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.

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.

Some methods aim to correct or test for relationships or to reconstruct the pedigree, or family tree. We show that these methods cannot resolve ties for correct relationships due to identifiability of the pedigree likelihood which is the probability of inheriting the data under the pedigree model. This means that no likelihood-based method can produce a correct pedigree inference with high probability. This lack of reliability is critical both for health and forensics applications. In this paper we present the first discussion of multiple typed individuals in non-isomorphic pedigrees, $\mathcal{P}$ and $\mathcal{Q}$, where the likelihoods are non-identifiable, $Pr[G~|~\mathcal{P},θ] = Pr[G~|~\mathcal{Q},θ]$, for all input data $G$ and all recombination rate parameters $θ$. While there were previously known non-identifiable pairs, we give an example having data for multiple individuals. Additionally, deeper understanding of the general discrete structures driving these non-identifiability examples has been provided, as well as results to guide algorithms that wish to examine only identifiable pedigrees. This paper introduces a general criteria for establishing whether a pair of pedigrees is non-identifiable and two easy-to-compute criteria guaranteeing identifiability. Finally, we suggest a method for dealing with non-identifiable likelihoods: use Bayes rule to obtain the posterior from the likelihood and prior. We propose a prior guaranteeing that the posterior distinguishes all pairs of pedigrees. Shortened version published as: B. Kirkpatrick. Non-identifiable pedigrees and a Bayesian solution. Int. Symp. on Bioinformatics Res. and Appl. (ISBRA), 7292:139-152 2012.

We show how recent results of Lieb and Seiringer [math-ph/0412009; Phys. Rev. A 71, 062329 (2005)] can be obtained from repeated use of the monotonicity of relative entropy under partial traces, and explain how to use their approach to obtain tighter bounds in many situations.

An emended and improved version of the present paper has been archived in math-ph/0505057, and a preliminary account of its content has been published in Phys.Rev.Lett. 92, 60601, (2004). Moreover, in order to prove the relevance of topology for phase transition phenomena in a broad domain of physically interesting cases, we have proved another theorem which is reported in math-ph/0505058 and which is crucially based on the result of the paper archived in math-ph/0505057.

In this lecture I discuss some aspects of MKM, Mathematical Knowledge Management, with particuar emphasis on information storage and information retrieval.