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A Tuttminx is a Rubik's Cube-like twisty puzzle, in the shape of a truncated icosahedron. It was invented by Lee Tutt in 2005. It has a total of 150 movable pieces to rearrange, compared to 20 movable pieces of the Rubik’s Cube.

DescriptionEdit

The Tuttminx has a total of 32 face centre pieces (12 pentagon and 20 hexagon), 60 corner pieces and 90 edge pieces. The face centres each have a single colour, which identifies the colour of that face in the solved state. The edge pieces have two colours and the corner pieces have three colours. Each hexagonal face contains a centre piece, 6 corner pieces and 6 edge pieces, while each pentagonal face contains a centre piece, 5 corner pieces and 5 edge pieces.

The puzzle twists around the faces: each twist rotates one face centre piece and moves all edge and corner pieces surrounding it. The pentagonal faces can be twisted 72° in either direction, while the hexagonal faces can be rotated 120°.

The purpose of the puzzle is to scramble the colours, and then restore it to its original state of having one colour per face.

Number of combinationsEdit

The puzzle has 150 movable pieces: 60 corner pieces, 60 edge pieces that are adjacent to a pentagonal face (so-called pentagonal edges) and 30 edge pieces that are not (non-pentagonal edges). Only even permutations of all three types of pieces are possible (i.e. it is impossible to have only one pair of identical pieces swapped). Thus, there are 60!/2 possible ways to arrange the corner pieces, 60!/2 ways to arrange the pentagonal edges and 30!/2 ways to arrange the non-pentagonal edges.

All corner pieces have only one possible orientation, as do all pentagonal edge pieces. The non-pentagonal edge pieces all have 2 possible orientations each. Only even orientations of those are possible (meaning that it is impossible to have only one edge piece flipped over). This means there are 229 ways to orientate the edge pieces.

The number of possible combinations on the Tuttminx is therefore equal to

$\frac{60! \times 60! \times 30! \times 2^{29}}{8} \approx 1.2325 \times 10^{204}$

The full number is 1 232 507 756 161 568 013 733 174 639 895 750 813 761 087 074 840 896 182 396 140 424 396 146 760 158 229 902 239 889 099 665 575 990 049 299 860 175 851 176 152 712 039 950 335 697 389 221 704 074 672 278 055 758 253 470 515 200 000 000 000 000 000 000 000 000 000 000 000 (about 1.2325 septensexagintillion on the short scale and 1.2325 quattuortrigintillion on the long scale).

VariationsEdit

There are a few variations of the Tuttminx that have been made. The most popular ones include:

• Void Tuttminx, which is a regular Tuttminx but without the face centre pieces;
• Rayminx (also called Giga Tuttminx), which is a higher-order version of the Tuttminx;
• Futtminx, which was invented by Oskar van Deventer and has been designed so that the hexagonal faces can be rotated 60° and mix up with the pentagonal faces.

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# If you did not already know

Modular, Optimal Learning Testing Environment (MOLTE)
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Beholder-GAN
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Multiple Graph Optimized Convolutional Network (M-GOCN)
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Hierarchical Planning and Reinforcement Learning (HIP-RL)
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Definition Extraction Tool (DefExt)
We present DefExt, an easy to use semi supervised Definition Extraction Tool. DefExt is designed to extract from a target corpus those textual fragments where a term is explicitly mentioned together with its core features, i.e. its definition. It works on the back of a Conditional Random Fields based sequential labeling algorithm and a bootstrapping approach. Bootstrapping enables the model to gradually become more aware of the idiosyncrasies of the target corpus. In this paper we describe the main components of the toolkit as well as experimental results stemming from both automatic and manual evaluation. We release DefExt as open source along with the necessary files to run it in any Unix machine. We also provide access to training and test data for immediate use. …

Acumos
Applying Machine Learning (ML) to business applications for automation usually faces difficulties when integrating diverse ML dependencies and services, mainly because of the lack of a common ML framework. In most cases, the ML models are developed for applications which are targeted for specific business domain use cases, leading to duplicated effort, and making reuse impossible. This paper presents Acumos, an open platform capable of packaging ML models into portable containerized microservices which can be easily shared via the platform’s catalog, and can be integrated into various business applications. We present a case study of packaging sentiment analysis and classification ML models via the Acumos platform, permitting easy sharing with others. We demonstrate that the Acumos platform reduces the technical burden on application developers when applying machine learning models to their business applications. Furthermore, the platform allows the reuse of readily available ML microservices in various business domains. …

DeceptionNet
We present a novel approach to tackle domain adaptation between synthetic and real data. Instead of employing ‘blind’ domain randomization, i.e. augmenting synthetic renderings with random backgrounds or changing illumination and colorization, we leverage the task network as its own adversarial guide towards useful augmentations that maximize the uncertainty of the output. To this end, we design a min-max optimization scheme where a given task competes against a special deception network, with the goal of minimizing the task error subject to specific constraints enforced by the deceiver. The deception network samples from a family of differentiable pixel-level perturbations and exploits the task architecture to find the most destructive augmentations. Unlike GAN-based approaches that require unlabeled data from the target domain, our method achieves robust mappings that scale well to multiple target distributions from source data alone. We apply our framework to the tasks of digit recognition on enhanced MNIST variants as well as classification and object pose estimation on the Cropped LineMOD dataset and compare to a number of domain adaptation approaches, demonstrating similar results with superior generalization capabilities. …

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Density-based spatial clustering of applications with noise (DBSCAN) is a data clustering algorithm which has the high-performance rate for dataset where clusters have the constant density of data points. One of the significant attributes of this algorithm is noise cancellation. However, DBSCAN demonstrates reduced performances for clusters with different densities. Therefore, in this paper, an adaptive DBSCAN is proposed which can work significantly well for identifying clusters with varying densities. …

Imitation Learning
Learning from Demonstration’: Imitation learning, a.k.a behavioral cloning, is learning from demonstration. In other words, in imitation learning, a machine learns how to behave by looking at what a teacher (or expert) does and then mimics that behavior. An example can be when we collect driving data from human and then use that data for a self driving car.
Imitation Learning in Tensorflow

LP-3DCNN
Traditional 3D Convolutional Neural Networks (CNNs) are computationally expensive, memory intensive, prone to overfit, and most importantly, there is a need to improve their feature learning capabilities. To address these issues, we propose Rectified Local Phase Volume (ReLPV) block, an efficient alternative to the standard 3D convolutional layer. The ReLPV block extracts the phase in a 3D local neighborhood (e.g., 3x3x3) of each position of the input map to obtain the feature maps. The phase is extracted by computing 3D Short Term Fourier Transform (STFT) at multiple fixed low frequency points in the 3D local neighborhood of each position. These feature maps at different frequency points are then linearly combined after passing them through an activation function. The ReLPV block provides significant parameter savings of at least, 3^3 to 13^3 times compared to the standard 3D convolutional layer with the filter sizes 3x3x3 to 13x13x13, respectively. We show that the feature learning capabilities of the ReLPV block are significantly better than the standard 3D convolutional layer. Furthermore, it produces consistently better results across different 3D data representations. We achieve state-of-the-art accuracy on the volumetric ModelNet10 and ModelNet40 datasets while utilizing only 11% parameters of the current state-of-the-art. We also improve the state-of-the-art on the UCF-101 split-1 action recognition dataset by 5.68% (when trained from scratch) while using only 15% of the parameters of the state-of-the-art. The project webpage is available at https://…/home.

DAG Variational Autoencoder (D-VAE)
Graph structured data are abundant in the real world. Among different graph types, directed acyclic graphs (DAGs) are of particular interests to machine learning researchers, as many machine learning models are realized as computations on DAGs, including neural networks and Bayesian networks. In this paper, we study deep generative models for DAGs, and propose a novel DAG variational autoencoder (D-VAE). To encode DAGs into the latent space, we leverage graph neural networks. We propose a DAG-style asynchronous message passing scheme that allows encoding the computations defined by DAGs, rather than using existing simultaneous message passing schemes to encode the graph structures. We demonstrate the effectiveness of our proposed D-VAE through two tasks: neural architecture search and Bayesian network structure learning. Experiments show that our model not only generates novel and valid DAGs, but also produces a smooth latent space that facilitates searching for DAGs with better performance through Bayesian optimization. …

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SGD (Stochastic Gradient Descent) is a popular algorithm for large scale optimization problems due to its low iterative cost. However, SGD can not achieve linear convergence rate as FGD (Full Gradient Descent) because of the inherent gradient variance. To attack the problem, mini-batch SGD was proposed to get a trade-off in terms of convergence rate and iteration cost. In this paper, a general CVI (Convergence-Variance Inequality) equation is presented to state formally the interaction of convergence rate and gradient variance. Then a novel algorithm named SSAG (Stochastic Stratified Average Gradient) is introduced to reduce gradient variance based on two techniques, stratified sampling and averaging over iterations that is a key idea in SAG (Stochastic Average Gradient). Furthermore, SSAG can achieve linear convergence rate of $\mathcal {O}((1-\frac{\mu}{8CL})^k)$ at smaller storage and iterative costs, where $C\geq 2$ is the category number of training data. This convergence rate depends mainly on the variance between classes, but not on the variance within the classes. In the case of $C\ll N$ ($N$ is the training data size), SSAG’s convergence rate is much better than SAG’s convergence rate of $\mathcal {O}((1-\frac{\mu}{8NL})^k)$. Our experimental results show SSAG outperforms SAG and many other algorithms. …

