Multiple Integrals Research Articles

Co-Fermentation of Glucose–Xylose–Cellobiose–XOS Mixtures Using a Synthetic Consortium of Recombinant Saccharomyces cerevisiae Strains

The efficient conversion of cellulosic sugars is vital for the economically viable production of biofuels/biochemicals from lignocellulosic biomass hydrolysates. Based on comprehensive screening, Saccharomyces cerevisiae RC212 was chosen as the chassis strain for multiple integrations of heterologous β-glucosidase and β-xylosidase genes in the present study. The resulting recombinant BLN26 and LF1 form a binary synthetic consortium, and this co-culture system achieved partial fermentation of four sugars (glucose, xylose, cellobiose, and xylo-oligosaccharides). Then, we developed a ternary S. cerevisiae consortium consisting of LF1, BSGIBX, and 102SB. Almost all four sugars were efficiently fermented to ethanol within 24 h, and the ethanol yield is 0.482 g g−1 based on the consumed sugar. To our knowledge, this study represents the first exploration of the conversion of mixtures of glucose, xylose, cellobiose, and xylo-oligosaccharides by a synthetic consortium of recombinant S. cerevisiae strains. This synthetic consortium and subsequent improved ones have the potential to be used as microbial platforms to produce a wide array of biochemicals from lignocellulosic hydrolysates.

Nonlinear Integral Inequalities Involving Tempered Ψ-Hilfer Fractional Integral and Fractional Equations with Tempered Ψ-Caputo Fractional Derivative

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered Ψ-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered Ψ-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered Ψ-Caputo fractional derivative of order 0<α<1, where the right side may depend on time, the solution, or its tempered Ψ-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the Ψ-exponential stability of trivial solutions are proven for these kinds of differential equations.

Simulation and Prediction Algorithm for the Whole Process of Debris Flow Based on Multiple Data Integration

In order to solve the problems of large errors and low accuracy in debris-flow forecasting, the simulation and prediction algorithm for the whole process of debris flow based on multiple data integrations is studied. The middleware method is used to integrate multiple GIS data sets, and the GIS spatial database after multiple data integrations is used to provide the basis of data for the whole process simulation and prediction of debris flow. The spatial cellular simulation model of debris flow is built using the cellular automatic mechanism. The improved kernel principal component analysis method is used to reduce the dimension of debris-flow prediction index data. The reduced dimension index data is input into the support vector machine, and the support vector machine is used to output the prediction results of debris flow in the space cell simulation model of debris flow. Through the simulation visualization technology, the dynamic display of the simulation prediction of the whole process of debris flow is carried out. The experimental results show that the algorithm can realize the simulation of the whole process of debris-flow changes, that the prediction results of debris flow are close to the actual results, and that the error is less than 5%, which improves the prediction accuracy of debris flow and can be used as the auxiliary basis for relevant decision-making departments.

New Clebsch–Gordan‐type integrals involving threefold products of complete elliptic integrals

Multiple elliptic integrals related to the generalized Clebsch–Gordan (CG) integral are of importance in many areas in physics and special functions theory. Zhou has introduced and applied Legendre function‐based techniques to prove symbolic evaluations for integrals of CG form involving twofold and threefold products of complete elliptic integral expressions, and this includes Zhou's remarkable proof of an open problem due to Wan. The foregoing considerations motivate the results introduced in this article, in which we prove closed‐form evaluations for new CG‐type integrals that involve threefold products of the complete elliptic integrals and . Our methods are based on the use of fractional derivative operators, via a variant of a technique we had previously referred to as semi‐integration by parts.

Strong error analysis of Euler methods for overdamped generalized Langevin equations with fractional noise: Nonlinear case

This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the fractional Brownian motion and the singular kernel, which is overcome by means of the Malliavin calculus and fine estimates of several multiple singular integrals. Consequently, these two methods are proved to be strongly convergent with order nearly min{2(H + α − 1), α}, where H ∈ (1/2, 1) and α ∈ (1 − H, 1) respectively characterize the singularity levels of fractional noises and singular kernels in the underlying equation. This result improves the existing convergence order H + α − 1 of Euler methods for the nonlinear case, and gives a positive answer to the open problem raised in Fang and Li [ESAIM Math. Model. Numer. Anal. 54 (2020) 431–463]. As an application of the theoretical findings, we further investigate the complexity of the multilevel Monte Carlo simulation based on the fast Euler method, which turns out to behave better performance than the standard Monte Carlo simulation when computing the expectation of functionals of the considered equation. Finally, numerical experiments are carried out to support the theoretical results.

