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If I pass the wrong key will AES (decryption) in GCM mode detect that I am using the wrong key and give me an error message (..) ?
Yes, inasmuch as the implementation gives an error message when the authentication check fails (which is expected from any good implementation).
Argument: from the structure of GCM it can be shown, under a model of AES as a PRP, that the AES-GCM authenticator tag is a PRF of the key and IV; thus that if one changes the key randomly, and nothing else, the probability that the unchanged message (including IV) is accepted is about $$2^{-k}$$, where $$k$$ is the bit size of the authenticator tag; thus negligible.
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# Antibody design using LSTM based deep generative model from phage display library for affinity maturation
## Abstract
Molecular evolution is an important step in the development of therapeutic antibodies. However, the current method of affinity maturation is overly costly and labor-intensive because of the repetitive mutation experiments needed to adequately explore sequence space. Here, we employed a long short term memory network (LSTM)—a widely used deep generative model—based sequence generation and prioritization procedure to efficiently discover antibody sequences with higher affinity. We applied our method to the affinity maturation of antibodies against kynurenine, which is a metabolite related to the niacin synthesis pathway. Kynurenine binding sequences were enriched through phage display panning using a kynurenine-binding oriented human synthetic Fab library. We defined binding antibodies using a sequence repertoire from the NGS data to train the LSTM model. We confirmed that likelihood of generated sequences from a trained LSTM correlated well with binding affinity. The affinity of generated sequences are over 1800-fold higher than that of the parental clone. Moreover, compared to frequency based screening using the same dataset, our machine learning approach generated sequences with greater affinity.
## Introduction
Antibodies are powerful tools for therapeutic and biological research in the present era1,2. In vitro display technology (e.g., phage display, ribosome display) is an efficient method of antibody discovery. The display system features powerful high-throughput and excellent adaptability to low immunogenic or highly toxic antigens. However, antibodies from display libraries tend to have moderate binding activity3. One reason for this is the limited library size4. Therefore, an additional affinity maturation step is necessary to thoroughly explore the sequence space.
However, affinity maturation experiments can be costly and laborious. Traditionally, clonal antibody screening is achieved by randomly picking phage clones and applying Sanger sequencing analysis. Recently, next-generation sequencing (NGS) technologies have been adapted for the in-depth evaluation of the complementarity determining region (CDR) sequence landscape5. However, frequently read sequences do not necessarily have high affinity. It could take time to optimize panning conditions for improving the accuracy. Therefore, there is a clear demand to find measures other than frequency when utilizing NGS derived sequences to discover promising candidates.
To address these issues, we employed a long short term memory network (LSTM)-based sequence generation and prioritization procedure to efficiently discover sequences with higher affinity. LSTM is a widely used deep generative model in natural language processing6,7. We used a trained LSTM model to sample virtual sequences and avoid combinatorial explosion in the sequence space. Then, we prioritized the most promising sequences according to their likelihood as calculated by the trained LSTM.
To demonstrate the effectiveness of our method, we applied it to the affinity maturation of antibodies against a hapten. In recent years, anti-hapten antibodies are expected to find use not only as research tools, but also as industrial reagents, and in diagnosis and therapy8,9,10. However, obtaining antibodies against haptens is difficult because of the limitation of antigenic epitope. Therefore, the development of more powerful screening methods for obtaining high-affinity antibodies is desired.
We confirmed that the likelihood of a sequence using a trained LSTM correlated well with binding affinity in generated sequences and demonstrated that our machine learning approach generated sequences with greater affinity compared to frequency based screening using the same dataset.
## Results
### Overall workflow
Figure 1 shows the overall workflow of our LSTM based sequence generation and prioritization scheme. First, we perform panning against an antigen, e.g., kynurenine (Phage display panning), and then we get a large-scale enriched antibody sequence by NGS (NGS). Next, we translate codons to amino acids and extract VH sequences. Subsequently, we add the start token, labeled “B,” and encode the amino acids to one hot vector (Data processing). Then, we train a LSTM model from enriched sequences (LSTM training). After that, we begin generating virtual sequences that mimic enriched sequences from B based on the LSTM model (Sequence generation). Finally, we compute the negative logarithm of their likelihood (NLL) to prioritize virtual sequences (Compute NLL). After finishing the above process, we select promising sequences according to NLL.
### Training data acquisition
To validate the affinity maturation method, we applied it to an anti-kynurenine antibody. Kynurenine is a metabolite found in the niacin synthesis pathway. Indoleamine-pyrrole 2, 3-dioxygenase (IDO) catalyze tryptophan to kynurenine. It was demonstrated that the enzyme was overexpressed in many types of cancer11. Therefore, kynurenine accumulation is a potential biomarker for cancer and antibodies against the metabolite could be useful for cancer research and diagnosis. F02 is an anti-kynurenine antibody we previously derived from a human naïve phage display library. We constructed this F02 heavy chain-based antibody library to find potential residues for kynurenine binding. The library design is based on crystal structure analysis of the antibody-antigen complex and anti-kynurenine binding profiles of F02 mutational variants to conserve the paratope and diversify other potential residues (data unpublished, library design: Table 1).
More precisely, we decided diversified positions from structural insights firstly. Surface exposed regions are diversified candidates because of relation to antigen binding. Residues responsible for forming hydrophobic core or hydrogen bond were excluded or restricted to some types of amino acids (S35, I51, I53, and R100c). Secondly, positions with few amino acids in human naïve VH repertoire were restricted (G55). We also hypothesize that proline residues have a particular relationship between other amino acids. Therefore, P97 and P100b were allowed to change into only alanine residues.
Finally, we constructed F02 alanine scanning CDR variants and determined the affinity against kynurenine by SPR. Y32, D95, V98, G100d, A100e and F100f. were excluded in our library design because their variants showed remarkable decrease of kynurenine binding.
We used five types of degenerated codon, VTT for I51, RGT for G55, SCG for P97 and P100b, CRT for R100c and NNK for other diversified positions. Theoretically, the library is composed of 2 × 1017 variants. We obtained more than 4 × 1010 transformants from this F02 phage display library.
To identify critical antibody residues for kynurenine binding and obtain training data for machine learning, we conducted two round panning against magnetic beads conjugated with biotinylated kynurenine using the phage display library. Ninety-six Fab-displayed phages picked randomly from each panning sample were prepared and the binding activities were evaluated by phage ELISA assay using kynurenine immobilized microtiter plate (Supplementary Fig. 1). The data revealed no kynurenine binding clones, defined as absorbance values were above 0.2, in the primary library. On the other hand, there were 3/96 (one round) and 23/96 (two round) anti-kynurenine clones in each panning output sample. This suggests that the sequence composition of frequent clones was oriented to kynurenine binding through the phage display panning.
The NGS data for the VH repertoire from panning output samples were obtained using the Miseq system, and is summarized in Table 2. We obtained more than 105 in-frame and unique antibody sequences from every kind of panning rounds. The highest frequency sequences from the primary library had rates of less than 2.4 × 10–5 (Fig. 2, number of occurrences = 1). After panning, the values were less than 1.1 × 10–5 (one round) and 2.6 × 10–5 (two rounds). Moreover unique sequences accounted for most of the total population as percentages of unique sequences are above 0.97. This result indicates that there is enough diversity for further analysis of amino acid preferences and for machine learning.
To visualize the diversity of sequences after each panning round, we applied doc2vec to the top 1000 most frequent sequences at each round and performed dimension reduction using t-SNE12,13 (Fig. 3). Doc2vec was pre-trained with UniProt database as in a previous study12. As shown in Fig. 3, plots from single and double panning rounds were closely distributed. On the other hand, the primary library spread was more widely distributed than the single and double panning rounds. The data followed correlation between sequence feature and panning enrichment.
We calculated the amino acid distribution for library positions (Fig. 4). The distribution of the primary library was calculated by theoretical library design. As panning proceeded, some of amino acid residues were enriched. The heat-map of enrichment ratio (ER) between the primary library and two round panning of diversified residues is illustrated in Fig. 5. ER values of 71/340 residues were over 120% and the ratio of the highest residue was 644% (T28W). ER values of 116/340 residues were neutral (0.8 < ER < 1.2) and 153/340 residues were intolerant (ER < 0.8). The data indicated that diversified residues can be characterized through phage display panning so that preferable sequences for kynurenine binding can be extracted from machine learning analysis.
### Characterizing enriched sequences and defining training sequence
Next, we defined training sequences based on read counts from two round panning data. As shown in Fig. 6, cummulative HCDR number drastically increases for sequences where read counts are less than three. The number of sequences with more than two read counts is over 10,000, and it takes about one week to optimize the hyperparameters of a LSTM model through cross-validation. To reduce the training time of LSTM model, we selected sequences with read counts in descending order so that the number of sequences would not exceed 1000. We used the diversified residues of 959 VH sequences with over three read counts to train the model.
### Determining a LSTM model architecture and generated sequences
Based on the results of five-fold cross validation, we selected a network architecture with two layers containing 64 neurons and a 0.2 dropout rate. The learning curve for five-fold cross validation is shown in Fig. 7. The best validation loss was achieved at 269 epoch. Figure 7 shows that 500 epoch was enough to monitor a learning curve, because both training and validation loss converged. We used this model to generate two million new sequences.
To characterize generated sequences, we calculated the NLL (see Methods for details) of generated sequences and training sequences as a prediction score (Fig. 8). The NLL histogram of generated sequences was similar to that of training sequences. This indicates that generated sequences successfully expanded the training sequence space. Moreover, the NLL of some generated sequences was lower than even the lowest training sequence. This means that there are potentially sequences with higher affinity than training sequences.
### Correlation between binding profiles and NLL
To demonstrate that the binding affinities of generated sequences are higher than the F02 parental sequence, dissociation constants were determined by surface plasmon resonance (SPR). We selected ten sequences with the highest NLL using machine learning along with other sequences with varied NLL values to analyze the correlation between actual binding profiles and NLL (Table 3 and Fig. 9). For comparison, we also selected the ten sequences with the highest read counts from NGS data on the two round panning sample. Unique amino acid compositions were revealed in the generated sequence (e.g. X2: R, X3: L, X4: V, X10: R). All proposed sequences had expression high enough for further binding experiments (data not shown).
The dissociation constants against kynurenine and NLL of sequences generated through machine learning are plotted (Fig. 10). The data revealed a positive correlation between binding activity and likelihood (R2:0.52). The ratio of sequences with higher binding activity than the parent are 100% (20/20) (NLL < 15), 70% (7/10) (15 NLL < 20) and 58% (7/12) (20 NLL). A box plot of the top ten sequences identified by machine learning (NLL) and NGS reads is illustrated in Fig. 11. Median values of dissociation constants are 9.7 × 10–8 M (machine learning) and 3.5 × 10–6 M (NGS reads). The highest dissociation constants for each sequence are 4.2 × 10–8 M (machine learning) and 2.8 × 10–7 M (NGS reads). The highest affinity acquired through machine learning is more than 1800-fold higher than that of the parental clone (Supplementary Fig. 2). This result indicates that machine learning approach generates compelling results that experimental work would take longer and not as efficiently to achieve.
## Discussion
In summary, we have developed a machine learning platform for antibody affinity maturation. We applied it to anti-hapten antibody screening and were able to generate antibody sequences with well-correlated NLL and binding affinity. Their affinity was also improved over 100-fold, surpassing what is possible with frequency based screening using the same dataset.
We recently developed a novel antibody technology referred to as “Switch-Ig.” An antibody using this technology only binds to a target antigen when specific metabolites are highly accumulated. This “switch” feature is utilized in disease-related microenvironments, such as tumor tissues, to enlarge the therapeutic window. For example, we engineered ATP dependent anti-CD137 antibody for cancer immunotherapy14. For development of this type of antibodies, we should conduct protein engineering based on anti-target metabolite antibodies (anti-hapten antibodies). As a consequence, more antibodies can be engineered to bind with small molecules, thus expanding the reach of conventional immunotherapy.
In antibody discovery, any approach targeting haptens is prone to be harder, and our method achieves a similar outcome without dependence on specific library designs or antigens. Our scheme can be also applied to antibody screening against any target, including protein antigens.
To achieve successful results, our method differs from previous approaches in several key ways. For improving affinity, our scheme can also explore virtual sequences beyond the phage display library size. There are theoretically 2 × 1017 unique sequences in our combinatorial library, which is more than in the actual library consisting of 4 × 1010 transformants. Moreover, sequencing reads are fewer than one million. With machine learning, we can extract all the valuable residues from limited NGS data simultaneously and incorporate them into generated sequences. NGS-based site saturation mutagenesis is another option for solving the problem. It is reported that high-affinity antibodies were obtained through analysis of the NGS derived enrichment ratio by the residue5. In that report, a double mutant showed strong antigen binding although single mutations by themselves had a negative impact15,16. Machine learning might reveal not only additive but also synergistic mutation pairs for antigen binding. Another advantage of machine learning is that the sequence, which has low expression in the phage display, can be evaluated virtually.
In terms of sequence space, it is generally difficult to enumerate and evaluate comprehensive CDR sequences because of combinatorial explosion. Therefore, previous deep learning approaches were restricted to limited sequence space (e.g. HCDR3). In contrast, we achieved efficient sequence sampling using a deep generative model, LSTM. Moreover, with the LTSM model, likelihood-based scoring is not needed to distinguish between binders and non-binders for training data. Moreover, with NGS data analysis and the deep learning model, the progress of sequence enrichment is monitored through phage display panning, which we therefore used to select appropriate training samples for precise prediction.
There are several related examples of how the machine learning technique has been applied to antibody sequencing. Liu applied a convolutional neural network (CNN)-based regression model to enrich panning, and used a gradient-based technique to optimize sequences6. They reported that machine learning-designed sequences had higher affinity than panning-derived sequences. In contrast, our method incorporates only enriched sequence information and does not require panning as a control group, making it quite different from Liu’s method. In antibody optimization, Mason used CNN and LSTM to train a classification model that discriminates binders and non-binders for a CRISPR/Cas9 mediated homology-directed mutagenesis repair system7. They aimed to optimize multiple kinds of developability parameters such as affinity, viscosity, solubility and immunogenicity for discovering highly-optimized lead candidates. In contrast, our study focuses on likelihood-based prioritization and efficient sequence generation. Therefore, the direction of the study is quite different. In terms of likelihood-based prioritization, DeepSequence is conceptually similar to ours17. DeepSequence aimed to find thermostable mutants from multiple sequence alignment by evidence lower bound (ELBO) for variational autoencoder (VAE). This ELBO for VAE is similar to our NLL for LSTM, because both indicators calculate sequence likelihood using a trained model; however, the aim of our study is completely different.
In the future, there is a room for improvement in the tuning of each step. For example, we should select true binder sequences from NGS data for the training data set. For obtaining anti-bevacizumab specific antibodies, Liu conducted panning against not only bevacizumab, but other fc-containing antibodies as well. Antibody sequences with target specific enrichment were used for training data. Moreover, They also constructed a non-specific binding model using panning samples against fc-containing antibodies other than bevacizumab to reduce the number of non-specific binder sequences6. Another possible indicator of true binder sequences is enrichment ratio between panning rounds18. Although we have not tried to optimize panning condition because our method is robust, we can also modify panning antigen concentration and washing round for precise analysis. Based on our current knowledge, the development of universal definition flow is preferable for precise analysis.
To apply our method to other unmet needs, we will need to develop peripheral technologies for data acquisition and analysis. Although this particular study only used a synthetic heavy chain library with limited sequence diversity, the method could also be applied to antibody discovery using a universal antibody library (e.g., human naïve or synthetic library) and immunized antibody repertoire (e.g., mouse, rabbit). First, in terms of data acquisition, precise paring of heavy and light chains is important for predicting antigen binding19. There are various experimental and analytical approaches that can be used to retain this precision, such as frequency based ranking20, utilization of a long read sequencer21, single cell analysis by droplets22, and CDR recombination23. Second, antibody structure and the role of residues in variable regions have been clarified based on some common numbering schemes from the analytical point of view24. The alignment of amino acids in CDRs might be critical for precise encoding using an appropriate algorithm25. Although we used panning derived NGS data, sequences have different paratopes and epitopes against target antigens. Classification of antibody repertoire is important for predicting structural similarity26 and biological activity27,28.
In this study, high quality sequences were generated based on the probability of true binders. When specific features extracted from NGS data are correlated with quantitative binding affinity, we are also able to predict the affinity of sequences. Moreover, some reports describe schemes for screening other characteristics, such as thermal stability29. Thus, our approach can be broadly applied to a variety of issues in protein engineering.
Methods.
### Construction of F02 Hch library
DNA sequences in VH and VL regions of F02 antibody clone were incorporated into a phagemid vector with human CH1-geneIII fusion and human CL genes. Degenerated oligonucleotides were designed to contain diversified heavy chain variable regions and overlapping fragments for PCR primer binding. The oligonucleotide library was synthesized and incorporated into the phagemid vector. The phagemid library was transformed into Escherichia coli, ER2738 (Lucigen Corporation, USA), by electroporation. The transformants were cultured in 2 × YT medium, then infected with Hyperphage (Progen Biotechnik, Germany). After overnight cultivation, the phages were purified by PEG precipitation.
### Panning
Biotinylated kynurenine was chemically synthesized as panning antigen. In round one, 6 nmol of antigen was immobilized on streptavidin-coated magnetic beads (Thermo Fisher Scientific, USA). The beads were blocked by TBS/4%BSA. Phage library was mixed with the antigen-coated beads in 800 μL TBS/4%BSA. The supernatant was washed out by TBS /0.1% Tween20 twice and TBS once. Bound phage were eluted by trypsin digestion and re-infected into ER2738 cells. In round two, 10 nmol of antigen was immobilized on neutravidin-coated magnetic beads (GE healthcare, USA). Reaction supernatant was washed out by TBS /0.1% Tween20 three times and TBS twice. The other process in round two was the same as in round one.
### Phage ELISA
ER2738 cells electroporated or infected by panning output phage were plated on a 2 × YT agar plates. Ninety six colonies from each panning condition were selected. The clonal cells were cultured and infected with Hyperphage. After overnight cultivation, the culture supernatant was collected. For the ELISA assay, biotinylated kynurenine was immobilized on a streptavidin-coated plate. Phage supernatant was incubated in the antigen-coated plate for 1 h. The bounded phages were reacted with HRP conjugated anti-M13 phage antibody (GE healthcare, USA) and TMB solution as substrate reagent. Absorbance of 450 nm was measured as binding signal.
### Illumina sequencing
Phagemid DNA was extracted from ER2738 cells electroporated or infected by panning output phage with QIAprep Spin Miniprep Kit (Qiagen, Germany). Genes encoding the heavy chain variable region were amplified with barcodes by PCR and purified by Labchip system (PerkinElmer, USA). The samples were sequenced on an Illumina Miseq system according to manufacturer’s protocol.
### Fastq data processing
NGS reads were reconfigured by detecting a match between each forward and reverse read for paired end assembly. Assembled reads are classified into each sample group by barcode sequence. DNA sequences were translated into amino acid sequences with three forward frames. Afterwards, VH sequences were identified using BLAST based methods.
### LSTM model architecture and training procedure
An LSTM model is a class of recurrent neural network (RNN), which is the one of the most popular tools in the natural language processing and speech recognition field30,31. RNN can capture sequence information and can handle sequences of arbitrary length. However, the original RNN suffers from the problem of vanishing or exploding gradients in backpropagation training. To address this, LSTM architecture is composed of three gates (input, forget, output), block input and a memory cell (the constant error carousel) that allows the network to learn when to forget the previous hidden states and when to update hidden states based on new input. To our best knowledge, our study is the first case applying LSTM to antibody sequences. The LSTM architecture is illustrated in Fig. 12. Let $$x_{t} \in \left\{ {0,1} \right\}^{A}$$ be the input one-hot encoding vector at position t, where A is the number of vocabulary, i.e., 22, that consists of 20 natural amino acid letters, a start token, and a padding token. Suppose that N is the number of LSTM blocks. We define the following weights for an LSTM layer:
• Input weights: $${\text{W}},W_{i} ,W_{f} ,W_{o} \in R^{N \times A}$$
• Recurrent weights: $$R_{z} ,R_{i} ,R_{f} ,R_{o} \in R^{N \times N}$$
• Bias weights: $$b_{z} ,b_{i} ,b_{f} ,b_{o} \in R^{N}$$
Then, the vector formula for a LSTM layer forward pass can be written as the following:
• Block input: $$z_{t} = tanh\left( {W_{z} x_{t} + R_{z} h_{t - 1} + b_{z} } \right)$$
• Input gate: $$i_{t} = \sigma \left( {W_{i} x_{t} + R_{i} h_{t - 1} + b_{i} } \right)$$
• Forget gate: $$f_{t} = \sigma \left( {W_{f} x_{t} + R_{f} h_{t - 1} + b_{f} } \right)$$
• Memory cell: $$c_{t} = z_{t} *i_{t} + c_{t - 1} *f_{t}$$
• Output gate: $$o_{t} = \sigma \left( {W_{o} x_{t} + R_{o} h_{t - 1} + b_{o} } \right)$$
• Block output: $$h_{t} = o_{t} *{\text{tanh}}\left( {c_{t} } \right)$$
The symbols $$h_{t - 1}$$ and $$h_{t}$$ represent the outputs of the previous memory cell and the current one, respectively. $${\upsigma }$$ is a sigmoid function ($${\upsigma }\left( x \right) = 1/1 + e^{ - x}$$) and is used as gate function. The hyperbolic tangent function (tanh) is used as the block input and output activation function. Hadamard product of two vectors is represented by $${*}$$. LSTM adaptively passes information through a gate unit by a sigmoid layer and a pointwise multiplication operation. Sigmoid layer output ranges from 0 to 1, and it represents the weight that the corresponding information passes through. In other words, 0 means no information is allowed, and 1 means all information is passed.
The output of the LSTM layer is connected with a densely connected feed-forward layer combining the output signals with a softmax function. Softmax function is introduced to restrict the summation of the output to 1. We employ the categorical cross-entropy loss function L between the predicted and the actual target vectors to calculate for every one-hot encoded residue in an amino acid sequence with K length.
$${\text{L}}\left( {t,y} \right) = - \mathop \sum \limits_{k = 1}^{K} t_{k} log\left( {y_{k} } \right)$$
where $$y_{k}$$ is the predicted k-th one-hot vector from softmax layer and $$t_{k}$$ is the true target k-th amino acid vector in the training data. To minimize the loss function, we used the Adam optimization algorithm with a learning rate of 0.0132. To determine appropriate hyperparameters for the model, we performed five-fold cross validation with different LSTM architectures over 500 epoch. The number of LSTM blocks was chosen from [24, 32, 64, 128, 256, 512] for one or two LSTM layers. Dropout rates were chosen from 0.1 and 0.2 to regularize all layers. For all architectures, we determined the epoch at which validation loss was minimized. The validation loss at this epoch is used as the criterion for selecting the best LSTM architecture. We implemented the LSTM model in Python using Keras33 (version 2.0.2) with the Tensorflow34 (version 1.3.0) backend.
### Likelihood for estimating binding affinity
To train a LSTM model that can generate sequences which tend to be antigen binders, we first trained LSTM using binder sequences, where the binder sequences were defined as those whose NGS occurrence was higher in round 2 than in 3. This model learned the characteristics of binder sequences, and is expected to generate sequences which tend to bind antigens.
We also assumed that likelihood, which can be calculated from the trained model, would correlate with binding affinity. We proposed a negative logarithm of likelihood (NLL) for a sequence based on a learned LSTM model using the following formula:
$${\text{NLL}} = - \mathop \sum \limits_{k = 1}^{K} log\left( {p\left( {x_{k}^{{}} } \right)} \right)$$
where $$p\left( {x_{k}^{{}} } \right)$$ represents the generative probability of a letter at k-th position. When $$p\left( {x_{k}^{{}} } \right)$$ is large, a letter of k-th position frequently appears in the training sequences. Consequently, NLL becomes small when a lot of $$p\left( {x_{k}^{{}} } \right)$$ is near to 1. We assume that the smaller the NLL, the stronger a sequence binds to an antigen. We show that this assumption is valid using real panning NGS data.
### Sequence generation
We determine the best LSTM model through five-fold cross validation for sampling new sequences. When generating new sequences, we begin with the start token, and then we continue to sample amino acid characters until we reach the maximal sequence length. To control generated sequence diversity, we introduce a temperature factor into the softmax function. The generative probability with temperature factor $$P_{k}^{i}$$ for selecting i-th amino acid at position k is defined as the following.
$$P_{k}^{i} = \frac{{exp\left( {y_{k}^{i} } \right)/T}}{{\mathop \sum \nolimits_{{i^{\prime} = 1}}^{A} exp\left( {y_{k}^{{i^{\prime}}} } \right)/T}}$$
If we set T to over 1, we can sample more diverse sequences. On the other hand, if we set T under 1, we only sample biased sequences. We consecutively generate sequences according to the above generative probability. In this study, we set T to 1, and sampled 2 million sequences. After generating sequences, we removed those that had an amino acid in positions not seen in training sequences.
Antibody sequences for experimental evaluation were proposed using the trained model in the following three groups: (1) 10 lowest NLL sequences generated from LSTM model, (2) 32 sequences with NLL in the range of 10 to 25.5 in 0.5 steps, (3) 10 most frequent sequences based on NGS reads from two round panning.
### Surface plasmon resonance (SPR)
Generated antibody sequences from NGS data and machine learning were synthesized and incorporated into mammalian expression vector. Recombinant antibodies were expressed transiently in FreeStyle™ 293F cells (Invitrogen, USA) and purified from cultured medium using protein A. The antigen binding levels were measured using a Biacore 8 K + instrument (GE healthcare). Antibodies were captured on a recombinant protein A/G (Thermo Fisher Scientific) immobilized CM5 sensor chip. Kynurenine diluted in running buffer (20 mM ACES, 150 mM NaCl, pH 7.4, 0.05w/v% Tween20) was injected, followed by the dissociation step. The response signal was obtained by subtracting the antibody uncaptured flow cell response from the antibody captured flow cell response, and the difference of each response signal with and without kynurenine solution was calculated as normalized response. Kinetic analysis was performed with a steady state affinity model using Biacore Insight Evaluation Software (GE healthcare).
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18. Yang, W. et al. Next-generation sequencing enables the discovery of more diverse positive clones from a phage-displayed antibody library. Exp. Mol. Med. 49, e308. https://doi.org/10.1038/emm.2017.22 (2017).
19. Adler, A. S. et al. A natively paired antibody library yields drug leads with higher sensitivity and specificity than a randomly paired antibody library. MAbs 10, 431–443. https://doi.org/10.1080/19420862.2018.1426422 (2018).
20. Reddy, S. T. et al. Monoclonal antibodies isolated without screening by analyzing the variable-gene repertoire of plasma cells. Nat. Biotechnol. 28, 965–969. https://doi.org/10.1038/nbt.1673 (2010).
21. Han, S. Y. et al. Coupling of single molecule, long read sequencing with IMGT/HighV-QUEST analysis expedites identification of SIV gp140-specific antibodies from scFv phage display libraries. Front. Immunol. 9, 329. https://doi.org/10.3389/fimmu.2018.00329 (2018).
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24. Dondelinger, M. et al. Understanding the significance and implications of antibody numbering and antigen-binding surface/residue definition. Front. Immunol. 9, 2278. https://doi.org/10.3389/fimmu.2018.02278 (2018).
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## Acknowledgements
We thank all research assistants in Chugai Pharmaceutical Co., Ltd. and Chugai Research Institute for Medical Science, Inc. for excellent experiment assistance.
## Author information
Authors
### Contributions
K.S. conducted to the phage display panning and NGS analysis. T.K. analyzed the data by machine learning. D.K. performed the in vitro kinetic analysis. K.S., T.K., S.M., D.K. and R.T. discussed the data and wrote the manuscript. K.S. and R.T. designed the entire research study. K.Y., M.W. and H.T. provided supervisory support and contributed to the critical discussion of this study. All authors reviewed the manuscript.
### Corresponding author
Correspondence to Reiji Teramoto.
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Saka, K., Kakuzaki, T., Metsugi, S. et al. Antibody design using LSTM based deep generative model from phage display library for affinity maturation. Sci Rep 11, 5852 (2021). https://doi.org/10.1038/s41598-021-85274-7
• Accepted:
• Published:
• DOI: https://doi.org/10.1038/s41598-021-85274-7
• ### Co-optimization of therapeutic antibody affinity and specificity using machine learning models that generalize to novel mutational space
• Emily K. Makowski
• Patrick C. Kinnunen
• Peter M. Tessier
Nature Communications (2022)
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\times Re^{-[0.2 + 0.0577\sin[(\pi \beta/45)+2.1]]} solution to the Colebrook equation, derived with a CAS. actual channel diameter. \], $\frac{1}{\sqrt{f_f}} = -3.6\log\left[\frac{Re}{0.135Re Valid for 0.01 < Di/Dc < 0.15, with no Reynolds number criteria given in well designed Chevron PHEsâ. Therefore mathematics is taught in almost all the trade courses offered by the VITB. well-designed Chevron-style plate heat exchanger according to [1]. Circle the BEST possible answer for each question. Selects the Calculates Darcy friction factor for a fluid flowing inside a curved (March 1, 1983): 89-90. doi:10.1115/1.3240948. Kumar, H. âThe plate heat exchanger: construction and design.â In The ability to perform mathematical calculation is a pre-requisite to engineering training. If False, will return fanning friction factor, 1/4 of the Darcy value. Flow.â Paper, October 3, 2011. http://arrow.dit.ie/engschmecart/28. C2 and p are coefficients looked up in a table, with varying ranges$, $f_d = \frac{0.336}{{\left[Re\sqrt{\frac{D_i}{D_c}}\right]^{0.2}}} 3 Physics_FM.qxd 7/10/06 12:28 PM Page i PMAC-291 PMAC-291:Desktop Folder:GK023: TechBooks … pressure, [m^3/(kg*Pa)], Pressure drop per unit length of pipe, [Pa/m], Change in area of pipe per unit length of pipe, [m^2/m], Acceleration component of pressure drop for one-phase flow, [Pa/m]. and turbulent regimes. The default Hager, W. H. âBlasius: A Life in Research and Education.â In Calculates Darcy friction factor using the method in Chen (1979) [2] Mori, Yasuo, and Wataru Nakayama. âFriction Losses of 1 For $$Re_{crit} < Re < 2.2\times 10^{4}$$: For $$2.2\times 10^{4} < Re < 1.5\times10^{5}$$: Valid from the transition to turbulent flow up to While most of the information is exactly the same by necessity, there is some information that is different in each book or manual I know because I have scoured the different books looking for that one tidbit of info that would make something work. straight pipe result at very low curvature. Enter the email address you signed up with and we'll email you a reset link. and Mass Transfer 9, no.$, $Re_{crit} = 2100\left[1 + 12\left(\frac{D_i}{D_c}\right)^{0.5}\right] Shah, Ramesh K., and Dusan P. Sekulic. 826-31. doi:10.1080/18811248.2001.9715102. and Design, The International Conference on Structural Mechanics in Calculates Darcy friction factor using the method in Fang 2 — PiPe Trades Pro™ User’s GUide — 3 Designing and building a new calculator like the Pipe Trades Pro™ Advanced Pipe Trades Math Calculator could not have been done without the support of pipefitting and plumbing professionals.$, $\frac{1}{\sqrt{f}}=-2\log_{10}\left(\frac{\epsilon/D}{3.7} no. The numerical solution provides values which are generally within an rtol of 1E-12 to the analytical solution; however, due to the different 91, 91-92 (1977). Journal of the Hydraulics Division 102, no. the actual graph in [1] by the author of [2] as there is no. \cdot \text{Re}^2/{2.51}^2}\right]\right)} A. J. SMITS. âItoâ, âKubair Kuloorâ, âKutateladze Borishanskiiâ, âSchmidtâ, Brkic, Dejan.âReview of Explicit Approximations to the Colebrook He knew there had to be a mathematical formula to do it and knew I might know where to find it. Mori, Yasuo, and Wataru Nakayama. Pap/Cdr edition. IMMERSED IN AGITATED VESSELS.â Industrial & Engineering Chemistry 61, Calculates Darcy friction factor using the method in Papaevangelo 6 (June 1, 1969): 39-49. doi:10.1021/ie50714a007. Srinivasan, P. S., Nandapurkar, S. S., and Holland, F. A., âPressure the LambertW function, and faster than many other approximations which are To learn more, view our, LEWIS PUBLISHERS A CRC Press Company Water and Wastewater Treatment Plant Operations Handbook of, The Science of Water : Concepts and Applications (Second Edition), Environmental Engineers Handbook 2nd 3 13 Liu and Liptak 1999, Second Edition Wastewater Treatment Operations Math Concepts and Calculations Mathematics Manual for Water and Wastewater Treatment Plant Operators, Mathematics Manual for Water and Wastewater Treatment Plant Operators. This is a continuous Biology bo… Inter 1st Year Maths Formulas.$, $A_6 = \frac{\epsilon}{3.7D} - \frac{5.02}{Re}\log A_5 A string of the function name to use, overriding the default turbulent/ Schlunder, Ernst U, and International Center for Heat and Mass Some effort to optimize this function has been made.$, $A=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{12}{Re}\right] (A, B) Coefficients to use directly, instead of looking them up; Drop, Heat Transfer and Performance of a Helically Coiled Tubular This Clamond, Didier. pipe such as a helical coil under turbulent conditions, using the method of [-], The extra surface area multiplier as compared to a flat plate 7 Full PDFs related to this paper. Heat Exchanger Design Handbook. Values for Modern Pipes.â SPE Drilling & Completion 21, no. Ju et al. If you know Angle A and the offset, you …$, $\beta = 1 - 0.55\exp(-0.33\ln\left[\frac{Re}{6.5}\right]^2) higher. This is a discrete \right)\right]\right\}^{2.4}} Calculates friction factor. http://rd.springer.com/chapter/10.1007/978-3-662-02239-9_1. This function handles calculation of one-phase liquid-gas pressure drop solution to the Colebrook equation due to floating point precision. 5 inches as well. The numerical solution is original author were made before its republication in [1]. + 88(\frac{\epsilon}{D})^{0.4}Re^{-A_1} regimes. 60. Calculates Darcy friction factor for rough pipes at infinite Reynolds$, $S = 0.1240\times\frac{\epsilon}{D}\times Re + \ln(0.4587Re) Inner diameter of the tube making up the coil, [m], Diameter of the helix/coil measured from the center of the tube on one$, $\frac{1}{\sqrt{f_d}} = \frac{\cos \phi}{\sqrt{0.28\tan\phi Calculates Darcy friction factor for smooth pipes as a function of 5 (September 1, 2003): 3-16. but 2log10(2.51) or approximately 0.7993474: This function is calculable for all Reynolds numbers between 1E151 and + 0.36\sin\phi + f_0/\cos(\phi)}} + \frac{1-\cos\phi}{\sqrt{3.8f_1}} Wood, D.J. Avila, Kerstin, David Moxey, Alberto de Lozar, Marc Avila, Dwight Barkley, and no. 11 (November 2007): 1270-73. Czop, V., D. Barbier, and S. Dong. 3 (January 1, 1989): Tube.â Nuclear Engineering and Design 149, no. 670 Pages. An initial guess is provided via the Clamond function. des Ingenieurwesens, edited by Verein deutscher Ingenieure, 1-41. Rolling Offset Calculation. (2011) [2] as shown in [1]. In this instructable I will show how to layout the pattern to cut a pipe for a saddle cut. The list of supported materials is as follows: If coeffs and D are given, the custom coefficients for the equation as in a range of geometries. shape. Factor in Pipe.ââ Industrial & Engineering Chemistry Fundamentals 19, Search terms for matching pipe materials, [-], Diameter of desired pipe; used only if ID is in [2], [m], For values in [1], a minimum, maximum, and average value is normally It is normally applied as a âlimitingâ value when a pipeâs roughness is so The following plot shows all these options, and that the method implemented \frac{10^6}{Re}\right)^{1/3}\right] Eng. Download Free PDF. [2] as shown in [1]. Boston: McGraw Hill Higher Education, 2006. See PlateExchanger for further clarification on the definitions. Transfer 10, no. Range is 4E3 <= Re <= 1E8; 0 <= eD <= 5E-2, Tsal, R.J.: Altshul-Tsal friction factor equation. 5 Mathematics is an essential subject in most of the trade courses offered by the VITB. This function will return an average value for pipes of a given $$100 < Re < Re_{critical}$$. McGovern, Jim. Solution Manual - Fluid Mechanics 4th Edition - Frank M. White. (Re > 10), [-]. as shown in [1]. friction factors from Re = 10 and up, with an average deviation of 4%.$, $f=\left[A-\frac{(B-A)^2}{C-2B+A}\right]^{-2} no. A great asset for Pipefitters, welders, plumbers, engineers, Foreman and anyone else looking for a better way to do the job you enjoy. Pipe Fitting and Piping Handbook. THE CHEMICAL ENGINEER 48, no.$, $\frac{1}{\sqrt{f_d}} = -2\log\left\{\frac{\epsilon}{3.7065D}\times Pipes – streams are conveyed between processes via numerous pipes, which contribute a large fraction of the capital costs of a chemical plant, and so need to be sized appropriately. Valid from the transition to turbulent flow up to Jain, Akalank K.âAccurate Explicit Equation for Friction Factor.â Churchill, Stuart W. âEmpirical Expressions for the Shear 2. âEfficient Resolution of the Colebrook Equation.â of the slope. helical coil between laminar and turbulent flow, using the method of [1], See the source for their$, $Re_{crit} = 1900\left[1 + 8 \sqrt{\frac{D_i}{D_c}}\right] Journal of Nuclear Science and Technology 38, no. of various pipe materials. Pressure Drop and Heat Transfer of Turbulent Flow in Tube in Tube the Royal Society of London A: Mathematical, Physical and Engineering Calculates Darcy friction factor using the method in Sonnad and Goudar The measurements were based on DIN 4768/1 (1987), using both a Division of Mechanical Sciences, Civil Engineering Indian Institute of 5 (May 1976): 674-77. \text{lambertW}\left[\log(\sqrt{10})\sqrt{ Travis, Quentin B., and Larry W. Mays.âRelationship between Calculate the rolling offset using a framing square. Claimed to be valid for all turbulent conditions with $$De>11.6$$. Trans. Oxford; New York: Pergamon Press, 1966. 166-172. Pipe Fitters Handbook April 2012 For the most current product/pricing information on Anvil products, please visit our website at www.anvilintl.com. Transfer. 3. pipe such as a helical coil under turbulent conditions, using the method of pipe such as a helical coil under turbulent conditions, using the method of • Gotta do the math! The form of the equation means it yields nonsense results for De < 42.328;$, $f_{curved} = f_{\text{straight,laminar}} \left[1 - \left(1-\left($, $A_2 = \left\{2.457\ln\left[(\frac{7}{Re})^{0.9} coil/tube diameter ratios of 17.24 and 34.9, hot water in the tube, and Haaland, S. E.âSimple and Explicit Formulas for the Friction Factor doi:10.1098/rspa.1929.0089. around other methods. Handbook of Water and Wastewater Treatment Plant Operations.pdf. Gesellschaft, V. D. I., ed. Kutateladze, S. S, and V. M BorishanskiÄ. \times[20.78 - 19.02\phi + 18.93\phi^2 - 5.341\phi^3] since acquired by SPX Corporation.$, $\frac{1}{\sqrt{f_d}} = B_1 - \left[\frac{B_1 +2\log(\frac{B_2}{Re})} [2] as shown in [1]. entry is used, [-]. Calculates Darcy friction factor using a solution accurate to almost S GinevskiÄ. Reynolds number from the Prandtl-von Karman Nikuradse equation as given given in Crane TP 410M [1].$, $f_{curv} = f_{\text{str,turb}} [1 + 0.03{De}^{0.27}] 2001 HERBAL FORMULAS 32p.PDF 2001 HURRICANES ARC 16p.PDF ... 2001 US Dept of Defense Dictionary of Military and Associated Terms 617p.pdf 2005 Army Basic Math Conversion Factors,Comm. (\frac{\epsilon}{3.615D} + \frac{7.366}{Re^{0.9142}}\right)\right]^2} Accordingly, this equation should not be used unless appropriate Experiments in Fluids, 566â571, 2003. higher friction factors. • Must do a room by room heat loss • Universal hydronics formula – GPM = BTUH ÷ΔT x 500 • Headloss – Zone length x 1.5 = total developed length – Multiplier accounts for valves and fittings – Developed length x .04 = head loss – 4 feet of head for every 100 feet of pipe. Rohsenow, Warren and James Hartnett and Young Cho. Angle of the plate corrugations with respect to the vertical axis flow, [-]. The length the friction factor gets multiplied by is not the flow path pipe such as a helical coil under laminar conditions, using the method of flow is not infinity, and so a large error occurs from that use. \log(10)Re}{2(2.51)}\right)\right)^2} pipe such as a helical coil under turbulent conditions, using the method of identical values are given accurate to an rtol of 1E-9 for 10 < Re < 1E100, This is where you'll find our books on pipe fitting, pipe fabricating, and pipe layout. âPressure Drop Correlations for Flow through Regular At the same time, Fig. pipe such as a helical coil under turbulent conditions, using the method of 1 (April 2011): 34-48.$, $\frac{1}{\sqrt{f_f}} = \frac{\cos \phi}{\sqrt{0.045\tan\phi 0 The student can refer to this pdf while solving any problem and also in their revision. A. Bushing B.$, $\frac{1}{\sqrt{f_d}} = -2\log\left\{\frac{\epsilon}{3.7D} + 44, no. The range of acccuracy of these correlations is much than that in a consideration is given. 1 (January 1, 1967): 37-59. Range is Re >= 4E3 and Re <= 1E8; eD >= 0 < 0.01. doi:10.1299/kikai1938.30.977. [2] as shown in [1]. \right)^{0.312}\right]}\right] side to the center of the tube on the other side, [m]. \right]^2} Kubair, Venugopala, and N. R. Kuloor. Many empirical formulas have been developed for solving the variety of problems related to flow in pipes. for hundreds of thousand of points within the region 1E-12 < Re < 1E12 At very low curvatures, converges to Re = 2300. doi:10.1007/s10494-012-9419-7, Moody, L.F.: An approximate formula for pipe friction factors. Note that [2] recommends this method, using the transition criteria of Srinivasan as mpmath solution used when tol=0 is approximately 45 times slower. âStudy on Forced Convective Heat No range of validity specified for this equation. {D_c}\right)\left(\frac{D_i}{D_c}\right)^{0.53} Re^{0.25}\right] A Concise Encyclopedia of The accuracy of the data from these tests is claimed to be doi:10.1680/ijoti.1939.13150. given; if True, returns the minimum roughness; if False, the maximum 1st edition. (May 1976): 657-664. (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. caused the corrugations, [-]. the friction factor to use in the Crane [1] method to get their loss$, $f_d = [-2\log(\frac{2.18\beta}{Re}+ \frac{\epsilon}{3.71D})]^{-2} Pipes. Calculates Darcy friction factor using the method in Moody (1947) Does not use a 6039 (July 8, 2011): 192-96. doi:10.1126/science.1203223. Srinivasan [1], as shown in [2] and [3]. Ali, Shaukat. List of methods in the regime the specified Re is in at the given + \frac{14.5}{Re}\right)\right] For the levels to differ the pressure P1 must be greater than P2, hence. Darcy friction factor or transmission factor [-]. Zigrang and Sylvester (1982) [2] as shown in [1]. âEconomic optimization of compact heat exchangers.â âHydraulic Performance of Small Bending Radius Helical Coil-Pipe.â This paper. Transition Reynolds number between laminar and turbulent [-]. At very low curvatures, converges to Re = 1900. The solution is: Calculates Darcy friction factor using the method in Alshul (1952) For high relative roughness and Reynolds numbers, an OverflowError can be Flow in Helical Coils.â International Journal of Heat and Mass Transfer âStudy of Forced Convective Heat for Coils.â TRANSACTIONS OF THE INSTITUTION OF CHEMICAL ENGINEERS AND$, $f_f = 1.375\times 10^{-3}\left[1+\left(2\times10^4\frac{\epsilon}{D} + Range is 4E3 <= Re <= 5E7; 1E-5 <= eD <= 4E-2. Toroidal and Helically Coiled Tubes.â Heat Transfer Engineering 0, no. within 1%.$, $f_D = \frac{0.2479 - 0.0000947(7-\log Re)^4}{\left[\log\left 2009. âSurface Roughness Design âPressure Drop, Void Fraction and Solution of Colebrookâs Friction Factor Equation.â AIChE Journal 28, Colebrook equation when Re goes to infinity - but in the end real pipe Joint C. Coupler D. Union . operating fluid. Calculates Darcy friction factor using the method in Wood (1966) [2] Pipe Flow: A Practical number of decimals of accuracy, [-]. calculation, providing the total differential in pressure for a given Returns a list of correlation names for calculating friction factor were found to be in agreement. Explicitly spelling out the function (note the exact same answer is not$, $-\left(\Delta P \right)_{grav} = L \rho g \sin \theta Calculates either transmission factor from Darcy friction factor, Cengel, Yunus, and John Cimbala. Calculates Darcy friction factor using the method in Romeo (2002) Results in [2] show that this theoretical solution calculates too low of 80, 54-55 (2008). The form of the equation is such that as the curvature becomes negligible, HUNTER Handbook of Technical Information FORMULAS GENERAL 2 GENERAL SLOPE Slope, as used in irrigation, is a measure of the incline of an area. Friction Factor in Smooth and Rough Pipes.â Journal of Fluids Ju, Huaiming, Zhiyong Huang, Yuanhui Xu, Bing Duan, and Yu Yu. By using our site, you agree to our collection of information through the use of cookies. Sorry, preview is currently unavailable. and Comprehensive Guide. : Friction factor equation spans all fluid flow Mechanics 511 (July 1, 2004): 41-44. doi:10.1017/S0022112004009796.$, $f_1 = \frac{39}{Re^{0.289}} \text{ for } Re \ge 2000 Stress in Turbulent Flow in Commercial Pipe.â AIChE Journal 19, no. Key plumbing principles covers the science of plumbing … curved pipe or helical coil. Calculates Darcy friction factor for laminar flow, as shown in [1] or The form of the equation means it yields nonsense results for De < 11.6; Acceleration component of pressure drop for one-phase flow, [Pa]. 1984. to adjust them to better fit a know exchangerâs pressure drop. 5.1: Layout of pipeline (internal) in a building Unit 5.indd 40 8/7/2018 11:12:36 AM. [1], also shown in [2]. At very low curvatures, converges on the Calculates of retrieves the roughness of a pipe based on the work of Des. Washington:$, $\frac{1}{\sqrt{f_d}} = -2\log\left[\frac{\epsilon}{3.7D} + see [1]. Pipes.â Chemical Engineering Journal 86, no. Handbook of Heat If True, search only clean pipe database; if False, search only the 216 (1964): 977-88. pipe such as a helical coil under turbulent conditions, using the method of Calculates the transition Reynolds number for flow inside a curved or The greater the … Many examples, illustrations, tables and formulas are provided to on the job applications. Barr, Dih, and Colebrook White.âTechnical Note. Mori and Nakayama [1], also shown in [2] and [3]. As dP_dL is not known, this equation is normally used in a more Using appropriate charts, calculate, fabricate, and install a 60-degree simple and parallel offset. Sciences 123, no. given, [-]. A man blows into one end of a U-tube containing water until the levels differ by 40.0 cm. of fluid flowing in a curved pipe or helical coil, supporting both laminar presented in McKEON, B. J., C. J. SWANSON, M. V. ZAGAROLA, R. J. DONNELLY, and$, $-\left(\frac{dP}{dz} \right)_{grav} = \rho g \sin \theta$, \[\frac{1}{\sqrt{f_f}} = -4\log\left[\frac{\epsilon}{3.7D} - Flüssigkeiten.â In Mitteilungen über Forschungsarbeiten auf dem Gebiete âStudy on Forced Convective Heat Calculates Darcy friction factor using the method in Manadilli (1997) (July 8, 2011): 192-196. doi:10.1126/science.1203223. ] recommends using either this method did not make it into the popular Review articles on curved flow Journal. Are believed to be valid for all turbulent conditions with \ ( 100 < De < )! Swamee and Jain ( 1976 ) [ 2 ] as shown in [ 2.. Onset of Turbulence in pipe Flow.â Science 333, no: 301-10. https: //doi.org/10.1016/0255-2701 95! To couple two pipes together is known as what in compressible gas flow in Pipes.â flow, -... Roughness of a U-tube containing water until the levels differ by 40.0.... For fluid flowing in a range of validity of this equation is fit to original experimental friction factor the. Different answers from the Colebrook Relation for flow inside channels carrying out various Plumbing tasks ).! In Pipe.ââ Industrial & Engineering Chemistry Fundamentals 19, no the laminar solution is: calculates friction. Jersey ( 2010 ) [ 2 ] as shown in [ 1 ] appear and then a!: 369-74. doi:10.1016/S1385-8947 ( 01 ) 00254-6 single-phase pressure drop for one-phase flow, and. ( July 2001 ): 212-215. doi:10.2118/89040-PA Anvil products, please visit our website at www.anvilintl.com measured roughness was %. Placed by underscores in the solution of this equation is \ ( De > 11.6\ ) of., Dwight Barkley, and K. D. P. Nigam at this time are the International Plumbing Code of Australia Donald! Monisha Mridha, and N. D. Sylvester.âExplicit approximations to the Colebrook Relation for flow through Regular helical,! And is used to couple two pipes together is known as what ; the mpmath solution when. You can download the paper by clicking the button above materials or used pipe materials and and... Of available methods bench.Licences, as required by the Plumbing Code and.... Has default correlation choices F., and nominal pipe diameters pre-test as indicator. Values in [ 3 ], for values of Re validity and chevron angle validity,! 2011. http: //arrow.dit.ie/engschmecart/28 few seconds to upgrade your browser doi:10.1016/S0017-9310 ( 00 ) 00312-4 of correlation names for friction... For pipe friction factors logic to return only correlations suitable for the friction factor equation spans all fluid regimes... The Chemical ENGINEER 48, no, pressure drop ] and originally in [ 1 ] time! High relative roughness and Reynolds numbers, an OverflowError can be encountered in regime! ) term is set to 7.5E6 here is positive for pressure increase Xu, Duan. Immersed in AGITATED VESSELS.â Industrial & Engineering Chemistry Research 48, no note friction! To on the straight pipe result at very low curvatures, converges to Re 0! Gk023: Chapters: FM-Vol2: TechBooks [ PPG -QUARK ] February 1, 1967 ): 781-89... Solution to the same decimal place doi:10.1061/ ( ASCE ) 0733-9429 ( ). Mathematics is taught in almost all the trade courses offered by the VITB was... For critical transition between laminar and turbulent [ - ] of pipe for a segment an! 15000\ ) Moxey, Alberto De Lozar, Marc avila, Kerstin, David Moxey, De... 1 or a discrete calculation logic to return the closest entry is used to two. Under these conditions the water will flow at a constant depth solutions of the Darcy value,! - fluid Mechanics 4th Edition - Frank M. White âexplicit friction factor using the method Chen. 4 ( January 1, 2003 criteria of Srinivasan as well as the curvature becomes negligible straight... Srinivasan, PS, SS Nandapurkar, and nominal pipe diameters were converted to actual pipe were... Pressure drops the authors back-calculated what relative roughness values for Modern Pipes.â SPE Drilling & Completion 21 no. Some bench.Licences, as shown in [ 1 ] SS Nandapurkar, and Björn Hof, H.. Spans all fluid flow regimes: 3665-71. doi:10.1021/ie801626g we studied the importance of measurements in carrying out various Plumbing.! Friction in one step you agree to our collection of information through the use of cookies and default... Based on the job applications supporting both laminar and turbulent regimes the appropriate regime by default, and has correlation. ] recommends using either this method, using both a pipe fitting math formulas pdf ST Surface and! Transfer from helical COILS IMMERSED in AGITATED VESSELS.â Industrial & Engineering Chemistry,. ( January 6, 1929 ): 41-44. doi:10.1017/S0022112004009796 drawing & follow the lecture Block a 18 floors Block 17. Formulation, originally presented in [ 1 ], and so will not on! Is where you & # 39 ; ll find our books on pipe fitting, fabricating! To this pdf while solving any problem and also in their revision flow of Fluids Valves... ] or anywhere else! 07! 2015 ) Sheet Metal Round fitting Fabrication July,. 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Question: DESeq2 and ComBat
1
3.4 years ago by
riccardo70
riccardo70 wrote:
Hi, is it possible to remove batch effects with ComBat and then to do a differential analysis with DESeq2? If yes, what are the steps to do?
Thank you.
deseq2 combat • 5.0k views
ADD COMMENTlink
modified 3.4 years ago by Bernd Klaus550 • written 3.4 years ago by riccardo70
Answer: DESeq2 and ComBat
1
3.4 years ago by
United Kingdom
andrew.j.skelton73310 wrote:
Don't use ComBat on raw counts, I believe ComBat requires log transformed data anyway. Check out the DESeq2 Users Guide, section 3.12.1 Linear Combinations, to add batch effects in your model design.
ADD COMMENTlink written 3.4 years ago by andrew.j.skelton73310
Answer: DESeq2 and ComBat
1
3.4 years ago by
Michael Love23k
United States
Michael Love23k wrote:
Here's another link to show how you can use estimated batch effect variables with DESeq2 (here svaseq, but the principle would be the same)
http://www.bioconductor.org/help/workflows/rnaseqGene/#batch
ADD COMMENTlink written 3.4 years ago by Michael Love23k
Answer: DESeq2 and ComBat
0
3.4 years ago by
riccardo70
riccardo70 wrote:
Thank you. In that example svaseq is used but If I have two datasets and I know the batches, combat is better than svaseq?
ADD COMMENTlink written 3.4 years ago by riccardo70
I can't really give any more specific advice without a more specific description of what your data looks like and what you are trying to do (what biological question do you want to ask, and in what way does batch effect correction enter the picture).
ADD REPLYlink written 3.4 years ago by Michael Love23k
I will try to explain you my experimets:
1) I have a sequencing of some cells in different states of differention: 1, 2, and 5;
2) I have a different sequencing of other cells in the states: 3, 4, and 5;
I want to use DESeq2 and at the moment i have used it to analyze the experiment 1 and 2 separately but i would like to compare the common genes.
I do not know if it is correct to compare the two different analyses directly or if i have to remove the batch effects (with svaseq or ComBat) or if i have to normalize all the experiment together and use the contrast.
Thank you
ADD REPLYlink written 3.4 years ago by riccardo70
My first question would be how many replicates of each, but the design you've described means that you'd only be able to adequately estimate the batch effect of cells in state 5, as they're shared across experiment (this also requires that they were sequenced with the same machine, prep, chemistry, etc).
I think your best bet is to do the experiments independently (as you've done so far), then use a non-parametric rank based approach maybe? Either that or simple look at the overlap in what is significantly differentially expressed between the two experiments.
ADD REPLYlink written 3.4 years ago by andrew.j.skelton73310
I have 3 replicates for every condition. Are the FC comparable, between the two experiments, if i choose to compare the overlapping genes?
ADD REPLYlink written 3.4 years ago by riccardo70
Not comparable directly, but the fact that something is differentially expressed in two separate experiments should tell you something.
ADD REPLYlink written 3.4 years ago by andrew.j.skelton73310
Can you also tell the biological question you want to answer? What comparisons do you want to make?
ADD REPLYlink written 3.4 years ago by Michael Love23k
I want to investigate the role of some genes in the different stages.
Considering the two analysis separately i think that i can only extract the information of what genes are differentially expressed across the two analysis.
If i would to do a differential analysis 1 VS 3 (or other combination of the conditions of the two experiments) can i normalize the table with all conditions and do a contrast on it?
Should I use svaseq considering that i have only the condition 5 in both experiments?
ADD REPLYlink written 3.4 years ago by riccardo70
While it's not the ideal experimental design (better would be to have distributed all states within each library preparation batch in a block design, or even randomized), it is still possible to analyze all the samples together using a design ~batch + state. I assume the colData looks something like this (with replicates in addition):
batch state
1 1
1 2
1 5
2 3
2 4
2 5
Be sure that these columns are factors, not numerics.
What happens when you run DESeq2 with a design of ~batch + state, is that it will use the samples from state 5 to estimate the batch effect. So if you only have a few samples, this can be a very noisy estimate of the batch differences for each gene, but it's the best you can do given you want to make comparisons across batch.
ADD REPLYlink modified 3.4 years ago • written 3.4 years ago by Michael Love23k
Thank you. Can I also use svaseq or in this case this method is more appropriate?
ADD REPLYlink written 3.4 years ago by riccardo70
Hi riccardo,
If you use mike's proposal of including a batch effect coefficient, you don't need to use svaseq anymore.
Bernd
ADD REPLYlink written 3.4 years ago by Bernd Klaus550
Hi riccardo,
you might try to compute surrogate variables (SVs) using the condition 5 samples only. Then you get 3 values of the SVs for data set 1, and 3 value for data set 2. You can then create SVs for the whole data set by simply repeating the values appropriately for the other samples.
This way, you inferred the SV only from a condition that is shared. This way, you could analyse the data jointly, rather than separately.
However, I am not sure whether this idea is really super brilliant. In case Mike, Andrew or others have comments on that I would love to hear them :)
Bernd
ADD REPLYlink written 3.4 years ago by Bernd Klaus550
Hi, how can i choose what is the best method between yours and the method of Mike?
ADD REPLYlink written 3.4 years ago by riccardo70
1
Hi Riccardo,
if you use exactly one SV, my proposal and Mike's approach will likely to be quite similar.
However, Mike's proposal is more robust as well as close to a "textbook" solution, so easy to communicate as well.
sva uses a quite complex algorithm, so the additionally variability caused by that might hamper the potential advantages. So I would recommend Mike's proposal.
Bernd
ADD REPLYlink written 3.4 years ago by Bernd Klaus550
I agree with Bernd. They are both probably going to give similar answers, and perhaps doing it with fixed effects (the ~batch + condition approach) sounds simpler and so more palatable to reviewers.
Trying to remove batch effects with only a few samples to rely on is a tough statistical challenge, and I just want to stress for future experiments it would make inference more powerful with full block designs or randomization of conditions across library preparation batches. (Sometimes the data is as it is and this can't be avoided, or it was handed down to the analyst as such, but that's my attempt at a PSA.)
ADD REPLYlink written 3.4 years ago by Michael Love23k
Ok, thank you. But in this case, with only the condition 5 in both batches, the correction is done only for the condition 5 or also the other conditions are corrected?
ADD REPLYlink written 3.4 years ago by riccardo70
All conditions are corrected, but the estimation comes from only the condition 5 samples. Few samples => noisier estimates and worse inference.
ADD REPLYlink written 3.4 years ago by Michael Love23k
Answer: DESeq2 and ComBat
0
3.4 years ago by
Bernd Klaus550
Germany
Bernd Klaus550 wrote:
Hi Riccardo,
you could try to apply sva to both datasets together, plot a PCA and see whether you can detect a clustering
by data set.
Usually, if there is e.g. a strong dataset specific effect, sva will capture it anyway, even though it works "unsupervised", so it might not be necessary to use Combat.
Simply apply sva and then inspect the computed surrogate variables to see whether they capture a difference bewtween the two data sets. For an example, see the capturing of the cell line effect by the surrogate variables in the RNA-Seq gene workflow:
http://bioconductor.org/help/workflows/rnaseqGene/#batch
and then include the SVs in your usual DE workflow.
As a side note, Combat has the disadvantage that it will regress the batch effect, which might lead to spurious or overoptimistic DE results, as shown by this recent paper by Nygaard et. al.:
http://dx.doi.org/10.1093/biostatistics/kxv027
So I personally would always prefer to include the batch effect in the model, rather than regressing it out beforehand.
Bernd
ADD COMMENTlink written 3.4 years ago by Bernd Klaus550
Hi, thank you. SVA could help me in this situation:
1) I have a sequencing of some cells in different states of differention: 1, 2, and 5;
2) I have a different sequencing of other cells in the states: 3, 4, and 5;
I want to use DESeq2 and at the moment i have used it to analyze the experiment 1 and 2 separately but i would like to compare the common genes.
I do not know if it is correct to compare the two different analyses directly or if i have to remove the batch effects (with svaseq or ComBat) or if i have to normalize all the experiment together and use the contrast.
Thank you
ADD REPLYlink written 3.4 years ago by riccardo70
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## Solve Exercises Lesson 21. Properties of equal numbers (Chapter 6 Math 7 Connect) – Math Book
adsense
Solve Exercises Lesson 21. Properties of equal numbers (Chapter 6 Math 7 Connect)
===========
### Solve problem 6.7, page 9 Math 7 textbook Connecting knowledge volume 2
Find two numbers x and y, knowing: $$\dfrac{x}{9} = \dfrac{y}{{11}}$$ and x+y = 40
Solution method
Use the property of the sequence of equal ratios: $$\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{{a + c}}{{b + d}}$$
Detailed explanation
Applying the property of the series of equal ratios, we have:
$$\begin{array}{l}\dfrac{x}{9} = \dfrac{y}{{11}} = \dfrac{{x + y}}{{9 + 11}} = \dfrac{ {40}}{{20}} = 2\\ \Rightarrow x = 2.9 = 18\\y = 2.11 = 22\end{array}$$
So x= 18, y = 22.
### Solve lesson 6.8, page 9 Math 7 Textbook Connecting knowledge volume 2
Find two numbers x and y, knowing: $$\dfrac{x}{{17}} = \dfrac{y}{{21}}$$ and x – y= 8
Solution method
Use the property of the series of equal ratios: $$\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{{a – c}}{{b – d}}$$
Detailed explanation
$$\begin{array}{l}\dfrac{x}{{17}} = \dfrac{y}{{21}} = \dfrac{{x – y}}{{17 – 21}} = \ dfrac{8}{{ – 4}} = – 2\\ \Rightarrow x = ( – 2).17 = – 34\\y = ( – 2).21 = – 42\end{array}$$
So x= -34; y = -42
### Solve lesson 6.9, page 9 Math 7 textbook Connecting knowledge volume 2
The ratio of products made by the two workers is 0.95. How many products does each person make, knowing that one person makes 10 more products than the other?
Solution method
Use the property of the series of equal ratios: $$\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{{a – c}}{{b – d}}$$
Detailed explanation
adsense
Let the number of products that 2 people can make, respectively, x, y (product) (x, y > 0)
Since one person makes 10 more products than the other, x – y = 10
Since the ratio of product made by the two workers is 0.95, $$\dfrac{y}{x} = 0.95 \Rightarrow \dfrac{y}{{0.95}} = \dfrac{x} {first}$$
Applying the property of the series of equal ratios, we have:
$$\begin{array}{l}\dfrac{x}{1} = \dfrac{y}{{0.95}} = \dfrac{{x – y}}{{1 – 0.95}} = \dfrac{{10}}{{0.05}} = 200\\ \Rightarrow x = 200.1 = 200\\y = 200.0.95 = 190\end{array}$$
So 2 people can make 200 and 190 products respectively.
### Solve lesson 6.10, page 9 Math textbook 7 Connecting knowledge volume 2
Three classes 7A, 7B, 7C were assigned the task of planting 120 trees to green the barren hills. Calculate the number of plants that can be planted in each layer, knowing that the number of plants planted in the three classes 7A, 7B, 7C is proportional to 7; 8; 9.
Solution method
Let the number of trees of 3 classes 7A, 7B, 7C be planted as x, y, z respectively (x,y,z > 0)
Use the property of the series of equal ratios: $$\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = \dfrac{{a + c + e} }{{b + d + f}}$$
Detailed explanation
Let the number of trees of 3 classes 7A, 7B, 7C be planted as x, y, z respectively (x,y,z > 0)
Since the total number of plants in 3 classes is 120 trees, x+y+z = 120
Since the number of crops grown in the three classes 7A, 7B, 7C is proportional to 7;8;9, $$\dfrac{x}{7} = \dfrac{y}{8} = \dfrac{z}{9}$$
Applying the property of the series of equal ratios, we have:
$$\begin{array}{l}\dfrac{x}{7} = \dfrac{y}{8} = \dfrac{z}{9} = \dfrac{{x + y + z}}{{ 7 + 8 + 9}} = \dfrac{{120}}{{24}} = 5\\ \Rightarrow x = 5.7 = 35\\y = 5.8 = 40\\z = 5.9 = 45\end{array}$$
So the number of trees of 3 classes 7A, 7B, 7C can be planted, respectively, 35; 40; 45 trees.
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# Do two embeddings of a Euclidean space into a higher dimensional one only differ by a diffeomorphism?
Let $d\le n$ and $$f,g\colon\mathbb{R}^d\hookrightarrow\mathbb{R}^n$$ be two smooth embeddings. Is there a diffeomorphism $$\phi\colon\mathbb{R}^n\rightarrow \mathbb{R}^n,$$ such that $$f=\phi\circ g$$ holds?
In other words, does the diffeomorpshism group $\operatorname{Diff}{(\mathbb{R}^n)}$ act transitively on the set of embeddings $\operatorname{Emb}(\mathbb{R}^d,\mathbb{R}^n)$ by postcomposition?
As the other answer suggests, there is an obstruction in the form of properness. If $f$ is a closed (synonymously, proper) embedding, then so is $\phi \circ f$. But even restricting to closed embeddings this is not true.
Consider the case of long knots. That is, embeddings of $\mathbb R^1 \hookrightarrow \mathbb R^3$ that are the standard embedding outside $[-n,n]$ for some $n$. Then any two long knots can be mapped from one to the other by a diffeomorphism if and only if they are (compactly supported) isotopic. In other words, the same obstructions as in knot theory - not all knots $K,L: S^1 \hookrightarrow S^3$ have a diffeomorphism taking $K$ to $L$ (for instance, the trefoil and the unknot), the same is true for long knots.
This same obstruction is going to be true for long embeddings $\mathbb R^{n-2} \hookrightarrow \mathbb R^n$; every long embedding $\mathbb R^{n-1} \hookrightarrow \mathbb R^n$ is diffeomorphic to the standard one (except possibly for $n=4$? but in this case we still have that they're PL homeomorphic) by the Schoenflies theorem; all other long embeddings $\mathbb R^{n-k} \hookrightarrow \mathbb R^n$ are PL homeomorphic to the standard one but not necessarily diffeomorphic.
To avoid dealing with long knots and proper embeddings or whatever you might prefer to work with smooth embeddings $S^k \hookrightarrow S^n$. This removes technical complications. See Levine for the classification of high-codimension smooth knots. See Zeeman for the proof that all high-codimension knots are PL-standard.
If $g$ is onto and $f$ is not that cannot be possible. That can't be either if $d < n$.
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SOLUTION 12: Compute the area of the region enclosed by the graphs of the equations $y = \sin \sqrt{x}$ and $y=0$ on the interval $[0, \pi^2]$. Now see the given graph of the enclosed region.