BitSplit-Net
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Vehicle Transfer Generative Adversarial Network (VTGAN)
Vehicle re-identification (reID) is to identify a target vehicle in different cameras with non-overlapping views. When deploy the well-trained model to a new dataset directly, there is a severe performance drop because of differences among datasets named domain bias. To address this problem, this paper proposes an domain adaptation framework which contains an image-to-image translation network named vehicle transfer generative adversarial network (VTGAN) and an attention-based feature learning network (ATTNet). VTGAN could make images from the source domain (well-labeled) have the style of target domain (unlabeled) and preserve identity information of source domain. To further improve the domain adaptation ability for various backgrounds, ATTNet is proposed to train generated images with the attention structure for vehicle reID. Comprehensive experimental results clearly demonstrate that our method achieves excellent performance on VehicleID dataset. …

Condition Monitoring (CM)
Condition monitoring (or, colloquially, CM) is the process of monitoring a parameter of condition in machinery (vibration, temperature etc.), in order to identify a significant change which is indicative of a developing fault. It is a major component of “Predictive Maintenance”. The use of condition monitoring allows maintenance to be scheduled, or other actions to be taken to prevent failure and avoid its consequences. Condition monitoring has a unique benefit in that conditions that would shorten normal lifespan can be addressed before they develop into a major failure. Condition monitoring techniques are normally used on rotating equipment and other machinery (pumps, electric motors, internal combustion engines, presses), while periodic inspection using non-destructive testing techniques and fit for service (FFS) evaluation are used for stationary plant equipment such as steam boilers, piping and heat exchangers.
http://…/9781466584051

# If you did not already know

Skip-GANomaly
Despite inherent ill-definition, anomaly detection is a research endeavor of great interest within machine learning and visual scene understanding alike. Most commonly, anomaly detection is considered as the detection of outliers within a given data distribution based on some measure of normality. The most significant challenge in real-world anomaly detection problems is that available data is highly imbalanced towards normality (i.e. non-anomalous) and contains a most a subset of all possible anomalous samples – hence limiting the use of well-established supervised learning methods. By contrast, we introduce an unsupervised anomaly detection model, trained only on the normal (non-anomalous, plentiful) samples in order to learn the normality distribution of the domain and hence detect abnormality based on deviation from this model. Our proposed approach employs an encoder-decoder convolutional neural network with skip connections to thoroughly capture the multi-scale distribution of the normal data distribution in high-dimensional image space. Furthermore, utilizing an adversarial training scheme for this chosen architecture provides superior reconstruction both within high-dimensional image space and a lower-dimensional latent vector space encoding. Minimizing the reconstruction error metric within both the image and hidden vector spaces during training aids the model to learn the distribution of normality as required. Higher reconstruction metrics during subsequent test and deployment are thus indicative of a deviation from this normal distribution, hence indicative of an anomaly. Experimentation over established anomaly detection benchmarks and challenging real-world datasets, within the context of X-ray security screening, shows the unique promise of such a proposed approach. …

What-If Tool
What If… you could inspect a machine learning model, with no coding required? Building effective machine learning systems means asking a lot of questions. It’s not enough to train a model and walk away. Instead, good practitioners act as detectives, probing to understand their model better. But answering these kinds of questions isn’t easy. Probing ‘what if’ scenarios often means writing custom, one-off code to analyze a specific model. Not only is this process inefficient, it makes it hard for non-programmers to participate in the process of shaping and improving machine learning models. For us, making it easier for a broad set of people to examine, evaluate, and debug machine learning systems is a key concern. That’s why we built the What-If Tool. Built into the open-source TensorBoard web application – a standard part of the TensorFlow platform – the tool allows users to analyze an machine learning model without the need for writing any further code. Given pointers to a TensorFlow model and a dataset, the What-If Tool offers an interactive visual interface for exploring model results. …

Merged-Averaged Classifiers via Hashing (MACH)
We present Merged-Averaged Classifiers via Hashing (MACH) for K-classification with ultra-large values of K. Compared to traditional one-vs-all classifiers that require O(Kd) memory and inference cost, MACH only need O(d log K) (d is dimensionality )memory while only requiring O(K log K + d log K) operation for inference. MACH is a generic K-classification algorithm, with provably theoretical guarantees, which requires O(log K) memory without any assumption on the relationship between classes. MACH uses universal hashing to reduce classification with a large number of classes to few independent classification tasks with small (constant) number of classes. We provide theoretical quantification of discriminability-memory tradeoff. With MACH we can train ODP dataset with 100,000 classes and 400,000 features on a single Titan X GPU, with the classification accuracy of 19.28%, which is the best-reported accuracy on this dataset. Before this work, the best performing baseline is a one-vs-all classifier that requires 40 billion parameters (160 GB model size) and achieves 9% accuracy. In contrast, MACH can achieve 9% accuracy with 480x reduction in the model size (of mere 0.3GB). With MACH, we also demonstrate complete training of fine-grained imagenet dataset (compressed size 104GB), with 21,000 classes, on a single GPU. To the best of our knowledge, this is the first work to demonstrate complete training of these extreme-class datasets on a single Titan X. …

Spatially-Preserved Doubly-Injected Object Detection CNN (S-DOD-CNN)
We present a novel event recognition approach called Spatially-preserved Doubly-injected Object Detection CNN (S-DOD-CNN), which incorporates the spatially preserved object detection information in both a direct and an indirect way. Indirect injection is carried out by simply sharing the weights between the object detection modules and the event recognition module. Meanwhile, our novelty lies in the fact that we have preserved the spatial information for the direct injection. Once multiple regions-of-intereset (RoIs) are acquired, their feature maps are computed and then projected onto a spatially-preserving combined feature map using one of the four RoI Projection approaches we present. In our architecture, combined feature maps are generated for object detection which are directly injected to the event recognition module. Our method provides the state-of-the-art accuracy for malicious event recognition. …

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Tarjan’s Strongly Connected Components Algorithm
Tarjan’s Algorithm (named for its discoverer, Robert Tarjan) is a graph theory algorithm for finding the strongly connected components of a graph. Although it precedes it chronologically, it can be seen as an improved version of Kosaraju’s algorithm, and is comparable in efficiency to the path-based strong component algorithm. …

OpenBLAS
OpenBLAS is an optimized BLAS library based on GotoBLAS2 1.13 BSD version. …

When observing the actions of others, humans carry out inferences about why the others acted as they did, and what this implies about their view of the world. Humans also use the fact that their actions will be interpreted in this manner when observed by others, allowing them to act informatively and thereby communicate efficiently with others. Although learning algorithms have recently achieved superhuman performance in a number of two-player, zero-sum games, scalable multi-agent reinforcement learning algorithms that can discover effective strategies and conventions in complex, partially observable settings have proven elusive. We present the Bayesian action decoder (BAD), a new multi-agent learning method that uses an approximate Bayesian update to obtain a public belief that conditions on the actions taken by all agents in the environment. Together with the public belief, this Bayesian update effectively defines a new Markov decision process, the public belief MDP, in which the action space consists of deterministic partial policies, parameterised by deep neural networks, that can be sampled for a given public state. It exploits the fact that an agent acting only on this public belief state can still learn to use its private information if the action space is augmented to be over partial policies mapping private information into environment actions. The Bayesian update is also closely related to the theory of mind reasoning that humans carry out when observing others’ actions. We first validate BAD on a proof-of-principle two-step matrix game, where it outperforms traditional policy gradient methods. We then evaluate BAD on the challenging, cooperative partial-information card game Hanabi, where in the two-player setting the method surpasses all previously published learning and hand-coded approaches. …

RuSentRel
In this paper we present the RuSentRel corpus including analytical texts in the sphere of international relations. For each document we annotated sentiments from the author to mentioned named entities, and sentiments of relations between mentioned entities. In the current experiments, we considered the problem of extracting sentiment relations between entities for the whole documents as a three-class machine learning task. We experimented with conventional machine-learning methods (Naive Bayes, SVM, Random Forest). …

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Multi-Directional Recurrent Neural Network (M-RNN)
Missing data is a ubiquitous problem. It is especially challenging in medical settings because many streams of measurements are collected at different – and often irregular – times. Accurate estimation of those missing measurements is critical for many reasons, including diagnosis, prognosis and treatment. Existing methods address this estimation problem by interpolating within data streams or imputing across data streams (both of which ignore important information) or ignoring the temporal aspect of the data and imposing strong assumptions about the nature of the data-generating process and/or the pattern of missing data (both of which are especially problematic for medical data). We propose a new approach, based on a novel deep learning architecture that we call a Multi-directional Recurrent Neural Network (M-RNN) that interpolates within data streams and imputes across data streams. We demonstrate the power of our approach by applying it to five real-world medical datasets. We show that it provides dramatically improved estimation of missing measurements in comparison to 11 state-of-the-art benchmarks (including Spline and Cubic Interpolations, MICE, MissForest, matrix completion and several RNN methods); typical improvements in Root Mean Square Error are between 35% – 50%. Additional experiments based on the same five datasets demonstrate that the improvements provided by our method are extremely robust. …

StackExchange
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CODED
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Maler
In this paper, we study adaptive online convex optimization, and aim to design a universal algorithm that achieves optimal regret bounds for multiple common types of loss functions. Existing universal methods are limited in the sense that they are optimal for only a subclass of loss functions. To address this limitation, we propose a novel online method, namely Maler, which enjoys the optimal $O(\sqrt{T})$, $O(d\log T)$ and $O(\log T)$ regret bounds for general convex, exponentially concave, and strongly convex functions respectively. The essential idea is to run multiple types of learning algorithms with different learning rates in parallel, and utilize a meta algorithm to track the best one on the fly. Empirical results demonstrate the effectiveness of our method. …

