Product Formula Research Articles

Application of Citrus maxima Burm. Essential Oil in Shower Gel Products

Nowadays, the trend of cosmetics containing natural ingredients is becoming one of the most popular trends in the beauty industries. More and more people are interested and looking for cosmetic products made from natural ingredients such as herbs, essential oils, fruits, vegetables, etc. Essential oils are widely incorporated in cosmetic, perfumery and related household products due to their diverse properties but mainly due to their pleasant aroma and biological activity. Citrus maxima essential oil contains D-limonene as the main compound (85-95%) and other compounds such as α-pinene, sabinene, β-mycrene, α-phellandrene, σ-cymene, β-ocimene, γ- terpinene, β-caryophyllen, β-copaene. These compounds further diversify the applications of the added product. In this study, the factors affecting the production process of Citrus maxima essential oil shower gel including main detergents, moisturizers, emulsifiers, preservatives were investigated to propose production formulas. The results showed that the main detergent accounted for 12%, co- detergent 2.0%, emulsifier 1.5%, foaming agent 7.0%, surfactant, emulsifier 3.0%, moisturizer 2.0%, preservative management 2.0% and Citrus maxima essential oil account 1.0% in the formula. The evaluations of the basic physico-chemical properties are in the permissible range. Thus, it shows that the shower gel products from Citrus maxima essential oil can be produced and marketed.

Some product formulas

Some product formulas

Perverse filtrations, Chern filtrations, and refined BPS invariants for local [formula omitted

We explore connections between three structures associated with the cohomology of the moduli of 1-dimensional stable sheaves on P2: perverse filtrations, tautological classes, and refined BPS invariants for local P2. We formulate the P=C conjecture identifying the perverse filtration with the Chern filtration for the free part of the cohomology. This can be viewed as an analog of de Cataldo–Hausel–Migliorini's P=W conjecture for Hitchin systems. Our conjecture is compatible with the enumerative invariants of local P2 calculated by refined Pandharipande–Thomas theory or Nekrasov partition functions. It provides a cohomological lift of a conjectural product formula of the asymptotic refined BPS invariants. We prove the P=C conjecture for degrees ≤4.

Skew Howe duality and limit shapes of Young diagrams

AbstractWe consider the skew Howe duality for the action of certain dual pairs of Lie groups on the exterior algebra as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe duality for the pairs , , , and using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The ‐representation multiplicity is given as a determinant formula using the Lindström–Gessel–Viennot lemma and as a product formula. These admit natural ‐analogs that we show equals the ‐dimension of a ‐representation (up to an overall factor of ), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at ), we take the infinite rank limit and prove that the diagrams converge uniformly to the limit shape.

Geometric error calibration and analysis of rotary axes on a five-axis dispensing machine

Geometric rotary axis errors are one of the main factors affecting the dispensing positioning accuracy of five-axis vision dispensing machines. Hence, the accurate detection and compensation of these error terms is an important and challenging problem. In this study, a decoupling identification method for both position-dependent geometric errors and position-independent geometric errors of a rotary dispensing table is proposed based on vision measurement technology. According to this strategy, the screw theory and exponential product formula are used to establish a kinematic model of the end-effector, followed by the definition of the mapping relationship between position-independent geometric errors and axis posture. Then, to separate the coupling error terms of the rotating axes, a step-by-step identification strategy is proposed based on the least squares method. Furthermore, a coordinate search-based error compensation algorithm is developed, which avoids the inverse singular value problem and is insensitive to the error model. Finally, the effectiveness of the proposed identification algorithm and compensation algorithm is experimentally validated.

Nature-convergent Principles and Rational Use of Resources in Water Treatment and Processing

New ways justified to improve technological nature-convergent principles and schemes for the rational use of resources in water treatment and processing. Main principles for the rational use of resources, nature-convergency and ressources substitution were developped. The impact of ultrasound on the production of highly reactive sodium hypochlorite was revealed. The know-how of an active chlorine-containing reagent from underground mineral waters was studied and its effectiveness in disinfecting natural and waste waters of pulp and paper and fish processing industries was proven. A new formula for fertile soil production based on tap water sediment was developed.