Concept Images of University Students for Geometric Representation of the Double Integral Concept

This study has focused on how university students understand the geometric representation of double integral. For this purpose, six participants have been asked six questions. Later, semi-structured interviews were conducted with the participants. In this study, the data obtained from questionnaire form and interviews were analyzed with open and axial coding. As a result of this research, it was observed that the concept images of university students were grouped into two categories as “area” and “volume”. It was determined that the participants acted with an intuitive approach without having to establish a relationship between the concept definition and the concept image, the ∬ in the symbol of the double integral caused the participants to think of it as a two-dimensional geometric structure and their image of the concept of the single integral was very active. The findings obtained in this research shows that there are problems in understanding the concept of the double integral, which is the first step of generalizing to multiple integrals, and that educator should produce solutions for this subject.

On isometric embeddability of $S_q^m$ into $S_p^n$ as non-commutative quasi-Banach spaces

The existence of isometric embedding of $S_q^m$ into $S_p^n$, where $1\leq p\neq q\leq \infty$ and $m,n\geq 2$, has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above-mentioned range of $p$ and $q$. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space $\ell _q^m(\mathbb {R})$ into $\ell _p^n(\mathbb {R})$, where $(q,p)\in (0,\infty )\times (0,1)$ and $p\neq q$. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$$\cup \, \{1\}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$$\cup \, \{\infty \}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ and $p\neq q$. Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in [2, \infty )\times (0,1)$. A new tool in our paper is the non-commutative Clarkson's inequality for Schatten class operators. Other tools involved are the Kato–Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

Ladder and zig-zag Feynman diagrams, operator formalism and conformal triangles

We develop an operator approach to the evaluation of multiple integrals for multiloop Feynman massless diagrams. A commutative family of graph building operators Hα for ladder diagrams is constructed and investigated. The complete set of eigenfunctions and the corresponding eigenvalues for the operators Hα are found. This enables us to explicitly express a wide class of four-point ladder diagrams and a general two-loop propagator-type master diagram (with arbitrary indices on the lines) as Mellin-Barnes-type integrals. Special cases of these integrals are explicitly evaluated. A certain class of zig-zag four-point and two-point planar Feynman diagrams (relevant to the bi-scalar D-dimensional “fishnet” field theory and to the calculation of the β-function in ϕ4-theory) is considered. The graph building operators and convenient integral representations for these Feynman diagrams are obtained. The explicit form of the eigenfunctions for the graph building operators of the zig-zag diagrams is fixed by conformal symmetry and these eigenfunctions coincide with the 3-point correlation functions in D-dimensional conformal field theories. By means of this approach, we exactly evaluate the diagrams of the zig-zag series in special cases. In particular, we find a fairly simple derivation of the values for the zig-zag multi-loop two-point diagrams for D = 4. The role of conformal symmetry in this approach, especially a connection of the considered graph building operators with conformal invariant solutions of the Yang-Baxter equation is investigated in detail.

Federal Data for Conducting Patient-centered Outcomes Research on Economic Outcomes.

Patients are increasingly interested in data on the economic burdens and impacts of health care choices; caregivers, employers, and payers are also interested in these costs. Although there have been various federal investments into patient-centered outcomes research (PCOR), an assessment of the coverage and gaps in federally funded data for PCOR economic evaluations has not been produced to date. To classify relevant categories of PCOR economic costs, to assess current federally funded data for coverage of these categories, and to identify gaps for future research and collection. A targeted internet search was conducted to identify a list of relevant outcomes and data sources. The study team assessed data sources for coverage of economic outcomes. A technical panel and key informant interviews were used for evaluation and feedback. Four types of formal health care sector costs, 3 types of informal health care sector costs, and 10 types of non-health care sector costs were identified as relevant for PCOR economic evaluations. Twenty-nine federally funded data sources were identified. Most contained elements on formal costs. Data on informal costs (eg, transportation) were less common, and non-health care sector costs (eg, productivity) were the least common. Most data sources were annual, cross-sectional, nationally representative individual-level surveys. The existing federal data infrastructure captures many areas of the economic burden of health and health care, but gaps remain. Research from multiple data sources and potential future integrations may offset gaps in individual data sources. Linkages are promising strategies for future research on patient-centered economic outcomes.