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Binary Stochastic Filtering (BSF)
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k-PDTM
Analyzing the sub-level sets of the distance to a compact sub-manifold of R d is a common method in TDA to understand its topology. The distance to measure (DTM) was introduced by Chazal, Cohen-Steiner and M{\’e}rigot in [7] to face the non-robustness of the distance to a compact set to noise and outliers. This function makes possible the inference of the topology of a compact subset of R d from a noisy cloud of n points lying nearby in the Wasserstein sense. In practice, these sub-level sets may be computed using approximations of the DTM such as the q-witnessed distance [10] or other power distance [6]. These approaches lead eventually to compute the homology of unions of n growing balls, that might become intractable whenever n is large. To simultaneously face the two problems of large number of points and noise, we introduce the k-power distance to measure (k-PDTM). This new approximation of the distance to measure may be thought of as a k-coreset based approximation of the DTM. Its sublevel sets consist in union of k-balls, k << n, and this distance is also proved robust to noise. We assess the quality of this approximation for k possibly dramatically smaller than n, for instance k = n 1 3 is proved to be optimal for 2-dimensional shapes. We also provide an algorithm to compute this k-PDTM. …

Iris
Today’s conversational agents are restricted to simple standalone commands. In this paper, we present Iris, an agent that draws on human conversational strategies to combine commands, allowing it to perform more complex tasks that it has not been explicitly designed to support: for example, composing one command to ‘plot a histogram’ with another to first ‘log-transform the data’. To enable this complexity, we introduce a domain specific language that transforms commands into automata that Iris can compose, sequence, and execute dynamically by interacting with a user through natural language, as well as a conversational type system that manages what kinds of commands can be combined. We have designed Iris to help users with data science tasks, a domain that requires support for command combination. In evaluation, we find that data scientists complete a predictive modeling task significantly faster (2.6 times speedup) with Iris than a modern non-conversational programming environment. Iris supports the same kinds of commands as today’s agents, but empowers users to weave together these commands to accomplish complex goals. …

Evolving Graph Convolutional Network (EvolveGCN)
Graph representation learning resurges as a trending research subject owing to the widespread use of deep learning for Euclidean data, which inspire various creative designs of neural networks in the non-Euclidean domain, particularly graphs. With the success of these graph neural networks (GNN) in the static setting, we approach further practical scenarios where the graph dynamically evolves. For this case, combining the GNN with a recurrent neural network (RNN, broadly speaking) is a natural idea. Existing approaches typically learn one single graph model for all the graphs, by using the RNN to capture the dynamism of the output node embeddings and to implicitly regulate the graph model. In this work, we propose a different approach, coined EvolveGCN, that uses the RNN to evolve the graph model itself over time. This model adaptation approach is model oriented rather than node oriented, and hence is advantageous in the flexibility on the input. For example, in the extreme case, the model can handle at a new time step, a completely new set of nodes whose historical information is unknown, because the dynamism has been carried over to the GNN parameters. We evaluate the proposed approach on tasks including node classification, edge classification, and link prediction. The experimental results indicate a generally higher performance of EvolveGCN compared with related approaches. …

# If you did not already know

Distilled-Exposition Enhanced Matching Network (DEMN)
This paper proposes a Distilled-Exposition Enhanced Matching Network (DEMN) for story-cloze test, which is still a challenging task in story comprehension. We divide a complete story into three narrative segments: an \textit{exposition}, a \textit{climax}, and an \textit{ending}. The model consists of three modules: input module, matching module, and distillation module. The input module provides semantic representations for the three segments and then feeds them into the other two modules. The matching module collects interaction features between the ending and the climax. The distillation module distills the crucial semantic information in the exposition and infuses it into the matching module in two different ways. We evaluate our single and ensemble model on ROCStories Corpus \cite{Mostafazadeh2016ACA}, achieving an accuracy of 80.1\% and 81.2\% on the test set respectively. The experimental results demonstrate that our DEMN model achieves a state-of-the-art performance. …

Truncated Variance Reduction (TruVaR)
We present a new algorithm, truncated variance reduction (TruVaR), that treats Bayesian optimization (BO) and level-set estimation (LSE) with Gaussian processes in a unified fashion. The algorithm greedily shrinks a sum of truncated variances within a set of potential maximizers (BO) or unclassified points (LSE), which is updated based on confidence bounds. TruVaR is effective in several important settings that are typically non-trivial to incorporate into myopic algorithms, including pointwise costs and heteroscedastic noise. We provide a general theoretical guarantee for TruVaR covering these aspects, and use it to recover and strengthen existing results on BO and LSE. Moreover, we provide a new result for a setting where one can select from a number of noise levels having associated costs. We demonstrate the effectiveness of the algorithm on both synthetic and real-world data sets. …

Proportional Degree
Several algorithms have been proposed to filter information on a complete graph of correlations across stocks to build a stock-correlation network. Among them the planar maximally filtered graph (PMFG) algorithm uses $3n-6$ edges to build a graph whose features include a high frequency of small cliques and a good clustering of stocks. We propose a new algorithm which we call proportional degree (PD) to filter information on the complete graph of normalised mutual information (NMI) across stocks. Our results show that the PD algorithm produces a network showing better homogeneity with respect to cliques, as compared to economic sectoral classification than its PMFG counterpart. We also show that the partition of the PD network obtained through normalised spectral clustering (NSC) agrees better with the NSC of the complete graph than the corresponding one obtained from PMFG. Finally, we show that the clusters in the PD network are more robust with respect to the removal of random sets of edges than those in the PMFG network. …

Non-Stationary Streaming PCA
We consider the problem of streaming principal component analysis (PCA) when the observations are noisy and generated in a non-stationary environment. Given $T$, $p$-dimensional noisy observations sampled from a non-stationary variant of the spiked covariance model, our goal is to construct the best linear $k$-dimensional subspace of the terminal observations. We study the effect of non-stationarity by establishing a lower bound on the number of samples and the corresponding recovery error obtained by any algorithm. We establish the convergence behaviour of the noisy power method using a novel proof technique which maybe of independent interest. We conclude that the recovery guarantee of the noisy power method matches the fundamental limit, thereby generalizing existing results on streaming PCA to a non-stationary setting. …

# If you did not already know

Residual Sum of Squares (RSS, SSR, SSE)
In statistics, the residual sum of squares (RSS) is the sum of squares of residuals. It is also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE). It is a measure of the discrepancy between the data and an estimation model. A small RSS indicates a tight fit of the model to the data.
In general, total sum of squares = explained sum of squares + residual sum of squares. For a proof of this in the multivariate ordinary least squares (OLS) case, see partitioning in the general OLS model. …

Inferential Model (IM)
Probability is a useful tool for describing uncertainty, so it is natural to strive for a system of statistical inference based on probabilities for or against various hypotheses. But existing probabilistic inference methods struggle to provide a meaningful interpretation of the probabilities across experiments in sufficient generality. In this paper we further develop a promising new approach based on what are called inferential models (IMs). The fundamental idea behind IMs is that there is an unobservable auxiliary variable that itself describes the inherent uncertainty about the parameter of interest, and that posterior probabilistic inference can be accomplished by predicting this unobserved quantity. We describe a simple and intuitive threestep construction of a random set of candidate parameter values, each being consistent with the model, the observed data, and a auxiliary variable prediction. Then prior-free posterior summaries of the available statistical evidence for and against a hypothesis of interest are obtained by calculating the probability that this random set falls completely in and completely out of the hypothesis, respectively. We prove that these IM-based measures of evidence are calibrated in a frequentist sense, showing that IMs give easily-interpretable results both within and across experiments. …

Golomb Ruler Problem
The Golomb ruler problem is defined as follows: Given a positive integer n, locate n marks on a ruler such that the distance between any two distinct pair of marks are different from each other and the total length of the ruler is minimized. The Golomb ruler problem has applications in information theory, astronomy and communications, and it can be seen as a challenge for combinatorial optimization algorithms. Although constructing high quality rulers is well-studied, proving optimality is a far more challenging task. …

Origraph
Data wrangling is widely acknowledged to be a critical part of the data analysis pipeline. Nevertheless, there are currently no techniques to efficiently wrangle network datasets. Here we introduce a set of interaction techniques that enable analysts to carry out complex network wrangling operations. These operations include deriving attributes across connected classes, converting nodes to edges and vice-versa, and faceting nodes and edges based on attributes. We implement these operations in a web-based, open-source system, Origraph, which provides interfaces to execute the operations and investigate the results. Designed for wrangling, rather than analysis, Origraph can be used to load data in many forms, wrangle and transform the network, and export it in formats compatible with common network visualization tools. We demonstrate Origraph’s usefulness in a series of examples with different datasets from a variety of sources. …


text:
# Chapter 4 - Section 4.1 - Maximum and Minimum Values - 4.1 Exercises - Page 283: 4

Absolute maximum at $r$, absolute minimum at $a$, local maxima at $b$ and $r$, local minimum at $d$, neither a maximum nor a minimum at $c$ and $s$.

#### Work Step by Step

$r$ and $a$ are the absolute extrema of the function because they are the highest and lowest value of the function, respectively. $b$ and $r$ are local maxima due to the fact they are the greatest value in their neighborhoods, and similarly $d$ is a local minima since it is the smallest value in it's neighborhood. $c$ and $s$ do not fall into any of these descriptions, so they are neither.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# physconst

Physical constants

## Description

example

const = physconst(name) returns the value of the physical constant const specified by the name argument.

## Examples

collapse all

Determine the wavelength of a 1 GHz electromagnetic wave.

freq = 1e9;
lambda = physconst('LightSpeed')/freq
lambda = 0.2998

Determine the thermal noise power per unit bandwidth in the in-phase (I) and quadrature (Q) channels of a receiver. Specify a receiver temperature of 290 K.

T = 290;
k = physconst('Boltzmann');

Compute the noise power per unit bandwidth, split evenly between the I and Q channels. Units are in dB.

Noise_power = 10*log10(k*T/2)
Noise_power = -206.9855

## Input Arguments

collapse all

Name of physical constant, specified as 'LightSpeed', 'Boltzmann', or 'EarthRadius'. See Physical Constants for a list of values for physical constants used in Phased Array System Toolbox™.

