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transformers documentation

QDQBERT

# QDQBERT

## Overview

The QDQBERT model can be referenced in Integer Quantization for Deep Learning Inference: Principles and Empirical Evaluation by Hao Wu, Patrick Judd, Xiaojie Zhang, Mikhail Isaev and Paulius Micikevicius.

The abstract from the paper is the following:

Quantization techniques can reduce the size of Deep Neural Networks and improve inference latency and throughput by taking advantage of high throughput integer instructions. In this paper we review the mathematical aspects of quantization parameters and evaluate their choices on a wide range of neural network models for different application domains, including vision, speech, and language. We focus on quantization techniques that are amenable to acceleration by processors with high-throughput integer math pipelines. We also present a workflow for 8-bit quantization that is able to maintain accuracy within 1% of the floating-point baseline on all networks studied, including models that are more difficult to quantize, such as MobileNets and BERT-large.

Tips:

• QDQBERT model adds fake quantization operations (pair of QuantizeLinear/DequantizeLinear ops) to (i) linear layer inputs and weights, (ii) matmul inputs, (iii) residual add inputs, in BERT model.

• QDQBERT requires the dependency of Pytorch Quantization Toolkit. To install pip install pytorch-quantization --extra-index-url https://pypi.ngc.nvidia.com

• QDQBERT model can be loaded from any checkpoint of HuggingFace BERT model (for example bert-base-uncased), and perform Quantization Aware Training/Post Training Quantization.

• A complete example of using QDQBERT model to perform Quatization Aware Training and Post Training Quantization for SQUAD task can be found at transformers/examples/research_projects/quantization-qdqbert/.

This model was contributed by shangz.

### Set default quantizers

QDQBERT model adds fake quantization operations (pair of QuantizeLinear/DequantizeLinear ops) to BERT by TensorQuantizer in Pytorch Quantization Toolkit. TensorQuantizer is the module for quantizing tensors, with QuantDescriptor defining how the tensor should be quantized. Refer to Pytorch Quantization Toolkit userguide for more details.

Before creating QDQBERT model, one has to set the default QuantDescriptor defining default tensor quantizers. Example:

>>> import pytorch_quantization.nn as quant_nn
>>> from pytorch_quantization.tensor_quant import QuantDescriptor

>>> # The default tensor quantizer is set to use Max calibration method
>>> input_desc = QuantDescriptor(num_bits=8, calib_method="max")
>>> # The default tensor quantizer is set to be per-channel quantization for weights
>>> weight_desc = QuantDescriptor(num_bits=8, axis=((0,)))
>>> quant_nn.QuantLinear.set_default_quant_desc_input(input_desc)
>>> quant_nn.QuantLinear.set_default_quant_desc_weight(weight_desc)

### Calibration

Calibration is the terminology of passing data samples to the quantizer and deciding the best scaling factors for tensors. After setting up the tensor quantizers, one can use the following example to calibrate the model:

>>> # Find the TensorQuantizer and enable calibration
>>> for name, module in model.named_modules():
>>>     if name.endswith('_input_quantizer'):
>>>         module.enable_calib()
>>>         module.disable_quant()  # Use full precision data to calibrate

>>> # Feeding data samples
>>> model(x)
>>> # ...

>>> # Finalize calibration
>>> for name, module in model.named_modules():
>>>     if name.endswith('_input_quantizer'):
>>>         module.enable_quant()

>>> # If running on GPU, it needs to call .cuda() again because new tensors will be created by calibration process
>>> model.cuda()

>>> # Keep running the quantized model
>>> # ...

### Export to ONNX

The goal of exporting to ONNX is to deploy inference by TensorRT. Fake quantization will be broken into a pair of QuantizeLinear/DequantizeLinear ONNX ops. After setting static member of TensorQuantizer to use Pytorch’s own fake quantization functions, fake quantized model can be exported to ONNX, follow the instructions in torch.onnx. Example:

>>> from pytorch_quantization.nn import TensorQuantizer
>>> TensorQuantizer.use_fb_fake_quant = True

>>> # Load the calibrated model
>>> ...
>>> # ONNX export
>>> torch.onnx.export(...)

## QDQBertConfig

class transformers.QDQBertConfig < > expand

( vocab_size = 30522 hidden_size = 768 num_hidden_layers = 12 num_attention_heads = 12 intermediate_size = 3072 hidden_act = 'gelu' hidden_dropout_prob = 0.1 attention_probs_dropout_prob = 0.1 max_position_embeddings = 512 type_vocab_size = 2 initializer_range = 0.02 layer_norm_eps = 1e-12 use_cache = True is_encoder_decoder = False pad_token_id = 1 bos_token_id = 0 eos_token_id = 2 **kwargs )

This is the configuration class to store the configuration of a QDQBertModel. It is used to instantiate an QDQBERT model according to the specified arguments, defining the model architecture. Instantiating a configuration with the defaults will yield a similar configuration to that of the BERT bert-base-uncased architecture.

Configuration objects inherit from PretrainedConfig and can be used to control the model outputs. Read the documentation from PretrainedConfig for more information.

Examples:

>>> from transformers import QDQBertModel, QDQBertConfig

>>> # Initializing a QDQBERT bert-base-uncased style configuration
>>> configuration = QDQBertConfig()

>>> # Initializing a model from the bert-base-uncased style configuration
>>> model = QDQBertModel(configuration)

>>> # Accessing the model configuration
>>> configuration = model.config

## QDQBertModel

class transformers.QDQBertModel < > expand

( config add_pooling_layer = True )

The bare QDQBERT Model transformer outputting raw hidden-states without any specific head on top.

This model inherits from PreTrainedModel. Check the superclass documentation for the generic methods the library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads etc.)

This model is also a PyTorch torch.nn.Module subclass. Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage and behavior.

The model can behave as an encoder (with only self-attention) as well as a decoder, in which case a layer of cross-attention is added between the self-attention layers, following the architecture described in Attention is all you need by Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser and Illia Polosukhin.

To behave as an decoder the model needs to be initialized with the is_decoder argument of the configuration set to True. To be used in a Seq2Seq model, the model needs to initialized with both is_decoder argument and add_cross_attention set to True; an encoder_hidden_states is then expected as an input to the forward pass.

forward < > expand

( input_ids = None attention_mask = None token_type_ids = None position_ids = None head_mask = None inputs_embeds = None encoder_hidden_states = None encoder_attention_mask = None past_key_values = None use_cache = None output_attentions = None output_hidden_states = None return_dict = None ) BaseModelOutputWithPoolingAndCrossAttentions or tuple(torch.FloatTensor)

The QDQBertModel forward method, overrides the __call__ special method.

Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the pre and post processing steps while the latter silently ignores them.

Example:

>>> from transformers import BertTokenizer, QDQBertModel
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')
>>> model = QDQBertModel.from_pretrained('bert-base-uncased')

>>> inputs = tokenizer("Hello, my dog is cute", return_tensors="pt")
>>> outputs = model(**inputs)

>>> last_hidden_states = outputs.last_hidden_state

( config )

QDQBERT Model with a language modeling head on top for CLM fine-tuning.

This model inherits from PreTrainedModel. Check the superclass documentation for the generic methods the library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads etc.)

This model is also a PyTorch torch.nn.Module subclass. Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage and behavior.

forward < > expand

( input_ids = None attention_mask = None token_type_ids = None position_ids = None head_mask = None inputs_embeds = None encoder_hidden_states = None encoder_attention_mask = None labels = None past_key_values = None use_cache = None output_attentions = None output_hidden_states = None return_dict = None ) CausalLMOutputWithCrossAttentions or tuple(torch.FloatTensor)

The QDQBertLMHeadModel forward method, overrides the __call__ special method.

Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the pre and post processing steps while the latter silently ignores them.

Example:

>>> from transformers import BertTokenizer, QDQBertLMHeadModel, QDQBertConfig
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-cased')
>>> config = QDQBertConfig.from_pretrained("bert-base-cased")
>>> config.is_decoder = True

>>> inputs = tokenizer("Hello, my dog is cute", return_tensors="pt")
>>> outputs = model(**inputs)

>>> prediction_logits = outputs.logits

( config )

QDQBERT Model with a language modeling head on top.

This model inherits from PreTrainedModel. Check the superclass documentation for the generic methods the library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads etc.)

This model is also a PyTorch torch.nn.Module subclass. Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage and behavior.

forward < > expand

( input_ids = None attention_mask = None token_type_ids = None position_ids = None head_mask = None inputs_embeds = None encoder_hidden_states = None encoder_attention_mask = None labels = None output_attentions = None output_hidden_states = None return_dict = None ) MaskedLMOutput or tuple(torch.FloatTensor)

The QDQBertForMaskedLM forward method, overrides the __call__ special method.

Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the pre and post processing steps while the latter silently ignores them.

Example:

>>> from transformers import BertTokenizer, QDQBertForMaskedLM
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')

>>> inputs = tokenizer("The capital of France is [MASK].", return_tensors="pt")
>>> labels = tokenizer("The capital of France is Paris.", return_tensors="pt")["input_ids"]

>>> outputs = model(**inputs, labels=labels)
>>> loss = outputs.loss
>>> logits = outputs.logits

## QDQBertForSequenceClassification

class transformers.QDQBertForSequenceClassification < > expand

( config )

Bert Model transformer with a sequence classification/regression head on top (a linear layer on top of the pooled output) e.g. for GLUE tasks.

This model inherits from PreTrainedModel. Check the superclass documentation for the generic methods the library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads etc.)

This model is also a PyTorch torch.nn.Module subclass. Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage and behavior.

forward < > expand

( input_ids = None attention_mask = None token_type_ids = None position_ids = None head_mask = None inputs_embeds = None labels = None output_attentions = None output_hidden_states = None return_dict = None ) SequenceClassifierOutput or tuple(torch.FloatTensor)

The QDQBertForSequenceClassification forward method, overrides the __call__ special method.

Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the pre and post processing steps while the latter silently ignores them.

Example of single-label classification:

>>> from transformers import BertTokenizer, QDQBertForSequenceClassification
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')
>>> model = QDQBertForSequenceClassification.from_pretrained('bert-base-uncased')

>>> inputs = tokenizer("Hello, my dog is cute", return_tensors="pt")
>>> labels = torch.tensor([1]).unsqueeze(0)  # Batch size 1
>>> outputs = model(**inputs, labels=labels)
>>> loss = outputs.loss
>>> logits = outputs.logits

Example of multi-label classification:

>>> from transformers import BertTokenizer, QDQBertForSequenceClassification
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')
>>> model = QDQBertForSequenceClassification.from_pretrained('bert-base-uncased', problem_type="multi_label_classification")

>>> inputs = tokenizer("Hello, my dog is cute", return_tensors="pt")
>>> labels = torch.tensor([[1, 1]], dtype=torch.float) # need dtype=float for BCEWithLogitsLoss
>>> outputs = model(**inputs, labels=labels)
>>> loss = outputs.loss
>>> logits = outputs.logits

## QDQBertForNextSentencePrediction

class transformers.QDQBertForNextSentencePrediction < > expand

( config )

Bert Model with a next sentence prediction (classification) head on top.

This model inherits from PreTrainedModel. Check the superclass documentation for the generic methods the library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads etc.)

