The Identity Problem in the special affine group of Z2\mathbb{Z}^2

We consider semigroup algorithmic problems in the Special Affine group $\mathsf{SA}(2, \mathbb{Z}) = \mathbb{Z}^2 \rtimes \mathsf{SL}(2, \mathbb{Z})$, which is the group of affine transformations of the lattice $\mathbb{Z}^2$ that preserve orientation. Our paper focuses on two decision problems introduced by Choffrut and Karhum\"{a}ki (2005): the Identity Problem (does a semigroup contain a neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of $\mathsf{SA}(2, \mathbb{Z})$. We show that both problems are decidable and NP-complete. Since $\mathsf{SL}(2, \mathbb{Z}) \leq \mathsf{SA}(2, \mathbb{Z}) \leq \mathsf{SL}(3, \mathbb{Z})$, our result extends that of Bell, Hirvensalo and Potapov (SODA 2017) on the NP-completeness of both problems in $\mathsf{SL}(2, \mathbb{Z})$, and contributes a first step towards the open problems in $\mathsf{SL}(3, \mathbb{Z})$.Comment: 17 pages, 10 figure

Recent advances in algorithmic problems for semigroups

In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite group $G$, often represented as a matrix group. Such problems might not be decidable in general. In fact, they gave rise to some of the earliest undecidability results in algorithmic theory. However, the situation changes when the group $G$ satisfies additional constraints. In this survey, we give an overview of the decidability and the complexity of several algorithmic problems in the cases where $G$ is a low-dimensional matrix group, or a group with additional structures such as commutativity, nilpotency and solvability.Comment: survey article for SIGLOG New

The Identity Problem in nilpotent groups of bounded class

Let $G$ be a unitriangular matrix group of nilpotency class at most ten. We show that the Identity Problem (does a semigroup contain the identity matrix?) and the Group Problem (is a semigroup a group?) are decidable in polynomial time for finitely generated subsemigroups of $G$. Our decidability results also hold when $G$ is an arbitrary finitely generated nilpotent group of class at most ten. This extends earlier work of Babai et al. on commutative matrix groups (SODA'96) and work of Bell et al. on $\mathsf{SL}(2, \mathbb{Z})$ (SODA'17). Furthermore, we formulate a sufficient condition for the generalization of our results to nilpotent groups of class $d > 10$. For every such $d$, we exhibit an effective procedure that verifies this condition in case it is true.Comment: 48 pages, title change

Termination of linear loops under commutative updates

We consider the following problem: given $d \times d$ rational matrices $A_1, \ldots, A_k$ and a polyhedral cone $\mathcal{C} \subset \mathbb{R}^d$, decide whether there exists a non-zero vector whose orbit under multiplication by $A_1, \ldots, A_k$ is contained in $\mathcal{C}$. This problem can be interpreted as verifying the termination of multi-path while loops with linear updates and linear guard conditions. We show that this problem is decidable for commuting invertible matrices $A_1, \ldots, A_k$. The key to our decision procedure is to reinterpret this problem in a purely algebraic manner. Namely, we discover its connection with modules over the polynomial ring $\mathbb{R}[X_1, \ldots, X_k]$ as well as the polynomial semiring $\mathbb{R}_{\geq 0}[X_1, \ldots, X_k]$. The loop termination problem is then reduced to deciding whether a submodule of $\left(\mathbb{R}[X_1, \ldots, X_k]\right)^n$ contains a ``positive'' element.Comment: 6 page

Semigroup intersection problems in the Heisenberg groups

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups $\operatorname{H}_{n}(\mathbb{K})$ over any algebraic number field $\mathbb{K}$, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group $\operatorname{H}_{3}(\mathbb{Q})$.Comment: 18 pages including appendix, 2 figure

Differential elimination for dynamical models via projections with applications to structural identifiability

Elimination of unknowns in a system of differential equations is often required when analysing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a natural prerequisite for meaningful parameter estimation, is often prohibitively expensive for medium to large systems due to the computationally expensive task of elimination. We propose an algorithm that computes a description of the set of differential-algebraic relations between the input and output variables of a dynamical system model. The resulting algorithm outperforms general-purpose software for differential elimination on a set of benchmark models from literature. We use the designed elimination algorithm to build a new randomized algorithm for assessing structural identifiability of a parameter in a parametric model. A parameter is said to be identifiable if its value can be uniquely determined from input-output data assuming the absence of noise and sufficiently exciting inputs. Our new algorithm allows the identification of models that could not be tackled before. Our implementation is publicly available as a Julia package at https://github.com/SciML/StructuralIdentifiability.jl