Example: 'LightSpeed'

## Output Arguments

collapse all

Value of physical constant, returned as a real-valued scalar. All values are in SI units.

collapse all

### Physical Constants

This table lists the supported constants and their values in SI units.

ConstantDescriptionValue
'LightSpeed'Speed of light in vacuum299,792,458 m/s. Most commonly denoted by c.
'Boltzmann'Boltzmann constant relating kinetic energy to temperature$1.\text{3806504}×{10}^{-23}$ J/K. 2006 NIST value, most commonly denoted by k.


text:
# American Institute of Mathematical Sciences

January  2012, 8(1): 41-49. doi: 10.3934/jimo.2012.8.41

## A note on the subtree ordered median problem in networks based on nestedness property

 1 Faculty of Management and Administration, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau SAR, China 2 Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China 3 Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China

Received  January 2011 Revised  May 2011 Published  November 2011

The nestedness property has become an increasingly important means for devising efficient algorithms for network location problems. In this paper we prove that the nestedness property holds for the tactical continuous, and strategic discrete and continuous subtree location problems in a tree network with the ordered median objective, where the $\lambda$-weights take at most two different values. These results extend some existing results in the literature. With these nestedness results, we solve the problems in polynomial time. Finally we pose an open problem on identifying the nestedness property for the $(k_1,k_2)$-trimmed problem.
Citation: Huajun Tang, T. C. Edwin Cheng, Chi To Ng. A note on the subtree ordered median problem in networks based on nestedness property. Journal of Industrial & Management Optimization, 2012, 8 (1) : 41-49. doi: 10.3934/jimo.2012.8.41
##### References:
 [1] J. N. Hooker, R. S. Garfinkel and C. K. Chen, Finite dominating sets for network location problems,, Operations Research, 39 (1991), 100.  doi: 10.1287/opre.39.1.100.  Google Scholar [2] J. Kalcsics, S. Nickel and J. Puerto, Multi-facility ordered median problems: A further analysis,, Networks, 41 (2003), 1.  doi: 10.1002/net.10053.  Google Scholar [3] E. Minieka, The optimal location of a path or tree in a tree network,, Networks, 15 (1985), 309.  doi: 10.1002/net.3230150304.  Google Scholar [4] S. Nickel, and J. Puerto, "Location Theory: A Unified Approach,", 1st edition, (2005).   Google Scholar [5] W. Ogryczak and A. Tamir, Minimizing the sum of $k$ largest functions in linear time,, Information Processing Letters, 85 (2003), 117.  doi: 10.1016/S0020-0190(02)00370-8.  Google Scholar [6] J. Puerto, and A. Tamir, Locating tree-shaped facilities using the ordered median objective,, Mathematical Programming, 102 (2005), 313.  doi: 10.1007/s10107-004-0547-2.  Google Scholar [7] A. Tamir, J. Puerto and D. Pérez-Brito, The centdian subtree on tree networks,, Discrete Applied Mathematics, 18 (2002), 263.  doi: 10.1016/S0166-218X(01)00199-8.  Google Scholar [8] H. J. Tang, T. C. E. Cheng and C. T. Ng, Finite dominating sets for the multi-facility ordered median problem in networks and algorithmic applications,, Computers & Industrial Engineering, 57 (2009), 707.  doi: 10.1016/j.cie.2009.01.015.  Google Scholar [9] H. J. Tang, T. C. E. Cheng and C. T. Ng, Multi-facility ordered median problems in directed networks,, Journal of Systems Science and Complexity, 24 (2011), 61.  doi: 10.1007/s11424-011-9327-2.  Google Scholar [10] P. M. Vaidya, An algorithm for linear programming which requires $O((m+n)n^2+(m+n)^{1.5}nL)$ arithmetic operations,, Mathematical Programming, 47 (1990), 175.  doi: 10.1007/BF01580859.  Google Scholar [11] B.-F. Wang, Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network,, Journal of Algorithms, 34 (2000), 90.  doi: 10.1006/jagm.1999.1020.  Google Scholar

show all references

##### References:
 [1] J. N. Hooker, R. S. Garfinkel and C. K. Chen, Finite dominating sets for network location problems,, Operations Research, 39 (1991), 100.  doi: 10.1287/opre.39.1.100.  Google Scholar [2] J. Kalcsics, S. Nickel and J. Puerto, Multi-facility ordered median problems: A further analysis,, Networks, 41 (2003), 1.  doi: 10.1002/net.10053.  Google Scholar [3] E. Minieka, The optimal location of a path or tree in a tree network,, Networks, 15 (1985), 309.  doi: 10.1002/net.3230150304.  Google Scholar [4] S. Nickel, and J. Puerto, "Location Theory: A Unified Approach,", 1st edition, (2005).   Google Scholar [5] W. Ogryczak and A. Tamir, Minimizing the sum of $k$ largest functions in linear time,, Information Processing Letters, 85 (2003), 117.  doi: 10.1016/S0020-0190(02)00370-8.  Google Scholar [6] J. Puerto, and A. Tamir, Locating tree-shaped facilities using the ordered median objective,, Mathematical Programming, 102 (2005), 313.  doi: 10.1007/s10107-004-0547-2.  Google Scholar [7] A. Tamir, J. Puerto and D. Pérez-Brito, The centdian subtree on tree networks,, Discrete Applied Mathematics, 18 (2002), 263.  doi: 10.1016/S0166-218X(01)00199-8.  Google Scholar [8] H. J. Tang, T. C. E. Cheng and C. T. Ng, Finite dominating sets for the multi-facility ordered median problem in networks and algorithmic applications,, Computers & Industrial Engineering, 57 (2009), 707.  doi: 10.1016/j.cie.2009.01.015.  Google Scholar [9] H. J. Tang, T. C. E. Cheng and C. T. Ng, Multi-facility ordered median problems in directed networks,, Journal of Systems Science and Complexity, 24 (2011), 61.  doi: 10.1007/s11424-011-9327-2.  Google Scholar [10] P. M. Vaidya, An algorithm for linear programming which requires $O((m+n)n^2+(m+n)^{1.5}nL)$ arithmetic operations,, Mathematical Programming, 47 (1990), 175.  doi: 10.1007/BF01580859.  Google Scholar [11] B.-F. Wang, Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network,, Journal of Algorithms, 34 (2000), 90.  doi: 10.1006/jagm.1999.1020.  Google Scholar
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text:
The change of state from a gas to a liquid. Single choice. (5 Points) water cycle evaporation precipitation condensation

Question:

the change of state from a gas to a liquid. Single choice.
(5 Points)
water cycle
evaporation
precipitation
condensation

How does the graph of g(x)=-(x+3)^4 compare to the parent function of f(x)= x^4

how does the graph of g(x)=-(x+3)^4 compare to the parent function of f(x)= x^4...

Thermal energy from the surface of the Earth is transferred to the atmosphere by _________________________.

Thermal energy from the surface of the Earth is transferred to the atmosphere by _________________________....

What is the contents of the vector names after the following statements? vector names; names.push_back(“Ann”); names.push_back(“Bob”); names.pop_back(); names.push_back(“Cal”);

What is the contents of the vector names after the following statements? vector names; names.push_back(“Ann”); names.push_back(“Bob”); names.pop_back(); names.push_back(“Cal”);...

A sound wave (a periodic longitudinal wave) from a loudspeaker travels from air into water. The frequency of the wave does not change, because the loudspeaker producing the sound determines the frequency. The speed of sound in air is 343 m/s, whereas the speed in fresh water is 1482 m/s. When the sound wave enters the water, does its wavelength increase, decrease, or remain the same

A sound wave (a periodic longitudinal wave) from a loudspeaker travels from air into water. The frequency of the wave does not change, because the loudspeaker producing the sound determines the frequency. The speed of sound in air is 343 m/s, whereas the speed in fresh water is 1482 m/s. When the so...

What was the first represenative goverment created in the new world?

What was the first represenative goverment created in the new world?...

Simplify 1+3+5+7+...199/2+4+6+8+...+200

Simplify 1+3+5+7+...199/2+4+6+8+...+200...

What is the recursiveformula for this geometric sequence? 4,-12,36,108

what is the recursiveformula for this geometric sequence? 4,-12,36,108...

When the Wicked Witch of the West appears (time code 29:17), the scene has two witches present. How does the film distinguish between the "wicked" witch and the "good" witch? Consider aspects such as costume, dialogue, body language, the relationship of the character to others in the scene, and so on.

When the Wicked Witch of the West appears (time code 29:17), the scene has two witches present. How does the film distinguish between the "wicked" witch and the "good" witch? Consider aspects such as costume, dialogue, body language, the relationship of the character to others in the scene, and so o...

Use the model to find 3x126

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f(x) = 3 4 x + 12, find the value of x when f(x) = x....

A survey of 4,000 of Ron’s customers showed that 60% of the customers buying chewing gum would buy organic chewing gum if it was placed near the register. In the first month, 1,680 out of 4,000 customers bought the organic gum. How many people were expected to buy the gum? 1,600 1,680 2,320 2,400

A survey of 4,000 of Ron’s customers showed that 60% of the customers buying chewing gum would buy organic chewing gum if it was placed near the register. In the first month, 1,680 out of 4,000 customers bought the organic gum. How many people were expected to buy the gum? 1,600 1,680 2,320 2,400...

Make an expression with the given table

Make an expression with the given table...

