational numbers.

We use standard interval notation; e.g.,  $(-3,7] := \{x:x\in\Re\wedge-3<x\le7\}$. Except where more particularly specified, the dimension $d\ge2$ of the real $d$-space $\Re^d$ of our isometries is arbitrary.

For $d=2$, the rotational axes are points in $\Re^2$. However, when $d\ge3$, the axes are directed lines in $\Re^d$.

$r_{\sf A}:\Re^d\rightarrow \Re^d$ is a rotation about the axis {\sf A}, and is a {\em directed} isometry of $\Re^d$. By this we mean that a nonzero rotation has a sign. We deem $r_{\sf A}$ positive iff we see
it as counterclockwise. For $d\ge3$, we judge $r_{\sf A}$ to be counterclockwise if we view it as counterclockwise when we look in the direction accorded to line {\sf A}. Let $F$ be a finite set, and
R $:=\{r_{{\sf A}_i}:i\in F\}$ a set of {\em generator rotations}, with exactly one generator $r_{{\sf A}_i}$ per axis ${\sf A}_i$. For each $i\in F$, let ${\cal G}_i$ be the cyclic
group $\{r_{{\sf A}_i}^z: z\in{\Bbb Z}\}$ of rotations about the axis ${\sf A}_i$.

We write Rad$(r)=\rho\pi$ to indicate the radian measure, modulo $2\pi$, of the angle through which $r$ rotates; here $\rho\in(-1,1]$ unless otherwise specified. \ Size$(r)$ denotes $|$Rad$(r)|$. We call
an angle $\angle PVQ$ rational iff Rad$(\angle PVQ)/\pi \in {\Bbb Q}$, and we call $r_{\sf A}$ rational if it rotates through a rational angle; i.e., if $\rho\in{\Bbb Q}$. Obviously $|{\cal G}_i| < \infty$ if and
only if $r_{{\sf A}_i}$ is rational or, equivalently, is of finite order. Finally, we call two rotations {\em equal} iff they have both the same rotational axis and also the same radian measure modulo $2\pi$.

$f:P\mapsto Pf$ presents the isometry $f$ for $P\in\Re^d$. When ${\sf M}\subseteq\Re^d$, we write $f:{\sf M}\mapsto{\sf M}f := \{Vf:V\in{\sf M}\}$.

The binary operation of each of the two sorts of groups in this paper is its own sort of left-to-right composition of isometries; that for a ``stationary'' group is the conventional $P(f\circ g) = (Pf)g$. But, as
we will explain later, a more peculiar relationship, $P(f\star g)=(Pf)(fg)$, holds for a ``peripatetic'' group.

The expression  $f^-$ denotes the inverse of the isometry $f$, and so $r_i^-$ is the reverse-sense rotation of $r_i$. The identity isometry on $\Re^d$ is $\iota$.\vspace{.5em}

\subsection{Stationary groups}

Let R $:=\{r_i:i\in F\}$ for $2\le |F|<\infty$, where the axis of $r_i$ is ${\sf A}_i$ for $i\in F$. The following stipulates a specializing name for that entity which most would  take to be ``the''
R-generated rotational isometry group.

\begin{defn} \label{Stationary}  For $\{f,h\}\subseteq{\cal S}$ and all $P\in\Re^d$, we let $P(f\circ h) = (Pf)h := (P(f\circ h_1))g_i$ where $h=h_1\circ g_i$ with $g_i\in{\cal G}_i$ for some  $i\in F$
and some $h_1\in{\cal S}$. The {\em stationary} group is $\langle{\cal S},\circ\rangle$ where ${\cal S := S}$(R). \end{defn}

\noindent{\bf Stationary Example.} To illustrate the compositional algorithm of the group ${\cal S:=S}$(R), we will compute the stationary product $s := g_1\circ g_2\circ g'_1\circ g_3\in {\cal S}$ of the
rotational sequence $\langle g_1,r_2,g'_1,g_3\rangle\in{\cal E}_1\times{\cal E}_2\times{\cal E}_1\times{\cal E}_3$. For $P\in\Re^d$, we will produce $s:P\mapsto Ps$ in four steps:

\underline{(1)}. We enact $g_1:P\mapsto Pg_1$ by rotating $P$ about the axis ${\sf A}_1$ with $g_1$.

\underline{(2)}.  $g_2:Pg_1\mapsto (Pg_1)g_2 = P(g_1\circ g_2)$ is realized by rotating the point $Pg_1$ about ${\sf A}_2$ with $g_2$.

\underline{(3)}. We realize $g'_1:P(g_1\circ g_2)\mapsto P(g_1\circ g_2\circ g'_1)$ by rotating $P(g_1\circ g_2)$ about ${\sf A}_1$ with $g'_1$.