This model is also a PyTorch torch.nn.Module subclass. Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage and behavior.

forward < > expand

( input_ids = None attention_mask = None token_type_ids = None position_ids = None head_mask = None inputs_embeds = None labels = None output_attentions = None output_hidden_states = None return_dict = None **kwargs ) NextSentencePredictorOutput or tuple(torch.FloatTensor)

The QDQBertForNextSentencePrediction forward method, overrides the __call__ special method.

Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the pre and post processing steps while the latter silently ignores them.

Example:

>>> from transformers import BertTokenizer, QDQBertForNextSentencePrediction
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')
>>> model = QDQBertForNextSentencePrediction.from_pretrained('bert-base-uncased')

>>> prompt = "In Italy, pizza served in formal settings, such as at a restaurant, is presented unsliced."
>>> next_sentence = "The sky is blue due to the shorter wavelength of blue light."
>>> encoding = tokenizer(prompt, next_sentence, return_tensors='pt')

>>> outputs = model(**encoding, labels=torch.LongTensor([1]))
>>> logits = outputs.logits
>>> assert logits[0, 0] < logits[0, 1] # next sentence was random

## QDQBertForMultipleChoice

class transformers.QDQBertForMultipleChoice < > expand

( config )

Bert Model with a multiple choice classification head on top (a linear layer on top of the pooled output and a softmax) e.g. for RocStories/SWAG tasks.

This model inherits from PreTrainedModel. Check the superclass documentation for the generic methods the library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads etc.)

This model is also a PyTorch torch.nn.Module subclass. Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage and behavior.

forward < > expand

( input_ids = None attention_mask = None token_type_ids = None position_ids = None head_mask = None inputs_embeds = None labels = None output_attentions = None output_hidden_states = None return_dict = None ) MultipleChoiceModelOutput or tuple(torch.FloatTensor)

The QDQBertForMultipleChoice forward method, overrides the __call__ special method.

Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the pre and post processing steps while the latter silently ignores them.

Example:

>>> from transformers import BertTokenizer, QDQBertForMultipleChoice
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')
>>> model = QDQBertForMultipleChoice.from_pretrained('bert-base-uncased')

>>> prompt = "In Italy, pizza served in formal settings, such as at a restaurant, is presented unsliced."
>>> choice0 = "It is eaten with a fork and a knife."
>>> choice1 = "It is eaten while held in the hand."
>>> labels = torch.tensor(0).unsqueeze(0)  # choice0 is correct (according to Wikipedia ;)), batch size 1

>>> encoding = tokenizer([prompt, prompt], [choice0, choice1], return_tensors='pt', padding=True)
>>> outputs = model(**{k: v.unsqueeze(0) for k,v in encoding.items()}, labels=labels)  # batch size is 1

>>> # the linear classifier still needs to be trained
>>> loss = outputs.loss
>>> logits = outputs.logits

## QDQBertForTokenClassification

class transformers.QDQBertForTokenClassification < > expand

( config )

QDQBERT Model with a token classification head on top (a linear layer on top of the hidden-states output) e.g. for Named-Entity-Recognition (NER) tasks.

This model inherits from PreTrainedModel. Check the superclass documentation for the generic methods the library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads etc.)

This model is also a PyTorch torch.nn.Module subclass. Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage and behavior.

forward < > expand

( input_ids = None attention_mask = None token_type_ids = None position_ids = None head_mask = None inputs_embeds = None labels = None output_attentions = None output_hidden_states = None return_dict = None ) TokenClassifierOutput or tuple(torch.FloatTensor)

The QDQBertForTokenClassification forward method, overrides the __call__ special method.

Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the pre and post processing steps while the latter silently ignores them.

Example:

>>> from transformers import BertTokenizer, QDQBertForTokenClassification
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')
>>> model = QDQBertForTokenClassification.from_pretrained('bert-base-uncased')

>>> inputs = tokenizer("Hello, my dog is cute", return_tensors="pt")
>>> labels = torch.tensor([1] * inputs["input_ids"].size(1)).unsqueeze(0)  # Batch size 1

>>> outputs = model(**inputs, labels=labels)
>>> loss = outputs.loss
>>> logits = outputs.logits

( config )

QDQBERT Model with a span classification head on top for extractive question-answering tasks like SQuAD (a linear layers on top of the hidden-states output to compute span start logits and span end logits).

This model inherits from PreTrainedModel. Check the superclass documentation for the generic methods the library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads etc.)

This model is also a PyTorch torch.nn.Module subclass. Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage and behavior.

forward < > expand

( input_ids = None attention_mask = None token_type_ids = None position_ids = None head_mask = None inputs_embeds = None start_positions = None end_positions = None output_attentions = None output_hidden_states = None return_dict = None ) QuestionAnsweringModelOutput or tuple(torch.FloatTensor)

The QDQBertForQuestionAnswering forward method, overrides the __call__ special method.

Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the pre and post processing steps while the latter silently ignores them.

Example:

>>> from transformers import BertTokenizer, QDQBertForQuestionAnswering
>>> import torch

>>> tokenizer = BertTokenizer.from_pretrained('bert-base-uncased')

>>> question, text = "Who was Jim Henson?", "Jim Henson was a nice puppet"
>>> inputs = tokenizer(question, text, return_tensors='pt')
>>> start_positions = torch.tensor([1])
>>> end_positions = torch.tensor([3])

>>> outputs = model(**inputs, start_positions=start_positions, end_positions=end_positions)
>>> loss = outputs.loss
>>> start_scores = outputs.start_logits
>>> end_scores = outputs.end_logits


text:
# A little baking recipe

My first question on Puzzling.SE, I don't think it should be too hard.

Take a bowl and add one.
At last mix it all together and bake it at 270 degrees.

What will you get?

• A pan with 8 sticks of melted butter? – SirParselot Aug 10 '15 at 12:52
• I am not a native English Speaker and i Accidently used the wrong Word – Cryol Aug 10 '15 at 13:09
• I have edited the question a little to make it easier to read. If my edit has changed the intent of the question, feel free to rollback the edits. – CodeNewbie Aug 10 '15 at 13:15

0 (bowl) + 1 + 7 = 8 which, when rotated 270 degrees, looks like $\infty$

format: plain


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# Center for Analysis and Design of Intelligent Agents

### Site Tools

public:rem4:rem4-18:related_work_references

# Differences

This shows you the differences between two versions of the page.

public:rem4:rem4-18:related_work_references [2018/01/19 10:44]
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---------- ----------

-====Related Work and References====+=====Related Work and References=====

-===Typical Structure of a Scientific Paper===+====Typical Structure of a Scientific Paper====
|  Abstract  | This section is key - it's a mini-summary of your paper, intended to allow others to decide whether your work is relevant to their work (and whether they should read on) | |  Abstract  | This section is key - it's a mini-summary of your paper, intended to allow others to decide whether your work is relevant to their work (and whether they should read on) |
|  Introduction  | Overall context of the work, short summary of related work and a presentation of the motivation for the work - the problems that are to be addressed. Last paragraph: Explain the structure of the paper.  | |  Introduction  | Overall context of the work, short summary of related work and a presentation of the motivation for the work - the problems that are to be addressed. Last paragraph: Explain the structure of the paper.  |
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\\ \\

-===Remember the Audience===+====Remember the Audience====
|  Ask before you start your research  | This will determine your research context, experimental paradigm and the emphasis or slant you choose for your work. \\ This is especially important if you are working in interdisciplinary research or on projects that can appeal to more than one scientific community.  | |  Ask before you start your research  | This will determine your research context, experimental paradigm and the emphasis or slant you choose for your work. \\ This is especially important if you are working in interdisciplinary research or on projects that can appeal to more than one scientific community.  |
|  Ask before you start writing your paper  | Select the journal / conference first. \\ Do a background search on papers recently published there, to verify that your background section and description of work fits into their context (less important for journals).  | |  Ask before you start writing your paper  | Select the journal / conference first. \\ Do a background search on papers recently published there, to verify that your background section and description of work fits into their context (less important for journals).  |
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\\ \\

-===Related Work Section Format===+====Related Work Section Format====
|  Pick your style - be consistent !  | | |  Pick your style - be consistent !  | |
|  The fewer words the better  | As few words as possible, but not fewer (to paraphrase Einstein).  | |  The fewer words the better  | As few words as possible, but not fewer (to paraphrase Einstein).  |
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\\ \\

-===The Potatostamp Method™===+====The Potatostamp Method™====

|  What is it?  | A handy method to help you write a nice Related Work section  | |  What is it?  | A handy method to help you write a nice Related Work section  |
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\\ \\

-===Finding Related Work===+====Finding Related Work====
|  When have I searched enough?  | That depends on how "green" you are in your field of study. The bottom line is: You can be sure you missed at least one paper that is highly relevant to your work. Ergo: Keep looking until the last minute. Just don't miss the deadline.  | |  When have I searched enough?  | That depends on how "green" you are in your field of study. The bottom line is: You can be sure you missed at least one paper that is highly relevant to your work. Ergo: Keep looking until the last minute. Just don't miss the deadline.  |
| Cited work: Is there a maximum?  | No. Most journals and conferences put no limitations on the number of references one can have in a paper. \\ If the paper calls for a lot of references then you should try to include them all. \\ Using the rule of proportions: It is strange to see more than 30% of the words in a paper devoted to references (typically it will be between 5% and 10%).  | | Cited work: Is there a maximum?  | No. Most journals and conferences put no limitations on the number of references one can have in a paper. \\ If the paper calls for a lot of references then you should try to include them all. \\ Using the rule of proportions: It is strange to see more than 30% of the words in a paper devoted to references (typically it will be between 5% and 10%).  |
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-===Structure of the References Section===+====Structure of the References Section=====
|  Name-Year system   | **Name of author and year listed; alphabetical in reference section.** \\ Jones, J. P. (2002). Bass Playing Through the Ages. International Musician, 12(8): 232-234. \\ Pullman, J. (1999). The Effects of Toasters on Human Health. J. of Toasterology, 11(12): 11-22. | |  Name-Year system   | **Name of author and year listed; alphabetical in reference section.** \\ Jones, J. P. (2002). Bass Playing Through the Ages. International Musician, 12(8): 232-234. \\ Pullman, J. (1999). The Effects of Toasters on Human Health. J. of Toasterology, 11(12): 11-22. |
|  Citation-sequence system  | **Publications are numbered in the order they are cited.** \\ [1] Pullman, J. (1999). The Effects of Toasters on Human Health. J. of Toasterology, 11(12): 11-22. \\ [2] Jones, J. P. (2002). Bass Playing Through the Ages. International Musician, 12(8): 232-234. \\ [n] ...  | |  Citation-sequence system  | **Publications are numbered in the order they are cited.** \\ [1] Pullman, J. (1999). The Effects of Toasters on Human Health. J. of Toasterology, 11(12): 11-22. \\ [2] Jones, J. P. (2002). Bass Playing Through the Ages. International Musician, 12(8): 232-234. \\ [n] ...  |
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\\ \\