Using vertical cross-sections to describe this region, we get that $$0 \le x \le \pi^2 \ \ and \ \ 0 \le y \le \sin \sqrt{x} \ ,$$ so that the area of this region is $$AREA = \displaystyle{ \int_{0}^{\pi^2 } (Top \ - \ Bottom) \ dx }$$ $$= \displaystyle { \int_{0}^{\pi^2} \sin \sqrt{x} \ dx }$$ Use a power" u-substitution. Let $u^2=x$ (or $u= \sqrt{x}$) so that $2u \ du = dx$. Then $$\displaystyle { \int \sin \sqrt{x} \ dx } = { 2 \int u \cdot \sin u \ du }$$ (Now use integration by parts. Let $w=u$ and $dv = \sin u \cdot du$ so that $dw = du$ and $v= -\cos u$.) $$= 2 \Big( -u \cos u - \displaystyle { \int - \cos u \ du } \Big)$$ $$= -2 u \cos u + 2 \displaystyle { \int \cos u \ du }$$ $$= -2 u \cos u + 2 \sin u + C$$ $$= -2 \sqrt{x} \cos \sqrt{x} + 2 \sin \sqrt{x} + C$$ Thus, continuing with the definite integral, we get $$\displaystyle { \int_{0}^{\pi^2} \sin \sqrt{x} \ dx } = (-2 \sqrt{x} \cos \sqrt{x} + 2 \sin \sqrt{x}) \Big\vert_{0}^{\pi^2}$$ $$= (-2 \sqrt{{\pi}^2} \cos \sqrt{{\pi}^2} + 2 \sin \sqrt{{\pi}^2} ) - (-2 \sqrt{0} \cos \sqrt{0} + 2 \sin \sqrt{0} )$$ $$= (-2 \pi \cos \pi + 2 \sin \pi ) - (-2 \sqrt{0} \cos 0 + 2 \sin 0 )$$ $$= (-2 \pi (-1) + 2(0)) - (-2(0)(1)+2(0))$$ $$= 2 \pi$$
Click HERE to return to the list of problems.
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/ hep-ex arXiv:1510.03720
Production of $\Lambda$ hyperons in inelastic p+p interactions at 158 GeV/$c$
Pages: 19
Abstract: Inclusive production of $\Lambda$-hyperons was measured with the large acceptance NA61/SHINE spectrometer at the CERN SPS in inelastic p+p interactions at beam momentum of 158~\GeVc. Spectra of transverse momentum and transverse mass as well as distributions of rapidity and x$_{_F}$ are presented. The mean multiplicity was estimated to be $0.120\,\pm0.006\;(stat.)\,\pm 0.010\;(sys.)$. The results are compared with previous measurements and predictions of the EPOS, UrQMD and FRITIOF models.
Note: 19 pages, 20 figures, 6 tables
Total numbers of views: 717
Numbers of unique views: 309
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### Home > CCG > Chapter 1 > Lesson 1.2.6 > Problem1-106
1-106.
The length of a side of a square is $5x+2$ units. If the perimeter is $48$ units, complete the following.
1. Write an equation to represent this information.
To find the perimeter, add the side lengths. In this case we have a square which has four equal side lengths.
One possibility is $4(5x+2)=48$.
2. Solve for $x$.
$x=2$
3. What is the area of the square?
Remember to substitute $x=2$ back into the original equation.
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# Graph the following function and then find the specified limits. When necessaX-1 x<4 flx) = 4sxs8 find Iim f(x) and
###### Question:
Graph the following function and then find the specified limits. When necessa X-1 x<4 flx) = 4sxs8 find Iim f(x) and lim f(x) 48 X+2 ifx>8 Choose the correct graph below: 0A 0 B. Select the correct choice and, ifnecessary; fill in the answer box to complete 0A. Iim f(x) = X-4 0 B. The limit is not 0 Or 00 and does not exist
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# Tolman–Oppenheimer–Volkoff equation
In astrophysics, the Tolman–Oppenheimer–Volkoff (TOV) equation constrains the structure of a spherically symmetric body of isotropic material which is in static gravitational equilibrium, as modelled by general relativity. The equation[1] is
${\displaystyle {\frac {dP(r)}{dr}}=-{\frac {G}{r^{2}}}\left[\rho (r)+{\frac {P(r)}{c^{2}}}\right]\left[M(r)+4\pi r^{3}{\frac {P(r)}{c^{2}}}\right]\left[1-{\frac {2GM(r)}{c^{2}r}}\right]^{-1}\;}$
Here, r is a radial coordinate, and ρ(r0) and P(r0) are the density and pressure, respectively, of the material at r = r0.
The equation is derived by solving the Einstein equations for a general time-invariant, spherically symmetric metric. For a solution to the Tolman–Oppenheimer–Volkoff equation, this metric will take the form[1]
${\displaystyle ds^{2}=e^{\nu (r)}c^{2}dt^{2}-(1-2GM(r)/rc^{2})^{-1}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\;}$
where ν(r) is determined by the constraint[1]
${\displaystyle {\frac {d\nu (r)}{dr}}=-\left({\frac {2}{P(r)+\rho (r)c^{2}}}\right){\frac {dP(r)}{dr}}\;}$
When supplemented with an equation of state, F(ρ, P) = 0, which relates density to pressure, the Tolman–Oppenheimer–Volkoff equation completely determines the structure of a spherically symmetric body of isotropic material in equilibrium. If terms of order 1/c2 are neglected, the Tolman–Oppenheimer–Volkoff equation becomes the Newtonian hydrostatic equation, used to find the equilibrium structure of a spherically symmetric body of isotropic material when general-relativistic corrections are not important.
If the equation is used to model a bounded sphere of material in a vacuum, the zero-pressure condition P(r) = 0 and the condition exp[ν(r)] = 1 − 2GM(r)/rc2 should be imposed at the boundary. The second boundary condition is imposed so that the metric at the boundary is continuous with the unique static spherically symmetric solution to the vacuum field equations, the Schwarzschild metric:
${\displaystyle ds^{2}=(1-2GM_{0}/rc^{2})c^{2}dt^{2}-(1-2GM_{0}/rc^{2})^{-1}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\;}$
## Total mass
M(r0) is the total mass inside radius r = r0, as measured by the gravitational field felt by a distant observer, it satisfies M(0) = 0.[1]
${\displaystyle {\frac {dM(r)}{dr}}=4\pi \rho (r)r^{2}\;}$
Here, M0 is the total mass of the object, again, as measured by the gravitational field felt by a distant observer. If the boundary is at r = rB, continuity of the metric and the definition of M(r) require that
${\displaystyle M_{0}=M(r_{B})=\int _{0}^{r_{B}}4\pi \rho (r)r^{2}\;dr\;}$
Computing the mass by integrating the density of the object over its volume, on the other hand, will yield the larger value
${\displaystyle M_{1}=\int _{0}^{r_{B}}{\frac {4\pi \rho (r)r^{2}}{\sqrt {1-2GM(r)/rc^{2}}}}\;dr\;}$
The difference between these two quantities,
${\displaystyle \delta M=\int _{0}^{r_{B}}4\pi \rho (r)r^{2}(1-(1-2GM(r)/rc^{2})^{-1/2})\;dr\;}$
will be the gravitational binding energy of the object divided by c2 and it is negative.
## Derivation from general relativity
Let us assume a static, spherically symmetric perfect fluid. The metric components are similar to those for the Schwarzschild metric:
${\displaystyle c^{2}d\tau ^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }=e^{\nu (r)}c^{2}dt^{2}-e^{\lambda (r)}dr^{2}-r^{2}d\theta ^{2}-r^{2}\sin ^{2}\theta d\phi ^{2}\;}$
By the perfect fluid assumption, the stress-energy tensor is diagonal (in the central spherical coordinate system), with eigenvalues of energy density and pressure:
${\displaystyle T_{0}^{0}=\rho (r)c^{2}\;}$
and
${\displaystyle T_{i}^{j}=-P(r)\delta _{i}^{j}\;}$
Where ${\displaystyle \rho (r)}$ is the fluid density and ${\displaystyle P(r)}$ is the fluid pressure.
To proceed further, we solve Einstein's field equations:
${\displaystyle {\frac {8\pi G}{c^{4}}}T_{\mu \nu }=G_{\mu \nu }\;}$
Let us first consider the ${\displaystyle G_{00}}$ component:
${\displaystyle {\frac {8\pi G}{c^{4}}}\rho c^{2}e^{\nu }={e^{\nu } \over r^{2}}\left(1-{\frac {d}{dr}}re^{-\lambda }\right).\;}$
Integrating this expression from 0 to ${\displaystyle r}$, we obtain
${\displaystyle e^{-\lambda }=1-{2GM(r) \over rc^{2}},\;}$
where ${\displaystyle M(r)}$ is as defined in the previous section. Next, consider the ${\displaystyle G_{11}}$ component. Explicitly, we have:
${\displaystyle -{\frac {8\pi G}{c^{4}}}P(r)e^{\lambda (r)}={\frac {-r\nu '(r)+e^{\lambda (r)}-1}{r^{2}}}\;}$
which we can simplify (using our expression for ${\displaystyle e^{\lambda }}$) to
${\displaystyle {d\nu \over dr}={1 \over r}\left(1-{2GM(r) \over c^{2}r}\right)^{-1}\left({2GM(r) \over c^{2}r}+{8\pi G \over c^{4}}r^{2}P\right).\;}$
We obtain a second equation by demanding continuity of the stress-energy tensor: ${\displaystyle \nabla _{\mu }T_{\nu }^{\mu }=0}$. Observing that ${\displaystyle \partial _{t}\rho =\partial _{t}P=0}$ (since the configuration is assumed to be static) and that ${\displaystyle \partial _{\phi }P=\partial _{\theta }P=0}$ (since the configuration is also isotropic), we obtain in particular
${\displaystyle 0=\nabla _{\mu }T_{1}^{\mu }=-{\mathrm {d} P \over \mathrm {d} r}-{1 \over 2}\left(P+\rho c^{2}\right){\mathrm {d} \nu \over \mathrm {d} r}\;}$
Rearranging terms yields:[2]
${\displaystyle {\frac {dP(r)}{dr}}=-\left({\frac {\rho (r)c^{2}+P(r)}{2}}\right){\frac {d\nu (r)}{dr}}\;}$
This gives us two expressions, both containing ${\displaystyle {\mathrm {d} \nu \over \mathrm {d} r}}$. Eliminating ${\displaystyle {\mathrm {d} \nu \over \mathrm {d} r}}$, we obtain:
${\displaystyle {\frac {dP(r)}{dr}}=-{1 \over r}\left({\frac {\rho (r)c^{2}+P(r)}{2}}\right)\left({2GM(r) \over c^{2}r}+{8\pi G \over c^{4}}r^{2}P\right)\left(1-{2GM(r) \over c^{2}r}\right)^{-1}\;}$
Pulling out a factor of ${\displaystyle G \over r}$ and rearranging factors of 2 and ${\displaystyle c^{2}}$ results in the Tolman–Oppenheimer–Volkoff equation:
${\displaystyle {\frac {dP(r)}{dr}}=-{\frac {G}{r^{2}}}\left(\rho (r)+{\frac {P(r)}{c^{2}}}\right)\left(M(r)+4\pi r^{3}{\frac {P(r)}{c^{2}}}\right)\left(1-{\frac {2GM(r)}{c^{2}r}}\right)^{-1}\;}$
## History
Richard C. Tolman analyzed spherically symmetric metrics in 1934 and 1939.[3][4] The form of the equation given here was derived by J. Robert Oppenheimer and George Volkoff in their 1939 paper, "On Massive Neutron Cores".[1] In this paper, the equation of state for a degenerate Fermi gas of neutrons was used to calculate an upper limit of ~0.7 solar masses for the gravitational mass of a neutron star. Since this equation of state is not realistic for a neutron star, this limiting mass is likewise incorrect. Modern estimates for this limit range from 1.5 to 3.0 solar masses.[5]
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# Preimage of Intersection under Mapping/Family of Sets/Proof 2
## Theorem
Let $S$ and $T$ be sets.
Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$.
Let $f: S \to T$ be a mapping.
Then:
$\displaystyle f^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i} = \bigcap_{i \mathop \in I} f^{-1} \sqbrk {T_i}$
where:
$\displaystyle \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\family {T_i}_{i \mathop \in I}$.
$f^{-1} \sqbrk {T_i}$ denotes the preimage of $T_i$ under $f$.
## Proof
$\displaystyle x$ $\in$ $\displaystyle f^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i}$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \map f x$ $\in$ $\displaystyle \bigcap_{i \mathop \in I} T_i$ $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \forall i \in I: \,$ $\displaystyle \map f x$ $\in$ $\displaystyle T_i$ $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \forall i \in I: \,$ $\displaystyle x$ $\in$ $\displaystyle f^{-1} \sqbrk {T_i}$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{i \mathop \in I} f^{-1} \sqbrk {T_i}$
$\blacksquare$
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Enter the force (N), the distance (m), and the time (s) into the calculator to determine the Rate of Work.
## Rate of Work Formula
The following formula is used to calculate the Rate of Work.
ROW = F*d / t
• Where ROW is the Rate of Work (J/s)
• F is the force (N)
• d is the distance (m)
• t is the time (s)
## How to Calculate Rate of Work?
The following example problems outline how to calculate the Rate of Work.
Example Problem #1
1. First, determine the force (N). In this example, the force (N) is given as 15 .
2. Next, determine the distance (m). For this problem, the distance (m) is given as 6 .
3. Next, determine the time (s). In this case, the time (s) is found to be 7.
4. Finally, calculate the Rate of Work using the formula above:
ROW = F*d / t
Inserting the values from above and solving yields:
ROW = 15*6 / 7 = 12.857 (J/s)
Example Problem #2
Using the same method as above, determine the variables required by the equation. For this example problem, these are provided as:
force (N) = 8
distance (m) = 4
time (s) = 5
Enter these given values yields:
ROW = 8*4 / 5 = 6.4 (J/s)
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# Asteroid Deflection
During the nineteen eighties it has become clear that the K/T mass extinction, 65 million years ago, has been caused by the impact of an asteroid or comet with a diameter of roughly ten kilometers. The first clear evidence was found in the form of an world-wide iridium layer, pointing to the presence of extraterrestrial material from the body of the impactor. The final proof came when the actual crater was found, with a diameter of almost 200 kilometer, buried under thick layers of sediment and located near Chicxulub on the Yucatan peninsula.
This impact coincided with the demise of the dinosaurs, as well as a significant fraction of species of animals and plants that were then populating the Earth. While we do not know yet many of the details of the effects of such a huge impact, it is clear that human civilization and perhaps even humanity itself would be threatened with extinction, were a similar impact to happen in the near future. Even a much smaller impact would wreak enormous havoc on local scales. For estimates of some of the effects of impacts, see e.g. a recent paper by O'Keefe, Lyons, and Ahrens (click on Full Printable Article' to retrieve the texts, and see the table at the end of the paper). For an up-to-date list of current impact risks is maintained at NASA's Jet Propulsion Laboratory.
Fortunately, we are able in principle to avoid such a future impact, as long as we have sufficient early warning. One possible approach is to position a plasma engine on the surface of an asteroid, and to let it burn for a year or longer. There is no need to change the orbit of the asteroid very much, given that the Earth is such a small target within the vast space of the solar system. Even a change in velocity of 10 cm/sec will be enough to prevent an impact, if that change is made a few years before the predicted collision between the asteroid and the Earth. In contrast, it will not be possible to prevent an impact at the last moment, contrary to some recent movies, such as Deep Impact (which was quite realistic in a number of other aspects).
For up-to-date information about surveys and other developments, see the Asteroid and Comet Impact Hazards NASA web site, and the Spaceguard web site. For the first mission to an asteroid that actually landed on its surface, see NEAR Shoemaker's mission to Eros. For detailed information and a database of impact structures on Earth, see the Earth Impact Database.
### Project B612
Astrophysicist/astronaut Ed Lu and I organized an informal workshop on Deflecting Asteroids in October 2001, at the NASA Johnson Space Center in Houston. We explored the possibility of sending a plasma engine, powered by a nuclear reactor, to an asteroid in order to test the ability to alter the orbit of that asteroid.
We called our project "B612", which was the name of the small asteroid on which the Little Prince lived (from the novel Le Petit Prince by Antoine de St. Exupery).
We discussed various ways in which we could demonstrate the technology of altering the dynamics of an asteroid. Although the main goal will be to change the orbit sufficiently to avoid an impact on Earth, a first test could be the easier task of chaning the spin of a small orbit: either by despinning an asteroid, or letting it come apart (if it is a rubble pile) by spinning it up. The latter would form a very literal science spin-off', which could provide enough information about internal structure to justify a mission in its own right. Or we could experiment with the companion of a double asteroid, in which we could alter the spin as well as the orbit of the smaller of the two asteroids. The two main component of such a mission would be formed by a plasma engine and a nuclear energy generator.
A year after we held our first workshop, we founded the B612 Foundation, with the stated goal to significantly alter the orbit of an asteroid in a controlled manner by 2015.'' While ambitious, we believe this to be possible. See our web site for announcements concerning further events and activities, and also our article:
• The Asteroid Tugboat, by Schweickart R.L., Lu, E.T., Hut, P. & Chapman, C.; 2003, Scientific American, 289, Number 5, pp. 54-61.
For a brief statement on the goal of asteroid deflection, see my contribution:
and a related opinion piece:
See also the report of the conference on Scientific Requirements for Mitigation of Hazardous Comets and Asteroids.
Here is the abstract of a presentation we made about 2004 MN4 Apophis:
At the NASA NEO Workshop in Vail, Colorado, in 2006, June 26-28, we presented several white papers, among others:
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# DIS 2015 - XXIII. International Workshop on Deep-Inelastic Scattering and Related Subjects
27 April 2015 to 1 May 2015
US/Central timezone
## Parity Violation Deep Inelastic Scattering Experiments at JLab
29 Apr 2015, 10:45
20m
THEATER ()
### THEATER
WG3 Electroweak Physics and Beyond the Standard Model
### Speaker
Vincent Sulkosky (MIT)
### Description
We report on a measurement of parity-violating (PV) asymmetries in the deep inelastic scattering (DIS) and nucleon resonance regions using inclusive scattering of longitudinally polarized electrons from unpolarized deuterons. The effective weak couplings $C_{2q}$ are accessible through these PV-DIS asymmetries. This measurement of the PV asymmetry yielded a determination of $2C_{2u} - C_{2d}$ with an improved precision of a factor of five relative to previous results. These results indicate evidence with 95% confidence that the $2C_{2u} - C_{2d}$ is non-zero. This experiment also provides the first parity-violation data covering the whole resonance region, which provide constraints on nucleon resonance models. The program to measure PV-DIS at JLab in the 12 GeV era will also be briefly discussed.
Slides
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## January 28, 2006
### Ghost D-Branes and Renormalization
Catch a very interesting discussion over at Cosmic Variance of a paper by Evans, Morris and Rosten, relating Morris’s Exact Renormalization Group for large-$N$ $SU(N)$ Yang Mills to the “Ghost D-brane” proposal of Okuda and Takayanagi.
The claim is that, by embedding $SU(N)$ Yang-Mills in a larger (nonunitary) theory, whose gauge groups is the supergroup $SU(N|N)$, spontaneously broken to $SU(N)\times SU(N)$, one can produce a gauge-invariant Pauli-Villars regulator, with which to implement the Exact RG. The latter theory, in turn, is what Okuda and Takyanagi argue is the world-volume theory of a stack of D-branes and ghost D-branes.
When the gauge symmetry is unbroken, the $SU(N|M)$ theory is equivalent to $SU(N-M)$, as far as computing gauge-invariant observables. In particular, there is a perfect cancellation of diagrams for $N=M$.
Turning on a nonzero Higgs VEV (separating the D-branes from the ghost D-branes) provides a cutoff for the original $SU(N)$ theory. Above the scale of the Higgs VEV, you get zero; far below it, the “original” $SU(N)$ degrees of freedom decouple from the ghost $SU(N)$.
Evans et al propose and AdS/CFT geometry realization of this idea, with the hope of connecting, in a explicit way, the “holographic RG” (evolution in the radial coordinate of AdS) with the “exact RG” of Morris.
Anyway, Takuya Okuda is over there, fielding questions, so take advantage …
Posted by distler at 1:29 AM | Permalink | Followups (13)
## January 20, 2006
### Gravity is Weak
I wasn’t going to post anything about the recent paper by Arkani-Hamed et al, figuring that Luboš is perfectly capable of explaining himself. But, after discussions with various people, it became clear that a few comments are in order.
They argue for two propositions which “must” be true in any theory of quantum gravity, but which are not obvious, at all, from the point of view of low-energy effective field theory. They both concern theories in which there is an unbroken $U(1)$ gauge symmetry in the low-energy theory.
First, they argue that there must exist charged particle(s) in the theory, whose charge-to-mass ratio exceeds1 the extremal bound on the charge-to-mass ratio of blackholes. A-priori, you could imagine that one could twiddle the masses and charges of fundamental particles arbitrarily. However, as they make clear, in the absence of particles which exceed the bound, charged blackholes cannot radiate away all their charge, and one is left with a large (possibly infinite) number of charged remnants.
This is a perfectly solid result, and one which can be understood quite clearly, once one takes account of blackholes and their evaporation. From it, they abstract away the slogan, “Gravity is the weakest force.” Which leads them to their second conjecture.
The strength of the effective gravitational force grows like a power-law in the UV. The strength of gauge-interactions vary only logarithmically (growing in the UV for abelian gauge theories and falling in the UV for asymptotically-free nonabelian gauge theories). If we go to high enough energies, the gravitational force, therefore, comes to dominate, or would do so if the theory were not cut off.
If we ignore the slow logarithmic running of the gauge coupling, demanding that gauge-interactions dominate over gravitational ones puts a cutoff on effective field theory, not at $M_{\text{pl}}$, but at a lower scale, $g M_{\text{pl}}$.
Now, it’s certainly true in all known string theories, 4D effective field theory breaks down below (often, well below) the 4D Planck scale. Indeed, as Arkani-Hamed knows well, the scale at which 4D effective field theory breaks down could be as low as several TeV. I firmly believe (along with the authors) that this is a general principle. But, to put a precise upper bound on the cutoff, at which 4D effective field theory must break down, does require taking account of the running of the gauge coupling.
To sharpen the conjecture, the authors assert that “$g$” in the above formula is the low-energy value of the gauge coupling below the mass of the lightest charged particle. This is not directly related to the “high-energy” value of the coupling (close to the cutoff scale). The rate at which the coupling runs depends on massive charged species at intermediate scales. Not just the magnitude, but even the sign of the $\beta$-function could change (if the abelian gauge theory is un-higgsed into a nonabelian one). So the scale at which gravity and gauge interactions become comparable in strength cannot be determined from low-energy data alone; it could be higher or lower than the “naïve” estimate of $g_{\text{IR}}M_{\pl}$.
Indeed, in many string backgrounds, $SU(2)\times U(1)$ is unbroken, and the quarks and leptons are massless. (In the same approximation, supersymmetry is, frequently, also unbroken.) In that case, the $U(1)$ gauge coupling is driven all the way to zero in the IR2. But that does not mean that effective field theory has zero range of validity.
One can imagine a self-consistent bound of the form $\Lambda = g(\Lambda) M_{\text{pl}}$. That’s the form of the bound that they actually check in examples. (It’s $g(\Lambda)$ that is directly related to $g_{\text{st}}$, not $g_{\text{IR}}$.) In that form, as they verify, the bound holds3. But it’s not a form that depends solely on low-energy data.
1 In the BPS case, saturates the bound.
2 The fact that a massless electron causes the gauge coupling to flow to zero in the IR does not contradict the previous argument about the charge-to-mass ratio. Even though both $m$ and $(\log(m))^{-1/2}$ vanish as $m\to 0$, the latter vanishes more slowly, so we preserve the fact that we have particles whose charge-to-mass ratio exceeds the extremal bound.
3 After some back-and-forth over email, Nima Arkani-Hamed agrees that this, rather than $\Lambda = g_{\text{IR}}M_{\text{pl}}$, is the bound. My argument is that it directly expresses the idea that $\Lambda$ is the scale at which gravity and gauge interactions become comparable in strength. Nima had a more sophisticated argument, involving the evaporation of magnetically-charged blackholes.
Posted by distler at 10:26 PM | Permalink | Followups (2)
## January 15, 2006
### More on Accesskeys
In my previous post, I argued that accesskey keyboard shortcuts, to be usable, need to be
1. discoverable
2. modifiable
and I suggested some techniques for achieving that. To make them discoverable, I suggested including on each page, in some artful way, a definition-list of the form
<dl id="AccessKeyList">
<dt>0</dt><dd><a href="/~distler/blog/accessibility.html" accesskey="0">Accessibility Statement</a></dd>
<dt>1</dt><dd>Main Page</dd>
<dt>3</dt><dd>List of Posts</dd>
<dt>4</dt><dd>Search</dd>
<dt>p</dt><dd>Previous (individual/monthly archive page)</dd>
<dt>n</dt><dd>Next (individual/monthly archive page)</dd>
</dl>
with the accesskeys for your site. To make them modifiable, I proposed a Javascript, which turns that definition-list into a template for editing the keybindings, and stores your modified keybindings for future visits.
But what about all those millions of sites, with accesskeys defined, which haven’t gone to this trouble? Since Gez Lemon inspired me to write the above Javascript, it was my turn to inspire Gez. Firefox users can install this Greasemonkey script which provides an (admittedly slightly crude) approximation to the same functionality on any site with accesskeys.
Without an author-supplied definition-list, Gez’s script needs to guess at the meaning of each accesskey. Usually, that works passably well. But, on some sites, the result can be downright comical. Obviously, it would be better if website authors went to the trouble of providing the aforementioned definition list. And with the aforementioned Javascript, all their users (not just those with Firefox and Greasemonkey installed) could customize those keybindings.
But still, it’s remarkable that, with a little clever scripting, Firefox users can now customize the keybindings for the accesskeys on any website. I’d call that progress.
(Update: The latest version of my Javascript restores compatibility with Gez’s Greasemonkey script.)
Looking into the (distant) future, XHTML 2 will offer a self-documenting reformulation of accesskeys. Instead of
<a accesskey="0" href="/accessibility.html" />
XHTML 2 authors will use the self-documenting <access> element. They’ll write
<access key="0"
title="Accessibility Statement"
targetrole="foo:accessibility" />
followed by
<a role="foo:accessibility" href="/accessibility.html" />
The WAI offer their own set of 8 predefined roles. Other roles need to be defined in their own namespace
xmlns:foo="http://mysite.com/roles/foo"
which points to an RDF Schema defining the role(s). Alternatively, there’s a
<access key="c"
targetid="comment" />
for those instances when a custom-defined role seems like overkill.
The definition-list we’ve been talking about here is a crude stand-in for the collection of <access> elements in an XHTML 2 document. With that substitution, the same techniques for customizing keybindings will work in XHTML 2. Hopefully, that capability will be built into XHTML 2 User Agents, rather than being an afterthought, as with today’s accesskey attribute.
#### Update (1/18/2006):
Gez’s script has been updated. The new features include
1. You can toggle the Accesskey menu by hitting Shift -Esc (to reclaim some screen real-estate, and avoid obscuring the bottom of the page).
2. The CSS styling is now rolled into the script, rather than loaded from Gez’s site.
For obvious reasons, if you’ve installed the previous version, you want to update. My Javascript has also been updated to detect the presence of Gez’s script and ensure that they play nicely together.
Posted by distler at 4:12 PM | Permalink | Followups (5)
## January 11, 2006
### Dem Bones
I was admiring the “doggiesaurus” dinosaur picture my 5 year old created in preschool today, as he cheerfully explained to me about their habits (they’re plant-eaters, apparently). So I turned to him and asked, “Son, when you grow up, would you like to be a scientist, who digs up dinosaur bones?” “Daddy,” he says to me, witheringly, “You mean a palæontologist.”
“Yes,” I said meekly, “a palæontologist.” “Sure.” he says, “And then, when I’m done, I can give the bones to a museum.”
Posted by distler at 11:06 PM | Permalink | Followups (1)
## January 8, 2006
### Editable Accesskeys
Accesskeys are a very nice mechanism for making your website more accessible. You can define a set of keyboard shortcuts for common navigational tasks, making it easier for visitors, who can’t (or don’t wish to) use a mouse, to find their way around your site. For instance, on this site, hitting Alt -4 (Cntrl -4 on a Mac) takes you to the search form on the current page; Alt -1 (Cntrl -1) takes you to the main page of this blog, etc.
Unfortunately, accesskeys suffer from two big drawbacks
1. There’s no easy way for visitors to discover what accesskeys are defined on a particular site.
2. The accesskeys you define may conflict with the user’s existing keyboard shortcuts. Even if they don’t, accesskey assignments vary from site to site, so there’s no “muscle memory” advantage to them.
The first problem is easy to fix. Long ago, I put a listing of the accesskeys, defined here, in the footer of each page. Scroll down to take a look at it. The markup that generated the footer is
<div id="footer">
<h2>Access Keys:</h2>
<dl id="AccessKeyList">
<dt>0</dt><dd><a href="/~distler/blog/accessibility.html" accesskey="0">Accessibility Statement</a></dd>
<dt>1</dt><dd>Main Page</dd>
<dt>3</dt><dd>List of Posts</dd>
<dt>4</dt><dd>Search</dd>
<dt>p</dt><dd>Previous (individual/monthly archive page)</dd>
<dt>n</dt><dd>Next (individual/monthly archive page)</dd>
</dl>
<a href="/~distler/blog/archives.html" accesskey="3"></a>
</div>
and it can be included at the bottom of every page, using SSI, PHP or, in my case MovableType’s <MTInclude> directive. Of course, this definition-list didn’t have to be in the footer of the page. It could have been in a CSS drop-down menu at the top of the page or … The point is to make it unobtrusive, and yet easily-discoverable. Someone, with better design skills than I, could have found a more artful solution.
The second drawback seemed harder to surmount. Somehow, we need to let the user customize the accesskey assignments. But how?
Recently, Gez Lemon found a solution using server-side scripting. Unfortunately, it was more than a little cumbersome. As a web-author, it requires mucking with the markup of your pages1. As a user, it involves thoroughly unnecessary back-and-forth interaction with the server. Still, it was inspirational. Once you realize it can be done, you start to think about other ways to achieve the same effect.
I wanted something that would not involve mucking with the existing markup of my pages, and which would work client-side (so as to be as fast and smooth as possible). If you’ve read this far, you probably want to see it in action. Scroll down2 to the bottom of the page and hit the, hitherto mysterious, button. You can assign any keyboard character you want to an accesskey, though most browsers do not distinguish between uppercase and lowercase letters. If you want to disable one (or all) of the accesskeys, simply delete the corresponding character. When you’re happy, hit . You can now use your new accesskey assignments anywhere on this blog. And they’ve been saved in a cookie, for your future surfing pleasure. If, instead, the current accesskey assignments suit you, just hit .
No markup was butchered to achieve this effect. All I needed to do was include this Javascript file and add initializeAccessKeys(); to the onload-handler for my pages. In fact, the Javascript uses the existing markup (a definition-list whose id="AccessKeyList")3 as an editing template for customizing the accesskeys.
I rather enjoyed crafting this, my first somewhat nontrivial bit of Javascript programming. There’s something delightfully recursive about using DOM-scripting to add event-handler attributes which, when activated, trigger other DOM-scripting operations.
I’ve only tested it in a few browsers (Mozilla, Safari and Opera), but expect it to work in others as well. Suggestions for improvements are welcome.
#### Update (1/16/2006):
Gez Lemon has crafted a Greasemonkey script to obtain similar (but not quite as good) functionality on any website with accesskeys. I’ve written a followup post about it. I’ve also revised my Javascript to play more nicely with Gez’s script.
#### Update (4/2/2006):
Rich Pedley wins the prize for catching a bug in the previous version of this Javascript. I’ve updated it to fix the bug.
#### Postscript: Event Handlers
Getting keyboard navigation to work right in the presence of certain event-handlers seems to be a black art. All the sources I’ve looked at give incorrect advice. As a consequence, I, myself, have been doing it wrong for years now.
onclick="..."
by itself doesn’t work right. The user should be able to trigger this handler by hitting Return. But that doesn’t happen in Safari (and, perhaps, other browsers).
onclick="..."
onkeypress="..."
will cause the handler to be triggered by hitting any key. Which is not what you want: that breaks Tab-navigation. What does work is
onclick="..."
onkeypress="if(window.event.keyCode == 13){...; return false;}"
Since I haven’t seen this explained anywhere4, I figured I’d spell it out here, and save someone else from tearing their hair out, too.
1 Among other drawbacks, Gez’s implementation is limited to the creation of <a href="" accesskey=""> and <label for="" accesskey=""> elements. No other attributes are allowed, and you can’t use it with <area>, <button>, <input>, <legend> or <textarea>, all of which are permitted to carry accesskey attributes.
2 You can also use keyboard navigation: hit Shift -Tab, to move to the bottom of the page, highlighting the button. You should be able to hit return to activate editing of the accesskeys for this site (you can navigate between the editing field using Tab and Shift -Tab).
3 The script assumes a <dl id="AccessKeyList">. If you want to give the definition-list containing the accesskeys and their meanings a different id, change the value of the accesskeylistid variable in the script.
4 Peter-Paul Koch’s otherwise-excellent table of Javascript Event support is, I’m afraid, incorrect on the status of onclick support in Safari.
Posted by distler at 11:54 PM | Permalink | Followups (17)
## January 7, 2006
### It’s Time to Speak of Serious Matters
The nonpartisan Congressional Research Service has issued their report, looking into the legality of the domestic spying program initiated 4 years ago by the President, in the wake of 9/11. Unsurprisingly, their conclusion is that it violated the Foreign Intelligence Surveillance Act and they, quietly but thoroughly, demolishes the Administration’s argument that the Sept. 2001 Authorization for Use of Military Force authorized the Administration to circumvent FISA1 at their pleasure.
Anyone with even a vestigial interest in the Rule-of-Law in this country ought to read it. It’s heavily laden with footnotes, but if you skip those, not too difficult to make your way through.
Administration loyalists have, predictably, decried the New York Times for compromising National Security with this revelation2. As Glen Greenwalt convincingly demonstrates, they don’t know what they’re talking about. (He’s been doing some excellent blogging on the NSA spying scandal.)
There’s no getting around the fact that the Adminstration has flagrantly, unrepentantly violated the Law of the land, and vows to continue to do so. They’ve had 4 years to propose changes to FISA (beyond those of the USA Patriot Act), if they felt there was an argument (which would pass Constitutional muster) for doing so. Instead, they decided to simply ignore FISA. Evidently, the Rule-of-Law is not on their radar screen. Everyone wants the President to have the tools necessary to fight the War on Terror. No one but a partisan hack would conclude that shredding the Constitution is the way to provide those tools.
So, where does that leave us? The current Majority in the Senate is, alas, of the opinion that Impeachment is a partisan club, to be wielded against one’s political opponents, rather than a solemn tool to ensure that the Executive adheres to the Rule-of-Law. Without tremendous public pressure, we’re unlikely to see substantive hearings, let alone Articles of Impeachment.
Perhaps a bumper sticker campaign is in order
WHO DO YOU HAVE TO BLOW TO GET A PRESIDENT IMPEACHED AROUND HERE?!
KING GEORGE? DIDN’T WE FIGHT A REVOLUTION ABOUT THAT?
1 And engage in torture and arbitrary detention without trial and …. But let’s not stray from the current subject.
2 Arguably, by sitting on the story for over a year, the New York Times compromised our National Security by ensuring the reelection of George W. Bush. But that’s probably not what Hindraker had in mind.
Posted by distler at 2:33 PM | Permalink | Followups (3)
## January 5, 2006
### 41-38
I’m, normally, not much of a football fan, but that was an amazing game. Vince Young is a God.
And besides, how often do I get to tweak my colleague, Clifford?
Posted by distler at 12:08 AM | Permalink | Followups (1)
## January 2, 2006
### CDT
After writing my two previous posts on approaches to quantum gravity, various people asked me to write something about Causal Dynamical Triangulations, a lattice model that has enjoyed a certain amount of favourable ‘buzz’ recently. I’ve been procrastinating about following up on that request because, to do a halfway decent job would require at least two post, one about generalities about lattice models of quantum gravity and one specifically about CDT. Alas, I’m not that interested in the subject, so I’ve always been able to find something else I’d rather write about …
Anyway, as a bit of New Years resolve, here’s a stab at such a post which will, alas, fall far short of what’s really required.
Posted by distler at 3:35 AM | Permalink | Followups (15)
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# Correctly using MOSFETs
I'm using a P-Channel (DMP1045UQ) and an N-Channel (BSS138K) FET to pull a 12v Logic line (12v @ 5mA) high (from GND) from a Logic Level(3.3v) MCU I/O pin.
Here is the schematic, where VIN = 12v, MCU_TEL_ON = MCU Pin and TEL_ON_IO is the line that is being pulled high.
My questions are:
• Have I done anything egregious here?
• Is R17 even necessary? Seems to work fine with it in the circuit, though.
Additionally, the circuit does not need to switch quickly -- TEL_ON_IO is meant to be kept high for long periods of time.
Your PMOS gate must tolerate the entire swing of Vin. Do not assume that Vgs_max is the same as Vds_max because it does 0% of the time from what I've seen.
The PMOS you have chosen has Vds_max = 12V, but Vgs_max = 8V and you plan to have Vin = 12V which will blow your PMOS the first time you pull its gate low.
R17 is necessary to turn the PMOS off by discharging the gate capacitance since the NMOS can't do anything to turn the PMOS off on its own.
• Thanks! I also forgot to add a resistor from the NMOS gate to GND to discharge its gate. – t3ddftw May 10 at 20:25
• Small follow-up the BSS54 would be a better PMOS choice, with a -20v Vgs, right? – t3ddftw May 10 at 20:43
• @t3ddftw Yes. Just note it's resistance is 10 ohm which is rather high, but what you get in return is low gate charge so it will turn on faster. – DKNguyen May 10 at 22:01
• RdsON is okay as long as it's less than ~1% of the load resistance, which in this case is 47 Ohms (given R18 @ 4.7k,) right? That's what I was basing my search on :) – t3ddftw May 10 at 22:17
• Yeah, it's not a big deal in this case since you're just talking about a signal (almost no current) and it's small compared to the other resistors in the circuit. – DKNguyen May 10 at 23:03
3 errors:
1. Pch, DS is reversed
2. Pch Vgs +/-8 max is exceeded
3. Item 2 above is due to extremely low RdsOn 31mΩ which is a poor choice considering 4k7 load. Usually below 1% of load is adequate
err # 2 "could be fixed" with a series R divider.
• For #1, I don't think these are reversed. Should positive voltage not be applied to the "Source"? In my testing, if I connect PchD to +12v and PchS to my DMM, I always have at least 12v. With PchS to 12v and PchD to my DMM, I get 12v only when the NchG is pulled high. Thanks for pointing out the inadequacy of DMP1045UQ! – t3ddftw May 10 at 20:23
• Yes but Diode is always conducting in Pch and source is on ground side. – Sunnyskyguy EE75 May 10 at 21:04
• I realized the error of my schematic -- my source is what the diode is pointing towards. – t3ddftw May 10 at 21:48
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## anonymous one year ago Decide how many solutions this equation has: x^2 + 3 = 0
1. anonymous
@twistnflip
2. ganeshie8
may be just try solving it : $$x^2+3=0 \implies x^2=-3 \implies \cdots$$
3. anonymous
Okay can you explain to me how to solve it. I came uo with x=6 so one solution but i dont know if thats correct
4. ganeshie8
plugin $$x=6$$ does it make the given equation true ?
5. anonymous
I think not
6. ganeshie8
x^2 + 3 = 0 6^2 + 3 = 0 36 + 3 = 0 39 = 0 which is clearly false, so nope, x=6 is not a solution.
7. anonymous
What about a square root of negative 3? Does such thing exist?
8. anonymous
I think there is no solution right?
9. anonymous
What if I try putting the negative sign before the bracket surrounding square root of 3?
10. anonymous
Like -(sqrt(3)) Probably I am wrong.
11. anonymous
Nah forget it I don't think that even exists.
12. anonymous
So yeah, I think x would have to be of square root value but with the negative sign I think that makes the equation unsolvable. Is that correct?
13. freckles
depends on the set we are solving over but no sqrt(-3) doesn't equal -sqrt(3) sqrt(-3) is imaginary while -sqrt(3) is real
14. anonymous
Right. Imaginary root shuold be used as the solution. So I belive minus square root of three might be more relevant? Forgive me if I am wrong.
15. freckles
$(-\sqrt{3})^2=((-1)\sqrt{3})^2= (-1)^2 (\sqrt{3})^2=1(3)=3 \neq -3$
16. anonymous
So that's the full solution for x apparently. @abriannaridley I guess that's your take home message from todayXD
17. anonymous
lol thanks for everyones help! :P
18. freckles
what conclusion did you draw from this @abriannaridley ?
19. ganeshie8
im liking jokes that cortana tells me when i feel bored... so let me tell you a joke :) gimme one moment to type..
20. anonymous
okay so im not 100% if this is right but there is only solution to that problem? @freckles
21. freckles
only solution you mean no real solution right?
22. freckles
x^2=negative number will never happen under the real numbers
23. anonymous
Does cortana make no comment like Siri when you talk like a you know person ?
24. anonymous
Yes lol! should've just said no solution
25. freckles
x^2=negative number can happen under the complex numbers
26. anonymous
Find the x-intercepts of the graph of the equation: y = x2 - 3x + 2 im not sure how to do this one either should i just post it as a new question?
27. freckles
x-intercepts can found by setting y to 0 and solve for x
28. anonymous
ohh okay!
29. freckles
and yeah to prevent things from getting to long sometimes it is best to post a new question
30. freckles
but you can post a question I guess if you have trouble solving 0=x^2-3x+2
31. ganeshie8
sry for the delay, here it is : A foreman is yelling down into an excavation : "How many of you men down there ?" the reply came back, "three." and he said "Okay, half of you come on up."
32. freckles
but you can post a question I guess if you have trouble solving 0=x^2-3x+2
33. ganeshie8
that joke refers to solving the equation $$2x=3$$. does it really have a solution ? If it has a solution, what does it represent ?
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Installation
Configuring SMTP server post-installation?
by Alasdair McAndrew -
Number of replies: 2
I have installed WeBWorK on a linux-based VSP, using the excellent installation script, which basically took care of everything for me. I didn't configure email at the time, as I had no email server installed or configured.
Do I need to go into WeBWorK config files to configure a server (I intend to use the external public google server smtp.gmail.com), or can I just configure that for my VSP and assume WeBWorK will use whatever server it can find?
Thanks,
Alasdair
Re: Configuring SMTP server post-installation?
by Michael Gage -
You'll need to specify your smtp server in the file
webwork2/conf/site.conf
There is a section on configuring the mail capabilities.
It should be pretty easy.
-- Mike
$mail{smtpSender}='mailserver.mail.com --login=myname --pass=mypasswod --port=11111' or, if I'm using nullmailer, something like $mail{smtpSender}=nullmailer
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# Get page numbers to the outer side in twoside mode
I´d like to use the twoside mode for my print but unfortunately the page numbers are on the right side on left pages and vice versa. I´d like to see the page numbers on the outer side of each page.
Since I´m not using any fancy packages so far I´d like to know which options I have to set to get this.
\documentclass[a4paper, 12pt, parskip=full-, listof=totoc, bibliography=totoc, headsepline, twoside]{scrartcl}
\usepackage[utf8]{inputenc}
\usepackage{lipsum}
\ofoot{\pagemark}
\begin{document}
\newpage
\section{Introduction}
\lipsum[1-20]
\end{document}
• The code you've posted doesn't generate the issue of "the page numbers are on the right side on left pages and vice versa". Instead, on recto (i.e., odd-numbered) pages, the page numbers are on the right, and vice versa for verso (i.e., even-numbered) pages. – Mico Feb 10 '18 at 17:34
• @Mico sorry for my bad I´ve updated a working code sample – Moka Feb 10 '18 at 17:59
• @The 2-page screenshot is maybe a bit misleading, or at least confusing. If you were to print the document two-sided (#2 behind #1, #4 behind #3, etc), you'd notice that the page numbers are always on the outside, exactly as desired. – Mico Feb 10 '18 at 19:51
• May it be that i´ve mixed up the desired order? Should the fist page always start as a right page? In this case i´m still wonderin why the larger side-border is on the outer and not at the inner side. – Moka Feb 10 '18 at 20:35
• The screenshot you posted doesn't actually show the pages in the way they'd appear if you opened a book that's been printed two-sided. In a book, when you're looking at facing pages, the (near-universal) convention is that even-numbered pages (aka "verso") should be on the left and odd-numbered pages (aka "recto") should be on the right. Another typographic convention is for the "outer" margins (those facing outward, away from the book's spine) should be larger than the "inner" margins. Again, that's exactly what you're getting with your current setup. – Mico Feb 10 '18 at 20:44
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# Homework Help: Time Dependent Current in a wire
1. Mar 7, 2012
### kjlchem
1. The problem statement, all variables and given/known data
An infinite straight wire carries a current I that varies with time as shown above. It increases from 0 at t = 0 to a maximum value I1 = 2.1 A at t = t1 = 14 s, remains constant at this value until t = t2 when it decreases linearly to a value I4 = -2.1 A at t = t4 = 24 s, passing through zero at t = t3 = 21.5 s. A conducting loop with sides W = 20 cm and L = 57 cm is fixed in the x-y plane at a distance d = 49 cm from the wire as shown.
What is ε1, the induced emf in the loop at time t = 7 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.
2. Relevant equations
B = μI/2∏r
Flux = B*A
-dflux/dt = ε
3. The attempt at a solution
I don't understand what I'm doing wrong with this problem.
This is what I have so far...
(dB*A)/dt= ε, A = L(W)
μ(dI)(L)W/(2∏rdt) = ε
μ=12.566*10^-7
dI = 2.1 A
L = .57 m
W = .2 m
dt=14 s.
On the left side of the box, r = .49 m and the current is negative, so the emf is positive.
On the right side of the box, r = 1.06 m and the current is positive, so the emf is negative.
Putting the 2 emf's together by subtracting the right side from the left side, I get an emf of -3.753*10^-9V.
What am I doing wrong?
File size:
2.3 KB
Views:
99
2. Mar 7, 2012
### Redbelly98
Staff Emeritus
I'm not sure about the method you are using. For calculating EMF for a straight section of wire, I am only familiar with doing that for the wire moving through a magnetic field.
Since the wire loop is not moving, I think you have to use ε=-dflux/dt instead. So first I would first calculate the flux through the loop -- as a function of time, during the time interval that contains 7 s.
3. Mar 7, 2012
### kjlchem
Yeah, I used ε = -dflux/dt. The flux as a function of time = μ(dI)(L)W/(2∏rdt)
4. Mar 8, 2012
### Redbelly98
Staff Emeritus
That won't work here; for one thing, there is no r given in this problem. I think I see your problem though:
That only works if B is uniform over the whole area. It isn't; B is stronger at the side of the rectangle closest to the wire, and weaker at the far side.
Instead, you'll need to do an integral to calculate the flux:
Flux = $\int B \cdot dA$
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# Vertical alignment in a tabular environment with asymptote image
In a tabular environment, I would like to align text vertically at the top of one cell while placing an Asymptote image in an adjacent cell. There are numerous posts on vertical alignment with images in a tabular environment with "normal" image placement using \includegraphics. See here for instance:
Vertical alignment of text and figures in a table
However, the two most commonly suggested techniques -- using \raisebox or \adjustbox -- don't seem to work with asymptote images. Here is a minimal example showing the effect I would like to get, and what actually happens when I use an asymptote image:
\documentclass{article}
\usepackage{mwe}
\usepackage{asymptote}
\begin{document}
\begin{tabular}{c c}
Text &
\raisebox{-.9\height}{\includegraphics[scale=0.25]{example-image}}
\\ Text &
\begin{asy}
import graph;
unitsize(1inch);
draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0));
\end{asy}
\end{tabular}
\end{document}
Any idea how to get top-aligned text next to an asymptote image?
With adjustbox it's really easy:
\documentclass{article}
\usepackage{asymptote}
\begin{document}
\begin{tabular}{c c}
Text &
Text &
I used the “command form” in the first case, because it's handier. Around the asy environment it's better instead using the “environment form”.
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Change Perturbation Level of Blocks Perturbed During Linearization
Blocks that do not have preprogrammed analytic Jacobians linearize using numerical perturbation. You can modify the size of the state and input signal perturbation levels for your application.
Change Block Perturbation Level
This example shows how to change the perturbation level to the Magnetic Ball Plant block in the magball model. Changing the perturbation level changes the linearization results.
For this model, the state and input signal values are double-precision values. The default perturbation size for the state and input signals in this model is ${10}^{-5}\left(1+|\mathit{x}|\right)$, where $\mathit{x}$ is the operating point value of the perturbed state or input signal.
Open the model before changing the perturbation level.
open_system('magball')
Change the perturbation level of the states to ${10}^{-7}\left(1+|\mathit{x}|\right)$, where $\mathit{x}$ is the state value.
blockname = 'magball/Magnetic Ball Plant';
set_param(blockname,'StatePerturbationForJacobian','1e-7');
To change the perturbation level of the input signal for this block to ${10}^{-3}\left(1+|\mathit{x}|\right)$, where $\mathit{x}$ is the input signal value, first obtain the block port handles and get the handle to the input port value.
ph = get_param(blockname,'PortHandles');
p_in = ph.Inport(1);
Then, set the input port perturbation level.
set_param(p_in,'PerturbationForJacobian','1e-3');
To obtain the current perturbation level for block states, use the following code.
statePerturb = get_param(blockname,'StatePerturbationForJacobian');
To obtain the current perturbation level for block input signals, use the following code.
inputPerturb = get_param(p_in,'PerturbationForJacobian');
When the corresponding state or input signal perturbation level is at its default value, both statePerturb and inputPerturb are 'auto'.
Default Perturbation Levels
The default perturbation size for double-precision states and input signals is ${10}^{-5}\left(1+|\mathit{x}|\right)$, where x is the operating point value of the perturbed state or input signal. For single-precision states and input signals, the default perturbation size is $0.005\left(1+|\mathit{x}|\right)$.
To restore the default perturbation level for block states, use the following code.
set_param(blockname,'StatePerturbationForJacobian','auto');
To restore the default perturbation level for block input signals, use the following code.
set_param(p_in,'PerturbationForJacobian','auto');
Perturbation Levels of Integer-Valued Blocks
A custom block that requires integer input ports for indexing might have linearization issues when the block does not support small perturbations in the input value. To fix the problem, try setting the perturbation level of such a block to zero, which sets the block linearization to a gain of 1.
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You can choose the layout that better suits your document, even if the equations are really long, or if you have to include several equations in the same line. If you wanted to just split in two parts, then multlined (notice the extra d) from the mathtools package would be a simpler solution: All the above works with elsarticle class in your updated question. You need to specify an alignment point on each line with & and separate lines with \\. Equations on Multiple Lines. Mathematical modes LaTeX allows two writing modes for mathematical expressions: the inline mode and the display mode. How to answer a reviewer asking for the methodology code of the paper? Anyway, I'm glad to know that removing the extra empty lines worked. Open an example of the amsmath package in Overleaf In fact, your example is probably best with the cases environment. 2). It is therefore up to you to format the equation appropriately (if they overrun the margin.) Missing $inserted.$ ...l N(\sigma;\lambda;\theta;t)}{\partial t} over multiple lines, @PaulGessler although as far as I can see the give s me a better visual result, that is why i am keen of using that one. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. This is because LaTeX typesets maths notation differently from normal text. Relationship between Cholesky decomposition and matrix inversion? Is Mr. Biden the first to create an "Office of the President-Elect" set? Line breaks are straightforward, a double backslash does the trick This is not the only command to insert line breaks, in the next sectiontwo more will be presented. Robotics & Space Missions; Why is the physical presence of people in spacecraft still necessary? Also tried this example as kindly suggested. The first one is used to write formulas that are part of a text. Currently I just have a sequence of align environments, with each equation inside in order to align the pieces of each equations. Latex equation in multiple lines. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. When numbering is allowed you can label each row individually. I am a new Latex user,I have loaded these two math equation in my Latex documents and i want to split an equation i have into multiple line, I am reading this topic for breaking the lines over multiple, although what ever i do it does not allow me to compile. @PaulGessler I had a look in the document, thank you for the link. Asking for help, clarification, or responding to other answers. I suggest you use a split environment, which in contrast to multline may be used a subenvironment of equation. In this case the first line should be move left relative to the others and the package mathtools provides a convenient command for this: Latex equation in multiple lines. I am using the package math of \usepackage{amssymb} \usepackage{amsthm}, @George I posted the self-contained code example above so that you could see one of doing this that works. The \overbrace command places a brace above the expression (or variables) and the command \underbrace places a brace below the expression. Usually, the eqnarray environment is only used if authors cannot use the amsmath package because the environments included in this package are easier to use (refer to Section . What location in Europe is known for its pipe organs? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Schematically, this is what its supposed to look like: You need to specify an alignment point on each line with & and separate lines with \\. I think the use of multiple lines structured environment can work. The first part of the equation is here separated from the rest of the formula. This typically requires some creative use of an eqnarrayto get elements shifted to a new line to align nicely. two âcolumnsâ, one with the equation, one with the annotations. 3 Special Characterulti Line Equations. Understanding the zero current in a simple circuit. I am trying to align a set of long equations, that are themselves align environments as most of them are spreading on multiple lines.. Subscribe to our newsletter to get notification about new updates, information, etc.. I am trying to learn Latex at the same time , thank you for your patience. Therefore, special environments have been declared for this purpose. It only takes a minute to sign up. How would one justify public funding for non-STEM (or unprofitable) college majors to a non college educated taxpayer? thank you for your time, Podcast 300: Welcome to 2021 with Joel Spolsky, Split numbered equation over multiple pages, Defining new split environment with reduced spacing, Can't generate png with Error: Erroneous nesting of equation structures, Formula too long and \split fails | contains “sqrt” (square root). As a test, when the 2nd backslash was deleted at the end of the 3rd line of code (\dot{x} & = \sigma(y-x) \), it produced the single line equation you provided as an example. Check out what we are up to! 9. How to start equation environment inside a custom environment? You have to put up with [\\\\] a line break at the desired location in your equation. ! Also worked properly when code was typed in. E.g. Here also, the amsmath package is required. I want to place them/break the equation in two lines and retain the format of numbering and equal placing beneath. In another practical tip we show you how to in your, LaTeX: equation in multiple rows - how to, d^2 = a^2 + b^2 + c^2 \geq a^2 + b^2 = a^2 + b^2 + 2ab - 2ab = (a+b)^2 - 2ab \\\ \geq 2ab. If we have two lines, then _{xxx} should be at the end of the first line (or at the start of the second line). Find all positive integer solutions for the following equation: Allow bash script to be run as root, but not sudo. Art Of Problem Solving. The command \overline and \underline places a line above or below the expression. When numbering is allowed, you can label each row individually. These environments provide pairs of left- and right-aligned columns. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Open an example in Overleaf Adding right brace and equation number. Also that each equation is separated from the one before by an. rev 2020.12.18.38240, The best answers are voted up and rise to the top, TeX - LaTeX Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, @PaulGessler I used it alone but it gave more errors in the compilers most of them saying ! The Moodle XML file can then be used to bu Contents 1 Introduction 2 Including the amsmath package 3 Writing a single equation 4 Displaying long equations You have to put up with [\\\\] a line break at the desired location in your equation. Multiple Lines and Multiple Equations. The % sign tells LATEX L A T E X to ignore the rest of the current line. LATEX doesnât break long equations to make them ï¬t within the margins as it does with normal text. With a normal line break, you can't write in LaTeX equations multiple lines. Then there is an equation and some conditions for the equation, i.e. â Charlie Jan 16 '14 at 3:38 Auto Latex Equations Google Workspace Marketplace. Unlike the tabular environment, there is no argument as the ⦠Latex allows two writing modes for mathematical expressions. LaTeX equation editing supports most of the common LaTeX mathematical keywords. How is HTTPS protected against MITM attacks by other countries? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. We will show you how you can format the equation-environment is still suitable. As with the tabular environment, use & to separate columns and \\ to separate rows. Thanks for contributing an answer to TeX - LaTeX Stack Exchange! In that version I was able to open an Equation Editor and enter multiple lines of math equations in a the editor, pressing return at the end of each line to move to the next line. Latex Support In Pages Macs Chemistry. The starred version doesn't number the equations. Find news, promotions, and other information pertaining to our diverse lineup of innovative brands as well as newsworthy headlines about our company and culture. Add Mathematical Equations In Keynote On Mac Apple Support. To make use of the inline math feature, simply write your text and if you need to typeset a single math symbol or formula, surround it with dollar signs:Output equation: This formula f(x)=x2 is an example.This formula f(x)=x2 is an example. In this case the first line should be move left relative to the others and the package mathtools provides a convenient command for this: I have split across three lines for clarity. LUXCO NEWS. That can be achieve in plain LaTeX without any specific package. formulas, graphs). Use instead the equation - just the multline environment, you can work as usual with line breaks. \label {eq:Maxwell}, which will reference the main equation (1.1 above), or adding a label at the end of each line, before the \\ command, which will reference the sub-equation (1.1a or 1.1b above). Math equation in LaTeX provides three stretchable lines/arrows that appear above or below the equation: braces, bars and arrows. The next lines contain only the equations and annotations, I would like them aligned with the equation and annotation of the second line. To put a pay attention, when you Create a new line in front of the comparison operator [ & ]. LaTeX assumes that each equation consists of two parts separated by a &; also that each equation is separated from the one before by an &. There are several ways to format multiple equations and the amsmath package adds several more. The { and } characters are used to surround multiple characters that LATEX L A T E X should treat as a single character. What might happen to a laser printer if you print fewer pages than is recommended? The description of your first image says "On Windows, rstudio renders the equations correctly", but the first image shows a broken equation. The amsmath package provides the align and align* environments for aligned equations. Referencing subordinate equations can be done using either of two methods: adding a label after the \begin {subequations} command, viz. I suggest you use a split environment, which in contrast to multline may be used a subenvironment of equation. I am attaching a screenshot of the result: 3 Multiple Lines of Displayed Maths . Andrew,@Andrew,i used \begin{multline} after the , i pasted your solution and i get an error which i have given in the question as a new edit. That can be achieve in plain LaTeX without any specific package. If we have three lines then _{xxx} should be in the middle of the second line. View PDF âºâº LaTeX needs to know when text is mathematical. LaTeX Error: \begin{document} ended by \end{multline}.See the LaTeX manual or LaTeX Companion for explanation.Type H for immediate help.... \end{multline}, Though i get the equation broken it is not numbered, or has a one line distance from my text as the To learn more, see our tips on writing great answers. Now I realize your second screenshot shows what you saw in the RStudio IDE (in the source R Markdown document). They can be distinguished into two categories depending on how they are presented: 1. text â text formulas are displayed inline, that is, within the body of text where it is declared, for example, I can say that a + a = 2 a {\displaystyle a+a=2a} within this sentence. The placement of _{xxx} depends on the number of lines that underbrace symbol should occupy, and in general it should be somewhere in the middle. Aligning Multiline Equation To The Left With Only One. Is my Connection is really encrypted through vpn? Post by Alirezakn » Thu Nov 22, 2018 7:02 am . By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. the first version becomes: Not sure what you tried doing with multine but this seems OK: Btw, you seem to be missing a bracket on the RHS -- and I deleted an extraneous comma after the . These environments provide pairs of left- and right-aligned columns to be run as root but. Not sudo learn more, see our tips on writing great answers and arrows attaching a screenshot of formula! To specify an alignment point on each line with & and separate lines with \\, which in contrast multline... Printer if you print fewer pages than is recommended Office of the formula formulas that are of! Need to specify an alignment point on each line with & and separate with! Meter app be used a subenvironment of equation can i ask if multiple packages can be achieve in plain without... Same time, thank you for the link this is best used at desired! One before by an the common LaTeX mathematical keywords or personal experience which will help to solve your (... Equal placing beneath something which looks what you saw in the middle of the comparison operator [ &.. At 3:38 3 Special Characterulti line equations in the RStudio IDE ( in the majority situations. In fact, your example is probably best with the split environment, use * to the. If multiple packages can be achieve in plain LaTeX without any specific package two writing modes for expressions! At the start of a text they overrun the margin. two lines retain... You need to use this Feature, the amsmath-package had a look in the majority situations. Single character Create an Office of the paper environment inside a custom environment printer! Back them up with [ \\\\ ] a line above or below the expression ( or unprofitable ) college to... The document, thank you for the equation is longer than one line or several formulas must grouped... Script to be run as root, but not sudo logo © 2021 Stack Exchange our newsletter to get about. To get notification about new updates, information, etc your answer ”, you can work as with... Eqnarray environment could be used the right which sets the same multiple equations and annotations, i 'm to. The same time, thank you for your patience declared for this purpose that each equation inside in order align! Latex mathematical keywords a reviewer asking for the following equation: braces, bars and.! Much, although if i may ask can i ask if multiple packages can be achieve plain... View PDF âºâº the % sign tells LaTeX L a T E X should as! Supposed to look like: LaTeX needs to know when text is mathematical specify an alignment point on each with... Thanks for contributing an answer to TeX - LaTeX Stack Exchange Inc ; user contributions licensed under cc.... Declared for this purpose attention, when you Create a new line in front of the current line multline. Biden the first to Create an Office of the result: that can achieve... For this purpose public funding for non-STEM ( or variables ) and command!, bars and arrows realize your second screenshot shows what you seems to need, LaTeX, ConTeXt and. Little ( or unprofitable ) college majors to a laser printer if you print fewer pages is! Editing supports most of the paper your little ( or not little technical. } \end { document } this code produces something which looks what you seems to need can a smartphone meter... The common LaTeX mathematical keywords line in front of the comparison operator [ ]! Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting.... Other countries this URL into your RSS reader LaTeX Dealing long you right-aligned columns is best used at start. Long you probably do n't want the align and align * environments aligned. Of a line break at the start of a text new line mark... Notification about new updates, information, etc of options for displaying.... Subenvironment of equation then _ { xxx } should be in the document, thank you for your patience,. Mathematical keywords empty lines worked MITM attacks by other countries licensed under by-sa. Reviewer asking for the following equation: braces, bars and arrows or... And tax breaks two writing modes for mathematical expressions: the inline and!, with each equation is longer than one line or several formulas must grouped... Environment if you have to put a pay attention, when you Create a new line front. Multline may be used a subenvironment of equation contributions licensed under cc by-sa second screenshot shows you. Part of a text and right-aligned columns margins as it does with normal.. Options for displaying equations LaTeX without any specific package { math } \end { document } this code something! Environment inside a custom environment how you can label each row individually {. There is no argument as the amsmath package provides a handful of options for displaying equations of equation environment. Back them up with [ \\\\ ] a line above or below the expression ( or not little technical... Could be used look in the middle of the second line sets the same time, thank you very,. Mathematical equations in Keynote on Mac Apple Support equation to the Left with only one column of.... Like them aligned with the cases environment this URL into your RSS reader an... Logo © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa document ) R Markdown document ) annotation... Script to be run as root, but it 's not as mature as the LUXCO. Second screenshot shows what you saw in the document, thank you very much, although if i may can! A text for your patience your second screenshot shows what you seems to.... Which sets the same multiple equations in several ways to format the equation and of... Here separated from the rest of the equation - just the multline environment, use & to separate columns \\... Document, thank you for your patience one with the cases environment line in front the. } this code produces something which looks what you saw in the document thank! Only the equations and the command \underbrace places a brace above the expression,... Result: that can be achieve in plain LaTeX without any specific package the code. Is the physical presence of people in spacecraft still necessary pages latex equation multiple lines is recommended what you saw in middle. Overleaf then there is no argument as the amsmath package have been declared for purpose! Custom environment methodology code of the paper the amsmath-package the same time, thank you for the equation: bash!, use * to toggle the equation: braces, bars and arrows of line! For help, clarification, or responding to other answers to write formulas that are part of a line or! Equation numbering making statements based on opinion ; back them up with [ \\\\ ] a to. The command \overline and \underline places a line to align nicely RSS feed, and. Which in contrast to multline may be used to write formulas that are part of a.! Educated taxpayer are used to surround multiple characters that LaTeX L a E... Linebreak a equation in several lines use instead the equation - just the multline environment, you probably n't! Make them ï¬t within the margins as it does with normal text break long equations to make them within. To this RSS feed, copy and paste this URL into your RSS.... Updates, information, etc LaTeX equations multiple lines lines structured environment can work ] a break! You to format multiple equations in Keynote on Mac Apple Support formulas must be grouped,. Can a smartphone light meter app be used for 120 format cameras bash to. Known for its pipe organs have to put a pay attention, when you Create a line..., bars and arrows provides a handful of options for displaying equations or personal experience currently i just have sequence. Use instead the equation in two lines and retain the format of numbering and equal placing beneath any. Of multiple lines you ca n't write in LaTeX equations multiple lines ConTeXt, and related typesetting systems well the! Environment could be used each equation is here separated from the rest of the line. When text is mathematical try the example on the right which sets same... Declared for this purpose of people in spacecraft still necessary maths notation differently from text. Should be in the document, thank you for the following equation: braces, and... Rstudio IDE ( in the middle of the President-Elect '' set brace above the expression does normal... Logo © 2021 Stack Exchange been declared for this purpose aligning Multiline equation the. Multiple lines structured environment can work as usual with line breaks be the... Create an Office of the current line point on each line with & separate. Website which will help to solve your little ( or unprofitable ) majors. Could be used a subenvironment of equation best used at the desired location in Europe is known its! To separate columns and \\ to separate columns and \\ to separate rows as mature as the ⦠NEWS... Which in contrast to multline may be used equations automatically for this purpose how you can work this,! Needs to know when text is mathematical to a new line in front of result... It works very well in the majority of situations, but it 's not as mature as â¦... On multiple lines of people in spacecraft still necessary ca n't write LaTeX. As the amsmath package each equation is longer than one line or several formulas must be grouped together the. At 3:38 3 Special Characterulti line equations eqnarray * } Post by Alirezakn Thu...
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# zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The Kerr spacetime in generalized Bondi-Sachs coordinates. (English) Zbl 1048.83014
Summary: We define a generalized Bondi-Sachs (GBS) coordinate system. We utilize the null geodesics with zero angular momentum in the Kerr spacetime to transform the metric into GBS form. We write down the Kerr metric components explicitly in these new coordinates. The flat ($M = 0$) subcase of the Kerr spacetime can be shown to be coordinatized by null geodesics emanating from a ring. The curved ($M \ne 0$) subcase with $a = 0$ is the Schwarzschild spacetime in outgoing Eddington-Finkelstein coordinates. We analyse the important physical properties of the Kerr spacetime in the new coordinates. We conclude with the suggestion that a radiating generalization of the Kerr spacetime is most likely to be found in GBS coordinates.
##### MSC:
83C57 Black holes 53Z05 Applications of differential geometry to physics
Full Text:
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Translator Disclaimer
VOL. 61 | 2011 Quantum groups and quantization of Weyl group symmetries of Painlevé systems
Gen Kuroki
Editor(s) Koji Hasegawa, Takahiro Hayashi, Shinobu Hosono, Yasuhiko Yamada
## Abstract
We shall construct the quantized $q$-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the Bäcklund transformations for Painlevé equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra for any symmetrizable generalized Cartan matrix. Then non-integral powers of the image of the Chevalley generators generate the quantized $q$-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized Bäcklund transformations of $q$-Painlevé equations constructed by Hasegawa. We shall also prove that any subquotient integral domain of a quantized universal enveloping algebra of finite or affine type is an Ore domain.
## Information
Published: 1 January 2011
First available in Project Euclid: 24 November 2018
zbMATH: 1247.81213
MathSciNet: MR2867150
Digital Object Identifier: 10.2969/aspm/06110289
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# Strict Haskell
What would a strict version of Haskell look like? That is, a pure language with syntax as close to Haskell as possible but where function application and data constructors are evaluated strictly. Laziness should be accessible through an explicit thunk datatype. (Nikita Volkov has written a good explanation of the explicit thunk datatype.)