\underline{(4)}. Finally, $g_3$ rotates $P(g_1\circ g_2\circ g'_1)$ about ${\sf A}_3$ to reach $P(g_1\circ g_2\circ g'_1\circ g_3) = Ps$.\vspace{.5em}

\noindent{\bf Caveat.} In the stationary context, neither rotational axes, nor points comprising them, are moved by group actions. However, a mobile point $P$ may own the same coordinate address as an
(immobile) axis point $U$; \ i.e., ``$P=U$'' in the address-sharing sense. But whereas $U$ is unmoved by $f$, we have $f:P\mapsto Pf\ne P$. That is, $P$ and $U$ are distinct entities at the same location
in $\Re^d$.\vspace{.5em}

We call a product $g_{j_1}\circ g_{j_2}\circ\cdots\circ g_{j_k}$ of $g_{j_i}\in{\cal E}_{j_i}$ {\em reduced} iff $j_i\ne j_{i+1}$ for every $i<k$.

\begin{lem}\label{StatFree} Let  $f\in{\cal S}$. Then there is exactly one reduced sequence ${\bf g} :=\langle g_{j_1},g_{j_2},\ldots,g_{j_k}\rangle$ in $\bigcup\{{\cal G}_i:i\in F\}$ for which
$f = g_{j_1}\circ g_{j_2}\circ\cdots\circ g_{j_k}$. So ${\cal S}$ is a free group on the generator set {\rm R}, modulo for those $i\in F$ with $|{\cal G}_i|<\infty$, to congruences $\mod2\pi$ which
select the $|{\cal G}_i|$ representatives in $(-\pi,\pi]\cap{\cal G}_i$ from $\{r_i^z:z\in{\Bbb Z}\}$.  \end{lem}

\begin{proof} The lemma holds for $k=1$. Suppose for all $i<k$ that, if $h\in{\cal S}$ is a product  $h=g_{j_1}\circ g_{j_2}\circ\cdots\circ g_{j_i}$ of a reduced sequence in $\bigcup\{{\cal G}_t:t\in F\}$, then
this factorization of $h$ is unique. Let $h := g_{j_1}\circ g_{j_2}\circ\cdots\circ g_{j_{k-1}}$, and let $f := h\circ g_{j_k}$. Pretend that $f = h\circ g_t$ for some $g_t \ne g_{j_k}$. Then, since both $f$ and $h$
are bijective transformations of $\Re^d$, we must  infer that $g_t = h^-\circ f = g_{j_k}$, a contradiction. The lemma follows.\end{proof}

\subsection{Peripatetic groups}

\begin{defn}\label{Peripatetic} For $g_i\in{\cal G}_i$ and  $f:\Re^d\rightarrow\Re^d$ an isometry, $fg_i$ denotes the rotation about the axis ${\sf A}_if$ with Rad$(fg_i) =$ Rad$(g_i)$. The binary operation
$\star$ of the group ${\cal W := W}$(R) is defined recursively by $P(f\star g_i) := (Pf)(fg_i)$ for all $P\in\Re^d$, and by $\iota\star f=f=f\star\iota$. We call the group
$\langle{\cal W}$(R)$;\star\rangle$ {\em peripatetic}.\end{defn}

${\cal W}$ is called ``peripatetic'' as a reminder that ${\sf A}_jg_i\not={\sf A}_j$ for all $\{i,j\}\subseteq F$ with $i\ne j$ unless $g_i=\iota$.\vspace{.5em}

\noindent{\bf Peripatetic Example.} Here $\langle g_1,g_2,g'_1,g_3\rangle$ is the same four-term sequence we employed in the Stationary Example above. Again let $P\in\Re^d$. We will illustrate the peripatetic
group's computational algorithm by showing how to obtain $w:P\mapsto Pw$ via the peripatetic group product $w := g_1\star g_2\star g'_1\star g_3$.\vspace{.3em}

Our calculation realizing $w:P\mapsto Pw$ proceeds in the following four steps.

\underline{(1)}.  $g_1:P\mapsto Pg_1$ swings $\Re^d$ about axis ${\sf A}_1$, while rotating the axis ${\sf A}_2$ into the position ${\sf A}_2g_1$, and ${\sf A}_3$ into the position ${\sf A}_3g_1$; \  i.e.,
when ${\cal W}$ is the group, then $g_1:{\sf A}_i\mapsto{\sf A}_ig_1$ if $i\in\{2,3\}$.  Of course ${\sf A}_1g_1={\sf A}_1$.

\underline{(2)}. The composite rotation $g_1g_2$ moves $Pg_1$ to $(Pg_1)(g_1g_2) = P(g_1\star g_2)$, by rotating $Pg_1$ about the new axis ${\sf A}_2g_1$. Simultaneously,
$g_1g_2:\langle {\sf A}_1g_1,{\sf A}_2g_1,{\sf A}_3g_1\rangle\mapsto\langle {\sf A}_1(g_1\star g_2),{\sf A}_2(g_1\star g_2),{\sf A}_3(g_1\star g_2)\rangle$.

\underline{(3)}. The composite rotation $(g_1\star g_2)g_1'$ rotates the point $P(g_1\star g_2)$ about the axis ${\sf A}_1(g_1\star g_2)$, thus effecting
$P(g_1\star g_2)\mapsto (P(g_1\star g_2))g_1' = P((g_1\star g_2)\star g_1') = P(g_1\star g_2\star g_1')$. Concomitantly  moving axes, we get
$(g_1\star g_2)g_1':\langle {\sf A}_1(g_1\star g_2),{\sf A}_2(g_1\star g_2),{\sf A}_3(g_1\star g_2)\rangle\mapsto
\langle {\sf A}_1(g_1\star g_2\star g_1'),{\sf A}_2(g_1\star g_2 \star g_1'),{\sf A}_3(g_1\star g_2\star g_1')\rangle$.