-===Structure of a Reference===+===Structure of a Reference====
|  APA Style (Amer. Psychological Assoc.)  | Very common -- possibly the most common reference style; used in many fields. The one we will use. \\ **Book:**  Molich, Rolf (2003). //Brugervenligt webdesign.// Copenhagen: Teknisk Forlag. \\ **Journal:**  Thórisson, K. R., H. Benko, A. Arnold, D. Abramov, S. Maskey, A. Vaseekaran (2004). Constructionist Design Methodology for Interactive Intelligences. //A.I. Magazine//, 25(4): 77-90. [OPTIONAL:] Menlo Park, CA: American Association for Artificial Intelligence. \\ **Conference:** Melson, G. F., Kahn, Jr., Peter H., Beck, A. M., Friedman, B., Roberts, T. and Garrett, E. (2005). Robots as Dogs? Children's Interactions with the Robotic Dog AIBO and a Live Australian Shepherd. //Proceedings of CHI 2005//, Philadelphia, PA, April 2-7, 33-39.  | |  APA Style (Amer. Psychological Assoc.)  | Very common -- possibly the most common reference style; used in many fields. The one we will use. \\ **Book:**  Molich, Rolf (2003). //Brugervenligt webdesign.// Copenhagen: Teknisk Forlag. \\ **Journal:**  Thórisson, K. R., H. Benko, A. Arnold, D. Abramov, S. Maskey, A. Vaseekaran (2004). Constructionist Design Methodology for Interactive Intelligences. //A.I. Magazine//, 25(4): 77-90. [OPTIONAL:] Menlo Park, CA: American Association for Artificial Intelligence. \\ **Conference:** Melson, G. F., Kahn, Jr., Peter H., Beck, A. M., Friedman, B., Roberts, T. and Garrett, E. (2005). Robots as Dogs? Children's Interactions with the Robotic Dog AIBO and a Live Australian Shepherd. //Proceedings of CHI 2005//, Philadelphia, PA, April 2-7, 33-39.  |
|  Other styles  | see e.g.: http://dal.ca.libguides.com/content.php?pid=860&sid=11818  | |  Other styles  | see e.g.: http://dal.ca.libguides.com/content.php?pid=860&sid=11818  |


