The theory of <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-rung orthopair fuzzy sets (<inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFSs) proposed by Yager effectively describes fuzzy information in the real world. Because <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFSs contain the parameter <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula> and can adjust the range of expressed fuzzy information, they are superior to both intuitionistic and Pythagorean fuzzy sets. Archimedean T-norm and T-conorm (ATT) is an important tool used to generate operational rules based on the <italic>q</italic>-rung orthopair fuzzy numbers (<inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFNs). In comparison, the Bonferroni mean (BM) operator has an advantage because it considers the interrelationships between the different attributes. Therefore, it is an important and meaningful innovation to extend the BM operator to the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFNs based upon the ATT. In this paper, we first discuss <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-rung orthopair fuzzy operational rules by using ATT. Furthermore, we extend BM operator to the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-ROFNs and propose the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-rung orthopair fuzzy Archimedean BM <inline-formula><tex-math notation="LaTeX">$(q\hbox{-}{ROFABM})$</tex-math></inline-formula> operator and the <italic>q</italic>-rung orthopair fuzzy weighted Archimedean BM <inline-formula><tex-math notation="LaTeX">$(q\hbox{-}{ROFWABM})$</tex-math></inline-formula> operator and study their desirable properties. Then, a new multiple-attribute decision-making (MADM) method is developed based on <inline-formula><tex-math notation="LaTeX">$q\hbox{-}{ROFWABM}$</tex-math></inline-formula> operator. Finally, we use a practical example to verify effectiveness and superiority by comparing to other existing methods.

On a probability space $(Ω,\mathcal{A},\mathbb{Q})$ we consider two filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time $θ$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$ predictable processes on $(0,θ]$. In this setup it is well-known that, for any $\mathbb{F}$ semimartingale $X$, the process $X^{θ-}$ ($X$ stopped "right before $θ$") is a $\mathbb{G}$ semimartingale.Given a positive constant $T$, we call $θ$ an invariance time if there exists a probability measure $\mathbb{P}$ equivalent to $\mathbb{Q}$ on $\mathcal{F}\_T$ such that, for any $(\mathbb{F},\mathbb{P})$ local martingale $X$, $X^{θ-}$ is a $(\mathbb{G},\mathbb{Q})$ local martingale. We characterize invariance times in terms of the $(\mathbb{F},\mathbb{Q})$ Azéma supermartingale of $θ$ and we derive a mild and tractable invariance time sufficiency condition. We discuss invariance times in mathematical finance and BSDE applications.

In this comment on"Origin of correlated isolated flat bands in copper-substituted lead phosphate apatite"(arXiv:2307.16892 [cond-mat.supr-con]), we discuss the flat half-occupied two-band manifold that appears in $\mathrm{Pb}_9\mathrm{Cu}(\mathrm{PO}_4)_6(\mathrm{OH})_2$ when using a semilocal DFT functional. We argue that the flat band is an artifact of the functional's overestimation of the energy of the oxygen p states in the valence band. When using the HSE hybrid functional, the energy of the oxygen p states is reduced, and the copper-derived manifold splits into one fully occupied and one empty band. While these results do not rule out the possibility of superconductivity in doped LK-99, they do predict that stoichiometric $\mathrm{Pb}_9\mathrm{Cu}(\mathrm{PO}_4)_6(\mathrm{OH})_2$ is an insulator, not a superconductor. Furthermore, we have shown that future first-principles studies of these materials should employ hybrid functionals or other advanced methods to ensure that the oxygen-derived valence-band energies are correctly described.

We explore the use of the stochastic resolution-of-the-identity (sRI) with the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) method. sRI is combined with four existing local energy evaluation strategies in ph-AFQMC, namely (1) the half-rotated electron repulsion integral tensor (HR), (2) Cholesky decomposition (CD), (3) tensor hypercontraction (THC), or (4) low-rank factorization (LR). We demonstrate that HR-sRI achieves no scaling reduction, CD-sRI scales as $\mathcal O(N^3)$, and THC-sRI and LR-sRI scale as $\mathcal O(N^2)$, albeit with a potentially large prefactor. Furthermore, the walker-specific extra memory requirement in CD is reduced from $\mathcal O(N^3)$ to $\mathcal O(N^2)$ with sRI, while sRI-based THC and LR algorithms lead to a reduction from $\mathcal O(N^2)$ extra memory to $\mathcal O(N)$. Based on numerical results for one-dimensional hydrogen chains and water clusters, we demonstrated that, along with the use of a variance reduction technique, CD-sRI achieves cubic-scaling {\it without overhead}. In particular, we find for the systems studied the observed scaling of standard CD is $\mathcal O(N^{3-4})$ while for CD-sRI it is reduced to $\mathcal O(N^{2-3})$. Once a memory bottleneck is reached, we expect THC-sRI and LR-sRI to be preferred methods due to their quadratic-scaling memory requirements and their quadratic-scaling of the local energy evaluation (with a potentially large prefactor). The theoretical framework developed here should facilitate large-scale ph-AFQMC applications that were previously difficult or impossible to carry out with standard computational resources.

An error in the paper [J. Math. Phys. 43, 6343 (2002); math-ph/0207009] is corrected. Further explanation is given.