\underline{(4)}.  $(g_1\star g_2\star g_1')g_3:P(g_1\star g_2\star g_1')\mapsto P(g_1\star g_2\star g_1'\star g_3)$. Thus does the group ${\cal W}$(R) produce $w:P\mapsto Pw$.\vspace{.3em}

We ask our readers to note the difference between this $Pw$ and the $Ps$ in the stationary example.\vspace{.5em}

The tumbling {\sf T} issue is more naturally treated by ${\cal W}$ than by ${\cal S}$, since by design ${\cal W}$ maintains the axes of rotation inside {\sf T}.  Overall, ${\cal W}$ groups may lend themselves
more readily to navigational strategies in a spaceship than do ${\cal S}$ groups, since a rotational coordinate system rooted in ${\cal W}$(R) travels with the traveler.\vspace{.5em}

We now firm up the foundation on which rest plausible but unproven assumptions about ${\cal W}$.

\begin{lem}\label{StarFree} Let  $f\in{\cal W}$. Then there is exactly one reduced sequence ${\bf g} :=\langle g_{j_1},g_{j_2},\ldots,g_{j_k}\rangle$ in $\bigcup\{{\cal G}_i:i\in F\}$ for which
$f = g_{j_1}\star g_{j_2}\star\cdots\star g_{j_k}$. Therefore ${\cal W}$ is a free group on the generator set {\rm R}, subject to the congruences modulo $2\pi$ of $|{\cal G}_i|$ for those $i\in F$ with
$|{\cal G}_i|<\infty$.   \end{lem}

\begin{proof} Uniformly substituting ``$\star$'' for ``$\circ$'' in the proof of Lemma \ref{StatFree}, we obtain a proof of Lemma \ref{StarFree}.  \end{proof}

\begin{cor}\label{Isomorphic} The groups ${\cal S}${\rm(R)} and ${\cal W}${\rm (R)} are isomorphic. \end{cor}

\begin{proof} This corollary is immediate from the two Lemmas, \ref{StatFree} and \ref{StarFree}.  \end{proof}

Although as abstract algebraic structures the groups ${\cal S}$(R) and ${\cal W}$(R) are indistinguishable, they differ as specific subgroups of the group of all isometries on $\Re^d$. Given $P\in\Re^d$ and R,
we ask our readers for an efficient algorithm that expresses some isomorphism $\varphi:{\cal S}$(R)$\rightarrow{\cal W}$(R).

For $P\in\Re^d$ what relationships obtain between the sets $P{\cal S}$ and $P{\cal W}$?

\section{The tumbling tetrahedron and peripatetic orbital densities}

When $x\in\Re$ then the expression $x{\Bbb Z}/{\Bbb Z}$ denotes the set $[0,1)\cap\{xa-b:\{a,b\}\subseteq{\Bbb Z}\}$.

Our applications of ${\cal W}$(R) to density issues depend upon the following well-known consequence\footnote{pointed out to us by Wies{\l}aw Dziobiak} of a theorem of Kronecker, \cite{apostol}. A theorem of
Hurwitz, \cite{hardy} \cite{hurwitz} \cite{leveque} \cite{niven1}, also provides a proof.\footnote{We will email a PDF of three very short proofs of this fact to those who request us to do so.}

\begin{lem}\label{Tool} Let $x\in\Re\setminus{\mathbb Q}$. Then the set $x{\mathbb Z}/{\mathbb Z}$ is dense in $[0,1)$.  \end{lem}

We need a corollary which is immediate from Lemma \ref{Tool}:

\begin{cor}\label{Circle} Let $P$ be a point on a circle {\sf C} with centerpoint $U$, and let $r$ be a rotation about $U$. Then these three assertions are equivalent:

{\bf 1.} \ The rotation $r$ is of infinite order.

{\bf 2.}  \ The size of $r$ is an irrational multiple of $\pi$ radians.

{\bf 3.}  \ The set $\{Pr^z:z\in{\mathbb Z}\}$ is dense in {\sf C}.\end{cor}

The generating sets R := $\{r_{{\sf A}_1},r_{{\sf A}_2}\}$ we will be using are two-membered, except when we deal with the tumbling tetrahedron {\sf T}, for which $|$R$_{\sf T}|=6$. Under the action of
${\cal W}$, a copy of $\Re^d$ moves against a fixed $\Re^d$ background, whose points are not budged by ${\cal W}$ actions.

Recall that in both the stationary and the peripatetic contexts, ${\cal G}_i$ is the cyclic subgroup $\{r_i^z:i\in{\Bbb Z}\}$.