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It should no longer be necessary rigourously to use the ADIC-model, described inCalculus 1c and Calculus 2c, because we now assume that the reader can do this himself. Solution. Rewriting the formulas for $${{a_n}},$$ $${{b_n}},$$ we can write the final expressions for the Fourier coefficients: ${{a_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\cos nxdx} ,\;\;\;}\kern-0.3pt{{b_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\sin nxdx} . {f\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {{d_n}\sin \left( {nx + {\varphi _n}} \right)} \;\;}\kern-0.3pt{\text{or}\;\;} {f\left( x \right) = \frac{1}{2} }+{ \frac{{1 – \left( { – 1} \right)}}{\pi }\sin x } The reasons for As before, only odd harmonics (1, 3, 5, ...) are needed to approximate the function; this is because of the, Since this function doesn't look as much like a sinusoid as. Part 1. {f\left( x \right) \text{ = }}\kern0pt Baron Jean Baptiste Joseph Fourier $$\left( 1768-1830 \right)$$ introduced the idea that any periodic function can be represented by a series of sines and cosines which are harmonically related. 1, & \text{if} & \frac{\pi }{2} \lt x \le \pi Fourier Series… + {\frac{{1 – {{\left( { – 1} \right)}^5}}}{{5\pi }}\sin 5x + \ldots } changes, or details, (i.e., the discontinuity) of the original function These cookies do not store any personal information. \[\int\limits_{ – \pi }^\pi {\left| {f\left( x \right)} \right|dx} \lt \infty ;$, ${f\left( x \right) = \frac{{{a_0}}}{2} \text{ + }}\kern0pt{ \sum\limits_{n = 1}^\infty {\left\{ {{a_n}\cos nx + {b_n}\sin nx} \right\}}}$, , The first term on the right side is zero. 1, & \text{if} & 0 < x \le \pi Since this function is odd (Figure. Using complex form find the Fourier series of the function $$f\left( x \right) = {x^2},$$ defined on the interval $$\left[ { – 1,1} \right].$$ Example 3 Using complex form find the Fourier series of the function The addition of higher frequencies better approximates the rapid Then, using the well-known trigonometric identities, we have, ${\int\limits_{ – \pi }^\pi {\sin nx\cos mxdx} }= {\frac{1}{2}\int\limits_{ – \pi }^\pi {\left[ {\sin{\left( {n + m} \right)x} }\right.}+{\left. The Fourier Series for an odd function is: f(t)=sum_(n=1)^oo\ b_n\ sin{:(n pi t)/L:} An odd function has only sine terms in its Fourier expansion. Suppose also that the function $$f\left( x \right)$$ is a single valued, piecewise continuous (must have a finite number of jump discontinuities), and piecewise monotonic (must have a finite number of maxima and minima). -periodic and suppose that it is presented by the Fourier series: {f\left ( x \right) = \frac { { {a_0}}} {2} \text { + }}\kern0pt { \sum\limits_ {n = 1}^\infty {\left\ { { {a_n}\cos nx + {b_n}\sin nx} \right\}}} f ( x) = a 0 2 + ∞ ∑ n = 1 { a n cos n x + b n sin n x } Calculate the coefficients. be. 2\pi. P = 1. Computing the complex exponential Fourier series coefficients for a square wave. Let's add a lot more sine waves. \end{cases},} The next couple of examples are here so we can make a nice observation about some Fourier series and their relation to Fourier sine/cosine series … Example 3. To define $${{a_0}},$$ we integrate the Fourier series on the interval $$\left[ { – \pi ,\pi } \right]:$$, \[{\int\limits_{ – \pi }^\pi {f\left( x \right)dx} }= {\pi {a_0} }+{ \sum\limits_{n = 1}^\infty {\left[ {{a_n}\int\limits_{ – \pi }^\pi {\cos nxdx} }\right.}+{\left. {{\int\limits_{ – \pi }^\pi {\sin nxdx} }={ \left. {\left( { – \frac{{\cos nx}}{n}} \right)} \right|_{ – \pi }^\pi }={ 0.}} ), At a discontinuity $${x_0}$$, the Fourier Series converges to, \[\lim\limits_{\varepsilon \to 0} \frac{1}{2}\left[ {f\left( {{x_0} – \varepsilon } \right) – f\left( {{x_0} + \varepsilon } \right)} \right].$, The Fourier series of the function $$f\left( x \right)$$ is given by, ${f\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {\left\{ {{a_n}\cos nx + {b_n}\sin nx} \right\}} ,}$, where the Fourier coefficients $${{a_0}},$$ $${{a_n}},$$ and $${{b_n}}$$ are defined by the integrals, ${{a_0} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)dx} ,\;\;\;}\kern-0.3pt{{a_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\cos nx dx} ,\;\;\;}\kern-0.3pt{{b_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\sin nx dx} . (in this case, the square wave). In an earlier module, we showed that a square wave could be expressed as a superposition of pulses. Fourier Series Examples. Find the Fourier Series for the function for which the graph is given by: So let us now develop the concept about the Fourier series, what does this series represent, why there is a need to represent the periodic signal in the form of its Fourier series. {\displaystyle P=1.} 11. There is Gibb's overshoot caused by the discontinuity. {\begin{cases} 1. A function $$f\left( x \right)$$ is said to have period $$P$$ if $$f\left( {x + P} \right) = f\left( x \right)$$ for all $$x.$$ Let the function $$f\left( x \right)$$ has period $$2\pi.$$ In this case, it is enough to consider behavior of the function on the interval $$\left[ { – \pi ,\pi } \right].$$, If the conditions $$1$$ and $$2$$ are satisfied, the Fourier series for the function $$f\left( x \right)$$ exists and converges to the given function (see also the Convergence of Fourier Series page about convergence conditions. Their representation in terms of simple periodic functions such as sine function … In order to incorporate general initial or boundaryconditions into oursolutions, it will be necessary to have some understanding of Fourier series. It is mandatory to procure user consent prior to running these cookies on your website. }$, First we calculate the constant $${{a_0}}:$$, ${{a_0} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)dx} }= {\frac{1}{\pi }\int\limits_0^\pi {1dx} }= {\frac{1}{\pi } \cdot \pi }={ 1. {\left( { – \frac{{\cos nx}}{n}} \right)} \right|_0^\pi } \right] }= { – \frac{1}{{\pi n}} \cdot \left( {\cos n\pi – \cos 0} \right) }= {\frac{{1 – \cos n\pi }}{{\pi n}}.}$. Fourier series calculation example Due to numerous requests on the web, we will make an example of calculation of the Fourier series of a piecewise defined function … Fourier series In the following chapters, we will look at methods for solving the PDEs described in Chapter 1. + {\frac{{1 – {{\left( { – 1} \right)}^3}}}{{3\pi }}\sin 3x } This might seem stupid, but it will work for all reasonable periodic functions, which makes Fourier Series a very useful tool. This example shows how to use the fit function to fit a Fourier model to data.. 5, ...) are needed to approximate the function. Fourier Series. Square waves (1 or 0 or −1) are great examples, with delta functions in the derivative. = {\frac{{{a_0}}}{2}\int\limits_{ – \pi }^\pi {\cos mxdx} } And we see that the Fourier Representation g(t) yields exactly what we were trying to reproduce, f(t). EEL3135: Discrete-Time Signals and Systems Fourier Series Examples - 1 - Fourier Series Examples 1. Find the constant a 0 of the Fourier series for function f (x)= x in 0 £ x £ 2 p. The given function f (x ) = | x | is an even function. In particular harmonics between 7 and 21 are not shown. By setting, for example, $$n = 5,$$ we get, $Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. This section contains a selection of about 50 problems on Fourier series with full solutions. 14. Now take sin(5x)/5: Add it also, to make sin(x)+sin(3x)/3+sin(5x)/5: Getting better! -1, & \text{if} & – \pi \le x \le – \frac{\pi }{2} \\ The Fourier Series also includes a constant, and hence can be written as: As $$\cos n\pi = {\left( { – 1} \right)^n},$$ we can write: \[{b_n} = \frac{{1 – {{\left( { – 1} \right)}^n}}}{{\pi n}}.$, Thus, the Fourier series for the square wave is, ${f\left( x \right) = \frac{1}{2} }+{ \sum\limits_{n = 1}^\infty {\frac{{1 – {{\left( { – 1} \right)}^n}}}{{\pi n}}\sin nx} . A Fourier series is nothing but the expansion of a periodic function f(x) with the terms of an infinite sum of sins and cosine values. ion discussed with half-wave symmetry was, the relationship between the Trigonometric and Exponential Fourier Series, the coefficients of the Trigonometric Series, calculate those of the Exponential Series. Our target is this square wave: Start with sin(x): Then take sin(3x)/3: And add it to make sin(x)+sin(3x)/3: Can you see how it starts to look a little like a square wave? Below we consider expansions of $$2\pi$$-periodic functions into their Fourier series, assuming that these expansions exist and are convergent. Specify the model type fourier followed by the number of terms, e.g., 'fourier1' to 'fourier8'.. The first zeros away from the origin occur when. In 1822 he made the claim, seemingly preposterous at the time, that any function of t, continuous or discontinuous, could be …$. Definition of the complex Fourier series. Introduction In these notes, we derive in detail the Fourier series representation of several continuous-time periodic wave-forms. solved examples in fourier series. 0/2 in the Fourier series. Fourier series: Solved problems °c pHabala 2012 Alternative: It is possible not to memorize the special formula for sine/cosine Fourier, but apply the usual Fourier series to that extended basic shape of f to an odd function (see picture on the left). FOURIER SERIES MOHAMMAD IMRAN JAHANGIRABAD INSTITUTE OF TECHNOLOGY [Jahangirabad Educational Trust Group of Institutions] www.jit.edu.in MOHAMMAD IMRAN SEMESTER-II TOPIC- SOLVED NUMERICAL PROBLEMS OF … }\], ${\int\limits_{ – \pi }^\pi {f\left( x \right)\cos mxdx} = {a_m}\pi ,\;\;}\Rightarrow{{a_m} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\cos mxdx} ,\;\;}\kern-0.3pt{m = 1,2,3, \ldots }$, Similarly, multiplying the Fourier series by $$\sin mx$$ and integrating term by term, we obtain the expression for $${{b_m}}:$$, ${{b_m} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\sin mxdx} ,\;\;\;}\kern-0.3pt{m = 1,2,3, \ldots }$. + {\frac{{1 – {{\left( { – 1} \right)}^4}}}{{4\pi }}\sin 4x } Calculate the Fourier coefficients for the sawtooth wave. \frac{\pi }{2} + x, & \text{if} & – \pi \le x \le 0 \\ \]. {\int\limits_{ – \pi }^\pi {f\left( x \right)\cos mxdx} } 2 π. Gibb's overshoot exists on either side of the discontinuity. b n = 1 π π ∫ − π f ( x) sin n x d x = 1 π π ∫ − π x sin n x d x. \end{cases}.} {\begin{cases} This website uses cookies to improve your experience. Replacing $${{a_n}}$$ and $${{b_n}}$$ by the new variables $${{d_n}}$$ and $${{\varphi_n}}$$ or $${{d_n}}$$ and $${{\theta_n}},$$ where, ${{d_n} = \sqrt {a_n^2 + b_n^2} ,\;\;\;}\kern-0.3pt{\tan {\varphi _n} = \frac{{{a_n}}}{{{b_n}}},\;\;\;}\kern-0.3pt{\tan {\theta _n} = \frac{{{b_n}}}{{{a_n}}},}$, Examples of Fourier series Last time, we set up the sawtooth wave as an example of a periodic function: The equation describing this curve is \begin {aligned} x (t) = 2A\frac {t} {\tau},\ -\frac {\tau} {2} \leq t < \frac {\tau} {2} \end {aligned} x(t) = 2Aτ t Here we present a collection of examples of applications of the theory of Fourier series. These cookies will be stored in your browser only with your consent. + {\sum\limits_{n = 1}^\infty {\left[ {{a_n}\int\limits_{ – \pi }^\pi {\cos nx\cos mxdx} }\right.}}+{{\left. Accordingly, the Fourier series expansion of an odd $$2\pi$$-periodic function $$f\left( x \right)$$ consists of sine terms only and has the form: \[f\left( x \right) = \sum\limits_{n = 1}^\infty {{b_n}\sin nx} ,, ${b_n} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\sin nxdx} .$. Contents. Even Pulse Function (Cosine Series) Aside: the periodic pulse function. 2\pi 2 π. }\], Find now the Fourier coefficients for $$n \ne 0:$$, ${{a_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\cos nxdx} }= {\frac{1}{\pi }\int\limits_0^\pi {1 \cdot \cos nxdx} }= {\frac{1}{\pi }\left[ {\left. There is no discontinuity, so no Gibb's overshoot. This website uses cookies to improve your experience while you navigate through the website. This example fits the El … In this section we define the Fourier Sine Series, i.e. { {b_n}\int\limits_{ – \pi }^\pi {\sin nx\cos mxdx} } \right]} .} Since this function is the function of the example above minus the constant . Example 1: Special case, Duty Cycle = 50%. Recall that we can write almost any periodic, continuous-time signal as an inﬁnite sum of harmoni-cally 1. This allows us to represent functions that are, for example, entirely above the x−axis. The Fourier library model is an input argument to the fit and fittype functions. Common examples of analysis intervals are: x ∈ [ 0 , 1 ] , {\displaystyle x\in [0,1],} and. Periodic Signals and Fourier series: As described in the precious discussion that the Periodic Signals can be represented in the form of the Fourier series. approximation improves. Fourier series is a very powerful and versatile tool in connection with the partial differential equations. \frac{\pi }{2} – x, & \text{if} & 0 \lt x \le \pi Necessary cookies are absolutely essential for the website to function properly.$, Therefore, all the terms on the right of the summation sign are zero, so we obtain, ${\int\limits_{ – \pi }^\pi {f\left( x \right)dx} = \pi {a_0}\;\;\text{or}\;\;\;}\kern-0.3pt{{a_0} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)dx} .}$. Can we use sine waves to make a square wave? As useful as this decomposition was in this example, it does not generalize well to other periodic signals: How can a superposition of pulses equal a smooth signal like a sinusoid? Example. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_0^\pi } \right] }= {\frac{1}{{\pi n}} \cdot 0 }={ 0,}\], ${{b_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\sin nxdx} }= {\frac{1}{\pi }\int\limits_0^\pi {1 \cdot \sin nxdx} }= {\frac{1}{\pi }\left[ {\left. {f\left( x \right) \text{ = }}\kern0pt Example of Rectangular Wave. Signal and System: Solved Question on Trigonometric Fourier Series ExpansionTopics Discussed:1. The Fourier series expansion of an even function $$f\left( x \right)$$ with the period of $$2\pi$$ does not involve the terms with sines and has the form: \[{f\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {{a_n}\cos nx} ,}$, where the Fourier coefficients are given by the formulas, ${{a_0} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)dx} ,\;\;\;}\kern-0.3pt{{a_n} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\cos nxdx} .}$. 0, & \text{if} & – \frac{\pi }{2} \lt x \le \frac{\pi }{2} \\ Figure 1 Thevenin equivalent source network. {f\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {{d_n}\cos\left( {nx + {\theta _n}} \right)} .} }\], Sometimes alternative forms of the Fourier series are used. This category only includes cookies that ensures basic functionalities and security features of the website. }\], We can easily find the first few terms of the series. Click or tap a problem to see the solution. But opting out of some of these cookies may affect your browsing experience. The signal x (t) can be expressed as an infinite summation of sinusoidal components, known as a Fourier series, using either of the following two representations. An example of a periodic signal is shown in Figure 1. { {b_n}\int\limits_{ – \pi }^\pi {\sin nxdx} } \right]}}\], $\end{cases},} Let’s go through the Fourier series notes and a few fourier series examples.. Definition of Fourier Series and Typical Examples, Fourier Series of Functions with an Arbitrary Period, Applications of Fourier Series to Differential Equations, Suppose that the function $$f\left( x \right)$$ with period $$2\pi$$ is absolutely integrable on $$\left[ { – \pi ,\pi } \right]$$ so that the following so-called. Exercises. Start with sinx.Ithasperiod2π since sin(x+2π)=sinx. {\begin{cases} We also use third-party cookies that help us analyze and understand how you use this website. { {b_n} }= { \frac {1} {\pi }\int\limits_ { – \pi }^\pi {f\left ( x \right)\sin nxdx} } = {\frac {1} {\pi }\int\limits_ { – \pi }^\pi {x\sin nxdx} .} 0, & \text{if} & – \pi \le x \le 0 \\ Find b n in the expansion of x 2 as a Fourier series in (-p, p). We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function. There is Gibb's overshoot caused by the discontinuities. Find the Fourier series of the function function Answer. harmonic, but not all of the individual sinusoids are explicitly shown on the plot. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. You also have the option to opt-out of these cookies. Periodic functions occur frequently in the problems studied through engineering education. A Fourier Series, with period T, is an infinite sum of sinusoidal functions (cosine and sine), each with a frequency that is an integer multiple of 1/T (the inverse of the fundamental period).$, The graph of the function and the Fourier series expansion for $$n = 10$$ is shown below in Figure $$2.$$. {\left( {\frac{{\sin 2mx}}{{2m}}} \right)} \right|_{ – \pi }^\pi + 2\pi } \right] }= {\frac{1}{{4m}}\left[ {\sin \left( {2m\pi } \right) }\right.}-{\left. P. {\displaystyle P} , which will be the period of the Fourier series. We look at a spike, a step function, and a ramp—and smoother functions too. As you add sine waves of increasingly higher frequency, the approximation gets better and better, and these higher frequencies better approximate the details, (i.e., the change in slope) in the original function. + {\frac{2}{{3\pi }}\sin 3x } { {\sin \left( {n – m} \right)x}} \right]dx} }={ 0,}\], ${\int\limits_{ – \pi }^\pi {\cos nx\cos mxdx} }= {\frac{1}{2}\int\limits_{ – \pi }^\pi {\left[ {\cos {\left( {n + m} \right)x} }\right.}+{\left. There are several important features to note as Tp is varied. Solved problem on Trigonometric Fourier Series,2. As you add sine waves of increasingly higher frequency, the Using 20 sine waves we get sin(x)+sin(3x)/3+sin(5x)/5 + ... + sin(39x)/39: Using 100 sine waves we ge… this are discussed. Fourier Series Jean Baptiste Joseph Fourier (1768-1830) was a French mathematician, physi-cist and engineer, and the founder of Fourier analysis. The amplitudes of the harmonics for this example drop off much more rapidly (in this case they go as. + {\frac{2}{{5\pi }}\sin 5x + \ldots }$, $= {\frac{1}{2} + \frac{2}{\pi }\sin x } To consider this idea in more detail, we need to introduce some definitions and common terms. A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.Fourier series make use of the orthogonality relationships of the sine and cosine functions. With a suﬃcient number of harmonics included, our ap- proximate series can exactly represent a given function f(x) f(x) = a 0/2 + a In order to find the coefficients $${{a_n}},$$ we multiply both sides of the Fourier series by $$\cos mx$$ and integrate term by term: \[$, ${\left( { – \frac{{\cos 2mx}}{{2m}}} \right)} \right|_{ – \pi }^\pi } \right] }= {\frac{1}{{4m}}\left[ { – \cancel{\cos \left( {2m\pi } \right)} }\right.}+{\left. So Therefore, the Fourier series of f(x) is Remark. {a_0} = {a_n} = 0. a 0 = a n = 0. Find the constant term a 0 in the Fourier series … As an example, let us find the exponential series for the following rectangular wave, given by The reader is also referred toCalculus 4b as well as toCalculus 3c-2. This section explains three Fourier series: sines, cosines, and exponentials eikx. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_{ – \pi }^\pi }={ 0\;\;}{\text{and}\;\;\;}}\kern-0.3pt Tp/T=1 or n=T/Tp (note this is not an integer values of Tp). This version of the Fourier series is called the exponential Fourier series and is generally easier to obtain because only one set of coefficients needs to be evaluated. + {\frac{{1 – {{\left( { – 1} \right)}^2}}}{{2\pi }}\sin 2x } 15. We defined the Fourier series for functions which are -periodic, one would wonder how to define a similar notion for functions which are L-periodic. { \sin \left( {2m\left( { – \pi } \right)} \right)} \right] + \pi }={ \pi . We'll assume you're ok with this, but you can opt-out if you wish. Because of the symmetry of the waveform, only odd harmonics (1, 3, representing a function with a series in the form Sum( B_n sin(n pi x / L) ) from n=1 to n=infinity. { \cancel{\cos \left( {2m\left( { – \pi } \right)} \right)}} \right] }={ 0;}$, ${\int\limits_{ – \pi }^\pi {\cos nx\cos mxdx} }= {\frac{1}{2}\int\limits_{ – \pi }^\pi {\left[ {\cos 2mx + \cos 0} \right]dx} ,\;\;}\Rightarrow{\int\limits_{ – \pi }^\pi {{\cos^2}mxdx} }= {\frac{1}{2}\left[ {\left. {f\left( x \right) \text{ = }}\kern0pt {{\int\limits_{ – \pi }^\pi {\cos nxdx} }={ \left. Fourier Cosine Series for even functions and Sine Series for odd functions The continuous limit: the Fourier transform (and its inverse) The spectrum Some examples and theorems F() () exp()ωωft i t dt 1 () ()exp() 2 ft F i tdω ωω π x ∈ [ … In the next section, we'll look at a more complicated example, the saw function. { {\cos \left( {n – m} \right)x}} \right]dx} }={ 0,}$, \[\require{cancel}{\int\limits_{ – \pi }^\pi {\sin nx\cos mxdx} }= {\frac{1}{2}\int\limits_{ – \pi }^\pi {\left[ {\sin 2mx + \sin 0} \right]dx} ,\;\;}\Rightarrow{\int\limits_{ – \pi }^\pi {{\sin^2}mxdx} }={ \frac{1}{2}\left[ {\left. The rightmost button shows the sum of all harmonics up to the 21st Since f ( x) = x 2 is an even function, the value of b n = 0. Saw function problems studied through engineering education that ensures basic functionalities and features! Into oursolutions, it will work for all reasonable periodic functions such as sine function ….! You wish your consent not shown introduction in these notes, we can easily the! Website to function properly us analyze and understand how you use this website mxdx }! 'Fourier8 ' for a function problems studied through engineering education as you add sine to!, e.g., 'fourier1 ' to 'fourier8 ' trying to reproduce, f ( x is... As well as toCalculus 3c-2 values of Tp ) procure user consent to. } and can easily find the first zeros away from the origin occur when complicated,! That are, for example, entirely above the x−axis 1 ], } and for reasonable... X ∈ [ 0, 1 ], } and series for a function Figure.... Mandatory to procure user consent prior to running these cookies on your website function is the function Answer! Expansions of \ ( 2\pi\ ) -periodic functions into their Fourier series Jean Baptiste Fourier... Function, the first zeros away from the origin occur when to reproduce, (... Waves to make a square wave square waves ( 1 or 0 or −1 are. Series ) Aside: the periodic Pulse function model is an input argument to the and... Functions in the next section, we derive in detail the Fourier representation g t! An earlier module, we can easily find the first zeros away from the origin occur when Fourier are! Example of a periodic signal is shown in Figure 1 you navigate through the to. We consider expansions of \ ( 2\pi\ ) -periodic functions into their series. Consider expansions of \ ( 2\pi\ ) -periodic functions into their Fourier series... ( in this case they go as periodic wave-forms several continuous-time periodic wave-forms cookies! Problem to see the solution caused by the discontinuity problem to fourier series examples the solution term on the right side zero. This is not an integer values of Tp ) Fourier series representation several! Understand how you use this website uses cookies to improve your experience while navigate... Your experience while you navigate through the website Trigonometric Fourier series make a wave... { a_0 } = { a_n } = 0. a 0 = a n = 0, showed... Sine series, assuming that these expansions exist and are convergent of several continuous-time periodic.... A very powerful and versatile tool in connection with the partial differential equations French,! In particular harmonics between 7 and 21 are not shown example of a periodic signal shown. = { a_n } = 0. a 0 = a n = 0 option! Use sine waves to make a square wave } \right ] }. useful tool work for all periodic... 1: Special case, Duty Cycle = 50 % sine waves to make square! 0 or −1 ) are great examples, with delta functions in the derivative is the function Answer. This case they go as exist and are convergent of Fourier series examples occur frequently the! = x 2 as a Fourier series notes and a ramp—and smoother functions too shown in 1! 'Fourier8 ' series of the Fourier series representation of several continuous-time periodic wave-forms off much more rapidly ( this! Term on the right side is zero ’ s go through the.... It will work for all reasonable periodic functions, which makes Fourier series of series... \Right ] }. Jean Baptiste Joseph Fourier ( 1768-1830 ) was a French mathematician, physi-cist and,! Forms of the series help us analyze and understand how you use this.... As Tp is varied followed by the discontinuities 1768-1830 ) was a mathematician. Security features of the function function Answer terms of the series detail, we need introduce. Reader is also referred toCalculus 4b as well as toCalculus 3c-2 \displaystyle }! You wish the value of b n = 0 if you wish for example, entirely above the.! You also have the option to opt-out of these cookies may affect your browsing experience we use sine of! Mandatory to procure user consent prior to running these cookies may affect your browsing experience fit and fittype.! Series, i.e great examples, with delta functions in the derivative can we sine... Also define the odd extension for a function and work several examples finding the Fourier series of the discontinuity order. N = 0 is zero an input argument to the fit and functions. 'S overshoot caused by the number of terms, e.g., 'fourier1 ' to '. Cosine series ) Aside: the periodic Pulse function ( Cosine series Aside... Is mandatory to procure user consent prior to running these cookies on your website great. Detail, we need to introduce some definitions and common terms will also define the Fourier of. Series notes and a few Fourier series examples to see the solution functions into their Fourier series Baptiste... Sine waves to make a square wave could be expressed as a series. Right side is zero or n=T/Tp ( note this is not an integer values Tp... 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# Dnes jsou cílem k trestání Maďarsko a Polsko, zítra může dojít na nás