Sucrose, Lactose, Thermogravimetry, and Differential Thermal Analysis: The Estimation of the Moisture Bond Types in Lactose-Containing Ingredients for Confectionery Products with Reduced Glycemic Index.

This research is aimed at conducting a comparative analysis of the dehydration process of lactose-containing ingredients and sucrose. Differential scanning calorimetry (DSC) and thermogravimetry (TG) methods were used to investigate the moisture bond types and the process of thermal degradation in native whey, dried skimmed milk, dried whole milk, dried milk whey demineralized, purified lactose, and sucrose. There were several peaks on the DSC and TG curves for lactose-containing ingredients. They determine the loss of the physically absorbed water on surfaces, physically occluded water and hydrate-forming water (up to 180°C), anomerization of lactose (160-220°C), and melting followed by decomposition (above 230°C). The multiple peaks on the dDSC curves from 135 to 170°C indicate the course of the Maillard reaction in the mix with proteins. For the native whey, the amount of chemically bound water was 56.11 ± 1%; for the dried whole milk, it was 59.86 ± 1%; for the milk whey demineralized, it was 64.56 ± 1%; and for the dried skimmed milk, it was67.17 ± 1%. The results are presented as mean values ± SD (n = 3) and were considered statistically significant when P < 0.05. The similar to some extent form of the DSC and TG curves to lactose-containing ingredients and granulated sugar confirms the possibility of using them as the component of bakery and confectionery product formulas to reduce the glycemic index and improve organoleptic properties of products as well as correct their mineral compositions. Considering the constant changing of technologies for obtaining various dairy ingredients as well as the occurrence of their new kinds, such studies contribute to the expansion of existing knowledge and reference data.

Higher cup products on hypercubic lattices: Application to lattice models of topological phases

In this paper, we derive the explicit formula for higher cup products on hypercubic lattices based on the recently developed geometrical interpretation on the simplicial complexes. We illustrate how this formalism can elucidate lattice constructions on hypercubic lattices for various models and derive them from spacetime actions. In particular, we demonstrate explicitly that the (3 + 1)D SPT S=12∫w22+w14 (where w1 and w2 are the first and second Stiefel-Whitney classes) is dual to the 3-fermion Walker-Wang model constructed on the cubic lattice. Other examples include the double-semion model and also the “fermionic” toric code in arbitrary dimensions on hypercubic lattices. In addition, we extend previous constructions of exact boson-fermion dualities and the Gu-Wen Grassmann integral to arbitrary dimensions. Another result that may be of independent interest is a derivation of a cochain-level action for the generalized double-semion model, reproducing a recently derived action on the cohomology level.

Monk rules for type GLn Macdonald polynomials

In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta [6], but our method of derivation is to use Cherednik's intertwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip [19]. Specializations of the Monk formula's at q=0 and/or t=0 provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials.

Applications of Euler Sums and Series Involving the Zeta Functions

A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis’s infinite product formula for π, Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums.

Generation of a coherent squeezed like state defined with the Lie--Trotter product formula using a nonlinear photonic crystal

Abstract In this paper, we investigate how to generate coherent squeezed like light using a nonlinear photonic crystal. Because the photonic crystal reduces the group velocity of the incident light, if it is composed of a material with a second-order nonlinear optical susceptibility $\chi^{(2)}$, the interaction between the nonlinear material and the light passing through it strengthens and the quantum state of the emitted light is largely squeezed. Thus, we can generate a coherent squeezed like light with a resonating cavity in which the nonlinear photonic crystal is placed. This coherent squeezed like state is defined with the Lie--Trotter product formula and its mathematical expression is different from those of conventional squeezed coherent states. We show that we can obtain this coherent squeezed like state with a squeezing level $15.9$ dB practically by adjusting physical parameters for our proposed method. Feeding the squeezed light whose average number of photons is given by one or two into a beam splitter and splitting the flow of the squeezed light into a pair of entangled light beams, we estimate their entanglement quantitatively.&amp;#xD;This paper is a sequel to H.~Azuma, J. Phys. D: Appl. Phys. \textbf{55}, 315106 (2022).

