Optimisation of the ultrasound-assisted extraction of betalains and polyphenols from Amaranthus hypochondriacus var. Nutrisol

The present study optimised the ultrasound-assisted extraction (UAE) of bioactive compounds from Amaranthus hypochondriacus var. Nutrisol. Influence of temperature (25.86–54.14&nbsp;°C) and ultrasonic power densities (UPD) (76.01–273.99 mW/mL) on total betalains (BT), betacyanins (BC), betaxanthins (BX), total polyphenols (TP), antioxidant activity (AA), colour parameters (L*, a*, and b*), amaranthine (A), and isoamaranthine (IA) were evaluated using response surface methodology. Moreover, betalain extraction kinetics and mass transfer coefficients (K<sub>Lsub>a) were determined for each experimental condition. BT, BC, BX, TP, AA, b*, K<sub>Lsub>a, and A were significantly affected (p&nbsp;<&nbsp;0.05) by temperature extraction and UPD, whereas L*, a*, and IA were only affected (p&nbsp;<&nbsp;0.05) by temperature. All response models were significantly validated with regression coefficients (R<sup>2sup>) ranging from 87.46 to 99.29%. BT, A, IA, and K<sub>Lsub>a in UAE were 1.38, 1.65, 1.50, and 29.93 times higher than determined using conventional extraction, respectively. Optimal UAE conditions were obtained at 41.80&nbsp;°C and 188.84 mW/mL using the desired function methodology. Under these conditions, the experimental values for BC, BX, BT, TP, AA, L*, a*, b*, K<sub>Lsub>a, A, and IA were closely related to the predicted values, indicating the suitability of the developed quadratic models. This study proposes a simple and efficient UAE method to obtain betalains and polyphenols with high antioxidant activity, which can be used in several applications within the food industry.     

Pure tetrahedral quadruple systems with index two

An oriented tetrahedron defined on four vertices is a set of four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order $n$ with index $\lambda$, denoted by , is a pair , where $X$ is an $n$‐set and is a set of oriented tetrahedra (blocks) such that every cyclic triple on $X$ is contained in exactly $\lambda$ members of . A is pure if there do not exist two blocks with the same vertex set. When $\lambda =1$, the spectrum of a pure TQS has been completely determined by Ji. In this paper, we show that there exists a pure if and only if and $n\ge 7$. A corollary is that a simple also exists if and only if and $n\ge 7$.     