## What problems are we trying to solve?
• There are a lot of implicit thunks floating around in Haskell. It is anathema to treat A -> IO B as if it were simply A -> B so it seems odd for A to mean a type of computations (returning values of type A) rather than itself a type of values. Strict Haskell would be more honest since thunks would be represented by an explicit type.
• The need for newtype is symptomatic of the confusion introduced by implicit thunks.
• The need for evaluate :: a -> IO a is a wart which indicates that we do not understand how to mix IO and forcing thunks.
• When we have a function which releases some memory how can we ensure that the release happens before the rest of the program continues execution? In the pure case it seems possible to use CPS and in IO it seems possible to use evaluate (and probably the CPS form too) but is there a unified, principled way? It seems unlikely that we will find one whilst thunks are implicit.
• Lots of libraries (Lens, ST, Map, ByteString, Text) have strict and lazy versions of functions or data structures. Lots of monads have strict and lazy versions. This is very confusing. Is it really necessary?
• For a Functor f, a value x :: f a often contains one or several thunks of type a. However, there is no general way to force these thunks. If we apply a large number of fmaps to x then even if we force x it doesn’t mean the thunks inside have been forced. We can experience a space leak this way. This is a demonstration seq is implicitly part of the interface of every data type, but it is generally not treated that way by datatype authors.
## Drawbacks I anticipate with strictness
Note that monadic bind does not have an obvious drawback in a strict regime. With m a -> (a -> m b) -> m b the m b need not actually run anything if the m a short circuits, because it’s already hidden behind a function (imagine Maybe for a prototypical case).
The Applicative instances will be potentially more problematic. With (<*>) :: f (a -> b) -> f a -> f b it will be hard (syntactically at the very least) to stop the f a computation from running, even if the f (a -> b) short circuits. It will also only make sense to use lazy Traversible instances. Mapping an Applicative value creating function over an entire Traversible is a waste when it is followed by a sequence which can short circuit. Perhaps the true nature of (<*>) is LazyTuple (f (a -> b), f a) -> f b.
Is Haskell’s IO itself lazy out of necessity? Maybe. Consider a pure function containing an expression let x = f y :: IO A. Then x is an IO action ready to be run, but not run yet! IO certainly contains some sort of delaying and I don’t understand the importance of this to the whole strict Haskell issue.
## Hughes
Would this language be worse than Haskell? Hughes talks about the importance of laziness for modularity but I think that the issues he raises only actually apply to lazy data structures.
## Augustsson
Augustsson raises some important points, though a lot of his examples are based on the behaviour of error which in my opinion is just wrong. error is an effect and has no place in pure code.
### Lazy bindings
The “error” example is just wrong. The \a -> a + expensive example can be solved in a strict language through an explicit thunk data structures.
### Lazy functions
Augustsson’s charge of lazy functions is a harder one to answer. Again he levels the false criticism of error, but the point still stands. I think his idea of using {exp} for a thunk that when called evaluates exp is a good one.
### Lazy constructors
I don’t understand Augustsson’s point about lazy constructors. Does “lazy constructor” mean “lazy datatype”?
### Cyclic data structures
I think these are solvable with explicit thunks.
### Reuse
I don’t understand Augustsson’s point here at all. His “biggest gripe” is about composability, but it seems trivially solvable with lazy data structures. He does seem to realise this: in the comments he says “It doesn’t matter if ‘or’ and ‘map’ are overloaded for different kinds of collections, it’s still wrong for strict collections”.
Bob Harper makes an interesting point in the comments:
‘In my mind however the conditional branch is not an example of [laziness]. The expression “if e then e1 else e2” is short-hand for a case analysis, which binds a variable (of unit type) in each branch. So the conditional is not “lazy” at all; rather, it is an instance of the general rule that we do not evaluate under binders (ie, evaluate only closed code).’
## Kmett
Edward Kmett raises some good points and some that I don’t understand.
I think one thing that he is hinting at is that in m >>= f the tail call is not f, rather (>>=). Thus if m is evaluated strictly (in our setup, that corresponds to “not wrapped in a thunk”) and is a large chain of calls of the same structure, then a lot of stack space will be consumed. (Kmett mentions traversable rather than a chain of monadic binds, but I suspect its the same issue).
Kmett later reiterated the issue and mentioned similar ones. See the examples starting with
• “Monads are toys due to the aforementioned restriction.”
• “You wind up with issues like SI-3295”
### Equational reasoning
I don’t really understand how laziness (or probably more accurately, non-strictness) is relevant to equational reasoning. I would like to find some explicit examples. Perhaps “Theorems for Free”?
### Value restriction
I really don’t understand the relevance of this at all.
### Code that can be understood in pieces
This is basically just lazy bindings. I get why it’s good, but would hope to find a syntax which makes it less relevant.
### Elegant expression of certain algorithms
This is a lazy data structures only comment, as far as I can tell.
### Control structures
This is exactly the same as Augustsson’s point above.
### Monads
Kmett claims monads only work with laziness. I don’t understand this.
## Being Lazy with Class
The “Being Lazy With Class” paper mentions two important uses of laziness:
• Recursive datastructures – this is the same as Augustsson’s “cyclic data structures argument” but better made. Is this easily replaceable with a lightweight thunking syntax?
• Unusual control flow – this is the same as Augusstson’s point, and Kmett’s “Control Structures” point
## Other
Don Syme notes in “Initializing Mutually Referential Abstract Objects: The Value Recursion Challenge” that “Wadler et al. describe techniques to add on-demand computations to strict languages.”
A CUFP paper notes that Mu has demonstrated strictness to be harmful to modularity.
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JACOBI OPERATORS ALONG THE STRUCTURE FLOW ON REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM II
Title & Authors
JACOBI OPERATORS ALONG THE STRUCTURE FLOW ON REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM II
Ki, U-Hang; Kurihara, Hiroyuki;
Abstract
Let M be a real hypersurface of a complex space form with almost contact metric structure ($\small{{\phi}}$, $\small{{\xi}}$, $\small{{\eta}}$, g). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $\small{R_{\xi}=R({\cdot},\;{\xi}){\xi}}$ is $\small{{\xi}}$-parallel. In particular, we prove that the condition $\small{{\nabla}_{\xi}R_{\xi}=0}$ characterizes the homogeneous real hypersurfaces of type A in a complex projective space or a complex hyperbolic space when $\small{R_{\xi}{\phi}S=R_{\xi}S{\phi}}$ holds on M, where S denotes the Ricci tensor of type (1,1) on M.
Keywords
complex space form;real hypersurface;structure Jacobi operator;Ricci tensor;
Language
English
Cited by
1.
REAL HYPERSURFACES OF NON-FLAT COMPLEX SPACE FORMS WITH GENERALIZED ξ-PARALLEL JACOBI STRUCTURE OPERATOR, Glasgow Mathematical Journal, 2016, 58, 03, 677
References
1.
J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic spaces, J. Reine Angew. Math. 395 (1989), 132-141.
2.
T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), no. 2, 481-499.
3.
U.-H. Ki and H. Kurihara, Real hypersurfaces with cyclic-parallel structure Jacobi operators in a nonflat complex space form, Bull. Aust. Math. Soc. 81 (2010), no. 2, 260-273.
4.
U.-H. Ki, H. Kurihara, S. Nagai and R. Takagi, Characterizations of real hypersurfaces of type A in a complex space form in terms of the structure Jacobi operator, Toyama Math. J. 32 (2009), 5-23.
5.
U.-H. Ki, H. Kurihara, and R. Takagi, Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form, Tsukuba J. Math. 33 (2009), no. 1, 39-56.
6.
U.-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990), 207-221.
7.
M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137-149.
8.
H. Kurihara, The structure Jacobi operator for real hypersurfaces in the complex projective plane and the complex hyperbolic plane, Tsukuba J. Math. 35 (2011), 53-66.
9.
M. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20 (1986), no. 2, 245-261.
10.
M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355-364.
11.
M. Ortega, J. D. Perez, and F. G. Santos, Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms, Rocky Mountain J. Math. 36 (2006), no. 5, 1603-1613.
12.
J. D. Perez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex projective spaces whose structure Jacobi operator is D-parallel, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 3, 459-469.
13.
R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495-506.
14.
R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures I, II, J. Math. Soc. Japan 27 (1975), 43-53, 507-516.
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International Association for Cryptologic Research
# IACR News Central
You can also access the full news archive.
Further sources to find out about changes are CryptoDB, ePrint RSS, ePrint Web, Event calender (iCal).
2015-02-23
22:17 [Pub][ePrint]
Authenticated encryption schemes guarantee both privacy and integrity, and have become the default level of encryption in modern protocols. One of the most popular authenticated encryption schemes today is AES-GCM due to its impressive speed. The current CAESAR competition is considering new modes for authenticated encryption that will improve on existing methods. One property of importance that is being considered more today is due to the fact that the nonce or IV repeats, then this can have disastrous effects on security. A (full) nonce misuse-resistant authenticated encryption scheme has the property that if the \\emph{same} nonce is used to encrypt the \\emph{same} message twice, then the same ciphertext is obtained and so the fact that the same message was encrypted is detected. Otherwise, full security is obtained -- even if the same nonce is used for different messages.
In this paper, we present a new fully nonce misuse-resistant authenticated encryption scheme that is based on carefully combining the GCM building blocks into the SIV paradigm of Rogaway and Shrimpton. We provide a full proof of security of our scheme, and an optimized implementation using the AES-NI and PCLMULQDQ instruction sets. We compare our performance to the highly optimized OpenSSL 1.0.2 implementation of GCM and show that our \\emph{nonce misuse-resistant} scheme is only 14\\% slower on Haswell architecture and 19\\% slower on Broadwell architecture. On Broadwell, GCM-SIV encryption takes only {\\em 0.92 cycles per byte}, and GCM-SIV decryption is exactly the same as GCM decryption taking only 0.77 cycles per byte. Beyond being very fast, our new mode of operation uses the same building blocks as GCM and so existing hardware and software can be utilized to easily deploy GCM-SIV. We conclude that GCM-SIV is a viable alternative to GCM, providing full nonce misuse-resistance at little cost.
22:17 [Pub][ePrint]
In recent years, there has been great interest in Functional Encryption (FE), a generalization of traditional encryption where a token enables a user to learn a specific function of the encrypted data and nothing else. In this paper we put forward a new generalization of FE that we call Mergeable FE (mFE). In a mFE system, given a ciphertext $c_1$ encrypting $m_1$ and a ciphertext $c_2$ encrypting $m_2$, it is possible to produce in an oblivious way (i.e., given only the public-key and without knowledge of the messages, master secret-key or any other auxiliary information) a ciphertext encrypting the string $m_1||m_2$ under the security constraint that this new ciphertext does not leak more information about the original messages than what may be leaked from the new ciphertext using the tokens. For instance, suppose that the adversary is given the token for the function $f(\\cdot)$ defined so that for strings $x\\in\\zu^n$, $f(x)=g(x)$ for some function $g:\\zu^n\\rightarrow\\zu$ and for strings $y=(x_1||x_2)\\in\\zu^{2n}$, $f(x_1||x_2)=g(x_1) \\vee g(x_2)$. Furthermore, suppose that the adversary gets a ciphertext $c$ encrypting $(x_1||x_2)$ that is the result of merging some ciphertexts $c_1$ and $c_2$ encrypting respectively $x_1$ and $x_2$, and suppose that the token for $f$ evaluates to $1$ on $c$. Then, the security of mFE guarantees that the adversary only learns the output $f(x_1,x_2) = g(x_1) OR g(x_2)=1$ and nothing else (e.g., the adversary should not learn whether $g(x_1)=1 or g(x_2)=1$). This primitive is in some sense FE with the best possible homomorphic properties and, besides being interesting in itself, it offers wide applications. For instance, it has as special case multi-inputs FE and thus indistinguishability obfuscation (iO) and extends the latter to support more efficiently homomorphic and re-randomizable properties. We construct mFE schemes supporting a single merging operation, one from indistinguishability obfuscation for Turing machines and one for messages of unbounded length from public-coin differing-inputs obfuscation. Finally, we discuss a construction supporting unbounded merging operations from new assumptions.
22:17 [Pub][ePrint]
Recent results have shown the usefulness of tamper-proof hardware tokens as a setup assumption for building UC-secure two-party computation protocols, thus providing broad security guarantees and allowing the use of such protocols as buildings blocks in the modular design of complex cryptography protocols. All these works have in common that they assume the tokens to be completely isolated from their creator, but this is a strong assumption. In this work we investigate the feasibility of cryptographic protocols in the setting where the isolation of the hardware token is weakened.
We consider two cases: (1) the token can relay messages to its creator, or (2) the creator can send messages to the token after it is sent to the receiver. We provide a detailed characterization for both settings, presenting both impossibilities and information-theoretically secure solutions.
03:08 [PhD][New]
Name: Jerzy Jaworski
Category: (no category)
03:08 [PhD][New]
Name: Przemyslaw Sokolowski
Topic: Contributions to cryptanalysis: design and analysis of cryptographic hash functions
Category: secret-key cryptography
Description:
A cryptographic hash function is a mechanism producing a fixed-length output of a message of arbitrary length. It fulfills a collection of security requirements guaranteeing that a hash function does not introduce any weakness into the system to which it is applied. The example applications of cryptographic hash functions include digital signatures and message authentication codes. This thesis analyzes cryptographic hash functions and studies the design principles in the construction of secure cryptographic hash functions.
We investigate the problem of building hash functions from block ciphers and the security properties of different structures used to design compression functions. We show that we can build open-key differential distinguishers for Crypton, Hierocrypt-3, SAFER++ and Square. We know that our attack on SAFER++ is the first rebound attack with standard differentials. To demonstrate the efficiency of proposed distinguishers, we provide formal proof of a lower bound for finding a differential pair that follows a truncated differential in the case of a random permutation. Our analysis shows that block ciphers used as the underlying primitive should also be analyzed in the open-key model to prevent possible collision attacks.
We analyze the IDEA-based hash functions in a variety of cipher modes. We present practical complexity collision search attacks and preimage attacks, where we exploit a null weak-key and a new non-trivial property of IDEA. We prove that even if a cipher is considered secure in the secret-key model, one has to be very careful when using it as a building block in the hashing modes.
Finally, we investigate the recent rotational analysis. We show how to extend the rotational analysis to subtractions, shifts, bit-wise Boolean functions, multi additions and multi subtractions. In particular, we develop formulae for calculation of probabilities of preserving the rotation property for multiple modular additions and subtra[...]
2015-02-22
13:11 [Event][New]
Submission: 20 March 2015
From June 11 to June 7
Location: Bucharest, Romania
05:37 [Event][New]
Submission: 20 March 2015
From June 15 to June 17
Location: Paris, France
05:37 [Event][New]
Submission: 20 March 2015
From June 15 to June 17
Location: Paris, France
2015-02-21
13:07 [Event][New]
Submission: 10 April 2015
From August 24 to August 25
Location: Heraklion, Crete, Greece
2015-02-20
08:08 [Event][New]
From October 12 to October 16
Location: Paris, France
MAC striping has been suggested as a technique to authenticate encrypted payloads using short tags. For an idealized MAC scheme, the probability of a selective forgery has been estimated as $\\binom{\\ell+m}{m}^{-1}\\cdot 2^{-m}$, when utilizing MAC striping with $\\ell$-bit payloads and $m$-bit tags. We show that this estimate is too optimistic. For $m\\le\\ell$ and any payload, we achieve a selective forgery with probability $\\ge \\binom{\\ell+m}{m}^{-1}$, and usually many orders of magnitude more than that.
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# Effective emissivity of surface
Some typical values for the effective emissivity of surface materials are listed in the table below. Also refer to the table in $\epsilon_{rad}$.
Symbol
$K_t$
Related
$\epsilon_{rad}$
Used in
$\alpha_{st}$
$T_{st}$
Choices
IdComponentValue
1Jacket/protective cover0.9
2Conductor, unpainted0.29
3Conductor, black color painting (by brush)0.9
4Enclosure, white color painting (by brush)0.7
5Enclosure, black color painting (by spray)0.95
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# 5. Identifying the limiting reagent: Given the reaction 2Cu + S → Cu2S, if you had 3 moles of both Cu and S, which would you run out of first?
###### Question:
5. Identifying the limiting reagent: Given the reaction 2Cu + S → Cu2S, if you had 3 moles of both Cu and S, which would you run out of first?
### Help!!! Will give brainliest!!!!
Help!!! Will give brainliest!!!!...
### The brazilian free-tailed bat can travel 99 miles per hour. after sunset, a colony of bats emerges from a cave and spreads out in a circular pattern. how long before these bats cover an area of 80,000 square miles? use pi = 3.14. 0.9 hours 1.6 hours 2.6 hours 5.1 hours
the brazilian free-tailed bat can travel 99 miles per hour. after sunset, a colony of bats emerges from a cave and spreads out in a circular pattern. how long before these bats cover an area of 80,000 square miles? use pi = 3.14. 0.9 hours 1.6 hours 2.6 hours 5.1 hours...
### Four year old robert tells his friend that he is the fastest runner in his preschool, that he will be the bet srudent in class when he starrts first grade, and that he is going to grow up to be rich. robert's statment reflects his
Four year old robert tells his friend that he is the fastest runner in his preschool, that he will be the bet srudent in class when he starrts first grade, and that he is going to grow up to be rich. robert's statment reflects his...
### 19. Find the value of X (In the picture) (giving points to best answer/brainlest)
19. Find the value of X (In the picture) (giving points to best answer/brainlest)...
### Why did Alfred the Great pay the Danes to leave england alone?
Why did Alfred the Great pay the Danes to leave england alone?...
### Conjugate the the verb in parentheses in the affirmative tú command form. Elisa, (encender) la luz. Question 3 options: encendía enciendas encienda enciende
Conjugate the the verb in parentheses in the affirmative tú command form. Elisa, (encender) la luz. Question 3 options: encendía enciendas encienda enciende...
### In the 1820s and 1830s, what kind of relationship did Americans have with the people of the Far West?\
In the 1820s and 1830s, what kind of relationship did Americans have with the people of the Far West?\...
### Roquan, a single taxpayer, is an attorney and practices as a sole proprietor. This year, Roquan had net business income of $90,000 from his law practice (net of the associated for AGI self-employment tax deduction). Assume that Roquan pays$40,000 in wages to his employees, has $10,000 of property (unadjusted basis of equipment he purchased last year), and has no capital gains or qualified dividends. His taxable income before the deduction for qualified business income is$100,000. (Leave no ans
Roquan, a single taxpayer, is an attorney and practices as a sole proprietor. This year, Roquan had net business income of $90,000 from his law practice (net of the associated for AGI self-employment tax deduction). Assume that Roquan pays$40,000 in wages to his employees, has \$10,000 of property (...
### Evaluate the function f(x)=4.7x for x=-1 and x=2. show your work
evaluate the function f(x)=4.7x for x=-1 and x=2. show your work...
### What is another term for controllable risk factors such as physical activity and diet
what is another term for controllable risk factors such as physical activity and diet...
### How does the U.S. Constitution reflect the principle of democracy? (1 point) It creates a republic government, which is the same as democracy. Most of the people can elect leaders and petition the government. It creates a republic form of government run by only a few officials. Most of the people have no role or effect on the legislative branch
How does the U.S. Constitution reflect the principle of democracy? (1 point) It creates a republic government, which is the same as democracy. Most of the people can elect leaders and petition the government. It creates a republic form of government run by only a few officials. Most of the people ha...
### Which of the following inequalities is true?
Which of the following inequalities is true?...
### The Constitution set up how many branches in the United States Government? A. 1 B. 2 C. 3 D. 4
The Constitution set up how many branches in the United States Government? A. 1 B. 2 C. 3 D. 4...
### During your research on a topic, you run across a blog site where several people posted that fuel prices have gone up 10% in the last six months. You can provide this data in your presentation as which of the following? Typical of opinions across the country, users on blahblah-blog.com have mentioned that gas prices have gone up 10% in the last six months. Although there are rumors about increased gas prices, there is no factual data. People across the country are reporting that gas prices have
During your research on a topic, you run across a blog site where several people posted that fuel prices have gone up 10% in the last six months. You can provide this data in your presentation as which of the following? Typical of opinions across the country, users on blahblah-blog.com have mentione...
### It takes Alexander 2/3 weeks to read one book. If Alexander reads for 1 1/2 weeks, how many books will he read?
it takes Alexander 2/3 weeks to read one book. If Alexander reads for 1 1/2 weeks, how many books will he read?...
### Mary wants to hang a mirror in her room. The mirror and frame must have an area of 7 square feet. The mirror is 2 feet wide and 3 feet long. Which quadratic equation can be used to determine the thickness of the frame, x? x2 + 14x − 2 = 0 2x2 + 10x − 7 = 0 3x2 + 12x − 7 = 0 4x2 + 10x − 1 = 0
Mary wants to hang a mirror in her room. The mirror and frame must have an area of 7 square feet. The mirror is 2 feet wide and 3 feet long. Which quadratic equation can be used to determine the thickness of the frame, x? x2 + 14x − 2 = 0 2x2 + 10x − 7 = 0 3x2 + 12x − 7 = 0 4x2 + 10x − 1...
### What was the new england colonial economy based upon?
What was the new england colonial economy based upon?...
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# Is a LTI filter completely characterized by its frequency response
I know that we use the frequency response to determine the properties gain and phase delay of a filter. However, I was wondering if this is enough for a complet characterization of the the filter for e.g. in terms of noise attenuation.
For example: I have a low pass filter. Am I guaranteed to attenuate e.g. colored noise?
Cheers, rot8
• but i would say that initial conditions are not properties of the filter but are property of the initial states of the filter. in general, i would say that the frequency response (you need both gain and phase) completely characterize the input/output relationship of the filter, just as the impulse response does in the time domain. But if there is pole/zero cancellation there could be internal states that don't show up in the output. – robert bristow-johnson Jun 21 '19 at 19:41
• @StanleyPawlukiewicz we had a similar discussion with MattL about initial conditions and LCCDE and LTI ness, however once the term LTI system is triggered then it shall also imply initial rest and therefore its frequency response is uniquely specifying its impulse response and that latter completely describes the (observable part of the) LTI system... – Fat32 Jun 21 '19 at 21:18
• @StanleyPawlukiewicz yes overlap save (block based) processing requires initial conditions (or previous block outputs) but it's not about characterizing the system...anyway not a big deal... – Fat32 Jun 21 '19 at 21:51
• @Fat32 what about if the filter is causal, anti causal, or no causal – user28715 Jun 21 '19 at 23:47
• @StanleyPawlukiewicz unlike Z-transform which requires a region of convergence to distinguish between causal, anti-causal seqeunces, Fourier transform alone can distinguish between causal or anti causal sequences. That's why it's said to uniquely representing $h[n]$. In particular if $h[n]$ is causal with $H(w)$ then $h[-n]$ will be anticausal with $H(-w)$ and you will have two different Fourier trasforms... – Fat32 Jun 22 '19 at 9:12
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[NTG-context] First document — problems with bib module and layout
Taco Hoekwater taco at elvenkind.com
Thu Dec 27 16:46:28 CET 2007
Matija Šuklje wrote:
> Dne četrtek 27. decembra 2007 je Matija Šuklje napisal(a):
>> From what I can tell the main problem is that in legal circles it's
>> considered bad form to cite (the short one) right next to the cited text.
>> The much preferred way is "below the line" in the footer.
>
> I think I solved the problem with footnotes now by using
> \footnote{\cite[extras={, p. 12}][ref]} ...but I'm still not happy with too
> little information produced by \cite and not having a working
> \placepublications
>
>
> Any help appreciated.
The solution is probably along the lines of
\def\shortcitation#1{%
\nocite[#1]%
\begingroup
\getcitedata[kratko][#1] to \shorttitle
\getcitedata[avtor][#1] to \shortauthor %define this bibfield first!
\expanded{\footnote{\shortitle, \shortauthor ...}}%
\endgroup}
More tomorrow, maybe. You have picked the worst possible time to ask
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Subjects -> MATHEMATICS (Total: 1082 journals) - APPLIED MATHEMATICS (86 journals) - GEOMETRY AND TOPOLOGY (23 journals) - MATHEMATICS (800 journals) - MATHEMATICS (GENERAL) (43 journals) - NUMERICAL ANALYSIS (24 journals) - PROBABILITIES AND MATH STATISTICS (106 journals) MATHEMATICS (800 journals) 1 2 3 4 | Last
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Similar Journals
Czechoslovak Mathematical JournalJournal Prestige (SJR): 0.307 Number of Followers: 1 Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572-9141 - ISSN (Online) 0011-4642 Published by Springer-Verlag [2626 journals]
• Annihilators of Local Homology Modules
• Abstract: Abstract Let $$(R,\mathfrak{m})$$ be a local ring, a an ideal of R and M a nonzero Artinian R-module of Noetherian dimension n with hd(a, M) = n. We determine the annihilator of the top local homology module $$H^{\mathfrak{a}}_{n}(M)$$ . In fact, we prove that $$\text{Ann}_R(H^{\mathfrak{a}}_{n}(M))=\text{Ann}(N(\mathfrak{a},M)),$$ where $$N(\mathfrak{a},M)$$ denotes the smallest submodule of M such that $$\text{hd}(\mathfrak{a},M/N(\mathfrak{a},M))<n$$ . As a consequence, it follows that for a complete local ring $$(R,\mathfrak{m})$$ all associated primes of $$H^{\mathfrak{a}}_{n}(M)$$ are minimal.
PubDate: 2019-03-01
• On Weakly-Supplemented Subgroups of Finite Groups
• Abstract: Abstract A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In this paper, some interesting results with weakly-supplemented minimal subgroups to a smaller subgroup of G are obtained.
PubDate: 2019-03-01
• Sums of Multiplicative Function in Special Arithmetic Progressions
• Abstract: Abstract We find, via the Selberg-Delange method, an asymptotic formula for the mean of arithmetic functions on certain APs. It generalizes a result due to Cui and Wu (2014).
PubDate: 2019-03-01
• A Note on a Conjecture on Niche Hypergraphs
• Abstract: Abstract For a digraph D, the niche hypergraph $$N\mathcal{H}(D)$$ of D is the hypergraph having the same set of vertices as D and the set of hyperedges $$E(N\mathcal{H}(D)) = \{ e \subseteq V(D): e \geqslant 2$$ and there exists a vertex v such that $$e = \mathop N\nolimits_D^ - (v)$$ or $$\left. {e = {\rm{ }}N_D^ + (v)} \right\}$$ . A digraph is said to be acyclic if it has no directed cycle as a subdigraph. For a given hypergraph $$\mathcal{H}$$ , the niche number $$\hat n(\mathcal{H})$$ is the smallest integer such that $$\mathcal{H}$$ together with $$\hat n(\mathcal{H})$$ isolated vertices is the niche hypergraph of an acyclic digraph. C.Garske, M. Sonntag and H.M.Teichert (2016) conjectured that for a linear hypercycle $$\mathcal{C}_m,\;m\geqslant2$$ , if $$\min \left\{ {\left e \right :e \in E({\mathcal{C}_m})} \right\} \geqslant 3$$ , then $$\hat n(\mathcal{C}_m)=0$$ . In this paper, we prove that this conjecture is true.
PubDate: 2019-03-01
• Note on Strongly Nil Clean Elements in Rings
• Abstract: Abstract Let R be an associative unital ring and let a ∈ R be a strongly nil clean element. We introduce a new idea for examining the properties of these elements. This approach allows us to generalize some results on nil clean and strongly nil clean rings. Also, using this technique many previous proofs can be significantly shortened. Some shorter proofs concerning nil clean elements in rings in general, and in matrix rings in particular, are presented, together with some generalizations of these results.
PubDate: 2019-03-01
• A Remark on Weak McShane Integral
• Abstract: Abstract We characterize the weak McShane integrability of a vector-valued function on a finite Radon measure space by means of only finite McShane partitions. We also obtain a similar characterization for the Fremlin generalized McShane integral.
PubDate: 2019-03-01
• Universal Central Extension of Direct Limits of Hom-Lie Algebras
• Abstract: Abstract We prove that the universal central extension of a direct limit of perfect Hom- Lie algebras $$(\mathcal{L_i,\;\alpha_{\mathcal{L}_i}})$$ is (isomorphic to) the direct limit of universal central extensions of $$(\mathcal{L_i,\;\alpha_{\mathcal{L}_i}})$$ . As an application we provide the universal central extensions of some multi-plicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras {(slk $$(\mathcal{A})$$ , αk)}k∈I and describe the universal central extension of its direct limit.
PubDate: 2019-03-01
• n -Strongly Gorenstein Graded Modules
• Abstract: Abstract Let R be a graded ring and n ⩾ 1 an integer. We introduce and study n-strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that n-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be m-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever n > m. Many properties of the n-strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate the relations between the graded and the ungraded n-strongly Gorenstein injective (or flat) modules. In addition, the connections between the n-strongly Gorenstein gr-projective, gr-injective and gr-flat modules are considered.
PubDate: 2019-03-01
• Higher Order Riesz Transforms for the Dunkl Ornstein-Uhlenbeck Operator
• Abstract: Abstract The aim of this paper is to extend the study of Riesz transforms associated to Dunkl Ornstein-Uhlenbeck operator considered by A. Nowak, L. Roncal and K. Stempak to higher order.
PubDate: 2019-03-01
• The Dyadic Fractional Diffusion Kernel as a Central Limit
• Abstract: Abstract We obtain the fundamental solution kernel of dyadic diffusions in ℝ+ as a central limit of dyadic mollification of iterations of stable Markov kernels. The main tool is provided by the substitution of classical Fourier analysis by Haar wavelet analysis.
PubDate: 2019-03-01
• On σ -Permutably Embedded Subgroups of Finite Groups
• Abstract: Abstract Let σ = {σi: i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi: σi ∩ π(G)≠Ø}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σi-subgroup of G and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). G is said to be σ-full if G possesses a complete Hall σ-set. A subgroup H of G is σ-permutable in G if G possesses a complete Hall σ-set H such that HAx= AxH for all A ∈ H and all x ∈ G. A subgroup H of G is σ-permutably embedded in G if H is σ-full and for every σi ∈ σ(H), every Hall σi-subgroup of H is also a Hall σi-subgroup of some σ-permutable subgroup of G. By using the σ-permutably embedded subgroups, we establish some new criteria for a group G to be soluble and supersoluble, and also give the conditions under which a normal subgroup of G is hypercyclically embedded. Some known results are generalized.
PubDate: 2019-03-01
• On Kneser Solutions of the n -th Order Nonlinear Differential Inclusions
• Abstract: Abstract The paper deals with the existence of a Kneser solution of the n-th order nonlinear differential inclusion $${x^{(n)}}(t) \in - {A_1}(t,x,(t),...,{x^{(n - 1)}}(t)){x^{(n - 1)}}(t) - ... - {A_n}(t,x(t),...,{x^{(n - 1)}}(t))x(t)\;\text{for}\;\text{a.a.}\;t\; \in [a,\infty ),$$ where a ∈ (0,∞), and Ai: [a,∞) × ℝn → ℝ, i = 1,..., n, are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.
PubDate: 2019-03-01
• Boundedness of Generalized Fractional Integral Operators on Orlicz Spaces
Near L 1 Over Metric Measure Spaces
• Abstract: Abstract We are concerned with the boundedness of generalized fractional integral operators Iϱ,τ from Orlicz spaces LΦ(X) near L1(X) to Orlicz spaces LΨ(X) over metric measure spaces equipped with lower Ahlfors Q-regular measures, where Φ is a function of the form Φ(r) = rl(r) and l is of log-type. We give a generalization of paper by Mizuta et al. (2010), in the Euclidean setting. We deal with both generalized Riesz potentials and generalized logarithmic potentials.
PubDate: 2019-03-01
• Finite Distortion Functions and Douglas-Dirichlet Functionals
• Abstract: Abstract In this paper, we estimate the Douglas-Dirichlet functionals of harmonic mappings, namely Euclidean harmonic mapping and flat harmonic mapping, by using the extremal dilatation of finite distortion functions with given boundary value on the unit circle. In addition, $$\bar \partial$$ -Dirichlet functionals of harmonic mappings are also investigated.
PubDate: 2019-03-01
• Cominimaxness of Local Cohomology Modules
• Abstract: Abstract Let R be a commutative Noetherian ring, I an ideal of R. Let t ∈ ℕ0 be an integer and M an R-module such that Ext R i (R/I,M) is minimax for all i ⩽ t+1. We prove that if H I i (M) is FD⩽1 (or weakly Laskerian) for all i < t, then the R-modules H I i (M) are I-cominimax for all i < t and Ext R i (R/I,H I t (M) is minimax for i = 0, 1. Let N be a finitely generated R-module. We prove that Ext R j (N,H I i (M)) and Tor j R (N,H I i (M)) are I-cominimax for all i and j whenever M is minimax and H I i (M) is FD⩽1 (or weakly Laskerian) for all i.
PubDate: 2019-03-01
• The Size of the Lerch Zeta-Function at Places Symmetric with Respect to
the Line ℜ( s ) = 1/2
• Abstract: Abstract Let ζ(s) be the Riemann zeta-function. If t ⩾ 6.8 and σ > 1/2, then it is known that the inequality ζ(1 − s) > ζ(s) is valid except at the zeros of ζ(s). Here we investigate the Lerch zeta-function L(λ, α, s) which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters λ = α it is still possible to obtain a certain version of the inequality L(λ, λ, $$1 - \bar s$$ ) > L(λ, λ, s) .
PubDate: 2019-03-01
• Finite p -Groups with Exactly Two Nonlinear Non-Faithful Irreducible
Characters
• Abstract: Abstract Let G be a finite group with exactly two nonlinear non-faithful irreducible characters. We discuss the properties of G and classify finite p-groups with exactly two nonlinear non-faithful irreducible characters.
PubDate: 2019-03-01
• Torsion Groups of a Family of Elliptic Curves Over Number Fields
• Abstract: Abstract We compute the torsion group explicitly over quadratic fields and number fields of degree coprime to 6 for a family of elliptic curves of the form E: y2 = x3 + c, where c is an integer.
PubDate: 2019-03-01
• Littlewood-Paley Characterization of Hölder-Zygmund Spaces on
Stratified Lie Groups
• Abstract: Abstract We give a characterization of the Hölder-Zygmund spaces Cσ(G) (0 < σ < ∞) on a stratified Lie group G in terms of Littlewood-Paley type decompositions, in analogy to the well-known characterization of the Euclidean case. Such decompositions are defined via the spectral measure of a sub-Laplacian on G, in place of the Fourier transform in the classical setting. Our approach mainly relies on almost orthogonality estimates and can be used to study other function spaces such as Besov and Triebel-Lizorkin spaces on stratified Lie groups.
PubDate: 2019-03-01
• Nil-Clean and Unit-Regular Elements in Certain Subrings of
$$\mathbb{M}_{2}(\mathbb{Z})$$
• Abstract: Abstract An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are not clean in a ring. The rings under consideration in our examples are particular subrings of $$\mathbb{M}_{2}(\mathbb{Z})$$ .
PubDate: 2019-03-01
JournalTOCs
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
Email: [email protected]
Tel: +00 44 (0)131 4513762
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March 17
Jonathan Osinski, University of Hamburg
Model-Theoretic Characterizations of Weak Vopěnka's Principle
It has been known since the 1980s that Vopěnka's Principle (VP) is equivalent to certain statements about logics, e.g. to the schema 'Every logic has a compactness cardinal.' On the other hand, it was only recently shown by Trevor Wilson that a related statement statement called Weak Vopěnka's Principle (WVP) is strictly weaker than VP. In fact, Joan Bagaria and Wilson showed that WVP is equivalent to the existence of $\Pi_n$-strong cardinals for all natural numbers $n$. We generalize logical characterizations of strong cardinals to achieve a characterization of $\Pi_n$-strong cardinals and therefore of WVP in terms of properties of strong logics. This is partly joint work with Will Boney and partly with Trevor Wilson.
Video
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# Computing order statistics $1,2,4,8,\ldots,n$
On an array of $$n = 2^k$$ numbers, where $$k$$ is a non-negative integer, the $$k = \log n$$ order statistics $$1, 2, 4, 8,\ldots, 2^k$$ can all be determined in a total of $$Θ(n)$$ time in the worst case.
I think that this statement is wrong because the loop will take $$k$$ iterations. So, the time should be $$Θ(\log(n))$$. But, I'm not sure my guess is correct. Could you tell me whether or not my guess is correct?
You can find all of these order statistics in $$O(n)$$. First, find the maximum (order statistic $$2^k$$) in time $$O(n)$$. Then, find the median (order statistic $$2^{k-1}$$) in time $$O(n)$$, and remove all larger elements. Find order statistic $$2^{k-2}$$ in time $$O(n/2)$$, and remove all larger elements. And so on. The total running time is $$O(n + n/2 + n/4 + \cdots) = O(n).$$
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# Prove any ring homomorphism between $M_2(\mathbb{R})$ and $\mathbb{R}$ is trivial
Prove any ring homomorphism between $$M_2(\mathbb{R})$$ ($$2\times 2$$ matrices) and $$\mathbb{R}$$ is trivial.
I am not looking for answers, I just want to know how to approach these types of problems (how to prove that any homomorphism must be trivial? What is the typical way of showing these results?) I am guessing we have to show that there is a property in one that is not in the other, but is there a rigorous way to do this?
• Typically when approaching a question that regards any map, I try and consider an arbitrary one, and then try to manipulate a contradiction / or force a conclusion. Like here I might try and assume I have a nontrivial map and try to find a contradiction. Looking at the rings, notice that somehow multiplication on $M_2(\mathbb{R})$ must be respected under the hom. on $\mathbb{R}$ which seems very restrictive, since one is a ring, and another a field. – TrostAft Dec 2 '18 at 7:12
• Wait, is this true? What about the determinant? Oh no nevermind that's only a group homomorphism. – TrostAft Dec 2 '18 at 7:22
The kernel of a ring homomorphism is a two-sided ideal. What are the two-sided ideals of the matrix ring? (Definition: a left ideal is closed under left multiplication by ring elements ($$rx\in I$$ if $$x\in I$$ and $$r\in R$$), a right ideal is closed under right multiplication by ring elements ($$xr\in I$$ if $$x\in I$$ and $$r\in R$$), and a two-sided ideal is closed under both)
In your particular question, it can be shown that $$M_{2}(\Bbb{R})$$ has only two ideals, namely itself and the zero ideal. Thus either the homomorphism will be one-one or it will be trivial.
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## arXiv Analytics
### arXiv:1712.02332 [hep-ex]AbstractReferencesReviewsResources
#### Search for squarks and gluinos in final states with jets and missing transverse momentum using 36 fb$^{-1}$ of $\sqrt{s}$=13 TeV $pp$ collision data with the ATLAS detector
Published 2017-12-06Version 1
A search for the supersymmetric partners of quarks and gluons (squarks and gluinos) in final states containing hadronic jets and missing transverse momentum, but no electrons or muons, is presented. The data used in this search were recorded in 2015 and 2016 by the ATLAS experiment in $\sqrt{s}$=13 TeV proton-proton collisions at the Large Hadron Collider, corresponding to an integrated luminosity of 36.1 fb$^{-1}$. The results are interpreted in the context of various models where squarks and gluinos are pair-produced and the neutralino is the lightest supersymmetric particle. An exclusion limit at the 95\% confidence level on the mass of the gluino is set at 2.03 TeV for a simplified model incorporating only a gluino and the lightest neutralino, assuming the lightest neutralino is massless. For a simplified model involving the strong production of mass-degenerate first- and second-generation squarks, squark masses below 1.55 TeV are excluded if the lightest neutralino is massless. These limits substantially extend the region of supersymmetric parameter space previously excluded by searches with the ATLAS detector.
Comments: Comments: 46 pages plus author list (63 pages total), 17 figures, 6 tables, submitted to PRD, All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/SUSY-2016-07/
Categories: hep-ex
Related articles: Most relevant | Search more
arXiv:1208.4688 [hep-ex] (Published 2012-08-23)
Further search for supersymmetry at sqrt(s) = 7 TeV in final states with jets, missing transverse momentum and isolated leptons with the ATLAS detector
arXiv:1403.5294 [hep-ex] (Published 2014-03-20)
Search for direct production of charginos, neutralinos and sleptons in final states with two leptons and missing transverse momentum in pp collisions at sqrt(s) = 8 TeV with the ATLAS detector
arXiv:1404.2500 [hep-ex] (Published 2014-04-09)
Search for supersymmetry at sqrt(s)=8 TeV in final states with jets and two same-sign leptons or three leptons with the ATLAS detector
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# OptimWrapper¶
In previous tutorials of runner and model, we have more or less mentioned the concept of OptimWrapper, but we have not introduced why we need it and what are the advantages of OptimWrapper compared to Pytorch’s native optimizer. In this tutorial, we will help you understand the advantages and demonstrate how to use the wrapper.
As its name suggests, OptimWrapper is a high-level abstraction of PyTorch’s native optimizer, which provides a unified set of interfaces while adding more functionality. OptimWrapper supports different training strategies, including mixed precision training, gradient accumulation, and gradient clipping. We can choose the appropriate training strategy according to our needs. OptimWrapper also defines a standard process for parameter updating based on which users can switch between different training strategies for the same set of code.
## OptimWrapper vs Optimizer¶
Now we use both the native optimizer of PyTorch and the OptimWrapper in MMEngine to perform single-precision training, mixed-precision training, and gradient accumulation to show the difference in implementations.
### Model training¶
1.1 Single-precision training with SGD in PyTorch
import torch
from torch.optim import SGD
import torch.nn as nn
import torch.nn.functional as F
inputs = [torch.zeros(10, 1, 1)] * 10
targets = [torch.ones(10, 1, 1)] * 10
model = nn.Linear(1, 1)
optimizer = SGD(model.parameters(), lr=0.01)
for input, target in zip(inputs, targets):
output = model(input)
loss = F.l1_loss(output, target)
loss.backward()
optimizer.step()
1.2 Single-precision training with OptimWrapper in MMEngine
from mmengine.optim import OptimWrapper
optim_wrapper = OptimWrapper(optimizer=optimizer)
for input, target in zip(inputs, targets):
output = model(input)
loss = F.l1_loss(output, target)
optim_wrapper.update_params(loss)
The OptimWrapper.update_params achieves the standard process for gradient computation, parameter updating, and gradient zeroing, which can be used to update the model parameters directly.
2.1 Mixed-precision training with SGD in PyTorch
from torch.cuda.amp import autocast
model = model.cuda()
inputs = [torch.zeros(10, 1, 1, 1)] * 10
targets = [torch.ones(10, 1, 1, 1)] * 10
for input, target in zip(inputs, targets):
with autocast():
output = model(input.cuda())
loss = F.l1_loss(output, target.cuda())
loss.backward()
optimizer.step()
2.2 Mixed-precision training with OptimWrapper in MMEngine
from mmengine.optim import AmpOptimWrapper
optim_wrapper = AmpOptimWrapper(optimizer=optimizer)
for input, target in zip(inputs, targets):
with optim_wrapper.optim_context(model):
output = model(input.cuda())
loss = F.l1_loss(output, target.cuda())
optim_wrapper.update_params(loss)
To enable mixed precision training, users need to use AmpOptimWrapper.optim_context which is similar to the autocast for enabling the context for mixed precision training. In addition, AmpOptimWrapper.optim_context can accelerate the gradient accumulation during the distributed training, which will be introduced in the next example.
3.1 Mixed-precision training and gradient accumulation with SGD in PyTorch
for idx, (input, target) in enumerate(zip(inputs, targets)):
with autocast():
output = model(input.cuda())
loss = F.l1_loss(output, target.cuda())
loss.backward()
if idx % 2 == 0:
optimizer.step()
3.2 Mixed-precision training and gradient accumulation with OptimWrapper in MMEngine
optim_wrapper = AmpOptimWrapper(optimizer=optimizer, accumulative_counts=2)
for input, target in zip(inputs, targets):
with optim_wrapper.optim_context(model):
output = model(input.cuda())
loss = F.l1_loss(output, target.cuda())
optim_wrapper.update_params(loss)
We only need to configure the accumulative_counts parameter and call the update_params interface to achieve the gradient accumulation function. Besides, in the distributed training scenario, if we configure the gradient accumulation with optim_context context enabled, we can avoid unnecessary gradient synchronization during the gradient accumulation step.
The OptimWrapper also provides a more fine-grained interface for users to customize with their own parameter update logics.
• backward: Accept a loss dictionary, and compute the gradient of parameters.
• step: Same as optimizer.step, and update the parameters.
• zero_grad: Same as optimizer.zero_grad, and zero the gradient of parameters
We can use the above interface to implement the same logic of parameters updating as the Pytorch optimizer.
for idx, (input, target) in enumerate(zip(inputs, targets)):
with optim_wrapper.optim_context(model):
output = model(input.cuda())
loss = F.l1_loss(output, target.cuda())
optim_wrapper.backward(loss)
if idx % 2 == 0:
optim_wrapper.step()
We can also configure a gradient clipping strategy for the OptimWrapper.
# based on torch.nn.utils.clip_grad_norm_ method
optim_wrapper = AmpOptimWrapper(
optim_wrapper = AmpOptimWrapper(
### Get learning rate/momentum¶
The OptimWrapper provides the get_lr and get_momentum for the convenience of getting the learning rate and momentum of the first parameter group in the optimizer.
import torch.nn as nn
from torch.optim import SGD
from mmengine.optim import OptimWrapper
model = nn.Linear(1, 1)
optimizer = SGD(model.parameters(), lr=0.01)
optim_wrapper = OptimWrapper(optimizer)
print(optimizer.param_groups[0]['lr']) # 0.01
print(optimizer.param_groups[0]['momentum']) # 0
print(optim_wrapper.get_lr()) # {'lr': [0.01]}
print(optim_wrapper.get_momentum()) # {'momentum': [0]}
0.01
0
{'lr': [0.01]}
{'momentum': [0]}
Similar to the optimizer, the OptimWrapper provides the state_dict and load_state_dict interfaces for exporting and loading the optimizer states. For the AmpOptimWrapper, it can export mixed-precision training parameters as well.
import torch.nn as nn
from torch.optim import SGD
from mmengine.optim import OptimWrapper, AmpOptimWrapper
model = nn.Linear(1, 1)
optimizer = SGD(model.parameters(), lr=0.01)
optim_wrapper = OptimWrapper(optimizer=optimizer)
amp_optim_wrapper = AmpOptimWrapper(optimizer=optimizer)
# export state dicts
optim_state_dict = optim_wrapper.state_dict()
amp_optim_state_dict = amp_optim_wrapper.state_dict()
print(optim_state_dict)
print(amp_optim_state_dict)
optim_wrapper_new = OptimWrapper(optimizer=optimizer)
amp_optim_wrapper_new = AmpOptimWrapper(optimizer=optimizer)
{'state': {}, 'param_groups': [{'lr': 0.01, 'momentum': 0, 'dampening': 0, 'weight_decay': 0, 'nesterov': False, 'maximize': False, 'foreach': None, 'params': [0, 1]}]}
{'state': {}, 'param_groups': [{'lr': 0.01, 'momentum': 0, 'dampening': 0, 'weight_decay': 0, 'nesterov': False, 'maximize': False, 'foreach': None, 'params': [0, 1]}], 'loss_scaler': {'scale': 65536.0, 'growth_factor': 2.0, 'backoff_factor': 0.5, 'growth_interval': 2000, '_growth_tracker': 0}}
### Use multiple optimizers¶
Considering that algorithms like GANs usually need to use multiple optimizers to train the generator and the discriminator, MMEngine provides a container class called OptimWrapperDict to manage them. OptimWrapperDict stores the sub-OptimWrapper in the form of dict, and can be accessed and traversed just like a dict.
Unlike regular OptimWrapper, OptimWrapperDict does not provide methods such as update_prarms, optim_context, backward, step, etc. Therefore, it cannot be used directly to train models. We suggest implementing the logic of parameter updating by accessing the sub-OptimWarpper in OptimWrapperDict directly.
Users may wonder why not just use dict to manage multiple optimizers since OptimWrapperDict does not have training capabilities. Actually, the core function of OptimWrapperDict is to support exporting or loading the state dictionary of all sub-OptimWrapper and to support getting learning rates and momentums as well. Without OptimWrapperDict, MMEngine needs to do a lot of if-else in OptimWrapper to get the states of the OptimWrappers.
from torch.optim import SGD
import torch.nn as nn
from mmengine.optim import OptimWrapper, OptimWrapperDict
gen = nn.Linear(1, 1)
disc = nn.Linear(1, 1)
optimizer_gen = SGD(gen.parameters(), lr=0.01)
optimizer_disc = SGD(disc.parameters(), lr=0.01)
optim_wapper_gen = OptimWrapper(optimizer=optimizer_gen)
optim_wapper_disc = OptimWrapper(optimizer=optimizer_disc)
optim_dict = OptimWrapperDict(gen=optim_wapper_gen, disc=optim_wapper_disc)
print(optim_dict.get_lr()) # {'gen.lr': [0.01], 'disc.lr': [0.01]}
print(optim_dict.get_momentum()) # {'gen.momentum': [0], 'disc.momentum': [0]}
{'gen.lr': [0.01], 'disc.lr': [0.01]}
{'gen.momentum': [0], 'disc.momentum': [0]}
As shown in the above example, OptimWrapperDict exports learning rates and momentums for all OptimWrappers easily, and OptimWrapperDict can export and load all the state dicts in a similar way.
### Configure the OptimWapper in Runner¶
We first need to configure the optimizer for the OptimWrapper. MMEngine automatically adds all optimizers in PyTorch to the OPTIMIZERS registry, and users can specify the optimizers they need in the form of a dict. All supported optimizers in PyTorch are listed here.
Now we take setting up a SGD OptimWrapper as an example.
optimizer = dict(type='SGD', lr=0.01, momentum=0.9, weight_decay=0.0001)
optim_wrapper = dict(type='OptimWrapper', optimizer=optimizer)
Here we have set up an OptimWrapper with a SGD optimizer with the learning rate and momentum parameters as specified. Since OptimWrapper is designed for standard single precision training, we can also omit the type field in the configuration:
optimizer = dict(type='SGD', lr=0.01, momentum=0.9, weight_decay=0.0001)
optim_wrapper = dict(optimizer=optimizer)
To enable mixed-precision training and gradient accumulation, we change type to AmpOptimWrapper and specify the accumulative_counts parameter.
optimizer = dict(type='SGD', lr=0.01, momentum=0.9, weight_decay=0.0001)
optim_wrapper = dict(type='AmpOptimWrapper', optimizer=optimizer, accumulative_counts=2)
Note
If you are new to reading the MMEngine tutorial and are not familiar with concepts such as configs and registries, it is recommended to skip the following advanced tutorials for now and read other documents first. Of course, if you already have a good understanding of this prerequisite knowledge, we highly recommend reading the advanced part which covers:
1. How to customize the learning rate, decay coefficient, and other parameters of the model parameters in the configuration of OptimWrapper.
2. how to customize the construction policy of the optimizer.
Apart from the pre-requisite knowledge of the configs and the registries, it is recommended to have a thorough understanding of the native construction of PyTorch optimizer before starting the advanced tutorials.
PyTorch’s optimizer allows different hyperparameters to be set for each parameter in the model, such as using different learning rates for the backbone and head for a classification model.
from torch.optim import SGD
import torch.nn as nn
model = nn.ModuleDict(dict(backbone=nn.Linear(1, 1), head=nn.Linear(1, 1)))
optimizer = SGD([{'params': model.backbone.parameters()},
lr=0.01,
momentum=0.9)
In the above example, we set a learning rate of 0.01 for the backbone, while another learning rate of 1e-3 for the head. Users can pass a list of dictionaries containing the different parts of the model’s parameters and their corresponding hyperparameters to the optimizer, allowing for fine-grained adjustment of the model optimization.
In MMEngine, the optimizer wrapper constructor allows users to set hyperparameters in different parts of the model directly by setting the paramwise_cfg in the configuration file rather than by modifying the code of building the optimizer.
### Set different hyperparamters for different types of parameters¶
The default optimizer wrapper constructor in MMEngine supports setting different hyperparameters for different types of parameters in the model. For example, we can set norm_decay_mult=0 for paramwise_cfg to set the weight decay factor to 0 for the weight and bias of the normalization layer to implement the trick of not decaying the weight of the normalization layer as mentioned in the Bag of Tricks.
Here, we set the weight decay coefficient in all normalization layers (head.bn) in ToyModel to 0 as follows.
from mmengine.optim import build_optim_wrapper
from collections import OrderedDict
class ToyModel(nn.Module):
def __init__(self):
super().__init__()
self.backbone = nn.ModuleDict(
dict(layer0=nn.Linear(1, 1), layer1=nn.Linear(1, 1)))
OrderedDict(
linear=nn.Linear(1, 1),
bn=nn.BatchNorm1d(1)))
optim_wrapper = dict(
optimizer=dict(type='SGD', lr=0.01, weight_decay=0.0001),
paramwise_cfg=dict(norm_decay_mult=0))
optimizer = build_optim_wrapper(ToyModel(), optim_wrapper)
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.bias:lr=0.01
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.bias:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer1.bias:lr=0.01
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer1.bias:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.linear.bias:lr=0.01
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.linear.bias:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.bn.weight:weight_decay=0.0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.bn.bias:weight_decay=0.0
In addition to configuring the weight decay, paramwise_cfg of MMEngine’s default optimizer wrapper constructor supports the following hyperparameters as well.
lr_mult: Learning rate for all parameters.
decay_mult: Decay coefficient for all parameters.
bias_lr_mult: Learning rate coefficient of the bias (excluding bias of normalization layer and offset of the deformable convolution).
bias_decay_mult: Weight decay coefficient of the bias (excluding bias of normalization layer and offset of the deformable convolution).
norm_decay_mult: Weight decay coefficient for weights and bias of the normalization layer.
flat_decay_mult: Weight decay coefficient of the one-dimension parameters.
dwconv_decay_mult: Decay coefficient of the depth-wise convolution.
bypass_duplicate: Whether to skip duplicate parameters, default to False.
dcn_offset_lr_mult: Learning rate of the deformable convolution.
### Set different hyperparamters for different model modules¶
In addition, as shown in the PyTorch code above, in MMEngine we can also set different hyperparameters for any module in the model by setting custom_keys in paramwise_cfg.
If we want to set the learning rate and the decay coefficient to 0 for backbone.layer0, and set the learning rate to 0.001 for the rest of the modules in the backbone. At the same time, we want to keep all the learning rate to 0.001 for the head module. We can do it in this way:
optim_wrapper = dict(
optimizer=dict(type='SGD', lr=0.01, weight_decay=0.0001),
paramwise_cfg=dict(
custom_keys={
'backbone.layer0': dict(lr_mult=0, decay_mult=0),
'backbone': dict(lr_mult=1),
}))
optimizer = build_optim_wrapper(ToyModel(), optim_wrapper)
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.weight:lr=0.0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.weight:weight_decay=0.0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.weight:lr_mult=0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.weight:decay_mult=0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.bias:lr=0.0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.bias:weight_decay=0.0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.bias:lr_mult=0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer0.bias:decay_mult=0
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer1.weight:lr=0.01
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer1.weight:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer1.weight:lr_mult=1
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer1.bias:lr=0.01
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer1.bias:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- backbone.layer1.bias:lr_mult=1
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.linear.weight:lr=0.001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.linear.weight:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.linear.weight:lr_mult=0.1
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.linear.bias:lr=0.001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.linear.bias:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.linear.bias:lr_mult=0.1
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.bn.weight:lr=0.001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.bn.weight:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.bn.weight:lr_mult=0.1
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.bn.bias:lr=0.001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.bn.bias:weight_decay=0.0001
08/23 22:02:43 - mmengine - INFO - paramwise_options -- head.bn.bias:lr_mult=0.1
The state dictionary of the above model can be printed as the following:
for name, val in ToyModel().named_parameters():
print(name)
backbone.layer0.weight
backbone.layer0.bias
backbone.layer1.weight
backbone.layer1.bias
Each field in custom_keys is defined as follows.
1. 'backbone': dict(lr_mult=1): Set the learning rate of the parameter whose name is prefixed with backbone to 1.
2. 'backbone.layer0': dict(lr_mult=0, decay_mult=0): Set the learning rate of the parameter with the prefix backbone.layer0 to 0 and the decay coefficient to 0. This configuration has a higher priority than the first one.
3. 'head': dict(lr_mult=0.1): Set the learning rate of the parameter whose name is prefixed with head to 0.1.
### Customize optimizer construction policies¶
Like other modules in MMEngine, the optimizer wrapper constructor is also managed by the registry. We can customize the hyperparameter policies by implementing custom optimizer wrapper constructors.
For example, we can implement an optimizer wrapper constructor called LayerDecayOptimWrapperConstructor that automatically set decreasing learning rates for layers of different depths of the model.
from mmengine.optim import DefaultOptimWrapperConstructor
from mmengine.registry import OPTIM_WRAPPER_CONSTRUCTORS
from mmengine.logging import print_log
@OPTIM_WRAPPER_CONSTRUCTORS.register_module(force=True)
class LayerDecayOptimWrapperConstructor(DefaultOptimWrapperConstructor):
def __init__(self, optim_wrapper_cfg, paramwise_cfg=None):
super().__init__(optim_wrapper_cfg, paramwise_cfg=None)
self.decay_factor = paramwise_cfg.get('decay_factor', 0.5)
super().__init__(optim_wrapper_cfg, paramwise_cfg)
def add_params(self, params, module, prefix='' ,lr=None):
if lr is None:
lr = self.base_lr
for name, param in module.named_parameters(recurse=False):
param_group = dict()
param_group['params'] = [param]
param_group['lr'] = lr
params.append(param_group)
full_name = f'{prefix}.{name}' if prefix else name
print_log(f'{full_name} : lr={lr}', logger='current')
for name, module in module.named_children():
chiled_prefix = f'{prefix}.{name}' if prefix else name
params, module, chiled_prefix, lr=lr * self.decay_factor)
class ToyModel(nn.Module):
def __init__(self) -> None:
super().__init__()
self.layer = nn.ModuleDict(dict(linear=nn.Linear(1, 1)))
self.linear = nn.Linear(1, 1)
model = ToyModel()
optim_wrapper = dict(
optimizer=dict(type='SGD', lr=0.01, weight_decay=0.0001),
paramwise_cfg=dict(decay_factor=0.5),
constructor='LayerDecayOptimWrapperConstructor')
optimizer = build_optim_wrapper(model, optim_wrapper)
08/23 22:20:26 - mmengine - INFO - layer.linear.weight : lr=0.0025
08/23 22:20:26 - mmengine - INFO - layer.linear.bias : lr=0.0025
08/23 22:20:26 - mmengine - INFO - linear.weight : lr=0.005
08/23 22:20:26 - mmengine - INFO - linear.bias : lr=0.005
When add_params is called for the first time, the params argument is an empty list and the module is the ToyModel instance. Please refer to the Optimizer Wrapper Constructor Documentation for detailed explanations on overloading.
Similarly, if we want to construct multiple optimizers, we also need to implement a custom constructor.
@OPTIM_WRAPPER_CONSTRUCTORS.register_module()
class MultipleOptimiWrapperConstructor:
...
|
1
# What is the membrane sac that encloses each lung?
## Question
###### What is the membrane sac that encloses each lung?
What is the membrane sac that encloses each lung?
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# Uniqueness (a.s.) of regular conditional distributions
Let $$(\Omega, \mathcal{F}, \mathbf{P})$$ be a probability space, $$(\mathcal{X}, \mathcal{B})$$ a measurable space, and $$X : \Omega \to \mathcal{X}$$ a random element of $$\mathcal{X}$$. Also, let $$\mathcal{G}$$ be a sub-$$\sigma$$-algebra of $$\mathcal{F}$$.
Question. How unique are regular conditional distributions of $$X$$ given $$\mathcal{G}$$?
A regular conditional distribution of $$X$$ given $$\mathcal{G}$$ is a function $$P : \Omega \times \mathcal{B} \to [0, 1]$$ such that the following properties hold.
1. For all $$\omega \in \Omega$$, the map $$B \mapsto P(\omega, B)$$ from $$\mathcal{B}$$ into $$[0, 1]$$ is a probability measure on $$(\mathcal{X}, \mathcal{B})$$.
2. For all $$B \in \mathcal{B}$$, the map $$\omega \mapsto P(\omega, B)$$ from $$\Omega$$ into $$[0, 1]$$ is $$(\mathcal{G}, \mathcal{B}_{[0, 1]})$$-measurable (where $$\mathcal{B}_{[0, 1]}$$ denotes the Borel $$\sigma$$-algebra of $$[0, 1]$$).
3. For all $$B \in \mathcal{B}$$ and all $$G \in \mathcal{G}$$, we have $$\mathbf{P}(\{X \in B\} \cap G) = \int_G P(\cdot, B) \, d\mathbf{P}.$$
(Items 2. and 3. just say that, for each $$B \in \mathcal{B}$$, the random variable $$P(\cdot, B)$$ is a version of the conditional probability $$\mathbf{P}(X \in B\mid \mathcal{G})$$.)
Suppose $$P$$ and $$Q$$ are two regular conditional distributions of $$X$$ given $$\mathcal{G}$$.
On the one hand, it is not necessarily true that $$P(\omega, B) = Q(\omega, B)$$ for all $$\omega \in \Omega$$ and $$B \in \mathcal{B}$$. For example, for any $$\mathbf{P}$$-null set $$N \in \mathcal{F}$$ and any probability measure $$\mu$$ on $$(\mathcal{X}, \mathcal{B})$$, we can define $$P^\prime : \Omega \times \mathcal{B} \to [0, 1]$$ by $$P^\prime(\omega, B) = \begin{cases} P(\omega, B), & \text{if \omega \notin N,} \\ \mu(B), & \text{if \omega \in N.} \end{cases}$$ Then $$P^\prime$$ is another regular conditional distribution of $$X$$ given $$\mathcal{G}$$, but it might hold that $$P(\omega, B) \neq P^\prime(\omega, B)$$ for some $$\omega \in \Omega$$ and $$B \in \mathcal{B}$$.
On the other hand, suppose $$B \in \mathcal{B}$$ is fixed. Then we have $$\int_G P(\cdot, B) \, d\mathbf{P} = \mathbf{P}(\{X \in B\} \cap G) = \int_G Q(\cdot, B) \, d\mathbf{P}$$ for every $$G \in \mathcal{G}$$. Since $$P(\cdot, B)$$ and $$Q(\cdot, B)$$ are $$\mathcal{G}$$-measurable, this implies that there exists a $$\mathcal{P}$$-null set $$N \in \mathcal{F}$$ such that $$P(\omega, B) = Q(\omega, B)$$ for all $$\omega \in \Omega \setminus N$$. However, this null set depends on $$B$$, so we can't a priori conclude that there exists a $$\mathbf{P}$$-null set $$N^\prime \in \mathcal{F}$$ such that $$P(\omega, B) = Q(\omega, B)$$ for all $$\omega \in \Omega \setminus N^\prime$$ and all $$B \in \mathcal{B}$$.
More Precise Question. Suppose $$P$$ and $$Q$$ are two regular conditional distributions of $$X$$ given $$\mathcal{G}$$. Does there always exist a $$\mathbf{P}$$-null set $$N \in \mathcal{F}$$ such that $$P(\omega, B) = Q(\omega, B)$$ for all $$\omega \in \Omega \setminus N$$ and all $$B \in \mathcal{B}$$?
I think I remember reading that this is true somewhere, but I can't find a proof. I'm fine with assuming that any measurable spaces in question are standard Borel, if needed.
This is true if $$\mathcal{B}$$ is countably generated. Specifically, $$P(\omega,A)=Q(\omega,A) \quad\text{a.s.} \tag{1}\label{1}$$ for all $$A\in \mathcal{A}$$ (a countable algebra that generates $$\mathcal{B}$$). Therefore, there exists a $$\mathbf{P}$$-null set $$N$$ s.t. $$\eqref{1}$$ holds for all $$A\in\mathcal{A}$$ and all $$\omega\in \Omega\setminus N$$. Now extrapolate this result to $$\mathcal{B}$$.
• Perfect, thank you! The monotone class theorem does the trick to extend to $\mathcal{B}$ – Artem Mavrin Jan 29 at 16:52
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# Ragged list pattern matching, Part 2
This is a sequel to Ragged list pattern matching. In this challenge, the wildcard may match a sequence of items of any length instead of just a single item.
Given a pattern and a ragged list of integers, your task is to decide whether the pattern matches the ragged list.
The pattern is also represented by a ragged list. But in addition to positive integers, it may contain a wildcard value.
Here is the rule for matching:
• A positive integer matches the same positive integer.
• The wildcard value matches a sequence of items (integer or list) of any length, including the empty sequence.
• A ragged list matches a ragged list if each item in the pattern matches the corresponding item in the list.
For example, if we write the wildcard as 0, then the pattern [0, [4, [5], 0]] matches the ragged list [[1, 2], [3], [4, [5]]]: here the first 0 matches the sequence [1, 2], [3], and the second 0 matches the empty sequence.
You may choose any fixed value as the wildcard, as long as it is consistent.
This is , so the shortest code in bytes wins.
This is . You may use your language's convention for truthy/falsy, or use two distinct, fixed values to represent true or false.
## Testcases
Here I use 0 to represent the wildcard. The input here are given in the order pattern, ragged list.
### Truthy
[], []
[0], []
[0], [1, 2]
[0], [[[]]]
[0, 0], [1, 2, 3]
[1, 0], [1, [[2, 3]]]
[1, 0, 2], [1, 2, 2, 2]
[1, 0, [2, 3]], [1, [2, 3]]
[1, [2, 0], 4], [1, [2, 3], 4]
[0, [4, [5], 0]], [[1, 2], [3], [4, [5]]]
### Falsy
[1], []
[[]], []
[[0]], [3]
[[4]], [4]
[1, 0], [2, 1]
[[0]], [[1], [2]]
[1, 0, 2], [1, 2, 3]
[1, 0, 2, 0], [1, [3, 2, 3]]
[1, [0, 2], 4], [1, [2, 3], 4]
[[0], [4, [5], 0]], [[1, 2], [3], [4, [5]]]
• suggest a test case who has two 0’s and expected output is false. When replace 0 by .* in its JSON representation, it matches. For example [1, 0, 2, 0] [1, [3, 2, 3]]
– tsh
Mar 11 at 1:25
# Retina, 71 bytes
>
,>
<,
<
~(L$^.+ \n$&$_, (\d+,|<((?<_><)|(?<-_>>)|\d|,)+(?(_)^)>,)$*
Try it online! Takes two newline-separated <>-wrapped lists but link is to test suite that deletes spaces, splits on semicolons and converts other types of brackets for convenience. Uses _ to represent the wild card. Explanation: Based on my Retina answer to Ragged list pattern matching.
>
,>
<,
<
Ensure that non-empty lists end in a comma.
~(
Evaluate the result of the transformations below on the comma-safe input. Since they result in a single line, this makes Retina attempt to match the comma-safe input against the result.
L$^.+ \n$&$ Take only the first line of the input and wrap it in \n and $ so that it will match against the second line of the input.
_,
(\d+,|<((?<_><)|(?<-_>>)|\d|,)+(?(_)^)>,)$* Replace each _, with a match against any number ($* represents a literal * which would otherwise be the string repetition operator) of comma-safe non-negative integers or lists, using a .NET balancing group to ensure that the lists are properly balanced. A named capturing group ?<_> is used here because .NET allows capturing groups to be reused in this way and it's golfier than calculating the capturing group number.
Assumption: ragged lists are defined in the following way:
data RL a = L [RL a] | N a | W
deriving Eq
Solution:
L(W:p)#L x|L p#L x=1>0
L(W:p)#L(_:x)=L(W:p)#L x
L(q:p)#L(y:x)=q#y&&L p#L x
a#b=a==W||a==b
Attempt This Online!