We now state and prove our more easily visualized peripatetic rotational density theorem.\footnote{Allan Silberger suggested Theorem \ref{AllanS}. }

\begin{thm}\label{AllanS} Let $U_1\not=U_2$ be points in $\Re^2$. Let $r_1$ and $r_2$ be rotations of infinite order about $U_1$ and $U_2$ respectively. Let ${\cal W:=W}${\rm(R)},
where {\rm R} $:= \{r_1,r_2\}$. Let $P\in\Re^2$. Then the orbit $P{\cal W}$ is dense in $\Re^2$.    \end{thm}

\begin{proof} Take it that $P\ne U_1$. We argue by contradiction. Pretend there is a (fixed) point $Q$, and a (fixed) disc ${\sf D}\subseteq\Re^2$ with center at $Q$, and whose radius $\epsilon>0$ is the
largest that allows ${\sf D}\cap P{\cal W}=\emptyset$. Since $\epsilon\le\|P-Q\|<\infty$, we have $Pf\in{\sf S}:=\overline{\sf D}\setminus{\sf D}$ for some $f\in{\cal W}$, where $\overline{\sf D}$ is the
closure of {\sf D}, and {\sf S} is a circle.

The points $Pf,\, (U_2f)(fg_1) =: U_2(f\star g_1)$ and $Q$ are noncollinear for some $g_1\in{\cal G}_1$. Let ${\sf C}_1\subseteq\Re^2$ be the circle with centerpoint $U_1$ and with $P\in{\sf C}_1$; by
hypothesis, the radius of ${\sf C}_1$ is $\|P-U_1\|>0$. Let ${\sf C}_2$ be the circle with centerpoint $U_2$ and with $P\in{\sf C}_2$; the radius of ${\sf C}_2$ is $\|P-U_2\|\ge 0$.

The circles ${\sf C}_1$ and ${\sf C}_2$ are mapped by $f$ isometrically onto the respective circles ${\sf C}_1f$ with centerpoint $U_1f$, and ${\sf C}_2f$ with centerpoint $U_2f$. We see that
${\sf C}_1f = {\sf C}_1(f\star g_1)$, since $f\star g_1$ merely rotates ${\sf C}_1f$ about the point $U_1f$. However, ${\sf C}_2(f\star g_1)\not={\sf C}_2f$; for, the rotation $f\star g_1$ swings $\Re^2$
around $U_1f$, thus mapping the circle ${\sf C}_2f$ isometrically onto the circle ${\sf C}_2(f\star g_1)$, whose centerpoint is $U_2fg_1\not=U_2f$. If $g_1$ is chosen prudently, then
${\sf C}_2(f\star g_1)\cap{\sf B}\not=\emptyset$. Now \ref{Circle} finishes the proof, since $P(f\star g_1)=(Pf)(fg_1)$ and $f\star g_1\in{\cal W}$. \end{proof}

The analogous next result will enable us to answer the Mycielski question which inspired this paper.

\begin{thm}\label{Main} Let $r_1$ and $r_2$ be infinite-order rotations about the respective skew directed lines ${\sf X}_1$ and ${\sf X}_2$ in $\Re^3$, let {\rm R} $:=\{r_1,r_2\}$, let ${\cal W:=W}${\rm (R)}, and
let $P\in\Re^3$. The orbit $P{\cal W} := \{Pf:f\in{\cal W}\}$ is dense in $\Re^3$.  \end{thm}

\begin{proof} We can take it that $P\notin{\sf X}_1$. For each $i\in\{1,2\}$ with $P\notin{\sf X}_i$, let $U_i\in{\sf X}_i$ be such that $PU_i\perp{\sf X}_i$, and let ${\sf C}_i$ be the circle in $\Re^3$ of radius
$\|P-U_i\|$ and with centerpoint $U_i$. If $P\notin{\sf X}_i$, then $\{U_i,P,Pg_i\}$ is a set of noncollinear points for $\iota \ne g_1\in{\cal G}_1$. But in the event that $P\in{\sf X}_i$, let $U_i:=P$ and
${\sf C}_i := \{P\}$.

Again arguing by contradiction, we pretend that there is a (fixed) point $Q$, and a $Q$-centered open ball ${\sf B}\subseteq\Re^3$ of radius $\epsilon>0$, such that ${\sf B}\cap P{\cal W}=\emptyset$,
but that if ${\sf B'}$ is a $Q$-centered ball with radius $\epsilon'>\epsilon$ then ${\sf B'}\cap P{\cal W}\not=\emptyset$. Since $\epsilon \le \|Q-P\| <\infty$, it follows that
$Pf\in{\sf S}:=\overline{\sf B}\setminus{\sf B}$ for some $f\in{\cal W}$,  where $\overline{\sf B}$ is the closure of {\sf B} and where {\sf S} is therefore a 2-sphere.

Since {\sf B} is open, if ${\sf B\cap C}_1f\not=\emptyset$ then ${\sf B\cap C}_1f$ is an arc of positive length in {\sf B}, whence Corollary \ref{Circle} concludes our proof. So we take the circle ${\sf C}_1f$ to be
tangent to the sphere {\sf S} at the point $Pf$. For $i\in\{1,2\}$, the point $U_if\in{\sf X}_if$ is the centerpoint of the circle ${\sf C}_if$.