The Chebotarev density theorem for function fields — Incomplete intervals

We prove a Pólya-Vinogradov type variation of the Chebotarev density theorem for function fields over finite fields valid for “incomplete intervals” s="true">s="true">s="true">Fs="true">s="true">psub><script type="math/mml" id="MathJax-Element-1">s="true">s="true">s="true">Fs="true">s="true">psub>script>, provided s="true">s="true">s="true">ps="true">s="true">1stretchy="false" is="true">/s="true">2sup>s="true">logs="true">⁡s="true">pstretchy="false" is="true">)stretchy="false" is="true">/stretchy="false" is="true">|s="true">Istretchy="false" is="true">|style="after" is="true">=s="true">ostretchy="false" is="true">(s="true">1stretchy="false" is="true">)<script type="math/mml" id="MathJax-Element-2">s="true">s="true">s="true">ps="true">s="true">1stretchy="false" is="true">/s="true">2sup>s="true">logs="true">⁡s="true">pstretchy="false" is="true">)stretchy="false" is="true">/stretchy="false" is="true">|s="true">Istretchy="false" is="true">|style="after" is="true">=s="true">ostretchy="false" is="true">(s="true">1stretchy="false" is="true">)script>. Applications include density results for irreducible trinomials in s="true">s="true">s="true">Fs="true">s="true">psub>stretchy="false" is="true">[s="true">xstretchy="false" is="true">]<script type="math/mml" id="MathJax-Element-3">s="true">s="true">s="true">Fs="true">s="true">psub>stretchy="false" is="true">[s="true">xstretchy="false" is="true">]script>, i.e. the number of irreducible polynomials in the set s="true">s="true">stretchy="false" is="true">{s="true">fstretchy="false" is="true">(s="true">xstretchy="false" is="true">)style="after" is="true">=sup is="true">s="true">s="true">xs="true">s="true">dsup>style="after" is="true">+sub is="true">s="true">s="true">as="true">s="true">1sub>s="true">xstyle="after" is="true">+sub is="true">s="true">s="true">as="true">s="true">0sub>s="true">∈sub is="true">s="true">s="true">Fs="true">s="true">psub>stretchy="false" is="true">[s="true">xstretchy="false" is="true">]stretchy="false" is="true">}s="true">sub is="true">s="true">s="true">as="true">s="true">0sub>s="true">∈sub is="true">s="true">s="true">Is="true">s="true">0sub>s="true">,sub is="true">s="true">s="true">as="true">s="true">1sub>s="true">∈sub is="true">s="true">s="true">Is="true">s="true">1sub>sub><script type="math/mml" id="MathJax-Element-4">s="true">s="true">stretchy="false" is="true">{s="true">fstretchy="false" is="true">(s="true">xstretchy="false" is="true">)style="after" is="true">=sup is="true">s="true">s="true">xs="true">s="true">dsup>style="after" is="true">+sub is="true">s="true">s="true">as="true">s="true">1sub>s="true">xstyle="after" is="true">+sub is="true">s="true">s="true">as="true">s="true">0sub>s="true">∈sub is="true">s="true">s="true">Fs="true">s="true">psub>stretchy="false" is="true">[s="true">xstretchy="false" is="true">]stretchy="false" is="true">}s="true">sub is="true">s="true">s="true">as="true">s="true">0sub>s="true">∈sub is="true">s="true">s="true">Is="true">s="true">0sub>s="true">,sub is="true">s="true">s="true">as="true">s="true">1sub>s="true">∈sub is="true">s="true">s="true">Is="true">s="true">1sub>sub>script> is s="true">s="true">s="true">Is="true">s="true">0sub>stretchy="false" is="true">|s="true">⋅stretchy="false" is="true">|sub is="true">s="true">s="true">Is="true">s="true">1sub>stretchy="false" is="true">|stretchy="false" is="true">/s="true">d<script type="math/mml" id="MathJax-Element-5">s="true">s="true">s="true">Is="true">s="true">0sub>stretchy="false" is="true">|s="true">⋅stretchy="false" is="true">|sub is="true">s="true">s="true">Is="true">s="true">1sub>stretchy="false" is="true">|stretchy="false" is="true">/s="true">dscript> provided s="true">s="true">s="true">Is="true">s="true">0sub>stretchy="false" is="true">|style="after" is="true">>sup is="true">s="true">s="true">ps="true">s="true">1stretchy="false" is="true">/s="true">2style="after" is="true">+s="true">ϵsup><script type="math/mml" id="MathJax-Element-6">s="true">s="true">s="true">Is="true">s="true">0sub>stretchy="false" is="true">|style="after" is="true">>sup is="true">s="true">s="true">ps="true">s="true">1stretchy="false" is="true">/s="true">2style="after" is="true">+s="true">ϵsup>script>, s="true">s="true">s="true">Is="true">s="true">1sub>stretchy="false" is="true">|style="after" is="true">>sup is="true">s="true">s="true">ps="true">s="true">ϵsup><script type="math/mml" id="MathJax-Element-7">s="true">s="true">s="true">Is="true">s="true">1sub>stretchy="false" is="true">|style="after" is="true">>sup is="true">s="true">s="true">ps="true">s="true">ϵsup>script>, or s="true">s="true">s="true">Is="true">s="true">1sub>stretchy="false" is="true">|style="after" is="true">>sup is="true">s="true">s="true">ps="true">s="true">1stretchy="false" is="true">/s="true">2style="after" is="true">+s="true">ϵsup><script type="math/mml" id="MathJax-Element-8">s="true">s="true">s="true">Is="true">s="true">1sub>stretchy="false" is="true">|style="after" is="true">>sup is="true">s="true">s="true">ps="true">s="true">1stretchy="false" is="true">/s="true">2style="after" is="true">+s="true">ϵsup>script>, s="true">s="true">s="true">Is="true">s="true">0sub>stretchy="false" is="true">|style="after" is="true">>sup is="true">s="true">s="true">ps="true">s="true">ϵsup><script type="math/mml" id="MathJax-Element-9">s="true">s="true">s="true">Is="true">s="true">0sub>stretchy="false" is="true">|style="after" is="true">>sup is="true">s="true">s="true">ps="true">s="true">ϵsup>script>, and similarly when s="true">s="true">s="true">xs="true">s="true">dsup><script type="math/mml" id="MathJax-Element-10">s="true">s="true">s="true">xs="true">s="true">dsup>script> is replaced by any monic degree d polynomial in s="true">s="true">s="true">Fs="true">s="true">psub>stretchy="false" is="true">[s="true">xstretchy="false" is="true">]<script type="math/mml" id="MathJax-Element-11">s="true">s="true">s="true">Fs="true">s="true">psub>stretchy="false" is="true">[s="true">xstretchy="false" is="true">]script>. Under the above assumptions we can also determine the distribution of factorization types, and find it to be consistent with the distribution of cycle types of permutations in the symmetric group s="true">s="true">s="true">Ss="true">s="true">dsub><script type="math/mml" id="MathJax-Element-12">s="true">s="true">s="true">Ss="true">s="true">dsub>script>.     