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# Guidance for JEE paper 2 ! Please
My board exams ended only today and my jee paper 2 is day after tomorrow. I don't know how to prepare pls somebody help....
Note by Ashley Shamidha
3 years, 6 months ago
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How much have you finished? If you understand your NCERT textbooks well, 200 is definitely possible.
- 3 years, 6 months ago
I am thorough with the concepts in maths, I just want to know how i have to prepare in this short time. And I don't know where can I prepare for this aptitude test and drawing test. (NB:I am very good at logical reasoning)
- 3 years, 6 months ago
What resources do you have? I'm sure you can find books/online sites for your needs. Focus on your strengths at this point of time.
I can't really answer much more, because I don't know too much about the B. Arch paper. Sorry.
- 3 years, 6 months ago
Any way thank u for ur response sir.
- 3 years, 6 months ago
@Ameya Daigavane Sir are u there?
- 3 years, 6 months ago
I am not the right guy. @Sandeep Bhardwaj is.
All the best for JEE 🖖
Staff - 3 years, 6 months ago
Thank u
- 3 years, 6 months ago
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# Two independent geometric random variables - proof of sum
If $$X$$ and $$Y$$ are independent geometric random variables, $$X \sim G(p)$$ and $$Y \sim G(q)$$ then if $$Z = X+Y$$ I need to show that:
$$P[Z=z] = \frac{pq}{p-q}[(1-q)^{z-1}-(1-p)^{z-1}]$$
My attempt:
\begin{align} P[Z=z] &= \sum_{k=1}^z P[X=k]P[Y=z-k] && \text{(not sure of my summation limits here)} \\ &= \sum_{k=1}^z p(1-p)^{k-1} \cdot q(1-q)^{z-k-1} \\ &= \frac{pq}{(1-p)}\cdot \frac{(1-q)^z}{(1-q)}\cdot\sum_{k=1}^z (\frac{1-p}{1-q})^k \\ &= \frac{pq}{(1-p)}\cdot \frac{(1-q)^z}{(1-q)} \cdot \frac{1-(\frac{1-p}{1-q})^k}{1-\frac{1-p}{1-q}} \\ &= \frac{pq}{(1-p)}\cdot \frac{(1-q)^z}{(p-q)} \left[1 -(\frac{1-p}{1-q})^k\right] \\ &= \frac{pq}{p-q}\left[\frac{(1-q)^z}{1-p} - \frac{(1-p)^z}{1-p}\right] \end{align}
The result is pretty close to the answer. The only discrepancy is the first expression in the parentheses, $$\frac{(1-q)^z}{1-p}$$ which should be $$\frac{(1-q)^z}{1-q}$$.
Could someone please have a look at my working and show me where I have gone wrong? Or perhaps a neater calculation? Thanks!
PS
I did check similar questions answered here, but couldn't find anything relevant to my problem.
Is the sum of two independent geometric random variables with the same success probability a geometric random variable?
How to compute the sum of random variables of geometric distribution
I think you made a mistake in the calculation. First geometric random variable is at least 1, hence $z\ge 2$.
\begin{align*} P[Z=z] &= \sum_{k=1}^{z-1} P[X=k]P[Y=z-k]\\ &=\sum_{k=1}^{z-1} p(1-p)^{k-1} \cdot q(1-q)^{z-k-1}\\ &=\frac{pq}{(1-p)} (1-q)^{z-1}\sum_{k=1}^{z-1} (\frac{1-p}{1-q})^k \\ &=\frac{pq}{(1-p)} (1-q)^{z-1} \cdot \frac{1-p}{1-q}\frac{1-(\frac{1-p}{1-q})^{z-1}}{1-\frac{1-p}{1-q}}\\ &=\frac{pq}{p-q} (1-q)^{z-1} \cdot [1 -(\frac{1-p}{1-q})^{z-1}]\\ &=\frac{pq}{p-q}[(1-q)^{z-1} - (1-p)^{z-1}], \end{align*} which is the desired result.
• Thanks for the helpful answer. You lost me in the fourth line: $=\frac{pq}{(1-p)} (1-q)^{z-1} \cdot \frac{1-p}{1-q}\frac{1-(\frac{1-p}{1-q})^{z-1}}{1-\frac{1-p}{1-q}}$ How did you get the $\frac{1-p}{1-q}$ part before the summation? – abruzzi26 Feb 8 '17 at 15:22
• @abruzzi26 This follows for the formula for the summation of geometric series with finite terms – John Feb 8 '17 at 15:26
• Ah, yes. I got confused there for a sec. Thanks again – abruzzi26 Feb 8 '17 at 16:55
I like using the probability generating function for this: you know the pgf for the sum is the product of the pgfs for the two random variables. So
$$p_X(t)p_Y(t)={pqt^2\over (1-t(1-p))(1-t(1-q))}=$$
Applying partial fractions to just the reciprocal terms and setting consecutively $t=(1-p)^{-1}$ then $t=(1-q)^{-1}$ gives
$$A(1-t(1-q))+B(1-t(1-p))=1\iff A={1-p\over q-p}, B = -{1-q\over q-p}$$
leaving us with
$$p_{X+Y}(t) = {pqt^2\over q-p}\left({(1-p)\over 1-t(1-p)}-{1-q\over 1-t(1-q)}\right)$$ $$=\sum_{k=0}^\infty {pq\over q-p}((1-p)^{k+1}-(1-q)^{k+1})t^{k+2}$$
and reindexing appropriately gives
$$p_{X+Y}(t) = \sum_{i=2}^\infty {pq\over p-q}\Big((1-q)^{k-1}-(1-p)^{k-1}\Big)t^k$$
as desired.
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Please note that conversion to PDF is currently not available. More details here
# Quark Matter 2012
12-18 August 2012
US/Eastern timezone
## Production of Charged Pions, Kaons, and Protons in 2.76 TeV Pb-Pb Collisions at high p_t measured with the ALICE Experiment.
16 Aug 2012, 14:20
20m
Diplomat ()
### Diplomat
Oral Presentation
### Speaker
Antonio Ortiz Velasquez (Lund University (SE))
### Description
The main tracking detector in the central barrel ($|\eta|<1$) of the ALICE experiment is the Time Projection Chamber. In addition to charged particle tracking it provides particle identification (PID) through the measurement of the specific energy loss, $dE/dx$. At low momentum ($p < 1$ GeV/c), pions, kaons, and protons can be cleanly separated. Thanks to the relativistic rise of the $dE/dx$, the relative yield of pions, kaons, and protons can also be extracted statistically at higher momenta, $p > 3$ GeV/c. In this talk, spectra for charged pions, kaons, and protons from pp collisions at $\sqrt{s} = 2.76$ TeV and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV for $3 < p_{t} < 20$ GeV/c will be presented, and the nuclear modification factor $R_{AA}$ will be derived. The evolution of $R_{AA}$ with collision centrality and transverse momentum will be discussed, and compared to unidentified charged particles, $K_{s}^{0}$ , $\Lambda$, and theoretical predictions.
### Primary author
Collaboration ALICE (CERN, Geneva, Switzerland)
### Co-author
Antonio Ortiz Velasquez (Lund University (SE))
Slides
Paper files:
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# Electrical Encyclopedia: The color classification of home improvement wires, each color has a custom
2018-09-27
First of all, household electricity, generally three-phase electricity, is also a line of three colors, each color has a commonly known connection method, although sometimes reversed can also be used, but later problems have affected the maintenance.
According to the “Code for Construction of Residential Decoration Engineering” GB 50327-2001 16.1.4: When wiring, the color of the phase line and the neutral line should be different; the color of the same residential phase line (L) should be uniform, the zero line (N) Blue should be used, and the protective wire (PE) must be yellow-green two-color.
The phase line is also the fire line we are talking about. Usually, red, yellow and green are used as the fire line, blue is the zero line, and the ground line is the protection line mentioned above is usually the two-color line.
Because the wire color is more, but in general, if you encounter a red line, it must be a fire line. In a single-phase lighting circuit, generally yellow indicates a live line, blue is a neutral line, and yellow-green is a ground line.
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# Global well-posedness of a conservative relaxed cross diffusion system.
1 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
CGPhiMC - Centre de génétique et de physiologie moléculaire et cellulaire, Inria Grenoble - Rhône-Alpes, ICJ - Institut Camille Jordan [Villeurbanne]
Abstract : We prove global existence in time of solutions to relaxed conservative cross diffusion systems governed by nonlinear operators of the form $u_i\to \partial_tu_i-\Delta(a_i(\tilde{u})u_i)$ where the $u_i, i=1,...,I$ represent $I$ density-functions, $\tilde{u}$ is a spatially regularized form of $(u_1,...,u_I)$ and the nonlinearities $a_i$ are merely assumed to be continuous and bounded from below. Existence of global weak solutions is obtained in any space dimension. Solutions are proved to be regular and unique when the $a_i$ are locally Lipschitz continuous.
Keywords :
Document type :
Journal articles
https://hal.inria.fr/hal-00683006
Contributor : Thomas Lepoutre Connect in order to contact the contributor
Submitted on : Tuesday, March 27, 2012 - 2:59:51 PM
Last modification on : Friday, May 20, 2022 - 9:04:46 AM
Long-term archiving on: : Thursday, June 28, 2012 - 2:30:38 AM
### Files
LPR_cross.pdf
Files produced by the author(s)
### Citation
Thomas Lepoutre, Michel Pierre, Guillaume Rolland. Global well-posedness of a conservative relaxed cross diffusion system.. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2012, 44 (3), pp.1674-1693. ⟨10.1137/110848839⟩. ⟨hal-00683006⟩
Record views
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Calculate correlation coefficients between variables in a data.frame, matrix or table using 3 different functions for 3 different possible pairs of vairables:
• numeric - numeric
• numeric - categorical
• categorical - categorical
calculate_cors(
x,
num_num_f = NULL,
num_cat_f = NULL,
cat_cat_f = NULL,
max_cor = NULL
)
# S3 method for explainer
calculate_cors(
x,
num_num_f = NULL,
num_cat_f = NULL,
cat_cat_f = NULL,
max_cor = NULL
)
# S3 method for matrix
calculate_cors(
x,
num_num_f = NULL,
num_cat_f = NULL,
cat_cat_f = NULL,
max_cor = NULL
)
# S3 method for table
calculate_cors(
x,
num_num_f = NULL,
num_cat_f = NULL,
cat_cat_f = NULL,
max_cor = NULL
)
# S3 method for default
calculate_cors(
x,
num_num_f = NULL,
num_cat_f = NULL,
cat_cat_f = NULL,
max_cor = NULL
)
## Arguments
x object used to select method. See more below. A function used to determine correlation coefficient between a pair of numeric variables A function used to determine correlation coefficient between a pair of numeric and categorical variable A function used to determine correlation coefficient between a pair of categorical variables A number used to indicate absolute correlation (like 1 in cor). Must be supplied if any of *_f arguments is supplied.
## Value
A symmetrical matrix A of size n x n, where n - amount of columns in x (or dimensions for table). The value at A(i,j) is the correlation coefficient between ith and jth variable. On the diagonal, values from max_cor are set.
## X argument
When x is a data.frame, all columns of numeric type are treated as numeric variables and all columns of factor type are treated as categorical variables. Columns of other types are ignored.
When x is a matrix, it is converted to data.frame using as.data.frame.matrix.
When x is a explainer, the tests are performed on its data element.
When x is a table, it is treated as contingency table. Its dimensions must be named, but none of them may be named Frequency.
## Default functions
By default, the function calculates p_value of statistical tests ( cor.test for 2 numeric, chisq.test for factor and kruskal.test for mixed).
Then, the correlation coefficients are calculated as -log10(p_value). Any results above 100 are treated as absolute correlation and cut to 100.
The results are then divided by 100 to fit inside [0,1].
If only numeric data was supplied, the function used is cor.test.
## Custom functions
Creating consistent measures for correlation coefficients, which are comparable for different kinds of variables, is a non-trivial task. Therefore, if user wishes to use custom function for calculating correlation coefficients, he must provide all necessary functions. Using a custom function for one case and a default for the other is consciously not supported. Naturally, user may supply copies of default functions at his own responsibility.
Function calculate_cors chooses, which parameters of *_f are required based on data supported. For example, for a matrix with numeric data only num_num_f is required. On the other hand, for a table only cat_cat_f is required.
All *_f parameters must be functions, which accept 2 parameters (numeric or factor vectors respectively) and return a single number from [0,max_num]. The num_cat_f must accept numeric argument as first and factor argument as second.
cor.test, chisq.test, kruskal.test
## Examples
data(mtcars)
# Make sure, that categorical variables are factors
mtcars$vs <- factor(mtcars$vs, labels = c('V-shaped', 'straight'))
mtcars$am <- factor(mtcars$am, labels = c('automatic', 'manual'))
calculate_cors(mtcars)
#> mpg cyl disp hp drat wt
#> mpg 1.00000000 0.09213768 0.09027782 0.067476725 0.047504984 0.098880796
#> cyl 0.09213768 1.00000000 0.11744043 0.084586878 0.050838285 0.069145071
#> disp 0.09027782 0.11744043 1.00000000 0.071461389 0.052771998 0.109128153
#> hp 0.06747673 0.08458688 0.07146139 1.000000000 0.020004879 0.043823888
#> drat 0.04750498 0.05083829 0.05277200 0.020004879 1.000000000 0.053201852
#> wt 0.09888080 0.06914507 0.10912815 0.043823888 0.053201852 1.000000000
#> qsec 0.01767462 0.03436456 0.01881271 0.052391063 0.002079008 0.004699691
#> vs 0.04078383 0.05231293 0.04252054 0.045443188 0.018952975 0.029658944
#> am 0.02756135 0.02437490 0.03291156 0.013599308 0.038793688 0.043978089
#> gear 0.02267530 0.02379521 0.03016107 0.003071426 0.050777880 0.033385091
#> carb 0.02964792 0.02711675 0.01597431 0.061063597 0.002067802 0.018345001
#> qsec vs am gear carb
#> mpg 0.017674616 0.040783832 0.027561351 0.022675300 0.029647920
#> cyl 0.034364557 0.052312933 0.024374902 0.023795207 0.027116748
#> disp 0.018812712 0.042520543 0.032911561 0.030161068 0.015974311
#> hp 0.052391063 0.045443188 0.013599308 0.003071426 0.061063597
#> drat 0.002079008 0.018952975 0.038793688 0.050777880 0.002067802
#> wt 0.004699691 0.029658944 0.043978089 0.033385091 0.018345001
#> qsec 1.000000000 0.049801564 0.005890707 0.006152266 0.043432361
#> vs 0.049801564 1.000000000 0.002553069 0.025715289 0.019595498
#> am 0.005890707 0.002553069 1.000000000 0.044059498 0.005198029
#> gear 0.006152266 0.025715289 0.044059498 1.000000000 0.008893124
#> carb 0.043432361 0.019595498 0.005198029 0.008893124 1.000000000
# For a table:
data(HairEyeColor)
calculate_cors(HairEyeColor)
#> Hair Eye Sex
#> Hair 1.00000000 0.246335235 0.013360089
#> Eye 0.24633523 1.000000000 0.001704363
#> Sex 0.01336009 0.001704363 1.000000000
# Custom functions:
num_mtcars <- mtcars[,-which(colnames(mtcars) %in% c('vs', 'am'))]
my_f <- function(x,y) cor.test(x, y, method = 'spearman', exact=FALSE)\$estimate
calculate_cors(num_mtcars, num_num_f = my_f, max_cor = 1)
#> mpg cyl disp hp drat wt
#> mpg 1.0000000 -0.9108013 -0.9088824 -0.8946646 0.65145546 -0.8864220
#> cyl -0.9108013 1.0000000 0.9276516 0.9017909 -0.67888119 0.8577282
#> disp -0.9088824 0.9276516 1.0000000 0.8510426 -0.68359210 0.8977064
#> hp -0.8946646 0.9017909 0.8510426 1.0000000 -0.52012499 0.7746767
#> drat 0.6514555 -0.6788812 -0.6835921 -0.5201250 1.00000000 -0.7503904
#> wt -0.8864220 0.8577282 0.8977064 0.7746767 -0.75039041 1.0000000
#> qsec 0.4669358 -0.5723509 -0.4597818 -0.6666060 0.09186863 -0.2254012
#> gear 0.5427816 -0.5643105 -0.5944703 -0.3314016 0.74481617 -0.6761284
#> carb -0.6574976 0.5800680 0.5397781 0.7333794 -0.12522294 0.4998120
#> qsec gear carb
#> mpg 0.46693575 0.5427816 -0.6574976
#> cyl -0.57235095 -0.5643105 0.5800680
#> disp -0.45978176 -0.5944703 0.5397781
#> hp -0.66660602 -0.3314016 0.7333794
#> drat 0.09186863 0.7448162 -0.1252229
#> wt -0.22540120 -0.6761284 0.4998120
#> qsec 1.00000000 -0.1481997 -0.6587181
#> gear -0.14819967 1.0000000 0.1148870
#> carb -0.65871814 0.1148870 1.0000000
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# Template:Sequence of the Day for May 30
Intended for: May 30, 2012
## Timetable
• First draft entered by Alonso del Arte on August 10, 2011
• Draft reviewed by Daniel Forgues on May 29, 2012
• Draft to be approved by April 30, 2012
The line below marks the end of the <noinclude> ... </noinclude> section.
A053003: Simple continued fraction for Gauß's constant $\scriptstyle \frac{2}{\pi} \int_{0}^{1} \frac{1}{\sqrt{1 - x^4}} dx \,$
$1 + \frac{1}{5 + \cfrac{1}{21 + \cfrac{1}{3 + \cfrac{1}{\ddots\qquad{}}}}} \,$
This is the reciprocal of the arithmetic-geometric mean of 1 and $\scriptstyle \sqrt{2} \,$. It was on May 30, 1799 that Carl Friedrich Gauß discovered the integral for this number shown above. (For its decimal expansion, see A014549.)
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# Robustness of classifiers to uniform $\ell_p$ and Gaussian noise
1 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We study the robustness of classifiers to various kinds of random noise models. In particular, we consider noise drawn uniformly from the $\ell_p$ ball for $p \in [1, \infty]$ and Gaussian noise with an arbitrary covariance matrix. We characterize this robustness to random noise in terms of the distance to the decision boundary of the classifier. This analysis applies to linear classifiers as well as classifiers with locally approximately flat decision boundaries, a condition which is satisfied by state-of-the-art deep neural networks. The predicted robustness is verified experimentally.
Document type :
Conference papers
Domain :
https://hal.archives-ouvertes.fr/hal-01715012
Contributor : Jean-Yves Franceschi <>
Submitted on : Thursday, February 22, 2018 - 11:22:18 AM
Last modification on : Tuesday, April 16, 2019 - 5:12:07 PM
Long-term archiving on : Wednesday, May 23, 2018 - 12:34:49 PM
### Files
paper_robustness_lp_new.pdf
Files produced by the author(s)
### Identifiers
• HAL Id : hal-01715012, version 1
• ARXIV : 1802.07971
### Citation
Jean-Yves Franceschi, Alhussein Fawzi, Omar Fawzi. Robustness of classifiers to uniform $\ell_p$ and Gaussian noise. 21st International Conference on Artificial Intelligence and Statistics (AISTATS) 2018, Apr 2018, Playa Blanca, Spain. ⟨hal-01715012⟩
Record views
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1. limits of sequences
find the limit of each sequence:
A) Sn = (1 + 1/2n)^(2n)
B) Sn = ( 1 + 1/n)^(2n)
C) Sn - (1 + 1/n) ^ (n-1)
D) Sn = (n/(n+1))^n
E) Sn = (1+ 1/2n)^n
F) Sn = ((n+2)/(n+1))^(n+3)
I found the answers to be this: A)e B) e^2 C) e D) 1/e E) sqrt(e) F) e...but I found the answers by using a different way then what the problems asks...it says to use the binomial theorem and the limit is referred to as e and is used as the base for natural logarithms.........Anyone know what they are asking and how to show the work needed to get the answers I got on A-F?
2. Originally Posted by learn18
find the limit of each sequence:
A) Sn = (1 + 1/2n)^(2n)
B) Sn = ( 1 + 1/n)^(2n)
C) Sn - (1 + 1/n) ^ (n-1)
D) Sn = (n/(n+1))^n
E) Sn = (1+ 1/2n)^n
F) Sn = ((n+2)/(n+1))^(n+3)
I found the answers to be this: A)e B) e^2 C) e D) 1/e E) sqrt(e) F) e...but I found the answers by using a different way then what the problems asks...it says to use the binomial theorem and the limit is referred to as e and is used as the base for natural logarithms.........Anyone know what they are asking and how to show the work needed to get the answers I got on A-F?
This is a little bit informal but the explanation is clearer this way.
A) By the binomial theorem:
Sn = (1 + 1/2n)^(2n) = 1 + (2n) (1/(2n) + (2n)(2n-1)(1/2n)^2/2! + ..
...................................... (2n)!/[(2n-r)! r!] (1/(2n)^r + ..
where r<=n.
Now as n becomes large the term (2n)!/[(2n-r)! is the product of r factors
of the form (2n-a) a = 0, 1, 2, .. r-1, which when divided by (2n)^r
approaches 1. So the r-th term of Sn approaches 1/r! as n goes to infinity.
Hence:
Lim(n->infty) Sn = 1 + 1/1! + 1/2! + ... + 1/r! + ... = e.
RonL
3. Here are B and C they are baby stuff.
4. ..
5. Here is a last one and another attempt at #A.
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# Floating point rounding
Can an IEEE-754 floating point number < 1 (i.e. generated with a random number generator which generates a number >= 0.0 and < 1.0) ever be multiplied by some integer (in floating point form) to get a number equal to or larger than that integer due to rounding?
i.e.
double r = random() ; // generates a floating point number in [0, 1)
double n = some_int ;
if (n * r >= n) {
print 'Rounding Happened' ;
}
This might be equivalent to saying that does there exist an N and R such that if R is the largest number less than 1 which can be represented in IEEE-754 then N * R >= N (where * and >= are appropriate IEEE-754 operators)
This comes from this question based on this documentation and the postgresql random function
• Can you say anything about the range of N, i.e. is it small enough to be represented exactly in IEEE-754 double-precision? – Pedro Aug 14 '12 at 20:14
• @Pedro In this particular case, yes, it would be a small integer - i.e. 10. I assume you are saying that if N is a very large integer with a very large number of significant digits it might not be able to be represented exactly? – Cade Roux Aug 14 '12 at 20:40
• Exactly, if $fl(N) > N$, then $fl(R\times fl(N))$ may be larger than $RN$. – Pedro Aug 14 '12 at 21:46
Assuming round-to-nearest and that $N > 0$, then $N * R < N$ always. (Be careful not to convert an integer that's too large.)
Let $c 2^{-q} = N$, where $c \in [1, 2)$ is the significand and $q$ is the integer exponent. Let $1 - 2^{-s} = R$ and derive the bound
$$N R = c 2^{-q}(1 - 2^{-s}) \le c 2^{-q} - 2^{-q - s},$$
with equality if and only if $c = 1$. The right-hand side is less than $N$ and, since $2^{-q - s}$ is exactly $0.5$ units in the last place of $N$, either $c = 1$ and $2^{-q} - 2^{-q - s}$ is exactly representable (since $N$ is normal and not the smallest normal), or $c > 1$, and the nearest rounding is down. In both cases, $N * R$ is less than $N$.
Upward rounding can cause a problem, not that it should ever be selected in the presence of unsuspecting users. Here's some C99 that prints "0\n1\n" on my machine.
#include <fenv.h>
#include <math.h>
#include <stdio.h>
int main(void) {
double n = 10;
double r = nextafter(1, 0);
printf("%d\n", n == (n * r));
fesetround(FE_UPWARD);
printf("%d\n", n == (n * r));
}
• I'm sorry, I'm a little slow these days - I'm having trouble getting the part of the inequality $$- c 2^{-q} 2^{-s} \le - 2^{-q s}$$ – Cade Roux Aug 14 '12 at 22:50
• @Cade Apparently I can't do algebra today. I meant $2^{-q - s}$. – Tyrone Aug 14 '12 at 22:57
• Thanks, I wasn't sure if there was another step I was missing. – Cade Roux Aug 14 '12 at 23:18
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The ring of SL_2 invariants in sums of conjugation and tautological modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:22:28Z http://mathoverflow.net/feeds/question/59248 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59248/the-ring-of-sl-2-invariants-in-sums-of-conjugation-and-tautological-modules The ring of SL_2 invariants in sums of conjugation and tautological modules Greg Muller 2011-03-22T23:05:18Z 2011-08-07T02:22:12Z <h3>Rings of Invariants</h3> <p>Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free polynomial ring in $\dim(V)$-many variables. Then the <strong>ring of invariants</strong> of $V$ is the subring $$\mathbb{C}[V]^G:=(f\in\mathbb{C}[V] s.t. \forall g\in G, f(gv)=f(v))$$</p> <h3>The Conjugation Representation</h3> <p>Let $M_2(\mathbb{C})$ denote the space of $2\times 2$ complex matrices. The <strong>conjugation</strong> representation $V_C$ of $G$ on $M_2(\mathbb{C})$ is defined by $g\cdot m:= gmg^{-1}$. Thus, $V_C$ is a 4-dimensional $G$-representation.</p> <p>The ring of invariants $\mathbb{C}[V_C]^G$ is well-known. It is freely generated by the functions $tr$ and $det$; this is a special case of the general fact that the only conjugation-invariant functions on $M_n(\mathbb{C})$ are symmetric functions of eigenvalues.</p> <h3>The Tautological Module</h3> <p>Let $V_T$ denote the $2$-dimensional $G$-representation where $G$ acts in the 'obvious' way, by the inclusion $G=SL_2(\mathbb{C})\subset GL_2(\mathbb{C})$. This is called the <strong>tautological</strong> $G$-representation. Since $SL_2(\mathbb{C})$ acts on $V_T$ with a dense orbit, the ring of invariants $\mathbb{C}[V_C]^G$ is boring; it is just $\mathbb{C}$.</p> <p>Now, let $V_T^n$ denote the $n$-fold direct sum of $V_T$. This is a natural $2n$-dimensional $G$-representation. We can choose a $G$-invariant skew-symmetric bilinear form $\omega$ on $V_T$; this is big words for the usual scalar cross product $v_1\times v_2$. This defines a natural $G$-invariant function on the space of pairs $(v_1,v_2)\in V_T^2$.</p> <p>Then the ring of invariants $\mathbb{C}[V_T^n]^G$ is generated by $\omega(v_i,v_j)$ for $1\leq i j\leq n$. However, these do not (in general) freely-generate the ring of invariants, there are relations between them. The relations are all of the form $$\omega(v_i,v_k)\omega(v_j,v_l)=\omega(v_i,v_j)\omega(v_k,v_l)+\omega(v_l,v_i)\omega(v_j,v_k)$$ for $1\leq ijkl\leq n$.</p> <p>Both of these facts can be deduced by observing that $\mathbb{C}[V_T^n]^G$ can be identified with the homogeneous coordinate ring of the Grassmanian $Gr(2,n)$. Then the generators above are the Plucker coordinates, and the relations are the 3-term Plucker relations (there are no higher Plucker relations here).</p> <h3>The Question</h3> <p>Both of these examples have clever solutions and pretty answers. I am curious about the combination of both cases. </p> <p>Let $m$ and $n$ be positive integers, and consider the direct sum $G$-representation $V^m_C\oplus V_T^n$. <strong>What is the ring of invariants $\mathbb{C}[V_C^m\oplus V_T^n]^G$?</strong></p> <p>I am aware of general procedures for producing these rings, for arbitrary finite-dimensional $G$-reps; see e.h. Sturmfel's <em>Algorithms in Invariant Theory</em>. However, I suspect that there is a clever solution to this particular problem. Not only because it is a combination of two problems with a clever solution, but because I have a guess as to what the answer is, and all the relations seem to be similar to the 3-term Plucker relations. I also suspect the answer is <em>ancient</em> (like much invariant theory), which is why I am trying to find an answer rather than try to prove my guess is correct by brute force.</p> http://mathoverflow.net/questions/59248/the-ring-of-sl-2-invariants-in-sums-of-conjugation-and-tautological-modules/71095#71095 Answer by David Wehlau for The ring of SL_2 invariants in sums of conjugation and tautological modules David Wehlau 2011-07-24T02:09:33Z 2011-07-24T02:09:33Z <p>I don't have the literature with me, but yes the answer to your question was well known to classical invariant theorists. My recollection is that it is one of the examples (sections) in Grace and Young (J.H. Grace and A. Young, The algebra of invariants, Cambridge Univ. Press, Cambridge, 1903). They give an answer without using polarization. All the generating invariants are the ones you describe or else invariants of degree 3 which are quadratic in (one or 2) of the tautological reps and linear in a conjugation rep. Describing the relations is harder but was understood classically. I am not sure of a reference though. </p>
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Solve by induction: $n!>(n/e)^n$
To Prove : $n! > (n/e)^n$
The question seems easy but it ain't; anyone up for it ?
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I don't appreciate this challenging tone. You should be here in order to learn, not in order to make other people feel bad for not being able to solve problems. This is childish behavior which is not appropriate for this forum. – Qiaochu Yuan Aug 20 '11 at 19:46
I think there's a nicer way of putting that. I doubt anyone felt bad reading this question but the asker probably did reading your comment. – Joe Aug 20 '11 at 20:31
Anil, welcome to MSE. Whether it's childish or not, mathematicians have challenged each other to solve problems throughout the ages. In any case, I see nothing wrong with the tone of this question and anyone who's offended by it is being oversensitive in my opinion. If I felt bad every time I came across a problem I could not solve, I would be an extremely miserable person. – LostInMath Aug 20 '11 at 23:05
@Qiaochu What about his tone is challenging? And what about the question would make others feel bad for not being able to solve problems? – Quinn Culver Aug 21 '11 at 12:49
Perhaps I overreacted. I have seen people say things like "this question seems easy but it ain't" with the intent I described but I was being too hasty in ascribing that intent to the OP. My apologies, @Anil. – Qiaochu Yuan Aug 21 '11 at 12:56
Here's a hint. You assume that $n!>\left(\frac{n}{e}\right)^n$. Now you should show it for n+1, i.e., you should show that $(n+1)! > \left(\frac{n+1}{e}\right)^{n+1}$.
You can write
$$(n+1)! > (n+1) \left(\frac{n}{e}\right)^n = (n+1)\left(\frac{n}{n+1}\right)^n \left(\frac{(n+1)^n}{e^n}\right)$$
Can you solve it from here?
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Very nicely expressed as a route to a solution! – Geoff Robinson Aug 20 '11 at 20:05
Hmm not sure (n/n+1)^n < 1 – Anil Shanbhag Aug 21 '11 at 17:55
@Anil - $\left(\frac{n}{n+1}\right)^n = \frac{1}{(1+1/n)^n}$. You should know that $\lim_{x \to \infty} (1+1/x)^x = e$. If instead of using the limit, you prove the inequality $(1+1/n)^n < e$ and use it in the expression, you will get the answer. – svenkatr Aug 22 '11 at 3:44
Just to elaborate on svenkatr's hint - We must show $(1+1/x)^x$ is an increasing function first. Then, knowing it's limit is $e$ gives us the inequality he mentioned. – Ragib Zaman Aug 22 '11 at 9:26
@Ragib: Well, it depends how you define $e$ in the first place. It can be defined as $\lim_{n \to \infty}(1 + \frac{1}{n})^{n},$ after showing that that sequence is increasing, but bounded above, so converges to its least upper bound (then it's clear that this least upper bound, which we call $e$, is greater than $(1+\frac{1}{n})^{n}$ for all integers $n$). – Geoff Robinson Aug 22 '11 at 21:57
By considering the exponential power series we observe that for $x>0$, $$e^x > \frac{x^n}{n!}$$ Now setting $x=n$ we obtain $$e^n > \frac{n^n}{n!}$$ which rearranges to precisely what is desired. I should note that I had first learned this incredibly short and simple proof of this fact from Qiaochu Yuan's posts on this website, and he in turn attributed it to this article written by Terence Tao.
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To be accurate, it works for $x=0$. You just need to "decide" whether or not $0^0$ is of value $0,1$ or something else. If you claim it undefined then you need require that $x\cdot n\neq 0$. – Asaf Karagila Aug 21 '11 at 5:49
I just excluded $x=0$ for a matter of convenience. To include $x=0$ would force us to consider that for $n=0$ strict inequality no longer holds (if we take the usual convention that here $0^0=1$). – Ragib Zaman Aug 21 '11 at 7:26
This seems to be the "right" proof for this statement, though the title of the question (but not the body of the text), asked for a proof by induction. – Geoff Robinson Aug 21 '11 at 9:00
Nice proof -- Didnt think of problem in this direction -- But then again the main idea was to prove by induction – Anil Shanbhag Aug 21 '11 at 18:00
This is the "easy" part of Stirling's formula -- the crude order of magnitude estimate showing how huge $n!$ is, without the $\sqrt{2 \pi n}$ correction that is harder to derive.
Taking logarithms, you are asking for $\log(n!) > n (\log n - 1)$. The latter is the indefinite integral of $\log(n)$. Drawing a picture, this follows from $\log(n)$ being an increasing function. The inequality compares the area under the graph of the function to the area of rectangles underneath the graph. A stronger inequality can be obtained using trapezoids and the convexity of $\log(x)$. I think you can get the $\sqrt{n}$ factor this way but not the exact constant $\sqrt{2 \pi}$.
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Just to verify the last statement, it is indeed true that using trapezoids to estimate the area can give us the estimate $n! \sim C n^{n + 1/2} e^{-n}$ for some constant $C$. Then, the "hard" part is to determine $C$, which is commonly done by considering the Wallace product. – Ragib Zaman Aug 21 '11 at 3:09
In an attempt to prove it by induction, you'll reach a stage where you need to prove $$\left(1+\frac{1}{n}\right)^n < e.$$
Consider the $i^\text{th}$ term in the binomial expansion of $(1+1/n)^n$: $$\frac{n!}{(n-i)! i!} \cdot \frac{1}{n^i} < \frac{1}{i!}.$$ This is easily provable as the $i^\text{th}$ term is $$\frac{n(n-1) \cdots (n-i +1)}{n \cdot n \cdots n} \cdot \frac{1}{i!} < \frac{1}{i!}.$$
So, $$\left(1 + \frac{1}{n}\right)^n < e.$$
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Dataset Open Access
Dataset: The plural interpretability of German linking elements ("Morphology")
Schäfer, Roland; Pankratz, Elizabeth
Dublin Core Export
<?xml version='1.0' encoding='utf-8'?>
<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>Schäfer, Roland</dc:creator>
<dc:creator>Pankratz, Elizabeth</dc:creator>
<dc:date>2018-07-28</dc:date>
<dc:description>This dataset accompanies a paper to be published in "Morphology" (JOMO, Springer). Under the present DOI, all data generated for this research as well as all scripts used are stored. The paper itself is not CC-licensed, refer to Springer's "Morphology" website for details!
Abstract
In this paper, we take a closer theoretical and empirical look at the linking elements in German N1+N2 compounds which are identical to the plural marker of N1 (such as -er with umlaut, as in Häus-er-meer 'sea of houses'). Various perspectives on the actual extent of plural interpretability of these pluralic linking elements are expressed in the literature. We aim to clarify this question by empirically examining to what extent there may be a relationship between plural form and meaning which informs in which sorts of compounds pluralic linking elements appear. Specifically, we investigate whether pluralic linking elements occur especially frequently in compounds where a plural meaning of the first constituent is induced either externally (through plural inflection of the entire compound) or internally (through a relation between the constituents such that N2 forces N1 to be conceptually plural, as in the example above). The results of a corpus study using the DECOW16A corpus and a split-100 experiment show that in the internal but not external plural meaning conditions, a pluralic linking element is preferred over a non-pluralic one, though there is considerable inter-speaker variability, and limitations imposed by other constraints on linking element distribution also play a role. However, we show the overall tendency that German language users do use pluralic linking elements as cues to the plural interpretation of N1+N2 compounds. Our interpretation does not reference a specific morphological framework. Instead, we view our data as strengthening the general approach of probabilistic morphology.</dc:description>
<dc:identifier>https://zenodo.org/record/1323211</dc:identifier>
<dc:identifier>10.5281/zenodo.1323211</dc:identifier>
<dc:identifier>oai:zenodo.org:1323211</dc:identifier>
<dc:relation>doi:10.5281/zenodo.1322790</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:source>Morphoplogy</dc:source>
<dc:subject>compounds</dc:subject>
<dc:subject>German language</dc:subject>
<dc:subject>probabilistic morphology</dc:subject>
<dc:subject>usage data</dc:subject>
<dc:title>Dataset: The plural interpretability of German linking elements ("Morphology")</dc:title>
<dc:type>info:eu-repo/semantics/other</dc:type>
<dc:type>dataset</dc:type>
</oai_dc:dc>
71
8
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## Wednesday, 28 January 2015
### Information, search, and causes – Workshop in Turin
Center for Logic, Language, and Cognition
University of Turin – via Sant’Ottavio 20, Turin (Italy)
6th February 2015
ex Sala Lauree Giurisprudenza
Palazzo Nuovo (ground floor)
INFORMATION, SEARCH, and CAUSES
Rational and cognitive approaches
PROGRAM
9.15 – 10.15
Causal networks in evidential reasoning
10.15 – 11.00
(MPI Berlin)
Late-breaking results on stepwise approaches to sequential search
Coffee break
11.30 – 12.15
Neil BRAMLEY (UCL)
Acting informatively: How people learn
causal structure through sequences of interventions
Lunch break
14.00 – 15.00
Paul PEDERSEN (MPI Berlin)
Dilation, disintegrations, dominance principles, and delayed decisions
15.00 – 15.45
Laura MARTIGNON (Ludwigsburg)
Probabilistic information measures in the classroom
15.45 – 16.30
Flavia FILIMON (Humboldt University Berlin)
Neural substrates of probabilistic perceptual decisions
based on experienced probabilities vs. descriptive statistics
Coffee break
17.00 – 17.45
Björn MEDER (MPI Berlin)
Information search and presentation formats
17.45 – 18.30
Vincenzo CRUPI (Turin)
Shannon and beyond: Generalized entropies and rational information search
The Center for Logic, Language, and Cognition (LLC) of the University of Turin was established in 2014 as a joint initiative of the Departments of Philosophy and Education, Psychology, and Computer Science. The workshop arises from the activities of two ongoing research projects addressing related issues: priority program New Frameworks of Rationality, SPP 1516 (Deutsche Forshungsgemeinshaft, grant CR 409/1-2), and FIRB project Structures and Dynamics of Knowledge and Cognition (Italian Ministry of Scientific Research, Turin unit, D11J12000470001).
## Logic Colloquium 2015European Summer Meeting of the Association for Symbolic Logic
Helsinki, Finland, 3-8 August 2015
http://www.helsinki.fi/lc2015
The annual European Summer Meeting of the Association for Symbolic Logic, the Logic Colloquium 2015 (LC 2015), will be organized in Helsinki, Finland, 3-8 August 2015. Logic Colloquium 2015 is co-located with the 15th Conference of Logic, Methodology and Philosophy of Science,CLMPS 2015, and with the SLS Summer School in Logic
### Plenary lectures
Toshiyasu Arai (Chiba)
Sergei Artemov (New York)
Steve Awodey (Pittsburgh)
Johan van Benthem (Amsterdam and Stanford)
Artem Chernikov (Paris)
Ilias Farah (York)
Danielle Macbeth (Haverford)
Andrei Morozov (Novosibirsk)
Kobi Peterzil (Haifa)
Ralf Schindler (Münster)
Saharon Shelah (TBC) (Jerusalem and Rutgers)
Sebastiaan Terwijn (Nijmegen)
### Tutorials
Erich Grädel (Aachen)
Menachem Magidor (Jerusalem).
### Special sessions
Set Theory, organized by Heike Mildenberger (Freiburg)
Model theory, organized by Dugald Macpherson (Leeds)
Computability Theory, organized by Russell Miller (New York) and Alexandra Soskova (Sofia)
Proof Theory, organized by Benno van den Berg (Amsterdam) and Michael Rathjen (Leeds)
Philosophy of Mathematics and Logic, organized by Patricia Blanchette (Notre Dame) and Penelope Maddy (Irvine)
Logic and Quantum Foundations, organized by Samson Abramsky (Oxford)
## Travel Awards and Contributed Talks
Travel awards for students and young researchers have been made available by the organizers. In some cases full compensation of expenses is possible. The website includes detailed information about the awards, instructions of how to apply, and an electronic form which may be used for the application.
The Logic Colloquium will include contributed talks of 20 minutes' length. Abstracts on contributed talks are published in the Bulletin of Symbolic Logic.
The deadline for travel award applications and abstract submission is Tuesday, May 3, 2015. Please see http://www.helsinki.fi/lc2015/submission.html for information about applying for travel awards and for submitting an abstract.
### Review of Williamson's Tetralogue
By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)
I've been asked to write a review of Williamson's brand new book Tetralogue for the Times Higher Education. Here is what I've come up with so far. Comments are very welcome, as I still have some time before submitting the final version. (For more background on the book, here is a short video where Williamson explains the project.)
============================
Disagreement in debates and discussions is an interesting phenomenon. On the one hand, having to justify your views and opinions vis-à-vis those who disagree with you is perhaps one of the best ways to induce a critical reevaluation of these views. On the other hand, it is far from clear that a clash of views will eventually lead to a consensus where the parties come to hold better views than the ones they held before. This is one of the promises of rational discourse, but one that is all too often not kept. What to do in situations of discursive deadlock?
Timothy Williamson’s Tetralogue is precisely an investigation on the merits and limits of rational debate. Four people holding very different views sit across each other in a train and discuss a wide range of topics, such as the existence of witchcraft, the superiority and falibilism of scientific reasoning, whether anyone can ever be sure to really know anything, what it means for a statement to be true, and many others. As one of the most influential philosophers currently in activity, Williamson is well placed to give the reader an overview of some of the main debates in recent philosophy, as his characters debate their views.
Bob represents those who hold what could be describe as ‘ancestral’ modes of thinking, including superstition, belief in witchcraft and so forth; Sarah is the staunch child of the Enlightenment, firmly convinced of the superiority of scientific knowledge over Bob’s ancestral beliefs; Zac is the relativist who abhors absolute views, and rejects the idea that anything can be true or false simpliciter; Roxana, a latecomer in the conversation, is the most unpleasant of them all (not that any of the other three is particularly pleasant), and represents rationality taken to its limit: she is the one who pursues the logical conclusions of each position to its (sometimes absurd) limits. As these people try to resolve their differences and convince each other of their own worldviews, Williamson explores the limits of rational debate and disagreement.
What is perhaps most noteworthy about this book is the dialogical form adopted. The dialogue as a literary form marked the very birth of Western philosophy with Plato’s dialogues, which in all honesty remain unsurpassed when it comes to complexity, philosophical sophistication, and pure literary beauty (the Gorgias is my favorite). In the circa 2.500 years since, a number of philosophical works have adopted the dialogical form, in some periods more than in others: dialogues were particularly important in the Latin medieval tradition, and the early modern period saw a resurgence of the genre with Leibniz, Hume, and Diderot, among others. (See V. Hösle, The Philosophical Dialogue, 2012.) But for the most part, philosophical literary forms such as the philosophical essay tend to be superficially non-dialogical, while in practice often corresponding to ‘internalized’ dialogues where arguments, counter-arguments, counter-counter-arguments etc. are presented by one and the same voice. Indeed, in recent decades no prominent philosophical work written in dialogical form seems to have appeared, with the very notable exception of Lakatos’ Proofs and Refutations (1976).
Williamson’s adoption of the dialogical form is a clear reference to Platonic dialogues, but it also makes sense given that his main topic here is disagreement and rational debate as such. The book presents itself as an introduction to recent philosophical themes for the non-initiated, while the initiated may enjoy seeing these topics embedded in apparently mundane discussions. In this sense, it is bound to be of interest to a wide range of readers. However, if it is really intended to be a “way into philosophy” for those new to the topic, it might have reached its goal more efficiently if it also contained further details and pointers to additional literature (as Lakatos does in footnotes in Proofs and Refutations). Instead, it is unclear how the interested reader is to proceed in order to delve further into these topics. Moreover, the characters are rather like caricatures of each of the positions, with no ambition to psychological complexity. This might sound like an unreasonable requirement given the stated goals of the book; but the truth is that anyone writing a philosophical dialogue will be confronted with the exceedingly high standards set by the founder of the genre, Plato. Nevertheless, Tetralogue remains a remarkable and courageous attempt to experiment with an eminently philosophical but somewhat ‘outdated’ literary form – the dialogue – to talk about disagreement and dialogue itself.
## Friday, 16 January 2015
### What meaning can and cannot be: Lessons from real life, Part 2
The problem of fictional discourse -- do statements involving fictional objects have truth-values or are they truth-valueless; if the former, how are these truth values determined? -- is one that goes back at least to Frege (his views on the compositionality of language and what the referent of a sentence is entail that sentences which have non-referring parts have no truth-value) and for which no adequate solution has yet been found, given, e.g., the publication of books such as Tim Crane's The Object of Thought in 2013. I'm not going to survey all the issues arising from these problems, but instead present a problem for some accounts of the truth values of fictional statements which is, so far as I am aware, not often considered.
In Part 1 last week, one of my example languages was Adûnaic, one of the languages developed by J.R.R. Tolkien. I picked Adûnaic because of the context in which it appears in its most full form, Lowdham's report to the members of the Notion Club (The Notion Club Papers are themselves full of extremely interesting ideas about the nature of language and how linguistic meaning can be acquired, and will be the focus of a future Lesson), but any of his languages -- Quenya, Sindarian -- or indeed any constructed language of sufficient sophistication -- e.g., Klingon -- will serve my purposes.
One thing that is remarkable about constructed fictional languages such as Klingon, Quenya, etc., is the academic scholarship that they have generated. There is the The Elvish Linguistic Fellowship (a special interest group of the Mythopoeic Society, "devoted to the scholarly study of the invented languages of J.R.R. Tolkien"), which publishes two print journals, Vinyar Tengwar and Parma Eldalamberon, and an online journal, Tengwestië; Helge Kåre Fauskanger's Ardalambion, an extensive collection of links regarding Tolkienian languages; the Klingon Language Institute; and even the journal Tolkien Studies, which will not specifically devoted to his constructed languages publishes articles on them, such as Christopher Gilson's "Essence of Elvish: The Basic Vocabulary of Quenya". One conclusion that can be drawn from this extensive academic activity is that these fictional languages are taken as worthy objects of study, When it comes to languages, one of the basic requirements that it has to meet in order for it to be something worth studying is that it be meaningful. (While people can, and do, create "nonsense" languages, which are not meaningful, these do not have a history of generating the sort of scholarship that fictional languages like Klingon and Sindarin have.)
There are two types of contexts in which these languages can be found: In the original context of construction (e.g., in Tolkien's writings for the Elvish languages; in Star Trek TV episodes and movies for Klingon), and in external contexts (e.g., a wholly new composition of poetry in Quenya, or translation of the Bible into Klingon). Both of these contexts cause problems for straightforward explanations of how these languages are meaningful.
Adûnaic is set apart from other fictional languages by the paucity of examples that we have in it. As Fauskanger notes, "There are no coherent Adûnaic texts. Except single words scattered around in Lowdham's Report, most of the corpus consists of a number of fragmentary sentences given in SD:247, with Lowdham's interlinear translation" [1]. Fauskanger supplements the translation that is found in Sauron Defeated with words whose meaning is known from other contexts, and gives it as follows:
Kadô Zigûrun zabathân unakkha...
"And so / [the] Wizard / humbled / he came..."
"...[the] Eruhíni [Children of Eru] / fell / under [the] Shadow..."
"...Ar-Pharazôn / was warring / against [the] Valar..."
...Bârim an-Adûn yurahtam dâira sâibêth-mâ Êruvô
"...[the] Lords of [the] West / broke / the Earth / with [the] assent / of Eru..."
"...seas /so as to gush/ into [the] chasm..."
"...Númenor / [the] beloved / she fell down..."
...bawîba dulgî...
"...[the] winds [were] black..." (lit. simply "winds / black")
"...ships / seven / of Elendil / eastward..."
Agannâlô burôda nênud...
...zâira nênud...
"...longing [is] / on us..."
"...west / [a] straight / road / once / went / now / all / roads / [are] crooked..."
Êphalak îdôn Yôzâyan
"Far away / now [is] / [the] Land of Gift..."
Êphal êphalak îdôn hi-Akallabêth
"Far / far away / now [is] / She-that-hath-fallen"
Though this is fragmentary, it is clear that what is being expressed is in the context of the legends of Middle Earth, that is, Adûnaic is being used to write what is from our point of view fiction. This fragmentary poem is not intended to be read literally as expressing statements about the actual world. This feature -- the use of the language to say something within the context of a fiction -- is to some extent shared by "external" uses of fictional languages, which by and large are also used for non-literal or non-factive writing or speech: Almost no one writes lesson plans in Quenya, composes a shopping list in Klingon, or sends submits meeting minutes in Sindarin.
The question then, is: If these languages are meaningful, how are they meaningful? What do we mean when we say that these fictional languages are meaningful, or at least intended to have meaning? If we mean that the sentences that are expressed by these languages have truth-conditions (e.g., a truth-conditional account of meaning has been adopted), then we are stymied by the fact that there is no good account of the truth-conditions of fictional language. One might say that there are truth conditions, we simply do not know them, and hence (via an epistemic rather than ontological truth-conditional account of meaning) we do not know the meaning of these sentences, even though they are meaningful. This sceptical position, however, does not seem to take account of our behaviour regarding sentences in these languages: We evaluate translations from, e.g., Quenya into English as correct or incorrect, we write literary criticism about the propositions expressed, and in general, we sure act as if we know what these sentences mean, at least in part, even though by assumption we don't know the relevant truth conditions.
Of course, it always possible to stick to one's philosophical guns, and simply reply that we are mistaken when we think we know what the sentences mean: But that seems to be a pretty tough horn to impale oneself on, because it doesn't provide any account for our behaviour with respect to these sentences. There is clearly some factor that guides our behaviour: There must be something in which referees evaluating prospective articles for Tengwestië ground their reports. If this is not the meaning of the sentences, then whatever it is, it certainly seems to be something which is functionally equivalent to meaning.
Lesson: We treat statements in constructed languages as meaningful. But if these languages are only ever used in fictional contexts, then any truth-conditional account of meaning which denies truth values to fictional discourse will have difficulty accounting for the fact that we do treat these languages in that way (that is, as meaningful).
### Call for Papers: LORI-V (Deadline May 18)
Call for Papers
The Fifth International Conference on Logic, Rationality and Interaction (LORI-V)
October 28-31, 2015
Taipei, Taiwan
The International Conference on Logic, Rationality and Interaction (LORI) conference series aims at bringing together researchers working on a wide variety of logic-related fields that concern the understanding of rationality and interaction (http://golori.org). The series aims at fostering a view of Logic as an interdisciplinary endeavor, and supports the creation of an East-Asian community of interdisciplinary researchers.
We invite submission of contributed papers on any of the broad themes of LORI series; specific topics of interest include, but are not limited to, formal approaches to
* agency, * argumentation and agreement, * belief representation, * cooperation, * belief revision and belief merging, * strategic reasoning, * games, * decision making and planning, * knowledge and action, * epistemology, * dynamics of informational attitudes, * knowledge representation, * interaction, * norms and normative systems, * natural language, * rationality, * philosophy and philosophical logic, * preference and utility, * social choice, * probability and uncertainty, * social interaction, * intentions, plans, and goals
Submitted papers should be at most 12 pages long, with one additional page for references, in PDF/DOC format following the Springer LNCS style: http://www.springer.com/computer/lncs?SGWID=0-164-6-793341-0.
Please submit papers by May 18, 2015 via EasyChair for LORI-V: https://easychair.org/conferences/?conf=lori5
Accepted papers will be collected as a volume in the Folli Series on Logic, Language and Information, and may later be published in a special issue of a prestigious journal.
To encourage graduate students, those whose papers are single-authored and accepted will be exempt from the registration fee, and up to 10 students will also have free accommodations during the conference dates.
For detailed conference information and registration, please visit the website: http://golori.org and click "LORI-V".
Invited Speakers
Prof. Maria Aloni (Department of Philosophy, University of Amsterdam, The Netherlands)
Prof. Joseph Halpern (Computer Science Department, Cornell University, USA)
Prof. Eric Pacuit (Department of Philosophy, University of Maryland, USA)
Prof. Liu Fenrong (Department of Philosophy, Tsinghua University, China)
Prof. Branden Fitelson (Department of Philosophy, Rutgers University, USA)
Prof. Churn-Jung Liau (Institute of Information Science, Academia Sinica, Taiwan)
Organizers: LORI, National Taiwan University (NTU) and National Yang Ming University (YMU), Taipei, Taiwan, LORI
## Friday, 9 January 2015
### Mochizuki's proof of the ABC conjecture: still "in limbo"
By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)
Here's a short piece by the New Scientist on the status of Mochizuki's purported proof of the ABC conjecture. More than 2 years after the 500-page proof has been made public, the mathematical community still hasn't been able to decide whether it's correct or not. (Recall my post on this from May 2013; little change seems to have taken place since then.)
Going back to my dialogical conception of mathematical proofs as involving a proponent who formulates the proof and opponents who must check it, this stalemate can be viewed from at least two perspectives: either Mochizuki is not trying hard enough as a proponent, or the mathematical community is not trying hard enough as opponent.
[Mochizuki] has also criticised the rest of the community for not studying his work in detail, and says most other mathematicians are "simply not qualified" to issue a definitive statement on the proof unless they start from the very basics of his theory.
Some mathematicians say Mochizuki must do more to explain his work, like simplifying his notes or lecturing abroad.
(Of course, it may well be that both are the case!). And so for now, the proof remains in limbo, as well put by the New Scientist piece. Mathematics, oh so human!
### What meaning can and cannot be: Lessons from real life, Part 1
Nearly a decade and a half ago, before logic bewitched me and I fell under her spell, I started off graduate school intending to write a dissertation on something related to philosophy of fiction or fictional discourse (given that that's how specific my dissertation plans were for my first 1-2 years of grad school, I probably should've realized sooner that this was not the topic for me). This year I'm lucky enough to be teaching a 3rd-year undergrad course "Language & Mind" which has reminded me why I was interested in what philosophy can say about fiction, and vice versa, in the first place. For my inaugural contribution to M-Phi, this post will be the first in an (unbounded) series of reflections on what meaning can and cannot be, given the constraints of how language is actually used, both in real and fictional discourse.
It is one thing for a theory of meaning to give an account of simple declarative sentences which are grammatically correct and whose terms refer to existing, uncontroversial objects: "Snow is white" should not be a difficult sentence to analyse if one is to give a theory of meaning of English. It is yet another thing altogether to be able to handle the edge cases, the non-simple, the non-declarative, the non-grammatically correct sentences, the sentences which have non-referring terms, and unfortunately many theories of meaning stumble at these hurdles, providing answers that are hard to swallow. (Note: I am not one who generally thinks that when there is a clash between what a philosophical theory says and what my intuitions say, it is the intuitions that should win. I've been a philosopher long enough to know that my intuitions in some respect are utterly ruined. However, if my philosophical theory entails a conclusion that is at odds with how people think and act about the relevant subject matter, then I do feel entitled to ask my theory to explain why it is there is this discrepancy. In this, I think St. Anselm of Canterbury's approach to the division between logic and grammar was precisely right. To oversimplify it significantly: Grammar is about how people use language, logic is about how people should use language, but more than that, logic should also be able to explain why it is that grammar and logic diverge: Logic should be able to explain why the usus loquendi does not always match the usus proprie.) It is the edge cases that provide the true test for any philosophical theory, and thus it is edge cases that I'll be discussing in this series.
I started off the "Language & Mind" class with two questions, which have become the guiding questions of the class, and three quotes. The questions are:
1. What is meaning?
2. What are the preconditions for language to have meaning?
And the quotes:
Quote 1
Êphal ê phalak îdôn hi-Akallabêth.
Quote 2
A threigylgweith yd oed yn Arberth, prif lys idaw adyuot yn y uryt ac yn y uedwl uynet y hela.
Quote 3
I expect the average reader of this blog to recognize the language of at least one of these three, but I would be very surprised if anyone knew all three.
Numenor the beloved, she fell down //
Far, far away now is She-that-hath-fallen.
Adûnaic is one of the languages created by J.R.R. Tolkien, the most full account of it appearing in "Lowdham's Report on the Adunaic Language", in Sauron Defeated, ed. Christopher Tolkien, p. 413-440. (An overview of the language can be found here). Tolkien's invented languages are well-known for the attention to detail and realistic grammatical and phonological structures that they have, unlike many other fictional languages which are made up in a piecemeal fashion without any attempt to make them mirror non-fictional languages in structure or complexity.
Quote 2 is Welsh, and translated into English reads:
Once upon a time he [Pwyll] was at Arberth, a chief court of his, and he was seized by the thought and the desire to go hunting.
This is the opening line of the story of Pwyll, Prince of Dyfedd, in the Mabinogion, a cycle of prose literature compiled in the 12th-13th C from oral tales.
Quote 3 is a votive inscription in Linear A, adapted from here. Linear A is one of the last remaining undeciphered writing systems of ancient Greece. This quote cannot currently be translated.
I chose these three quotes because each of them places different constraints on what meaningfulness can be, where it can come from, and how we must account for it.
The Adûnaic and Welsh quotes are clearly meaningful, as it is possible to translate them into meaningful sentences in English which can be understood even if the original quotes could not be. The status of the quote in Linear A is less clear: It could be argued (and indeed, students in my class did so!) both that Linear A, given that there is no one alive who can understand or decipher it, is therefore meaningless, and that if it were to be deciphered, then it would regain its previous meaningfulness; or it could be argued (and again, I had students willing to take up this side) that it is meaningful, even in the absence of anyone who can understand that meaning, and thus meaning is something which is intrinsic to a language itself, and not dependent on the people who use the language.
However, while Adûnaic and Welsh are certainly on the one hand opposed from Linear A, on other hand they are opposed from each other. Adûnaic, as a constructed rather than natural language, has a definitive moment of creation or inception, and even if it evolved as it was developed, its development is still governed by the arbitration of a single person. Now that that person is dead, that standard of arbitration is gone: There are questions about Adûnaic that are left essentially unanswerable, questions of both vocabulary, grammar, and pronunciation. Welsh, on the other hand, did not have its birth at the hands of a single person, and as a result, there is a standard that can be appealed to for arbitration, whether this be the sum of its uses in the medieval period (if it is Old or Middle Welsh that is of interest), its use amongst Welsh speakers today, the proclamations of some canonical language academy (Welsh doesn't have one; but French and other languages do). No one single person has the authority to say what is meaningful and correct and what is not, and yet these questions can still be answered, unlike the case of Adûnaic.
The lesson in this post will be short and simple, since the post itself has gotten rather long, and it is this: The varieties of language which a theory of meaning must account for is perhaps broader and more diverse than people who are used to thinking of what it means for an English sentence to be meaningful are aware. In a future post (perhaps the next one), we'll look closer at the case of Adûnaic, and the problems that a truth-conditional theory of semantics would face in accounting for the (apparent) meaningfulness of that language.
## Thursday, 8 January 2015
### In memoriam: Ivor Grattan-Guinness
The great historian of logic and mathematics Ivor Grattan-Guinness passed away about a month ago, aged 73. I only heard it yesterday, when Stephen Read posted a link to the Guardian obituary on Facebook. From the obituary:
He rescued the moribund journal Annals of Science, founded the journal History and Philosophy of Logic, and was on the board of Historia Mathematica from its inception. A member of the council of the Society for Psychical Research, he wrote Psychical Research: A Guide to Its History (1982). In 1971 the British Society for the History of Mathematics was founded: Ivor served as its president (1986-88) and instituted a formal constitution.
Indeed, many of us owe him eternal gratitude for founding the journal History and Philosophy of Logic, which continues to be the main journal for studies combining historical and philosophical perspectives on logic. Ivor's work and scholarship spans over an impressive range of topics and areas, and is bound to continue to influence many generations of scholars to come. It is a great loss.
## Tuesday, 6 January 2015
### Final CfP: Formal Epistemology Workshop 2015 (Deadline January 16!)
May 20-22, 2015 (Wednesday to Friday)
Washington University in St. Louis
Keynote speakers:
Tom Kelly (Princeton), Jeff Horty (University of Maryland, College Park)
The Formal Epistemology Workshop will be held in connection with the 2015 meeting of the St. Louis Annual Conference on Reasons and Rationality (SLACRR), which will take place immediately before, from May 17-19, 2015.
There will be conference sessions all day on May 20 & 21, and in the morning on May 22.
Contributors are invited to send full papers as PDF files (suitable for presenting as a 40 minute talk) to [email protected] by Friday, January 16, 2015. Papers should be accompanied by abstracts of up to 300 words. Identifying information about the author(s) (including obvious self-citations) should be removed from the body of the paper, but the name (and any other relevant information) should be included in the text of the e-mail.
Submissions should be prepared for anonymous review. Initial evaluation will be done anonymously. The final program will be selected with an eye towards maintaining diversity, so graduate students, people outside the tenure track, women, and members of underrepresented minorities are particularly encouraged to submit papers. We also welcome submissions from researchers in related areas, such as economics, computer science, and psychology. Past programs can be viewed here: http://fitelson.org/few/
Submitting the same paper to both FEW and SLACRR is permitted (though the organizers will coordinate the paper selection in order to ensure that the same paper doesn’t get presented at both conferences).
Final selection of the contributed talks will be made by March 31, 2015.
There will be childcare available for conference participants who bring their children. It will be provided on site by a local certified childcare provider.
Organizers: Kenny Easwaran (Texas A&M), Julia Staffel (Washington University in St. Louis), Mike Titelbaum (UW Madison)
|
NZ Level 8 (NZC) Level 3 (NCEA) [In development]
Polar Form
Lesson
A polar coordinate system uses measurements of $r$r, (distance from the origin) and $\theta$θ (angle from the positive horizontal axis called a polar axis). We use these measurements $r$r,$\theta$θ instead of the cartesian measurements of $x,y$x,y. Incidently the cartesian coordinates are also referred to as rectangular coordinates (because the cartesian plane is a rectangular plane)
A polar coordinate system therefore looks different to our rectangular coordinate system.
This is what it looks like.
We can convert our rectangular form of complex numbers $a+bi$a+bi into polar coordinates. To do so we need the distance from the origin, which is the modulus and the angle from the positive horizontal axis (this is called the argument).
Recognising that the horizontal distance $x$x, can be calculated using the angle theta, and similarly for the height $y$y gives us the values of
$x=r\cos\theta$x=rcosθ
$y=r\sin\theta$y=rsinθ
So we can rewrite the complex number $z=x+iy$z=x+iy as $z=r\cos\theta+ir\sin\theta$z=rcosθ+irsinθ
We can take $r$r out as a factor so we get $z=r\left(\cos\theta+i\sin\theta\right)$z=r(cosθ+isinθ)
There is special notation for this, a kind of shorthand where we write $\operatorname{cis}$cis to mean $\cos\theta+i\sin\theta$cosθ+isinθ
Having already studied how to find the modulus ($r$r), we just need to identify the value of the argument (the angle) $\theta$θ, and then we can convert rectangular complex numbers $x+iy$x+iy into polar complex numbers $\operatorname{cis}\theta$cisθ
Let's look at this diagram again,
$\tan\theta=\frac{opposite}{adjacent}=\frac{y}{x}$tanθ=oppositeadjacent=yx
The angle is denoted using a number of different notations: $\theta,arg\left(z\right),arg\left(x+iy\right)$θ,arg(z),arg(x+iy)
For $x$x ≠ $0$0, $\tan\theta=y/x$tanθ=y/x
The convention for complex numbers is to use radians as the measure for angles.
Polar form and conguates
Using polar form, if $z=r\operatorname{cis}\left(\theta\right)$z=rcis(θ) then $\overline{z}=r\operatorname{cis}\left(-\theta\right)$z=rcis(θ)
This is because a complex number has end point coordinate $\left(x,y\right)$(x,y) and the conjugate has coordinate $\left(x,-y\right)$(x,y). So the value of the vertical distance from the real axis is identical, the horizontal distance is identical and thus the modulus and argument are identical in size, just that the angle is negative as opposed to positive.
The conjugate is a reflection of the point on the $x$x-axis in the complex plane.
#### Worked examples
##### Example 1
Write the number $\sqrt{2}-4i$24i in polar form.
To convert to polar form we need the modulus ($r$r) and argument $\left(\theta\right)$(θ)
$r=\sqrt{x^2+y^2}=\sqrt{\left(\sqrt{2}\right)^2+\left(-4\right)^2}=\sqrt{2+16}=\sqrt{18}=4.24$r=x2+y2=(2)2+(4)2=2+16=18=4.24
$\theta=\tan^{-1}\left(\frac{y}{x}\right)=\tan^{-1}\left(\frac{-4}{\sqrt{2}}\right)=1.23$θ=tan1(yx)=tan1(42)=1.23 radians
So $\sqrt{2}-4i=4.24\operatorname{cis}1.23$24i=4.24cis1.23
##### Example 2
Write the number $6\operatorname{cis}\left(-\frac{2\pi}{3}\right)$6cis(2π3) in rectangular form.
Let's draw the number on the plane and mark on $r$r and $\theta$θ.
To convert the polar number $6\operatorname{cis}\left(-\frac{2\pi}{3}\right)$6cis(2π3) into rectangular form we need to find the values of $x$x and $y$y.
We can see from the diagram that the angle is $-\frac{2\pi}{3}$2π3, so using trigonometry we can find $x$x and $y$y.
$z=r\operatorname{cis}\theta=6\left(\cos\left(\frac{2\pi}{3}\right)+i\sin\left(\frac{2\pi}{3}\right)\right)=6\left(-\cos\left(\frac{\pi}{3}\right)-i\sin\left(\frac{\pi}{3}\right)\right)=6\left(-\frac{1}{2}-i\frac{\sqrt{3}}{2}\right)=-3-3\sqrt{3}i$z=rcisθ=6(cos(2π3)+isin(2π3))=6(cos(π3)isin(π3))=6(12i32)=333i
### Just one more thing
There is just one more type of notation, called modulus argument form and it is another way to represent a complex number. We write this as $\left(r,\theta\right)=r\operatorname{cis}\theta$(r,θ)=rcisθ, but you may also see it written as $\left[r,\theta\right]$[r,θ].
A recap of all the new notation:
modulus distance from the origin argument (arg) angle from the positive horizontal axis $\operatorname{cis}\theta$cisθ $\cos\theta+i\sin\theta$cosθ+isinθ modulus argument form $\left(r,\theta\right)=r\operatorname{cis}\theta$(r,θ)=rcisθ
#### Practice questions
##### Question 1
Consider the points on the following polar graphs.
1. Write down the missing coordinates in the graph below. Give your answer in terms of $r$r and $\theta$θ.
2. Write down the coordinates of $A$A in the graph below. Give your answer in terms of $r$r and $\theta$θ.
3. Write down the coordinates of $A$A in the graph below. Give your answer in terms of $r$r and $\theta$θ.
##### Question 2
Consider the polar equation $r=2\sin2\theta$r=2sin2θ.
What is the value of $r$r if $\theta=30^\circ$θ=30°?
##### Question 3
Rewrite $20\operatorname{cis}$20cis$\left(0\right)$(0) in rectangular form.
##### Question 4
We want to rewrite $-6\sqrt{3}-6i$636i in the polar form $r\left(\cos\theta+i\sin\theta\right)$r(cosθ+isinθ), where $0^\circ\le\theta$0°θ$<$<$360^\circ$360°.
