Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with A Natural Extension To Infinite Structures

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G$. Internally, there is also a canonical order for the elements of any finite group $G$, and we find equivalent objects. This allows us to find the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and a minimal set of independent equations that define the group is obtained. Examples are given, using all groups with less than ten elements, to illustrate the procedure for finding all groups of $n$ elements, and we order them externally and internally. The canonical block form of the symmetry group $\Delta_4$ is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, also. We make brief mention on the calculus of real numbers. In general, we are able to represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects of all types are well assigned to tree structures. We conclude with comments on type theory and future work on computational and physical aspects of these representations.

Feeding value of sorghum stover fed to tropical hair sheep as complete rations in chop, mash, pellet, and block forms

Background and Aim: Poor palatability, low bulk density, and low nutritive value restrict the utilization of the crop residues as animal feeds. Altering the physical characteristics of feed by blending the roughage and concentrates in the form of complete feed improves the nutrient use efficiency and reduces the feed wastage, feed cost, and labor cost. The study aims to determine suitable processing methods (mash, pellet, or block forms) for efficient utilization of sorghum stover-based complete rations vis-a-vis conventional feeding methods in sheep. Materials and Methods: The sorghum stover was incorporated in complete rations with roughage to concentrate ratio of 50:50 proportion in a growth trial of 120 days. The feed ingredients were chaffed, ground in a hammer mill, passed through expander-extruder, and compressed by feed block machine to prepare chop, mash, pellet, and block form of rations, respectively. Twenty-four male intact growing Nellore×Deccani cross lambs (3.5±0.5 months age, 14.50±0.41 kg (mean ± SD) at the start of the experiment) were divided into four experimental groups of six animals each in a complete randomized design. The experimental rations were randomly allotted to each group and evaluated for their intake, nutrient utilization, and growth performance. Results: The sheep fed on pellet-based ration consumed a higher (p<0.05) quantity of dry matter. The digestibility coefficients of organic matter, crude protein, and neutral detergent fiber were higher (p<0.05) in processed rations (mash, pellet, or block). Further, the nitrogen balance (g/d) was higher (p<0.05) in the mash, pellet, and block form of rations, compared to chopped ration. The processing method did not influence calcium or phosphorous balance parameters, except for their urinary losses, which showed an increasing trend (p=0.07). The ram lambs fed with pelleted ration showed higher (p<0.05) weight gain than mash, block, or chaff forms. The efficiency of feed utilization in gaining one kg body weight was higher (p<0.05) in lambs fed a pelleted diet. Feeding pelletized ration was more economical to gain one kg body weight. The bulk density was highest for block-based ration followed by pellet, mash, and conventional rations, and the carrying capacity of truck was highest with the least transport cost in block-form of rations. Conclusion: Physical processing (mash, pellet, and block) of sorghum stover-based complete rations increased the nutrient utilization and growth performance of sheep compared to conventional chopped form. Pelleting the mash with expander-extruder procedure was found to be more profitable. Nevertheless, the cost economics revealed blocks as more preferable forms for transporting the complete rations to larger distances.

Kristalinitas Abu Vulkanik Gunung Soputan dan Implikasinya Terhadap Perilaku Eksplosif pada Erupsi Tahun 2016 dan 2018