Correlation functions and polaron-molecule crossover in one-dimensional attractive Fermi gases

We study the correlation functions in a one-dimensional attractive Fermi polaron system of spin-1/2 Fermi gas with one flipped spin as an impurity immersed in the fully polarized spin-up media. By means of the exact Bethe ansatz wave function with string solutions for the momentum, we manage to explicitly deduce the asymptotic form of the correlation functions in the infinite attractive limit and show that the system undergoes a polaron-molecule crossover when the attraction grows. The correlation functions in the infinite attractive limit can be decomposed into the free fermion term with the number of particles reduced by one and the molecular term indicated by the emergence of a sharp peak or dip in the central area of the correlation. These analytical results are checked by the numerical Monte Carlo methods for multiple integrals. The calculation is further extended to the excited states, i.e., the super-Tonks-Girardeau gas, and there is no peak or dip found in the vicinity of the correlation center. Tan contact is found to be proportional to the cubic of the interaction strength and the total energy is verified to satisfy the Tan adiabatic theorem in the strong attraction regime.

A Multifunctional and Highly Adaptable Reporter System for CRISPR/Cas Editing.

CRISPR/Cas systems are some of the most promising tools for therapeutic genome editing. The use of these systems is contingent on the optimal designs of guides and homology-directed repair (HDR) templates. While this design can be achieved in silico, validation and further optimization are usually performed with the help of reporter systems. Here, we describe a novel reporter system, termed BETLE, that allows for the fast, sensitive, and cell-specific detection of genome editing and template-specific HDR by encoding multiple reporter proteins in different open-reading frames. Out-of-frame non-homologous end joining (NHEJ) leads to the expression of either secretable NanoLuc luciferase, enabling a highly sensitive and low-cost analysis of editing, or fluorescent mTagBFP2, allowing for the enumeration and tissue-specific localization of genome-edited cells. BETLE includes a site to validate CRISPR/Cas systems for a sequence-of-interest, making it broadly adaptable. We evaluated BETLE using a defective moxGFP with a 39-base-pair deletion and showed spCas9, saCas9, and asCas12a editing as well as sequence-specific HDR and the repair of moxGFP in cell lines with single and multiple reporter integrants. Taken together, these data show that BETLE allows for the rapid detection and optimization of CRISPR/Cas genome editing and HDR in vitro and represents a state-of the art tool for future applications in vivo.

Interaction effects of multiple input parameters on the integrity of safety instrumented systems with the k‐out‐of‐n redundancy arrangement under uncertainties

AbstractA safety instrumented system (SIS) is extensively used to prevent or reduce risk. The Probability of Failure to perform its intended functions on Demand (PFD) of a practical SIS may vary within multiple safety integrity levels (SIL) due to uncertainties relevant to input parameters. An SIS will be considered to be unsafe when its PFD is greater than a prescribed value, and unsafety probability (UP) is employed to measure the unsafety degree of the investigated SIS in this work. Redundancy architecture is commonly employed to improve the reliability of an SIS. This paper investigates the effects of multiple uncertain input parameters on the UP of an SIS with the k‐out‐of‐n redundancy arrangement. We derive the detailed formulations of the sensitivity for such effects and we also discuss the physical meaning of the proposed sensitivity. An example is employed to demonstrate the proposed sensitivity and the results show that the estimated sensitivity values will be kept to be unchanged with respect to the redundancy when the redundancy is high, whereas the results will vary with the redundancy when the redundancy is low. Meanwhile, we provide a comparison of the results for the truncated uncertain parameters and the non‐truncated uncertain parameters. The results for truncated uncertain parameters and non‐truncated uncertain parameters will approach the same values as the truncated region of the uncertain parameter reduces to zero. The results show that we can approximate the sensitivities of the truncated parameters by the ones of the non‐truncated parameters with less computation time when the truncated region is less than 10−3.

On convergence of form factor expansions in the infinite volume quantum Sinh-Gordon model in 1+1 dimensions

This paper develops a technique allowing one to prove the convergence of a class of series of multiple integrals which corresponds to the form factor expansion of space-like separated two-point functions in the 1+1 dimensional massive integrable Sinh-Gordon quantum field theory.