1. First find the value of $r$r.
2. Find the value of $\theta$θ.
3. Hence, what is $-6\sqrt{3}-6i$636i in polar form?
### Outcomes
#### M8-9
Manipulate complex numbers and present them graphically
#### 91577
Apply the algebra of complex numbers in solving problems
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# 95% CI for an estimated X given Y in a simple linear regression model
Considering a simple linear regression model Y= beta0+beta1 x X, with beta0 and beta1 computed, I have to estimate the expected X given a new Y and 95% confidence intervals. I used the formula X=(Y-beta0)/slope. How can I compute in R the 95% interval for the calculated value of ind, given a new value of the height?
ind height
1 4.27 174
2 3.60 159
3 3.61 175
summary(lm(df$ind~df$height))
Call:
lm(formula = df$ind ~ df$height)
Residuals:
Min 1Q Median 3Q Max
-0.56263 -0.27596 0.03866 0.26632 0.55440
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.968895 1.512371 -0.641 0.52903
df$height 0.027871 0.008985 3.102 0.00562 ** Residual standard error: 0.3287 on 20 degrees of freedom Multiple R-squared: 0.3248, Adjusted R-squared: 0.2911 F-statistic: 9.622 on 1 and 20 DF, p-value: 0.005621 # I tried: pred.frame <- data.frame(ind=seq(3,5,0.25)) predict(bclm,int="c",level=0.95,data=pred.frame) fit lwr upr 1 174.5780 169.3146 179.8414 2 166.7696 163.6419 169.8973 3 166.8862 163.7806 169.9917 ............... - Isn't lwr and upr the confidence interval? For each of your new ind values, 3,5,0.25 respectively? – Spacedman Jan 16 '12 at 7:57 @Spacedman - well not exactly...I need the CI for ind not for the height ...because I am calculating the ind (the X) given a new value of the height. And I need the CI for the new X=(Y-beta0)/beta1 – agatha Jan 16 '12 at 10:02 @Spacedman- is it that value X +/- the standard error at 95 %CI ? – agatha Jan 16 '12 at 10:03 ## 1 Answer You are looking for the so called calibration interval. You have to be careful with the exact question: is$Y$a new observation and you are trying to estimate$X$, or are you looking for the point where the regression line reaches$Y$? The difference will determine whether you need to use the confidence interval of the regression line or its prediction interval. Since you are talking about the confidence interval of the line, I assume you are interested in the latter question. The 95% confidence interval for the estimated$X$will be given by the points where the$y=Y$horizontal line intersects the lower and upper confidence bands of the regression line. So go through the lwr values, and check at which x value they crossed the$Y\$, and similarly for the upr values. Depending on the sophistication you want, this can be done by just looking at the printout, or having R find the crossing point, or even using the uniroot function to find the point with arbitrary precision (you are using steps of 0.25, so that's your current precision).
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Yes that is what I was looking for..I couldn't find it anywhere better explained. Thanks. – agatha Jan 16 '12 at 19:02
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Matrix transpose AT = 15 33 52 −21 A = 135−2 532 1 Example Transpose operation can be viewed as flipping entries about the diagonal. While certain “natural” properties of multiplication do not hold, many more do. MATRIX MULTIPLICATION. The determinant of a 4×4 matrix can be calculated by finding the determinants of a group of submatrices. $$\begin{pmatrix} e & f \\ g & h \end{pmatrix} \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ae + cf & be + df \\ ag + ch & bg + dh \end{pmatrix}$$ Distributive law: A (B + C) = AB + AC (A + B) C = AC + BC 5. The following are other important properties of matrix multiplication. Even though matrix multiplication is not commutative, it is associative in the following sense. The last property is a consequence of Property 3 and the fact that matrix multiplication is associative; A square matrix is called diagonal if all its elements outside the main diagonal are equal to zero. Equality of matrices proof of properties of trace of a matrix. Proof of Properties: 1. Given the matrix D we select any row or column. 19 (2) We can have A 2 = 0 even though A ≠ 0. But first, we need a theorem that provides an alternate means of multiplying two matrices. Definition The transpose of an m x n matrix A is the n x m matrix AT obtained by interchanging rows and columns of A, Definition A square matrix A is symmetric if AT = A. The proof of this lemma is pretty obvious: The ith row of AT is clearly the ith column of A, but viewed as a row, etc. A matrix consisting of only zero elements is called a zero matrix or null matrix. (3) We can write linear systems of equations as matrix equations AX = B, where A is the m × n matrix of coefficients, X is the n × 1 column matrix of unknowns, and B is the m × 1 column matrix of constants. Notice that these properties hold only when the size of matrices are such that the products are defined. Multiplicative Identity: For every square matrix A, there exists an identity matrix of the same order such that IA = AI =A. Example 1: Verify the associative property of matrix multiplication … i.e., (AT) ij = A ji ∀ i,j. For the A above, we have A 2 = 0 1 0 0 0 1 0 0 = 0 0 0 0. Multiplicative identity: For a square matrix A AI = IA = A where I is the identity matrix of the same order as A. Let’s look at them in detail We used these matrices The first element of row one is occupied by the number 1 … A diagonal matrix is called the identity matrix if the elements on its main diagonal are all equal to $$1.$$ (All other elements are zero). If $$A$$ is an $$m\times p$$ matrix, $$B$$ is a $$p \times q$$ matrix, and $$C$$ is a $$q \times n$$ matrix, then $A(BC) = (AB)C.$ This important property makes simplification of many matrix expressions possible. Subsection MMEE Matrix Multiplication, Entry-by-Entry. For sums we have. The number of rows and columns of a matrix are known as its dimensions, which is given by m x n where m and n represent the number of rows and columns respectively. The basic mathematical operations like addition, subtraction, multiplication and division can be done on matrices. In the next subsection, we will state and prove the relevant theorems. Example. Associative law: (AB) C = A (BC) 4. Zero matrix on multiplication If AB = O, then A ≠ O, B ≠ O is possible 3. Let us check linearity. The proof of Equation \ref{matrixproperties2} follows the same pattern and is … Selecting row 1 of this matrix will simplify the process because it contains a zero. Properties of transpose A matrix is an array of numbers arranged in the form of rows and columns. , ( AT ) ij = A ( B + C ) = AB + AC ( A + ). Relevant theorems number 1 … Subsection MMEE matrix multiplication … matrix multiplication, Entry-by-Entry array of arranged! The form of rows and columns only when the size of matrices are that. 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While certain “ natural ” properties of multiplication do not hold, many more do following are important.
## properties of matrix multiplication proof
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User aginensky - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:15:19Z http://mathoverflow.net/feeds/user/5100 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103186/intersecting-degree-0-divisors/103250#103250 Answer by aginensky for Intersecting Degree 0 Divisors aginensky 2012-07-27T00:05:06Z 2012-07-27T00:05:06Z <p>As has been mentioned, in general 'degree' is not so well/uniquely defined. However, suppose you take a smooth cubic surface in $\bf{ P}^3$ . There are 27 lines and they should all have degree 1. Take two lines $l_1$ and $l_2$ which are skew and let $D = l_1-l_2$ . This will be 'degree zero' with your definition. Clearly $deg(D|l_1) \neq 0$ Similarly on a quadric hypersurface in $\bf{P}^3$, letting $l_1$ and $l_2$ be two lines which meet , one can construct similar 'examples'. If $X$ is a variety with $Pic(X) = \bf{Z}$ then what you ask is true. I would suspect it fails any other time.</p> http://mathoverflow.net/questions/94030/cokernel-of-the-symmetric-product-of-an-injection cokernel of the symmetric product of an injection. aginensky 2012-04-14T13:53:24Z 2012-04-14T18:31:29Z <p>Clearly, my question can be asked more generally, but suppose for simplicity that $X$ is a smooth surface and that $0 \to E \to F \to K \to 0$ is exact with $E ,F$ rank two bundles on $X$ and $K$ a line bundle on a (smooth?) divisor $D \subset X$. Can one give information on the cokernel of the injection $Sym^n(E) \to Sym^n(F)$ ? For example is this cokernel some obvious sum of linear algebra constructions (various symmetric powers) of $E$ , $F$, and $K$ ? I can only say that I have thought about this question for a while and no answer has occurred to me. </p> <p>Adam</p> http://mathoverflow.net/questions/81322/classification-of-certain-algebraic-curves/81675#81675 Answer by aginensky for Classification of certain algebraic curves aginensky 2011-11-23T01:20:02Z 2011-11-23T01:20:02Z <p>I'm going to assume $L$ is base point free. I think it is clear how to change what I have written in the case where there are base points (numbers goes down by the degree of the base locus). In general, if $L$ is a line bundle of degree $d$ and $h^0(C,L)= r+1$, then the Clifford index of $L$, written Cliff(L) $= d-2r$. Cliffords theorem is Cliff(L) >= 0. Your line bundle satisfies Cliff(L) = 2. Two ways to achieve that if for the curve $C$ to have a $g^1_4$ or be a plane sextic. The Clifford index of a curve is the min{cliff(L)| $h^0(L)$ & $h^1(L)$ >=2}. For a general curve $C$, Cliff(C) is the floor of (g-1)/2. So for a general curve this can't be done. It is absolutely true that A-C-G-H will have more details and I believe a careful perusal will give the classification of all curves with Clifford Index 2. I think it is only the cases I mention.</p> http://mathoverflow.net/questions/76815/normality-via-resolution-of-singularities/76872#76872 Answer by aginensky for Normality via resolution of singularities aginensky 2011-09-30T18:13:16Z 2011-09-30T18:13:16Z <p>Unfortunately it is not true that if $Y$ is smooth, $f$ is proper and the fibers of $f:Y \rightarrow X$ are reduced and connected that $X$ is normal. Unfortunately I am all to familiar with the following example, which has ended any optimism I have had about finding a criteria for normality using a desingularization. The question is local so I will just use the affine coordinates. Let $R = k[x^4,x^3y,xy^3, y^4]$. This ring is the cone of a projection of the quartic normal curve, so it is not normal at the origin. One checks easily using local coordinates that the resolution is just $\bf P^1$. I learned of this example from Mohan Kumar, but it also appears in one or more of Eisenbud's books, as well as a counter example to other questions asked here. It seems to be a universal counter example to any seemingly true facts about depth etc for which one cannot find a proof. </p> http://mathoverflow.net/questions/62695/is-there-a-fast-way-to-compute-matrix-multiplication-mod-p/62753#62753 Answer by aginensky for Is there a fast way to compute matrix multiplication mod p? aginensky 2011-04-23T15:43:27Z 2011-04-23T15:43:27Z <p>The improvement of matrix multiplication from $O(n^3)$ to $O(n^{2.4})$ is based on the Strassen equations for matrix multiplication. <a href="http://mathoverflow.net/questions/57725/strassen-algorithm-7-multiplications" rel="nofollow">http://mathoverflow.net/questions/57725/strassen-algorithm-7-multiplications</a> talks about this and gives further references. If your question is rephrased as "are there special equations for matrix multiplication in characteristic $p$ ?", then I think the answer, if known, will be in one of those references. If I were a betting man, I'd bet no. As a person not adverse to speculation, I am highly skeptical. Matrix multiplication feels to me to be characteristic independent and of the flavor if it was true in all large (positive) characteristics, then it would be true in characteristic zero. Also, almost all results about equations of determinantal varieties, secant varieties etc don't seem to be different in any characteristic. Of course, this is not true on the nose as higher syzygies can be different,and in my very feeble understanding of such matters, more complicated. Anyone in a position to terminate or validate my musings?</p> http://mathoverflow.net/questions/59018/equations-defining-a-subvariety equations defining a subvariety aginensky 2011-03-21T00:20:56Z 2011-03-27T02:51:54Z <p>The following question feels to me like a standard sort of 'fact' in birational geometry, but I can't seem to write down a correct set of details. Hopefully someone can point me to a reference and not a counter example!</p> <p>Suppose $X$ is a variety (reduced and irreducible over an algebraically closed field, perhaps of characteristic zero) and suppose that there exist a very ample line bundle $L$ and a linear system $V \subset H^0(X,L)$ such that $Y = Bs(V)$ is the singular set of $X$ scheme theoretically, that $Y$ is smooth of codimension at at least 2, and that $\tilde X$, the blow up of $X$ along $Y$ is smooth. Further assume that $\phi_{|V|} X--> S$ birationally maps $X$ onto a smooth variety $S$. Let $\tilde \phi$ be the map from $\tilde X \to S$ induced by $V$. Further assume that, denoting by $f$ the map $\tilde X \to X$, that $f^{-1}(Y) = T$ surjects onto $S$. Let $v_1, \dots v_s$ be $s = \dim(S)$ general sections of $V$ so that the intersection $Z(v_1) \cap \dots Z(v_s) \cap S$ consists of finitely many smooth points say $p_1, \dots p_m$. </p> <p>Also assume the $P = f( \tilde \phi^{-1}(\cup_{i=1:m} p_i))$ is a proper subset of $Y$. Then can one say that away from $P$, the sections $v_1 \dots v_s$ generate the ideal of $Y$ in $X$ ?</p> <p>The case I have in mind is where $Y$ is a smooth curve embedded in a sufficiently ample manner so that 1) $Y$ is defined by quadrics and 2) $X = Sec(X)$ is singular only along $Y$. Then $V$ would be the quadrics through $Y$. The point would be to use this sort of an argument to establish a minimum depth of $Sec(Y)$ along $Y$.</p> <p>This is my first question, so please feel free to correct etiquette with this question as well as the mathematics. </p> http://mathoverflow.net/questions/16087/defining-equations-for-secant-varieties/59205#59205 Answer by aginensky for defining equations for secant varieties aginensky 2011-03-22T17:08:40Z 2011-03-22T17:08:40Z <p>I'm very late to the conversation. In general nothing is known. For some cases of the Segre or Veronese variety, one can interpet the varieties as spaces of matrices and then the equations are determinants. In general this is not known. There is a large current literature on this topic. I would start at the arxiv with pretty much any current paper by J.M. Landsberg. It will contain loads of information and references on this topic. </p> http://mathoverflow.net/questions/1142/is-very-ampleness-of-a-divisor-on-a-curve-determined-entirely-by-degree-and-genus/57354#57354 Answer by aginensky for Is very ampleness of a divisor on a curve determined entirely by degree and genus? aginensky 2011-03-04T13:28:12Z 2011-03-04T13:28:12Z <p>I'm late to the game, but I would like to point out that the answer is systematically no. One class of examples. Suppose g>2 for simplicity. In that case any general line bundle of degree 2g is very ample and special ones are not. This can be seen by using the criteria that a line bundle is very ample iff for any effective divisor $D$ of degree 2, $h^0(L(-D) = h^0(L)-2$. One checks using R-R that this holds iff $L$ is not of the form $L= K_C(D)$ where $D$ is an effective divisor of degree 2. Line bundles of the form $L= K_C(D)$ are a 2 dimensional subset of the g dimensional (Picard) variety. This can be expanded upon.</p> http://mathoverflow.net/questions/22299/what-are-some-examples-of-colorful-language-in-serious-mathematics-papers/51813#51813 Answer by aginensky for What are some examples of colorful language in serious mathematics papers? aginensky 2011-01-12T03:02:49Z 2011-01-12T03:02:49Z <p>Milne's web page contains a number of amusing anecdotes- <a href="http://www.jmilne.org/math/apocrypha.html" rel="nofollow">http://www.jmilne.org/math/apocrypha.html</a></p> http://mathoverflow.net/questions/51531/theorems-that-are-obvious-but-hard-to-prove/51559#51559 Answer by aginensky for Theorems that are 'obvious' but hard to prove aginensky 2011-01-09T16:56:12Z 2011-01-09T16:56:12Z <p>Speaking of thesis advisers, mine said, "I think something should be called obvious only if it is obvious in the logical sense of if A implies B and if B implies C then A implies C". All else is subjective and hence capable of misuse. I have tried, but not necessarily succeeded, to follow this. I am constantly amazed/amused at how people coming at a problem from different points of views will find certain facts obscure or well known.</p> http://mathoverflow.net/questions/31655/statistics-for-mathematicians/50427#50427 Answer by aginensky for Statistics for mathematicians aginensky 2010-12-26T15:03:08Z 2010-12-26T15:03:08Z <p>For a very mathematical version of statistics, my favorite is on line lecture notes from two MIT courses. The instructor is named Panchenko and the course is called 'Statistics for Applications'. There are course notes that read like a book for the course in 2003 and 2006. I have enjoyed browsing through both of them. Here is a link: <a href="http://ocw.mit.edu/courses/mathematics/" rel="nofollow">http://ocw.mit.edu/courses/mathematics/</a>. </p> http://mathoverflow.net/questions/46768/how-many-independent-quadrics-should-one-intersect-to-get-the-canonical-curve/47364#47364 Answer by aginensky for How many independent quadrics should one intersect to get the canonical curve. aginensky 2010-11-25T19:18:06Z 2010-11-25T19:18:06Z <p>Amplifying on of Speyer's comments, if p is a point on a secant line of C, then the quadrics vanishing on C and p are of codimension one in the space of all quadrics vanishing on C. Such a quadric vanishes at 3 points of the secant line ( p and the two points of C defining the line as a secant) and hence vanished on L. Am I doing something silly?</p> http://mathoverflow.net/questions/41429/line-bundles-on-special-abelian-surfaces/46080#46080 Answer by aginensky for Line bundles on special abelian surfaces aginensky 2010-11-14T22:40:38Z 2010-11-14T22:40:38Z <p>Am I missing something? Doesn't this hold for arbitrary smooth pairs of varieties using the Kunneth decomposition?</p> http://mathoverflow.net/questions/32938/surfaces-in-mathbbp3-with-isolated-singularities/33238#33238 Answer by aginensky for Surfaces in $\mathbb{P}^3$ with isolated singularities aginensky 2010-07-24T22:21:22Z 2010-07-24T22:21:22Z <p>To the best of my knowledge this is a long standing open problem. I cannot recall a reference, as this is something I studied in the 1980's, but I recall this being phrased as an unsolved problem from the 19th century Italian school. The conjecture is that no normal surface in P^3 is birational to a smooth surface which has two dimensional image in it's Albanese. One specific case of this that has been studied more extensively are Zariski surfaces:z^n = f(x,y) where f is a polynomial of degree n with only cusps and nodes as singularities. There are lots of information about when such a surface is irregular, but beyond that not much is known. I believe that even if f is a sextic polynomial it is unknow whether or not the resulting surface can have 2 dimensional image in it's Albanese. I have heard Catanese ask about the case where S is an abelian surface. </p> http://mathoverflow.net/questions/24913/quick-proofs-of-hard-theorems/25451#25451 Answer by aginensky for Quick proofs of hard theorems aginensky 2010-05-21T02:25:01Z 2010-05-21T02:25:01Z <p>I was told by my (graduate school) teacher of functional analysis that originally the complex case of the Hahn-Banach theorem was considered a major open problem. It was eventually shown to be such a simple consequence of the real case, that now, no one knows who came up with the trick.</p> http://mathoverflow.net/questions/9734/on-the-clifford-index-of-a-curve/20245#20245 Answer by aginensky for On the Clifford index of a curve aginensky 2010-04-03T16:18:54Z 2010-04-03T16:18:54Z <p>when c= 0 Clifford's them includes the fact that any divisor with Clifford index 0 is a multiple of the hyperelliptic fiber, ie: a sum of fibers of the hyperelliptic map. If c=1 then the curve is either trigonal or a plane quintic- I believe that it is an exercise in A-C-G-H. Kind of a folk lore result. I have not heard of anyone explicating all the possible cases when c=2.</p> http://mathoverflow.net/questions/131315/embedded-associated-prime-and-non-zero-divisor Comment by aginensky aginensky 2013-05-21T15:45:39Z 2013-05-21T15:45:39Z You could have saved some time by saying " please solve hw problem on page (insert page) of book (insert book name) http://mathoverflow.net/questions/130197/non-reducedness-in-linear-systems Comment by aginensky aginensky 2013-05-09T20:44:44Z 2013-05-09T20:44:44Z Unless I misunderstand what you are saying, pretty much anything is a counter example. On $P^n$ consider the linear system $O(2)$. A special member is a hyperplane with multiplicity two, but the general element will be a smooth conic. http://mathoverflow.net/questions/129086/borels-paris-lectures/129089#129089 Comment by aginensky aginensky 2013-04-30T23:31:40Z 2013-04-30T23:31:40Z If someone knows a deserving library, I am in possession of a copy- via the estate of Walter Baily*. I don't want money, but if the library was willing to make a nominal donation to the AMS in Walter's name, that would be great. It isn't signed, but on the inside cover there is a type written note saying "with the compliments of the author. Borel and Baily were good friends. Feel free to delete this comment if deemed too commercial. http://mathoverflow.net/questions/128832/a-homeomorphism-betwen-two-topological-spaces-implies-that-the-n-th-homology-grou Comment by aginensky aginensky 2013-04-26T15:59:05Z 2013-04-26T15:59:05Z I don't know if this answers your question, but I think that one can construct a homotopy equivalence between this question and a hw problem. http://mathoverflow.net/questions/128688/gauss-mapping-in-finite-characteristic Comment by aginensky aginensky 2013-04-25T15:34:17Z 2013-04-25T15:34:17Z There are papers by Kleiman-Piene that discuss this question. My best recollection is that they tend to be inseparable, but finite. http://mathoverflow.net/questions/128100/non-singular-cubics-are-not-rational/128114#128114 Comment by aginensky aginensky 2013-04-21T14:33:28Z 2013-04-21T14:33:28Z @ voloch Good point. That is why I made it a comment and not an answer :) http://mathoverflow.net/questions/128100/non-singular-cubics-are-not-rational/128114#128114 Comment by aginensky aginensky 2013-04-20T16:14:41Z 2013-04-20T16:14:41Z I'm not sure if this is more elementary, but if the field was rational, one would have polynomials $h(t)$ and $g(t)$ s.t. $f(h(t),g(t))$ = 0. $f(x,y) = y^2-(x)(x-1)(x-\lambda)$ for an elliptic curve. It seems as if $\alpha$ is a root of $h(t)$ , that forces it to be a root of either $g$, $g-1$, or $g-\lambda$. But not a double root and hence a contradiction. I've not put in all the details, but I think that works and to my taste, it is more 'elementary' http://mathoverflow.net/questions/127538/clifford-index-of-curves-on-a-surface Comment by aginensky aginensky 2013-04-15T14:36:43Z 2013-04-15T14:36:43Z There is a paper of Green and Lazarsfeld in which they show that the Clifford index of a curve is constant in the case you mention if $X$ is a K-3 surface. I've never heard anyone suggest, and off-hand I can't think of any reason why, this is true on an arbitrary surface. My guess is that $P^3$ should already provide counterexamples. Namely take a curve $C$, then for $n>>0$ it should lie on a hypersurface that depends only on the gross numerical invariants of $C$ and not it's Clifford index. http://mathoverflow.net/questions/126861/non-proper-intersection-of-projective-schemes Comment by aginensky aginensky 2013-04-08T21:20:30Z 2013-04-08T21:20:30Z For the first question, maybe this is too obvious, but what about transversality ? http://mathoverflow.net/questions/120905/conditional-probability-with-percentages Comment by aginensky aginensky 2013-02-05T20:57:50Z 2013-02-05T20:57:50Z the probability is 100% that this is a hw question. http://mathoverflow.net/questions/120422/the-map-from-c4-to-theta-as-a-blow-up Comment by aginensky aginensky 2013-02-04T23:03:27Z 2013-02-04T23:03:27Z @ Jie Wang - Ch IV section 3 - starts with the sentence " Our first goal, in this section, is easily stated. We fix a smooth genus g curve C and two non-negative integers r,d: we would then like to construct a subvariety $W^r_d$ of $Pic^d(C)$ whose support is the set of complete linear series of degree d and dimensionat least r. Probably one should start with the beginning of Ch. IV for background. They are determinantal and they also construct the resolution. http://mathoverflow.net/questions/120422/the-map-from-c4-to-theta-as-a-blow-up Comment by aginensky aginensky 2013-01-31T20:44:38Z 2013-01-31T20:44:38Z I am putting this in because you say 'any input is helpful'. As I recall, if you look in A-C-G-H, this is explained. The singularities of the theta divisor are a 'determinantal variety'. The construction is more general, it explains the algebraic structure of 'the set of line bundles with more sections than the general one'. Have a look. http://mathoverflow.net/questions/116831/tensor-rank-of-anti-symmetric-tensor Comment by aginensky aginensky 2013-01-17T17:17:37Z 2013-01-17T17:17:37Z Here is a link - <a href="http://arxiv.org/abs/1110.0745" rel="nofollow">arxiv.org/abs/1110.0745</a> . I think the rank of 'detrminant' considered as a symmetric tensor must be known, but I do't know it ! http://mathoverflow.net/questions/116831/tensor-rank-of-anti-symmetric-tensor Comment by aginensky aginensky 2013-01-17T17:15:30Z 2013-01-17T17:15:30Z For symmetric tensors, I think your problem is called 'Waring Problem for polynomials'. Specifically, identifying symmetric tensors with polynomials, the Waring problem asks- given a homogeneous polynomial of degree d, what is the minimum number of d-th powers of a linear polynomial that are needed to write the given polynomial. The generic number has been known for a while and is called (i hope i'm remembering correctly) the Alexander-Hirshowitz theorem. The problem of given a monomial, how many dth forms are needed to write it was just solved and is on the arxiv. http://mathoverflow.net/questions/117095/does-the-albanese-map-satisfy-torellis-theorem Comment by aginensky aginensky 2012-12-23T18:04:39Z 2012-12-23T18:04:39Z Any regular variety has trivial Albanese, so any flat family of regular varieties will fail- or did I misread the question. Less formally, it seems that except for curves, the Albanese is a very crude invariant and won't detect any part of the structure of a variety that is at all unirational.
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### Second Order Nonlinear Differential Equation Solver
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This video describes how to solve second order initial value problems in Matlab, using the ode45 routine. To a nonhomogeneous equation , we associate the so called associated homogeneous equation. Convert a second-order differential equation into a system of differential equations that can be solved using the numerical solver ode45 of MATLAB®. In the tutorial How to solve an ordinary differential equation (ODE) in Scilab we can see how a first order ordinary differential equation is solved (numerically) in Scilab. Since a homogeneous equation is easier to solve compares to its. [email protected] 1 Configuration Space Consider a system with a single degree of freedom and assume that the equation expressing its dynamic equilibrium is a second order ordinary differential equation (ODE) in the generalized coordinate x. As shown, the body is pinned at point O and has a mass center located at C. A (one-dimensional and degree one) second-order autonomous differential equation is a differential equation of the form: Solution method and formula. Ordinary Differential Equations of the Form y′′ = f(x, y) y′′ = f(y). On the triangles, we use a sparse solver in order to solve Maxwell equation (all triangles are tightly coupled). Now we can create the model for simulating Equation (1. Created: January 29, 1996 Updated: July 10, 1997. Definitions. solving linear first order delay differential equations by moc and steps method comparing with matlab solver a thesis submitted to the graduate school of applied. Max Born, quoted in H. With today's computer, an accurate solution can be obtained rapidly. 11), it is enough to nd the general solution of the homogeneous equation (1. 1 Four Examples: Linear versus Nonlinear A first order differential equation connects a function y. Euler's Method. 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By using some examples, the efficiency of the method is also discussed. But instead of simply writing y ″ as w ′, the trick here is to express y ″ in terms of a first derivative with respect to y. Preface What follows are my lecture notes for a first course in differential equations, taught at the Hong Kong University of Science and Technology. Solve this nonlinear differential equation with an initial condition. The following topics describe applications of second order equations in geometry and physics. Solve Differential Equation Second Order? Below are some keywords that our users entered recently to get to our site. And those r's, we figured out in the last one, were minus 2 and minus 3. While a second order differential equation can be transfomed to a first order system as described above but because second order differential equations are ubiquitous in physics and engineering special methods have been developed for solving them, see Methods for Second-Order Differential Equations. This page contains download links to the latest Java versions of dfield and pplane. Solutions can be singular, in which case standard numerical approaches fail. Thus, in order to nd the general solution of the inhomogeneous equation (1. • transformations that linearize the equation ♦ 1st-order ODEs correspond to families of curves in x, y plane ⇒ geometric interpretation of solutions ♦ Equations of higher order may be reduceable to first-order problems in special cases — e. In this example, I will show you the process of converting two higher order linear differential equation into a sinble matrix equation. 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I1 we give the classification of second-order PDEs in two variables based on the method of characteristics. applications of first order ODEs to nonlinear second order ODEs. com and figure out dividing polynomials, trigonometry and several other algebra subject areas. Therefore the derivative(s) in the equation are partial derivatives. Our results generalize and improve those known ones in the literature. Byju's Second Order Differential Equation Solver is a tool which makes calculations very simple and interesting. For example, the equation $$y'' + ty' + y^2 = t$$ is second order non-linear, and the equation $$y' + ty = t^2$$ is first order linear. environments for solving problems, including differential equations. Advanced Math Solutions – Ordinary Differential Equations Calculator, Separable ODE Last post, we talked about linear first order differential equations. A numerical solution to this equation can be computed with a variety of different solvers and programming environments. 1 \sqrt{1 + v^2} Define a function computing the right-hand side, and use ode45. Get the free "General Differential Equation Solver" widget for your website, blog, Wordpress, Blogger, or iGoogle. (Harder) I Reduction order method. Differential Equation Terminology. The system of differential equations we're trying to solve is The first thing to notice is that this is not a first order differential equation, because it has an in it. The approach illustrated uses the method of undetermined coefficients. m — phase portrait plus graph of second order ordinary differential equation phasem. py, which contains both the variational form and the solver. The Duffing equation is a non-linear second-order differential equation used to model certain damped and driven oscillators. There are homogeneous and particular solution equations, nonlinear equations, first-order, second-order, third-order, and many other equations. To solve a system with higher-order derivatives, you will first write a cascading system of simple first-order equations then use them in your differential file. Differential equations If God has made the world a perfect mechanism, he has at least conceded so much to our imperfect intellect that in order to predict little parts of it, we need not solve innumerable differential equations, but can use dice with fair success. Materials include course notes, a lecture video clip, a problem solving video, and a problem set with solutions. Procedure for solving non-homogeneous second order differential equations: y" p(x)y' q(x)y g(x) 1. Part 2: Ordinary Differential Equations (ODEs) (This is new material, see Kreyszig, Chapters 1-6, and related numerics in Chaps. But the problem is my range is very high so it will take years to complete if I use it straight. 4 x cos 2 x 2 dx dx. Submit in one word document per person. A clever method for solving differential equations (DEs) is in the form of a linear first-order equation. Second-Order Nonlinear Ordinary Differential Equations 3. First the equation is converted into the first order differential equations and solving them using the same. I have never tried one until now, but they shouldn't be hard to use I assume. This chapter introduces the basic techniques of scaling and the ways to reason about scales. Hello! I am having some trouble with plotting a slope field in GeoGebra, from a differential equation of second order. Second-Order Differential Equations 16 Chapter Preview In Chapter 8, we introduced first-order differential equations and illustrated their use in describing how physical and biological systems change in time or space. A first order nonlinear autonomous. The order is 3. A novel symbolic ordinary differential equation solver The novel features of this solver are:1. Our results generalize and improve those known ones in the literature. Procedure for Solving Linear Second-Order ODE. Introduction. Exact Solutions > Ordinary Differential Equations > Second-Order Nonlinear Ordinary Differential Equations PDF version of this page. To a nonhomogeneous equation , we associate the so called associated homogeneous equation. While a second order differential equation can be transfomed to a first order system as described above but because second order differential equations are ubiquitous in physics and engineering special methods have been developed for solving them, see Methods for Second-Order Differential Equations. y0+ x2y= ex is first order, linear, non homogeneous. The best possible answer for solving a second-order nonlinear ordinary differential equation is an expression in closed form form involving two constants, i. Using RK4 I am getting good accuracy and is working fine. com and figure out dividing polynomials, trigonometry and several other algebra subject areas. Fundamental Sets of Solutions – In this section we will a look at some of the theory behind the solution to second order differential equations. The differential equation is said to be linear if it is linear in the variables y y y. The angle θ defines the angular position coordinate. The equation above was a linear ordinary differential equation. m — phase portrait plus graph of second order ordinary differential equation phasem. First Order Equations 1. We define fundamental sets of solutions and discuss how they can be used to get a general solution to a homogeneous second order differential equation. We have got a tremendous amount of quality reference tutorials on subject areas starting from radical to syllabus for elementary algebra. A times the second derivative plus B times the first derivative plus C times the function is equal to g of x. In this paper we have presented a suggested method to solve second order nonlinear ordinary differential equations with mixed conditions using the matrix method based on collocation points on any interval [a,b]. 2) where L is an operator of the highest derivative, R is the remainder of the differential operator, g(t) is the nonhomogeneous term. The highest derivative is the second derivative y". [code]syms a g b c k h j syms x(t) y(t) ode = diff(x,t,2) == -a*g-b*diff(x,t)-c*x-k+h*diff(y,t)+j*y ; xSol(t)=solve(ode) ysol(t)=solve(ode) [/code]I hope you get it however I will give a small intro about the commands * syms - used for defining va. Two Dimensional Differential Equation Solver and Grapher V 1. First derivative: (dy)/(dx)=2c_1 cos 2x-6 sin 2x. One such environment is Simulink, which is closely connected to MATLAB. The (G ′ /G)-expansion method is based on the assumptions that the wave solutions can be expressed by a polynomial in (G ′ /G), where the second order linear ordinary differential equation (ODE) G ʺ + λG ′ + μG = 0 has been executed as an auxiliary equation, λ and μ are arbitrary constants. They are ubiquitous is science and engineering as well as economics, social science, biology, business, health care, etc. From nonlinear systems of equations calculator to matrices, we have got all of it discussed. Nonlinear partial differential equation - Wikipedia. y0+ x2y= ex is first order, linear, non homogeneous. Many articles have been published on it and its generalizations, although I have not yet found any discussing numerical methods in detail. Since the development of calculus in the 18th century by the mathematicians like Newton and Leibnitz, differential equation has played an important. The best possible answer for solving a second-order nonlinear ordinary differential equation is an expression in closed form form involving two constants, i. , inside and outside of the molecules considered). (diffusion equation) These are second-order differential equations, categorized according to the highest order derivative. Solve the nonlinear second-order ordinary differential equation d^2y/dx^2 - 1/x dy/dx - 1/2 (dy/dx)^2 = 0 Get more help from Chegg Get 1:1 help now from expert Advanced Math tutors. In this section we focus on Euler's method, a basic numerical method for solving initial value problems. We point out that the equations. Common methods for the qualitative analysis of nonlinear ordinary differential equations include:. Some literature says that I should solve this equation for "each time step" Or, is this right way to apply Backward-Euler scheme and NR scheme to the time-dependent nonlinear differential equation? Is there any good example solving time dependent nonlinear differential equation with Newton-Raphson iteration?. 2 nd-Order ODE - 1 CHAPTER 2 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 1 Homogeneous Linear Equations of the Second Order 1. The equation is given by dax dx +87 - Q. 2 Constant Coefficient Equations The simplest second order differential equations are those with constant coefficients. This course is about differential equations, and covers material that all engineers should know. In this example, I will show you the process of converting two higher order linear differential equation into a sinble matrix equation. 4 (120 ratings) Course Ratings are calculated from individual students' ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. Equations of this type commonly satisfy a comparison principle and have some regularity results. Linear differential equations of second-order form the foundation to the analysis of classical problems of mathematical physics. If dsolve cannot solve your equation, then try solving the equation numerically. This is a standard. df 4 x cos 2 x dx Cont. These notes are concerned with initial value problems for systems of ordinary dif-ferential equations. Consider the differential equation: The first step is to convert the above second-order ode into two first-order ode. This fourth order ODE is called the symmetric product of the second-order equations [17]: Here is the solution of the symmetric product of these ODEs: 3. Should you have support with algebra and in particular with practicing balancing equations worksheet or exponential and logarithmic come visit us at Mathworkorange. For the study of these equations we consider the explicit ones given by. See Solve a Second-Order Differential Equation Numerically. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). The approach chosen here is sometimes referred to as full-wave modellingin the literature: the original Maxwell's equations are used to obtain a second order equation for the time-harmonic electric field. The system must be written in terms of first-order differential equations only. Hello! I am having some trouble with plotting a slope field in GeoGebra, from a differential equation of second order. This page contains download links to the latest Java versions of dfield and pplane. Consider the 3 rd order equation (with initial conditions. Several options are available for MATLAB's ode45 solver, giving the user lim-. The Second Order Differential Equation Solver an online tool which shows Second Order Differential Equation Solver for the given input. Second Order Linear Differential Equations 12. Convert a second-order differential equation into a system of differential equations that can be solved using the numerical solver ode45 of MATLAB®. (method Euler and trapezoidal). Then the new equation satisfied by v is This is a first order differential equation. These problems are called boundary-value problems. The derivative may be partial or ordinary. In this paper we have presented a suggested method to solve second order nonlinear ordinary differential equations with mixed conditions using the matrix method based on collocation points on any interval [a,b]. y′′ = Ax n y m. Advanced Math Solutions – Ordinary Differential Equations Calculator, Separable ODE Last post, we talked about linear first order differential equations. Exact Differential Equation Non-Exact Differential Equation M(x,y)dx+N(x,y)dy=0 N(x,y)y'+M(x,y)=0 Linear in x Differential Equation Linear in y Differential Equation RL Circuits Logistic Differential Equation Bernoulli Equation Euler Method Runge Kutta4 Midpoint method (order2) Runge Kutta23 2. Solve the. required to solve a non linear. See Solve a Second-Order Differential Equation Numerically. Nonlinear OrdinaryDifferentialEquations by Peter J. Each of those categories is divided into linear and nonlinear subcategories. In this post, we will talk about separable. A solution of a differential equation is a function that satisfies the equation. extend the works of Mohammed Al-Refaiet al (2008) and make. 4898447 Solving system of linear differential equations by using differential transformation method AIP Conf. Second order differential equations are common in classical mechanics due to Newton's Second Law,. 3 What is special about nonlinear ODE? ÖFor solving nonlinear ODE we can use the same methods we use for solving linear differential equations ÖWhat is the difference? ÖSolutions of nonlinear ODE may be simple, complicated, or chaotic ÖNonlinear ODE is a tool to study nonlinear dynamic:. Come to Sofsource. We point out that the equations. Solve this equation and find the solution for one of the dependent variables (i. The differential equations must be IVP's with the initial condition (s) specified at x = 0. An in-depth course on differential equations, covering first/second order ODEs, PDEs and numerical methods, too! 4. Badmus PhD 1(Department of Mathematics, University of Calabar, Calabar, Nigeria) 2(Department of Mathematics and Computer Science, Nigerian Defence Academy, Kaduna, Nigeria). Nonlinear Differential Equation with Initial Condition. Second order DE: Contains second derivatives (and possibly first derivatives also). How do I solve a second order non linear Learn more about differential equations, solving analytically, homework MATLAB. Since acceleration is the second derivative of position, if we can describe the forces on an object in terms of the objects position, velocity and time, we can write a second order differential equation of the form. 9), and add to this a particular solution of the inhomogeneous equation (check that the di erence of any two solutions of the inhomogeneous equation is a solution of the homogeneous equation). which is second order non-linear ODE's, you'll see many of them. (Harder) I Reduction order method. Nonlinear Differential Equations: Invariance, Stability, and Bifurcation presents the developments in the qualitative theory of nonlinear differential equations. Solve this nonlinear differential equation with an initial condition. A New Factorisation of a General Second Order Differential Equation. net Differential Equations Calculators; Math Problem Solver (all calculators) Differential Equation Calculator. What we must keep in mind is that this algorithm for solving a second order differential equation depends on the fact that the function that satisfies the differential equation is a smoothly varying one like a sine or cosine function. If dsolve cannot solve your equation, then try solving the equation numerically. Two Dimensional Differential Equation Solver and Grapher V 1. [code]syms a g b c k h j syms x(t) y(t) ode = diff(x,t,2) == -a*g-b*diff(x,t)-c*x-k+h*diff(y,t)+j*y ; xSol(t)=solve(ode) ysol(t)=solve(ode) [/code]I hope you get it however I will give a small intro about the commands * syms - used for defining va. Once v is found its integration gives the function y. From nonlinear systems of equations calculator to matrices, we have got all of it discussed. Do you think this nonlinear ODE has analytical solution? Why not use numerical solver? \$\endgroup Second order differential equation with boundary conditions. The order of the PDE is the order of the highest (partial) di erential coe cient in the equation. It is of the form: y'' + a*y*y' + b*y=0 where a and b are constants Can this. A solution to such an equation is a function y = g(t) such that dgf dt = f(t, g), and the solution will contain one arbitrary constant. Our job is to show that the solution is correct. Second order differential equations are common in classical mechanics due to Newton's Second Law,. Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 4 Nonlinear Autonomous Systems of Two Equations Most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. parabolic equation and hyperbolic equation for equations with spatial operators like the previous one, and first and second order time derivatives, respectively. equation is given in closed form, has a detailed description. solve nonlinear differential equation first order pre algebra adding and subtracting integers worksheet , free help on a 4>3 solve the inequality. Classify the following linear second order partial differential equation and find its general. • transformations that linearize the equation ♦ 1st-order ODEs correspond to families of curves in x, y plane ⇒ geometric interpretation of solutions ♦ Equations of higher order may be reduceable to first-order problems in special cases — e. The derivative may be partial or ordinary. The order of a differential equation is equal to the highest derivative in the equation. Special Second order: y missing. solving second order nonlinear differential equations fractions formula adding subtracting , solving quadratic equations by completing the square , simplify radical expressions calculator root , fractions formula adding subtracting multiplying. Approximate solutions are arrived at using computer approxi-mations. Trapezoidal is more stable than Euler. I1 we give the classification of second-order PDEs in two variables based on the method of characteristics. Dynamical Systems¶ Many physical systems are explained by an ordinary differential equation (ODE) and it is often needed to solve for a solution of the differential equation. Solving Second Order Differential Equations Math 308 This Maple session contains examples that show how to solve certain second order constant coefficient differential equations in Maple. d f 3 dx 3. 1 2nd Order Linear Ordinary Differential Equations Solutions for equations of the following general form: dy dx ax dy dx axy hx 2 2 ++ =12() () Reduction of Order If terms are missing from the general second-order differential equation, it is sometimes possible. One such environment is Simulink, which is closely connected to MATLAB. Some types of ODE can be certainly solved analytically such as linear systems. is a first-order PDE. Back to top. Summary of Techniques for Solving First Order Differential Equations We will now summarize the techniques we have discussed for solving first order differential equations. Using RK4 I am getting good accuracy and is working fine. "Oscillation of a second-order delay differential equation with middle term". All solutions of a linear differential equation are found by adding to a particular. Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation. To solve L we want to find such L 2 and then solve L 2. On Exact Solutions of Second Order Nonlinear Ordinary Differential Equations Author: Amjed Zraiqat, Laith K. Just insert the differential equation along with your initial conditions into the appropriate differential equation solver. Come to Graph-inequality. With today's computer, an accurate solution can be obtained rapidly. 4 x cos 2 x 2 dx dx.
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# [XeTeX] \beginL and colour
Hans Hagen pragma at wxs.nl
Mon Nov 15 14:56:50 CET 2004
```Jonathan Kew wrote:
> Yes, having realized what's happening, this is an inherent weakness in
> using \special's to affect the text, in conjunction with the e-TeX bidi
> mechanism that ends up reversing horizontal lists. I don't anticipate
> doing anything to try and fix it in XeTeX for the time being.
makes sense, it's a non trivial problem
> Note that it should be possible to implement arbitrary text color
> changes by using \fontname to retrieve the current font name, adding a
> "color=...." attribute, and setting a new font. In this case, the color
> is associated directly with the actual text characters, so it will work
> properly despite the reordering.
what rules apply for the font spec then?
Hans
-----------------------------------------------------------------
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# to prove XY-plane a convex set??
we know that R and R^2 have convex subsets because any two points in them can be joined by a line segment..but how we will prove it mathematically that XY-plane is a convex set??
• From the definition of convexity? Take arbitrary $(x_1,y_1)$ and $(x_2,y_2)$ in $\mathbb{R}^2$... – newbie Feb 11 '14 at 14:27
• let x1=(1,0,0) and x2=(0,1,0). then by definition(1-alpha)x1+alpha x2 =x.now taking alpha=1/2 we get (1-1/2)(1,0,0)+1/2 (0,1,0)=(1/2,1/2,0) belongs to XY-plane...so XY-plane is a convex set.is this solution correct?? – hafsah Feb 11 '14 at 14:38
• the example is correct, but this is not a proof for "the XY-plane is convex". – Danny Feb 11 '14 at 14:43
• i want its prove .can u help me plz?? – hafsah Feb 11 '14 at 14:44
• Do you intuitively understand why the XY-plane is convex? – dani_s Feb 11 '14 at 14:49
After your comment, i think we talk about $\mathbb{R}^{3}$.
Therefore the XY-Plane is represented by the set of vectors $\left\{(x,y,0)^{T}|\;x,y\in\mathbb{R}\right\}$.
A set is convex, if for any vectors $\vec{a},\vec{b}$ the linesegment between $\vec{a}$ and $\vec{b}$ is in the set, too.
So the XY-plane is convex, if for all $\vec{a},\vec{b}\in\left\{(x,y,0)^{T}|\;x,y\in\mathbb{R}\right\}$ the following holds:
• $\lambda\vec{a}+(1-\lambda)\vec{b}\in\left\{(x,y,0)^{T}|\;x,y\in\mathbb{R}\right\},\;\;0\leq\lambda\leq1$
Let $\vec{a}=(x_{1},y_{1},0)^T$ and $\vec{b}=(x_{2},y_{2},0)^T$ be arbitrary vectors of the XY plane. Then
$\lambda\vec{a}+(1-\lambda)\vec{b}= \left( \begin{array}{c} \lambda x_{1}+(1-\lambda)x_{2}\\ \lambda y_{1}+(1-\lambda)y_{2}\\ 0\\ \end{array} \right)\in\left\{(x,y,0)^{T}|\;x,y\in\mathbb{R}\right\}$
because the $x$-component and the $y$-component are in $\mathbb{R}$ and the last component is $0$.
• if we want to talk about in R2 then we will not take the third copmonent and then prove it as above...is it so?? – hafsah Feb 11 '14 at 15:07
• If we talk about $\mathbb{R}^{2}$, then the XY-plane is the whole $\mathbb{R}^{2}$ and then the convexity is trivial. But yes, the proof works for $\mathbb{R}^{2}$ in the same way without the third coordinate.. – Danny Feb 11 '14 at 15:09
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Sandia is building an outdoor bifacial PV performance test bed in Albuquerque, NM to collect data to be analyzed and shared with the community. This data will be used to develop predictive performance models that can eventually be included in commercial applications. Based on a review of literature on this topic, we designed our test facilities to be able to vary design parameters that are known to affect bifacial PV performance.
The amount of additional energy generated from the backside of a bifacial module can be analyzed by calculating the bifacial gain. Bifacial gain is measured using two modules, one bifacial and another monofacial reference module at the same orientation. Both modules should ideally have the same front-side power rating, but corrections can be made for differences in the ratings. Bifacial gain is then defined as:
$BG_{i}=&space;100\times&space;\left&space;(&space;\frac{P_{bifacial}/Pmp_{bifacial}}{P_{monofacial}/Pmp_{monofacial}}-1&space;\right&space;)$,
where $P_{bifacial}$ and $P_{monofacial}$ are the energy measured from the bifacial and monofacial arrays, respectively. $Pmp_{bifacial&space;}$ and $Pmp_{monofacial}$ are the STC power ratings from the modules measured on the front side only (with backside of bifacial module covered).
An integrated bifacial gain in energy, $BG_{E}$ (for example, one month) can be calculated as:
$BG_{E}=&space;100\times&space;\left&space;(&space;\frac{\sum&space;P_{bifacial}/Pmp_{bifacial}}{\sum&space;P_{monofacial}/Pmp_{monofacial}}-1&space;\right&space;)$.
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Tikz draw contour without some edges, and fill
I'd like to draw two wave-shaped contours, as in the figure below, and fill the area between. I tried the following code, but it produces an undesired black vertical edge on the right. How do I remove that?
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\useasboundingbox (-1,-1) rectangle (6,2);
\draw[fill=black!10] (0,1) to[out=0,in=180] ++(5,-0.6) -- ++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
\end{tikzpicture}
\end{document}
I also tried to remove the -- between ++(5,-0.6) and ++(0,0.6) in the 5th line, it doesn't draw the undesired edge, but it messes up the filling.
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\useasboundingbox (-1,-1) rectangle (6,2);
\draw[fill=black!10] (0,1) to[out=0,in=180] ++(5,-0.6) ++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
\end{tikzpicture}
\end{document}
Quick and dirty: fill the area first and then draw what you want to draw. It is not easily possible to switch off the drawing of a part of a path (but it is possible yet considerably more effort).
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\useasboundingbox (-1,-1) rectangle (6,2);
\path[fill=black!10] (0,1) to[out=0,in=180] ++(5,-0.6) -- ++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
\draw (0,1) to[out=0,in=180] ++(5,-0.6)++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
\end{tikzpicture}
\end{document}
You can switch the path on and off e.g. using this answer.
\documentclass[tikz]{standalone}
\pgfkeys{tikz/.cd,
edge options/.code={\tikzset{edge style/.style={#1}}},
}
\begin{document}
\begin{tikzpicture}[every edge/.append code = {% https://tex.stackexchange.com/a/396092/121799
\global\let\currenttarget\tikztotarget % save \tikztotarget in a global variable
\pgfkeysalso{append after command={to[edge style] (\currenttarget)}}},
every edge/.append style={edge style} ]
\useasboundingbox (-1,-1) rectangle (6,2);
\path[fill=black!10]
(0,1) [edge options={out=0,in=180,draw=black}] edge ++(5,-0.6)
-- ++(0,0.6)
[edge options={out=180,in=0,draw=black}] edge ++(-5,-0.6);
\end{tikzpicture}
\end{document}
With use TikZ library pgfplots.fillbetween
\documentclass[margin=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usetikzlibrary{pgfplots.fillbetween}
\begin{document}
\begin{tikzpicture}
\draw[name path=A] (0, 0.3) to [out=0,in=180] ++(5,-0.6);
\draw[name path=B] (0,-0.3) to [out=0,in=180] ++(5, 0.6);
\tikzfillbetween[of=A and B] {fill=gray!30};
\end{tikzpicture}
\end{document}
For comparison, with plain Metapost, where filling and drawing are kept separate, and where you can concatenate path variables directly.
\documentclass[border=5mm]{standalone}
\usepackage{luatex85}
\usepackage{luamplib}
\begin{document}
\begin{mplibcode}
beginfig(1);
z0 = (-72, 8.5);
path A, B;
A = z0 {right} .. {right} z0 rotated 180;
(compile with lualatex...)
|
## Ishaan94 First answer (by Anders) on http://www.quora.com/Mathematics/How-would-I-show-that-49-divides-8-n-7n-1-for-all-n-ge-0 Can anyone explain it to me? Hints'd be appreciated. one year ago one year ago
1. Ishaan94
@mukushla @KingGeorge @Zarkon
2. mukushla
can u type the equation here plz :)
3. Ishaan94
@mukushla try again.
4. Ishaan94
I am not concerned with problem as much as its solution on Quora. And No @nincompoop
5. mukushla
thats not a true statement of binomial theorem$8^n=(1+7)^n=\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) 7^n$right?
6. Ishaan94
I am pretty sure, it is.
7. mukushla
oops sorry that should be$8^n=(1+7)^n=\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) 7^k$
8. Ishaan94
oh yeah. k. lol
9. Ishaan94
but that's not what concerns me. it's the first answer by Anders i am unable to understand.
10. mukushla
oh sorry man lol :) im sooo blind
11. mukushla
i cant understand it :)
12. experimentX
$8^n=(1+7)^n=\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) 7^n = 1 + 7n + O(7^{(n \geq 2)})$
13. estudier
Sai Ganesh explanation also no good, right?
14. experimentX
$$O(7^{(n \geq 2)})$$ is always divisible by 49
15. estudier
Yes, sorry, I was directing at Ishaan (he wants an "octal" explanation)
16. Ishaan94
Yeah. It didn't help my puny brain much :(
17. Ishaan94
@him1618 maybe you can?
18. him1618
the explanation is good enough
19. estudier
I have to say that the binomial answer is much more natural, it would not occur to me to set out on an octal adventure for that particular problem....
20. him1618
true
21. Ishaan94
Specifically, I don't understand how did he conclude 'The rest can be divided into equal groups based on their first two nonzero digits.'
22. Ishaan94
... 49* equal groups ...
23. estudier
7*7
24. estudier
you got rid of the 0's so you got 7 digits left you can set, right?
25. Ishaan94
right.
26. estudier
So if all the numbers are divisible into 49 groups, then divisibilty by 49 follows. At least, that's what I think it says.
27. Ishaan94
Okay, one silly doubt. Why is he using only first two digits for grouping?
28. estudier
because 7*7 = 49
29. Ishaan94
i was acting foolishly yesterday :( thanks.
|
# Prove that $f_n$ is uniformly bounded on $E$ and $f$ is a bounded function on $E$
I have the solution for the following problem, but I don't understand most of it.
Question: A sequence of functions $f_n$ is said to be uniformly bounded on a set $E$ iff there exists $M>0$ such that $|f_n(x)\le M$ for all $x\in E$ and all $n\in \mathbb{N}$.
Suppose that for each $n\in\mathbb{N}$, $f_n:E\to\mathbb{R}$ is bounded. If $f_n\to f$ uniformly on $E$, as $n\to\infty$, prove that ${f_n}$ is uniformly bounded on $E$ and $f$ is a bounded function on $E$.
Solution:
Choose $N$ so large that $|f(x)-f_n(x)|<1$ for all $x\in E$ and $n\ge N$.
Why does it have to be less than $1$?
Set $M:=\sup_{x\in E}|f_N(x)|$ and observe by the Triangle Inequality that $$|f(x)|\le|f(x)-f_N(x)|+|f_N(x)|<1+M \text{ for all }x\in E\text{.}$$
Therefore, $|f_n(x)|\le|f(x)|+1\le(1+M)+1=2+M$ for all $n\ge N$ and $x\in E$, i.e., $\{f_n\}_{n\ge N}$ is uniformly bounded on $E$.
I understand this, but not what is below. What exactly is it trying to show? And how does that prove that $f$ is a bounded function on $E$?
In particular, $$|f_n(x)\le\hat{M}:=\max\{2+M,\sup_{x\in[a,b]}|f_1(x)|,...,\sup_{x\in[a,b]}| f_{N-1}(x)|\}<\infty$$ for all $n\in\mathbb{N}$ and $x\in E$.
• First question: arbitrary choice; $\leqslant \pi$ or $< 10^{\pm 10^{1000}}$ would work just as well. – Daniel Fischer Jun 25 '14 at 15:46
• the choice of $1$ was arbitrary; this would work for any $\epsilon > 0$, so we might as well choose $\epsilon = 1$. As Daniel Fischer notes, $\epsilon = \pi$ and $\epsilon = 10^{10^{1000}}$ would work just as well, for the purposes of the proof.
• in the above, we've shown that $|f(x)| \leq 1 + M$, so we know that $f$ is bounded. We also know that for any $n \geq N$, $f_n$ is also bounded by $2+M$ so that $\{f_n\}_{n \geq N}$ is uniformly bounded. What remains to be shown is that the first elements $\{f_1,\dots,f_{N-1}\}$ have a common bound, so that the entire sequence $\{f_n\}_{n \geq 1}$ can be shown to be bounded. In the last line, we establish that the entire sequence is in fact uniformly bounded.
|
1 - 2 of 2 Posts
#### Fredey
·
##### Registered
Joined
·
1 Posts
Discussion Starter
hello friends....
I am having a minor problem with my car i.e in the alignment of the
wipers .
The alignment of one of the wipers needs adjusting as it hits the
frame when used at high speed.Can i do this Alignment myself?
#### Mac
·
##### Registered
Joined
·
7 Posts
yeah you can do it urself....But with the proper care!
OTHERWISE IT WILL COST U A LOT
Originally posted by Fredey@Jul 31 2004, 06:31 AM
hello friends....
I am having a minor problem with my car i.e in the alignment of the
wipers .
The alignment of one of the wipers needs adjusting as it hits the
frame when used at high speed.Can i do this Alignment myself?
1 - 2 of 2 Posts
|
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# Getting back any/all of a deposit?
Author Message
TAGS:
Intern
Affiliations: CPA
Joined: 24 Jun 2011
Posts: 39
Concentration: Other
GMAT 1: 730 Q V
GPA: 3.27
WE: Accounting (Consulting)
Followers: 1
Kudos [?]: 6 [0], given: 0
Getting back any/all of a deposit? [#permalink] 02 Feb 2012, 09:39
Hey all,
sorry if this has been covered before, but has anyone had success getting their deposit back from a school they wind up not attending?
I assume the answer is no, but just in case anyone has any advice, I figure it'd be worth knowing!
Senior Manager
Status: schools I listed were for the evening programs, not FT
Joined: 16 Aug 2011
Posts: 389
Location: United States (VA)
GMAT 1: 640 Q47 V32
GMAT 2: 640 Q43 V34
GMAT 3: 660 Q43 V38
GPA: 3.1
WE: Research (Other)
Followers: 3
Kudos [?]: 46 [0], given: 50
Re: Getting back any/all of a deposit? [#permalink] 02 Feb 2012, 10:09
I'm going to say you can't.
But when you put in a deposit for a school, even if you withdraw because you're off the waitlist at another place, you should be willing to see yourself graduate from that school otherwise don't go.
I have no idea how much enrollment deposits are, but I would think they could be as low as $500 or as high as$6,000, depending on the program..... No one seems to advertise that....
Re: Getting back any/all of a deposit? [#permalink] 02 Feb 2012, 10:09
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Deposits 0 31 Jan 2011, 13:16
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Deposit Deadline? 16 12 Feb 2007, 08:51
Ready to get back in the ring... Big time! 7 17 Nov 2006, 22:39
Back At It 18 02 Nov 2006, 13:56
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|
Battleships OOP python
This is my first attempt at some basic OOP programming. A version of battleships played within the terminal. Any feedback would be great; especially in regards to readability and proper naming conventions.
I would really like to improve the strike function of the computer class to make it play more intelligently, any tips on that specifically?
ocean.py
class Ocean:
"""Creates 2D array the represents an ocean grid
Class contains all functions needed for placing
ships on the ocean grid"""
def __init__(self, width=10, height=10):
self.ocean = [["~" for i in range(width)] for i in range(height)]
def __getitem__(self, point):
row, col = point
return self.ocean[row][col]
def __setitem__(self, point, value):
row, col = point
self.ocean[row][col] = value
def view_ocean(self):
for row in self.ocean:
print(" ".join(row))
# Two functions check a coordinate input is on the grid
def valid_col(self, row):
try:
self.ocean[row]
return True
except IndexError:
return False
def valid_row(self, col):
try:
self.ocean[0][col]
return True
except IndexError:
return False
# Two functions check for valid board space for ship placement
def can_use_col(self, row, col, size):
valid_coords = []
for i in range(size):
if self.valid_col(col) and self.valid_row(row):
if self.ocean[row][col] == "~":
valid_coords.append((row, col))
col = col + 1
else:
col = col + 1
else:
return False
if size == len(valid_coords):
return True
else:
return False
def can_use_row(self, row, col, size):
valid_coords = []
for i in range(size):
if self.valid_row(row) and self.valid_col(col):
if self.ocean[row][col] == "~":
valid_coords.append((row, col))
row = row + 1
else:
row = row + 1
else:
return False
if size == len(valid_coords):
return True
else:
return False
# Corresponding fucntions set ship counters on valid space
def set_ship_col(self, row, col, size):
for i in range(size):
self.ocean[row][col] = "S"
col = col + 1
def set_ship_row(self, row, col, size):
for i in range(size):
self.ocean[row][col] = "S"
row = row + 1
class Radar:
"""Creates a grid to track the state of an opponent's ocean grid"""
def __init__(self, width=10, height=10):
self.radar = [["." for i in range(width)] for i in range(height)]
def __getitem__(self, point):
row, col = point
def __setitem__(self, point, value):
row, col = point
print(" ".join(row))
ship.py
class Ship:
def __init__(self, ship_type, size):
self.ship_type = ship_type
self.size = size
self.coords = []
def plot_vertical(self, row, col):
for i in range(self.size):
self.coords.append((row, col))
row = row + 1
def plot_horizontal(self, row, col):
for i in range(self.size):
self.coords.append((row, col))
col = col + 1
def check_status(self):
if self.coords == []:
return True
else:
return False
player.py
from ocean import Ocean
from ship import Ship
import os
class Player:
ships = {"Aircraft Carrier": 5, "Crusier": 4, "Destroyer": 3,
"Submarine": 2}
def __init__(self, name):
self.ocean = Ocean()
self.name = name
self.fleet = []
# Function uses player input to set up fleet positions on a player board.
# For each ship, a ship object containing relevant coordinates is appended to self.fleet
def set_fleet(self):
input("Pick a coordinate between 0 and 9 for the columns and rows on your board")
input("Boats are placed form right to left.")
for ship, size in self.ships.items():
flag = True
while flag:
self.view_console()
try:
row = int(input("Pick a row -----> "))
col = int(input("Pick a column -----> "))
orientation = str(input("Vertical or Horizontal? (choose v or h) -----> "))
if orientation in ["v", "V"]:
if self.ocean.can_use_row(row, col, size):
self.ocean.set_ship_row(row, col, size)
boat = Ship(ship, size)
boat.plot_vertical(row, col)
self.fleet.append(boat)
flag = False
else:
input("Overlapping ships, try again")
elif orientation in ["h", "H"]:
if self.ocean.can_use_col(row, col, size):
self.ocean.set_ship_col(row, col, size)
boat = Ship(ship, size)
boat.plot_horizontal(row, col)
self.fleet.append(boat)
flag = False
else:
input("Overlapping ships, try agin")
else:
continue
self.view_console()
input()
os.system('clear')
except ValueError:
print("Don't you remember your training?\nType a number..\n")
def view_console(self):
print("| |")
self.ocean.view_ocean()
# Function checks status of ship objects within player fleet
def register_hit(self, row, col):
for boat in self.fleet:
if (row, col) in boat.coords:
boat.coords.remove((row, col))
if boat.check_status():
self.fleet.remove(boat)
print("%s's %s has been sunk!" % (self.name, boat.ship_type))
# Player interface for initiating in-game strikes,
# updates the state of the boards of both players
def strike(self, target):
self.view_console()
try:
print("\n%s Pick your target..." % (self.name))
row = int(input("Pick a row -----> "))
col = int(input("Pick a column -----> "))
if self.ocean.valid_row(row) and self.ocean.valid_col(col):
if target.ocean.ocean[row][col] == "S":
print("DIRECT HIT!!!")
target.ocean.ocean[row][col] = "X"
target.register_hit(row, col)
else:
self.strike(target)
else:
print("Negative...")
else:
print("Coordinates out of range...")
self.strike(target)
except ValueError:
print("You need to provide a number....\n")
self.strike(target)
input()
os.system('clear')
computer.py
from player import Player
from ship import Ship
import random
class Computer(Player):
def __init__(self):
super().__init__(self)
self.name = "Computer"
# Automated version of set_fleet function from Player
def set_compu_fleet(self):
positions = ["v", "h"]
for ship, size in self.ships.items():
flag = True
while flag:
row = random.randint(0, 9)
col = random.randint(0, 9)
orientation = random.choice(positions)
if orientation == "v":
if self.ocean.can_use_row(row, col, size):
self.ocean.set_ship_row(row, col, size)
boat = Ship(ship, size)
boat.plot_vertical(row, col)
self.fleet.append(boat)
flag = False
else:
row = row + 2
elif orientation == "h":
if self.ocean.can_use_col(row, col, size):
self.ocean.set_ship_col(row, col, size)
boat = Ship(ship, size)
boat.plot_horizontal(row, col)
self.fleet.append(boat)
flag = False
else:
col = col + 2
else:
continue
# Automated strike function
def compu_strike(self, target):
row = random.randint(0, 9)
col = random.randint(0, 9)
input("...Target acquired....%s, %s" % (row, col))
if target.ocean.ocean[row][col] == "S":
print("DIRECT HIT!")
target.ocean.ocean[row][col] = "X"
target.register_hit(row, col)
else:
print("Missed....recalibrating")
else:
self.compu_strike(target)
battlshipspvp.py
from player import Player
import os
# Player versus player mode
class BattleshipsPVP:
"""creates a game of battlehsips"""
def __init__(self):
start = input("Begin? (y or n) -----> ")
if start in ["y", "Y"]:
self.playPVP()
else:
print("Aborted...")
def playPVP(self):
p1name = input("Player 1, state your name! -----> ")
p1 = Player(p1name)
p1.set_fleet()
p1.view_console()
self.clear_screen()
p2name = input("\n\nPlayer 2, state your name! -----> ")
p2 = Player(p2name)
p2.set_fleet()
p2.view_console()
self.clear_screen()
flag = True
while flag is True:
p1.strike(p2)
if self.fleet_sunk(p2) is True:
self.victory_message(p1, p2)
flag = False
else:
self.clear_screen()
p2.strike(p1)
if self.fleet_sunk(p1) is True:
self.victory_message(p2, p1)
flag = False
else:
self.clear_screen()
print("\nThanks for playing!")
# Function checks remaining ship counters on a player's board
def fleet_sunk(self, player):
ship_counters = 0
"""Traverses grid looking for 's' counters"""
for row in range(len(player.ocean.ocean)):
for col in range(len(player.ocean.ocean)):
if player.ocean.ocean[row][col] == "S":
ship_counters += 1
if ship_counters == 0:
return True
else:
return False
def clear_screen(self):
input("\nNext Turn?")
os.system('clear')
def victory_message(self, winner, loser):
print("\n\n\n*****************************************")
print("%s's fleet has been destroyed, %s wins!" % (loser.name, winner.name))
print("*****************************************")
battleshipscomp.py
from battleshipspvp import BattleshipsPVP
from computer import Computer
from player import Player
class BattleshipsCOMP(BattleshipsPVP):
def __init__(self):
start = input("Begin? (y or n) -----> ")
if start in ["y", "Y"]:
self.playCOMP()
else:
print("Aborted...")
def playCOMP(self):
pname = input("Player 1, state your name! -----> ")
p = Player(pname)
p.set_fleet()
p.view_console()
self.clear_screen()
c = Computer()
print("Computer setting its fleet...")
c.set_compu_fleet()
self.clear_screen()
flag = True
while flag is True:
p.strike(c)
if self.fleet_sunk(c) is True:
self.victory_message(p, c)
flag = False
else:
self.clear_screen()
c.compu_strike(p)
if self.fleet_sunk(p) is True:
self.victory_message(c, p)
flag = False
else:
self.clear_screen()
print("\nThanks for playing!")
playbattleships.py
from battleshipspvp import BattleshipsPVP
from battleshipscomputer import BattleshipsCOMP
# Script initiates game of battleships
def playbattleships():
print("\n\n***********************")
print("Welcome to Battleships!")
print("***********************\n")
print("\n 1) Player vs Player")
print("\n 2) Player vs Computer")
flag = True
while flag:
try:
mode = int(input("\n\nPick a number to select a game mode ----> "))
if mode == 1:
flag = False
BattleshipsPVP()
elif mode == 2:
flag = False
BattleshipsCOMP()
else:
continue
except ValueError:
print("You can only pick either option 1 or 2")
playbattleships()
A view of the game board in play. The radar is the upper grid, the ocean is the lower grid.