Gunung Soputan adalah gunung basaltik dengan tipe erupsi eksplosif. Tipikal eksplosif berkaitan dengan karakter magma. Penelitian ini bertujuan untuk mengkaji kristalinitas abu vulkanik Gunung Soputan tahun 2016 dan 2018 dan implikasinya terhadap perilaku eksplosifnya. Kristalinitas abu vulkanik dikarakterisasi menggunakan peralatan difraktometer sinar X. Indeks kristalinitas dikalkulasi menggunakan persamaan Soltys dan ukuran kristal dikomputasi dengan persamaan Debye-Scherrer. Morfologi partikel abu dikarakterisasi menggunakan SEM. Hasil karakterisasi menunjukkan bahwa abu Soputan tahun 2016 dan 2018 memiliki indeks kristalinitas yang tinggi dan mengandung sejumlah kristal berukuran kecil serta memiliki dua bentuk morfologi yaitu bentuk padat dan blok bervesikular untuk abu hasil erupsi 4 Januari 2016, dan bentuk padat dan blok untuk abu hasil erupsi 6 Februari 2016 dan 3 Oktober 2018. Kristalinitas yang tinggi menyebabkan magma Soputan memiliki viskositas yang memadai untuk terjadinya erupsi yang eksplosif. Karakter eksplosif Gunung Soputan terekam dalam morfologi abu vulkaniknya.Kata kunci: Abu vulkanik; erupsi eksplosif; fragmentasi getas; indeks kristalinitasCristallinity of Volcanic Ash of Mount Soputan and Its Implication to Explosive Behaviour on Eruption of 2016 and 2018ABSTRACTMount Soputan is a basaltic mountain with an explosive eruption type. Typical explosives relate to the character of the magma. This study aims to examine the crystallinity of Mount Soputan's volcanic ash in 2016 and 2018 and its implications for its explosive behavior. The crystallinity of volcanic ash was characterized using an X-ray diffractometer. The crystallinity index was calculated using the Soltys equation and the crystal size computed using the Debye-Scherrer equation. The morphology of the ash particles was characterized using SEM. The characterization results show that the 2016 and 2018 Soputan ash has a high crystallinity index and contains a number of small crystals and has two morphological forms, namely solid form and vesicular block for ash from the eruption on January 4, 2016, and solid and block form for eruption ash 6 February 2016 and October 3 2018. High crystallinity causes the Soputan magma to have sufficient viscosity for explosive eruptions. The explosive character of Mount Soputan is recorded in the morphology of its volcanic ash.Keywords: Brittle fragmentation; crystallinity index; explosive eruption; volcanic ash

DISCRETE TRANSFORM IN BLOCK FORM AND INVARIANCE TO PROCESSING DIRECTION

Рассмотрен ряд вопросов инвариантности дискретных блочных преобразований. Показано, что смена направления обработки при выполнении обратимых преобразований приводит к изменению, получаемых частотных составляющих. Предложены математические модели блочных дискретных преобразований на основе матричного представления. Предлагается рассматривать дискретные преобразования в зависимости от количества блоков и направления обработки. Приводятся результаты моделирования на основе дискретного преобразования Фурье. Some questions of invariance of a discrete block transforms are considered. It is shown a processing direction change in a reversible transforms performing leads to a frequency components change. Mathematical models are proposed for block discrete transforms based on a matrix representation. The discrete transforms are proposed depending on a blocks number and direction of processing. Modeling results on the discrete Fourier transform are presented.

Towards Next-Generation Sustainable Composites Made of Recycled Rubber, Cenospheres, and Biobinder

The utilisation of industrial residual products to develop new value-added materials and reduce their environmental footprint is one of the critical challenges of science and industry. Development of new multifunctional and bio-based composite materials is an excellent opportunity for the effective utilisation of residual industrial products and a right step in the Green Deal’s direction as approved by the European Commission. Keeping the various issues in mind, we describe the manufacturing and characterisation of the three-component bio-based composites in this work. The key components are a bio-based binder made of peat, devulcanised crumb rubber (DCR) from used tyres, and part of the fly ash, i.e., the cenosphere (CS). The three-phase composites were prepared in the form of a block to investigate their mechanical properties and density, and in the form of granules for the determination of the sorption of water and oil products. We also investigated the properties’ dependence on the DCR and CS fraction. It was found that the maximum compression strength (in block form) observed for the composition without CS and DCR addition was 79.3 MPa, while the second-highest value of compression strength was 11.2 MPa for the composition with 27.3 wt.% of CS. For compositions with a bio-binder content from 17.4 to 55.8 wt.%, and with DCR contents ranging from 11.0 to 62.0 wt.%, the compressive strength was in the range from 1.1 to 2.0 MPa. Liquid-sorption analysis (water and diesel) showed that the maximum saturation of liquids, in both cases, was set after 35 min and ranged from 1.05 to 1.4 g·g −1 for water, and 0.77 to 1.25 g·g−1 for diesel. It was observed that 90% of the maximum saturation with diesel fuel came after 10 min and for water after 35 min.

Computational Treatment of First Order Delay Differential Equations Using Hybrid Extended Second Derivative Block Backward Differentiation Formulae

This paper considers the computational solution of first order delay differential equations (DDEs) using hybrid extended second derivative backward differentiation formulae method in block form without the implementation of interpolation techniques in estimating the delay term. By matrix inversion approach, the discrete schemes were obtained through the linear multistep collocation approach from the continuous form of each step number which after implementation strongly revealed the convergence and region of absolute stability of the proposed method. Computational results are presented and compared to the exact solutions and other existing method to demonstrate its efficiency and accuracy.