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„Pouze nezávislý soudní orgán může stanovit, co je vláda práva, nikoliv politická většina,“ napsal slovinský premiér Janša v úterním dopise předsedovi Evropské rady Charlesi Michelovi. Podpořil tak Polsko a Maďarsko a objevilo se tak třetí veto. Německo a zástupci Evropského parlamentu změnili mechanismus ochrany rozpočtu a spolu se zástupci vlád, které podporují spojení vyplácení peněz z fondů s dodržováním práva si myslí, že v nejbližších týdnech Polsko a Maďarsko přimějí změnit názor. Poláci a Maďaři si naopak myslí, že pod tlakem zemí nejvíce postižených Covid 19 změní názor Němci a zástupci evropského parlamentu.

Mechanismus veta je v Unii běžný. Na stejném zasedání, na kterém padlo polské a maďarské, vetovalo Bulharsko rozhovory o členství se Severní Makedonií. Jenže takový to druh veta je vnímán pokrčením ramen, principem je ale stejný jako to polské a maďarské.

Podle Smlouvy o EU je rozhodnutí o potrestání právního státu přijímáno jednomyslně Evropskou radou, a nikoli žádnou většinou Rady ministrů nebo Parlamentem (Na návrh jedné třetiny členských států nebo Evropské komise a po obdržení souhlasu Evropského parlamentu může Evropská rada jednomyslně rozhodnout, že došlo k závažnému a trvajícímu porušení hodnot uvedených ze strany členského státu). Polsko i Maďarsko tvrdí, že zavedení nové podmínky by vyžadovalo změnu unijních smluv. Když změny unijních smluv navrhoval v roce 2017 Jaroslaw Kaczyński Angele Merkelové (za účelem reformy EU), ta to při představě toho, co by to v praxi znamenalo, zásadně odmítla. Od té doby se s Jaroslawem Kaczyńskim oficiálně nesetkala. Rok se s rokem sešel a názor Angely Merkelové zůstal stejný – nesahat do traktátů, ale tak nějak je trochu, ve stylu dobrodruhů dobra ohnout, za účelem trestání neposlušných. Dnes jsou cílem k trestání Maďarsko a Polsko, zítra může dojít na nás třeba jen za to, že nepřijmeme dostatečný počet uprchlíků.

Čeští a slovenští ministři zahraničí považují dodržování práva za stěžejní a souhlasí s Angelou Merkelovou. Asi jim dochází, o co se Polsku a Maďarsku jedná, ale nechtějí si znepřátelit silné hráče v Unii. Pozice našeho pana premiéra je mírně řečeno omezena jeho problémy s podnikáním a se znalostí pevného názoru Morawieckého a Orbana nebude raději do vyhroceného sporu zasahovat ani jako případný mediátor kompromisu. S velkou pravděpodobností v Evropské radě v tomto tématu členy V4 nepodpoří, ale alespoň by jim to měl říci a vysvětlit proč. Aby prostě jen chlapsky věděli, na čem jsou a nebrali jeho postoj jako my, když onehdy překvapivě bývalá polská ministryně vnitra Teresa Piotrowska přerozdělovala uprchlíky.

Pochopit polskou politiku a polské priority by měli umět i čeští politici. České zájmy se s těmi polskými někde nepřekrývají, ale naše vztahy se vyvíjí velmi dobře a budou se vyvíjet doufejme, bez toho, že je by je manažerovali němečtí či holandští politici, kterým V4 leží v žaludku. Rozhádaná V4 je totiž přesně to, co by Angele Merkelové nejvíc vyhovovalo.

# Morawiecki: Hřbitovy budou na Dušičky uzavřeny

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V sobotu, neděli a v pondělí budou v Polsku uzavřeny hřbitovy – rozhodla polská vláda. Nechceme, aby se lidé shromažďovali na hřbitovech a ve veřejné dopravě, uvedl premiér Mateusz Morawiecki.

„S tímto rozhodnutím jsme čekali, protože jsme žili v naději, že počet případů nakažení se alespoň mírně sníží. Dnes je ale opět větší než včera, včera byl větší než předvčerejškem a nechceme zvyšovat riziko shromažďování lidí na hřbitovech, ve veřejné dopravě a před hřbitovy“. vysvětlil Morawiecki.

Dodal, že pro něj to je „velký smutek“, protože také chtěl navštívit hrob svého otce a sestry. Svátek zemřelých je hluboce zakořeněný v polské tradici, ale protože s sebou nese obrovské riziko, Morawiecki rozhodl, že život je důležitější než tradice.

# Poslankyně opozice atakovaly předsedu PiS

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Ochranná služba v Sejmu musela oddělit lavici, ve které sedí Jaroslaw Kaczyński od protestujících poslankyň.

„Je mi líto, že to musím říci, ale v sále mezi členy Levice a Občanské platformy jsou poslanci s rouškami se symboly, které připomínají znaky Hitlerjugent a SS. Chápu však, že totální opozice odkazuje na totalitní vzorce.“ řekl na začátku zasedání Sejmu místopředseda Sejmu Ryszard Terlecki.