Uniqueness in Harper’s vertex-isoperimetric theorem

For a set s="true">s="true">s="true">Qs="true">s="true">nsub>style="after" is="true">=sup is="true">s="true">se="}" open="{" is="true">s="true">s="true">0s="true">,s="true">1s="true">s="true">nsup> the -neighbourhood of is s="true">s="true">s="true">Ns="true">s="true">tsup>se=")" open="(" is="true">s="true">s="true">Astyle="after" is="true">=se="}" open="{" is="true">s="true">s="true">xspace width="0.16667em" is="true">space>s="true">:space width="0.16667em" is="true">space>s="true">dse=")" open="(" is="true">s="true">s="true">xs="true">,s="true">As="true">≤s="true">t, where denotes the usual graph distance on s="true">s="true">s="true">Qs="true">s="true">nsub>. Harper’s vertex-isoperimetric theorem states that among the subsets s="true">s="true">s="true">Qs="true">s="true">nsub> of given size, the size of the -neighbourhood is minimised when is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if s="true">s="true">s="true">Qs="true">s="true">nsub> is a subset of given size for which the sizes of both s="true">s="true">s="true">Ns="true">s="true">tsup>se=")" open="(" is="true">s="true">s="true">A and s="true">s="true">s="true">Ns="true">s="true">tsup>se=")" open="(" is="true">s="true">sup is="true">s="true">s="true">As="true">s="true">csup> are minimal for all , does it follow that is isomorphic to an initial segment of the simplicial order?Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e.&nbsp;a set with s="true">s="true">s="true">xs="true">,s="true">rstyle="after" is="true">⊆s="true">Astyle="after" is="true">⊆s="true">Bse=")" open="(" is="true">s="true">s="true">xs="true">,s="true">rs="true">+s="true">1 for some s="true">s="true">s="true">Qs="true">s="true">nsub>. We go further to classify all the sets s="true">s="true">s="true">Qs="true">s="true">nsub> for which the sizes of both s="true">s="true">s="true">Ns="true">s="true">tsup>se=")" open="(" is="true">s="true">s="true">A and s="true">s="true">s="true">Ns="true">s="true">tsup>se=")" open="(" is="true">s="true">sup is="true">s="true">s="true">As="true">s="true">csup> are minimal for all among the subsets of s="true">s="true">s="true">Qs="true">s="true">nsub> of given size. We also prove that, perhaps surprisingly, if s="true">s="true">s="true">Qs="true">s="true">nsub> for which the sizes of s="true">s="true">s="true">A and s="true">s="true">sup is="true">s="true">s="true">As="true">s="true">csup> are minimal among the subsets of s="true">s="true">s="true">Qs="true">s="true">nsub> of given size, then the sizes of both s="true">s="true">s="true">Ns="true">s="true">tsup>se=")" open="(" is="true">s="true">s="true">A and s="true">s="true">s="true">Ns="true">s="true">tsup>se=")" open="(" is="true">s="true">sup is="true">s="true">s="true">As="true">s="true">csup> are also minimal for all among the subsets of s="true">s="true">s="true">Qs="true">s="true">nsub> of given size. Hence the same classification also holds when we only require s="true">s="true">s="true">A and s="true">s="true">sup is="true">s="true">s="true">As="true">s="true">csup> to have minimal size among the subsets s="true">s="true">s="true">Qs="true">s="true">nsub> of given size.     

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.     

Nonnegative solutions to reaction–diffusion system with cross-diffusion and nonstandard growth conditions

We establish the existence of nonnegative weak solutions to nonlinear reaction–diffusion system with cross-diffusion and nonstandard growth conditions subject to the homogeneous Neumann boundary conditions. We assume that the diffusion operators satisfy certain monotonicity condition and nonstandard growth conditions and prove that the existence of weak solutions using Galerkin's approximation technique.     

安培环路定理的拓扑学描述

近年来拓扑学在量子力学中得到了广泛的运用.本文将安培环路定理积分式重新表达为一矢量场在轮胎参数面上的第一类陈数积分.数值模拟展示了该积分值为一整数即第一陈数,其代表矢量场的整体性质:当经历连续变换时,矢量场的局部数值发生改变但整体积分值即陈数仍保持不变;若陈数发生改变,则表明矢量场变换的连续性条件发生破坏,矢量场出现奇点.进一步通过高斯映射将该矢量场从参数轮胎面映射到单位球面上,并给出了第一陈数的直观几何意义.理论和数值结果揭示了安培环路定理的拓扑学本质,表明拓扑概念在经典物理学中也会有广泛应用.     