Since the axes ${\sf X}_1$ and ${\sf X}_2$ are skew and since $f$ is an isometry, ${\sf X}_1f$ is skew to ${\sf X}_2f$. Furthermore, Order$(r_2) = \infty$ by hypothesis. WeThere are two cases.

\underline{Case}: $P\notin{\sf X}_2$. Then we can infer from Corollary \ref{Circle} that there exists $g_2\in{\cal G}_2$ with $P(f\star g_2) := (Pf)(fg_2)\in{\sf B}$. Moreover, $f\star g_2 \in {\cal W}$. Thus
${\sf B}\cap P{\cal W} \ne \emptyset$, contrary to our choice of ${\sf B}$.

\underline{Case}: $P=U_2$. By Corollary \ref{Circle} there exists $g_2\in{\cal G}_2$ with ${\sf B}\cap{\sf C}_1(f\star g_2) \ne \emptyset$. So here too ${\sf B}\cap P{\cal W} \ne \emptyset$.

\noindent In both cases we reach a contradiction. \end{proof}

\noindent{\bf Remark.} If the hypotheses of Theorem \ref{Main} allowed ${\sf X}_1\cap{\sf X}_2=\{V\}$, then $P{\cal W}\subseteq{\sf Y}$ where ${\sf Y}\subseteq\Re^3$ is the sphere of radius $\|P-V\|$ and
centerpoint $V$; if also $P\not= V$ then $\|P-V\|>0$, and $P{\cal W}$ would be dense in {\sf Y}.\vspace{.7em}

In order that assure that $P{\cal W}({\rm R}_{\sf T})$ is dense in $\Re^3$, where \ R$_{\sf T}$ \ is the set of six edge rotations $r_{\sf E}$ of {\sf T}, Theorem \ref{Main} requires only two of the six to be irrational.
But  Size$(r_{\sf E}) := \pi-\theta$ for each $r_{\sf E}\in {\rm R}_{\sf T}$, where $\theta$ is the size of each dihedral angle of {\sf T}. So we will need to prove that $\theta$ is irrational.\vspace{.5em}

\noindent{\bf Quiz.} Show that $\sin(\theta)=2\sqrt{2}/3$, and that equivalently $\cos{\theta} = 1/3$.

\begin{lem}\label{Niven} Let $0<\theta<\pi/2$. If $\cos(\theta)=1/3$ then $\theta$ is irrational. \end{lem}

\begin{proof} By Corollary 3.12 of \cite{niven1}, both the angle $\pi/2-\phi$ and the real number $\cos(\phi)=\sin(\pi/2-\phi)$ are rational if and only if $\phi = \pi/3$. So, since $\cos(\pi/3)=1/2 \not=
1/3 = \cos(\theta)$, we see that $\theta$ is irrational. \end{proof}

The answer to Mycielski's half-century-old query about tumbling {\sf T} is now obvious from \ref{Main} with \ref{Niven}.

\begin{cor}\label{Tetrahed} Let {\sf T} be a regular tetrahedron in $\Re^3$, and let ${\cal W := W}({\rm R}_{\sf T})$ be the peripatetic rotational isometry group determined by the set ${\rm R}_{\sf T}$ of six
generating rotations $r_{\sf E}$ around the edges {\sf E} of {\sf T}, where each {\rm Size}$(r_{\sf E})$ is the supplement of the dihedral angle of {\sf T}. Then the orbit $P{\cal W}$ is dense in $\Re^3$ for each $P\in\Re^3$.   \end{cor}

Steve Silverman asks whether there exists a tetrahedron, all six of whose dihedral angles are rational. He provides an example in which four of the six are rational. Is four best possible? If ``yes'', then the tumblings
of an arbitrary tetrahedron {\sf K}, each edge {\sf E} of which is assigned a rotation $r_{\sf E}$ whose size is the supplement of the dihedral angle at {\sf E}, will trace out an orbit $P{\cal W}$ that is dense in
$\Re^3$ for each $P\in\Re^3$.

\section{Orbital density for stationary groups}

The following may be a duplication of a decades-old unpublished result of Jan Mycielski.

\begin{thm}\label{StaDense} Let $d\in\{2,3\}$. Let the generating set {\rm R} be as in Theorem \ref{AllanS} or \ref{Main}, and let both of the rotations $r_1$ and $r_2$ be of infinite order. Let
${\cal S:=S}${\rm(R)} be the stationary group determined  {\rm R}. Then the orbit $V{\cal S}$ is dense in $\Re^d$ for every $V\in\Re^d$. \end{thm}

\begin{proof} Since the proof for $d=3$ is essentially the same as that for $d=2$, we will argue only the $d=2$ case.

The axes of $r_1$ and $r_2$ are $U_1$ and $U_2$ respectively. However, unlike in Theorem \ref{AllanS}, here the points $U_i$ are fixed, and are unaffected by actions of ${\cal S}$. On the other hand,
the point $V$ is not fixed. Suppose without loss of generality that $V\not= U_2$; \ i.e., that the points $V$ and $U_2$ have different addresses in $\Re^2$.