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Why did the delegates draft a new plan instead of revising The Articles of Federation?...


text:
## Missouri Journal of Mathematical Sciences

### Generalized Pythagorean Triples and Pythagorean Triple Preserving Matrices

#### Abstract

Traditionally, Pythagorean triples (PT) consist of three positive integers, $(x, y, z) \in \mathbb{Z}^3_+$, such that $x^2 + y^2 = z^2$, and Pythagorean triple preserving matrices (PTPM) $A$ are $3 \times 3$ matrices with entries in the real numbers $\R$, such that the product $(x, y, z)A$ is also a Pythagorean triple. In this paper, we study PT and PTPM from the view of projective geometry, and extend the results concerning PT and PTPM from integers to any commutative ring with identity. In particular, we use the method of polynomial parametrization for projective conics to obtain the general form of PT over any commutative ring with identity. In addition, we view the PTPM as projective transformations and formulate the general form of a PTPM over any commutative ring with identity.

#### Article information

Source
Missouri J. Math. Sci., Volume 21, Issue 1 (2009), 3-12.

Dates
First available in Project Euclid: 14 September 2011

https://projecteuclid.org/euclid.mjms/1316032675

Digital Object Identifier
doi:10.35834/mjms/1316032675

Mathematical Reviews number (MathSciNet)
MR2503169

Zentralblatt MATH identifier
1210.11041

#### Citation

Tikoo, Mohan; Wang, Haohao. Generalized Pythagorean Triples and Pythagorean Triple Preserving Matrices. Missouri J. Math. Sci. 21 (2009), no. 1, 3--12. doi:10.35834/mjms/1316032675. https://projecteuclid.org/euclid.mjms/1316032675


text:
# Why are “heavier” particles harder to detect than “lighter” ones?

Something I have read multiple times that I've never intuitively understood is that "heavier" particles are harder to detect than "lighter" ones... For example, I quote from Stephen Hawking's "The Grand Design" in relation to supersymmetry:

But various calculations that physicists have performed indicate that the partner particles corresponding to the particles we observe ought to be a thousand times as massive as a proton, if not even heavier. That is too heavy for such particles to have been seen in any experiments to date, but there is hope that such particles will eventually be created in the Large Hadron Collider in Geneva.

Could someone please explain, in simple terms, why heavy particles are harder to detect? Intuitively (to me, a non-physicist) it would seem that it should be the other way around, because a particle with more mass should interact more strongly with other matter.

-

They are harder to detect for mainly two reasons:

First, because they decay very quickly into lighter particles. Infact the heaviest elementary particle we know of ( top quark ) decay's so rapidly it is theoretically and technically impossible to measure it in any other way than indirectly through its decay products.

This poses another problem as to what statistical and theoretical basis one can with some certainty claim detection of such particles.

Another problem is that of production, since they have more mass, they require much more energy to produce, thus larger and more expensive accelerators.

-
@kalle43, your answer is wrong. Light neutrino is harder to detect than heavier proton, light neutral pion is harder to detect than heavier omega-hyperon. – voix Dec 18 '10 at 9:29
@voix This question has a pretty clear context. – user1708 Dec 18 '10 at 10:18
Let's just say it is incomplete. It's indeed not that heavier particles are harder to detect than lighter ones. What makes a particle harder to detect is how weakly it interacts. But, heavy particles are indeed unstable, because they pack a lot of energy in a tiny space, and precisely because of their potential to interact they will decay into lighter particles. So, it's more the instability of heavy particles which makes them inaccessible than their possibly low interactivity. – Raskolnikov Dec 18 '10 at 10:23
@Raskolnikov, there are many unstable light particles with a short half-life too. So-called resonances. – voix Dec 18 '10 at 11:29
I think that part of the problem here is that the OP wrote "detect" when "create as on shell particles" might have been better. It is much harder to make on on-shell top quark than it is a on-shell charm quark because of the energy-in-one-place requirement. And it takes more energy to make on-shells tau than on-shell muons. Contrast this with neutrinos which are easy to make (in all flavors if you are willing to stand back and let oscillation happen), but a real bear to detect. Detecting heavy quark states is complicated mostly by the abundant backgrounds. – dmckee Dec 18 '10 at 23:49

Hawking's phrase "seen in any experiments" really means the following. First you have to produce the heavy particle. Ordinary matter, consisting of electrons, protons and neutrons (the latter two themselves bound states of quarks), does not contain these new heavy particles, so they must be produced in high-energy collisions. By Einstein's famous formula $E=mc^2$, the production of very heavy particles (say with mass in the $TeV/c^2$ range) requires energies of order a $TeV$. Hence the LHC. Then there is the separate problem of detecting the particles once you have produced them. Your expectation that particles with more mass interact more strongly with other particles is true for gravity, but it is not true for the strong, weak or electromagnetic interactions which are relevant for the detection of elementary particles (the gravitational interaction is much too weak for these purposes). The strength of electromagnetic interactions depends on the electric charge and there are analogous "charges" for the weak and strong interactions. Detecting the particles is extremely complicated and depends on their mass, their lifetime, whether they have strong, weak or electromagnetic interactions or not, and what kinds of Standard Model background effects might have the same experimental signal as the particle you are trying to detect. This link from the LHC website has some information about particle detectors: http://public.web.cern.ch/public/en/LHC/LHCExperiments-en.html

-
Nice overview and I guess it's hard to add anything else without going into messy details of interactions. For now I'll link to the related question of detecting the Higgs boson which has pretty neat answers (of various technical levels) that elaborate a little on the stuff you mention. physics.stackexchange.com/questions/185/… – Marek Dec 19 '10 at 20:26


text:
• Volume 115, Issue 1

February 2005,   pages  1-116

• Zeta function of the projective curveaY21 =bX21 +cZ21 over a class of finite fields, for odd primesl

Letp andl be rational primes such thatl is odd and the order ofp modulol is even. For such primesp andl, and fore = l, 2l, we consider the non-singular projective curvesaY21 =bX21 +cZ21 defined over finite fields Fq such thatq = pα? l(mode).We see that the Fermat curves correspond precisely to those curves among each class (fore = l, 2l), that are maximal or minimal over Fq. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. Fore = 2l, we explicitly determine the ζ -function(s) for this class of curves, over Fq, as rational functions in the variablet, for distinct cases ofa, b, andc, in Fq*. Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves.

Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over Fq. As a consequence, when the field of definition of the curve(s) is fixed, this provides concrete information on the growth of class numbers for constant field extensions of the function field(s) of the curve(s).

• Moduli for decorated tuples of sheaves and representation spaces for quivers

We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.

• Localization of tight closure in two-dimensional rings

It is shown that tight closure commutes with localization in any two-dimensional ringR of prime characteristic if eitherR is a Nagata ring orR possesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic.

• Fields and forms on ρ-algebras

In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: ρ-algebras. We also define the Frölicher-Nijenhuis bracket in the non-commutative geometry on ρ-algebras.

• On the structure ofp-zero-sum free sequences and its application to a variant of Erdös-Ginzburg-Ziv theorem

Letp be any odd prime number. Letk be any positive integer such that $$2 \leqslant k \leqslant \left[ {\frac{{p + 1}}{3}} \right] + 1$$. LetS = (a1,a2,...,a2p−k) be any sequence in ℤp such that there is no subsequence of lengthp of S whose sum is zero in ℤp. Then we prove that we can arrange the sequence S as follows: $$S = (\underbrace {a,a,...,a,}_{u times}\underbrace {b,b,...,b,}_{v times}a'_1 ,a'_2 ,...,a'_{2p - k - u - v} )$$whereuv,u +v ≥ 2p - 2k + 2 anda -b generates ℤp. This extends a result in [13] to all primesp andk satisfying (p + 1)/4 + 3 ≤k ≤ (p + 1)/3 + 1. Also, we prove that ifg denotes the number of distinct residue classes modulop appearing in the sequenceS in ℤp of length 2p -k (2≤k ≤ [(p + 1)/4]+1), and $$g \geqslant 2\sqrt 2 \sqrt {k - 2}$$, then there exists a subsequence of S of lengthp whose sum is zero in ℤp.

• Inequalities for dual quermassintegrals of mixed intersection bodies

In this paper, we first introduce a new concept ofdual quermassintegral sum function of two star bodies and establish Minkowski’s type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov-Fenchel inequality and the Brunn-Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak’s mixed prosection bodies inequalities.

• On two functionals connected to the Laplacian in a class of doubly connected domains in space-forms

LetB1 be a ball of radiusr1 inSn (ℍn), and letB0 be a smaller ball of radiusr0 such thatB0B1. ForSn we considerr1π. Let u be a solution of the problem- δm = 1 in Ω :=B1 /B0 vanishing on the boundary. It is shown that the associated functionalJ (Ω) is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on Ω is maximal if and only if the balls are concentric.

• Isometric multipliers ofLp(G, X)

Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetLp(G, X) be the space of X-valued measurable functions whose norm-functions are in the usualLp. A left multiplier ofLp(G, X) is a bounded linear operator onBp(G, X) which commutes with all left translations. We use the characterization of isometries ofLp(G, X) onto itself to characterize the isometric, invertible, left multipliers ofLp(G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thelp-direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofLp(G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U(Ryf)(x). As an application, we determine the isometric left multipliers of L1Lp(G, X) and L1C0(G, X) whereG is non-compact andX is not the lp-direct sum of two non-zero subspaces. If G is a locally compact abelian group andH is a separable Hubert space, we define $$A^p (G,H) = \{ f \in L^1 (G,H):\hat f \in L^p (\Gamma ,H)\}$$ where г is the dual group of G. We characterize the isometric, invertible, left multipliers ofAp(G, H), provided G is non-compact. Finally, we use the characterization of isometries ofC(G, X) for G compact to determine the isometric left multipliers ofC(G, X) providedX* is strictly convex.

• Measure free martingales

We give a necessary and sufficient condition on a sequence of functions on a set Ω under which there is a measure on Ω which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019


text:
# How to efficiently compute the order of a prime number mod $l$ [duplicate]

Possible Duplicate:
Quick algorithm for computing orders mod n?

Let $l$ be an odd prime number. Let $p$ be a prime number such that $p \neq l$. I'd like to compute efficiently the order of $p$ mod $l$ using a calculator. For example, how can I compute efficiently the order of 2 mod 53? Thanks in advance.