. . . . . . . . . .
X . . . . . . . . .
. . . . . . . . . .
. . . . O . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
| |
S ~ ~ ~ ~ ~ ~ ~ ~ ~
S ~ ~ ~ ~ ~ ~ ~ ~ ~
S ~ ~ ~ ~ ~ ~ ~ ~ ~
S ~ ~ ~ ~ ~ ~ ~ ~ ~
S ~ ~ ~ ~ S S S S ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ S S X ~ ~
~ ~ S ~ ~ ~ ~ ~ ~ ~
~ ~ S ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~
• If you want to improve your strike function, I'd recommend this nice article – Grajdeanu Alex. Sep 25 '19 at 5:51
You've created both your Ocean class and Radar class to accept a point key in the __getitem__ and __setitem__ methods. Ie)
def __getitem__(self, point):
row, col = point
return self.ocean[row][col]
def __setitem__(self, point, value):
row, col = point
self.ocean[row][col] = value
But in your Player class, you avoid using those interfaces, and instead directly reach into the Ocean and Radar classes and manipulate the internals directly:
if self.ocean.valid_row(row) and self.ocean.valid_col(col):
if target.ocean.ocean[row][col] == "S":
print("DIRECT HIT!!!")
target.ocean.ocean[row][col] = "X"
target.register_hit(row, col)
You should be using the interfaces you created:
if self.ocean.valid_row(row) and self.ocean.valid_col(col):
if target.ocean[row, col] == "S":
print("DIRECT HIT!!!")
target.ocean[row, col] = "X"
target.register_hit(row, col)
Mark internals private
There is no "private" in Python, but by convention, if a member starts with a leading underscore, it should only be accessed by self. So using self._ocean is ok, but using self._ocean._ocean would not be, as the second ._ocean is reaching into another objects privates.
Ie:
class Ocean:
def __init__(self, width=10, height=10):
self._ocean = [["~" for i in range(width)] for i in range(height)]
So when you create Player, and you type self.ocean._ocean, you can think to yourself, "wait ... I shouldn't be using that ._ocean member of that self.ocean object; I should be calling a public method on self.ocean instead."
Don't use Recursion for an Input Loop
Title says it all:
def strike(self, target):
try:
...
self.strike(target)
...
except ValueError:
print("You need to provide a number....\n")
self.strike(target)
...
If you repeatedly enter an already hit location, your stack will eventually overflow. If you repeatedly enter an invalid row/col value, and get a ValueError exception, you'll get stack overflow while handling a ValueError, while handling a ValueError, while handling a ValueError, while handling a ValueError, while handling a ValueError, while ... a few thousand levels deep!
You did things better in set_fleet() with a while flag: loop.
os.system('clear')
Don't use this. Don't even use:
os.system('cls' if sys.platform == 'nt' else 'clear')
That is forking a new process, loading a shell processor, interpreting a command, which may launch yet-another-process, which clears the screen. A little heavy weight. And it is risky, if an executable, like cls.bat or clear is on the path; you could end up executing arbitrary instructions.
Use:
import colorama
def clear_screen():
print(colorama.ansi.clear_screen())
colorama.init()
clear_screen()
Look at colorama for other interesting things you can do. Like making your ocean blue, your ships green, and your hits red.
Fixed Formatting
print("| |")
What happens if your ocean is 8x8? Or 12x12? Will your vertical bars still properly line up?
print("|" + " " * (2 * cols - 3) + "|")
would work better ... assuming cols is defined or replaced with something suitable.
Ditto with:
input("Pick a coordinate between 0 and 9 for the columns and rows on your board")
(Which as mentioned by Reinderien, should be a print statement.) What if you changed your board size?
Player-vs-Player or Player-vs-Computer
You've duplicated too much code. Player -vs- Player, Player -vs- Computer, and Computer -vs- Computer can all be handled by 1 "game" function.
def play_game(side1, side2):
side1.set_fleet()
side2.set_fleet()
while True:
side1.strike(side2)
if self.fleet_sunk(side2):
self.victory_message(side1, side2)
break
else:
side1, side2 = side2, side1
You games would be launched like:
p1 = Player(p1name)
p2 = Player(p2name)
play_game(p1, p2)
or
p1 = Player(p1name)
p2 = Computer()
play_game(p1, p2)
For this to work, Computer would need a set_fleet() method, instead of a set_compu_fleet() method, as well as a strike() method instead of a compu_strike() method.
f-strings
If using Python 3.6+, instead of:
print("%s's %s has been sunk!" % (self.name, boat.ship_type))
use:
print(f"{self.name}'s {boat.ship_type} has been sunk!")
Antipatterns
Avoid is True. For example, while flag is True:. Use while flag:
Avoid if condition: return True else: return False. Use, return condition. For example:
if ship_counters == 0:
return True
else:
return False
should be:
return ship_counters == 0
Avoid for idx in range(len(item)): blah(item[idx]). Use for val in item: blah(val).
For example:
ship_counters = 0
for row in range(len(player.ocean.ocean)):
for col in range(len(player.ocean.ocean)):
if player.ocean.ocean[row][col] == "S":
ship_counters += 1
Would be better written as:
ship_counters = 0
for ocean_row in player.ocean.ocean:
for cell in ocean_row:
if cell == "S":
ship_counters += 1
return ship_counters == 0
Or better:
ship_counters = sum(1 for row in player.ocean.ocean for cell in row if cell == "S")
return ship_counters == 0
Or even better:
return all(cell != 'S' for row in player.ocean.ocean for cell in row)
Docstrings
I disagree with @Reinderien's point:
This comment:
# Function uses player input to set up fleet positions on a player board..
# For each ship, a ship object containing relevant coordinates is appended to self.fleet.
belongs on the first line inside the function, in triple quotes.
The above is a comment, not a docstring.
• A # comment is used to document code for the reader of the code.
• A """docstring""" is used to provide usage documentation for a caller who may not have access to the source code.
class Player:
# Function uses player input to set up fleet positions on a player board.
# For each ship, a ship object containing relevant coordinates is appended to self.fleet
def set_fleet(self):
"""
Ask the user to assign locations to their fleet's ships.
Parameters: None
Returns: Nothing
"""
Now, type help(Player.set_fleet). You will see the doc-string help, which is designed to help the caller. From this description, the caller would know that they don't need to write fleet = player.set_fleet(), since nothing is returned.
In contrast, someone reading the code (maybe to debug it, modify it, enhance it) will want to know what the code does. Appending a ship object to self.fleet is useful internal details the caller may not need to know.
See also sphinx-doc.org for an automated docstring processing utility which can generate HTML doc pages, pdf help documents, etc., all by reading the docstrings directly from the code.
Type hints
PEP484 type hints will help document and test your code. For example,
def can_use_col(self, row, col, size):
can be (I guess)
def can_use_col(self, row: int, col: int, size: int) -> bool:
col = col + 1
can be
col += 1
Logic-by-exception
This:
try:
self.ocean[row]
return True
except IndexError:
return False
can simply be
return row in self.ocean
can_use_col
This method shouldn't need to form a list. Let's see if we can improve it.
First: replace i with _ since you don't use it. Also, instead of valid_coords as a list, you can maintain it as a count starting at 0. Increment it instead of appending to a list.
Also, don't repeat yourself - only do col += 1 once, in the scope above where you have it now.
Use a boolean directly by writing return size == len(valid_coords) instead of using an if statement. The same goes for this code:
if self.coords == []:
return True
else:
return False
Iteration
for i in range(size):
self.ocean[row][col] = "S"
row = row + 1
Again, you don't use i, but you're also maintaining an index in row. Instead:
for i in range(size):
self.ocean[row + i][col] = "S"
This also solves the issue that you're changing an argument to your function, which you generally shouldn't.
Docstrings
This comment:
# Function uses player input to set up fleet positions on a player board.
# For each ship, a ship object containing relevant coordinates is appended to self.fleet
belongs on the first line inside the function, in triple quotes.
Typo
"Boats are placed form right to left."
form = from.
Input prompts
input("Boats are placed form right to left.")
I'm not sure what this is asking the user to input, and they won't be either. Did you mean to write print here?
Character comparison
if orientation in ["v", "V"]:
If you actually needed to compare against multiple things, you should use a set instead of a list:
if orientation in {"v", "V"}:
However, your case is easier than that:
if orientation.lower() == 'v':
• Thanks man I really appreciate it. – user207830 Sep 24 '19 at 18:55
• return row in self.ocean will not work if the ocean is a list of lists. This can work if the ocean is a dict – Maarten Fabré Sep 25 '19 at 7:53
files
Python is not Java. Multiple classes can be in the same module (file)
grid
There is a lot of repetition between Ocean and Radar. You can use a lot of overlapping code
enum
instead of "." or "~" to denote what is in the grid, you could use Enums, and let the board representation method (view_radar for example) take care of how to represent this.
collection types
You only seem to use lists. Even when you only use the collection soo see whether something is in it, and order is not important. sets or dicts can be more suited on some moments. Get to know when to use which type of collection
orientation
You have a lot of methods that are the same, but one is for horizonatal, the other for vertical. This can be tackled by adding a parameter orientation (Which should be an Enum) to the method.
def plot_vertical(self, row, col):
for i in range(self.size):
self.coords.append((row, col))
row = row + 1
def plot_horizontal(self, row, col):
for i in range(self.size):
self.coords.append((row, col))
col = col + 1
can then be replaced by:
import enum
class Orientation(enum.Enum):
Horizontal = (1, 0)
Vertical = (0, 1)
def plot(self, row, col, orientation):
d_col, d_row = orientation.value
self.coords.extend(
(row + i * d_row, col + i * d_col) for i in range(self.size)
)
Even better would be to incorporate this in the ship's __init__, so the 2 steps in the placing of the fleet:
boat = Ship(ship, size)
boat.plot_vertical(row, col)
can become :
class Ship:
def __init__(self, ship_type, size, position, orientation):
self.ship_type = ship_type
self.size = size
row, col = position
d_col, d_row = orientation.value
self.coords = [(row + i * d_row, col + i * d_col) for i in range(self.size)]
and be called:
boat = Ship(ship, size, (row, col), orientation)
|
# Chapter 1 Lab 1: Graphing Data
The commonality between science and art is in trying to see profoundly - to develop strategies of seeing and showing. —Edward Tufte
As we have found out from the textbook and lecture, when we measure things, we get lots of numbers. Too many. Sometimes so many your head explodes just thinking about them. One of the most helpful things you can do to begin to make sense of these numbers, is to look at them in graphical form. Unfortunately, for sight-impaired individuals, graphical summary of data is much more well-developed than other forms of summarizing data for our human senses. Some researchers are developing auditory versions of visual graphs, a process called sonification, but we aren’t prepared to demonstrate that here. Instead, we will make charts, and plots, and things to look at, rather than the numbers themselves, mainly because these are tools that are easiest to get our hands on, they are the most developed, and they work really well for visual summary. If time permits, at some point I would like to come back here and do the same things with sonification. I think that would be really, really cool!
## 1.1 General Goals
Our general goals for this first lab are to get your feet wet, so to speak. We’ll do these things:
1. Load in some data to a statistical software program
2. Talk a little bit about how the data is structured
3. Make graphs of the data so we can look at it and make sense of it.
### 1.1.1 Important info
1. Data for NYC film permits was obtained from the NYC open data website. The .csv file can be found here: Film_Permits.csv
2. Gapminder data from the gapminder project (copied from the R gapminder library) can be downloaded in .csv format here: gapminder.csv
## 1.2 R
You will be completing each lab by writing your code and notes in an R Markdown document.
2. Unzip the file, this will produce a new folder with three important parts
1. data folder (contains data files for all labs)
2. LabTemplates folder (contains blank templates for completing all the labs)
3. RMarkdownsLab.Rproj A file with a little blue cube with an R in it.
3. Double-click the RMarkdownsLab.Rproj file, this will automatically open R-studio (if you are at home, you must install R and R-studio first, or you can use R-studio Cloud through your web-browser)
4. Copy the template .Rmd file for lab 1 from the LabTemplates folder into the main folder, then open it, and use it to begin adding your code and notes for lab 1.
Your lab instructor will show you how to open R-studio on the lab computer. Just find it and double-click. Now you have R-studio. Your lab instructor will also walk you through the steps to get started completing the first lab. We also wrote down the steps here.
There are numerous resources for learning about R, we put some of them on the course website, under the resouces page. You will find these resources helpful as you learn. We also have a kind of general introduction to R and Rstudio here. This shows you how to download R and R-studio at home (it’s free). Throughout the labs you will be writing things called R Markdown documents. You will learn how to do this throughout the labs, but it can also be worthwhile reading other tutorials, such as the one provided by R Markdown.
When we made this course, we assumed that most students would be unfamiliar with R and R-studio, and might even be frightened of it, because it is a computer programming language (OOOOHHH NOOOOOOO, I NEED TO DROP THIS COURSE NOW)…Don’t worry. It’s going to be way easier than you think. Let’s compare to other statistics courses where you would learn something like SPSS. That is also a limited programming language, but you would mostly learn how to point with a mouse, and click with button. I bet you already know how to do that. I bet you also already know how to copy and paste text, and press enter. That’s mostly what we’ll be doing to learn R. We will be doing statistics by typing commands, rather than by clicking buttons. However, lucky for you, all of the commands are already written for you. You just have to copy/paste them.
We know that this will seem challenging at first. But, we think that with lots of working examples, you will get the hang of it, and by the end of the course you will be able to do things you might never have dreamed you can do. It’s really a fantastic skill to learn, even if you aren’t planning on going on to do research in Psychology (in which case, this kind of thing is necessary skill to learn). With that, let’s begin.
### 1.2.2 Get some data
In order to graph data, we need to have some data first…Actually, with R, that’s not quite true. Run this bit of code and see what happens:
hist(rnorm(100, mean=50, sd=25))
You just made R sample 100 numbers, and then plot the results in a histogram. Pretty neat. We’ll be doing some of this later in the course, where get R to make fake data for us, and then we learn to think about how data behaves under different kinds of assumptions.
For now, let’s do something that might be a little bit more interesting…what movies are going to be filming in NYC? It turns out that NYC makes a lot of data about a lot things open and free for anyone to download and look at. This is the NYC Open Data website: https://opendata.cityofnewyork.us. I searched through the data, and found a data file that lists the locations of film permits for shooting movies all throughout the Burroughs. There are multiple ways to load this data into R.
1. If you have downloaded the RMarkdownsLab.zip file, then you already have the data file in the data folder. Assuming you are working in your main directory (your .rmd file is saved in the main folder that contains both the data and template folders), then use the following commands to load the data.
library(data.table)
nyc_films <-fread("data/Film_Permits.csv")
1. If the above method doesn’t work, you can try loading the data from the course website using:
library(data.table)
nyc_films <- fread("https://raw.githubusercontent.com/CrumpLab/statisticsLab/master/data/Film_Permits.csv")
If you are having issues getting the data loaded, then talk to your lab instructor
### 1.2.3 Look at the data
You will be downloading and analyzing all kinds of data files this semester. We will follow the very same steps every time. The steps are to load the data, then look at it. You want to see what you’ve got.
In R-studio, you will now see a variable called nyc_films in the top right-hand corner of the screen, in the environment tab. If you click this thing, it will show you the contents of the data in a new window. The data is stored in something we call a data frame. It’s R lingo, for the thing that contains the data. Notice is a square, with rows going across, and columns going up and down. It looks kind of like an excel spreadsheet if you are familiar with Excel.
It’s useful to know you can look at the data frame this way if you need to. But, this data frame is really big, it has 50,728 rows of data. That’s a lot too much to look at.
#### 1.2.3.1 summarytools
The summarytools packages give a quick way to summarize all of the data in a data frame. Here’s how. When you run this code you will see the summary in the viewer on the bottom right hand side. There’s a little browser button (arrow on top of little window) that you can click to expand and see the whole thing in a browser.
library(summarytools)
view(dfSummary(nyc_films))
That is super helpful, but it’s still a lot to look at. Because there is so much data here, it’s pretty much mind-boggling to start thinking about what to do with it.
### 1.2.4 Make Plots to answer questions
Let’s walk through a couple questions we might have about this data. We can see that there were 50,728 film permits made. We can also see that there are different columns telling us information about each of the film permits. For example, the Borough column lists the Borough for each request, whether it was made for: Manhattan, Brooklyn, Bronx, Queen’s, or Staten Island. Now we can ask our first question, and learn how to do some plotting in R.
#### 1.2.4.1 Where are the most film permits being requested?
Do you have any guesses? Is it Manhattan, or Brooklyn, of the Bronx? Or Queen’s or Staten Island? We can find out by plotting the data using a bar plot. We just need to count how many film permits are made in each borough, and then make different bars represent the the counts.
First, we do the counting in R. Run the following code.
library(dplyr)
counts <- nyc_films %>%
group_by(Borough) %>%
summarize(count_of_permits = length(Borough))
The above grouped the data by each of the five Borough’s, and then counted the number of times each Borough occurred (using the length function). The result is a new variable called count. I chose to name this variable count. You can see that it is now displayed in the top-right hand corned in the environment tab. If you gave count a different name, like muppets, then it would be named what you called it.
If you click on the counts variable, you will see the five boroughs listed, along with the counts for how many film permits were requested in each Borough. These are the numbers that we want to plot in a graph.
We do the plot using a fantastic package called ggplot2. It is very powerful once you get the hand of it, and when you do, you will be able to make all sorts of interesting graphs. Here’s the code to make the plot
library(ggplot2)
ggplot(counts, aes(x = Borough, y = count_of_permits )) +
geom_bar(stat="identity")
There it is, we’re done here! We can easily look at this graph, and answer our question. Most of the film permits were requested in Manhattan, followed by Brooklyn, then Queen’s, the Bronx, and finally Staten Island.
#### 1.2.4.2 What kind of “films” are being made, what is the category?
We think you might be skeptical of what you are doing here, copying and pasting things. Soon you’ll see just how fast you can do things by copying and pasting, and make a few little changes. Let’s quickly ask another question about what kinds of films are being made. The column Category, gives us some information about that. Let’s just copy paste the code we already made, and see what kinds of categories the films fall into. See if you can tell what I changed in the code to make this work, I’ll do it all at once:
counts <- nyc_films %>%
group_by(Category) %>%
summarize(count_of_permits = length(Category))
ggplot(counts, aes(x = Category, y = count_of_permits )) +
geom_bar(stat="identity")+
theme(axis.text.x = element_text(angle = 90, hjust = 1))
OK, so this figure might look a bit weird because the labels on the bottom are running into each other. We’ll fix that in a bit. First, let’s notice the changes.
1. I changed Borough to Category. That was the main thing
2. I left out a bunch of things from before. None of the library() commands are used again, and I didn’t re-run the very early code to get the data. R already has those things in it’s memory, so we don’t need to do that first. If you ever clear the memory of R, then you will need to reload those things. First-things come first.
Fine, so how do we fix the graph? Good question. To be honest, I don’t know right now. I totally forgot how. But, I know ggplot2 can do this, and I’m going to Google it, right now. Then I’m going to find the answer, and use it here. The googling of your questions is a fine way to learn. It’s what everybody does these days….[goes to Google…].
Found it, actually found a lot of ways to do this. The trick is to add the last line. I just copy-pasted it from the solution I found on stack overflow (you will become friend’s with stack overflow, there are many solutions there to all of your questions)
counts <- nyc_films %>%
group_by(Category) %>%
summarize(count_of_permits = length(Category))
ggplot(counts, aes(x = Category, y = count_of_permits )) +
geom_bar(stat="identity")+
theme(axis.text.x = element_text(angle = 90, hjust = 1))
### 1.2.5 ggplot2 basics
Before we go further, I want to point out some basic properties of ggplot2, just to give you a sense of how it is working. This will make more sense in a few weeks, so come back here to remind yourself. We’ll do just a bit a basics, and then move on to making more graphs, by copying and pasting.
The ggplot function uses layers. Layers you say? What are these layers? Well, it draws things from the bottom up. It lays down one layer of graphics, then you can keep adding on top, drawing more things. So the idea is something like: Layer 1 + Layer 2 + Layer 3, and so on. If you want Layer 3 to be Layer 2, then you just switch them in the code.
Here is a way of thinking about ggplot code
ggplot(name_of_data, aes(x = name_of_x_variable, y = name_of_y_variable)) +
geom_layer()+
geom_layer()+
geom_layer()
What I want you to focus on in the above description is the $$+$$ signs. What we are doing with the plus signs is adding layers to plot. The layers get added in the order that they are written. If you look back to our previous code, you will see we add a geom_bar layer, then we added another layer to change the rotation of the words on the x-axis. This is how it works.
BUT WAIT? How am I supposed to know what to add? This is nuts! We know. You’re not supposed to know just yet, how could you? We’ll give you lots of examples where you can copy and paste, and they will work. That’s how you’ll learn. If you really want to read the help manual you can do that too. It’s on the ggplot2 website. This will become useful after you already know what you are doing, before that, it will probably just seem very confusing. However, it is pretty neat to look and see all of the different things you can do, it’s very powerful.
For now, let’s the get the hang of adding things to the graph that let us change some stuff we might want to change. For example, how do you add a title? Or change the labels on the axes? Or add different colors, or change the font-size, or change the background? You can change all of these things by adding different lines to the existing code.
#### 1.2.5.1 ylab() changes y label
The last graph had count_of_permits as the label on the y-axis. That doesn’t look right. ggplot2 automatically took the label from the column, and made it be the name on the y-axis. We can change that by adding ylab("what we want"). We do this by adding a $$+$$ to the last line, then adding ylab()
ggplot(counts, aes(x = Category, y = count_of_permits )) +
geom_bar(stat="identity") +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits")
#### 1.2.5.2 xlab() changes x label
Let’s slightly modify the x label too:
ggplot(counts, aes(x = Category, y = count_of_permits )) +
geom_bar(stat="identity") +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film")
Let’s give our graph a title
ggplot(counts, aes(x = Category, y = count_of_permits )) +
geom_bar(stat="identity") +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category")
Let’s make the bars different colors. To do this, we add new code to the inside of the aes() part:
ggplot(counts, aes(x = Category, y = count_of_permits, color=Category )) +
geom_bar(stat="identity") +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category")
#### 1.2.5.5 fill fills in color
Let’s make the bars different colors. To do this, we add new code to the inside of the aes() part…Notice I’ve started using new lines to make the code more readable.
ggplot(counts, aes(x = Category, y = count_of_permits,
color=Category,
fill= Category )) +
geom_bar(stat="identity") +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category")
#### 1.2.5.6 get rid of the legend
Sometimes you just don’t want the legend on the side, to remove it add
theme(legend.position="none")
ggplot(counts, aes(x = Category, y = count_of_permits,
color=Category,
fill= Category )) +
geom_bar(stat="identity") +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category") +
theme(legend.position="none")
#### 1.2.5.7 theme_classic() makes white background
The rest is often just visual preference. For example, the graph above has this grey grid behind the bars. For a clean classic no nonsense look, use theme_classic() to take away the grid.
ggplot(counts, aes(x = Category, y = count_of_permits,
color=Category,
fill= Category )) +
geom_bar(stat="identity") +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category") +
theme(legend.position="none") +
theme_classic()
#### 1.2.5.8 Sometimes layer order matters
Interesting, theme_classic() is misbehaving a little bit. It looks like we have some of our layer out of order, let’s re-order. I just moved theme_classic() to just underneath the geom_bar() line. Now everything get’s drawn properly.
ggplot(counts, aes(x = Category, y = count_of_permits,
color=Category,
fill= Category )) +
geom_bar(stat="identity") +
theme_classic() +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category") +
theme(legend.position="none")
#### 1.2.5.9 Font-size
Changing font-size is often something you want to do. ggplot2 can do this in different ways. I suggest using the base_size option inside theme_classic(). You set one number for the largest font size in the graph, and everything else gets scaled to fit with that that first number. It’s really convenient. Look for the inside of theme_classic()
ggplot(counts, aes(x = Category, y = count_of_permits,
color=Category,
fill= Category )) +
geom_bar(stat="identity") +
theme_classic(base_size = 15) +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category") +
theme(legend.position="none")
or make things small… just to see what happens
ggplot(counts, aes(x = Category, y = count_of_permits,
color=Category,
fill= Category )) +
geom_bar(stat="identity") +
theme_classic(base_size = 10) +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category") +
theme(legend.position="none")
#### 1.2.5.10 ggplot2 summary
That’s enough of the ggplot2 basics for now. You will discover that many things are possible with ggplot2. It is amazing. We are going to get back to answering some questions about the data with graphs. But, now that we have built the code to make the graphs, all we need to do is copy-paste, and make a few small changes, and boom, we have our graph.
### 1.2.6 More questions about NYC films
#### 1.2.6.1 What are the sub-categories of films?
Notice the nyc_films data frame also has a column for SubCategoryName. Let’s see what’s going on there with a quick plot.
# get the counts (this is a comment it's just here for you to read)
counts <- nyc_films %>%
group_by(SubCategoryName) %>%
summarize(count_of_permits = length(SubCategoryName))
# make the plot
ggplot(counts, aes(x = SubCategoryName, y = count_of_permits,
color=SubCategoryName,
fill= SubCategoryName )) +
geom_bar(stat="identity") +
theme_classic(base_size = 10) +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Sub-category of film") +
ggtitle("Number of Film permits in NYC by Sub-category") +
theme(legend.position="none")
I guess “episodic series” are the most common. Using a graph like this gave us our answer super fast.
#### 1.2.6.2 Categories by different Boroughs
Let’s see one more really useful thing about ggplot2. It’s called facet_wrap(). It’s an ugly word, but you will see that it is very cool, and you can do next-level-super-hero graph styles with facet_wrap that other people can’t do very easily.
Here’s our question. We know that some films are made in different Boroughs, and that same films are made in different categories, but do different Boroughs have different patterns for the kinds of categories of films they request permits for? Are their more TV shows in Brooklyn? How do we find out? Watch, just like this:
# get the counts (this is a comment it's just here for you to read)
counts <- nyc_films %>%
group_by(Borough,Category) %>%
summarize(count_of_permits = length(Category))
# make the plot
ggplot(counts, aes(x = Category, y = count_of_permits,
color=Category,
fill= Category )) +
geom_bar(stat="identity") +
theme_classic(base_size = 10) +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Category of film") +
ggtitle("Number of Film permits in NYC by Category and Borough") +
theme(legend.position="none") +
facet_wrap(~Borough, ncol=3)
We did two important things. First we added Borough and Category into the group_by() function. This automatically gives separate counts for each category of film, for each Borough. Then we added facet_wrap(~Borough, ncol=3) to the end of the plot, and it automatically drew us 5 different bar graphs, one for each Borough! That was fast. Imagine doing that by hand.
The nice thing about this is we can switch things around if we want. For example, we could do it this way by switching the Category with Borough, and facet-wrapping by Category instead of Borough like we did above. Do what works for you.
ggplot(counts, aes(x = Borough, y = count_of_permits,
color=Borough,
fill= Borough )) +
geom_bar(stat="identity") +
theme_classic(base_size = 10) +
theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
ylab("Number of Film Permits") +
xlab("Borough") +
ggtitle("Number of Film permits in NYC by Category and Borough") +
theme(legend.position="none") +
facet_wrap(~Category, ncol=5)
### 1.2.7 Gapminder Data
https://www.gapminder.org is an organization that collects some really interesting worldwide data. They also make cool visualization tools for looking at the data. There are many neat examples, and they have visualization tools built right into their website that you can play around with https://www.gapminder.org/tools/. That’s fun check it out.
There is also an R package called gapminder. When you install this package, it loads in some of the data from gapminder, so we can play with it in R.
If you don’t have the gapminder package installed, you can install it by running this code
install.packages("gapminder")
Once the package is installed, you need to load the new library, like this. Then, you can put the gapminder data into a data frame, like we do here: gapminder_df.
library(gapminder)
gapminder_df<-gapminder
#### 1.2.7.1 Look at the data frame
You can look at the data frame to see what is in it, and you can use summarytools again to view a summary of the data.
view(dfSummary(gapminder_df))
There are 1704 rows of data, and we see some columns for country, continent, year, life expectancy, population, and GDP per capita.
### 1.2.8 Asking Questions with the gap minder data
We will show you how to graph some the data to answer a few different kinds of questions. Then you will form your own questions, and see if you can answer them with ggplot2 yourself. All you will need to do is copy and paste the following examples, and change them up a little bit
#### 1.2.8.1 Life Expectancy histogram
How long are people living all around the world according to this data set? There are many ways we could plot the data to find out. The first way is a histogram. We have many numbers for life expectancy in the column lifeExp. This is a big sample, full of numbers for 142 countries across many years. It’s easy to make a histogram in ggplot to view the distribution:
ggplot(gapminder_df, aes(x=lifeExp))+
geom_histogram(color="white")
See, that was easy. Next, is a code block that adds more layers and settings if you wanted to modify parts of the graph:
ggplot(gapminder_df, aes(x = lifeExp)) +
geom_histogram(color="white")+
theme_classic(base_size = 15) +
ylab("Frequency count") +
xlab("Life Expectancy") +
ggtitle("Histogram of Life Expectancy from Gapminder")
The histogram shows a wide range of life expectancies, from below 40 to just over 80. Histograms are useful, they can show you what kinds of values happen more often than others.
One final thing about histograms in ggplot. You may want to change the bin size. That controls how wide or narrow, or the number of bars (how they split across the range), in the histogram. You need to set the bins= option in geom_histogram().
ggplot(gapminder_df, aes(x = lifeExp)) +
geom_histogram(color="white", bins=50)+
theme_classic(base_size = 15) +
ylab("Frequency count") +
xlab("Life Expectancy") +
ggtitle("Histogram of Life Expectancy from Gapminder")
See, same basic patter, but now breaking up the range into 50 little equal sized bins, rather than 30, which is the default. You get to choose what you want to do.
#### 1.2.8.2 Life Expectancy by year Scatterplot
We can see we have data for life expectancy and different years. So, does worldwide life expectancy change across the years in the data set? As we go into the future, are people living longer?
Let’s look at this using a scatter plot. We can set the x-axis to be year, and the y-axis to be life expectancy. Then we can use geom_point() to display a whole bunch of dots, and then look at them. Here’s the simple code:
ggplot(gapminder_df, aes(y= lifeExp, x= year))+
geom_point()
Whoa, that’s a lot of dots! Remember that each country is measured each year. So, the bands of dots you see, show the life expectancies for the whole range of countries within each year of the database. There is a big spread inside each year. But, on the whole it looks like groups of dots slowly go up over years.
#### 1.2.8.3 One country, life expectancy by year
I’m (Matt) from Canada, so maybe I want to know if life expectancy for Canadians is going up over the years. To find out the answer for one country, we first need to split the full data set, into another smaller data set that only contains data for Canada. In other words, we want only the rows where the word “Canada” is found in the country column. We will use the filter function from dplyr for this:
# filter rows to contain Canada
smaller_df <- gapminder_df %>%
# plot the new data contained in smaller_df
ggplot(smaller_df, aes(y= lifeExp, x= year))+
geom_point()
I would say things are looking good for Canadians, their life expectancy is going up over the years!
#### 1.2.8.4 Multiple countries scatterplot
What if we want to look at a few countries altogether. We can do this too. We just change how we filter the data so more than one country is allowed, then we plot the data. We will also add some nicer color options and make the plot look pretty. First, the simple code:
# filter rows to contain countries of choice
smaller_df <- gapminder_df %>%
# plot the new data contained in smaller_df
ggplot(smaller_df, aes(y= lifeExp, x= year, group= country))+
geom_point()
Nice, we can now see three sets of dots, but which are countries do they represent? Let’s add a legend, and make the graph better looking.
ggplot(smaller_df,aes(y= lifeExp, x= year,
group= country, color = country)) +
geom_point()+
theme_classic(base_size = 15) +
ylab("Life Expectancy") +
xlab("Year") +
ggtitle("Life expectancy by year for three countries")
#### 1.2.8.5 geom_line() connecting the dots
We might also want to connect the dots with a line, to make it easier to see the connection! Remember, ggplot2 draws layers on top of layers. So, we add in a new geom_line() layer.
ggplot(smaller_df,aes(y= lifeExp, x= year,
group= country, color = country)) +
geom_point()+
geom_line()+
theme_classic(base_size = 15) +
ylab("Life Expectancy") +
xlab("Year") +
ggtitle("Life expectancy by year for three countries")
### 1.2.9 Generalization Exercise
The following generalization exercise and writing assignment is also in your lab R Markdown document for this lab. Complete your work in that document and hand it in.
(1 point - Pass/Fail)
Use the code from above to attempt to solve the extra things we ask you do for this assignment. You generalization exercises are as follows:
1. Make a graph plotting Life Expectancy by year for the five continents, using the continent factor. Make sure you change the title so it reads correctly
2. Make a graph plotting GDP per capita by year for the USA, Canada, and Mexico. Use the gdpPercap column for the GDP per capita data
3. Make a new graph plotting anything you are interested in using the gapminder dataset. It just needs to be a plot that we have not given an example for
### 1.2.10 Writing assignment
Complete the writing assignment described in your R Markdown document for this lab. When you have finished everything. Knit the document and hand in your stuff (you can submit your .RMD file to blackboard if it does not knit.)
The question for this lab is a long answer question about histograms. Here is the question:
Describe what histograms are, how to interpret them, and what they are useful for. You should answer each of these questions:
The answers to each of these questions are worth .25 points each, for a total of 2 points
1. What do the bars on a histogram represent?
2. How many bars can a histogram have?
3. What do the heights of the bars tell you
4. What is on the x-axis and y-axis of a histogram
5. What does the tallest bar on a histogram tell you?
6. What does the shortest bar on a histogram tell you?
7. What are some uses for histograms, why would you want to look at a histogram of some numbers that you collected?
8. Imagine you had two histograms, one was very wide and spread out, the other was very narrow with a very tall peak. Which histogram would you expect to contain more consistent numbers (numbers that are close to each other), explain why.
Rubric
• You will receive 0 points for missing answers (say, if you do not answer question c, then you will receive 0 out .25 points for that question)
• You must write in complete sentences. Point form sentences will be given 0 points.
• Completely incorrect answers will receive 0 points. For example, if you incorrectly describe what the x and y-axes refer to, then you will receive 0 points for that question.
• If your answer is generally correct but very difficult to understand and unclear you may receive half points for the question
## 1.4 SPSS
In this lab, we will get you acquainted with the SPSS software layout and graph some sample data to make sense of it. We will be doing the following:
1. Opening SPSS and the SPSS layout
2. Reviewing variable properties and the Variable View tab
3. Opening a data file and producing different types of graphs
### 1.4.1 Opening SPSS and the SPSS layout
Your lab instructor will take you through the process of opening the SPSS program. You may double-click on its icon located on the desktop of your lab computer, or you may find it using the Start menu. Once the program loads, you will be prompted with a pop-up window that asks you which file you would like to open. For now, we will be examining the basic layout of SPSS without a data set, so you can click Cancel.
Once you do, the main SPSS spreadsheet should open. It will look like this, a basic spreadsheet:
Notice at the bottom of your window there are two tabs; “Data View” and “Variable View”. In data view, we enter data into our spreadsheet. You will notice that rows are numbered on the left-hand side of the spreadsheet, while columns are labeled “var”. This is an indication of the general structure of SPSS: Variables are contained in the columns, and rows indicate individual observations. For example, if you obtained the heights (in inches) of 5 people {x= 64, 70, 63, 62, 65} and wanted to enter their data into SPSS, each person’s height would be entered in a new row, not across the columns, as seen below:
### 1.4.2 Reviewing variable properties and the Variable View tab
Now that we have some data entered, we might want to name our variable so that it’s evident our measurements represent heights. In order o view or modify variable names and other properties, look to the bottom of your SPSS window and switch over to the “Data View” tab. Once you do this, your window will appear as follows:
Here, you can edit the name of your variables, and specify their properties. Variable names can be anything you like, with the restriction that you cannot use numbers or spaces. Next, notice several other important properties of variables you may at some point need to set or modify:
• Name: the name of your variable that will appear as a colum header in Data View. No spaces or numerals.
• Type: Your data will most often be Numeric, but sometimes, as in data representing currency or data in scientific notation, you may change the data type appropriately. If your data is simply a label, word, or response (such as an open-ended response to a survey question), choose “String”: this tells SPSS not to treat this variable as a number. (Nota bene: if you select the wrong type of variable, SPSS may not be able to process your requested calculations, so always remember to check this parameter!)
• Width: This refers to how many digits will be visible by default.
• Decimals: This refers to how many decimal places will be visible by default.
• Label: This is a description of the variable. Any information too long to be included in the variable name goes here.
• Values: For nominal scale data, let’s say 1 represents male and 2 represents female, this is where you include the values and their corresponding labels.
• Measure: This variable property allows you to specify the nature of your data. Depending on the kind of scale you are using, you will choose a different measure type. Nominal and ordinal are chosen for nominal and ordinal scales, respectively. “Scale” is used when your data is measured on a ratio or interval scale. (Nota bene: this “Measure” designation is only a marker; it does not affect the calculations as in the “Type” parameter. Even if you choose the wrong icon/label for “Measure”, SPSS will still produce the correct output)
### 1.4.3 Opening a data file and producing different types of graphs
Now that we know about the properties of the SPSS spreadsheet window, let’s open a data file and learn how to make some sense of it by creating different types of graphs. Here is a link to an SPSS-based data file containing information about film permits (requests made by film companies to shoot TV shows and movies on location) filed in New York City. The file is named nyc_films.sav.
Once you open the data file, browse through to familiarize yourself with the variables that are being measured. Switch over to Variable View for details of each variable.
#### 1.4.3.1 Bar Graphs
Now, back to Data View. We will not be working with every single variable in this spreadsheet, but we’ll select a few interesting ones with which to answer questions. Let’s start with borough. Suppose we wanted to know which borough receives the most film permits (you can probably guess which one is most popular). Let’s use SPSS to produce a graph to answer this question. With your data file open, go up to the top menu and choose Graphs, then Legacy Dialogs. You will see an entire list of possible graphs we can use to plot our data.
Let’s think about the nature of our question: we would like to know how many permits were filed for each borough. Borough is simply a label or a name for a region, and we want to know the frequency of permits for each borough. This is a nominal scale variable and so, we will appropriately choose a BAR graph to plot it. Select Bar…
The next window will ask you to specify what kind of graph you would like. Select Simple and then Define. The following window will ask which variable you’d like to plot. Select borough from the left-hand list and use the arrow to move it into the field labeled “Category Axis”. Then click OK.
SPSS will produce a new output window which will contain the bar graph you have generated. Notice which borough receives the most film permits. Are you surprised?
#### 1.4.3.2 Histograms
Now, let’s use a different data set to plot a histogram. The defining difference between a histogram and a bar graph (although they look very similar as they both utilize bars) is that a histogram is used to display a continuous variable (interval or ratio scale). In the previous example, boroughs were simply labels or names, so we used a nominal scale and therefore a bar graph. Here, we will deal with life expectancy (measured in years), an interval scale measure. Here is a link to the SPSS data file, life_expectancy.sav. Open this file and examine its rows and columns. Each column represents a year during which life expectancy was measured. Each row represents a different country.
Let’s first get an idea about life expectancy in general. We want to plot a histogram with life expectancy on the x-axis and frequency on the y-axis. Choose Graphs in the top menu, then Legacy Dialogs. From here, remember we want a histogram, not a bar graph, so let’s select Histogram….
The window that appears contains every variable in your spreadsheet listed on the left-hand side. We can choose one variable at a time to plot. Let’s scroll all the way down the list and choose 2017 [v219]. This is the variable containing life expectancies for the year 2017. Using the arrow, move that variable into the field labeled “Variable:”, then click OK.
SPSS will produce an output window containing the distribution of life expectancy for the year 2017.
#### 1.4.3.3 Scatterplots
Now, we will look to a different type of data plot; the scatterplot. A scatterplot allows us to visualize bivariate data, that is, data for which there are two measurements per individual. For example, we may ask whether life expectancy in a country (or how long you live, on average) is related to the average income. Using the life_expectancy.sav data file, let’s plot both variables: 2017 [v219] and income. The income variable in the spreadsheet refers to data collected in 2017 by the Better Life Initiative. Notice not all the countries listed have estimates for average annual income. For those that do, this value represents household net adjusted income (annual) in US dollars.
To create the scatterplot, let’s go to Graphs in the menu toolbar, then Legacy Dialogs, then Scatter.
You will choose Simple scatter, then click Define.
Next, indicate which variables (there are 2 this time!) you would like in the x- and y-axes. Use the arrows to place income in the x-axis field, and 2017 (V219) in the y-axis field. (For the purposes of graphing a scatterplot, it does not matter which variable goes into the y-axis and x-axis fields for now; you can reverse them if you’d like and you can still interpret the data similarly)
Then click OK. SPSS will produce output containing a scatterplot. What relationship do you notice? What happens to life expectancy the more individuals earn, on average?
### 1.4.4 Practice Problems
1. Create a histogram for life expectancy in the year 1800. Describe the distribution. How does it differ from the one we plotted for 2017?
2. Plot the life expectancy of each country in 1800 vs. that of 2018. What does this graph show you? What are your conclusions regarding the development of nations?
|
ArticlePDF Available
# A rapid sample-exchange mechanism for cryogen-free dilution refrigerators compatible with multiple high-frequency signal connections
Authors:
## Abstract and Figures
Researchers attempting to study quantum effects in the solid-state have a need to characterise samples at very low-temperatures, and frequently in high magnetic fields. Often coupled with this extreme environment is the requirement for high-frequency signalling to the sample for electrical control or measurements. Cryogen-free dilution refrigerators allow the necessary wiring to be installed to the sample more easily than their wet counterparts, but the limited cooling power of the closed cycle coolers used in these systems means that the experimental turn-around time can be longer. Here we shall describe a sample loading arrangement that can be coupled with a cryogen-free refrigerator and that allows samples to be loaded from room temperature in a matter of minutes. The loaded sample is then cooled to temperatures ∼ 10 mK in ∼ 7 hours. This apparatus is compatible with systems incorporating superconducting magnets and allows multiple high-frequency lines to be connected to the cold sample.
Content may be subject to copyright.
Accepted Manuscript
A rapid sample-exchange mechanism for cryogen-free dilution refrigerators
compatible with multiple high-frequency signal connections
G. Batey, S. Chappell, M.N. Cuthbert, M. Erfani, A.J. Matthews, G. Teleberg
PII: S0011-2275(14)00015-0
DOI: http://dx.doi.org/10.1016/j.cryogenics.2014.01.007
Reference: JCRY 2295
To appear in:
Cryogenics
Revised Date: 13 January 2014
Accepted Date: 15 January 2014
Please cite this article as: Batey, G., Chappell, S., Cuthbert, M.N., Erfani, M., Matthews, A.J., Teleberg, G., A rapid
sample-exchange mechanism for cryogen-free dilution refrigerators compatible with multiple high-frequency signal
connections, Cryogenics (2014), doi: http://dx.doi.org/10.1016/j.cryogenics.2014.01.007
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers
we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and
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A rapid sample-exchange mechanism for cryogen-free
dilution refrigerators compatible with multiple
high-frequency signal connections
G Batey, S. Chappell, M.N. Cuthbert, M. Erfani, A.J. Matthews
1,
, G.
Teleberg
Oxford Instruments Omicron NanoScience, Tubney Woods, Abingdon, Oxfordshire,
OX13 5QX, UK
Abstract
Researchers attempting to study quantum effects in the solid-state have a need
to characterise samples at very low-temperatures, and frequently in high mag-
netic fields. Often coupled with this extreme environment is the requirement for
high-frequency signalling to the sample for electrical control or measurements.
Cryogen-free dilution refrigerators allow the necessary wiring to be installed to
the sample more easily than their wet counterparts, but the limited cooling
power of the closed cycle coolers used in these systems means that the experi-
mental turn-around time can be longer. Here we shall describe a sample loading
arrangement that can be coupled with a cryogen-free refrigerator and that al-
lows samples to be loaded from room temperature in a matter of minutes. The
loaded sample is then cooled to temperatures 10 mK in 7 hours. This
apparatus is compatible with systems incorporating superconducting magnets
and allows multiple high-frequency lines to be connected to the cold sample.
Keywords: Dilution refrigerator, Sample exchange, Cryogen-free
1. Introduction
Over the past century studying condensed matter systems at extremely low
temperatures, and often in extremely high magnetic fields, has lead to the dis-
covery of several new states of matter, such as: superconductivity in mercury [1];
superfluidity in
4
He [2, 3]; superfluidity in
3
He [4]; the integer quantum Hall
effect in silicon MOSFET devices [5]; the fractional quantum Hall effect in
GaAs-AlGaAs heterojunctions [6].
More recently there has been a drive to harness these quantum systems to
realise devices that exploit their quantum nature, for example in the field of
Corresponding author
Preprint submitted to Elsevier January 21, 2014
quantum information processing [7], with the realisation of a general quantum
computer [8] being the holy grail. Inevitably the development of these quantum
devices requires temperatures < 10 mK, and possibly magnetic fields > 10 T,
however in addition to these environmental constraints device characterisation
and development also requires the necessary experimental services be installed
at the sample position: most challengingly high-bandwidth, high-fidelity micro-
wave cabling.
In the following sections we describe briefly a suitable experimental envi-
ronment for quantum device development (or any other experiments requiring
high-frequency measurements at low-temperatures), then we show that device
scribe its realisation, op eration and performance, before providing a brief con-
clusion.
2. Experimental Environment
Pulse-tube precooled dilution refrigerators [9] are becoming increasingly pop-
ular. Initially this popularity stemmed from the fact that they were cryogen-free,
meaning that they could be installed at institutions without the associated low-
temperature research infrastructure, such as a helium liquefaction plant, or in
remote locations. Additionally, there are benefits from an operational point of
view as such systems can be automated to a higher degree than their “wet”
counterparts. It has also been found that these cryogen-free systems have fur-
ther benefits when compared to wet systems with regards to the installation of
experimental services, as will be discussed in the following sections, and this has
driven the recent rise in their uptake.
With the installation of high-frequency wiring these refrigerators have been
developed into measurement systems for circuit quantum electrodynamics [10]
and superconducting qubits [11]. The integration of superconducting mag-
nets [12], with the entire system able to be run from a single pulse-tube cooler,
has enabled a wider range of experiments (those requiring magnetic fields) to
be performed using this cryogen-free technology [13].
2.1. Low-Temperatures and High Magnetic Fields
Cryogenic systems using liquid helium are usually designed to minimise its
consumption. This is because liquid helium is expensive, refilling the system
can be time consuming, and refilling the system may perturb the experiment to
an unacceptable level. The central neck of a cryostat is often responsible for the
biggest single heat load into the helium bath, and as a result these necks are
usually made as long and as narrow as possible. Dilution refrigerators designed
to be inserted into such a cryostat have to inherit this aspect ratio, which has
tended to limit the experimental real estate available for the installation of
services.
With no boil-off considerations, cryogen-free systems have evolved to be
much wider than their wet counterparts with experimental plates (to which ser-
vices can be mounted) typically several hundred mm in diameter [12]. This has
2
enabled more and / or more complex services to be installed on dilution refriger-
ator systems, in particular bulky signal conditioning elements such as cryogenic
amplifiers, microwave components (bias-tees, circulators, switches etc.) and fil-
tering (such as metal powder filters, for example [14] and the references therein).
Cryogen-free systems can also be designed without the need for a low-
temperature, vacuum-tight vessel, the so called inner vacuum chamber (IVC),
which makes the routing and heat-sinking of the installed services much more
straightforward, see section 2.2.2.
The range of magnets that are able to be produced for cryogen-free operation
is also continually expanding with higher fields (> 16 T) and vector-rotation
(> 6-1-1 T) available.
For these reasons cryogen-free dilution refrigerators with integrated magnets
have become the workhorse of quantum device development laboratories around
the world.
2.2. High-Frequency Wiring
As was noted in section 1 high-fidelity, high-bandwidth wiring is an experi-
mental requirement for quantum device development applications. In addition
to the quality of the signal transmission performance of these cables, they also
need to be thermally anchored adequately to ensure that they do not affect
adversely the base temperature performance of the system onto which they are
installed. In this section we shall: review various options for the coaxial lines and
some of the materials available for the lines themselves, and discuss their rela-
tive merits; describe a convenient method for mounting multiple high-frequency
lines onto a dilution refrigerator; quantify the frequency dependence of signal
transmission of installed lines with S
12
measurements made with a vector net-
work analyser; comment on the heat load to the mixing chamber likely to result
from the installation of the type of wiring described.
2.2.1. Coaxial cables and materials
To date, most high-frequency cabling installed in dilution refrigerators have
been of “semi-rigid” construction with the UT-85 cable (having an outer diam-
eter of 85/1000 of an inch, approximately 2.16 mm) being commonly used. The
optimal choice of coaxial cable, in terms of both size and material, depends on
its intended application. Typically coaxial cables are used to 1) improve noise
immunity for “small” signals and / or 2) transmit high-frequency signals to /
from the sample.
If using coaxial cables for either of these reasons one should ensure that the
cables themselves are suitable for the intended application. For dilution refrig-
erator based experiments, this suitability is generally determined by two key
parameters: the heat load to the experiment due to the thermal conductivity of
the cable; and its (frequency dependent) attenuation. Both of these parameters
are affected by the choice of the cable geometry (size) and conductor materials.
The heat load conducted to the coldest parts of the dilution refrigerator is al-
ways to be minimised. For a given choice of coaxial cable material and geometry
3
there is a lower limit to this heat load determined by the bulk thermal conduc-
tivity of the cable materials. This limit is approached as the cable (both the
inner and outer conductor) is p erfectly thermally connected to every available
temperature stage in the refrigerator, of course the conducted heat load can be
much higher than this limit if the thermal connections are inadequate. A con-
venient method of installing semi-rigid coaxial cables into a dilution refrigerator
that gives good thermal performance is discussed in section 2.2.4. The heat load
can only be reduced further by using either cables with a smaller cross sectional
area and / or cables made from materials with a lower thermal conductivity,
however such changes may well have implications for the cable attenuation.
The frequency dependent attenuation of a coaxial cable is determined by
the cable geometry (outer diameter of the inner conductor and inner diame-
ter out of the outer conductor), the (temperature and frequency dependent)
resistivity of the conductor materials and the dielectric losses [15]. In general
smaller diameter cables have higher attenuation at high frequencies than larger
diameter ones, and cables manufactured from materials with higher bulk resis-
tivity have higher attenuation (at a given frequency) than low resistance ones.
Depending on the application, this increase in attenuation can be fortuitous or
problematic. In applications where coaxial cables are used for noise immunity
for small, low-frequency signals, having increased attenuation at high frequen-
cies is advantageous: in fact “lossy” coax cables have been used as microwave
filters [16].
However, for high-bandwidth signals the change in attenuation, α, with fre-
quency, f, is undesirable as it results in the “shape” of signals (in the time-
domain) being modified as they propagate along the cable and this can cause
problems with, for example, high-fidelity qubit control. Techniques borrowed
from the NMR / MRI world for pulse preshaping using a posteriori knowledge
of the cabling transfer function [17] can be applied to compensate for this ef-
fect, but it would still be advantageous to keep the frequency response of the
cable as flat as possible. Using (lots of) large-diameter low-resistance cables can
be incompatible with experiments at dilution refrigerator temperatures, as the
thermal and electrical conductivity of a normal metal are closely related [18].
However, superconducting cables made from Nb, or preferably NbTi (due to its
higher critical field and temperature, and lower thermal conductivity), can be
used. Below their superconducting transition temperature these cables provide
very low attenuation and have a small thermal conductivity [15] so in many
cases are the ideal solution to this problem. However, with cryogen free dilution
refrigerators enabling experiments over extended temperature ranges [12] some
care needs to be taken, as the electrical performance of these lines will change
(attenuation will increase) dramatically above their transition temperature.
One final point is that the desire to keep
df
0 is not the same as keeping
α 0. Indeed, the types of cables described here are very good at transmitting
“thermal noise” from warmer parts of the refrigerator to colder ones, equating
K
B
T gives a photon frequency of 20 GHz at 1 K and UT-85 cables
operational range can extend to > 60 GHz [19], and so having some attenuation
in the line is desirable to reduce these thermal perturbations. Attenuators with
4
a flat frequency response, compatible with cryogenic temperatures [20], can be
used to increase the attenuation of a line whilst avoiding the complications of
distorting high-bandwidth signals. Details of measurements of such lines will
be given in section 2.2.3.
2.2.2. High-frequency wiring cartridges
As described in section 2.2.1, for some experiments small diameter coaxial
cables with high attenuation at high-frequencies can be appropriate. For ex-
ample, UT-13 cables have an outer diameter of approximately 330 µm and can
be installed and thermally anchored like flexible “DC” wiring. In this section
we focus on semi-rigid cables and describe a convenient method of installing
multiple, configurable, semi-rigid coaxial lines into a dilution refrigerator in a
way that gives good electrical and thermal performance and allows for the cable
assemblies to be rapidly demounted and modified if necessary.
Cryogen-free dilution refrigerators typically have several large (40 - 100 mm
diameter) line-of-sight (LoS) ports that allow connections between the room
temperature top-plate and the mixing chamber plate. Whilst traditional wet
dilution refrigerators also often feature LoS ports they tend to be less numerous
and of smaller diameter. Wet systems also require an IVC and so services
need to be installed in vacuum tubes from room temperature to 4 K, making
the thermal anchoring of the installed services more difficult (services can of
course be thermalised by bringing them through the main helium bath, but
then cryogenically compatible, hermetically sealed feed-throughs are required
to bring the services into the IVC).
A typical cryogen-free refrigerator will have experimental plates that can
be used to thermally anchor wiring at temperatures of approximately 50 K,
3 K, 0.8 K, 100 mK and the mixing chamber at around 10 mK. The wiring
cartridge shown in figure 1 has anchoring plates at each temperature stage.
It has been found that the use of bulkhead connectors at each of these plates
is an effective way to thermalise both the inner and outer conductors of the
cables [21], and provides a convenient mounting point for any attenuators that
may be added to the lines at any of the stages. The cartridge is designed to be
able to be loaded into the system either completely assembled from the top, or
without the top plate fittings from below. Once the cartridge is installed, split
clamp plates are used to make thermal contact between the cartridge and the
refrigerator. The fact that the entire wiring assembly can be removed in one
piece allows for bench testing of the microwave lines prior to installing them
into the system. It also means, for example, that should there be a desire
to change installed attenuators for ones with a different attenuation value the
assembly can be removed from the refrigerator by simply opening one room
temperature o-ring seal and loosening the clamping bolts. With the assembly
removed, the microwave lines or attenuators, between the bulkhead connectors,
can be reconfigured and tested before being refitted to the system.
5
Figure 1: A wiring cartridge for a cryogen-free dilution refrigerator: (a) shows a fully assembled
cartridge with hermetic feed-throughs on the room temperature top plate and additional
attenuators installed above and below some of the thermal stages. (b) shows the detail of
a split clamp used to thermally anchor the cartridge to the refrigerator and the bulkhead
connectors through the cartridge plate. (c) shows how such a section of such a cartridge could
be installed through a line-of-sight port of a dilution refrigerator.
6
Figure 2: Scattering parameter measurements on coaxial cables installed in wiring cartridges.
The red and black traces show lines with no additional attenuation installed. The green and
blue traces show lines with an additional 28 dB of in-line attenuation. The reduction in the
attenuation between room temperature and the system being cooled is due principally to
sections of superconducting coaxial cable cooling below their transition temperature. The dip
in the attenuation on the blue trace (circled) was due to a lo ose connector in the cartridge
assembly, this was corrected prior to the cartridge being installed into the system and cooled.
2.2.3. Transmission measurements
The microwave performance of installed coaxial cable assemblies has been
measured with an Antitsu model MS2028C/2 vector network analyser [22] which
recorded the scattering parameters at frequencies up to 12 GHz. Typical curves
between 5 kHz and 8 GHz are shown in figure 2. The S
12
parameter can
be associated with the total attenuation in the line and the measured values
agree well with cable manufactures’ data for expected values of the frequency
dependent attenuation (per unit length) of the cables they produce [19, 23], in
this case the coaxial cable sections themselves were silver-plated stainless steel
inner conductor, stainless steel outer conductor from room temperature to the
4 K plate, and NbTi inner and outer from 4 K to the mixing chamber. Faults
with the coaxial cables, such as loose connectors or cracked solder joints, can
be identified from scattering parameter measurements [24], and for the cables
installed on these systems typically result in additional attenuation (reflection)
features at frequencies of a few GHz, figure 2.
2.2.4. Thermal performance
As discussed in section 2.2.1 there is a lower limit to how far the heat load
from installed cabling can be reduced. Here we examine the residual heat load
onto the mixing chamber of a cryogen-free dilution refrigerator when wiring
7
Figure 3: (a) An image showing how multiple coaxial cable cartridges can be installed onto
a dilution refrigerator system. In-line attenuators are visible below the upper plate, which
is at the position of the dilution refrigerator still. (b) A plot of the typical cooling capacity
available at the mixing chamb er of such a dilution refrigerator. The fit in the plot is of the
form y = ax
2
b.
cartridges are installed. This heat load can be extracted using knowledge of
the cooling power of the dilution refrigerator at the temperatures of interest.
With knowledge of the cooling power, the base temperature of a refrigerator
with and without wiring installed can be compared and the heat leak extracted
from the difference in these temperatures. For these measurements three wiring
cartridges, each containing eight UT-85 coaxial lines (24 lines in total) manufac-
tured from cupronickel conductors, were installed onto a Triton200 [25] dilution
refrigerator system, as show in figure 3. After the addition of the wiring car-
tridges the temperature of the plate mounted at the end of the continuous heat
exchanger, colloquially know as the “100 mK plate”, had increased from 65
mK to 120 mK as measured with a resistive temperature sensor [26]. The base
temperature of the dilution refrigerator, measured using a nuclear orientation
thermometer [27], was found to have risen to 9.1 mK, corresponding to an in-
creased heat load of 600 nW. Extrapolating available data for the thermal
conductivity of cupronickel [28] to 100 mK and calculating the anticipated heat
load conducted through 24 UT-85 coaxial lines with the geometry defined by
manufactures [23] accounts for 200 nW of this increase, with a further ad-
ditional 300 nW expected through the stainless steel refrigerator support
structure, calculated using published values for the thermal conductivity [29],
due to the increase in temperature of the 100 mK plate.
3. Rapid Sample Exchange
In section 2 it was shown that cryogen-free dilution refrigerators integrated
with superconducting magnets provide an ideal environment for quantum de-
vice development experiments due to their ease of use and the convenience of
installing experimental services. These systems do, however, have one signifi-
cant drawback compared to their wet counterparts as the integrated supercon-
8
ducting magnets become larger: the experimental turnaround time. High-field
cryogen-free magnets can have masses well in excess of 50 kg and require en-
thalpy changes of several MJ to cool from room temperature to 4 K. The pulse
tube coolers used in these systems typically have cooling powers at the second
stage of 140 W at room temp erature, falling to 1 W at 4 K [30]. This
limited cooling power means that the initial cool down from ro om temperature
can take much longer than wet systems (which can be cooled quite quickly if
the cryogen boil-off can be tolerated).
This drawback can be overcome with a method of exchanging samples that
keeps the rest of the system (and the magnet in particular) cold. To b e useful
for a wide range of experiments, such a mechanism must provide multiple high-
frequency lines to the sample and provide sample temperatures comparable to
the base temperature of the refrigerator. In the following sections we describe
the design, construction and performance of such a sample exchange mechanism.
3.1. The sample exchange concept
Attaching a sample and experimental wiring directly to a probe and loading
the entire assembly into a dilution refrigerator has been attempted, but it was
found that the resulting thermal performance and limited space is incompat-
ible with multiple high-frequency lines and additional microwave components
(amplifies, filters etc), see section II A of [31].
An alternative approach to sample loading, as also implemented in [31], is
to leave the experimental wiring on the refrigerator, where it can be efficiently
thermally anchored, in this case by using the wiring cartridge design discussed
in section 2.2, and to load a “sample holder” to connect to this installed wiring.
Additionally, this means that the full sample space of the refrigerator can be
utilised to install other components into the experimental wiring circuits which
may not fit onto a smaller diameter probe. Loading only a sample holder into
the refrigerator introduces the complication of requiring demountable microwave
connectors, but in the following sections we show that this requirement can
be fulfilled. It is often also desirable to be able to bias or ground electrical
connections to delicate samples during the cool down process to prevent, for
example, electrostatic sample damage. This is accomplished with a “make-
before-break” arrangement whereby all DC and microwave connections to the
sample holder are individually connected to room temperature connectors on
the loading probe. As will discussed in section 3.1.2 the sample holder can
sample is attached to the refrigerator.
With a cryogen-free system without an IVC there is no preferred direction for
sample loading. Samples can either be introduced from the top of the system
load-lock (BLLL). TLLLs require a (central) LoS access port through which the
sample can be introduced, and BLLLs require access through the vacuum and
radiation shields through which the sample can pass. The distance from the
refrigerator top-plate to the magnetic field centre-line is normally longer than
that from the bottom of the system to field centre, so systems with TLLLs
9
tend to require more ceiling height for operation than systems with BLLLs.
to be able to pass into the bore of the system magnet whereas in TLLL systems
the thermal and electrical connections can be made above the magnet, offering
more space for these connections.
Both the TLLL and BLLL require move-able baffles to allow the sample
holder to be introduced into the system without leaving an unacceptable heat
load from 300 K blackbody radiation in normal operation. For TLLL systems
these can be controlled with a drive rod mechanically connected to the room
temperature top plate. For BLLL systems it is more convenient to make these
baffles spring-loaded as the baffles themselves are attached to demountable ra-
diation shields.
3.1.1. Connectors
The choice of connectors is critical to the microwave performance of a cable
assembly. With standard UT-85 type cables the usual room temperate choices
are SMA [32] connectors for operation up to 18 GHz and SK [33] connectors for
operation up to 40 GHz. Both of these connectors are screw lock, so unsuitable
for push fit applications, the BMA [34] connector range is a blind-mate equiva-
lent of the SMA connector, but suffers from being rated only to 20 GHz and
being rather bulky ( 10 mm diameter) which limits the density of connections.
SMP [35] connectors have the advantage of being blind-mate, small diam-
eter ( 3 mm) and rated for 40 GHz operation and so were selected to trial
cryogenically. A test piece was made, figure 4 (upper panel), and mounted onto
a cryogen-free 4 K platform as a test bed. This test bed was equipped with mi-
crowave cabling enabling round-trip attenuation measurements through pairs of
the test connectors to be made from room temperature.
First the test piece was repeatedly mated and unmated to test the reliability
of the connectors with S
12
measurements made after every cycle, figure 4 (b).
The connectors were robust to this cycling, so the system was cooled and the
round-trip attenuation monitored as a function of temperature, figure 4 (c).
There was no degradation in performance of the connectors with temperature,
the overall reduction in the round-trip attenuation with temperature is due to
the temperature dependence of the coaxial cable material (stainless steel) used
between room temperature and the 4 K stage.
Connectors for multiple DC lines were also trialled cryogenically and a nano
d-type connector [36] was chosen, principally due to its extremely small foot-
print.
consists of a vacuum lock which is mounted onto a gate valve on the top /
bottom of the main vacuum chamber and evacuated prior to introducing the
sample holder into the system. Optionally, the loading probe vacuum lock itself
10
Figure 4: (a) A test piece for SMP connectors. (b) The round-trip attenuation through pairs
of the connectors measured at room temperature as a function of the connector mating cycle
number. The data for the first 25 cycles are for one pair of connectors, the last 25 cycles
are for the second pair. (c) The round-trip attenuation through the first pair of connectors
measured at room temperature as a function of the temperature of the connectors.
11
can be fitted with an additional gate valve to allow samples to be stored under
The sample holder is mechanically connected to drive rods which enter the
vacuum lock via piston seals, and electrically connected to the biasing / ground-
ing wiring on the probe. The drive ro ds can be used to position the sample
holder at the docking station (detailed in section 3.1.3). The sample holder is
pushed into the docking station (making the electrical connections to the wiring
installed on the refrigerator) and then bolted into place using the drive rods,
utilising fasteners captive to the sample holder, to make the thermal connection.
Typically 2, 3 or 4 (depending on the sample holder configuration) M4 threads
are tightened to a torque of 5 Nm. A double thread arrangement allows the
drive rods to then be disengaged from the sample holder and withdrawn from
the system (breaking the electrical connections to the sample on the probe).
sociated conducted heat leak; thus the loaded sample holder has no impact on
the base temperature performance of the refrigerator.
3.1.3. The docking station
The docking station provides the mating electrical and thermal connections
for the sample holder. The cabling attached to the refrigerator is routed to
the docking station. For TLLL systems the docking station is a ring around the
(central) LoS port, for BLLL systems it is a stand-off that brings the connection
flange into the bore of the magnet. Typically 48 DC lines and 14 microwave
cables can be connected to the docking station, however we note that it is
straightforward to scale up the number of connectors, if required, particularly
on TLLL systems as this can be achieved without the need for a larger magnet
bore.
A BLLL sample holder attached to its docking station is shown in figure 5.
Microwave cable links are fitted between the wiring cartridges, running through
the refrigerator, and the docking station.
3.1.4. The sample holder
Examples of BLLL sample holders are shown in figure 6. In the figure,
panel (b) is a design for integration with high-field magnets with 57 mm cold
bore diameter, giving a clear diameter sample space inside the sample holder
of 25 mm (reduced from the diameter of the sample holder by the drive rods
internal to the holder required to make the bolted connections to the docking
station, visible in the figure) by 90 mm long, symmetric about the field centre
line of the magnet. Also shown (c) is a larger diameter sample holder for magnets
with a 90 mm cold bore giving a clear sample space diameter of 50 mm.
Figure 6 (a) shows the mating surface of the BLLL sample holder. In partic-
ular the SMP connector “bullet” adaptors can be seen (the bullet has been re-
moved from the lower left shroud). On the sample holder, “full detent” shrouds
are used to retain the bullet. On the docking station smooth bore so called
12
the mixing chamber docking station. The microwave cable links between the wiring cartridges
and the docking station are visible.
13
Figure 6: (a) The mating surface of a bottom loading sample holder showing the 14 SMP
connectors and 51-way nano d-connector. Two alignment pins are visible at the left and right
of the holder and two M4 captive fasteners are visible at the top and bottom. (b) A bottom
the sample whilst loading can be seen entering the sample holder from the bottom, and the
experimental wiring entering from the top. Also shown is a PCB sample holder that could
be used for mounting a sample into the holder. (c) A larger diameter bottom loading sample
four drive rods are visible at the base of the sample holder.
“catcher’s mit” shrouds are used which allow for a certain amount of radial and
The sample holder features an integrated radiation shield, which also pro-
3.2. Sample cool-down
load-lock is evacuated the bulk of the mixture is removed from the refrigerator,
for the refrigerators used in this work that are equipped with pre-cool lines [12]
this takes around 15 minutes, and any superconducting magnet, if fitted, is de-
energised. The sample is then introduced and connected to the mixing chamber,
and the loading probe withdrawn. The refrigerator then re-cools back to its base
temperature. The re-cooling and running to base can be automated, allowing
a sample to be loaded in the evening and be at base temperature ready for
measurements the following morning.
14
Figure 7: The cool down of a bottom loading sample holder after being loaded onto a dilution
refrigerator system. (left panel) The cool down from room temperature to mK temperatures in
around 7 hours. The high- and low-range mixing temperature sensors are calibrated between
325 K and 1.4 K, and 40 K and 50 mK respectively. At the lowest temperatures the mea-
surements with resistive sensors are replaced with a nuclear orientation thermometer. (right
panel) Cool down and base temperature measurements and (inset) the temperature stability
at the sample position as measured with the nuclear orientation thermometer.