By hypothesis, for each $i\in\{1,2\}$ we have that Size$(r_i) = \rho_i\pi$ for an irrational $\rho_i$. We can take it that $\rho_i\in(0,1)$. Let $k_i$ be the positive integer for which $k_i\rho_i< 1 < (k_i+1)\rho_i$.

Each set ${\sf C}^*_{i,V} := {\sf C}_{i,V}\cap V{\cal S}$ is dense in the unique circle ${\sf C}_{i,V}$ with centerpoint $U_i$ and with $V\in{\sf C}_i$, provided only that ${\sf C}_i\not=\{U_i\}$. Let ${\sf O}_1$
be the finite set  $\{Vr_1^j: 0\le j\le 2k_1\}\subset{\sf C}_{1,V}^*$.

For each $A\in{\sf O}_1$, we create the finite set $\{Ar_2^j:0\le j\le 4k_2\}\subset {\sf C}_{2,A}^*$, where ${\sf C}_{2,A}$ is the unique circle having centerpoint $U_2$ and such that $A\in{\sf C}_{2,A}$.
Define ${\sf O}_2 :=\bigcup\{\{Ar_2^j:0\le j\le 4k_2\}:A\in{\sf O}_1\}$.

Similarly, using the points $A\in{\sf O}_2$, define ${\sf O}_3 := \bigcup\{\{Ar_1^j:0\le j\le 2^3k_1\}:A\in{\sf O}_2\}$.

Next, define ${\sf O}_4 := \bigcup\{\{Ar_2^j:0\le j\le 2^4k_2\}:A\in{\sf O}_3\}$. Continue thus, back and forth between the fixed rotational axes $U_1$ and $U_2$. This recursion produces an infinite sequence
$\langle{\sf O}_b\rangle_{b=1}^\infty$ of finite subsets of $V{\cal S}$ with ${\sf O}_b\subset{\sf O}_{b+1}$, where eventually the convex closures of the ${\sf O}_b$ approximate elliptical regions whose minimal diameters
increase without bound, and such that the meshes of the sets ${\sf O}_b$ diminish to zero.\footnote{It is natural to define the mesh of a subset ${\sf E}\subseteq V{\cal S}$ as the minimum $\epsilon>0$ for which no
disc ${\sf D}$ of radius $\epsilon$ inside the convex closure of {\sf E} can fail to contain points in {\sf E}. }  Ultimately we obtain a denumerable subset $\bigcup\{{\sf O}_b:b\in{\mathbb N}\}\subseteq V{\cal S}$,
where $\bigcup\{{\sf O}_b:b\in{\mathbb N}\}$ is dense in $\Re^2$. Therefore, the orbit $V{\cal S}$ of $V$ under the actions of the stationary rotational isometry group ${\cal S}$(R) is itself dense in $\Re^2$. \vspace{.5em}

\noindent The three-dimensional case involves a reiteration of the argument above, modulo these five observations:

(1). The axes of rotation are now two fixed directed skew lines instead of two fixed points.

(2). The circle ${\sf C}_{1,V}$ is not coplanar with the circle ${\sf C}_{2,V}$.

(3). For $d=3$, the convex closures of the ${\sf O}_b$, approximate ellipsoids or spheroids rather than ellipses.

(4). Re ``mesh'': In $\Re^3$ we talk of $\epsilon$-radius balls instead of the $\epsilon$-radius discs that make sense in $\Re^2$.

(5). In $\Re^2$ the convex closure of $V{\cal S}$ is two-dimensional. But it is three-dimensional in $\Re^3$.  \end{proof}

\section{Related issues}

The classes of stationary and peripatetic rotational groups in $\Re^d$ for $2\le d<\infty$ are antipodal in regards to the modes of their rotational generators $r_i$. This suggests the possible fruitfulness of studying
rotational groups of sorts that lie between those antipodal classes; i.e., rotational groups some of whose given axes are stationary while others are peripatetic.\vspace{1em}

We believe Theorem \ref{Main} extends in a natural way to $\Re^d$ for $d\in\{4,5,6\ldots\}$, given an appropriate finite collection ${\cal X}$ of lines ${\sf X}_i\subseteq\Re^d$ and of infinite-order rotations $r_i$
of 2-planes about those lines. Our readers will expect that the claim in such a generalization of Theorem \ref{Main} would be that the orbit $P{\cal W}$ is dense in $\Re^d$ for every $P\in \Re^d$.

We ask this question of our readers: Which, if any, of the following three hypotheses about the collection ${\cal X}$ is both necessary and sufficient to guarantee the density in $\Re^d$ of $P{\cal W}$ for all $P$?

{\bf 1.} \ $|{\cal X}|=d-1$ and no proper hyperplane in $\Re^d$ is a superset of $\bigcup{\cal X}$.

{\bf 2.} \ $|{\cal X}|=d-1$ and ${\cal X}$ is pairwise skew.