My motivation came from this.

-

## marked as duplicate by Gerry Myerson, LVK, Old John, William, wentaway Sep 2 '12 at 21:40

The multiplicative group of integers mod $l$. Namely $(\mathbb{Z}/l\mathbb{Z})^*$. The order of an integer mod $l$ is standard in elementary number theory. –  Makoto Kato Jul 23 '12 at 23:43
It is cheap in general whether $p$ is a quadratic residue of $\ell$. In this case $2$ is a NR of $53$. The order divides $52$. The only interesting possibilities are $13$, $26$, $52$. It cannot be $13$, else $2$ would be a QR. So calculate $2^{26}\pmod{53}$, or maybe $2^{27}$. –  André Nicolas Jul 24 '12 at 0:12
In your case, and using Legrende's symbol, we get $$\left(\frac{2}{53}\right)=-1$$ since $\,53\neq \pm 1\mod 8\,$ , so $\,2\,$ has a chance to be a primitive root modulo $\,53\,$
Now, and doing operations modulo $\,53\,$ , we get: $$2^6=64=11\,\,,\,11^2=15\,\,,\,11^3=6\Longrightarrow 2^{18}=(2^6)^3=11^3=6\Longrightarrow$$ $$\Longrightarrow2^{26}=2^{18}\cdot 2^8=6\cdot 11\cdot 4=-1\Longrightarrow$$ $\,2\,$ is a primitive root in $\,\left(\Bbb Z/53\Bbb Z\right)^*\,$


text:
# Simple extensions of local fields

Let $L/K/\mathbb Q_p$ be finite extensions of local fields and let $v_L$ and $v_K$ be normalised discrete valuations on $L$ and $K$ respectively.

My question is quite a general one:

If the valuation rings are given by $\mathcal O_L=\mathcal O_K[\alpha]$ for some $\alpha\in \mathcal O_L$, then is it true that $L=K(\alpha)$ as fields?

Can this also be true in the case where $\mathcal O_K$ is a Dedekind domain, $K$ its quotient field, $L$ a finite separable extension of $K$ and $\mathcal O_L$ the integral closure of $\mathcal O_K$ in $L$?

$L$ is the fraction field of $\mathcal{O}_L$, so it certainly contains $\mathcal{O}_K$, so it contains $K$, and it contains $\alpha$, so $L\supset K(\alpha)$.
Conversely, $K(\alpha)$ contains $\mathcal{O}_K$, and it contains $\alpha$, so it contains $\mathcal{O}_L$, so it contains $Frac(\mathcal{O}_L) = L$.
Note that this doesn't use anything about the existence of a discrete valuation. It only relies on the fact that $L = Frac(\mathcal{O}_L)$ and $K = Frac(\mathcal{O}_K)$.


text:
This is a detailed discussion for Eratosthenes’s Issue #21.

The primary feature we are trying to implement on Eratosthenes is to include library version in its various libraries. Currently, Eratosthenes will always hold the latest release of external library, and make sure the archive of each library is up-to-date. However, this is not a desired feature since Arduino now has so many different configurations, and it is not always a good idea to provide the latest library due to the compatibility issue. So the best approach is we provide different versions in our system, and let user to choose what version of library they want (just like what we do in npm’s package.json or composer’s composer.json). If they do not specify a version, then by default the latest version will be provided. This is a good practice as we can enhance user’s usability and system’s functionality, and meanwhile most likely user’s current experience will not be affected as well.

So how do we do that? Currently there are several consideration in our design:

1. There may be multiple versions for each library
2. For each version, the supporting architectures may also be different
3. The version may not always be perfect semantic versioning (weird name such as ‘@first-version.1.0’ are expected)
4. In future, each user of the system (currently only codebender) may update their default supported version for each library

First, in each library, there may be more than one version, so we need to find a easy way to maintain them. The solution we proposed is to create a new table called Version. In this table, it holds a foreign key library_id that refer to the library id, and we use a ManyToOne relationship with the Library table to describe it. In this way, we can easily find which library a version belongs to, and also from a given library, find all the versions it has.

Secondly, as Arduino is an open source platform, people can create their own Arduino controller with different hardware architecture. Accordingly, there may be different library for different architecture as well. The solution we proposed is firstly create an Architecture table to store the architectures used by various Arduino. Then we use a relationship to connect architecture with the version of the library. So for a library given, we can find out all versions it has; by using this table, we know what is the supported architectures for this version.

Next, since there is no compulsory standard to name the version (although the semantic versioning is recommended), we need to find a way to unify the naming. Also, it the name uses some special characters, the file system may not be able to display them correctly.

One possible solution is we hash all the version name, so we will always get unique and fixed-length name. But on the other hand, the system admin will have a hard time to do the maintaining work because it is not human readable.

Another approach is we will make use the version id in the database, plus the adjusted version number. e.g. 130-1.1 (the id in the Version table). In this way, the filename is always compatible with the file system, and it is also easier to maintain things as the name is human readable. However, that would means you will always need to insert the library into the database first, then generate the name. In future, if the database has some problem, then basically we need to rebuild the whole file system!

Finally, our decided approach is we will manually adjust the version name when we are creating this version for the system. As we have discussed, we discovered that only a small amount of libraries are using special characters in their version names. Therefore, when we are generating the version to be used on file system, we will escape those characters. e.g. A version named ‘1.3.?’ will be escaped into things like ‘1.3.-‘. In such way, we can decouple the database from the versioning, at the same time the readability is guaranteed as well.
So the folder structure for each library is as follows:

Finally, in order to store the default version used by each user, we need to create another two tables: Partner and Preference, where the authentication information for different partners(users) as well as their default version preference will be hold.

The Database diagram is as following:

We are creating Doctrine entities according to this design now, and next step we will start to write fixture as well as migration for our models.


text:
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Preparing Juniors for future

Everybody knows that in today's world competiton in academics has got a lot tougher.Students (primary,middle and high school) also give various olympiads,exams and tests to know their ability,misconceptions and other errors.As in Brilliant not only me but everybody knows that topics such as Mechanics,Calculus,Number Theory,Combinatorics etc get a lot tougher for higher levels(level3,level4,level5).Many users on Brilliant who are at school level are not able to solve these problems.what kind of preparation would you recommend for the school levels form clases 7 to 12 so that are able to solve these problems and understand the basic concepts properly ? (I mean any kind of books,resources,sites to help improve our concepts better.)

Note by Banfig Teure
4 years, 1 month ago

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Hi Banfig,

I'll see if I can link you to some other old advice on Brilliant regarding what approaches people on Brilliant take to improve at topics that they almost never receive formal instruction in.

This discussion about what classes people take and what they do outside of them is probably relevant to you.

For historic book recommendations, I would go to the search bar and search notes for "books" and see what other people have recommended.

In my opinion, additional resources such as books and websites are pretty useless unless you have the desire to actually use them. The most valuable part of my education growing up was not that I had great text-books and solid syllabi, but that I had friends and parents that helped me make them worth reading.

Staff - 4 years, 1 month ago

thanks

- 4 years, 1 month ago


text:
# An interior penalty discontinuous Galerkin method for the time-domain acoustic-elastic wave interaction problem

In this paper, we consider numerical solutions of a time domain acoustic-elastic wave interaction problem which occurs between a bounded penetrable elastic body and a compressible inviscid fluid. It is also called the fluid-solid interaction problem. Firstly, an artificial boundary is introduced to transform the original transmission problem into a problem in a bounded region, and then its well-posed results are presented. We then employ a symmetric interior penalty discontinuous Galerkin method for the solution of the reduced interaction problem consisting of second-order wave equations. A priori error estimate in the energy norm is presented, and numerical results confirm the accuracy and validity of the proposed method.

## Authors

• 8 publications
• 34 publications
• 8 publications
• 8 publications
• ### A time-dependent FEM-BEM coupling method for fluid-structure interaction in 3d

We consider the well-posedness and a priori error estimates of a 3d FEM-...
08/10/2020 ∙ by Heiko Gimperlein, et al. ∙ 0

• ### Influence of the SIPG penalisation on the numerical properties of linear systems for elastic wave propagation

Interior penalty discontinuous Galerkin discretisations (IPDG) and espec...
12/15/2017 ∙ by Uwe Köcher, et al. ∙ 0

• ### Convergence of the perfectly matched layer method for transient acoustic-elastic interaction above an unbounded rough surface

This paper is concerned with the time-dependent acoustic-elastic interac...
07/23/2019 ∙ by Changkun Wei, et al. ∙ 0

• ### Multipatch Discontinuous Galerkin IGA for the Biharmonic Problem On Surfaces

We present the analysis of interior penalty discontinuous Galerkin Isoge...
12/07/2020 ∙ by Stephen E. Moore, et al. ∙ 0

• ### A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form

We present a stable discontinuous Galerkin (DG) method with a perfectly ...
10/15/2019 ∙ by Kenneth Duru, et al. ∙ 0

• ### Time-Dependent Wave-Structure Interaction Revisited: Thermo-piezoelectric Scatterers

In this paper, we are concerned with a time-dependent transmission probl...
02/08/2021 ∙ by George C. Hsiao, et al. ∙ 0

• ### Discontinuous Galerkin methods for a dispersive wave hydro-sediment-morphodynamic model

A dispersive wave hydro-sediment-morphodynamic model developed by comple...
10/11/2020 ∙ by Kazbek Kazhyken, et al. ∙ 0

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## 1 Introduction

In a time-domain fluid-solid interaction (FSI) problem, an incident acoustic wave is scattered by a bounded elastic obstacle immersed in a homogeneous, compressible and inviscid fluid. The problem of determining the scattered wave field plays prominent roles in many scientific and engineering areas, such as detecting and identifying submerged objects, geophysical exploration, medical imaging, seismology, and oceanography (such as [30]). Because of the difficulties of dealing with the time dependence, the FSI problem is usually studied based on a time-harmonic setting. Various efficient and accurate numerical methods have been developed to solve the time-harmonic FSI problem. Some of them provide proper guidance also for time-domain problems. Popular methods include the boundary integral equation (BIE) method, see [33, 46] and coupling methods such as the so-called coupled FEM-BEM method, see [47]. An artificial boundary or an absorbing layer ([43]) can be introduced to reduce the original unbounded problem to a bounded problem which could be solved using field equation solvers such as the finite element method.