Zelená aktivistka a místopředsedkyně poslaneckého klubu Občanské koalice Małgorzata Tracz, která měla na sobě masku se symbolem protestu proti rozsudku Ústavního soudu – červený blesk: „Pane místopředsedo, nejvyšší sněmovno, před našimi očima se odehrává historie, 6 dní protestují tisíce mladých lidí v ulicích polských měst, protestují na obranu své důstojnosti, na obranu své svobody, na obranu práva volby, za právo na potrat. Toto je válka a tuto válku prohrajete. A kdo je za tuto válku zodpovědný? Pane ministře Kaczyński, to je vaše odpovědnost.“


text:
A Time-Dependent Wave-Thermoelastic Solid Interaction

# A Time-Dependent Wave-Thermoelastic Solid Interaction

G. C. Hsiao, T. Sánchez-Vizuet, F.–J.Sayas & R. J. Weinacht
Department of Mathematical Sciences, University of Delaware, USA.
{ghsiao, fjsayas, weinacht}@udel.edu
Courant Institute of Mathematical Sciences, New York University, USA.
tonatiuh@cims.nyu.edu
July 30, 2019
This paper is dedicated to Wolfgang L. Wendland
on the occasion of his 80th birthday.
###### Abstract

This paper presents a combined field and boundary integral equation method for solving the time-dependent scattering problem of a thermoelastic body immersed in a compressible, inviscid and homogeneous fluid. The approach here is a generalization of the coupling procedure employed by the authors for the treatment of the time-dependent fluid-structure interaction problem. Using an integral representation of the solution in the infinite exterior domain occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with nonlocal boundary conditions. The nonlocal boundary problem is analyzed with Lubich’s approach for time-dependent boundary integral equations. Existence and uniqueness results are established in terms of time-domain data with the aid of Laplace-domain techniques. Galerkin semi-discretization approximations are derived and error estimates are obtained. A full discretization based on the Convolution Quadrature method is also outlined. Some numerical experiments in 2D are also included in order to demonstrate the accuracy and efficiency of the procedure.

Key words: Fluid-structure interaction, Coupling BEM-FEM, Kirchhoff representation formula, Retarded potential, Time-domain boundary integral equation, Variational formulation, Wave scattering, Convolution quadrature.

Mathematics Subject Classifications: 35J20, 35L05, 45P05, 65N30, 65N38. âº

## 1 Introduction

The mathematical study of the thermodynamic response of a linearly elastic solid to mechanical strain dates back at least to Duhamel’s 1837 pioneering work [10] on thermoelastic materials where he proposed the constitutive relation linking the temperature variations and elastic strains with the thermoelastic stress now known as Duhamel-Neumann law [7, 32].

Kupradze’s encyclopedic works [23] can be considered the standard reference for a modern mathematical treatment of the purely thermoelastic problem. The dynamic problem is dealt with in more recent works like [34, 37] where the matrix of fundamental solutions for the dynamic equations is revisited, while [20, 21] provide generalized Kirchhoff-type formulas for thermoelastic solids.

In the case of the scattering of thermoelastic waves, major theoretical contributions have been made by Çakoni and Dassios in [6]. The unique solvability of a boundary integral formulation is established in [5] and the interaction of elastic and thermoelastic waves is explored for homogeneous materials in [8]. The study of time-harmonic interaction between a scalar field and a thermoelastic solid has been the subject of works like [27] where the interface is taken to be a plane, or [26, 22] where time-harmonic scattering by bounded obstacles is considered.

In this paper, we present a combined field and boundary integral method for a time-dependent fluid-thermoelastic solid interaction problem. The approach here is a generalization of the method employed by the authors for treating time-dependent fluid -structure interaction problems in [18, 17]. The present communication is an improvement over those previous efforts in the sense that it considers a more general constitutive law that accounts for the coupling between elastic and thermal effects. To our knowledge no attempt has been made to investigate with rigorous justifications the time-dependent acoustic scattering by a thermoelastic obstacle.

The setting is introduced in Section 2, along with the physical assumptions and the constitutive relation under consideration leading to the time-domain system of governing equations. The problem is then recast in Section 3 where the Laplace domain system is transformed into an equivalent integro-differential non-local problem that will be formulated variationally for discretization later on. The question of existence and uniqueness of the solutions to the non-local problem is dealt with on Section 4. The error analysis of the proposed discretization is addressed in Section 5, where semi-discrete error estimates for spatial discretization are provided.

The final Section 6 discusses the computational considerations related to the numerical solution of the discrete problem. A full discretization using Convolution Quadrature (CQ) in time is outlined and the coupling of boundary and finite elements for the spatial discretization is discussed. Convergence experiments in 2D are performed for test problems in both frequency and time domains as a demonstration of the applicability of the formulation which remains valid also in 3D. For time discretization both second order backward differentiation formula (BDF2) and Trapezoidal Rule CQ are used, providing evidence that the approximation is stable and of second order globally. Time-domain illustrative experiments using the proposed formulation are included.

In closing, we remark that for homogeneous thermoelastic solid medium, a pure boundary integral equation formulation may be adapted as in the fluid-structure interaction problem [17]. We will pursue these investigations in a separate communication.

## 2 Formulation of the problem

Consider a thermoelastic solid with constant density in an undeformed reference configuration and at thermal equilibrium at temperature . Under the action of external forces the body will be subject to internal stresses that will induce local variations of temperature. Reciprocally, if a heat source induces a change in temperature, the body will react by dilating or contracting and this will create internal stresses and deformations. We will denote by the elastic deformation with respect to the reference configuration and by the variation of temperature with respect to the equilibrium temperature. In the classical linear theory [23, 24], the coupling between the mechanic strain and the thermal gradient is modeled by the Duhamel-Neumann law which defines the thermoelastic stress and the thermoelastic heat flux (also known as free energy) as functions of the elastic displacement and the variation in temperature by

 σ:= Cε(U)−ζΘI, F:= −η∂U∂t+κ∇Θ.

In the previous expressions

 ε(U):=12(∇U+(∇U)t)

is the elastic strain tensor, is the identity matrix, is the thermal diffusivity coefficient, which from physical principles [14] is required to be positive, is the product of the volumetric thermal expansion coefficient and the bulk modulus of the material, and is given by the relation

 η=Θ0ζ/cvol.

Here the volumetric heat capacity is the ratio between the thermal diffusivity and the thermal conductivity, and it can also be expressed as the product of the mass density and the specific heat capacity. For the case of homogeneous isotropic material that we are considering, the elastic stiffness tensor is given by

 Cijkl:=λδijδkl+μ(δikδjl+δilδjk),

where the constants and are Lamé’s second parameter and the shear modulus respectively, and is Kronecker’s delta.

We are concerned with a time-dependent direct scattering problem in fluid-thermoelastic solid interaction, which can be simply described as follows: an acoustic wave propagates in a fluid domain of infinite extent in which a bounded thermoelastic body is immersed. Throughout the paper, we let be the bounded domain in occupied by the thermoelastic body with a Lipschitz boundary and we let be its exterior, occupied by a compressible fluid. The problem is then to determine the scattered velocity potential in the fluid domain, the deformation of the solid and the variation of the temperature in the obstacle. It is assumed that .

The governing equations of the displacement field and temperature field are the thermo-elastodynamic equations:

 ρΣ∂2U∂t2−Δ∗U+ζ∇Θ= 0 in Ω−×(0,T), (2.1) 1κ∂Θ∂t−ΔΘ+η∂∂t(∇⋅U) = 0 in Ω−×(0,T), (2.2)

where is a given positive final time, and as usual the symbol is the Lamé operator defined by

 Δ∗U:=μΔU+(λ+μ)∇(∇⋅U).

We remark that if the thermal effect is neglected () Duhamel-Neumann’s law reduces to the usual expression for Hooke’s law of the classical theory for an arbitrary isotropic medium (see, e.g. [7, 23]). In the thermoelastic medium, the given physical constants , are assumed to satisfy the inequalities:

 ρΣ>0,μ>0,3λ+2μ>0,ζη>0,κ>0.

In the fluid domain , we consider a barotropic and irrotational flow of an inviscid and compressible fluid with density as in [18]. The formulation can be presented in terms of a scalar potential such that the scattered velocity field and the pressure are given by

 V=−∇VandP=ρf∂V∂t.

Then we arrive at the wave equation

 1c2∂2V∂t2−ΔV=0inΩ+×(0,T) (2.3)

where is the sound speed.

On the interface between the solid and the fluid we have the transmission conditions

 σ(U,Θ)−n= −ρf(∂V∂t+∂Vinc∂t)+n on Γ×(0,T), (2.4a) ∂U−∂t⋅n= −(∂V∂n+∂Vinc∂n)+ on Γ×(0,T), (2.4b) ∂Θ∂n−= 0 on Γ×(0,T), (2.4c)

where is the exterior unit normal to , and denotes the given incident field, which is assumed to be supported away from at . Here and in the sequel, we adopt the notation that denotes the limit of the function on from respectively. Regarding the transmission conditions we remark that from the physical point of view, equation (2.4a) enforces the equilibrium of pressure at the solid-fluid interface, the condition (2.4b) expresses the continuity of the normal component of the velocity field, and (2.4c) refers to a thermally insulated body. We assume the causal initial conditions

 U(x,t)=∂U(x,t)∂t=0,Θ(x,t)=0 forx∈Ω−,t≤0, (2.5a) V(x,t)=∂V∂t(x,t)=0, forx∈Ω+,t≤0. (2.5b)

We will study the the time-dependent scattering problem consisting of the partial differential equations (2.1)-(2.3) together with the transmission conditions (2.4a)-(2.4c) and the homogeneous initial conditions (2.5a)-(2.5b).

## 3 Reduction to a nonlocal problem

In order to apply Lubich’s approach as in the case of fluid-structure interaction [18, 17], we first need to transform the initial-boundary transmission problem (2.1)-(2.4c) in the Laplace domain. Then we reduce the corresponding problem to a nonlocal boundary value problem. We begin with the Laplace transform for a restricted class of distributions. Let be a Banach space and denote the Schwartz class of functions. We say that is a causal tempered distribution with values in if it is a continuous linear map such that

 F(φ)=0∀φ∈S(R) such that supp\leavevmode\nobreak φ⊂(−∞,0).

For such a distribution and

 s∈C+:={s∈C:Re\leavevmode\nobreak s>0},

the Laplace transform of can be defined in a natural way by

 f(s)=L{F}(s):=∫∞0e−stF(t)dt,

where the integral must be understood in the sense of Bochner [35]. We remark that the Laplace transform can be defined for a much broader class of distributions [4, 9], but this restricted class suffices for the current application.

Let then

 u:=u(x,s)=L{U(x,t)},θ:=θ(x,s)=L{Θ(x,t)},v:=v(x,s)=L{V(x,t)}.

Then the initial-boundary transmission problem consisting of (2.1) - (2.4c) in the Laplace transformed domain becomes the following transmission boundary value problem:

 −Δ∗u+ρΣs2u+ζ∇θ= 0 inΩ−, (3.1a) −Δθ+sκ\leavevmode\nobreak θ+s\leavevmode\nobreak η\leavevmode\nobreak ∇⋅u= 0 inΩ−, (3.1b) −Δv+s2c2\leavevmode\nobreak v= 0 inΩ+, (3.1c) σ(u,θ)−n+ρfsv+n= −ρfsvincn onΓ, (3.1d) s\leavevmode\nobreak u−⋅n+∂v∂n+= −∂vinc∂n onΓ, (3.1e) ∂θ∂n−= 0 onΓ. (3.1f)

We remark that (3.1) is an exterior scattering problem for which normally a radiation condition is needed in order to guarantee the uniqueness of the solution. However, in the present case no additional radiation condition is required and global behavior at infinity suffices.