Effects of coexisting zinc and nickel ions on crystallization behaviour and properties of spinel synthesised in chromium-containing wastewater

ABSTRACT Electroplating wastewater contains chromium ions, zinc ions, and nickel ions due to the emergence of a multi-layer plating process and unified treatment of mixed wastewater. The effects of coexisting zinc and nickel ions on crystallization behaviour and properties of spinel synthesised in chromium-containing wastewater by ferrite method were investigated in the study. The results indicated that the quality of the supernatant and the stability of synthetic products were barely affected. However, the magnetic property of the synthetic products was significantly increased when the concentration ratio of zinc and nickel ions was appropriate. With the extension of stirring time, the stability of the synthetic product increased, but the crystallinity gradually decreased. Chromium and zinc ions entered the synthetic products preferentially over nickel ions during ferrite synthesis. In addition, the morphology of the synthetic products was irregular and dense, and the lustre distribution was uniform. The distributions of iron, chromium, zinc, nickel and oxygen were overlapped, owing to a single spinel, and the molecular formula of synthetic products was speculated as (3.53 < x < 3.99, 0.2 < y < 0.36, 0.04 < z < 0.05, 0.03 < n < 0.05). Finally, the transport model of metal ions in the mixed wastewater was established, and the synthesis mechanism was revealed.

Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation

Simulating the time-evolution of a Hamiltonian is one of the most promising applications of quantum computers. Multi-Product Formulas (MPFs) are well suited to replace standard product formulas since they scale better with respect to time and approximation errors. Hamiltonian simulation with MPFs was first proposed in a fully quantum setting using a linear combination of unitaries. Here, we analyze and demonstrate a hybrid quantum-classical approach to MPFs that classically combines expectation values evaluated with a quantum computer. This has the same approximation bounds as the fully quantum MPFs, but, in contrast, requires no additional qubits, no controlled operations, and is not probabilistic. We show how to design MPFs that do not amplify the hardware and sampling errors, and demonstrate their performance. In particular, we illustrate the potential of our work by theoretically analyzing the benefits when applied to a classically intractable spin-boson model, and by computing the dynamics of the transverse field Ising model using a classical simulator as well as quantum hardware. We observe an error reduction of up to an order of magnitude when compared to a product formula approach by suppressing hardware noise with Pauli Twirling, pulse efficient transpilation, and a novel zero-noise extrapolation based on scaled cross-resonance pulses. The MPF methodology reduces the circuit depth and may therefore represent an important step towards quantum advantage for Hamiltonian simulation on noisy hardware.

Local well-posedness of VPFP in hybrid modulation-Lebesgue space

In this paper, we are concerned about the local well-posedness of Vlasov-Poisson-Fokker-Planck (VPFP) in modulation-Lebesgue space for large initial data. To accomplish this goal, we establish convolution inequalities associated with the fundamental solution to Fokker-Planck equation. The product formula and off-diagonal estimates involving the electronic term are established as well.

Degenerations and multiplicity-free formulas for products of $$\psi $$ and $$\omega $$ classes on $$\overline{M}_{0,n}$$

Degenerations and multiplicity-free formulas for products of $$\psi $$ and $$\omega $$ classes on $$\overline{M}_{0,n}$$

Quantum simulation of quantum mechanical system with spatial noncommutativity

Quantum simulation has become a promising avenue of research that allows one to simulate and gain insight into the models of high energy physics whose experimental realizations are either complicated or inaccessible with current technology. We demonstrate the quantum simulation of such a model, a quantum mechanical system with spatial noncommutativity, which is inspired by the works in noncommutative geometry and noncommutative field theory for a universal quantum computer. We use the novel group theoretical formalism to map the Hamiltonian of such a noncommutative quantum system into the ordinary quantum mechanical Hamiltonian and then carry out the quantum simulation using the Trotter–Suzuki product formula. Furthermore, we distinguish the impact of the noncommutativity parameter on the quantum simulation, especially on the Trotter error, and point out how its sizable value affects the simulation.