Let $\mathbb{Q}$ and $\mathbb{P}$ be equivalent probability measures and let $ψ$ be a $J$-dimensional vector of random variables such that $\frac{d\mathbb{Q}}{d\mathbb{P}}$ and $ψ$ are defined in terms of a weak solution $X$ to a $d$-dimensional stochastic differential equation. Motivated by the problem of \emph{endogenous completeness} in financial economics we present conditions which guarantee that every local martingale under $\mathbb{Q}$ is a stochastic integral with respect to the $J$-dimensional martingale $S_t \set \mathbb{E}^{\mathbb{Q}}[ψ|\mathcal{F}_t]$. While the drift $b=b(t,x)$ and the volatility $σ= σ(t,x)$ coefficients for $X$ need to have only minimal regularity properties with respect to $x$, they are assumed to be analytic functions with respect to $t$. We provide a counter-example showing that this $t$-analyticity assumption for $σ$ cannot be removed.

Drug delivery systems represent a promising strategy to treat cancer and to overcome the side effects of chemotherapy. In particular, polymeric nanocontainers have attracted major interest because of their structural and morphological advantages and the variety of polymers that can be used, allowing the synthesis of materials capable of responding to the biochemical alterations of the tumour microenvironment. While experimental methodologies can provide much insight, the generation of experimental data across a wide parameter space is usually prohibitively time consuming and/or expensive. To better understand the influence of varying design parameters on the drug release profile and drug kinetics involved, appropriately-designed mathematical models are of great benefit. Here, we developed a novel mathematical model to describe drug transport within, and release from, a hollow nanocontainer consisting of a core and a pH-responsive polymeric shell. The two-layer mathematical model fully accounts for drug dissolution, diffusion and interaction with polymer. We generated experimental drug release profiles using daunorubicin and [Cu(TPMA)(Phenantroline)](ClO_4)_2 as model drugs, for which the nanocontainers exhibited excellent encapsulation ability. The in vitro drug release behaviour was studied under different conditions, where the system proved capable of responding to the selected pH stimuli by releasing a larger amount of drug in an acidic than in the physiological environments. By comparing the results of the mathematical model with our experimental data, we were able to identify the model parameter values that best-fit the data and demonstrate that the model is capable of describing the phenomena at hand. The proposed methodology can be used to describe and predict the release profiles for a variety of drug delivery systems.

In most cases, insurance contracts are linked to the financial markets, such as through interest rates or equity-linked insurance products. To motivate an evaluation rule in these hybrid markets, Artzner et al. (2022) introduced the notion of insurance-finance arbitrage. In this paper we extend their setting by incorporating model uncertainty. To this end, we allow statistical uncertainty in the underlying dynamics to be represented by a set of priors $\mathscr{P}$. Within this framework we introduce the notion of robust asymptotic insurance-finance arbitrage and characterize the absence of such strategies in terms of the concept of ${Q}\mathscr{P}$-evaluations. This is a nonlinear two-step evaluation which guarantees no robust asymptotic insurance-finance arbitrage. Moreover, the ${Q}\mathscr{P}$-evaluation dominates all two-step evaluations as long as we agree on the set of priors $\mathscr{P}$ which shows that those two-step evaluations do not allow for robust asymptotic insurance-finance arbitrages. Furthermore, we introduce a doubly stochastic model under uncertainty for surrender and survival. In this setting, we describe conditional dependence by means of copulas and illustrate how the ${Q}\mathscr{P}$-evaluation can be used for the pricing of hybrid insurance products.

Recent calculations further supports the premise that large-scale synchronous firings of neurons may affect molecular processes. The context is scalp electroencephalography (EEG) during short-term memory (STM) tasks. The mechanism considered is $\mathbfΠ = \mathbf{p} + q \mathbf{A}$ (SI units) coupling, where $\mathbf{p}$ is the momenta of free $\mathrm{Ca}^{2+}$ waves $q$ the charge of $\mathrm{Ca}^{2+}$ in units of the electron charge, and $\mathbf{A}$ the magnetic vector potential of current $\mathbf{I}$ from neuronal minicolumnar firings considered as wires, giving rise to EEG. Data has processed using multiple graphs to identify sections of data to which spline-Laplacian transformations are applied, to fit the statistical mechanics of neocortical interactions (SMNI) model to EEG data, sensitive to synaptic interactions subject to modification by $\mathrm{Ca}^{2+}$ waves.

The Kullback-Leibler cluster entropy $\mathcal{D_{C}}[P \| Q] $ is evaluated for the empirical and model probability distributions $P$ and $Q$ of the clusters formed in the realized volatility time series of five assets (SP\&500, NASDAQ, DJIA, DAX, FTSEMIB). The Kullback-Leibler functional $\mathcal{D_{C}}[P \| Q] $ provides complementary perspectives about the stochastic volatility process compared to the Shannon functional $\mathcal{S_{C}}[P]$. While $\mathcal{D_{C}}[P \| Q] $ is maximum at the short time scales, $\mathcal{S_{C}}[P]$ is maximum at the large time scales leading to complementary optimization criteria tracing back respectively to the maximum and minimum relative entropy evolution principles. The realized volatility is modelled as a time-dependent fractional stochastic process characterized by power-law decaying distributions with positive correlation ($H>1/2$). As a case study, a multiperiod portfolio built on diversity indexes derived from the Kullback-Leibler entropy measure of the realized volatility. The portfolio is robust and exhibits better performances over the horizon periods. A comparison with the portfolio built either according to the uniform distribution or in the framework of the Markowitz theory is also reported.