Effects of Tai Chi on the Executive Function and Physical Fitness of Female Methamphetamine Dependents: A Randomized Controlled Trial

Purpose: Exercise improves the health and mental status of drug dependents. The way by which Tai Chi (TC) as a special exercise treatment affects executive functions (EFs) of methamphetamine (MA) dependents is yet to be established. This study aimed to explore the effects of TC on the EFs and physical fitness of MA dependents.Methods: A total of 76 female MA dependents were randomly assigned to the exercise and control groups. The exercise group underwent three 60-min sessions of TC training per week for 12 weeks. The control group was trained with conventional exercises including the 9th Guang Bo Ti Cao and square dance. Physical fitness and EF assessments that evaluated inhibitory control (IC, go/no-go task), working memory (3-back task) and cognitive flexibility (switching task) were performed at baseline and at 12 weeks. A repeated-measures ANOVA was applied to analyze the differences of group and time.Results: The exercise group showed decreased response time (RT) with a significant main effect of time on the go/no-go task [F(1, 68) = 9.6, p < 0.05]. The interaction effect between time and group was significant on accuracy [F(1, 61) = 4.73, p < 0.05], and the main effect of time was significant on RT [F(1, 61) = 4.66, p < 0.05] in the 3-back task of the exercise group. Significant changes in BMI [F(1, 68) = 19.57, p < 0.05], vital capacity [F(1, 68) = 6.00, p < 0.05], and systolic blood pressure [F(1, 68) = 6.11, p < 0.05] were observed in the exercise group.Conclusion: These findings showed that 3 months of TC training can improve the IC and maintain the working memory and cognitive flexibility of MA dependents. Other data implied that TC may improve the physical fitness of MA dependents.Clinical Trial Registration:http://www.chictr.org.cn/, ChiCTR1900022091

Integrative Biomarker Assessment of the Influence of Saxitoxin on Marine Bivalves: A Comparative Study of the Two Bivalve Species Oysters, Crassostrea gigas, and Scallops, Chlamys farreri

Harmful algae blooms have expanded greatly in recent decades, and their secreted toxins pose a severe threat to human health and marine ecosystems. Saxitoxin (STX) is a main paralytic shellfish poison naturally produced by marine microalgae of the genus Alexandrium. Despite numerous studies have assessed the impacts of STX on marine bivalves, comparative in vivo study on the toxicity of STX on bivalves with distinct accumulation ability (such as oysters and scallops) has been seldom investigated. The aim of this study was to identify whether distinct sensitivity exists between oysters, Crassostrea gigas, and scallops, Chlamys farreri under the same amount of STX exposure using multiple biomarker responses. The responses of different biochemical markers including oxidative stress markers (catalase, superoxide dismutase, glutathione S-transferase, and lipid peroxidation) and immunotoxicity biomarkers (hemocyte phagocytosis rate, reactive oxidative species production, and DNA damages) were evaluated in bivalves after 12, 48, and 96 h of exposure to STX. The integrated biomarker responses value combined with two-way ANOVA analysis suggested that STX posed slightly severer stress on scallops than oysters for the extended period of time. This study provided preliminary results on the usefulness of a multi-biomarker approach to assess the toxicity associated with STX exposure in marine bivalves

Skin transcriptome profiles associated with coat color in sheep

Background Previous molecular genetic studies of physiology and pigmentation of sheep skin have focused primarily on a limited number of genes and proteins. To identify additional genes that may play important roles in coat color regulation, Illumina sequencing technology was used to catalog global gene expression profiles in skin of sheep with white versus black coat color. Results There were 90,006 and 74,533 unigenes assembled from the reads obtained from white and black sheep skin, respectively. Genes encoding for the ribosomal proteins and keratin associated proteins were most highly expressed. A total of 2,235 known genes were differentially expressed in black versus white sheep skin, with 479 genes up-regulated and 1,756 genes down-regulated. A total of 845 novel genes were differentially expressed in black versus white sheep skin, consisting of 107 genes which were up-regulated (including 2 highly expressed genes exclusively expressed in black sheep skin) and 738 genes that were down-regulated. There was also a total of 49 known coat color genes expressed in sheep skin, from which 13 genes showed higher expression in black sheep skin. Many of these up-regulated genes, such as DCT, MATP, TYR and TYRP1, are members of the components of melanosomes and their precursor ontology category. Conclusion The white and black sheep skin transcriptome profiles obtained provide a valuable resource for future research to understand the network of gene expression controlling skin physiology and melanogenesis in sheep