The studies of this time-domain scattering problems now gain more and more attention since the time-domain model gives more information about the wave, more general material, and nonlinearity, see [7, 39]. There are relatively fewer mathematical analysis and numerical studies for the time-domain wave scattering problems. As to the numerical studies towards the time-domain wave scattering problems, the main challenge is how to handle the problem defined on the unbounded domain. Many approaches attempted to solve the time-domain problems numerically are developed, such as coupling of boundary element and finite element with different time quadratures, see [17, 21], time-domain boundary elemment methods  [26, 25], and to name a few. The authors of [27] gave the exact non-reflective boundary condition for the three dimensional time-domain wave scattering problem in 1995. For basic isotropy wave equation with constant coefficients, a planar PML method in one space direction designed for some particular domain is considered in [29]. In [8], the mathematical analysis of a time-domain DtN operator and the convergence analysis of the PML method for acoustic wave scattering were given. In [39], the time-domain exact nonreflective boundary conditions in both two and three dimensions were computed and analyzed. For the time-domain wave FSI problem, the rigorous mathematical study is still an open challenge, and related work include [19, 31, 22, 6]. Concerning its numerical solutions, the authors of  [32] applied time-domain boundary integral equations to time-domain FSI problem and analyzed the resulting nonlocal initial-boundary problem, motivated by the time-harmonic FSI problems. Coupling methods are also utilized in  [17, 18, 20, 23, 24] for solving the time-domain FSI problems.

Instead of numerical methods mentioned above, in this paper, we focus on using discontinuous Galerkin (DG) methods, which has natural advantages for dealing with the time-domain FSI problem, see [16]. The original DG methods were proposed for the numerical solution of hyperbolic neutron transport equations, as well as for discontinuities in some elliptic and parabolic problems. For second order wave equations, various DG methods have been proposed, which first reformulate the original wave problem to a first-order hyperbolic system, such as the local discontinuous Galerkin (LDG) methods in [13, 42, 9], and we refer to [10, 11, 34, 40] for a review of some other DG methods for the first order wave equations. The first DG method for the original second-order formulation of acoustic wave equation was proposed in [36], which is based on a nonsymmetric formulation. Here, we propose and analyze a symmetric interior penalty DG (SIPDG) method to solve the time-domain FSI problem. It is the same method used in [28] for the spatial discretization of the second-order scalar wave equation. Compared to the nonsymmetric formulation in [36], the symmetric discretization of the second-order form wave equation offers extra benefits such as a positive definite stiffness matrix and hence is free of any (unnecessary) damping. One can refer to [3] for details of the DG methods for second order equations. Finally we note that another DG formulation for wave equations in second order for is the energy based method proposed in [4] and extended to the coupled acoustic-elastic problem in [5].

The remainder of the paper is organized as follows. We first describe the original time-domain FSI problem in Section 2. Then in Section 3, the unbounded problem is reduced to a bounded initial-boundary value problem. In Section 4, we establish a priori error estimates for the IPDG solution of the reduced problem. Numerical experiments are presented in Section 5 to confirm the theoretical results, and finally, a conclusion is made in Section 6. For the sake of completeness, we provide the mathematical analysis towards well-posedness of the reduced problem introduced in Section 3 in the appendix.

## 2 Model

Here we study the same model as in [31], where the authors gave mathematical analysis from the aspect of integral equation method. The statement of the model is as follows. Suppose is a bounded domain of a elastic body with boundary , which is enclosed by the unbounded homogenous compressible inviscid fluid domain (see Figure 1), and a finite time interval . Given an incident wave , the scattered wave is generated by the induced solid in the fluid domain . In the fluid domain , the governing equations are the linearized Euler equation and the linearized equation of continuity

 ρ1∂v∂t+∇p=0,∂p∂t+c2ρ1div v=0,(x,t)∈Ω+×J, (2.1)

where is the velocity field, , and are the pressure, the density and the speed of sound in the fluid respectively.

For an irrational flow, the velocity potential s be the velocity potential satisfies:

 v=−∇φ,andp=ρ1∂φ∂t

Then, the wave equation for takes the form:

 1c2∂2φ∂t2−Δφ=0,(x,t)∈Ω+×J. (2.2)

The elastic wave in is described by the displacement field , which satisfies the dynamic linear elastic equation:

 ρ2∂2u∂t2−div(σ(u))=0,(x,t)∈Ω×Jσ(u)=λtr(ε(u))I+2με(u)ε(u)=12(∇u+(∇u)T). (2.3)

Here is the constant density of the elastic body, which is assumed to be homogeneous and isotropic with the Lamé constants and such that and .

The velocity potential in equation (2.2) and the elastic displacement field in equation (2.3) are coupled by the transmission conditions on , together with the homogeneous initial conditions, we can get an initial-boundary value problem:

 ρ2∂2u∂t2−div(σ(u)) =0, in~{}Ω×J, (2.4) 1c2∂2φ∂t2−Δφ =0, in~{}Ω+×J, (2.5) σ(u)n =−ρ1(∂φ∂t+∂φi∂t)n, on~{}Γ×J, (2.6) ∂φ∂n =−∂φi∂n−∂u∂t⋅n, on~{}Γ×J, (2.7) r12(∂φ∂n+∂φ∂t) →0as~{}r=|x|→∞, a.e.~{}t∈J, (2.8) u|t=0=∂u∂t∣∣∣t=0=0, x∈Ω~{}and~{}φ|t=0=∂φ∂t∣∣∣t=0=0, x∈Ω+. (2.9)

Here

is the unit outward vector on

from towards , is the given incident field and equation (2.8) is the radiation condition for .

Before the discussion, we first introduce the definitions of some relevant Sobolev spaces and norms. Let be the function space consisting of all square integrable functions over equipped with the norm

 ∥u∥0,Ω=(∫Ω|u(x,t)|2dx)1/2.

For , the standard Sobolev space is denoted by

 Hs(Ω)={Dαu∈L2(Ω)~{}for all~{}|α|≤s}

with the norm

 ∥u∥s,Ω=⎛⎝∫Ω∑|α|

and the trace functional space for under the inner produce

 ⟨u,v⟩Γ=∫Γuvγ.

It is clear to note that and are the dual space of and . We also denote and . For any , the Frobenius norm is defined as:

 ∥∇u∥F(Ω)=(2∑j=1∫Ω|∇uj|2dx)1/2.

And a simple calculation gives:

 ∥∇u∥2F(Ω)+∥div~{}u∥20,Ω≤C∥u∥21,Ω, (2.10)

where is a positive constant.

Furthermore, for we will make use of the Bochner space , endowed with the norm:

 ∥w(x,t)∥Lq(J;H1(Ω))=⎧⎪ ⎪⎨⎪ ⎪⎩(∫J∥w∥q1,Ω)1/q1≤q<∞,esssup0≤t≤T∥w∥1,Ωdtq=∞,

and the Bochner space , endowed with norm:

 ∥w(x,t)∥H1(J;Hs(Ω))=(∫J(∥w∥2s,Ω+∥wt∥2s,Ωdt))1/2.

## 3 The reduced problem in bounded domain

To reduce this exterior problem to a problem in a bounded domain, we impose the first order approximate boundary condition on the artificial boundary (see Figure 2):

 ∂φ∂n=−∂φ∂t,~{}on~{}ΓR×J,

where is the outward unit vector from on . Then the reduced problem on the bounded domain reads:

 ρ2∂2u∂t2−div(σ(u)) =0, ~{}in~{}Ω×J, (3.1) 1c2∂2φ∂t2−Δφ =0, ~{}in~{}ΩR×J, (3.2) σ(u)n =−ρ1(∂φ∂t+∂φi∂t)n, ~{}on~{}Γ×J, (3.3) ∂φ∂n =−∂φi∂n−∂u∂t⋅n, ~{}on~{}Γ×J, (3.4) ∂φ∂n =−∂φ∂t, ~{}on~{}ΓR×J, (3.5) u|t=0=∂u∂t∣∣∣t=0 =0,x∈Ω,and φ|t=0=∂φ∂t∣∣∣t=0=0,x∈ΩR. (3.6)

Then, we present the well-posedness and stability of the reduced problem in the follows.

### 3.1 Well-posedness and stability

First of all, we need to show that the reduced interaction problem in the bounded domain is well-posed and stable.

###### Theorem 3.1.

Suppose the incident wave , and . Then (3.1)-(3.6) has a unique solution satisfying:

 u∈L2(J;H1(Ω))∩H1(J;L2(Ω)),
 φ∈L2(J;H1(ΩR))∩H1(J;L2(ΩR)),

and we have the following stability estimate:

 maxt∈[0,T](∥ut∥0,Ω+∥∇u∥F(Ω)+∥∇⋅u∥0,Ω) ≤ C(∥∥φit∥∥2L1(J;H−1/2(Γ))+∥∥∇φi∥∥2L1(J;H−1/2(Γ))+maxt∈[0,T]∥∥φitt∥∥2−12,Γ +maxt∈[0,T]∥∥∇φit∥∥2−12,Γ+∥φittt∥2L1(J;H−1/2(Γ))+∥∇φitt∥2L1(J;H−1/2(Γ))), (3.7)

and

 maxt∈[0,T](∥φt∥0,ΩR+∥∇φ∥0,ΩR) ≤ C(∥∥φit∥∥2L1(J;H−1/2(Γ))+∥∥∇φi∥∥2L1(J;H−1/2(Γ))+maxt∈[0,T]∥∥φitt∥∥2−12,Γ (3.8) +maxt∈[0,T]∥∥∇φit∥∥2−12,Γ+∥φittt∥2L1(J;H−1/2(Γ))+∥∇φitt∥2L1(J;H−1/2(Γ))). (3.9)

Although different boundary conditions are considered to develop the reduced problem, the proof of above theorem 3.1 is analogous to the proof of Theorem 3.2 in [6]. We omit to present the proof of Theorem 3.1 in this section, and report the detailed proof of Theorem 3.1 in the appendix for the sake of completeness.