To derive a proper nonlocal boundary problem, as usual, we begin via Green’s third identity with the representation of the solutions of (3.1c) in the form:

 v=D(s)ϕ−S(s)λinΩ+, (3.2)

where and are the Cauchy data for in (3.1c) and and are the simple-layer and double-layer potentials, respectively defined by

 S(s)λ(x):= ∫ΓEs/c(x,y)λ(y)dΓy, x∈R3∖Γ, (3.3) D(s)ϕ(x):= ∫Γ∂∂nyEs/c(x,y)ϕ(y)dΓy, x∈R3∖Γ. (3.4)

Here

 Es/c(x,y)=e−s/c\leavevmode\nobreak |x−y|4π|x−y|

is the fundamental solution of the operator in (3.1c). By standard arguments in potential theory, we have the relations for the the Cauchy data and :

 (ϕλ)=(12I+K(s)−V(s)−W(s)(12I−K(s))′)(ϕλ)onΓ. (3.5)

Here and are the four basic boundary integral operators familiar from potential theory [19] such that

 V(s)λ(x):= ∫ΓEs/c(x,y)λ(y)dΓy x∈Γ, K(s)ϕ(x):= ∫Γ∂∂nyEs/c(x,y)ϕ(y)dΓy x∈Γ, K′(s)λ(x):= ∫Γ∂∂nxEs/c(x,y)λ(y)dΓy x∈Γ, W(s)ϕ(x):= −∂∂nx∫Γ∂∂nyEs/c(x,y)ϕ(y)dΓy x∈Γ.

By using the transmission condition (3.1e), we obtain from the second boundary integral equation in (3.5),

 −su−⋅n+W(s)ϕ−(12I−K(s))′λ=∂vinc∂nonΓ (3.6)

while the first boundary integral equation in (3.5) is simply

 (12I−K(s))ϕ+V(s)λ=0onΓ. (3.7)

With the Cauchy data and as new unknowns, the partial differential equation (3.1c) in may be eliminated. This leads to a nonlocal boundary value problem in for the unknowns satisfying the partial differential equations (3.1a), (3.1b), and the boundary integral equations (3.6), (3.7) together with the conditions (3.1d) and (3.1f) on .

Here and in the sequel let and denote trace operators of the functions and their normal derivatives from inside and outside , respectively. We will use the symbol interchangeably to denote the scalar, vector, or Frobenius inner products of functions defined on the open set , while the angled brackets will be reserved for pairings between elements of the trace space and its dual. All the forms will be kept linear and conjugation will be done explicitly when needed. Finally, the space should be understood as the Cartesian product of copies of the standard scalar Sobolev space endowed with the natural product norm.

Let us first consider the unknowns . Then multiplying (3.1a) by the testing function and integrating by parts, we obtain the weak formulation of (3.1a):

 a(u,v;s)−⟨σ(u,θ)n,γ−v⟩Γ−ζ(θ,∇⋅v)Ω−=0, (3.8)

where is the bilinear form defined by

 a(u,v;s):=(Cε(u),ε(v))Ω−+s2ρΣ(u,v)Ω−. (3.9)

In terms of the transmission condition (3.1d), we obtain from (3.8)

 a(u,v;s)−ζ(θ,∇⋅v)Ω−+ρfs⟨ϕ\leavevmode\nobreak n,γ−v⟩Γ=−sρf⟨vincn,γ−v⟩Γ. (3.10)

Similarly, multiplying (3.1b) by the test function , integrating by parts and making use of the condition (3.1f), we have

 b(θ,ϑ;s)+s\leavevmode\nobreak η(∇⋅u,ϑ)Ω−=0 (3.11)

with

 b(θ,ϑ;s):=(∇θ,∇ϑ)Ω−+sκ(θ,ϑ)Ω−. (3.12)

Now let

 As: H1(Ω−) ⟶ (H1(Ω−))′ u ⟼ a(u,⋅;s) Bs: H1(Ω−) ⟶ (H1(Ω−))′ θ ⟼ b(θ,⋅;s)

be the operators associated to the bilinear forms (3.9) and (3.12), respectively. Then from (3.10), (3.12), (3.6), and (3.7), the nonlocal problem may be formulated as a system of operator equations: Given data find such that

 A⎛⎜ ⎜ ⎜⎝uθϕλ⎞⎟ ⎟ ⎟⎠:=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝As−ζ\leavevmode\nobreak (div)′s\leavevmode\nobreak ρfγ−′\leavevmode\nobreak n0s\leavevmode\nobreak η\leavevmode\nobreak divBs00−s\leavevmode\nobreak n⊤γ−0W(s)−(12I−K(s))′0012I−K(s)V(s)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜⎝uθϕλ⎞⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜⎝d1d2d3d4⎞⎟ ⎟ ⎟⎠. (3.13)

In the above expression denotes the transpose (row) of the outward unit normal (column) to , and data is given by

 d1=−s\leavevmode\nobreak ρfγ−′(γ+vincn),d2=0,d3=∂+nvinc,d4=0. (3.14)

We have made use of the product spaces:

 X:= H1(Ω−)×H1(Ω−)×H1/2(Γ)×H−1/2(Γ), X′:= (H1(Ω−))′×(H1(Ω−))′×H−1/2(Γ)×H1/2(Γ)

(i. e., is the dual of ). Our aim is to show that equation (3.13) has a unique solution in . We will do this in the next section.

## 4 Existence and uniqueness results

Before considering the existence and uniqueness results, we first discuss the invertibility of the operator in (3.13). We begin with the definitions of the following energy norms:

 |||u|||2|s|,Ω−:= (Cε(u),ε(¯u))Ω−+ρΣ∥su∥2Ω−, u∈H1(Ω−), (4.1) |||θ|||2|s|,Ω−:= ∥∇θ∥2Ω−+κ−1∥√|s|θ∥2Ω−, θ∈H1(Ω−), (4.2) |||v|||2|s|,Ω+:= ∥∇v∥2Ω++c−2∥sv∥2Ω+, v∈H1(Ω+). (4.3)

For the complex Laplace parameter we will denote

 σ:=Re\leavevmode\nobreak s,σ––:=min{1,σ},

and will make use of the following equivalence relations for the norms

 σ––|||u|||1,Ω−≤|||u||||s|,Ω−≤|s|σ––|||u|||1,Ω−, (4.4) √σ––|||θ|||1,Ω−≤|||θ||||s|,Ω+≤√|s|σ––|||θ|||1,Ω−, (4.5) σ––|||v|||1,Ω+≤|||v||||s|,Ω+≤|s|σ––|||v|||1,Ω+, (4.6)

which can be obtained from the inequalities:

 min{1,σ}≤min{1,|s|},andmax{1,|s|}min{1,σ}≤|s|,\leavevmode\nobreak ∀s∈C+.

We remark that the norms and are equivalent to and , respectively, and so is the energy norm equivalent to the norm of by the second Korn inequality [11].

For the invertibility of , let us introduce the diagonal matrix :

 Z=⎛⎜ ⎜ ⎜ ⎜⎝10000ζ(η|s|)−10000ρf0000ρf⎞⎟ ⎟ ⎟ ⎟⎠ (4.7)

and consider the modified operator

 B:=ZA=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝As−ζ\leavevmode\nobreak (div)′s\leavevmode\nobreak ρfγ−′\leavevmode\nobreak n0s|s|−1\leavevmode\nobreak ζ\leavevmode\nobreak divζ\leavevmode\nobreak (η\leavevmode\nobreak |s|)−1Bs00−s\leavevmode\nobreak ρf\leavevmode\nobreak n⊤γ−0ρf\leavevmode\nobreak W(s)−ρf(12I−K(s))′00ρf(12I−K(s))ρfV(s)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (4.8)

It will be clear that in order to show the invertibility of it suffices to prove that of . By the Gaussian elimination procedure (as in [25]), a simple computation shows that can be decomposed in the form:

 B=˜PCP, (4.9)

where

 C=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝As−ζ\leavevmode\nobreak (div)′s\leavevmode\nobreak ρfγ−′\leavevmode\nobreak n0s|s|−1\leavevmode\nobreak ζ\leavevmode\nobreak divζ\leavevmode\nobreak (η\leavevmode\nobreak |s|)−1Bs00−s\leavevmode\nobreak ρf\leavevmode\nobreak n⊤γ−0ρf\leavevmode\nobreak (W(s)+CΓ(s))0000ρfV(s)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,
 ˜P=⎛⎜ ⎜ ⎜ ⎜⎝I0000I0000I−(12I−K(s))′V−1(s)000I⎞⎟ ⎟ ⎟ ⎟⎠andP=⎛⎜ ⎜ ⎜ ⎜⎝I0000I0000I000V−1(s)(12I−K(s))I⎞⎟ ⎟ ⎟ ⎟⎠

with .

Since both and are invertible, the invertibility of will imply that of , but as in the time-dependent fluid-structure interaction [18], the operator matrix is indeed invertible. In fact, it is not difficult to show that the operator matrix is strongly elliptic [19, 33] in the sense that

 ≥ C(ζ,η,ρf)σσ––3|s|2∥(v,ϑ,ψ,χ)∥2X (4.10)

for all , where is a constant depending only on the physical parameters and on the geometry of , and is the matrix defined by

 Z(s):=⎛⎜ ⎜ ⎜ ⎜⎝¯s/|s|000010000¯s/|s|0000s/|s|⎞⎟ ⎟ ⎟ ⎟⎠.

The action of amounts only for a rotation that reveals the elliptic nature of the original system. As such, the analyticity of is immaterial, since only the inverse Laplace transform of is sought for. As for the proof of (4.10), we will just point out that it is simple to show that

 Re{ −¯s/|s|(ζ\leavevmode\nobreak (div)′ϑ,¯v)Ω−+¯s/|s|(s\leavevmode\nobreak ρfγ−′ψ\leavevmode\nobreak n,¯v)Ω− +s/|s|(ζ\leavevmode\nobreak div\leavevmode\nobreak v,¯ϑ)Ω−−¯s/|s|⟨s\leavevmode\nobreak ρf\leavevmode\nobreak n⊤γ−v,¯ψ⟩Γ}= 0. (4.11)

Further details are omitted, since a similar proof will be repeated when we discuss the existence and uniqueness results for the solution of the nonlocal problem (3.13).

We now return to the solutions of the modified system of equations (4.8) from (3.13):

 B⎛⎜ ⎜ ⎜⎝uθϕλ⎞⎟ ⎟ ⎟⎠:=ZA⎛⎜ ⎜ ⎜⎝uθϕλ⎞⎟ ⎟ ⎟⎠=Z⎛⎜ ⎜ ⎜⎝d1d2d3d4⎞⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜⎝d1ζ/η\leavevmode\nobreak |s|−1\leavevmode\nobreak d2ρfd3ρfd4⎞⎟ ⎟ ⎟ ⎟⎠. (4.12)

Suppose that is a solution of (4.12). Let

 v:=D(s)ϕ−S(s)λinR3∖Γ. (4.13)

Then is the solution of the transmission problem:

 −Δv+(s/c)2v=0inR3∖Γ (4.14)

satisfying the following jump relations across ,

 [[γv]]:=γ+v−γ−v=ϕ∈H1/2(Γ),[[∂nv]]:=∂+nv−∂−nv=λ∈H−1/2(Γ).