{\bf 3.} \ ${\cal X}=\{{\sf X}_1,{\sf X}_2\}$ is a set of skew lines.\vspace{.5em}

To illuminate the contention between the hypotheses, {\bf 1} and {\bf 2}, we offer for $4\le d\in{\mathbb N}$ this example:

For each $i\in{\mathbb N}$, let ${\sf Y}_i:=\{\langle i,ti,ti^2,0,0,\ldots 0\rangle:t\in\Re\}\subseteq\Re^d$. Then ${\cal Y}:=\{{\sf Y}_i:i\in{\mathbb N}\}$ is an infinite pairwise skew set of lines for which $\bigcup{\cal Y}$
is a subset of a copy of  $\Re^3$ that occurs as a proper subspace of $\Re^d$.\vspace{1em}

Both Theorem \ref{Main} and Corollary \ref{Tetrahed} provide orbits $P{\cal W}$, each of which is dense in $\Re^3$. But the orbit encountered in \ref{Tetrahed} is plainly richer than that encountered in \ref{Main}. This
invites an observation.

Let the regular tetrahedron {\sf T} have an edge length of $\sqrt{6}$, let $P$ be its barycenter, let ${\cal G}$ be the peripatetic rotational isometry group generated by the six edge rotations $r_{UV}$ of {\sf T}, let
$f\in{\cal G}$, and let the axes of rotations $r_{AB}$ and $r_{AC}$ share a single vertex, $A$. Then ${\sf H}\cap P{\cal G}$ contains the vertices of a tiling of {\sf H} by hexagons of edge length $1$, where {\sf H} is the
plane determined by the point set $\{Pf,Pfr_{AB},Pfr_{AC}\}$.\vspace{.5em}

\noindent{\bf Conjecture 1.} If plane ${\sf H}\supseteq\{Pf,Pfr_{AB},Pfr_{AC}\}$ then ${\sf H}\cap P{\cal G}$ is the vertex set of a hexagonal tiling of {\sf H}.\vspace{1em}

Let {\sf X} be the $x$-axis of $\Re^3$. Let {\sf Y} be parallel to the $y$-axis and through $\langle 0,0,1\rangle$.  Let ${\cal W := W}$(R) be the peripatetic group generated by ${\rm R} :=\{r_{\sf X},r_{\sf Y}\}$,
 where Order$(r_{\sf X})=$ Order$(r_{\sf Y})=4$.  Let $P\in\Re^3\setminus({\sf X\cup Y})$. Then $P{\cal W}$ is an infinite discrete set of vertices of rectangular parallelopipeds, and $P{\cal W}$  is nowhere dense.

This suggests a shift of focus from rotations to their axes of rotation.\vspace{.5em}

For $d\ge 2$, say that nonparallel lines {\sf X} and {\sf Y} in $\Re^d$ {\em conform rationally} iff some translate of {\sf Y} intersects {\sf X} to form a rational angle.

Let ${\sf X}_1$ and ${\sf X}_2$ be skew lines in $\Re^3$, and let both of their respective rotations $r_1$ and $r_2$ be of finite order. Let ${\cal F}$ be the peripatetic rotational isometry group generated by
F $:=\{r_1,r_2\}$. \vspace{.5em}

\noindent{\bf Conjecture 2.} \ $P{\cal F}$ is nowhere dense for each $P\in\Re^3$ if and only if ${\sf X}_1$ conforms rationally to ${\sf X}_2$.\vspace{1em}

\noindent{\bf Conjecture 3.} \ If ${\sf X}_1$ does not conform rationally to ${\sf X}_2$, then $P{\cal F}$ is dense in $\Re^3$ for all $P\in\Re^3\setminus({\sf X}_1\cup{\sf X}_2)$. \vspace{1em}

Suppose Conjecture 2 is true. Let ${\sf X}_1$ and ${\sf X}_2$ be rationally conforming skew lines in $\Re^3$, and let both $r_1$ and $r_2$ be rotations of finite order on those respective axes. Let $F :=
\{r_1,r_2\}$ be the generator set for the peripatetic group ${\cal F:=F}$(F).\vspace{.5em}

\noindent We call a polyhedron ${\sf K}\subseteq\Re^3$ a $\langle P,$F$\rangle$-{\em chamber} iff {\sf K} is maximal with respect to these four conditions:\vspace{.3em}

(1). \ {\sf K} is convex and of finite diameter.

(2). \ Every vertex of {\sf K} is an element in $P{\cal F}$.

(3). \ No element in $P{\cal F}$ is in the interior of {\sf K}.

(4). \ No three vertices of {\sf K} are collinear.\vspace{.3em}

\noindent What can one say about these $\langle P,$F$\rangle$-chambers?