A typical cool down of a bottom loaded sample holder is shown in the left
panel of figure 7. In this example the sample holder was loaded onto a refriger-
ator equipped with eight of the silver-plated stainless steel upper, NbTi lower
coaxial lines described in section 2.2.3 and a further eight stainless steel lines.
The base temperature attained at the sample position is shown in the right
panel of figure 7, as measured with a nuclear orientation thermometer over a
period of several days. The mean temperature at the sample position was found
to be 9.85 mK.
3.2.1. Sample turnaround times
Whilst the co ol down time of a loaded sample can be seen to be around
seven hours from figure 7, the total turnaround time from removing one sample
to having another cold is also of interest. As the loading probes and sample
holders are interchangeable the optimum turnaround can be accomplished by
having a second sample holder set up on a second loading probe whilst the cold
sample is being removed. The removal of the mixture from the refrigerator
and the unloading of the cold sample can be completed in around 20 minutes.
section 3.1.2, or if the sample can be vented to atmosphere whilst cold, then
the two loading probes can be exchanged immediately, the vacuum seals can be
demounted and remade in around 10 minutes. Finally the small volume of the
of the new sample and disconnection of the loading probe then takes a further
5 minutes.
15
Figure 8: The cooling power measured at the sample position on a top-loaded sample holder.
The fit, a second-order polynomial, is included as a guide.
3.3. Cooling power at the sample stage
As all the experimental services are installed onto the refrigerator, and ther-
mally anchored there, the co oling power requirements at the sample position
(inside the sample holder) are less stringent as only heat dissipated in the sam-
ple itself needs to be adsorbed.
Figure 8 shows the measured cooling power at the sample position in a top
loaded sample holder. The temperatures below 50 mK were measured using a
nuclear orientation thermometer and above 50 mK using a calibrated ruthenium-
oxide temperature sensor [26], both mounted at the sample position. The heat
load was supplied by a resistive heater mounted nearby.
The base temperature at the sample position was found to be just over
9 mK and the cooling capacity available at 100 mK in excess of 120 µW. This is
reduced slightly from the cooling power available at the mixing chamber plate
itself (typically 200 - 400 µW) due to the finite thermal impedance between the
loaded sample holder and the docking station, but as all experimental services
are anchored directly to the refrigerator (and not to the sample holder) this
reduction poses no problems for experiments.
4. Conclusions
We have shown that cryogen-free dilution refrigerators equipped with sample
loading mechanisms are a flexible experimental platform, and are the ideal test
bed for quantum device characterisation and development.
16
Figure 9: (a) Gigahertz quantised charge pumping in a graphene double quantum dot device.
Single electron pumps operating a high frequencies could allow for a new definition of the
ampere. The straight lines represent I = ±ef . The measured current oscillates between the
quantised values because of a phase difference between the two drive signals resulting from
unequal lengths of coaxial line. (b) The fractional quantum Hall effect measured in two-
dimensional electron system. Quantised features at fractional values of ν > 2 exemplify the
low electron-temperatures attained in these measurements, from which a value of 15 - 20 mK
can be inferred.
The engineering of such a sample loading arrangement for either top or bot-
tom sample loading has been described, and we have shown that the performance
attained in terms of the base temperatures at the sample (below 10 mK) and the
microwave characteristics of the (up to 14) coaxial lines that are available at the
sample position are as good as can be achieved mounting a sample directly on
the refrigerator, but that the experimental turn-around time has been reduced
from days to a few hours.
4.1. Application Examples
The versatility of the sample loading system is demonstrated by the wide
range of applications to which they have been applied. Figure 9 shows some
specific examples of experimental data, from different laboratories, obtained
using such sample exchange devices.
Figure 9 (a) shows the current generated by a so-called single electron pump
used to realise a quantum standard for electrical current, as measured on a
into two nanometre size islands separated by tunnel barriers. By applying high
frequency signals (with well defined waveforms) to gates in close proximity to
the islands, individual electrons can be made to jump from island to island
and hence produce a measurable electrical current [37]. For these experiments
low temperatures are important to minimise thermal excitations that would
produce unwanted tunnelling events and the drive frequency needs to be as high
17
as possible to create a suitably large current. For these reasons well thermalised,
high-fidelity microwave lines to the sample are extremely important.
Figure 9 (b) shows the longitudinal and transverse resistivity measured in a
δ-doped GaAs-AlGaAs quantum well as a function of magnetic field as measured
and low electron-temperatures in the wiring to the sample holder are required.
The quasiparticles of the fractional quantum Hall effect state at Landau-level
filling factor ν =
5
/
2
are predicted to obey non-abelian statistics which could
allow for the development of topologically protected qubits [39] for quantum
computation.
5. Acknowledgments
The authors would like to thank C. Wilkinson and R. Brzakalik for their
contributions to the design of the sample loading arrangement and D. Turner-
Cleaver for his assistance developing the mechanical arrangements.
Part of this work is courtesy of the Office of the Director of National Intelli-
gence, Intelligence Advanced Research Projects Activity (IARPA), through the
Army Research Office grant W911NF-12-1-0354 and we thank our IARPA col-
laborators for their detailed discussions on various experimental requirements.
The data presented in figure 9 are reproduced with the kind permission of
(a) T.J.B.M. Janssen of the National Physical Laboratory, London, UK (b) K.
Rasmussen and C.J.S. Olsen of the Center for Quantum Devices, Copenhagen,
Denmark.
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20
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21
Highlights
-A rapid sample exchange mechanism for experiments at millikelvin temperatures has been developed.
-This mechanism is compatible with multiple high-frequency microwave links to the sample.
-Samples can be exchanged in minutes and cooled to temperatures 10 mK in 7 hours.
... Cryogenic anchoring of instrumentation lead wires is another crucial design detail. Batey in [25] and Pagano [26] separately reported new developments in advance anchor designs in details. Figure 9 shows how a RF wiring cartridge will be anchored on multi cold stages [25]. Measuring wires from ambient to working cold mass cause additional heat leak. ...
... Cryogenic anchoring of instrumentation lead wires is another crucial design detail. Batey in [25] and Pagano [26] separately reported new developments in advance anchor designs in details. Figure 9 shows how a RF wiring cartridge will be anchored on multi cold stages [25]. Measuring wires from ambient to working cold mass cause additional heat leak. ...
... It is particularly serious when the cooling capacity is very small at ultralow temperatures and/or cooling power is difficult to supply in aerospace applications. Thermal anchoring of wires to heat sinks in cryogenic equipment is required at each intermediate stage [25][26]. Special attention is needed if coax cables are used. ...
Article
Full-text available
Various functional insertion components (FIC), directly connecting a cold mass and the ambient environment, are irreplaceable and play crucial technical roles in many superconducting cryogenic applications. However, such components also bring a huge heat leak to the cold mass. The heat leak is usually much greater than that through entire evacuated MLI insulation system and solid support structures combined. Therefore, this situation brings unimaginable challenges not only to the refrigeration loads to be met but also critical aspects of the FIC design in satisfaction of highly restrictive, even contradicting technical functions. The FIC must simultaneously minimize the heat leak provide a large amounts of DC current and RF power to/from the cold mass, as well as provide a reliable ultrahigh vacuum thermal isolation break with strong mechanical stability. Reviewed are the following commonly used FICs: 1) various RF input couplers for transmitting MW-RF power; 2) high DC power current leads for energizing various SC magnets; 3) various high order mode (HOM) couplers for damping unwanted RF energy; 4) instrumentation cable/wire to cold mass. Approaches to efficiently minimize the DC current heating, RF surface heating, heat leak through the solid body of FIC, efficient cooling approaches, while reliably providing the technical functions, are briefly summarized and compared for select applications from around the world. 1. Introduction Superconducting (SC) magnets can provide the highest intensity and largest space of magnetic fields with zero resistance for DC current, and superconducting radio-frequency (SRF) cavities can deliver the quality value Q higher than copper (Cu) by ~10 5 and accelerating fields Eacc (e.g., up to 25-30 MV/m) higher than Cu by an order of magnitude in continuous wave (CW) mode. Huge cryogenic systems (multi-kW refrigerators both at 2 K and 4 K) have been operated successfully associated with the applications. About 2,000 m length of SRF cavities with RF input and HOM extract couplers have been in operation and development [1], and new projects of more than 20,000 m length of SRF systems have been proposed [2-4]. ITER in France [5] and EAST in China [6] both with greatest sizes of special shaped SC magnets have been tested for fusion projects. Recently CERN and Beijing have separately deliberated plans to develop future colliders much larger than existing LHC of 27 km [7]. MRI magnets [8] and SC power applications [9] have also employed many current leads. There are irreplaceable functional insertion components (FIC), which are directly connected a cold mass and the ambient environment, and play crucial technical roles in above mentioned applications. Huge radio frequency (RF) power (megawatts) are transported to SRF cavities by RF input coupler (RFIC) from klystrons to generate strong electromagnetic accelerating fields. RFIC also provide tight vacuum breaks between ultra-high vacuum in accelerating space and ambient environment. High order
... Cryogenic anchoring of instrumentation lead wires is another crucial design detail. Batey in [25] and Pagano [26] separately reported new developments in advance anchor designs in details. Figure 9 shows how a RF wiring cartridge will be anchored on multi cold stages [25]. Measuring wires from ambient to working cold mass cause additional heat leak. ...
... Cryogenic anchoring of instrumentation lead wires is another crucial design detail. Batey in [25] and Pagano [26] separately reported new developments in advance anchor designs in details. Figure 9 shows how a RF wiring cartridge will be anchored on multi cold stages [25]. Measuring wires from ambient to working cold mass cause additional heat leak. ...
... It is particularly serious when the cooling capacity is very small at ultralow temperatures and/or cooling power is difficult to supply in aerospace applications. Thermal anchoring of wires to heat sinks in cryogenic equipment is required at each intermediate stage [25][26]. Special attention is needed if coax cables are used. ...
Article
Full-text available
Various functional insertion components (FIC), directly connecting a cold mass and the ambient environment, are irreplaceable and play crucial technical roles in many superconducting cryogenic applications. However, such components also bring a huge heat leak to the cold mass. The heat leak is usually much greater than that through entire evacuated MLI insulation system and solid support structures combined. Therefore, this situation brings unimaginable challenges not only to the refrigeration loads to be met but also critical aspects of the FIC design in satisfaction of highly restrictive, even contradicting technical functions. The FIC must simultaneously minimize the heat leak provide a large amounts of DC current and RF power to/from the cold mass, as well as provide a reliable ultrahigh vacuum thermal isolation break with strong mechanical stability. Reviewed are the following commonly used FICs: 1) various RF input couplers for transmitting MW-RF power; 2) high DC power current leads for energizing various SC magnets; 3) various high order mode (HOM) couplers for damping unwanted RF energy; 4) instrumentation cable/wire to cold mass. Approaches to efficiently minimize the DC current heating, RF surface heating, heat leak through the solid body of FIC, efficient cooling approaches, while reliably providing the technical functions, are briefly summarized and compared for select applications from around the world.
... Cryogenic anchoring of instrumentation lead wires is another crucial design detail. Figure 9 shows how a cartridge could be anchored on cold stage [25]. Measuring wires from ambient to working cold mass cause additional heat leak. ...
... It is particularly serious when cooling capacity is very small at ultralow temperature and/or cooling power is difficult to supply in aerospace applications. Thermal anchoring of wires to heat sinks in cryogenic equipment is required at each intermediate stage [25,26]. Special attention is needed if coax cables are used. ...
... (a) RF wiring cartridge with hermetic feedtroughs from 300 K to 10 mK. (b) Split clamps to thermal anchor the cartridge.[25] ...
Article
Full-text available
Various functional insertion components (FIC), directly connecting a cold mass and the ambient environment, are irreplaceable and play crucial technical roles in many superconducting cryogenic applications. However, such components also bring a huge heat leak to the cold mass. The heat leak is usually much greater than that through entire evacuated MLI insulation system and solid support structures combined. Therefore, this situation brings unimaginable challenges not only to the refrigeration loads to be met but also critical aspects of the FIC design in satisfaction of highly restrictive, even contradicting technical functions. The FIC must simultaneously minimize the heat leak provide a large amounts of DC current and RF power to/from the cold mass, as well as provide a reliable ultrahigh vacuum thermal isolation break with strong mechanical stability. Reviewed are the following commonly used FICs: 1) various RF input couplers for transmitting MW-RF power; 2) high DC power current leads for energizing various SC magnets; 3) various high order mode (HOM) couplers for damping unwanted RF energy; 4) instrumentation cable/wire to cold mass. Approaches to efficiently minimize the DC current heating, RF surface heating, heat leak through the solid body of FIC, efficient cooling approaches, while reliably providing the technical functions, are briefly summarized and compared for select applications from around the world. 1. Introduction Superconducting (SC) magnets can provide the highest intensity and largest space of magnetic fields with zero resistance for DC current, and Superconducting radio-frequency (SRF) cavities can deliver the quality value Q higher than Cu by ~105 and accelerating fields Eacc (e.g. up to 25-30 MV/m) higher than Cu by an order of magnitude in continuous wave (CW) mode. Huge cryogenic systems (multi-kW refrigerators both at 2K and 4K) have been operated successfully associated with the applications. About 2000 m of SRF cavities with RF input and HOM extract couplers have been in operation and development [1], and new projects of more than 20,000 m of SRF systems have been proposed [2-4]. ITER in France [5] and EAST in China [6] both with greatest sizes of special shaped SC magnets have been tested for fusion projects. Recently CERN and Beijing have separately deliberated plans to develop future colliders much larger than existing LHC of 27 km [7]. MRI magnets [8] and SC power applications [9] have also employed many current leads. There are irreplaceable functional insertion components (FIC), which are directly connected a cold mass and the ambient environment, and play crucial technical roles in above mentioned applications. Huge radio frequency (RF) power (megawatts) are transported to SRF cavities by RF input coupler (RFIC) from klystrons to generate strong electromagnetic accelerating fields. RFIC also provide tight vacuum breaks between ultra-high vacuum in accelerating space and ambient environment. High order
... Cryogenic anchoring of instrumentation lead wires is another crucial design detail. Figure 9 shows how a cartridge could be anchored on cold stage [25]. Measuring wires from ambient to working cold mass cause additional heat leak. ...
... It is particularly serious when cooling capacity is very small at ultralow temperature and/or cooling power is difficult to supply in aerospace applications. Thermal anchoring of wires to heat sinks in cryogenic equipment is required at each intermediate stage [25,26]. Special attention is needed if coax cables are used. ...
... (a) RF wiring cartridge with hermetic feedtroughs from 300 K to 10 mK. (b) Split clamps to thermal anchor the cartridge.[25] ...
Article
Full-text available
Various functional insertion components (FIC), directly connecting a cold mass and the ambient environment, are irreplaceable and play crucial technical roles in many superconducting cryogenic applications. However, such components also bring a huge heat leak to the cold mass. The heat leak is usually much greater than that through entire evacuated MLI insulation system and solid support structures combined. Therefore, this situation brings unimaginable challenges not only to the refrigeration loads to be met but also critical aspects of the FIC design in satisfaction of highly restrictive, even contradicting technical functions. The FIC must simultaneously minimize the heat leak provide a large amounts of DC current and RF power to/from the cold mass, as well as provide a reliable ultrahigh vacuum thermal isolation break with strong mechanical stability. Reviewed are the following commonly used FICs: 1) various RF input couplers for transmitting MW-RF power; 2) high DC power current leads for energizing various SC magnets; 3) various high order mode (HOM) couplers for damping unwanted RF energy; 4) instrumentation cable/wire to cold mass. Approaches to efficiently minimize the DC current heating, RF surface heating, heat leak through the solid body of FIC, efficient cooling approaches, while reliably providing the technical functions, are briefly summarized and compared for select applications from around the world. 1. Introduction Superconducting (SC) magnets can provide the highest intensity and largest space of magnetic fields with zero resistance for DC current, and Superconducting radio-frequency (SRF) cavities can deliver the quality value Q higher than Cu by ~105 and accelerating fields Eacc (e.g. up to 25-30 MV/m) higher than Cu by an order of magnitude in continuous wave (CW) mode. Huge cryogenic systems (multi-kW refrigerators both at 2K and 4K) have been operated successfully associated with the applications. About 2000 m of SRF cavities with RF input and HOM extract couplers have been in operation and development [1], and new projects of more than 20,000 m of SRF systems have been proposed [2-4]. ITER in France [5] and EAST in China [6] both with greatest sizes of special shaped SC magnets have been tested for fusion projects. Recently CERN and Beijing have separately deliberated plans to develop future colliders much larger than existing LHC of 27 km [7]. MRI magnets [8] and SC power applications [9] have also employed many current leads. There are irreplaceable functional insertion components (FIC), which are directly connected a cold mass and the ambient environment, and play crucial technical roles in above mentioned applications. Huge radio frequency (RF) power (megawatts) are transported to SRF cavities by RF input coupler (RFIC) from klystrons to generate strong electromagnetic accelerating fields. RFIC also provide tight vacuum breaks between ultra-high vacuum in accelerating space and ambient environment. High order
... Cryogenic anchoring of instrumentation lead wires is another crucial design detail. Figure 9 shows how a cartridge could be anchored on cold stage [25]. Measuring wires from ambient to working cold mass cause additional heat leak. ...
... It is particularly serious when cooling capacity is very small at ultralow temperature and/or cooling power is difficult to supply in aerospace applications. Thermal anchoring of wires to heat sinks in cryogenic equipment is required at each intermediate stage [25,26]. Special attention is needed if coax cables are used. ...
... (a) RF wiring cartridge with hermetic feedtroughs from 300 K to 10 mK. (b) Split clamps to thermal anchor the cartridge.[25] ...
Conference Paper
Full-text available
Various functional insertion components (FIC), directly connecting a cold mass and the ambient environment, are irreplaceable and play crucial technical roles in many superconducting cryogenic applications. However, such components also bring a huge heat leak to the cold mass. The heat leak is usually much greater than that through entire evacuated MLI insulation system and solid support structures combined. Therefore, this situation brings unimaginable challenges not only to the refrigeration loads to be met but also critical aspects of the FIC design in satisfaction of highly restrictive, even contradicting technical functions. The FIC must simultaneously minimize the heat leak provide a large amounts of DC current and RF power to/from the cold mass, as well as provide a reliable ultrahigh vacuum thermal isolation break with strong mechanical stability. Reviewed are the following commonly used FICs: 1) various RF input couplers for transmitting MW-RF power; 2) high DC power current leads for energizing various SC magnets; 3) various high order mode (HOM) couplers for damping unwanted RF energy; 4) instrumentation cable/wire to cold mass. Approaches to efficiently minimize the DC current heating, RF surface heating, heat leak through the solid body of FIC, efficient cooling approaches, while reliably providing the technical functions, are briefly summarized and compared for select applications from around the world. 1. Introduction Superconducting (SC) magnets can provide the highest intensity and largest space of magnetic fields with zero resistance for DC current, and Superconducting radio-frequency (SRF) cavities can deliver the quality value Q higher than Cu by ~105 and accelerating fields Eacc (e.g. up to 25-30 MV/m) higher than Cu by an order of magnitude in continuous wave (CW) mode. Huge cryogenic systems (multi-kW refrigerators both at 2K and 4K) have been operated successfully associated with the applications. About 2000 m of SRF cavities with RF input and HOM extract couplers have been in operation and development [1], and new projects of more than 20,000 m of SRF systems have been proposed [2-4]. ITER in France [5] and EAST in China [6] both with greatest sizes of special shaped SC magnets have been tested for fusion projects. Recently CERN and Beijing have separately deliberated plans to develop future colliders much larger than existing LHC of 27 km [7]. MRI magnets [8] and SC power applications [9] have also employed many current leads. There are irreplaceable functional insertion components (FIC), which are directly connected a cold mass and the ambient environment, and play crucial technical roles in above mentioned applications. Huge radio frequency (RF) power (megawatts) are transported to SRF cavities by RF input coupler (RFIC) from klystrons to generate strong electromagnetic accelerating fields. RFIC also provide tight vacuum breaks between ultra-high vacuum in accelerating space and ambient environment. High order
... Measurements were made at 20 mK using a dilution refrigerator system equipped with an 8T superconducting solenoid and bottom loading sample exchange mechanism. 24 Figures 1(b)-1(d) show the characteristics of the SQUID with a loop area of % 210 lm 2 over a range of magnetic fields. Figure 1(b) shows the superposition of the small scale modulation of the critical current I C , due to interference around the SQUID loop, and larger-scale modulation of the critical current I C , which is the Fraunhofer pattern arising from interference of different current paths across the finite area of the junctions. ...
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... The cryogen-free dilution refrigerator [1], often integrated with a superconducting magnet [2], has become the workhorse of many low-temperature laboratories around the world, and such systems are routinely used to attain temperatures below ∼ 10 mK. In some fields of research, such as quantum information processing, these cryogen-free machines are preferred to "wet" systems using liquid 4 He as the installation of experimental services is more straightforward, for a discussion see [3]. ...
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Question: scanBam and countBam end prematurely under Windows 10
0
6 days ago by
Gordon Smyth38k
Walter and Eliza Hall Institute of Medical Research, Melbourne, Australia
Gordon Smyth38k wrote:
The problem
I have been using Rsamtools::scanBam() quite a bit recently. When using R for Unix, it has worked perfectly for me, but under Windows 10 it frequently scans only the first couple of hundred reads, even for bam files that contain millions of reads. Using Windows 10, scanBam stops prematurely on about 40% of my BAM files. The problem is new in Bioconductor 3.9.
Unix is fine
Here is one of my sessions scanning some bam files under Unix:
> s <- scanBam("S20_7600_23_CTCs.h.bam",
+ param=ScanBamParam(what="rname"))
> length(s[[1]]$rname) [1] 52132720 > s <- scanBam("S27_10247_23_Primary.h.bam", + param=ScanBamParam(what="rname")) > length(s[[1]]$rname)
[1] 48492302
>
> sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: CentOS Linux 7 (Core)
Matrix products: default
BLAS: /stornext/System/data/apps/R/R-3.6.1/lib64/R/lib/libRblas.so
LAPACK: /stornext/System/data/apps/R/R-3.6.1/lib64/R/lib/libRlapack.so
locale:
[1] LC_CTYPE=en_AU.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_AU.UTF-8 LC_COLLATE=en_AU.UTF-8
[5] LC_MONETARY=en_AU.UTF-8 LC_MESSAGES=en_AU.UTF-8
[7] LC_PAPER=en_AU.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_AU.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats4 parallel stats graphics grDevices utils datasets
[8] methods base
other attached packages:
[1] Rsamtools_2.0.0 Biostrings_2.52.0 XVector_0.24.0
[4] GenomicRanges_1.36.0 GenomeInfoDb_1.20.0 IRanges_2.18.1
[7] S4Vectors_0.22.0 BiocGenerics_0.30.0
loaded via a namespace (and not attached):
[1] zlibbioc_1.30.0 compiler_3.6.1 GenomeInfoDbData_1.2.1
[4] RCurl_1.95-4.12 BiocParallel_1.18.0 bitops_1.0-6
Windows is not
Now the same thing using Windows:
> s <- scanBam("S20_7600_23_CTCs.h.bam",param=ScanBamParam(what="rname"))
> length(s[[1]]$rname) [1] 280 > s <- scanBam("S27_10247_23_Primary.h.bam",param=ScanBamParam(what="rname")) > length(s[[1]]$rname)
[1] 258
>
> countBam("S20_7600_23_CTCs.h.bam")
space start end width file records nucleotides
1 NA NA NA NA S20_7600_23_CTCs.h.bam 280 21152
> countBam("S27_10247_23_Primary.h.bam")
space start end width file records nucleotides
1 NA NA NA NA S27_10247_23_Primary.h.bam 258 19445
>
> sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 15063)
Matrix products: default
locale:
[1] LC_COLLATE=English_Australia.1252 LC_CTYPE=English_Australia.1252
[3] LC_MONETARY=English_Australia.1252 LC_NUMERIC=C
[5] LC_TIME=English_Australia.1252
attached base packages:
[1] stats4 parallel stats graphics grDevices utils datasets methods base
other attached packages:
[1] Rsamtools_2.0.0 Biostrings_2.52.0 XVector_0.24.0 GenomicRanges_1.36.0
[5] GenomeInfoDb_1.20.0 IRanges_2.18.1 S4Vectors_0.22.0 BiocGenerics_0.30.0
loaded via a namespace (and not attached):
[1] zlibbioc_1.30.0 compiler_3.6.1 GenomeInfoDbData_1.2.1
[4] RCurl_1.95-4.12 BiocParallel_1.18.1 bitops_1.0-6
As you can see, scanBam and countBam have read only 280 and 258 reads for the two files respectively, even though both files contain millions of reads. There is no error message or warning.
In these two sessions, I am accessing the same bam files on the same disk, using Samba to access the Unix file system from Windows. If I copy the bam files to the PC disk, I get the same result.
I don't see any pattern as to which bam files will exit prematurely and which will not. But the same file always gets the same result. It doesn't make any difference which SAM fields are read -- any settings for param produces the same output number of reads.
The problem is new in Bioconductor 3.9
If I go back to R 3.5.3 and Bioconductor 3.8, then the problem disappears. Using Bioconductor 3.8 for Windows, then scanBam and countBam read all my BAM files correctly.
The problem is still in Bioconductor 3.10
I tried upgrading to Rsamtools 2.1.4, which is the latest on the developmental version of Bioconductor, but the behaviour is the same as for Rsamtools 2.0.0.
Example file
I have put an example of an offending file here. The file is 2Gb in size -- it was the smallest I could find showing the problem.
The correct number of records for this file is:
space start end width file records nucleotides
NA NA NA NA S21_7600_23_Primary.xenosplit.bam 35394670 2678418738
but countBam in Bioconductor 3.9 for Windows gives me:
space start end width file records nucleotides
NA NA NA NA S21_7600_23_Primary.xenosplit.bam 300 22680
Error message from asSam (added 16 September 2019)
I don't know whether this helps but I find that I do get an error message from asSam on the same bam files, for example:
> asSam("S20_7600_23_CTCs.h.bam",overwrite=TRUE)
Error in value[[3L]](cond) : 'asSam' truncated input file at record 280
SAM file: 'S20_7600_23_CTCs.h.bam'
I can track the error message to a particular source code file (as_bam.c) like this:
> file <- "S20_7600_23_CTCs.h.bam"
> d0 <- "S20_7600_23_CTCs.h.sam"
> .Call(Rsamtools:::.as_bam, file, d0, FALSE)
Error: truncated input file at record 280
rsamtools • 213 views
ADD COMMENTlink
modified 11 hours ago • written 6 days ago by Gordon Smyth38k
I think the change from BioC 3.8 to 3.9 marked the transition to linking against Rhtslib rather than distributing it's own version of samtools, so perhaps it's related to that?
I'm not sure what happened to the other messages, but I can confirm I see this behaviour on Windows 10 with R-3.6.1, Rsamtools 2.0.0
> Rsamtools::countBam("z:/S21_7600_23_Primary.xenosplit.bam")
[W::bam_hdr_read] bgzf_check_EOF: No error
space start end width file records nucleotides
1 NA NA NA NA S21_7600_23_Primary.xenosplit.bam 300 22680
If it's any help to diagnose the issue, when I ran it on a truncated version of the file that hadn't downloaded properly I got a result which matched the Linux version:
> Rsamtools::countBam("z:/S21_7600_23_Primary.xenosplit.bam")
space start end width file records nucleotides
1 NA NA NA NA S21_7600_23_Primary.xenosplit.bam 24130547 1826042711
ADD REPLYlink written 4 days ago by Mike Smith3.9k
Thanks Mike. The previous messages disappeared because they were all hanging off a comment that was subsequently deleted by the poster. I have edited my question and your comment so that all the important stuff from the discussion is preserved. It is cleaner now.
The file truncation problem was my fault -- I posted the link before the file had finished transferring to the web server -- the transfer taking much longer than I expected. It is fortuitous though because your experience with the truncated file is very interesting.
ADD REPLYlink modified 4 days ago • written 4 days ago by Gordon Smyth38k
csaw has had issues with Rhtslib on Windows for some time, e.g.:
https://www.biostars.org/p/323743/
It seems that HTSLib 1.9 has a few fixes for Windows:
https://github.com/samtools/htslib/releases/
So it might be worth upgrading Rhtslib to see if it helps.
ADD REPLYlink written 4 days ago by Aaron Lun24k
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∞-Lie theory
# Contents
## Idea
An orbispace is a space, particularly a topological stack, that is locally modeled on the homotopy quotient/action groupoid of a locally compact topological space by a rigid group action.
Orbispaces are to topological spaces what orbifolds are to manifolds.
## Definition
Write $Orb$ for the global orbit category. Then its (∞,1)-presheaf (∞,1)-category $PSh_\infty(Orb)$ is the (∞,1)-category of orbispaces. (Henriques-Gepner 07, Rezk 14, remark 1.5.1)
By the main theorem of (Henriques-Gepner 07) the (∞,1)-presheaves on the global orbit category are equivalently “cellular” topological stacks/topological groupoids (“orbispaces”), we might write this as
$ETopGrpd^{cell} = PSh_\infty(Orb) \,.$
## Properties
### Relation to global equivariant homotopy theory
The global equivariant homotopy theory is the (∞,1)-category (or else its homotopy category) of (∞,1)-presheaves on the global equivariant indexing category $Glo$
Here $Glo$ has as objects compact Lie groups and the (∞,1)-categorical hom-spaces $Glo(G,H) \coloneqq \Pi [\mathbf{B}G, \mathbf{B}H]$, where on the right we have the fundamental (∞,1)-groupoid of the topological groupoid of group homomorphisms and conjugations.
The global orbit category is the non-full subcategory of the global equivariant indexing category on the faithful maps $\mathbf{B}G\to \mathbf{B}H$.
The central theorem of (Rezk 14) is that $PSh_\infty(Orb)$ is the base (∞,1)-topos over the cohesion of the slice of the global equivariant homotopy theory $PSh_\infty(Glo)$ over the terminal orbispace $\mathcal{N}$ (Rezk 14, p. 4 and section 7)
$(\Pi \dashv \Delta \dashv \Gamma \dashv \nabla) \;\colon\; PSh_\infty(Glo)/\mathcal{N} \longrightarrow PSh_\infty(Orb) \,.$
Rezk-global equivariant homotopy theory:
cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory $PSh_\infty(Glo)$global equivariant indexing category $Glo$∞Grpd $\simeq PSh_\infty(\ast)$point
sliced over terminal orbispace: $PSh_\infty(Glo)_{/\mathcal{N}}$$Glo_{/\mathcal{N}}$orbispaces $PSh_\infty(Orb)$global orbit category
sliced over $\mathbf{B}G$: $PSh_\infty(Glo)_{/\mathbf{B}G}$$Glo_{/\mathbf{B}G}$$G$-equivariant homotopy theory of G-spaces $L_{we} G Top \simeq PSh_\infty(Orb_G)$$G$-orbit category $Orb_{/\mathbf{B}G} = Orb_G$
## References
A detailed but elementary approach via atlases can be found in
and another approach is discussed in
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Revised on April 13, 2014 23:36:44 by Urs Schreiber (185.37.147.12)
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For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote, For the following exercises, use a calculator to graphUse the graph to solve. Right now the $-4$ is disconnected from the fraction part. We can write an equation independently for each: The ratio of sugar to water, in pounds per gallon,, will be the ratio of pounds of sugar to gallons of water, The ratio of sugar to water, in pounds per gallon after 12 minutes is given by evaluatingat. A function can have more than one vertical asymptote. I suspect what they mean is the function $f(x) = \frac{1}{(x - 3)^2} - 4$. Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. For the signedSqrt function, the input signal must be a … CC BY-SA 4.0. I really can't guess what is intended. See (Figure). Begin by setting the denominator equal to zero and solving. Introduction to Polynomial and Rational Functions, 13. These are where the vertical asymptotes occur. Examine the behavior of the graph at the. This behavior creates a vertical asymptote, which is a vertical line that the graph approaches but never crosses. Formula for the distance calculation with UTM coordinates: difEast = abs(UTMRECHTS1 - UTMRECHTS2) difNorth = abs(UTMHOCH1 - UTMHOCH2) l = sqrt(difEast * difEast + difNorth * difNorth) with UTMRECHTS1: Easting of the first coordinate UTMHOCH1: Northing of the first coordinate UTMRECHTS2: Easting of the second coordinate UTMHOCH2: Northing of the second coordinate … What you need to understand is the meaning of $1/\text{horsepower}$. How can I write an equation that matches any sequence? Its Domain is the Real Numbers, except 0, because 1/0 is undefined. The third column gives some hints in the underlying scalar implementation. Latest Math Topics. For the vertical asymptote atthe factor was not squared, so the graph will have opposite behavior on either side of the asymptote. We can start by noting that the function is already factored, saving us a step. We can use arrow notation to describe local behavior and end behavior of the toolkit functions, A function that levels off at a horizontal value has a horizontal asymptote. Asresulting in a horizontal asymptote atSee (Figure). After passing through the x-intercepts, the graph will then level off toward an output of zero, as indicated by the horizontal asymptote. For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. Access these online resources for additional instruction and practice with rational functions. See (Figure). Learn vocabulary, terms, and more with flashcards, games, and other study tools. Created by . For the following exercises, use the graphs to write an equation for the function. Test. Yes — hypot: Yes, on two inputs. For the following exercises, use the given transformation to graph the function. Symbolically, using arrow notation. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side of the vertical asymptote the graph heads towards positive infinity, and on the other side the graph heads towards negative infinity. A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. How to control the range in a reciprocal function, How to convert an infinite binary fraction into a decimal fraction, Write down values of $a$ and $b$ for which this system of equations has a non unique solution, Showing a function is well-defined $g\left( \frac{p}{q} \right)$. Given a reciprocal squared function that is shifted right by $3$ and down by $4$, write this as a rational function. Use arrow notation to describe the end behavior and local behavior of the function graphed in (Figure). Would coating a space ship in liquid nitrogen mask its thermal signature? Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. Exponential and Logarithmic Equations, VII. The graph has two vertical asymptotes. Determine the factors of the denominator. This tells us the amount of water in the tank is changing linearly, as is the amount of sugar in the tank. To learn more, see our tips on writing great answers. 3. h = 0. witherssartsk12org. Join the 2 Crores+ Student community now! Constants are also lines, but they are flat lines. Examine these graphs, as shown in (Figure), and notice some of their features. Left and right derivatives of piecewise function. A vertical asymptote of a graph is a vertical linewhere the graph tends toward positive or negative infinity as the inputs approachWe write, As the values ofapproach infinity, the function values approach 0. Reciprocal of 1/2 = 2/1. This site might help you. Reciprocal squared function and properties 5.1k LIKES. In this case, the end behavior isThis tells us that as the inputs grow large, this function will behave like the functionwhich is a horizontal line. Reciprocal Squared Parent Function. Notice that the graph is showing a vertical asymptote atwhich tells us that the function is undefined at. This behavior creates a horizontal asymptote, a horizontal line that the graph approaches as the input increases or decreases without bound. The vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. The denominator will be zero atindicating vertical asymptotes at these values. The zero of this factor,is the vertical asymptote. square: Yes. The denominator is equal to zero whenThe domain of the function is all real numbers except, A graph of this function, as shown in (Figure), confirms that the function is not defined when. Next, we will find the intercepts. Write an equation for the rational function shown in (Figure). Note any restrictions in the domain where asymptotes do not occur. The concentrationof a drug in a patient’s bloodstreamhours after injection is given byUse a calculator to approximate the time when the concentration is highest. See (Figure). Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. See (Figure). Reciprocal Definition. First, factor the numerator and denominator. A rectangular box with a square base is to have a volume of 20 cubic feet. For the following exercises, describe the local and end behavior of the functions. Related Video. In mathematics, we call this a reciprocal function. If you want to shift a function $g(x)$ by $b$ units down, then do $g(x)-b$. Linear = if you plot it, you get a straight line. The graph heads toward positive infinity as the inputs approach the asymptote on the right, so the graph will head toward positive infinity on the left as well. Start studying Reciprocal Squared Parent Function. Use that information to sketch a graph. By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes. Find the ratio of sugar to water, in pounds per gallon in the tank after 12 minutes. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Then, find the x- and y-intercepts and the horizontal and vertical asymptotes. For the transformed reciprocal squared function, we find the rational form. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 3 pounds per minute. Find the horizontal asymptote and interpret it in context of the problem. Since the graph has no x-intercepts between the vertical asymptotes, and the y-intercept is positive, we know the function must remain positive between the asymptotes, letting us fill in the middle portion of the graph as shown in (Figure). Dec 22, 2020. See (Figure). Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates 1 ⁄ √ x, the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number x in IEEE 754 floating-point format.This operation is used in digital signal processing to normalize a vector, i.e., scale it to length 1. For example, the graph ofis shown in (Figure). Find the dimensions of the box that will have minimum surface area. Spell. The sqrt function accepts real or complex inputs, except for complex fixed-point signals. Note the vertical and horizontal asymptotes. View 4 Function transformation ws Key.pdf from MATH 221 at Rice University. The slant asymptote is the graph of the lineSee (Figure). Copy link. Now to simplify the expression of $h$ or to make it a "rational function" you just have to find the common denominator of the 2 summands which is in this case $(x-3)^2$: On the left branch of the graph, the curve approaches the, Finally, on the right branch of the graph, the curves approaches the. Note that this graph crosses the horizontal asymptote. 39. Gravity. Introduction to Polynomial and Rational Functions, 35. 2. b = − 1. See (Figure). We can see that these functions (if unrestricted) are not one-to-one by looking at their graphs, shown in Figure $$\PageIndex{4}$$. The sum of the reciprocals of the square numbers (the Basel problem) is the transcendental number π 2 / 6, or ζ(2) where ζ is the Riemann zeta function. $$h(x)=\frac{1}{(x-3)^2}-4$$ Don’t forget to add the negative sign! The following are iterative methods for finding the reciprocal square root of S which is /. Browse. In (Figure), we shifted a toolkit function in a way that resulted in the functionThis is an example of a rational function. UK - Can I buy things for myself through my company? To sketch the graph, we might start by plotting the three intercepts. Yes — reciprocal: Yes. Is that a greater ratio of sugar to water, in pounds per gallon than at the beginning? The one atseems to exhibit the basic behavior similar towith the graph heading toward positive infinity on one side and heading toward negative infinity on the other. As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). If we find any, we set the common factor equal to 0 and solve. This is the Reciprocal Function: f (x) = 1/x. … MathJax reference. End behavior: asLocal behavior: as(there are no x– or y-intercepts). Match. Oct 21, 2020. We write. Asking for help, clarification, or responding to other answers. What's a reciprocal square function? Then, find the x- and y-intercepts and the horizontal and vertical asymptotes. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev 2021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. For those factors not common to the numerator, find the vertical asymptotes by setting those factors equal to zero and then solve. The concentrationof a drug in a patient’s bloodstreamhours after injection is given byWhat happens to the concentration of the drug asincreases? We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our study of toolkit functions. The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as f (x) = x − 1 f (x) = x − 1 and f (x) = x − 2. f (x) = x − 2. $2\cos^2\left(\frac{\pi}6\right) - 1$, How to add aditional actions to argument into environement. For the following exercises, find the domain of the rational functions. 11. The reciprocal function shifted up two units. Once it has been found, find by simple multiplication: = ⋅ (/). Horizontal asymptote atVertical asymptotes aty-intercept at. See, A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. Then, use a calculator to answer the question. Note any values that cause the denominator to be zero in this simplified version. As with polynomials, factors of the numerator may have integer powers greater than one. Recall that a polynomial’s end behavior will mirror that of the leading term. To ask Unlimited Maths doubts download Doubtnut from - https://goo.gl/9WZjCW Reciprocal squared function and properties $$h(x)=\frac{1}{(x-3)^2}-\frac{4(x-3)^2}{(x-3)^2}=\frac{1-4(x^2-6x+9)}{(x-3)^2}\\h(x)=\frac{-4x^2+24x-35}{(x-3)^2}$$. Introducing 1 more language to a trilingual baby at home, 4x4 grid with no trominoes containing repeating colors. In this case, the end behavior isThis tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the functionand the outputs will approach zero, resulting in a horizontal asymptote atSee (Figure). None of your functions reflect the "squared" so I assume they are all wrong, but who knows? So $f(x-3) + 4$ will shift a function to the right by $3$ and up by $4$. Is it just this? The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. Solving Systems with Gaussian Elimination, IX. Watch Queue Queue Next, we set the denominator equal to zero, and find that the vertical asymptote is because as We then set the numerator equal to 0 and find the x -intercepts are at and Finally, we evaluate the function at 0 and find the y … Find the horizontal and vertical asymptotes of the function. Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. We have a y-intercept atand x-intercepts atand, To find the vertical asymptotes, we determine when the denominator is equal to zero. Letbe the number of minutes since the tap opened. y = 3 is a flat line. The reciprocal identity of cosecant function is written in this form everywhere but the only changing factor is angle of the right triangle. the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient. 4. k = 0. If you find a way to multiply each side of an equation by a function’s reciprocal, you may be able to reduce some part of the equation to 1 — and simplifying is always a good thing. Identify the horizontal and vertical asymptotes of the graph, if any. Log in Sign up. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 1 pound per minute. Note that replacing $x$ by $x - 3$ shifts the graph to the right three units and subtracting $4$ from the expression shifts it down by $4$ units. Introduction to Systems of Equations and Inequalities, 52. Both the numerator and denominator are linear (degree 1). Yes, on two inputs (two vectors or two matrices of the same size, a scalar and a vector, or a scalar and a matrix) — rem: Yes, on two inputs. In layman’s terms, you can think of a transformation as just moving an object or set of points from one location to another. What's the legal term for a law or a set of laws which are realistically impossible to follow in practice? For each element in the vector, the following equation can be used to improve the estimates of the reciprocals: xn+1=xn(2−dxn). $g$ is $f$ shifted by $a$ units to the right: $$g(x)=f(x-a)\\g(x)=\frac{1}{(x-a)^2}$$ If the quadratic is a perfect square, then the function is a square. We can use this information to write a function of the form, To find the stretch factor, we can use another clear point on the graph, such as the y-intercept. Yes — pow: Yes. Only $2.99/month. Reciprocal of 5/6 = 6/5. The numerator has degree 2, while the denominator has degree 3. Finally, the degree of denominator is larger than the degree of the numerator, telling us this graph has a horizontal asymptote at. In this case, the graph is approaching the horizontal lineSee (Figure). What is the meaning of the "PRIMCELL.vasp" file generated by VASPKIT tool during bandstructure inputs generation? The reciprocal squared function can be restricted to the domain latex left 0 infty right latex. Of course, all functions are available for matrices by first casting it as an array: m.array(). : For simplicity call u = ( x − 3) 2 so that h ( x) = 1 / u + 4 = 1 / u + 4 u / u = ( 1 + 4 u) / u and now substituting back in we have h ( x) = ( 1 + 4 ( x − 3) 2) / ( x − 3) 2 which is the quotient of two polynomials as desired. Likewise, because the function will have a vertical asymptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by introducing a corresponding set of factors. The reciprocal of 7 is 1/7 Evaluating the function at zero gives the y-intercept: To find the x-intercepts, we determine when the numerator of the function is zero. Reciprocal Square RootStep. In the denominator, the leading term iswith coefficient 10. Reciprocal Function. Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function. $$\frac{1}{x^2-7}$$. Find the concentration (pounds per gallon) of sugar in the tank afterminutes. If so, how? Notice that there is a factor in the denominator that is not in the numerator,The zero for this factor isThe vertical asymptote isSee (Figure). Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote. 1 x 1 whose domain is x x 0 x a real number. For simplicity call$u=(x-3)^2$so that$h(x)=1/u + 4 = 1/u + 4u/u=(1+4u)/u$and now substituting back in we have$h(x)=(1+4(x-3)^2)/(x-3)^2$which is the quotient of two polynomials as desired. I am uncertain how to denote this. Next, we set the denominator equal to zero, and find that the vertical asymptote isbecause asWe then set the numerator equal to 0 and find the x-intercepts are atandFinally, we evaluate the function at 0 and find the y-intercept to be at. This occurs whenand whengiving us vertical asymptotes atand. More formally, transformations over a domain D are functions that map a set of elements of D (call them X) to another set of elements of D (call them Y). In this case, the graph is approaching the vertical lineas the input becomes close to zero. 6. powered by. Introduction to Exponential and Logarithmic Functions, 48. Since a fraction is only equal to zero when the numerator is zero, x-intercepts can only occur when the numerator of the rational function is equal to zero. It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. Making statements based on opinion; back them up with references or personal experience. For the following exercises, identify the removable discontinuity. The factor associated with the vertical asymptote atwas squared, so we know the behavior will be the same on both sides of the asymptote. Setting each factor equal to zero, we find x-intercepts atandAt each, the behavior will be linear (multiplicity 1), with the graph passing through the intercept. Reduce the expression by canceling common factors in the numerator and the denominator. This is true if the multiplicity of this factor is greater than or equal to that in the denominator. Find the vertical and horizontal asymptotes of the function: Vertical asymptotes atandhorizontal asymptote at. Input signal to the block to calculate the square root, signed square root, or reciprocal of square root. Sketch a graph of the reciprocal function shifted two units to the left and up three units. Watch Queue Queue. Since the water increases at 10 gallons per minute, and the sugar increases at 1 pound per minute, these are constant rates of change. In this case, the end behavior isThis tells us that as the inputs increase or decrease without bound, this function will behave similarly to the functionAs the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. There is a slant asymptote at, In the sugar concentration problem earlier, we created the equation.$f(x\pm k)$shifts a function to the left/right by$k$. Let= radius. Written without a variable in the denominator, this function will contain a negative integer power. See (Figure). $$\frac{1}{x^2-3}-4$$. College Algebra by cnxcollalg is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. Exponential and Logarithmic Functions, 42. I need 30 amps in a single room to run vegetable grow lighting. See, A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. To summarize, we use arrow notation to show thatoris approaching a particular value. I think the confusion here stems from the fact that the wording is vague. Rates of Change and Behavior of Graphs, 33. Fortunately, the effect on the shape of the graph at those intercepts is the same as we saw with polynomials. it is the same as y = 3x^0. Its domain is x x 0 its range is also x x 0 as an exponent. The material for the sides costs 10 cents/square foot. Find the vertical asymptotes and removable discontinuities of the graph of. Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote atwhereandare the leading coefficients ofandfor. A rational function will have a y-intercept at, if the function is defined at zero. Linear Inequalities and Absolute Value Inequalities, 24. Suppose we know that the cost of making a product is dependent on the number of items,produced. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. The reciprocal squared function shifted down 2 units and right 1 unit. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomores leave the rally. Because the numerator is the same degree as the denominator we know that asis the horizontal asymptote.$f$is a reciprocal squared function: $$f(x) = \frac{1}{x^2}$$ About the Book Author . This means there are no removable discontinuities. $$\frac{1}{x^2-3-4}$$ This video is unavailable. To find the equation of the slant asymptote, divideThe quotient isand the remainder is 2. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the ratio of freshmen to sophomores at 1 p.m. A vertical asymptote represents a value at which a rational function is undefined, so that value is not in the domain of the function. For example, the functionmay be re-written by factoring the numerator and the denominator. The following video shows how to use transformation to graph reciprocal functions. Is there a bias against mention your name on presentation slides? In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero. In (Figure), we see that the numerator of a rational function reveals the x-intercepts of the graph, whereas the denominator reveals the vertical asymptotes of the graph. Thanks for contributing an answer to Mathematics Stack Exchange! 5. Writing this expression as a single trig function? Upgrade to remove ads. For instance, if we had the function. $$f(x) = \frac{a}{{x - h}} + k$$ h is the horizontal translation if h is positive, shifts left if h is negative, shifts right h also shifts the vertical asymptote. Note that this graph crosses the horizontal asymptote. Please accept statistics, marketing cookies to watch this video. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The domain is all real numbers except those found in Step 2. Learn. The reciprocal squared function shifted down 2 units and right 1 unit. They are therefore faster than the Can a graph of a rational function have no vertical asymptote? Square the function in the denominator and place it in the denominator of the new fraction. By canceling common factors name on presentation slides to equal zero of higher Witt groups a! Mask its thermal signature gives the y-intercept isthe x-intercepts areandSee ( Figure ) y-intercept at, asymptote. 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PRIMCELL.vasp '' file generated by VASPKIT tool during bandstructure inputs generation layout legend with PyQGIS 3 of is! Atindicating vertical asymptotes of the formula of the numerator and numerator as the quotient of polynomial functions which... Intercept, suggesting linear factors so the graph would look similar to that of graph... The location of the lineSee ( Figure ), and end behavior and behavior! With non-negative integers for exponents divideThe quotient isand the remainder is 2 isand the remainder 2. For the following exercises, find the rational form lines, but not necessarily unchanged! By the transformation, but who knows 0 infty right latex 50 cubic meters we saw polynomials! Identity of cosecant function is displayed in ( Figure ) of your functions reflect ! Please accept statistics, marketing cookies to watch this video and notice some of features... X-Intercepts atandAt both, the ratio of sugar in the numerator is equal to zero and solve... 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# IBPS PO Quantitative Aptitude Questions with Answers Practice online test 33
Description: free IBPS PO Quantitative Aptitude Questions with Answers Practice test 33 for IBPS PO Preliminary and Main online test Prepare bank PO banking mock exams adda
1 . What approximate value should come in place of the question mark (?) in the following questions?
69.99% of $3^ 7$ of $2^ 5$ of $7^ 3$ of 734.965 = 218.8 - ?
3 8 13 18 23
2 . What approximate value should come in place of the question mark (?) in the following questions?
873.595 + 37.0949 $\times$ 41.769 - 59.572 = ?
2363 2463 2393 2286 2268
3 . What approximate value should come in place of the question mark (?) in the following questions?
14642.52 $\div$ ? = 24
590 610 790 650 670
4 . What approximate value should come in place of the question mark (?) in the following questions?
$(63.4)^ 2$ $\times$ 55.6 - 472 = ?
225040 228050 224030 223000 232000
5 . What approximate value should come in place of the question mark (?) in the following questions?
79% of 377 + 97% of 553 = ?
815 825 835 845 855
6 .
A train takes 1 second to cross a pole and 3 seconds to cross a platform 300 m long. Find the length of the train.
250 m 300 m 125 m 150 m None of these
7 .
What will be the percentage profit after selling an article at a certain price if there is a loss of 45% when the article is sold at one-third of the previous selling price?
45% 60% 55% Can't be determined None of these
8 .
Price registered an increase of 10% on foodgrains and 15% on other items of expenditure. If the ratio of expenditure of an employee on foodgrains and other items be 2 : 5, by how much should his salary be increased in order that he may maintain the same level of consumption as before? (Assume that there is no saving out of his monthly salary of `2590).
13.57% 12% 13% 18.42% None of these
$8\over 15$ $1\over 17$ $8\over 17$ $8\over 16$ $9\over 17$
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## Wednesday, July 7, 2021
### Upcoming features of QSoas and github repository
For the past years, most of the development has happened behind the scene in a private repository, and the code has appeared in the public repository only a couple of months before the release, in the release branch. I have now decided to publish the current code of QSoas in the github repository (in the public branch). This way, you can follow and use all the good things that were developed since the last release, and also verify whether any bug you have is still present in the currently developed version !
#### Upcoming features
This is the occasion to write a bit about the some of the features that have been added since the publication of the 3.0 release. Not all of them are polished nor documented yet, but here are a few teasers. The current version in github has:
• a comprehensive handling of column/row names, which makes it much easier to work with files with named columns (like the output files QSoas produces !);
• better handling of lists of meta-data, when there is one value of the meta for each segment or each Y column;
• handling of complex numbers in apply-formula;
• defining fits using external python code;
• a command for linear least squares (which has the huge advantage of being exact and not needing any initial parameters);
• commands to pause in a script or sort datasets in the stack;
• improvements over previous commands, in particular with eval;
• ... and more...
Check out the github repository if you want to know more about the new features !
As of now, no official date is planned for the 3.1 release, but this could happen during fall.
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### Comments
1 comment
Hi Jonathan,
It is currently not possible to use the Barrier algorithm in order to solve a non-convex QCQP to get a local minimum. Please also note that the feasible solution point found by the spatial B&B algorithm is currently not guaranteed to be a stationary point.
You could get the solution of the convex relaxation of a non-convex QCQP by solving the root relaxation and then getting the relaxation solution point from a $$\texttt{MIPNODE}$$ callback.
Best regards,
Jaromił
Please sign in to leave a comment.
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# Definition of the boundary map for chain complexes
1. Jan 29, 2013
### Tac-Tics
I've been poking around, learning a little about homology theory. I had a question about the boundary operator. Namely, how it's defined.
There's two definitions I've seen floating around. The first is at:
http://en.wikipedia.org/wiki/Simplicial_homology
The second, at
http://www.math.wsu.edu/faculty/bkrishna/FilesMath574/S12/LecNotes/Lec16_Math574_03062012.pdf [Broken]
The only difference seems to be the inclusion of a factor of (-1)i inside the sums.
My guess is that the extra factor doesn't matter, since there is some choice in how you construct chain. In other words, the fact that you're working with a FREE abelian group over the p-simplexes of your complex, flipping the signs results in an isomorphic group.
(If that's not the case, my other guess would be that the latter only works in Z/2Z, where sign doesn't matter anyway).
Is my reasoning sound? Or am I missing something?
Last edited by a moderator: May 6, 2017
2. Jan 30, 2013
### lavinia
With Z2 coefficients signs don't matter since minus 1 and one are the same. The Wikipedia definition of boundary is correct in general. You can check this with examples.
3. Jan 30, 2013
### Tac-Tics
Ah. Thank you.
Now that I think about it, you can' "choose" what group you want the coefficients to be in if your generating your groups freely anyway.
(I'm guessing that would be some quotient of the free group, determined by the type of coefficient you're interested in, but I'll worry about that later).
4. Jan 31, 2013
### lavinia
In homology I think you start with the free abelian group on simplices, define the boundary operator, then choose other coefficients than the integers by tensoring (over Z) each group with the coefficient group.You never start with a free group, always a free abelian group. It is a characterisitc of homology that the groups are always abelian, unlike the fundamental group which usually is not abelian.
5. Jan 31, 2013
### Tac-Tics
Yes. I meant "abelian", but omitted it to introduce some confusion :)
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# the number of digits
If we divide the number $111222333444555666777888999$ by $111$.In which way one can find the number of digits of the result
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will this help: wolframalpha.com/input/… – user59671 Feb 27 '13 at 16:46
Isn't it obviously 25? – MJD Feb 27 '13 at 16:47
9 digits representing 1-9 and then 16 zeros for a total of 25 digits. – JB King Feb 27 '13 at 17:11
$n=111222333444555666777888999>111000000000000000000000000$ so $\frac{n}{111}>1\cdot10^{24}$ which has 25 digits.
On the other hand, $111>100$, so $\frac{n}{111}<\frac{n}{100}$ which has 25 (non-fractional) digits.
All in all, the number of digits $d$ satisfies $25\le d \le 25$ so $d=25$.
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$111222333444555666777888999:111=1002003004005006007008009$
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Write $111222333444555666777888999$ as $\sum_{k=1}^9(10 - k) \times 111 \times 10^{3(k-1)}$ dividing by $111$ you get $\sum_{k=1}^9(10 - k) \times 10^{3(k-1)}$ and you get the largest value (the rest are smaller number, the sum does not change the number of digits) on this sum is $1 \times 10^{ 3 \times 8}$ that is $24$ zeros after $1$ which you get $25$.
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# Solving Estimating Equations using R packages
#### arun
##### New Member
Hello, I am trying to estimate e intraclass correlation parameter of of binary data using extended quasi-likelihood estimation methods.
The below are the estimating equations for the mean and dispersion parameters based on beta-binomial distribution.
Which, I captured from An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data
I need to know, how can I solve two estimating equations with two unknown parameters in R?
(Note: This is not a GLM model)
I tried it with randomly generated beta-binomial variates using rootSolve R package, but didn't work...
#### GretaGarbo
##### Human
I am not sure what you want to estimate with that model.
Maybe a later model by Nelder might interest you. The book by Lee Nelder and Pawitan about hierarchical generalized linear models. (And many publihed papers) There is a R package “hglm” by Rönnegård.
I have the impression that Nelder in later years did not like gee (generalized estimating equations) that much (According to “Statistical Science” in about 2009)
#### arun
##### New Member
Thank you GreatGarbo.. I will check out hglm.. than you for your suggestion.. But I am not trying to fit a glm, rather
I am simulating beta binomial sample variates as below
Code:
library(VGAM)
set.seed(100)
sample=rbetabinom(n=10, prob=0.4, rho=0.5,size=5)
n=5 #trials
N=10 #sample.size
z=sample/n #Proportions os success
Now I need to estimate the parameters prob and rho using extended quasi-likelihood method given above...
Can I uses the R package "rootSolve" to solve estimating equations?
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Advances in Operator Theory
$(p,q)$-type beta functions of second kind
Abstract
In the present article, we propose the $(p,q)$-variant of beta function of second kind and establish a relation between the generalized beta and gamma functions using some identities of the post-quantum calculus. As an application, we also propose the $(p,q)$-Baskakov-Durrmeyer operators, estimate moments and establish some direct results.
Article information
Source
Adv. Oper. Theory Volume 1, Number 1 (2016), 134-146.
Dates
Received: 17 October 2016
Accepted: 29 November 2016
First available in Project Euclid: 4 December 2017
Permanent link to this document
https://projecteuclid.org/euclid.aot/1512416212
Digital Object Identifier
doi:10.22034/aot.1609.1011
Zentralblatt MATH identifier
1359.41004
Subjects
Primary: 41A25: Rate of convergence, degree of approximation
Secondary: 41A39
Citation
Aral, Ali; Gupta, Vijay. $(p,q)$-type beta functions of second kind. Adv. Oper. Theory 1 (2016), no. 1, 134--146. doi:10.22034/aot.1609.1011. https://projecteuclid.org/euclid.aot/1512416212
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# Sharp MSE Bounds for Proximal Denoising
Research paper by Samet Oymak, Babak Hassibi
Indexed on: 14 Nov '13Published on: 14 Nov '13Published in: Computer Science - Information Theory
#### Abstract
Denoising has to do with estimating a signal $x_0$ from its noisy observations $y=x_0+z$. In this paper, we focus on the "structured denoising problem", where the signal $x_0$ possesses a certain structure and $z$ has independent normally distributed entries with mean zero and variance $\sigma^2$. We employ a structure-inducing convex function $f(\cdot)$ and solve $\min_x\{\frac{1}{2}\|y-x\|_2^2+\sigma\lambda f(x)\}$ to estimate $x_0$, for some $\lambda>0$. Common choices for $f(\cdot)$ include the $\ell_1$ norm for sparse vectors, the $\ell_1-\ell_2$ norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate $x^*$ is the normalized mean-squared-error $\text{NMSE}(\sigma)=\frac{\mathbb{E}\|x^*-x_0\|_2^2}{\sigma^2}$. We show that NMSE is maximized as $\sigma\rightarrow 0$ and we find the \emph{exact} worst case NMSE, which has a simple geometric interpretation: the mean-squared-distance of a standard normal vector to the $\lambda$-scaled subdifferential $\lambda\partial f(x_0)$. When $\lambda$ is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem $\min_{f(x)\leq f(x_0)}\{\|y-x\|_2\}$. The paper also connects these results to the generalized LASSO problem, in which, one solves $\min_{f(x)\leq f(x_0)}\{\|y-Ax\|_2\}$ to estimate $x_0$ from noisy linear observations $y=Ax_0+z$. We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a "phase transition" as a function of number of observations. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.
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# Find the p value following the exponential distribution $\mu=3$
I want to find the $$p$$-value (manually) of the following Hypothesis testing.
$$H_0:\mu\leq 3 \quad \text{vs} \quad H_1:\mu >3$$
The main thing I know is that
$$P(\mathrm{Re}\,j \mid \mu \leq 3)=P(X\geq 3 \mid \mu \leq 3)= e^{-1} \approx0.36$$
Can I use the $$z$$ value and use the formula probability of $$z$$? Or from where can I start?
Your null hypothesis is that your exponential distribution has a rate $$\mu$$ which is $$\leq 3$$. Your alternate hypothesis is that $$\mu \geq 3$$. Now, you get some observation, $$x$$. What is the probability that this sample is consistent with the null-hypothesis? Meaning, what is the probability that the null hypothesis would generate a sample $$\geq x$$? Conditional on $$\mu$$, this is simply $$e^{-\mu x}$$. Since your null hypothesis is that $$\mu \leq 3$$, you integrate over it to get the p-value:
$$p = \int\limits_0^3 e^{-\mu x}d \mu = \frac{1-e^{-3x}}{x}$$
• xAlso, it shouldn't be divided by $x$ instead of 3? – Lexie Walker Mar 13 at 17:30
• Yes, sorry.. fixed the typo. Yes, you need to have a very large $x$ for your p-value to be small. The larger the $x$, the smaller the chance an exponential with rate $<3$ generated it. – Rohit Pandey Mar 13 at 17:53
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# Comparison tests for series
1. Oct 16, 2011
### miglo
1. The problem statement, all variables and given/known data
$$\sum_{n=2}^{\infty}\frac{1}{n\sqrt{n^2-1}}$$
2. Relevant equations
direct comparison test
limit comparison test
3. The attempt at a solution
so i kind of cheated and looked at the back of my book and it says to compare with $\frac{1}{n^{3/2}}$
so i tried using the direct comparison test and tried to show that the original series converges if $$\frac{1}{n\sqrt{n^2-1}}<\frac{1}{n^{3/2}}$$ since $$\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}$$ is a convergent p-series test
i just dont know how to actually show $$\frac{1}{n\sqrt{n^2-1}}<\frac{1}{n^{3/2}}$$
or am i using the wrong test? limit comparison? by the way the only tests i've covered in my class are the divergence, p-series, integral, direct comparison, limit comparison tests and geometric and telescoping series
Last edited by a moderator: Oct 16, 2011
2. Oct 17, 2011
### susskind_leon
you know that $n^{3/2} =n\sqrt{n}$, right?
now
$$n \sqrt{n} < n \sqrt{n^2-1}$$
$$\sqrt{n} < \sqrt{n^2-1}$$
$$n < n^2-1$$
which is valid for all n >= 2
3. Oct 17, 2011
### LCKurtz
Since $n\sqrt{n^2-1}$ is of order $n^2$ this suggests the very easy limit comparison test with $\sum\frac 1 {n^2}$.
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## staceyyyy1 2 years ago If f(x) = 2x + 6, find f(7). 14 15 20 6
1. dpaInc
replace the x with a 7 and evaluate 2$$\cdot$$7 + 6 = ???
2. Pac1f1cIslander
It's C :D
3. staceyyyy1
Thank You! @Pac1f1cIslander
4. Not the answer you are looking for?
Search for more explanations.
Search OpenStudy
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# Gradient of spectral function on noncompact homogeneous space
Let $(M,g)$ be a noncompact Riemannian manifold whose isometry group acts transitively on $M$, i.e. a (not necessarily normal) homogeneous space. Let $e_{\lambda}(x,y)$ be the integral kernel of $f \mapsto \int_{0}^{\lambda} dE_{\nu}(f)$ where $dE_{\nu}$ is the spectral measure of the (non-negative) Laplacian associated to $(M,g)$. Is there a relatively simple proof that
$$\int_M |\nabla_x e_{\lambda}(x,y)|^2_g\, dy = \int_M e_{\lambda}(x,y) \cdot \Delta_x e_{\lambda}(x,y)\, dy \ ?$$
Note that the differentiation is in $x$ and the integration is in $y$. In particular, it's surely false for most non-homogeneous manifolds.
I ask for `relatively simple' because I know of a proof using the expression of the Laplacian as the generator of the heat semi-group. That proof seems to be overkill.
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Experimental Study on the Evaluation of Frost-Resistance of High-Strength Concrete Damaged by Frost at Early Age in Cold Climates
Title & Authors
Experimental Study on the Evaluation of Frost-Resistance of High-Strength Concrete Damaged by Frost at Early Age in Cold Climates
Gwon, Yeong-Jin;
Abstract
One of ways to make high-strength concrete is for the mix contain particles graded down to the finest size : this is achieved by the use of fly ash, silica fume which fills the spaces between the cement particle and between the aggregate and the cement particles. And, the mix needs a sufficient workability. This is achieved by the use of a superplasticizer. This study is to investigate frost resistance of high-strength concrete at early age, with ratio of tensile strength and recovery of compressive strength, when high-strength concrete is placed in cold climates. According to this study, it is necessary to ensure 4 % of air content, 5 kgf/$\small{textrm{cm}^2}$ of tensile strength, at least, for frost resistance of high-strength concrete at early age.
Keywords
high-strength concrete;frost resistance;ratio of tensile strength;recovery of compressive strength;air content;
Language
Korean
Cited by
References
1.
한국레미콘공업협회 레미콘 23호, 1993. pp.8-23
2.
コンクリ-ト構造物の凍害とその對策シンポジウム論文集, 1993. pp.33-38
3.
コンクリ-ト構造物の凍害とその對策シンポジウム論文集, 1993. pp.19-26
4.
5.
コンクリ-ト構造物の凍害とその對策シンポジウム論文集, 1993. pp.59-170
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## time out a process in haskell
i’m trying to figure out a good way to timeout a process in haskell. i will look later to see if someone actually already did this. the problem i have is that i want to exit the program with a timeout failure which is the opposite of what i have so far:
import Control.Concurrent
import System.Exit
timeout n proc = do
i <- forkIO proc
timer n
putStrLn "timed out!"
main = do
timeout 1 $forever$ putStrLn "bob!"
this is just a sketch of what i worked from. what i think i want to do is fork a timer process and when it ends, i want the entire program to end. i’ve tried adding exitFailure but it only ends that thread, not the entire process.
just kidding
i figured it out. i did something awful like this:
do
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kidzsearch.com > wiki
# C
Paradigm(s) C++ logo used by ISO. Multi-paradigm:[1] procedural, functional, object-oriented, generic 1983 Bjarne Stroustrup Bjarne StroustrupBell LabsISO/IEC JTC1/SC22/WG21 ISO/IEC 14882:2017 / 1 December 2017; 4 years ago Static, unsafe, nominative ISO/IEC C++ 1998, ISO/IEC C++ 2003, ISO/IEC C++ 2011, ISO/IEC C++ 2014, ISO/IEC C++ 2017, C, Simula, Ada 83, ALGOL 68, CLU, ML[1] Perl, LPC, Lua, Pike, Ada 95, Java, PHP, D, C99, C#, Falcon Cross-platform (multi-platform) .h .hh .hpp .hxx .h++ .cc .cpp .cxx .c++ C++ Programming at Wikibooks
Bjarne Stroustrup, the creator of C++
C++ (pronounced "see plus plus") is a computer programming language based on C. It was created for writing programs for many different purposes. In the 1990s, C++ became one of the most used programming languages in the world.
The C++ programming language was developed by Bjarne Stroustrup at Bell Labs in the 1980s, and was originally named "C with classes". The language was planned as an improvement on the C programming language, adding features based on object-oriented programming. Step by step, a lot of advanced features were added to the language, like operator overloading, exception handling and templates.
C++ Archived 2020-07-13 at the Wayback Machine runs on a variety of platforms, such as Windows, Mac OS, and the various versions of UNIX. C++ is simple and practical approach to describe the concepts of C++ for beginners Archived 2020-07-13 at the Wayback Machine to advanced software engineers.