Can Anti-Aufbau DFT Calculations Estimate Singlet Excited State Aromaticity? Correspondence on “Dibenzoarsepins: Planarization of 8π-Electron System in the Lowest Singlet Excited State”

The simple anti-aufbau DFT approach for estimating singlet excited state aromaticity suggested in a recent Communication published in this journal is shown to produce incorrect results because it targets a linear combination of the singlet and triplet configurations involving the HOMO and LUMO rather than the first singlet excited state. If the S<sub>1sub> state of a molecule is dominated by the HOMO→LUMO excitation, a comparably simple but theoretically consistent and qualitatively correct approximation to the S<sub>1sub> wavefunction can be achieved by performing a small “two electrons in two orbitals” CASSCF(2,2) calculation which can be followed by the evaluation of magnetic aromaticity criteria such as NICS.     

Random walk Metropolis algorithm in high dimension with non-Gaussian target distributions

High-dimensional asymptotics of the random walk Metropolis–Hastings algorithm are well understood for a class of light-tailed target distributions. Although this idealistic assumption is instructive, it may not always be appropriate, especially for complicated target distributions. We here study heavy-tailed target distributions for the random walk Metropolis algorithms. When the number of dimensions is <svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.216ex" height="1.971ex" viewBox="0 -747.2 523.5 848.5" role="img" focusable="false" style="vertical-align: -0.235ex;">stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">s="true">se xlink:href="#MJMATHI-64">se>svg><script type="math/mml" id="MathJax-Element-1">script>, the rate of consistency is s="true">s="true">s="true">ds="true">s="true">2sup><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.272ex" height="2.432ex" viewBox="0 -945.9 978.4 1047.3" role="img" focusable="false" style="vertical-align: -0.235ex;">stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">s="true">s="true">s="true">se xlink:href="#MJMATHI-64">se>s="true" transform="translate(524,421)">s="true">se transform="scale(0.707)" xlink:href="#MJMAIN-32">se>svg><script type="math/mml" id="MathJax-Element-2">s="true">s="true">s="true">ds="true">s="true">2sup>script> and the calculation cost is s="true">s="true">s="true">ds="true">s="true">3sup>s="true">)<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.563ex" height="3.24ex" viewBox="0 -945.9 2825.5 1395" role="img" focusable="false" style="vertical-align: -1.043ex;">stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)">s="true">se xlink:href="#MJMATHI-4F">se>s="true" transform="translate(930,0)">se xlink:href="#MJSZ1-28" is="true">se>s="true" transform="translate(458,0)">s="true">s="true">se xlink:href="#MJMATHI-64">se>s="true" transform="translate(524,421)">s="true">se transform="scale(0.707)" xlink:href="#MJMAIN-33">se>se xlink:href="#MJSZ1-29" is="true" x="1436" y="-1">se>svg><script type="math/mml" id="MathJax-Element-3">s="true">s="true">s="true">ds="true">s="true">3sup>s="true">)script>, which might be too expensive in high dimension.     

Boundedness of the maximal and potential operators in Herz-Morrey type spaces

Our aim in this paper is to discuss the boundedness of the maximal and potential operators in the central Herz-Morrey type space <span class="NLM_disp-formula-image inline-formula">script>src="/na101/home/literatum/publisher/tandf/journals/content/gcov20/2020/gcov20.v065.i09/17476933.2019.1669571/20200717/images/gcov_a_1669571_ilm0001.gif" alt="" />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gcov20/2020/gcov20.v065.i09/17476933.2019.1669571/20200717/images/gcov_a_1669571_ilm0001.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula inline-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />span> for the double phase functional φ such that <span class="NLM_disp-formula-image disp-formula">script>src="/na101/home/literatum/publisher/tandf/journals/content/gcov20/2020/gcov20.v065.i09/17476933.2019.1669571/20200717/images/gcov_a_1669571_um0001.gif" alt="" />script>src="//:0" alt="" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gcov20/2020/gcov20.v065.i09/17476933.2019.1669571/20200717/images/gcov_a_1669571_um0001.gif"}" /><span class="mml-formula">span>span><span class="NLM_disp-formula disp-formula">src="//:0" alt="" data-formula-source="{"type" : "mathjax"}" />span>