### 3.2 Variational formulation

The standard variational formulation of problem (3.1)-(3.6) is as follows: For given incident wave , find

 (u,φ)∈L2(J;H1(Ω))×L2(J;H1(ΩR)),

with

such that, :

 ρ2ρ1⟨utt,v⟩+1c2⟨φtt,ϕ⟩+a(u,φ;v,ϕ)=L(v,ϕ). (3.10)

with the initial conditions

 u|t=0=ut|t=0=0,φ|t=0=φt|t=0=0.

Here denotes the duality pairing between spaces and with the associated domain, and and are defined as:

 a(u,φ;v,ϕ) = (3.11) +(∇φ,∇ϕ)0,ΩR−(ut⋅n,ϕ)0,Γ+(φt,ϕ)0,ΓR, L(v,ϕ) = (∂φi∂n,ϕ)0,Γ−(φitn,v)0,Γ (3.12)

with the standard inner product in -space.

## 4 The IPDG method

### 4.1 Spaces, jumps and averages

Assume that with Lipschitz boundary is regularly divided into disjoint elements by mesh such that , where is a triangle or quadrilateral in 2D, or a tetrahedron or hexahedron in 3D. Similarly, The regular meshes partitions the fluid domain into disjoint elements such that . The diameter of element is denoted by , and is the mesh size given by .

The boundary of the elastic body , i.e., is approximated by the boundary edges of the subdivision: The set of boundary edges of is , since , the edges in are also the boundary edges of , so the sets of boundary edges of are , . We denote by and the set of interior edges of the subdivision and respectively.

For any real number , the broken Sobolev spaces and are defined as :

 Hm(Eh)={v∈L2(Ω):∀E∈Eh,v|E∈Hm(E)},

and

 Hm(Kh)={φ∈L2(ΩR):∀K∈Kh,φ|K∈Hm(K)},

with the broken Sobolev norms:

 ∥v∥m,Eh=⎛⎝∑E∈Eh∥v∥2m,E⎞⎠12,|φ∥m,Kh=⎛⎝∑K∈Kh∥φ∥2m,K⎞⎠12.

In particular, we also define the broken gradient seminorm:

 ∥∇v∥0,Eh=⎛⎝∑E∈Eh∥∇v∥20,E⎞⎠12, ∥∇φ∥0,Kh=⎛⎝∑K∈Kh∥∇φ∥20,K⎞⎠12.

If , the trace of along any side of each element is well defined. Let be the edge between the elements and , then the jump and average of a scalar function on are given by:

 {φ}=12(φ|K1+φ|K2),and[φ]=φ|K1nK1+φ|K2nK2,

respectively, where is the outward unit normal from to and likewise for . On the boundary edge or , we extend the definition: , where is the element that or .

Similarly, for a vector valued function , the jump and average of a vector function on are given by:

 {v}=12(v|E1+v|E2),%and[v]=v|E1⋅nE1+v|E2⋅nE2,

respectively, where is outward unit normal from to and likewise for . Similarly, on the boundary edge : , where is the element that .

### 4.2 Spatial discretization

For given partitions of , of and an approximation order , we can approximate the solution of (3.1)-(3.6) in the finite element subspaces

 Dk(Eh):={v∈L2(Ω): ∀E∈Eh, vi|E∈Pk(E),i=1,2},

and

 Dk(Kh):={φ∈L2(ΩR): ∀K∈Kh, φ|K∈Pk(K)},

of the broken Sobolev space and for . Here denotes the space of polynomials of degree at most .

Then we consider the following semidiscrete DG approximation of (3.1)-(3.6): find and such that

 ρ2ρ1⟨uhtt,v⟩+1c2⟨φhtt,ϕ⟩+ah(uh,φh;v,ϕ)=Lh(v,ϕ), (4.1)

holds for any and . Here, the discrete bilinear form and linear form are given by the IPDG discretization as:

 ah(uh,φh;v,ϕ) = ∑E∈Eh(λρ1(∇⋅uh,∇⋅v)0,E+2μρ1(ε(uh):ε(v))F(E)) (4.2) −∑e∈Γ1I∫e{σ(uh)⋅n}[v]dS−∑e∈Γ1I∫e{σ(v)⋅ν}[uh]dS +∑K∈Kh(∇φh,∇ϕ)0,K−∑e∈Γ2I∫e{∂φh∂n}[ϕ]dS−∑e∈Γ2I∫e{∂ϕ∂n}[φh]dS +∑e∈Γ2I∫eα|e|β[φh][ϕ]dS+∑e∈Γ1I∫eα|e|β[uh][v]dS+∑e∈Γh∫eφhtnvdS −∑e∈Γh∫euht⋅nϕdS+∑e∈ΓRh∫eφhtϕdS Lh(v,ϕ) = ∑e∈Γh∫e∂φi∂nϕdS−∑e∈Γh∫eφitnvdS. (4.3)

The interior penalty stabilization function penalizes the jumps over the edges of and , where is a positive parameter independent of the local mesh sizes. Here simply means the length of and we have

 |e|≤hE/K≤h,∀e∈∂E~{}or~{}∂K,

with , and when the method is superpenalized.

We define the space and . On and , we define the DG energy norm as

and

 ∥ϕ∥2h:=∑K∈Kh∥∇φ∥20,K+1h∑e∈Γh∪Γ2I∥[ϕ]∥20,e.

The consistency of the scheme is straightforward since the jumps at element boundaries vanishes when and for . Next we will discuss the property of the bilinear form . For more details about the IPDG method, one can refer to [28]

###### Lemma 4.1 (Coercivity).

There exists a positive constant independent of such that for all and

 ah(v,ϕ;v,ϕ) ≥ Ccoer⎛⎜⎝∑E∈Eh∥ε(v)∥2F(E)+∑e∈Γh∪Γ1I∥[v]∥20,e (4.4) +∑K∈Kh∥∇ϕ∥20,K+∑e∈Γh∪Γ2I∥[ϕ]∥20,e⎞⎟⎠.

Proof: By Cauchy-Schwarz inequality,

 ∑e∈Γ1I∫e{σ(v)⋅n}[v]dS ≤ ∑e∈Γ1I∥{σ(v)⋅n}∥0,e∥[v]∥0,e ≤ ∑e∈Γ1I∥{σ(v)⋅n}∥0,e(1|e|β)12−12∥[v]∥0,e.

Consider the average of the fluxes for an interior edge shared by and and apply the trace theorem, we have:

 ∥{σ(v)⋅n}∥0,e ≤ 12∥(σ(v)⋅n)|Ee1∥0,e+12∥(σ(v)⋅n)|Ee2∥0,e ≤ C+2h−12Ee1∥σ(v)∥F(Ee1)+C+2h−12Ee1∥σ(v)∥F(Ee2).

Assume and , we have:

 ∫e{σ(v)⋅n}[v]dS ≤ C+2|e|β2(h−12Ee1∥σ(v)∥F(Ee1)+h−12Ee1∥σ(v)∥F(Ee2))(1|e|β)12∥[v]∥0,e ≤ C+2(hβ2−12Ee1+hβ2−12Ee1)(∥σ(v)∥2F(Ee1)+∥σ(v)∥2F(Ee2))12(1|e|β)12∥[v]∥0,e ≤ C+(∥σ(v)∥2F(Ee1)+∥σ(v)∥2F(Ee2))12(1|e|β0)12∥[v]∥0,e.

We denote the maximum number of neighbors an element can have.

 ∑e∈Γ1I∫e{σ(v)⋅n}[v]dS ≤ ≤ C+√n0(∑E∈Eh∥σ(v)∥2F(E))12(∑e∈Γ1I1|e|β∥[v]∥20,e)12.

Using Young’s inequality, we have for

 ∑e∈Γ1I∫e{σ(v)⋅n}[v]dS ≤

Following a similar steps, we get:

 ∑e∈Γ2I∫e{∂ϕ∂n}[ϕ]dS ≤ δ2∑K∈Kh∥∇ϕ∥20,K+C2+n02δ∑e∈Γ2I1|e|β∥[ϕ]∥20,e.

Thus we obtain a lower bound for :

 ah(v,ϕ;v,ϕ) ≥ ∑E∈Eh(λρ1∥∇⋅v∥20,E+2μρ1∥ε(v)∥2F(E)−δ∥σ(v)∥2F(E)) +∑e∈Γ1Iαδ−C2+n0δ|e|β∥[v]∥20,e+∑K∈Kh(1−δ)∥∇ϕ∥20,K+∑e∈Γ2Iαδ−C2+n0δ|e|β∥[ϕ]∥20,e −C∑e∈Γh(∥ϕt∥20,e+∥v∥20,e−∥ϕ∥20,e−∥vt∥20,e)−C∑e∈ΓRh(∥ϕt∥20,e+∥ϕ∥20,e).

For any , using the Young’s inequality, similar as (38) we have

 ∥v∥20,e≤C∥vt∥20,e∥ϕ∥20,e≤C∥ϕt∥20,e.

Then, we have

 ah(v,ϕ;