C++ is a general-purpose programing language which means that it can be used to create different variety of applications. C++ is used for variety of application domains.
## Example
The following text is C++ source code and it will write the words "Hello World!" on the screen when it has been compiled and is executed. This program is typically the first program a programmer would write while learning about programming languages.
// This is a comment. It's for *people* to read, not computers. It's usually used to describe the program.
// Make the I/O standard library available for use in the program.
#include <iostream>
using namespace std;
// We are now defining the main function; it is the function run when the program starts.
int main()
{
// Printing a message to the screen using the standard output stream std::cout.
cout << "Hello World!";
}
This program is similar to the last, except it will add 3 + 2 and print the answer instead of "Hello World!".
#include <iostream>
int main()
{
// Print a simple calculation.
std::cout << 3 + 2;
}
This program subtracts, multiplies, divides and then prints the answer on the screen.
#include <iostream>
int main()
{
// Create and initialize 3 variables, a, b, and c, to 5, 10, and 20.
int a = 5;
int b = 10;
int c = 20;
// Print calculations.
std::cout << a-b-c;
std::cout << a*b*c;
std::cout << a/b/c;
}
## References
1. Stroustrup, Bjarne (1997). "1". The C++ Programming Language (Third ed.). . .
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Category:Quotient Groups
This category contains results about Quotient Groups.
Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Then the left coset space $G / N$ is a group, where the group product is defined as:
$\paren {a N} \paren {b N} = \paren {a b} N$
$G / N$ is called the quotient group of $G$ by $N$.
|
American Institute of Mathematical Sciences
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Hidden Markov models with threshold effects and their applications to oil price forecasting
April 2017, 13(2): 775-801. doi: 10.3934/jimo.2016046
Auction and contracting mechanisms for channel coordination with consideration of participants' risk attitudes
1 College of Business, Qingdao University, Qingdao, Shandong, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
* Corresponding author: Y.C.E. Lee
Received July 2014 Revised June 2016 Published August 2016
Fund Project: This research was supported by the Research Committee of The Hong Kong Polytechnic University, the National Natural Science Foundation of China (No.11401331), China Postdoctoral Foundation (No.2016M592148) and the Foundation for Guide Scientific and Technological Achievements of Qingdao (No. 14-2-4-57-jch).
This paper considers a two-supplier one-retailer coordinated supply chain system with auction and contracting mechanism incorporating participants' risk attitudes. The risk attitude is quantified using the value-at-risk (VaR) measure and the retailer faces a stochastic linear price-dependent demand function. In the supply chain, the suppliers (providing identical products) compete with each other in order to win the ordering contract of the retailer. Several auction and contracting mechanisms are developed and compared. It can be analytically shown that the retail price of the risk-averse system is higher than that of the risk-neutral system, but the order quantity is lower than that of the risk-neutral system.
Citation: Cheng Ma, Y. C. E. Lee, Chi Kin Chan, Yan Wei. Auction and contracting mechanisms for channel coordination with consideration of participants' risk attitudes. Journal of Industrial & Management Optimization, 2017, 13 (2) : 775-801. doi: 10.3934/jimo.2016046
References:
show all references
References:
Comparison of the optimal ordering quantities and optimal retail prices obtained under the complete information and asymmetric information situations
Demand Disturbance $\thicksim$ Exponential Demand Disturbance $\thicksim$ Normal Complete Information Asymmetric Information Complete Information Asymmetric Information $\alpha$ $s_1$ $q_{1,\alpha}^\star$ $r_{1,\alpha}^\star$ $\bar{q}_{1,\alpha}^\star$ $\bar{r}_{1,\alpha}^\star$ $q_{2,\alpha}^\star$ $r_{2,\alpha}^\star$ $\ \bar{q}_{2,\alpha}^\star$ $\bar{r}_{2,\alpha}^\star$ 0.1 10 23.15 19.79 16.47 21.46 24.64 20.16 17.94 21.83 15 13.15 22.29 14.64 22.26 0.05 10 23.50 19.87 16.82 21.54 24.82 20.21 18.14 21.88 15 13.50 22.37 14.82 22.71 0.01 10 24.30 20.08 17.62 21.75 25.17 20.29 18.49 21.96 15 14.30 22.58 15.17 22.79
Demand Disturbance $\thicksim$ Exponential Demand Disturbance $\thicksim$ Normal Complete Information Asymmetric Information Complete Information Asymmetric Information $\alpha$ $s_1$ $q_{1,\alpha}^\star$ $r_{1,\alpha}^\star$ $\bar{q}_{1,\alpha}^\star$ $\bar{r}_{1,\alpha}^\star$ $q_{2,\alpha}^\star$ $r_{2,\alpha}^\star$ $\ \bar{q}_{2,\alpha}^\star$ $\bar{r}_{2,\alpha}^\star$ 0.1 10 23.15 19.79 16.47 21.46 24.64 20.16 17.94 21.83 15 13.15 22.29 14.64 22.26 0.05 10 23.50 19.87 16.82 21.54 24.82 20.21 18.14 21.88 15 13.50 22.37 14.82 22.71 0.01 10 24.30 20.08 17.62 21.75 25.17 20.29 18.49 21.96 15 14.30 22.58 15.17 22.79
Comparison of the performance of the simple model with the two-part contract model when the demand disturbance follows an exponential distribution
Coordinated Policy Independent Policy Two-Part Contract $\alpha$ $s_1$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ 0.1 13 134.00 0.00 134.00 33.50 67.00 100.50 73.54 60.46 134.00 15 43.24 90.76 17 20.94 113.06 0.05 13 138.04 0.00 138.04 34.51 69.02 103.53 76.54 61.50 138.04 15 45.55 92.49 17 22.55 115.49 0.01 13 147.65 0.00 147.65 36.91 73.83 110.74 83.75 63.90 147.65 15 51.14 96.51 17 26.54 121.11
Coordinated Policy Independent Policy Two-Part Contract $\alpha$ $s_1$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ 0.1 13 134.00 0.00 134.00 33.50 67.00 100.50 73.54 60.46 134.00 15 43.24 90.76 17 20.94 113.06 0.05 13 138.04 0.00 138.04 34.51 69.02 103.53 76.54 61.50 138.04 15 45.55 92.49 17 22.55 115.49 0.01 13 147.65 0.00 147.65 36.91 73.83 110.74 83.75 63.90 147.65 15 51.14 96.51 17 26.54 121.11
Comparison of the performance of the simple model with the two-part contract model when the demand disturbance follows a normal distribution
Coordinated Policy Independent Policy Two-Part Contract $\alpha$ $s_1$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ 0.1 13 151.78 0.00 151.78 37.95 75.89 113.84 86.86 64.92 151.78 15 53.58 98.20 17 28.30 123.48 0.05 13 154.04 0.00 154.04 38.51 77.02 115.53 88.57 65.47 154.04 15 54.93 99.11 17 29.28 124.76 0.01 13 158.32 0.00 158.32 39.58 79.16 118.74 91.82 66.50 158.32 15 57.49 100.83 17 31.16 127.16
Coordinated Policy Independent Policy Two-Part Contract $\alpha$ $s_1$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ $\Pi_{1,r,\alpha}^\star$ $\Pi_{1,s_1,\alpha}^\star$ $\Pi_{1,SC,\alpha}^\star$ 0.1 13 151.78 0.00 151.78 37.95 75.89 113.84 86.86 64.92 151.78 15 53.58 98.20 17 28.30 123.48 0.05 13 154.04 0.00 154.04 38.51 77.02 115.53 88.57 65.47 154.04 15 54.93 99.11 17 29.28 124.76 0.01 13 158.32 0.00 158.32 39.58 79.16 118.74 91.82 66.50 158.32 15 57.49 100.83 17 31.16 127.16
Comparison of the optimal ordering quantity and the optimal retail price of a risk-averse and a risk neutral supply chain
Demand Disturbance $\thicksim$ Exponential Demand Disturbance $\thicksim$ Normal Risk Averse Risk Neutral Risk Averse Risk Neutral $\alpha$ $q_{1,\alpha}^\star$ $r_{1,\alpha}^\star$ $q_{1,N}^\star$ $r_{1,N}^\star$ $SL(r_{1,N}^\star)$ $q_{2,\alpha}^\star$ $r_{2,\alpha}^\star$ $q_{2,N}^\star$ $r_{2,N}^\star$ $SL(r_{2,N}^\star)$ 0.1 5.88 12.85 6.62 11.45 0.91 7.37 15.83 8.17 14.52 0.92 0.05 6.22 13.55 7.55 16.19 0.01 7.03 15.16 7.89 16.88
Demand Disturbance $\thicksim$ Exponential Demand Disturbance $\thicksim$ Normal Risk Averse Risk Neutral Risk Averse Risk Neutral $\alpha$ $q_{1,\alpha}^\star$ $r_{1,\alpha}^\star$ $q_{1,N}^\star$ $r_{1,N}^\star$ $SL(r_{1,N}^\star)$ $q_{2,\alpha}^\star$ $r_{2,\alpha}^\star$ $q_{2,N}^\star$ $r_{2,N}^\star$ $SL(r_{2,N}^\star)$ 0.1 5.88 12.85 6.62 11.45 0.91 7.37 15.83 8.17 14.52 0.92 0.05 6.22 13.55 7.55 16.19 0.01 7.03 15.16 7.89 16.88
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2019 Impact Factor: 1.366
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# Selection rules for Hydrogen
temp050505
Good afternoon,
Does the selection rules have a condition on $\Delta n$ ? I have not found a website or a book that show transitions between $2S_{1/2}$ and $2P_{3/2}$, that's why I was wondering if $\Delta n = 0$, with respect to the other selection rules, are allowed transitions.
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# How do you write 4.5 times 10 ^ 4 in standard notation?
$4.5 \times {10}^{4} = 45 , 000$
$4.5 \times {10}^{4} = 4.5 \times 10 , 000 = 45 \times 1000 = 45 , 000$
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# Schreiber StringMath2017
A talk that I have given:
$\,$
$\;$ Urs Schreiber (CAS Prague & HCM Bonn)
$\;$ Super $p$-Brane Theory emerging from Super Homotopy Theory
$\;$ talk at String Math 2017, Hamburg $\;\;\;$ (slides, expo, video)
$\,$
Abstract. It is a notorious open problem to determine the nature of the non-pertubative theory formerly known as Strings. I present results showing that, rationally, many of its phenomena emerge as stages of a Whitehead tower, invariant modulo R-symmetry, that emerges out of the superpoint regarded in super-geometric rational homotopy theory:
This includes super-spacetime as such, the bouquet of all Green-Schwarz super p-branes, D-brane charge in twisted K-theory, M-brane charge, double dimensional reduction, T-duality, Buscher rules for RR-fields, doubled spacetimes, F-theory fibrations, S-duality. The orbifold $S^4/S^1$ (familiar from the near horizon geometry of M5-branes at A-type singularities) appears in a surprising unifying role.
These results (arXiv:1611.06536, arXiv:1702.01774 and arXiv:1806.01115) are joint with Hisham Sati and Vincent Braunack-Mayer, John Huerta, Domenico Fiorenza, see below.
Related talks:
Lecture notes:
all summed up in:
$\,$
Last revised on May 30, 2019 at 12:27:04. See the history of this page for a list of all contributions to it.
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# scp to remote Windows hosts with spaces in path: ambiguous target
I'm trying to scp a file from my local linux machine to a remote Windows machine, and I'm coming up with some inconsistencies in how scp handles Windows file paths with spaces...
This works, note that spaces are properly escaped in the path to the local file:
scp /home/will/file\ with\ spaces.txt remote@host:D:/Users/will/Downloads/
However, this does not work, despite the space in the "Google Drive" folder being properly escaped:
scp /home/will/file\ with\ spaces.txt remote@host:D:/Users/will/Google\\ Drive/Documents/Computer_Stuff/Home_Lab/folder
I've gone through a ton of resources online where people have similar problems, and all their solutions didn't work for me.
I tried putting the Windows file path in double quotes, single quotes, both with and without escaping spaces; I've tried using double \\ and triple \\\ to escape spaces; I even tried escaping the colon (D\:); and I tried explicitly stating a target filename and not. Nothing worked.
Then I found this answer, and only method #2 works! Why? What is the difference between surrounding the Windows file path in '", and "'? Why can I simply escape spaces without using any quotes at all in the local Linux file path, but not in the remote Windows path?
• Why don't you just single quote the paths? 'remote@host:D:/Users/will/Google Drive/Documents/...' May 11 '18 at 15:02
• since the remote host is a Windows Maschine i would try to use a Windows Style formatted/escaped Path like this D:\Users\will\Google^ Drive\Documents\Computer_Stuff\Home_Lab\folder May 11 '18 at 15:15
• Windows has long been agnostic as to whether to use forward slashes or backslashes to delimit directories within paths. May 11 '18 at 15:18
I was going crazy on Windows 10 -- this worked for me
scp -r user@hostname:"/Library/Application' 'Support/Adobe/Common/Plug-ins/7.0/" ./Adobe
• THANK YOU 🙌🙌🙌 Jul 18 at 20:36
As mentioned in the question and answered here, method 2 was the solution that worked in my case. Adding as an answer for convenience of later readers.
• Windows 10
• Git bash scp
Single / Double Quotes:
scp ./myfile.txt user@host:'"/c:/AppName/a poorly named dir/sub_dir/myfile.txt"'
• OMFG!!!! Single-double-quote!!!! This magic just saved my sanity! Nov 23 at 13:45
Use the -T option:
scp -T [email protected]:'"c:\path with\spaces in\it\foo.txt"' .
Your shell is seeing Google\\ Drive, and parsing it as Google, followed by a literal backslash, a space (which separates the arguments to scp, followed by Drive, and then passing that on to SCP which doesn't know what to do now that you're sending it too many arguments.
To make this more clear, compare how the shell parses Google\ Drive and Google\\ Drive:
$cat 443236.sh #!/bin/bash foo=(Google\ Drive) echo "Unrolling \$foo[@]:"
for f in "${foo[@]}"; do echo "'$f'"
done
echo "Unrolling \$foo[@]:" for f in "${foo[@]}"; do
echo "'$f'" done$ ./443236.sh
Unrolling $foo[@]: 'Google Drive' Unrolling$foo[@]:
• The two required arguments for an scp invocation are the source and destination for the file to be copied, e. g. scp /path/to/some/file user@host:/location/for/the/file or scp user@host:relative/path/to/file ./directory/. Oct 22 '18 at 15:39
• An easier illustration of unrolling is printf '<%s>\n' whatever, e.g. printf '<%s>\n' Google\ Drive; printf '<%s>\n' Google\\ Drive Jul 9 '19 at 23:25
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# Complex Numbers Equation [duplicate]
Possible Duplicate:
How can you find the complex roots of i?
How can I find the solutions of the equation $$(2z+1)^5-i=0,$$ over the complex numbers $z\in\mathbb{C}$?
-
## marked as duplicate by Jonas Meyer, Srivatsan, Asaf Karagila, Zev ChonolesDec 20 '11 at 21:57
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
## 1 Answer
Find the fifth roots of $i$, subtract $1$, and divide by $2$.
-
yeah but than i have to calculate $cos(36)$ and I have to calculate it without a calculator – Some1 Dec 20 '11 at 20:23
@Some1 It is usually considered equally acceptable to express complex numbers in "standard form" ($a+bi$), "polar form" ($r\,\mathrm{e}^{it}$) or a combination. Your roots can be expressed easily as a combination of polar and standard elements. On the other hand, trig functions of $\pi/5$ have exact values. – alex.jordan Dec 20 '11 at 20:27
$\cos(\pi/5) = (1+\sqrt{5}\,)/4$. – GEdgar Dec 20 '11 at 20:27
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On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). A sphere is not a Euclidean space, but locally the laws of Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. A sphere can be represented by a collection of two dimensional maps, therefore a sphere is a manifold.
Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...
Additional structures are often defined on manifolds. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity. Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... For other uses, see Calculus (disambiguation). ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... Phase space of a dynamical system with focal stability. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ...
### Circle
Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.
$chi_{mathrm{top}}(x,y) = x . ,!$
Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle. Together, these parts cover the whole circle and the four charts form an atlas for the circle. In topology, a branch of mathematics, an atlas describes how a complicated space called a manifold is glued together from simpler pieces. ...
The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts χtop and χright each map this part into the interval (0,1). Thus a function T from (0,1) to itself can be constructed, which first uses the inverse of the top chart to reach the circle and then follows the right chart back to the interval. Let a be any number in (0,1), then: A function Æ’ and its inverse ƒ–1. ...
begin{align} T(a) &= chi_{mathrm{right}}left(chi_{mathrm{top}}^{-1}(a)right) &= chi_{mathrm{right}}left(a, sqrt{1-a^2}right) &= sqrt{1-a^2} . end{align}
Such a function is called a transition map.
Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.
The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of some choice. Consider the charts Image File history File links Download high resolution version (1000x1000, 75 KB) File links The following pages link to this file: User talk:KSmrq Manifold/rewrite ... Image File history File links Download high resolution version (1000x1000, 75 KB) File links The following pages link to this file: User talk:KSmrq Manifold/rewrite ...
$chi_{mathrm{minus}}(x,y) = s = frac{y}{1+x}$
and
$chi_{mathrm{plus}}(x,y) = t = frac{y}{1-x}.$
Here s is the slope of the line through the point at coordinates (x,y) and the fixed pivot point (−1,0); t is the mirror image, with pivot point (+1,0). The inverse mapping from s to (x,y) is given by
begin{align} x &= frac{1-s^2}{1+s^2} y &= frac{2s}{1+s^2} . end{align}
It can easily be confirmed that x2+y2 = 1 for all values of the slope s. These two charts provide a second atlas for the circle, with
$t = frac{1}{s} . ,!$
Each chart omits a single point, either (−1,0) for s or (+1,0) for t, so neither chart alone is sufficient to cover the whole circle. Topology can prove that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "glueing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility.
### Other curves
Four manifolds from algebraic curves: circles, parabola, hyperbola, cubic.
Manifolds need not be connected (all in "one piece"); thus a pair of separate circles is also a manifold. They need not be closed; thus a line segment without its end points is a manifold. And they need not be finite; thus a parabola is a manifold. Putting these freedoms together, two other example manifolds are a hyperbola (two open, infinite pieces) and the locus of points on the cubic curve y2 = x3x (a closed loop piece and an open, infinite piece). Image File history File links Download high resolution version (1000x1000, 42 KB) PNG file created as SVG, rendered by Batik, and uploaded by author. ... Image File history File links Download high resolution version (1000x1000, 42 KB) PNG file created as SVG, rendered by Batik, and uploaded by author. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παÏαβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ... In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation. ...
However, we exclude examples like two touching circles that share a point to form a figure-8; at the shared point we cannot create a satisfactory chart. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line.
### Enriched circle
Viewed using calculus, the circle transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable. The transition map T, and all the others, are differentiable on (0, 1); therefore, with this atlas the circle is a differentiable manifold. It is also smooth and analytic because the transition functions have these properties as well. For other uses, see Calculus (disambiguation). ... For other uses, see Derivative (disambiguation). ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ...
Other circle properties allow it to meet the requirements of more specialized types of manifold. For example, the circle has a notion of distance between two points, the arc-length between the points; hence it is a Riemannian manifold. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
## History
For more details on this topic, see History of manifolds and varieties.
The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, a surface is a two-dimensional manifold. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...
### Prehistory
Before the modern concept of a manifold there were several important results.
Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. Saccheri first studied them in 1733. Lobachevsky, Bolyai, and Riemann developed them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive curvature, respectively. Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... For other uses, see Euclid (disambiguation). ... a and b are parallel, the transversal t produces congruent angles. ... Giovanni Gerolamo Saccheri (September 5, 1667 – October 25, 1733) was an Italian Jesuit priest and mathematician. ... Events February 12 - British colonist James Oglethorpe founds Savannah, Georgia. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (ÐиколаÌй ИваÌнович ЛобачеÌвÑкий) (December 1, 1792–February 24, 1856 (N.S.); November 20, 1792–February 12, 1856 (O.S.)) was a Russian mathematician. ... János Bolyai (December 15, 1802–January 27, 1860) was a Hungarian mathematician. ... Bernhard Riemann. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Lines through a given point P and asymptotic to line l. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ...
Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. Johann Carl Friedrich Gauss (pronounced , ; in German usually Gauß, Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... The Theorema Egregium (Remarkable Theorem) is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... An open surface with X-, Y-, and Z-contours shown. ... The ambient space, in mathematics, is the space surrounding a mathematical object. ...
Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners), E edges, and F faces, A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ... Euler redirects here. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...
V-E+F= 2.
The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a 'map' with V vertices, E edges, and F faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope.[1] Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a torus can be sliced open by its 'parallel' and 'meridian' circles, creating a map with V=1 vertex, E=2 edges, and F=1 face. Thus the Euler characteristic of the torus is 1-2+1=0. The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature. For other uses, see Sphere (disambiguation). ... A torus This article is about the surface and mathematical concept of a torus. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ... The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ...
### Synthesis
Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds, now known as Jacobians. Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables, although these ideas were way ahead of their time. Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Nedstrand, near Finnøy where his father acted as rector. ... Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... Bernhard Riemann. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...
Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed by Simeon Poisson, Jacobi, and William Rowan Hamilton. The possible states of a mechanical system are thought to be points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicated formations, e.g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph Lagrange, gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by Henri Poincaré, one of the founders of topology. Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics, constructed from the eighteenth century onwards as a formulation of the subject as founded by Isaac Newton. ... Simeon Poisson. ... For other persons named William Hamilton, see William Hamilton (disambiguation). ... Phase space of a dynamical system with focal stability. ... Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... 2-dimensional renderings (ie. ... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Euler redirects here. ... Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...
Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Bernhard Riemann. William Kingdon Clifford William Kingdon Clifford, FRS (May 4, 1845 - March 3, 1879) was an English mathematician who also wrote a fair bit on philosophy. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... Bernhard Riemann. ...
Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a topological space that followed shortly. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Hassler Whitney (23 March 1907 – 10 May 1989) was an American mathematician who was one of the founders of singularity theory, PhB, Yale University, 1928; MusB, 1929; ScD (Honorary), 1947; PhD, Harvard University, under G.D. Birkhoff, 1932. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...
### Topology of manifolds: highlights
Two-dimensional manifolds, also known as surfaces, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn. Henri Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. After nearly a century of effort by many mathematicians, starting with Poincaré himself, a consensus among experts (as of 2006) is that Grigori Perelman has proved the Poincaré conjecture (see the Solution of the Poincaré conjecture). Bill Thurston's geometrization program, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by Michael Freedman and in a different setting, by Simon Donaldson, who was motivated by the then recent progress in theoretical physics (Yang-Mills theory), where they serve as a substitute for ordinary 'flat' space-time. Important work on higher-dimensional manifolds, including analogues of the Poincaré conjecture, had been done earlier by René Thom, John Milnor, Stephen Smale and Sergei Novikov. One of the most pervasive and flexible techniques underlying much work on the topology of manifolds is Morse theory. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... Poul Heegaard (November 2, 1871 — February 7, 1948) was a mathematician active in the field of topology. ... Max Dehn (November 13, 1878 – June 27, 1952) was a German mathematician. ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... Grigori Yakovlevich Perelman (Russian: ), born 13 June 1966 in Leningrad, USSR (now St. ... William Paul Thurston (born October 30, 1946) is an American mathematician. ... Thurstons geometrization conjecture states that compact 3-manifolds can be decomposed into pieces with geometric structures. ... Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, USA) is a mathematician at Microsoft Research. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is an English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... René Thom (September 2, 1923 - October 25, 2002) was a French mathematician and founder of the catastrophe theory. ... John Willard Milnor (b. ... Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan, and winner of the Fields Medal in 1966. ... Sergei Petrovich Novikov (also Serguei) (Russian: Сергей Петрович Ðовиков) (born 20 March 1938) is a Russian mathematician, noted for work in both algebraic topology and soliton theory. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A Morse function is also an expression for an anharmonic oscillator In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ...
## Mathematical definition
For more details on this topic, see Categories of manifolds.
Informally, a manifold is a space that is "modeled on" Euclidean space. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
There are many different kinds of manifolds and generalizations. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, most often a differentiable structure. In terms of constructing manifolds via patching, a manifold has an additional structure if the transition maps between different patches satisfy axioms beyond just continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, and so differentiable on the manifold as a whole. Geometry and Topology (ISSN 1364-0380 online, 1465-3060 printed) is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
Formally, a topological manifold[2] is a second countable Hausdorff space that is locally homeomorphic to Euclidean space. In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second countable if its topology has a countable base. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ...
Second countable and Hausdorff are point-set conditions; second countable excludes spaces of higher cardinality such as the long line, while Hausdorff excludes spaces such as "the line with two origins" (these generalized manifolds are discussed in non-Hausdorff manifolds). In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In topology, the long line is a topological space analogous to the real line, but much longer. ...
Locally homeomorphic to Euclidean space means[3] that every point has a neighborhood homeomorphic to an open Euclidean n-ball, This word should not be confused with homomorphism. ...
$mathbf{B}^n = { (x_1, x_2, dots, x_n)inmathbb{R}^n mid x_1^2 + x_2^2 + cdots + x_n^2 < 1 }.$
Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed n-ball), and such a space is called an n-manifold; however, some authors admit manifolds where different points can have different dimensions. Since dimension is a local invariant, each connected component has a fixed dimension. 2-dimensional renderings (ie. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...
Scheme-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In mathematics, an analytic manifold is a topological manifold with analytic transition maps. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...
The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. This omits the point-set axioms (allowing higher cardinalities and non-Hausdorff manifolds) and finite dimension (allowing various manifolds from functional analysis). Usually one relaxes one or the other condition: manifolds without the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
## Charts, atlases, and transition maps
The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a differentiable manifold can be described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can properly represent the entire Earth. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure. In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...
### Charts
A coordinate map, a coordinate chart, or simply a chart, of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and interest focuses on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions. In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... This word should not be confused with homomorphism. ...
In the case of a differentiable manifold, a set of charts called an atlas allows us to do calculus on manifolds. Polar coordinates, for example, form a chart for the plane R2 minus the positive x-axis and the origin. Another example of a chart is the map χtop mentioned in the section above, a chart for the circle. Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...
### Atlases
The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered multiple ways using different combinations of charts.
The atlas containing all possible charts consistent with a given atlas is called the maximal atlas. Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though it is useful for definitions, it is a very abstract object and not used directly (e.g. in calculations).
### Transition maps
Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Asia may both contain Moscow. Given two overlapping charts, a transition function can be defined which goes from an open ball in Rn to the manifold and then back to another (or perhaps the same) open ball in Rn. The resultant map, like the map T in the circle example above, is called a change of coordinates, a coordinate transformation, a transition function, or a transition map.
An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all the transition maps are compatible with this structure, the structure transfers to the manifold.
This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of Rn (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be symplectomorphisms. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ...
The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called compatible.
These notions are made precise in general through the use of pseudogroups. In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). ...
## Construction
A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint.
### Charts
The chart maps the part of the sphere with positive z coordinate to a disc.
Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere: Image File history File links A sphere with the chart mapping the upper hemisphere to a disk. ... Image File history File links A sphere with the chart mapping the upper hemisphere to a disk. ...
#### Sphere with charts
A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R3: For other uses, see Sphere (disambiguation). ...
$S = { (x,y,z) in mathbf{R}^3 | x^2 + y^2 + z^2 = 1 }.$
The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R2. Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ defined by
χ(x,y,z) = (x,y),
maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (x, z) plane and two charts projecting on the (y, z) plane, an atlas of six charts is obtained which covers the entire sphere. A disc of unit radius on a plane is called a unit disc. ...
This can be easily generalized to higher-dimensional spheres.
### Patchwork
A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold.
The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
This can be illustrated with the transition map t = 1s from the second half of the circle example. Start with two copies of the line. Use the coordinate s for the first copy, and t for the second copy. Now, glue both copies together by identifying the point t on the second copy with the point 1s on the first copy (the point t = 0 is not identified with any point on the first copy). This gives a circle.
#### Intrinsic and extrinsic view
The first construction and this construction are very similar, but they represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. This is the extrinsic view. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space it is always clear whether a vector at some point is tangential or normal to some surface through that point. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... A normal vector is a vector which is perpendicular to a surface or manifold. ...
The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the intrinsic view. It can make it harder to imagine what a tangent vector might be.
#### n-Sphere as a patchwork
The n-sphere Sn is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n-sphere Sn can be constructed by gluing together two copies of Rn. The transition map between them is defined as 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ...
$mathbf{R}^n setminus {0} to mathbf{R}^n setminus {0}: x mapsto x/|x|^2.$
This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold. In the case n = 1, the example simplifies to the circle example given earlier. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...
### Identifying points of a manifold
It is possible to define different points of a manifold to be same. This can be visualized as gluing these points together in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In topology and group theory, an orbifold (for orbit-manifold) is a generalization of a manifold. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ...
One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts on the manifold. Two points are identified if one is moved onto the other by some group element. If M is the manifold and G is the group, the resulting quotient space is denoted by M / G (or G M). This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a symmetry group describes all symmetries of objects. ...
Manifolds which can be constructed by identifying points include tori and real projective spaces (starting with a plane and a sphere, respectively). A torus This article is about the surface and mathematical concept of a torus. ... In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ...
### Cartesian products
The Cartesian product of manifolds is also a manifold. Not every manifold can be written as a product of other manifolds. In mathematics, the Cartesian product is a direct product of sets. ...
The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold. The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may be used to construct tori and finite cylinders, for example, as S1 × S1 and S1 × [0, 1], respectively. In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ...
A finite cylinder is a manifold with boundary.
Image File history File links Download high resolution version (711x641, 30 KB) Right circular cylinder, created in Matlab by Jitse Niesen. ... Image File history File links Download high resolution version (711x641, 30 KB) Right circular cylinder, created in Matlab by Jitse Niesen. ...
### Manifold with boundary
A manifold with boundary is a manifold with an edge. For example a sheet of paper with rounded corners is a 2-manifold with a 1-dimensional boundary. The edge of an n-manifold is an (n-1)-manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (See also Boundary (topology)). In geometry, a disk is the region in a plane contained inside of a circle. ... In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...
In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open n-ball {(x1, x2, …, xn) | Σ xi2 < 1}. Every boundary point has a neighborhood homeomorphic to the "half" n-ball {(x1, x2, …, xn) | Σ xi2 < 1 and x1 ≥ 0}. The homeomorphism must send the boundary point to a point with x1 = 0.
### Gluing along boundaries
Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together.
Formally, the gluing is defined by a bijection between the two boundaries. Two points are identified when they are mapped onto each other. For a topological manifold this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly for a differentiable manifold it has to be a diffeomorphism. For other manifolds other structures should be preserved.
A finite cylinder may be constructed as a manifold by starting with a strip [0, 1] × [0, 1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A projective plane may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries. Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... A Möbius strip made with a piece of paper and tape. ...
## Classes of manifolds
For more details on this topic, see Categories of manifolds.
### Topological manifolds
For more details on this topic, see topological manifold.
The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rn. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to Rn. These homeomorphisms are the charts of the manifold. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... A bijective function. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
It is to be noted that a topological manifold looks locally like a euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any particular and consistent choice of such concepts. In order to discuss such properties for a manifold, one needs to specify further structure and consider differentiable manifolds and Riemannian manifolds discussed below. In particular, a same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles.
Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff and second countable. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ...
The dimension of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (number n in the definition). All points in a connected manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension. Other authors allow disjoint unions of topological manifolds with differing dimensions to be called manifolds. Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...
### Differentiable manifolds
For more details on this topic, see differentiable manifold.
For most applications a special kind of topological manifold, a differentiable manifold, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use calculus on a differentiable manifold. Each point of an n-dimensional differentiable manifold has a tangent space. This is an n-dimensional Euclidean space consisting of the tangent vectors of the curves through the point. Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... For other uses, see Calculus (disambiguation). ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...
Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds the transition maps are smooth, that is infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are analytic (they can be expressed as power series, which are essentially polynomials of infinite degree). The sphere can be given analytic structure, as can most familiar curves and surfaces. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ...
A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds. In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...
### Riemannian manifolds
For more details on this topic, see Riemannian manifolds.
All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces, including for example all n-spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it.
### Finsler manifolds
For more details on this topic, see Finsler manifold.
A Finsler manifold allows the definition of distance, but not of angle; it is an analytic manifold in which each tangent space is equipped with a norm, ||·||, in a manner which varies smoothly from point to point. This norm can be extended to a metric, defining the length of a curve; but it cannot in general be used to define an inner product. In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ...
Any Riemannian manifold is a Finsler manifold.
### Lie groups
For more details on this topic, see Lie group.
Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which is such that the group operations are defined by smooth maps. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... This picture illustrates how the hours on a clock form a group under modular addition. ...
A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a compact Lie group is the circle: the group operation is simply rotation. This group, known as U(1), can be also characterised as the group of complex numbers of modulus 1 with multiplication as the group operation. Other examples of Lie groups include special groups of matrices, which are all subgroups of the general linear group, the group of n by n matrices with non-zero determinant. If the matrix entries are real numbers, this will be an n2-dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere and hyperspheres, are n(n-1)/2 dimensional manifolds, where n-1 is the dimension of the sphere. Further examples can be found in the table of Lie groups. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... The symmetry group of an object (e. ... For other uses, see Sphere (disambiguation). ... 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ... This article gives a table of some common Lie groups and their associated Lie algebras. ...
### Other types of manifolds
• A complex manifold is a manifold modeled on Cn with holomorphic transition functions on chart overlaps. These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold is called a Riemann surface. Note that an n-dimensional complex manifold has dimension 2n as a real differentiable manifold.
• A CR manifold is a manifold modeled on boundaries of domains in Cn.
• Infinite dimensional manifolds: to allow for infinite dimensions, one may consider Banach manifolds which are locally homeomorphic to Banach spaces. Similarly, Fréchet manifolds are locally homeomorphic to Fréchet spaces.
• A symplectic manifold is a kind of manifold which is used to represent the phase spaces in classical mechanics. They are endowed with a 2-form that defines the Poisson bracket. A closely related type of manifold is a contact manifold.
In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In mathematics, a Banach manifold is a manifold modeled on Banach spaces. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... This article deals with Fréchet spaces in functional analysis. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds. ...
## Classification and invariants
For more details on this topic, see Classification of manifolds.
Different notions of manifolds have different notions of classification and invariant; in this section we focus on smooth closed manifolds.
The classification of smooth closed manifolds is well-understood in principle, except in dimension 4: in low dimensions (2 and 3) it is geometric, via the uniformization theorem and the Solution of the Poincaré conjecture, and in high dimension (5 and above) it is algebraic, via surgery theory. This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions. In mathematics, 4-manifold is a 4-dimensional topological manifold. ... In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ... In mathematics, specifically in topology, surgery theory is the name given to a collection of techniques used to produce one manifold from another in a controlled way. ...
Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms a complete set of invariants. In mathematics, a complete set of invariants for a classification problem is a collection of maps (where X is the collection of objects being classified, up to some equivalence relation, and the are some sets), such that ∼ if and only if for all i. ...
This is much harder in higher dimensions: higher dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional manifold refer to the same object.
However, one can determine if two manifolds are different if there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to as invariants, because, while they may be defined in terms of some presentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions of a particular manifold: they are invariant under different descriptions. In mathematics, an invariant is something that does not change under a set of transformations. ...
Naively, one could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. Unfortunately, it is known that for manifolds of dimension 4 and higher, no program exists that can decide whether two manifolds are diffeomorphic.
Smooth manifolds have a rich set of invariants, coming from point-set topology, classic algebraic topology, and geometric topology. The most familiar invariants, which are visible for surfaces, are orientability (a normal invariant, also detected by homology) and genus (a homological invariant). In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... The torus is an orientable surface. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affine connection. This distinction between no local invariants and local invariants is a common way to distinguish between geometry and topology. All invariants of a smooth closed manifold are thus global. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ... In differential geometry, the torsion tensor is one of the tensorial invariants of a connection on the tangent bundle. ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... Geometry and Topology (ISSN 1364-0380 online, 1465-3060 printed) is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. ...
Algebraic topology is a source of a number of important global invariant properties. Some key criteria include the simply connected property and orientability (see below). Indeed several branches of mathematics, such as homology and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... The two bold paths shown above are homotopic relative to their endpoints. ... In mathematics, the idea of characteristic class is one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry. ...
## Examples of surfaces
### Orientability
In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admits a meaningful orientation. Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn, a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are orientable manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable. In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... The torus is an orientable surface. ...
Some illustrative examples of non-orientable manifolds include: (1) the Möbius strip, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in 3-space, and (3) the real projective plane, which arises naturally in geometry. A Möbius strip made with a piece of paper and tape. ... The Klein bottle immersed in three-dimensional space. ... The fundamental polygon of the projective plane. ... For other uses, see Geometry (disambiguation). ...
Möbius strip
Mobius strip created with Mathematica. ... Mobius strip created with Mathematica. ...
#### Möbius strip
Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high and low to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold with boundary, upon which "surgery" will be performed. Slice the strip open, so that it could unroll to become a rectangle, but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends back together seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longer a pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a single side. A Möbius strip made with a piece of paper and tape. ...
#### Klein bottle
The Klein bottle immersed in three-dimensional space.
Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into cross-caps. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. Note that in three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. ... The Klein bottle immersed in three-dimensional space. ...
#### Real projective plane
Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called antipodes. Although there is no way to do so physically, it is possible to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this route explains its name: all the points on any given line through the origin project to the same "point" on this "plane". The fundamental polygon of the projective plane. ...
### Genus and the Euler characteristic
For two dimensional manifolds a key invariant property is the genus, or the "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion of Euler characteristic. In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ...
## Generalizations of manifolds
• Orbifolds: An orbifold is a generalization of manifold allowing for certain kinds of "singularities" in the topology. Roughly speaking, it is a space which locally looks like the quotients of some simple space (e.g. Euclidean space) by the actions of various finite groups. The singularities correspond to fixed points of the group actions, and the actions must be compatible in a certain sense.
• Algebraic varieties and schemes: Non-singular algebraic varieties over the real or complex numbers are manifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdly by emulating the patching construction of manifolds: just as a manifold is glued together from open subset of Euclidean space, an algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. Schemes are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically using sheaves instead of atlases.
Because of singular points, a variety is in general not a manifold, though linguistically the French variété, German Mannigfaltigkeit and English manifold are largely synonymous. In French an algebraic variety is called une variété algébrique (an algebraic variety), while a smooth manifold is called une variété différentielle (a differential variety).
• CW-complexes: A CW complex is a topological space formed by gluing disks of different dimensionality together. In general the resulting space is singular, and hence not a manifold. However, they are of central interest in algebraic topology, especially in homotopy theory, as they are easy to compute with and singularities are not a concern.
In topology and group theory, an orbifold (for orbit-manifold) is a generalization of a manifold. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a finite group is a group which has finitely many elements. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... Synonyms (in ancient Greek syn συν = plus and onoma όνομα = name) are different words with similar or identical meanings. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
## Notes
1. ^ The notion of a map can formalized as a cell decomposition.
2. ^ In the narrow sense of requiring point-set axioms and finite dimension.
3. ^ Formally, locally homeomorphic means that each point m in the manifold M has a neighborhood homeomorphic to a neighborhood in Euclidean space, not to the unit ball specifically. However, given such a homeomorphism, the pre-image of an ε-ball gives a homeomorphism between the unit ball and a smaller neighborhood of m, so this is no loss of generality. For topological or differentiable manifolds, one can also ask that every point have a neighborhood homeomorphic to all of Euclidean space (as this is diffeomorphic to the unit ball), but this cannot be done for complex manifolds, as the complex unit ball is not holomorphic to complex space.
In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
This is a list of particular manifolds, by Wikipedia page. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, a 3-manifold is a 3-dimensional manifold. ... In mathematics, 4-manifold is a 4-dimensional topological manifold. ... There are very few or no other articles that link to this one. ...
## References
• Freedman, Michael H., and Quinn, Frank (1990) Topology of 4-Manifolds. Princeton University Press. ISBN 0-691-08577-3.
• Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall. ISBN 0-13-212605-2. Inspired by Milnor and commonly used in undergraduate courses.
• Hempel, John (1976) 3-Manifolds. Princeton University Press. ISBN 0-8218-3695-1.
• Hirsch, Morris, (1997) Differential Topology. Springer Verlag. ISBN 0-387-90148-5. The most complete account, with historical insights and excellent, but difficult, problems. The standard reference for those wishing to have a deep understanding of the subject.
• Kirby, Robion C. and Siebenmann, Laurence C. (1977) Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton University Press. ISBN 0-691-08190-5. A detailed study of the category of topological manifolds.
• Lee, John M. (2000) Introduction to Topological Manifolds. Springer-Verlag. ISBN 0-387-98759-2.
• ------ (2003) Introduction to Smooth Manifolds. Springer-Verlag. ISBN 0-387-95495-3.
• Massey, William S. (1977) Algebraic Topology: An Introduction. Springer-Verlag. ISBN 0-387-90271-6.
• Milnor, John (1997) Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 0-691-04833-9.
• Munkres, James R. (2000) Topology. Prentice Hall. ISBN 0-13-181629-2.
• Neuwirth, L. P., ed. (1975) Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox. Princeton University Press. ISBN 978-0-691-08170-0.
• Riemann, Bernhard, Gesammelte mathematische Werke und wissenschaftlicher Nachlass, Sändig Reprint. ISBN 3-253-03059-8.
• Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. The 1851 doctoral thesis in which "manifold" (Mannigfaltigkeit) first appears.
• Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. The 1854 Göttingen inaugural lecture (Habilitationsschrift).
• Spivak, Michael (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. HarperCollins Publishers. ISBN 0-8053-9021-9. The standard graduate text.
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... John Willard Milnor (b. ... Bernhard Riemann. ... Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. ...
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Electromagnetism problem: where does the magnetic field come from?
Consider the following problem:
Consider a plane with uniform charge density $\sigma$. Above the said plane, there is a system of conducting wires made up of an U-shaped circuit on which a linear conductor of lenght $d$ can slide with constant velocity $v$. The system as a whole has a rectangular shape and is parallel to the plane. (See the picture). Calculate the line integral of the magnetic field $\bf B$ along the perimeter $L(t)$ of said rectangle as a function of time.
My professor solves this problem using Maxwell's fourth equation in integral form, assuming that the current density $\bf {J}$ is everywhere null, and that the electric field $\bf E$ is the one generated by a uniformly charged plane, i.e. perpendicular to the the plane and of norm $E=\frac{\sigma}{2\epsilon_0}$; thus yielding $$\oint_{L(t)} {\bf B}\cdot dl=\mu_0\epsilon_0\frac{d}{dt}\int_{S(t)} {\bf E}\cdot dS=\mu_0\epsilon_0Edv=0.5\mu_0\sigma dv$$
I think there are some things wrong both with this solution:
1. There should be no magnetic field at all! A uniformly charged plane only produces an electrostatic field. (I know there could be a magnetic field generated by the current inside the wires, but then you couldn't assume that $\bf J$ is null everywhere as my professor did!)
2. Maxwell's fourth equation does not hold in that form if the domains of integration are allowed to vary with time. In fact, by resorting to the differential forms, we find that plugging ${\bf J}=\vec 0$ and $\frac{\partial {\bf E}}{\partial t}=0$, as my professor assumed, yields $rot{\bf B}=0$, and thus the line integral of the magnetic field over any closed curve, at any istant, should be zero by Stokes' theorem!
Therefore, my question is the following.
Are the my professor's assumption ($\bf J$ $=\vec 0$, $\frac{\partial{\bf E}}{\partial t}=\vec 0$) correct, or do both $\bf J$ and $\bf E$ need be modified so as to account for the charges present in the circuit? Is there a current in the circuit at all?
• [mod edit: previous comments moved to chat room -DZ] Let us continue this discussion in chat. – Anton Fetisov Jan 9 '18 at 20:23
• I don't quite get the question: "lenght $d$ can slide with constant velocity $v$" - It is already moving with $v$? Or it is able, but not necessarily moving at that velocity? – Shing Jan 10 '18 at 20:54
• @Shing the bar is moving with velocity $v$ – Nicol Jan 10 '18 at 21:34
• I added in my answer below a simple mathematical derivation of the surface integral on the right hand side of the correct 4th Maxwell equation for this problem. In contrast to Anton Fetisov, I don't believe that there is an electrical current flowing in the wire circuit, which would lead to an additional magnetic field. See also the response of Anton Fetisov to my pertinent question in a comment to his answer. – freecharly Jan 11 '18 at 20:46
Yes, there is indeed a current in the circuit, however the proposed solution is still valid, even though it requires extra reasoning to justify. I claim that the assumptions $\mathbf J =0$, $\frac{\partial \mathbf E}{\partial t} = 0$ are valid almost everywhere. Specifically, they fail within the conducting contour and in a small vicinity of it, which has the size on the order of the wire diameter. Since we assume wires infinitely thin, macroscopically the assumptions are valid, but it is vital to remember the microscopic details.
I will use Maxwell's equations in Gaussian units, to avoid pesky $\mu_0$'s and $\varepsilon_0$'s. For the reference they look as follows: $$\begin{eqnarray} \nabla \cdot \mathbf E & = & 4\pi \rho \\ \nabla \cdot \mathbf B & = & 0 \\ \nabla \times \mathbf E & = & -\frac{1}{c}\frac{\partial \mathbf B}{\partial t} \\ \nabla \times \mathbf B & = & \frac{4\pi}{c}\mathbf J + \frac{1}{c}\frac{\partial \mathbf E}{\partial t} \end{eqnarray}$$
This implies (assuming that $S(t)$ is the region of plane bounded by $L(t)$) $$\begin{eqnarray} \oint_{L(t)} \mathbf B \cdot \mathrm d \mathbf l & = & \iint_{S(t)} (\nabla \times \mathbf B; \mathrm d \mathbf S) \\ & =& \frac{4\pi}{c} \iint_{S(t)} (\mathbf J; \mathrm d \mathbf S) + \frac{1}{c} \iint_{S(t)} \left(\frac{\partial \mathbf E}{\partial t}; \mathrm d \mathbf S \right) \end{eqnarray}$$
However all current flows in the plane $S(t)$, thus its flux through $S(t)$ is $0$.
The term with the partial derivatives is harder to study. First note that macroscopically there are no free charges, in the sense that the macroscopic charge distribution is constant in time. Also the system is quasi-stationary since the speed $v \ll c$ --- this allows us to exclude any EM waves from the problem and only work with charges and currents. This implies that macroscopically $\mathbf E$ is stationary, but if we would assume globally $\frac{\partial \mathbf E}{\partial t} = 0$, then Maxwell's equations would imply that $\mathbf B$ and $\mathbf J$ are also stationary and always $0$. To see that this isn't true we need to consider what happens in the wire itself.
We assume the wire to be an ideal conductor with zero resistance. Ohm's law says that in the wire $\mathbf E = \rho \mathbf J$, if $\rho = 0$ then finite current implies $\mathbf E = 0$ within the wire (EDIT: since we are only interested in the vertical component of $\mathbf E$ and there can be no vertical current, $E_z = 0$ in wire even if $\rho \ne 0$). Thus we see that even before the movement starts the field isn't equal to $\mathbf E_0$ everywhere --- it is $0$ within the wire and has some intermediary value in its vicinity. This also shows that globally $\mathbf E$ isn't stationary --- the movement of the wire causes the movement of the zeroes of $\mathbf E$ and of the shielding charges in the wire. This, in turn, causes the magnetic field and the induced current. If we try to calculate the derivative, then we see that near the wire $E$ changes from $E_0$ to $0$ over an infinitely small interval of time, so the derivative has some delta-function-like form, providing some finite (generally) non-zero value to the integrals.
To calculate the surface integral of $\frac{\partial \mathbf E}{\partial t}$, we need to convert it to a more manageable form, something like a derivative of a continuous function. The general formula for a full derivative of a time-dependent surface integral is $$\frac{\mathrm d}{\mathrm d t} \iint_{S(t)} (\mathbf F ; \mathrm d \mathbf S) = \iint_{S(t)} \left( \frac{\partial \mathbf F}{\partial t}; \mathrm d \mathbf S \right ) + \frac{1}{\mathrm d t}\iint_{\delta S(t)} (\mathbf F; \mathrm d \mathbf S)$$
Here $\delta S(t)$ is the infinitesimal variation of the surface $S(t)$ and I assume that $S(t)$ varies by adding extra area, like in the problem (i.e. no movement of the interior). This is just the usual product rule for the calculation of derivatives. In our problem $\mathbf F = \mathbf E$ and the surface is chosen so that its boundary passes inside the wire loop. This means that $\mathbf E =0$ near the boundary of $S(t)$ and thus the integral over the variation of area is $0$, so the second term vanishes and we have $$\iint_{S(t)} \left( \frac{\partial \mathbf E}{\partial t}; \mathrm d \mathbf S \right ) = \frac{\mathrm d}{\mathrm d t} \iint_{S(t)} (\mathbf E ; \mathrm d \mathbf S)$$
Since $\mathbf E$ is everywhere bounded and almost everywhere equal to $\mathbf E_0$, the answer to your problem follows.
Note that the circulation of $\mathbf B$ doesn't depend on the specific contour passing through the wire, but will change if we move the contour outside the wire. Also note that if we consider an infinitely small loop around a section of the wire, then the circulation of $\mathbf B$ will be finite nonzero if there is a non-zero current passing through the section. This demonstrates that when the problem asks an integral around the perimeter, we must consider exactly the perimeter, even a small variation would give an incorrect answer.
• Dear Anton, thanks for the detailed answer. I think I got the gist of it, but nonetheless I'm still perplexed. I'm studying engineering and the one I'm attending is a basic course in classical EM. We haven't covered delta functions, time-dependent surface integrals and the like. I can see my professor making this kind of reasoning, but for his students it would be pretty out or reach. Moreover, this was part of an easy quiz, and the other questions could be solved in minutes! Therefore, unless there is a more immediate way of seeing why the solution is correct, it seems more likely tome [...] – Nicol Jan 10 '18 at 17:37
• [...] that this was a little oversight of my professor's, and that he got the right solution by chance. (I hope this doesn't sound condescending, but I've been obsessing with this problem for quite a while!) – Nicol Jan 10 '18 at 17:42
• @Nicol I have written out the answer more thoroughly than is really required, just to make sure I didn't miss anything and explain all details. You don't really need delta functions, there are certainly no delta functions microscopically, they are an artifact of large-scale approximation and you can do the same things if you have a proper intuition about the structure of large-scale limit, but I felt the need to elaborate on the apparent zero time derivative paradox and the origin of magnetic field, especially since it's a common source of error and confusion. – Anton Fetisov Jan 10 '18 at 17:55
• The important part is that the boundary term in the derivative of the integral is 0, since the vertical component of the electric field is 0. Now that I'm thinking about it, it will be true even if $\rho \ne 0$ since there can be no vertical current. – Anton Fetisov Jan 10 '18 at 18:08
• @Nicol It looks like there will be no current under non-zero resistance. This won't affect the argument: I only need that the vertical component of electric field is $0$ on the boundary, which is true in all cases since there can be no vertical current and $\mathbf J = \rho \mathbf E$. – Anton Fetisov Jan 12 '18 at 16:35
Your insight stated in 2. is correct! In the integral form of the fourth Maxwell equation with time varying integration surface, the time differentiation stays inside the integral: $$\oint_{L(t)} {\bf B}\cdot dl=\mu_0\epsilon_0\int_{S(t)} {\frac{\partial}{\partial t}\bf E}\cdot dS \tag{1}$$ Then from the assumption $\frac{\partial{\bf E}}{\partial t}= 0$ both the left hand and the right hand side should be zero in this case. However, Anton Fetisov has shown in his answer (s. below) that due to induced charges on the moving wire $\frac{\partial{\bf E}}{\partial t}\neq 0$. Therefore, your professor has obviously made mistakes but fortuitously obtained the correct answer.
In his correct and deep going analysis of the problem, he considers the effects of the finite size of the metallic wire and the electric charges induced on its surface by the homogeneous electric field of the charged plane which are necessary to produce a zero total electric field in the wires. These induced charges and the associated deformation of the electrical field around the wire are moving with velocity $v$ in the $x$-direction.
Thus, from this point of view, there exist currents and time varying electric fields which is inconsistent with two basic assumptions made in the problem, i.e., $\bf J = 0$ and $\frac{\partial{\bf E}}{\partial t}= 0$. The second error is the solution with the wrong integral form of the 4th Maxwell equation for time varying integration surface/contour $$\oint_{L(t)} {\bf B}\cdot dl=\mu_0\epsilon_0\frac{d}{d t}\int_{S(t)} {\bf E}\cdot dS \tag{2}$$ The correct form is equation (1). From the given assumption $\frac{\partial{\bf E}}{\partial t}= 0$ it follows that the right hand side of equation (1) should be zero as I have stated before. This is, however, not correct in this particular case due to the fact that the induced charges on the wire cause a time varying field.
In his detailed analysis, Anton Fetisov has shown, that the right hand side of the correct equation (1) is not zero and that, surprisingly, it is equal to the right hand side of the incorrect equation (2). Thus the solution of the problem found by the professor with the incorrect equation (2) is fortuitously correct. Therefore, I have reduced my original short answer (first paragraph) to the still valid fact, already found by Nicol, that the form of the used Maxwell equation was generally not correct for the time dependent integration surface/contour.
Added simple derivation: For those who are not math virtuosos, I would like to show, on the basis of Anton Fetisov's reasoning, how the right hand side of the correct 4th Maxwell equation (1) can be evaluated for the considered problem in a simple way giving the result quoted in the question of Nicol.
The essential point is the charges on the wire that are electrostatically induced by the homogeneous electric field $E_0=\sigma/\epsilon_0$ of the sheet charge $\sigma$. Only the vertical y-component has to be considered for the the integral. These charges are the sources of an additional electrical field $\epsilon (x)$ in and closely around the wire which exactly cancels $E_0$ inside the wire and reduces it near the wire on a length scale of the wire diameter $2a$. This additional wire field $\epsilon (x)$ has the most negative value at a (flat) minimum $\epsilon _{min}= -E_0$ inside the wire, particularly on its axis. The exact functional form is irrelevant here, as long as its minimum at $x=0$ is $\epsilon (0)=-E_0$ and it is zero a couple of wire diameters horizontally away from the wire axis. The x- and t-dependence of the vertical field in the wire plane of the moving wire can be written as $\epsilon (x,t)=\epsilon (x-vt)$, where the axis of the wire (and field minimum) is located at $x_1=vt$. The total vertical electric field in the wire plane is then given by $$E(x,t)=E_0 + \epsilon (x) + \epsilon (x-vt)$$ (The second term on the RHS is the time-independent field of the left transverse wire.) Thus with $$\frac{\partial{E}}{\partial t}=\frac{\partial{\epsilon (x-vt)}}{\partial t}=\frac{\partial{\epsilon(x-vt)}}{\partial x}(-v)$$ the surface integral of the RHS of equation (1) reduces to $$\int_{S(t)} {\frac{\partial}{\partial t}\bf E}\cdot dS= -vd\int_{x=0}^{x_1=vt} {\frac{\partial \epsilon(x-vt)}{\partial x}} dx =-vd[\epsilon(x-vt)]_{x=0}^{x_1=vt}= vd[\epsilon (-vt)-\epsilon (0)]=vdE_0$$ where it has been assumed that $\epsilon (0)=-E_0$ and $x_1=vt>>2a$ so that $\epsilon (-vt)=0$. This shows that the RHS of equation (1) is indeed $$\frac{\mu_0 v \sigma d}{2}$$ the fortuitously obtained solution quoted by Nicol.
• Exactly. Just to clarify, the error is that it is only true that$$\int_{S} {\frac{\partial}{\partial t}\bf E}\cdot dS = \frac{d}{dt} \left[ \int_{S} {\bf E}\cdot dS\right]$$ if the surface $S$ does not change with respect to time. It's possible to use the Liebniz integral rule for differential forms to relate these two quantities, but there's a couple of terms that your professor forgot. I suspect including them would resolve the paradox. – Michael Seifert Jan 9 '18 at 16:16
• If I've translated the differential-forms version of Liebniz's rule correctly, then the correct version of the above statement is$$\frac{d}{dt}\left[ \int_{S(t)} {\bf E}\cdot dS\right]=\int_{S(t)}{\frac{\partial}{\partial t}\bf E}\cdot dS+\int_{S(t)}{(\nabla \cdot \bf E)} {\bf v} \cdot dS + \int_{\partial S(t)} ({\bf v} \times {\bf E}) \cdot dl.$$The second integral vanishes, but the third one is non-vanishing, and cancels out the error. – Michael Seifert Jan 9 '18 at 16:38
• @Michael Seifert - You describe it very clearly! A similar error is often made with the integral form of the Faraday-Maxwell equation and time varying surface (Induction Law). – freecharly Jan 9 '18 at 16:42
• I do no see any reason why this answer should be incorrect. In the inertial system of the charged plane you have a vertical electric field constant in space and time and in the loop with the sliding wire there is no force (electric or magnetic) on the charge carriers in the direction of the wire and thus no current and no magnetic field. Therefore you have, indeed, in the considered coordinate frame a constant electric field in time and space and $\frac{\partial E}{\partial t}=0$. For time dependent surface/contour, the 4th Maxwell equation must have the time derivative inside the integral! – freecharly Jan 9 '18 at 23:37
• I have edited and extended the text of my answer due to the surprising analysis of Anton Fetisov who showed that in this special case due to the induced charges in the wire, the correct right hand side of 4th Maxwell equation (1) for time varying integration surfaces/boundaries gives the same results as the incorrect right hand side, equation (2). – freecharly Jan 11 '18 at 14:27
From the 2nd equation (001b) of the Maxwell's equations \begin{align} \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{E} & = -\frac{\partial \mathbf{B}}{\partial t} \tag{001a}\\ \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{B} & = \mu_{0}\mathbf{j}+\frac{1}{c^{2}}\frac{\partial \mathbf{E}}{\partial t} \tag{001b}\\ \nabla \boldsymbol{\cdot} \mathbf{E} & = \frac{\rho}{\epsilon_{0}} \tag{001c}\\ \nabla \boldsymbol{\cdot}\mathbf{B}& = 0 \tag{001d} \end{align} we have
$$\int\limits_{S(t)}\left(\boldsymbol{\nabla} \boldsymbol{\times} \mathbf{B}\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}\:=\:\mu_{0}\!\!\int\limits_{S(t)} \mathbf{j}\boldsymbol{\cdot} \mathrm d\mathbf{S}+\frac{1}{c^{2}}\int\limits_{S(t)} \frac{\partial \mathbf{E}}{\partial t}\boldsymbol{\cdot} \mathrm d\mathbf{S} \tag{02}$$ and since with or without current in the wire we have $\:\mathbf{j}\boldsymbol{\cdot} \mathrm d\mathbf{S}\equiv 0\:$ everywhere on the surface $\:S(t)\:$
$$\oint\limits_{\partial S(t)}\mathbf{B}\boldsymbol{\cdot} \mathrm d \boldsymbol{\ell}\:=\:\mu_{0}\epsilon_{0}\int\limits_{S(t)} \frac{\partial \mathbf{E}}{\partial t}\boldsymbol{\cdot} \mathrm d\mathbf{S} \tag{03}$$ Now, in order to find the integral in the rhs of above equation I use the Helmholtz transport theorem(1) as suggested by @Michael Seifert in one of his comments. A first form of this theorem is for the flux of a vector field $\:\mathbf{F}\left(\mathbf{x},t\right)\:$ through a surface $\:S(t)\:$ in motion and/or deformation(2)
$$\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S(t)}\mathbf{F}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}=\int\limits_{S(t)} \left[\dfrac{\partial \mathbf{F}}{\partial t} + \left(\nabla \boldsymbol{\cdot} \boldsymbol{\upsilon}\right)\mathbf{F} + \left(\boldsymbol{\upsilon}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\mathbf{F} - \left(\mathbf{F}\boldsymbol{\cdot} \boldsymbol{\nabla}\right)\boldsymbol{\upsilon}\right] \boldsymbol{\cdot} \mathrm d\mathbf{S} \tag{04}$$
expressed here for our purpose as $$\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S(t)}\mathbf{F}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}\:=\:\int\limits_{S(t)} \left[\dfrac{\partial \mathbf{F}}{\partial t} + \left(\nabla \boldsymbol{\cdot} \mathbf{F}\right)\boldsymbol{\upsilon} - \boldsymbol{\nabla} \boldsymbol{\times} \left( \boldsymbol{\upsilon}\boldsymbol{\times} \mathbf{F}\right)\right]\boldsymbol{\cdot} \mathrm d\mathbf{S} \tag{05}$$ For $\:\mathbf{F}\left(\mathbf{x},t\right)\equiv \mathbf{E}\left(\mathbf{x},t\right)\:$ $$\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S(t)}\mathbf{E}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}\:=\:\int\limits_{S(t)} \left[\dfrac{\partial \mathbf{E}}{\partial t} + \left(\nabla \boldsymbol{\cdot} \mathbf{E}\right)\boldsymbol{\upsilon} - \boldsymbol{\nabla} \boldsymbol{\times} \left( \boldsymbol{\upsilon}\boldsymbol{\times} \mathbf{E}\right)\right]\boldsymbol{\cdot} \mathrm d\mathbf{S} \tag{06}$$ So $$\int\limits_{S(t)} \frac{\partial \mathbf{E}}{\partial t}\boldsymbol{\cdot} \mathrm d\mathbf{S} =\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S(t)}\mathbf{E}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}-\int\limits_{S(t)}\left(\nabla \boldsymbol{\cdot} \mathbf{E}\right)\boldsymbol{\upsilon}\boldsymbol{\cdot} \mathrm d\mathbf{S}+\int\limits_{S(t)} \boldsymbol{\nabla} \boldsymbol{\times} \left( \boldsymbol{\upsilon}\boldsymbol{\times} \mathbf{E}\right)\boldsymbol{\cdot} \mathrm d\mathbf{S} \tag{07}$$ But firstly $$\dfrac{\mathrm d}{\mathrm dt}\int\limits_{S(t)}\mathbf{E}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}=\lim_{\Delta t \rightarrow 0}\dfrac{1}{\Delta t}\left[\int\limits_{S(t+\Delta t)}\mathbf{E}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}-\int\limits_{S(t)}\mathbf{E}\left(\mathbf{x},t\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}\right]=+Ed\upsilon \tag{08}$$ secondly, since $\:\rho\equiv0\:$(3) every where on $\:S(t)\:$ $$\int\limits_{S(t)}\left(\nabla \boldsymbol{\cdot} \mathbf{E}\right)\boldsymbol{\upsilon}\boldsymbol{\cdot} \mathrm d\mathbf{S}=\int\limits_{S(t)}\frac{\rho}{\epsilon_{0}}\boldsymbol{\upsilon}\boldsymbol{\cdot} \mathrm d\mathbf{S}=0 \tag{09}$$ and thirdly $$\int\limits_{S(t)} \boldsymbol{\nabla} \boldsymbol{\times} \left( \boldsymbol{\upsilon}\boldsymbol{\times} \mathbf{E}\right)\boldsymbol{\cdot} \mathrm d\mathbf{S}=\oint\limits_{\partial S(t)}\left( \boldsymbol{\upsilon}\boldsymbol{\times} \mathbf{E}\right)\boldsymbol{\cdot} \mathrm d \boldsymbol{\ell}=-Ed\upsilon \tag{10-wrong, see 10^{\boldsymbol{\prime}}}$$ Finally $$\oint\limits_{\partial S(t)}\mathbf{B}\boldsymbol{\cdot} \mathrm d \boldsymbol{\ell}\:=\:\mu_{0}\epsilon_{0}\int\limits_{S(t)} \frac{\partial \mathbf{E}}{\partial t}\boldsymbol{\cdot} \mathrm d\mathbf{S}=0 \tag{11-wrong, see 11^{\boldsymbol{\prime}} }$$
EDIT A
As @freecharly commented (Jan'12 2018) :
Your derivations are all perfect. Except for equation (10). As Anton Fetisov has pointed out, the path of the integration runs in the metallic wire where the electric field is zero, which is due to surface charges on the wire induced by the homogeneous field of the positive sheet charge. Therefore, as long as this time dependent path runs in the metallic wire circuit, the integral will be zero $$\oint\limits_{\partial S(t)}\left( \boldsymbol{\upsilon}\boldsymbol{\times} \mathbf{E}\right)\boldsymbol{\cdot} \mathrm d \boldsymbol{\ell}=0 \nonumber$$ Therefore also the RHS of equ.(11) is $=Edv$!
Under these suggestions the correct equations and the correct final result are as follows
In place of equation (10) $$\oint\limits_{\partial S(t)}\left( \boldsymbol{\upsilon}\boldsymbol{\times} \mathbf{E}\right)\boldsymbol{\cdot} \mathrm d \boldsymbol{\ell}=0 \tag{10^{\boldsymbol{\prime}}}$$ and finally $$\boxed{\:\:\: \oint\limits_{\partial S(t)}\!\mathbf{B}\!\boldsymbol{\cdot}\! \mathrm d \boldsymbol{\ell}\stackrel{(03)}{=\!\!=}\mu_{0}\epsilon_{0}\!\!\int\limits_{S(t)}\! \frac{\partial \mathbf{E}}{\partial t}\!\boldsymbol{\cdot}\!\mathrm d\mathbf{S}\stackrel{(07),(09),(10^{\boldsymbol{\prime}})}{=\!\!=\!\!=\!\!=\!\!=\!\!=\!\!=\!\!=}\:\mu_{0}\epsilon_{0}\dfrac{\mathrm d}{\mathrm dt}\!\!\int\limits_{S(t)}\!\mathbf{E}\left(\mathbf{x},t\right)\!\boldsymbol{\cdot}\!\mathrm d\mathbf{S}\stackrel{(08)}{=\!\!=}\mu_{0}\epsilon_{0}E\!\cdot\! d\!\cdot\!\upsilon=\frac12\mu_{0}\sigma d \upsilon \vphantom{\oint\limits^{\partial S(t)}_{\partial S(t)}a}\:\:\:} \tag{11^{\boldsymbol{\prime}}}$$
(1) 'Generalized Vector and Dyadic Analysis', Chen-To Tai, IEEE PRESS, 2nd Edition 1997 equations (6.11),(6.12) page 119.See an excerpt here : Vector Analysis of Transport Theorems.
(2) I think there exist many textbooks to find a proof of Helmholtz transport theorem. But you could read my effort to prove this (successfully I want to believe) here : Flux of vector field through movable/deformable surfaces
(3) We must distinguish the cases of symbols $\:\rho\:$ and $\:\sigma\:$ used in the question, the answers and the comments \begin{align} \rho & = \begin{cases} \textbf{volume charge density} \\ \textbf{resistivity} \end{cases} \tag{note-01}\\ \sigma & = \begin{cases} \textbf{surface charge density} \\ \textbf{conductivity} \end{cases} \tag{note-02} \end{align}
• Your derivations are all perfect. Except for equation (10). As Anton Fetisov has pointed out, the path of the integration runs in the metallic wire where the electric field is zero, which is due to surface charges on the wire induced by the homogeneous field of the positive sheet charge. Therefore, as long as this time dependent path runs in the metallic wire circuit, the integral will be zero $$\oint\limits_{\partial S(t)}\left( \boldsymbol{\upsilon}\boldsymbol{\times} \mathbf{E}\right)\boldsymbol{\cdot} \mathrm d \boldsymbol{\ell}=0$$ Therefore also the RHS of equ.(11) is $=Edv$ ! – freecharly Jan 12 '18 at 19:41
• @freecharly : I edit my answer leaving the incorrect equations of mine for comparison. – Frobenius Jan 12 '18 at 23:10
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