Universal structures for adaptation in biochemical reaction networks

At the molecular level, the evolution of life is driven by the generation and diversification of adaptation mechanisms. A universal description of adaptation-capable chemical reaction network (CRN) structures has remained elusive until now, since currently-known criteria for adaptation apply only to a tiny subset of possible CRNs. Here we identify the definitive structural requirements that characterize all adaptation-capable collections of interacting molecules, however large or complex. We show that these network structures implement a form of integral control in which multiple independent integrals can collaborate to confer the capacity for adaptation on specific molecules. Using an algebraic algorithm informed by these findings, we demonstrate the existence of embedded integrals in a variety of biologically important CRNs that have eluded previous methods, and for which adaptation has been observed experimentally. This definitive picture of biological adaptation at the level of intermolecular interactions represents a blueprint for adaptation-capable signaling networks across all domains of life, and for the design of synthetic biosystems.

Order estimate of functionals related to fractional Brownian motion

Nualart and Yoshida (2019) presented a general scheme for asymptotic expansion of Skorohod integrals. However, when applying the general theory to a variation of a process related to a fractional Brownian motion, one repeatedly needs estimates of the order of Lp-norms and Sobolev norms of functionals that are complicated as a randomly weighted sum of products of multiple integrals of the fractional Brownian motion. To resolve the difficulties, in this paper, we construct a theory of exponents based on a graphical representation of the structure of the functionals, which enables us to tell an upper bound of the order of functionals by a simple rule. We also show how the exponents change by the actions of the Malliavin derivative and its projection. The exponents are useful in various studies of limit theorems as well as in applications to asymptotic expansion.

Bayesian instrumental variable estimation in linear measurement error models

AbstractIn this article, we study the problem of parameter estimation for measurement error models by combining the Bayes method with the instrumental variable approach, deriving the posterior distribution of parameters under different priors with known and unknown variance parameters, respectively, and calculating the Bayes estimator (BE) of the parameters under quadratic loss. However, it is difficult to obtain an explicit expression for BE because of the complex multiple integrals involved. Therefore, we adopt the linear Bayes method, which does not specify the form of the prior and avoids these complicated integral calculations, to obtain an expression for the linear Bayes estimator (LBE) for different priors. We prove that this LBE is superior to the two‐stage least squares estimator under the mean squared error matrix criterion. Numerical simulations show that our LBE is very close to the real parameter whether the variance parameters are known or unknown, and it gradually approaches BE as the sample size increases. Our results indicate that this instrumental variable approach is valid for measurement error models.

Some New Hermite–Hadamard Type Inequalities Pertaining to Generalized Multiplicative Fractional Integrals

There is significant interaction between the class of symmetric functions and other types of functions. The multiplicative convex function class, which is intimately related to the idea of symmetry, is one of them. In this paper, we obtain some new generalized multiplicative fractional Hermite–Hadamard type inequalities for multiplicative convex functions and for their product. Additionally, we derive a number of inequalities for multiplicative convex functions related to generalized multiplicative fractional integrals utilising a novel identity as an auxiliary result. We provide some examples for the appropriate selections of multiplicative convex functions and their graphical representations to verify the authenticity of our main results.

Hardy-Leindler Type Inequalities for Multiple Integrals on Time Scales

Abstract Hardy-Leindler type inequalities and their converses for multiple integrals on time scales are proved by using Fubini’s theorem and induction principle. Some generalized versions of Hardy, Wirtinger and Leindler inequalities in both continuous and discrete cases are also derived in seek of applications.

A wavelet method for large-deflection bending of irregular plates

A very rare sixth-order multiresolution method based on wavelet theory for solving irregular domain problems is proposed and applied to the large-deflection bending of irregular plates. High accuracy approximation of multiple integrals of functions defined in the irregular domain is realized by suggesting a new method of boundary extension of functions, which is associated with a boundary node removal technique to avoid singular matrices in the integration process. Each derivative of the unknown function is defined as a new function in the solution of the governing differential equations of the problem. The original equations containing derivatives of different orders can then be transformed into a system of algebraic equations with only discrete nodal values of the highest-order derivative, based on integral relations between these functions and excellent accuracy features on the wavelet approximation of multiple integrals. Various boundary conditions for irregular domains can be automatically and almost exactly included in the integration operation. The proposed method is applied to the solution of large-deflection bending of various irregular plates. The displacement and stress results demonstrate that the accuracy of the proposed method is still as high as the rare sixth order even when the problems are strongly nonlinear and defined in irregular regions, showing a much better accuracy and efficiency than those of classical methods such as the finite element method.

Singular multiple integrals and nonlinear potentials

We derive sharp partial regularity criteria of nonlinear potential theoretic nature for the Lebesgue-Serrin-Marcellini extension of nonhomogeneous singular multiple integrals featuring (p,q)-growth conditions.