Towards Next Generation Sustainable Rubber Composites from Biobinder Made of Homogenised Peat

The utilisation of the industrial residual products to develop new value-added materials and to reduce footprint is one of the critical challenges of science and industry. Development of the new multifunctional and bio-based composites materials is an excellent opportunity for the effective utilisation of industrial residual products. Keeping the different issues in mind, in this work, we describe the manufacturing and characterisation of the three-phases bio-based composites. The key components are bio-based binder made of peat, devulcanised crumb rubber (DCR) from used tires and part of the fly ash, i.e. the cenosphere (CS). The three-phase composite prepared in the form of a block were investigated for their mechanical properties and density. The three-phase composite was prepared in the form a) of a block were investigated for their mechanical properties, and density and b) a form of granules for determination of the water and oil products sorption were investigated. We have also investigated the dependence of the properties on the DCR and CS fraction. It has been found, that maximum compression strength (in block form) observed for composition without CS and DCR addition was- 79.3 MPa, while the second-highest value of compression strength is 11.2 MPa for composition with 27.3 wt% of CS. For compositions with bio binder content from 17.4 to 55.8 wt% and with DCR contents in range from 11.0 to 62.0 wt%, the compression strength is in the range from 1.1 to 2.0 MPa. Liquid sorption analysis (water and diesel) showed that the maximum saturation of liquids in both cases is set after 35 minutes and ranges from 1.05 to 1.4 g·g -1 for water and 0.77 to 1.25 g·g-1 for diesel. It was noted that 90% of the maximum saturation with diesel fuel comes after 10 minutes and for water after 35 minutes.

A quotient of the Lubin–Tate tower II

In this article we construct the quotient $$\mathcal {M}_\mathbf {1}/P(K)$$ M 1 / P ( K ) of the infinite-level Lubin–Tate space $$\mathcal {M}_\mathbf {1}$$ M 1 by the parabolic subgroup $$P(K) \subset \mathrm {GL} _n(K)$$ P ( K ) ⊂ GL n ( K ) of block form $$(n-1,1)$$ ( n - 1 , 1 ) as a perfectoid space, generalizing the results of Ludwig (Forum Math Sigma 5:e17, 41, 2017) to arbitrary n and $$K/{\mathbb {Q}} _p$$ K / Q p finite. For this we prove some perfectoidness results for certain Harris–Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze’s candidate for the mod p Jacquet–Langlands and mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of $$\mathcal {M}_\mathbf {1}/P(K)$$ M 1 / P ( K ) when $$n=2$$ n = 2 , and shows that $$\mathcal {M}_\mathbf {1}/Q(K)$$ M 1 / Q ( K ) is not perfectoid for maximal parabolics Q not conjugate to P.

Utilisation of Bio-binder Made of Homogenised Peat in Crumb Rubber-containing Composites

The utilisation of the industrial residual products to create new value-added materials and to reduce footprint is a modern challenge of science and industry. Development of the new multifunctional and bio-based composites is an excellent opportunity for complex utilisation of industrial residual products. The study describes the preparation and characterisation of the three-phases bio-based composites. The main components are bio-based binder made of peat, devulcanised crumb rubber (DCR) from used tires and part of the fly ash the cenosphere (CS). Three-phase composite prepared in the form of a block for investigation of the mechanical properties and density and a form of granules for determination of the water and oil products sorption was investigated. This work investigated the dependence of the properties on the main component DCR and CS fraction. Is found, that maximum compression strength (in block form) observed for composition without CS and DCR addition - 79.3 MPa, the second highest value of compression strength is 11.2 MPa for composition with 27.3 wt% of CS. For compositions with bio binder content from 17.4 to 55.8 wt% and with DCR contents in range from 11.0 to 62.0 wt% compression strength is in range 1.1 to 2.0 MPa. Liquid sorption analysis (water and diesel) showed that the maximum saturation of liquids in both cases is set after 35 minutes and ranges from 1.05 to 1.4 g&middot;g -1 for water and 0.77 to 1.25 g&middot;g-1 for diesel. It was noted that 90% of the maximum saturation with diesel fuel comes after 10 minutes and for water after 35 minutes.