On the [formula omitted]-enumeration of barely set-valued tableaux and plane partitions

Barely set-valued tableaux are a variant of Young tableaux in which one box contains two numbers as its entry. It has recently been discovered that there are product formulas enumerating certain classes of barely set-valued tableaux. We give some q-analogs of these product formulas by introducing a version of major index for these tableaux. We also give product formulas and q-analogs for barely set-valued plane partitions. Many of the results are stated in the generality of P-partitions that then specialize to particularly nice formulas for rectangles and minuscule posets. The proofs use several probability distributions on the set of order ideals of a poset, depending on the real parameter q>0, which we think could be of independent interest.

Productivity Formula of Horizontal Well in Low-Permeability Gas Reservoir considering Multiple Factors

For horizontal well in low-permeability gas reservoir, the effects of threshold pressure gradient, stress sensitivity, and gas slippage have significant impacts on the well productivity. At present, there are few productivity formulas for horizontal gas well considering all these factors. In this paper, based on the flow analysis of horizontal well in low-permeability gas reservoir, the whole flow field was divided into two parts, namely, far wellbore region and near wellbore region, among which the far wellbore region is composed of plane linear flow region and plane radial flow region and the near wellbore region is composed of vertical plane radial flow region and spherical plane central flow region. Then, a new productivity formula is established based on the steady-state seepage theory and the equivalent seepage resistance method, with the consideration of threshold pressure gradient, stress sensitivity, and gas slippage effect. The accuracy of the formula in this paper is verified through comparing with other classical models, and the influence of various factors on the well productivity is analyzed. The analysis results show that stress sensitivity has the most significant effect on horizontal gas well production, followed by threshold pressure gradient, and the gas slippage has the least effect. With the consideration of all influencing factors, the higher the formation pressure and reservoir thickness, the higher the productivity, and the increase of productivity increases with the increase of flow pressure difference. The increasing trend of gas productivity index per meter with the increase of reservoir permeability is first fast and then slow. When the reservoir permeability is greater than 1.2 mD, the increment of gas productivity index per meter (MGPI) decreases. When the length of horizontal well is greater than 1400 m, the increment of gas productivity index per meter decreases with the increase of gas reservoir thickness. Therefore, it is recommended to control the horizontal well length within 1400 m in low-permeability gas reservoir. In addition, the absolute open flow charts corresponding to reservoir thickness and horizontal well length under different reservoir permeability conditions were also given, which can provide theoretical guidance for the selection of horizontal well length during the development of low-permeability gas reservoirs.

REGRESSION ANALYSIS OF ENTREPRENEURIAL BEHAVIOUR OF SWEET ORANGE GROWERS

A sweet orange (Citrus sinensis L.) is an important fruit crop of Maharashtra state of India. It is grown on an area of 55,000 ha. The area under this fruit crop is increasing rapidly as a result of dynamic employment guarantee scheme launched by the Government of Maharashtra for fruit crops. Sweet Orange is a major fruit crop in Marathwada region of the State. 0.77 lakh hectares area is under this crop (Anonymous, 2014a). The development of any nation depends primarily on the important role played by entrepreneurs. Hence, the part played by entrepreneurs is of vital importance in a developing country like India. Thus in all economic development activities more attention is being given to entrepreneurship development. An entrepreneur is primarily concerned with changes in the formula of production over which he has full control. Further, it is commonly believed that an entrepreneur is basically an intelligent person and has a definite ability to create something new to prove its worthiness. The Ex-post-facto research design was used for the present study. The study was conducted in Aurangabad district. Three tahsils namely, Aurangabad, Paithan and Kannad were selected purposively on the basis of the maximum area under sweet orange cultivation. The main objective of this study is to study the regression analysis of entrepreneurial behaviour of sweet orange growers. It is revealed from that coefficient of determination (R2) of the independent variables was 0.5040. It means that 50.40 per cent of the total variation in entrepreneurial behavior of sweet orange growers was explained by the selected 10 independent variables i.e. Farming experience, Education, Family size, Occupation, Land holding, Annual income, Social participation, Mass media use, Economic motivation and Market orientation. The present study was used as a multistage sampling procedure. Collected data were classified, tabulated and analyzed by using multiple regression.

Successive vertex orderings of fully regular graphs

A graph G=(V,E) is called fully regular if for every independent set I⊂V, the number of vertices in V∖I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph.As an application of our results, we give alternative proofs of two theorems of Stanley and Gao & Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph.As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.