redpajama_set_name: RedPajamaArXiv


text:
by Callie Malvik
Callie Malvik2022-07-15T21:28:04+00:00September 14, 2020|
How Technology Enabled a Smooth and Swift Transition to a Fully Virtual Campus [Case Study]
The COVID-19 pandemic has been the driving force behind a whole slew of challenges society has faced so far in 2020. Its detrimental effects have hit every industry in some way, but higher education has been dealt some especially devastating blows.
When colleges and universities were called to abruptly close down campuses in the spring, they were forced to transition residential operations online virtually overnight. For many schools, this was a rude awakening that outdated technology ecosystems simply weren't able to deliver the quality online experience and support that today's students and faculty have come to expect.
But that wasn't the case for Cabrini University. The private institution located outside of Philadelphia, Pennsylvania had already established a strong partnership with Collegis Education to digitally transform their online presence. So when the governor of Pennsylvania ordered all higher ed institutions to go fully remote on March 13, Cabrini was up for the challenge.
Building on an established foundation in technology
Cabrini was aware of the importance of investing in technology long before disaster struck. Prior to the pandemic, the college had partnered with Collegis to help strengthen its technology infrastructure and online learning platforms, putting them a step ahead of most schools at the time.
So when the Cabrini campus was closed, the following key elements were already in place:
Online platform: An LMS platform with training materials was in place for all online programs.
24/7 virtual helpdesk: Technology support resources were available to help remote students and employees any time day or night.
Connectivity: Most technology systems were remotely accessible through cloud-based apps and VPN solutions.
Laptops: Most faculty and staff had laptops available to them, allowing for an easy transition to remote work.
Eric J. Olson, vice president of finance and treasurer at Cabrini University, believes the foundation that was put in place before COVID-19 helped the institution weather the storm that struck.
"The effort that went into building this infrastructure payed a lot of dividends for us when met with this crisis," Olson says.
"The processes we've put in place have really made a lot of difference," he adds.
Anticipating and addressing new challenges
Fortunately, when disaster hit, Cabrini already had access to the expanded technology support offered by Collegis's large IT organization to implement or scale new technology quickly. But even so, the team faced new challenges that required immediate action:
All programs required electronic course materials and exam proctoring.
Some students lacked the technology needed to access online courses.
Many faculty members were unfamiliar with online teaching.
Digital communication and outreach became far more important and required greater access and training.
In-person campus events had to be reimagined with virtual solutions.
The entire Cabrini staff had to transition into a fully remote workforce.
New information security vulnerabilities had to be anticipated and protected against.
Of these concerns, faculty adoption of online teaching was of utmost importance. Cabrini has a legacy of providing excellence in education and was determined to live up to that reputation even in a virtual environment.
With help from Collegis, training sessions were initiated around the clock along with the development of LMS user guides and materials for all faculty. Regular communication was established between Cabrini leadership and the Cabrini community via email, virtual townhall meetings and the campus portal.
New collaboration technology – such as Zoom and Microsoft Teams – deployed immediately to allow for convenient communication and synchronous learning options. 24/7 tech support was offered for students and staff, along with access to loaner laptops for those who needed one. Investments were also made in software programs that could help ensure exam integrity.
With the existing technology foundation and the support and guidance from Collegis to address new obstacles, Cabrini was able to transition to a fully virtual model in less than two weeks.
Empowering and supporting the virtual campus
Nearly every higher education institution across the country was faced with a similar timeline for quickly migrating residential classes to a virtual format. A handful of tactics were universally used to provide a temporary fix to finish out the term online, but Cabrini was determined to go a step further and provide the most enriching experience possible for their students.
Technology was the vital enabler in order to achieve this. With the help of Collegis, they were able to stream digital events via video platforms, provide virtual campus visits for new students and even host their highly anticipated annual Accepted Students Day in a remote format.
Collegis was also able to play an integral role in providing ongoing IT support for Cabrini through the crisis. This included providing the following:
Virtual CIO services
24/7 student and staff IT support (including prospective students)
Recommendations and guidance on use of technology platforms
Training and support on expanded use of technology
Assistance in processing CARES payments and refunds for students
Information security vigilance of remote computers and connections
Evaluation of enhanced classroom technology for the fall term
Moving forward with an uncertain future
With the uncertainty of the length and extent of the COVID-19 crisis, Cabrini and Collegis have continued to prepare for changes to teaching and learning for the remainder of the year and into 2021. This has involved planning for the increased usage of video learning options, scaling online learning platforms, embracing technology innovation and new features to platforms, and extending campus wi-fi coverage to include outdoor common areas.
Cabrini University's rapid transition to fully remote operations is certainly a success story. The investments made over the past few years to strengthen their technology infrastructure undoubtedly helped prepare them to persevere through these unprecedented times. The importance of such efforts is more evident than ever in today's higher education landscape.
But you don't have to go at it alone. Implementing a full digital transformation is a lofty task. This is especially true for smaller institutions, like Cabrini, that simply don't have the capacity to handle it all in-house.
"For schools like us that are small and resource-challenged, it doesn't mean your technology needs are any less. So, you're faced with issues of 'how are we going to meet those needs?'" Olson shares. But in partnering with Collegis, Cabrini was able to overcome the challenges brought by COVID-19 and continue pushing forward in their mission of academic excellence, leadership development and a commitment to social justice.
Check out this on-demand webinar to hear firsthand about this successful partnership from Cabrini and Collegis leaders.
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Tags: case study, COVID-19 Support, technology solutions
Callie Malvik
Callie Malvik is a Senior Content Manager with Collegis Education. She holds a Bachelor's Degree in Mass Communication and Advertising from Winona State University, Minn.
Jessica Conte2022-12-07T17:35:16+00:00September 27, 2022|
Knowledge Check: Do You Know How Technology Affects Your Higher Ed Institution?