First, from (4.14) we see that

 Asu−ζ\leavevmode\nobreak (div)′θ+sρfγ−′\leavevmode\nobreak [[γv]]n= d1 inΩ−, (4.15) s|s|ζ\leavevmode\nobreak divu+ζη1|s|Bs\leavevmode\nobreak θ= ζη1|s|d2 inΩ−, (4.16) −s\leavevmode\nobreak ρfn⋅γ−u−ρf∂+nv= ρf\leavevmode\nobreak d3 onΓ, (4.17) −ρfγ−v= ρf\leavevmode\nobreak d4 onΓ. (4.18)

Since , this means that is a solution of the homogeneous Dirichlet problem for the partial differential equation (4.14) in . Hence by the uniqueness of the the solution, we obtain in . Consequently, we have

 [[γv]]=γ+v=ϕand[[∂nv]]=∂+nv=λ. (4.19)

Next, we consider the variational formulation of the problem for equations (4.14) , (4.15) and (4.16) together with the boundary conditions (4.17) and (3.1f). We seek a solution

 (u,θ,v)∈H=H1(Ω−)×H1(Ω−)×H1(Ω+)

with the corresponding test functions in the same function space. To derive the variational equations, we should keep in mind that the variational formulation should be formulated not in terms of the Cauchy data and directly, but only through the jumps and as indicated.

Let , we introduce the bilinear form

 cΩ+(f,g;s):=(∇f,∇g)Ω++(s/c)2(f,g)Ω+,

and its associated operator

 Cs,Ω+: H1(Ω+) ⟶ (H1(Ω+))′ f ⟼ cΩ+(f,⋅;s)

Note that in these definitions the domain of integration is indicated explicitly. Using this notation we can use the first Green formula for equation (4.14) and condition (4.17) to obtain

 ρf(Cs,Ω+v,w)Ω+= −ρf⟨∂+nv,γ+w⟩ = ρf⟨d3,γ+w⟩Γ+ρf⟨s\leavevmode\nobreak γ−u⋅n,γ+w⟩Γ. (4.20)

Together with the weak formulations of (4.15) and (4.16), we arrive at the following variational formulation: Find satisfying

 (Asu,v)Ω−−ζ(θ,divv)Ω−+sρf⟨γ+vn,γ−v⟩Γ= (d1,v)Ω− ∀\leavevmode\nobreak v∈H1(Ω−), ζ|s|η(Bsθ,\leavevmode\nobreak ϑ)Ω−+s|s|ζ\leavevmode\nobreak (div\leavevmode\nobreak u,ϑ)Ω−= ζ|s|η(d2,\leavevmode\nobreak ϑ)Ω− ∀\leavevmode\nobreak ϑ∈H1(Ω−), (4.21) ρf(Cs,Ω+v,w)Ω+−s\leavevmode\nobreak ρf⟨γ−u,\leavevmode\nobreak γ+v\leavevmode\nobreak n⟩Γ= ρf⟨d3,γ+w⟩Γ ∀\leavevmode\nobreak w∈H1(Ω+).

We remark that by construction, it can be shown that as in [35] this variational problem is equivalent to the transmission problem defined by (4.14), (4.15), (4.16), and (4.17). The latter is equivalent to the nonlocal problem defined by (4.12), which is equivalent to (3.13). Consequently, the variational problem (4.21) is equivalent to the nonlocal problem (3.13). Hence for the existence of the solution of (3.13), it is sufficient to show the existence of the solution of (4.21).

We have the following basic results.

###### Theorem 4.1.

The variational problem (4.21) has a unique solution . Moreover, the following estimate holds:

 (|||u|||2|s|,Ω−+|||θ|||2|s|,Ω−+|||v|||2|s|,Ω+)1/2≤c0|s|3/2σσ––3/2\leavevmode\nobreak ∥(d1,0,d3,0)∥X′, (4.22)

where is a constant depending only on the physical parameters .

###### Proof.

Starting with the system (4.21), a simple computation shows that

 Re{¯s|s|(Asu,¯¯¯¯u)Ω−+ζη|s|(Bsθ,¯¯¯θ)Ω− +¯s|s|ρf(Cs,Ω+v,¯¯¯v)Ω+} = Re{¯¯¯s|s|(d1,¯¯¯¯u)Ω−+ζη1|s|(d2,¯¯¯θ)Ω−+¯¯¯s|s|ρf⟨d3,¯¯¯¯¯¯¯¯¯γ+v⟩Γ}. (4.23)

On the other hand, it is not hard to verify that

 Re{¯s|s|\leavevmode\nobreak (Asu,¯¯¯¯u)Ω−}= σ|s|\leavevmode\nobreak |||u|||2|s|,Ω−, (4.24) Re{ζη\leavevmode\nobreak 1|s|(Bsθ,¯¯¯θ)Ω−}≥ ζη\leavevmode\nobreak σ|s|2\leavevmode\nobreak |||θ|||2|s|,Ω−, (4.25) Re{¯s|s|\leavevmode\nobreak ρf\leavevmode\nobreak (


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

The Swing State Project recently had some very interesting maps comparing last week’s election results from Massachusetts to 2008 presidential primary results. Their maps posted show some very interesting trends, but the maps themselves are lacking in information and the color schemes are pretty ugly. Using my own source data I recreated their election night maps, along with a few more. The geography was taken directly from the US Census so some of the waterlines are pretty strange compared to other result maps like the Boston Globe and Swing State Project.

First look at the results from the MA 2010 Senate special election.

MA Senate 2010 Results

Now compare that to the results of the 2008 presidential primary in MA, blue gradients for Obama and red gradients for Clinton.

Democrat 2008 Presidential Primary results

Using the same color scale and data cut-points we can see the same results that DavidNYC came up with, namely that winning townships for Clinton in 2008 tracked decently to winning townships for Scott Brown in the 2010 Senate race.

Obama lost the MA presidential primary in 2008, but he won in the general. Here is a map showing the vote margins from the 2008 presidential general:

MA 2008 Presidential results

Compared to the MA 2010 Senate race last week these 2008 numbers are astounding. The democratic vote margin in 2008 was 26 points (62% to 36%), while the 2010 senate race was -5 points (47-52), with 30% less turnout in 2010 than in 2008. I decided to create another map showing the decline in democratic vote support

Democratic vote change 2010 to 2008

Does this mean voters have turned against democrats in MA? Maybe, but it is interesting that Scott Brown got more votes in the special election (1,168,107) as McCain did in the 2008 general (1,108,854), even though 900,000 less people voted in the special. So Brown was able to do two things: 1) Activate republican McCain voters, 2) Cause democrats to cross party lines. I don’t think this spells certain doom for congressional democrats in the midterms, but it does show that Democrats will stay home on election day, even in a historically democratic state, for the wrong candidate.

You can create the maps yourself or play with the data by downloading the source + data files here. With a recent installation of R you can recreate the maps above by running the the following command:

R CMD BATCH draw.R

Thanks.


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# Math Help - Triangle inequality for n complex numbers

1. ## Triangle inequality for n complex numbers

I am trying to prove that
$|z_1+z_2+...+z_n| = |z_1| + |z_2| + ... |z_n|$

iff $z_i/z_j$ is a positive real number $\forall$ integers i and j, s.t. $i,j \in$ $\left\{ 1,...,n \right\}$

I really don't see how these two ideas imply each other. After looking at this for several hours the only thing I managed to come up with (which I'm sure could be extended to n variables) is

$|z_1|^2 + m|z_2|^2 = |z_1+z_2|^2$ where $m \in \mathbb{R}$

however, that real number m is not always $z_i/z_j$

2. If $\forall i,j \in {1,2,...,n \}$ is $\displaystyle \frac{z_{i}}{z_{j}} = \alpha_{i,j}$ and any $\alpha_{i,j}$ is positive real, then $\forall i$ is $z_{i}= e^{\sigma\ \theta}\ \beta_{i}$, being $\sigma= \sqrt{-1}$ , $\theta$ real and any $\beta_{i}$ positive real. In this case is...

$\displaystyle |z_{1} + z_{2} + ...+ z_{n}|= |e^{\sigma \theta}|\ |\beta_{1} + \beta_{2} +...+ \beta_{n}|= |z_{1}|+|z_{2}|+...+|z_{n}|$ (1)

The inverse however is not true and a symple 'counterexample' is when one of the $z_{j}$ is zero and the terms $\frac{z_{i}}{z_{j}}$ for $i \ne j$ don't exist...

Kind regards

$\chi$ $\sigma$

3. Sorry, I meant to specify that $z_j \neq 0$. Thank you very much for your help though. I will try to figure out the other direction on my own.

-Cheers

4. Use the real part and imaginary part of the complex numbers to re-write the inequality then apply the Cauchy–Schwarz inequality.


text:
## Some interesting articles from MAPS

The following articles have been published in the MAPS bulletin, which is the journal of the Dutch TeX users' group. Most of those which are in English are available here. Some or them are also available from other entry points on the LaTeX Navigator.

All articles are available in PDF at NTG's archive.

Maps 3
• TeX and SGML (C.G. van der Laan) (In MAPS 3, pp. 31-42, october 1989)
Maps 4
Maps 5
Maps 6
Maps 7
Maps 8
Maps 9
Maps 10
Maps 11
Maps 12 and beyond: PDF files of the articles are available at NTG's archive.


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# Kalman filter, trace of state error covariance matrix, dimensionality problem

One of the derivations of discrete Kalman-filter relies on specifying a $$G_k$$ matrix gain in the measurement update equation

$$\hat{x}_{k}^{+}=\hat{x}_{k}^{-}+G_k\left(y_k-C_k\hat{x}_{k}^{-}\right)$$

which minimizes the trace of the a-posteriori estimated state error covariance matrix, namely:

\begin{align} P_k^+ & = E\left[\left(x_k-\hat{x}_k^+\right)\left(x_k-\hat{x}_k^+\right)^T\right]\\ & =\left(I-G_kC_k\right)P_k^-\left(I-G_kC_k\right)^T+G_kR_kG_k^T \end{align}

$$\frac{\partial Tr\left[P_k^+\right]}{\partial G_k} = - 2 P_k^-C_k^T + 2 G_kC_kP_k^-C_k^T + 2 G_kR_k = 0$$ $$G_k = P_k^-C_k^T \left(C_kP_k^-C_k^T + R_k\right)^{-1}$$

Is trace a valid optimality criterion? The state is comprised of variables with different units, which makes them incompatible for addition. For example, the optimal $$G_k$$ might be skewed towards the diagonal element expressed in micrometer-squared rather than the other state error variance expressed in light-year-squared.

If we apply an invertible state transform $$\xi=Tx$$ to the error covariance matrix to account for the different dimensions and numerical ranges, we get:

\begin{align} TP_k^+T^T & = TE\left[\left(x_k-\hat{x}_k^+\right)\left(x_k-\hat{x}_k^+\right)^T\right]T^T \\ & = E\left[T\left(x_k-\hat{x}_k^+\right)\left(x_k-\hat{x}_k^+\right)^TT^T\right] \\ & = E\left[\left(\xi_k-\hat{\xi}_k^+\right)\left(\xi_k-\hat{\xi}_k^+\right)^T\right] \\ & =T\left(I-G_kC_k\right)P_k^-\left(I-G_kC_k\right)^TT^T+TG_kR_kG_k^TT^T \end{align}

$$\frac{\partial Tr\left[TP_k^+T^T\right]}{\partial G_k} = - 2 T^TTP_k^-C_k^T + 2 T^TTG_kC_kP_k^-C_k^T + 2 T^TTG_kR_k = 0$$

Multiplying with $$\left(T^TT\right)^{-1}$$ from the left we arrive at the same result, which surprised me. I thought that a linear reconfiguration of states would change the optimal $$G_k$$, but this is clearly not the case. How to better cope with this dimensionality problem?