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# High Temperature Materials and Processes

Editor-in-Chief: Fukuyama, Hiroyuki

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Volume 36, Issue 1

# Modeling the Hot Tensile Flow Behaviors at Ultra-High-Strength Steel and Construction of Three-Dimensional Continuous Interaction Space for Forming Parameters

Guo-zheng Quan
• Corresponding author
• State Key Laboratory of Mechanical Transmission, School of Material Science and Engineering, Chongqing University, Chongqing 400044, China
• Email
• Other articles by this author:
/ Zong-yang Zhan
• State Key Laboratory of Mechanical Transmission, School of Material Science and Engineering, Chongqing University, Chongqing 400044, China
• Other articles by this author:
/ Tong Wang
• State Key Laboratory of Mechanical Transmission, School of Material Science and Engineering, Chongqing University, Chongqing 400044, China
• Other articles by this author:
/ Yu-feng Xia
• State Key Laboratory of Mechanical Transmission, School of Material Science and Engineering, Chongqing University, Chongqing 400044, China
• Other articles by this author:
Published Online: 2016-03-12 | DOI: https://doi.org/10.1515/htmp-2015-0156

## Abstract

The response of true stress to strain rate, temperature and strain is a complex three-dimensional (3D) issue, and the accurate description of such constitutive relationships significantly contributes to the optimum process design. To obtain the true stress–strain data of ultra-high-strength steel, BR1500HS, a series of isothermal hot tensile tests were conducted in a wide temperature range of 973–1,123 K and a strain rate range of 0.01–10 s−1 on a Gleeble 3800 testing machine. Then the constitutive relationships were modeled by an optimally constructed and well-trained backpropagation artificial neural network (BP-ANN). The evaluation of BP-ANN model revealed that it has admirable performance in characterizing and predicting the flow behaviors of BR1500HS. A comparison on improved Arrhenius-type constitutive equation and BP-ANN model shows that the latter has higher accuracy. Consequently, the developed BP-ANN model was used to predict abundant stress–strain data beyond the limited experimental conditions. Then a 3D continuous interaction space for temperature, strain rate, strain and stress was constructed based on these predicted data. The developed 3D continuous interaction space for hot working parameters contributes to fully revealing the intrinsic relationships of BR1500HS steel.

## Introduction

In recent years, the increased performance and safety requirements as well as a more rational and cost-effective manufacturing have broadened the interest in high-strength and ultra-high-strength steels in many industrial sectors such as commercial vehicles, railcars, containers, cranes, machinery, etc. The application of ultra-high-strength steels makes it possible to reduce material thicknesses and thus the weight by up to 70 %. Successful light-weight design improves the working efficiency, decreases fuel consumption and even provides cost savings from procurement to manipulation and processing. Nevertheless, it is known that the formability of a material with higher strength deteriorates, and even its springback degree grows rapidly at room temperature. A hot stamping process [1] in the austenite temperature region followed by room temperature die quenching is employed for fabricating structural components having strength of 1,500 MPa or more, by which the sheet is formed, and its strength is improved by the transformation from austenite to martensite [2]; meanwhile, the springback is eliminated. Hot stamping is a thermo-mechanical forming process with large deformation and intended phase transformation. So a realistic finite element model (FEM) for such process must consider the interaction among mechanical, thermal and microstructural fields. This requires several main process characteristics such as heat transfer coefficient, material flow behaviors and phase transformation kinetics under process-relevant conditions. Consequently, as the work shown here, it is valuable to deeply understand, accurately characterize and predict the flow behaviors of ultra-high-strength sheet material, which provides the basis stress–strain data for numerical simulation [3].

Over the past decades, great attentions have been paid to the characterization of the complex nonlinear relationships between true stress and deformation parameters such as strain, strain rate and temperature. Numerous efforts have been made to three types of constitutive models involving analytical, phenomenological [46] and empirical or semi-empirical ones [7]. The analytical constitutive model is considered as the simplest and most computationally efficient model for predicting metal flow behaviors. But each analytical model is based on certain assumptions, and if these assumptions are not met, the model may fail to predict correctly. The empirical or semi-empirical constitutive model is proposed on a range of assumptions, and it couples the effect of strain rate and temperature but treats strain in an exclusive way. Usually the effects of strain, strain rate and temperature are statistically analyzed in a mutually exclusive manner from the data collected in measurements or observations. The accuracy of this model mainly depends on the experimental data amount, regression tolerance and the authenticity of these assumptions. The phenomenological constitutive model is an accurate mathematical model and can have relatively many coefficients that need to be calibrated with experimental data. This constitutive model requires many computational and experimental efforts, and its accuracy mainly depends on the regression tolerance and the formula applicability.

Relative to the previous three types of constitutive models, an easier and more adaptable modeling approach, the artificial neural network (ANN), has been rapidly developed in recent years and popularly applied in material flow behaviors [8]. ANN is a model emulating some functions of biological neural networks with a data-driven black box structure [9]; thus, it merely needs a collection of some typical examples from the anticipant mapping functions for training regardless of explicit professional knowledge of deformation mechanisms. Most ANNs contain some different “learning rules” to modify the weights of the connections on the basis of the input patterns that it is presented with. Some different sorts of learning rules such as the Hebbian rule, delta rule, the competitive learning rule and the anti-Hebbian rule are utilized by neural networks, in which the delta rule is often used by the most widely and successfully used ANN called “backpropagational artificial neural network” (BP-ANN). With BP, “learning” is a supervised process that follows with each cycle through the feed-forward computation of activations and the backward propagation of error signals for weight adjustments via the generalized delta rule. Recently, a lot of efforts have been spent on the hot deformation behaviors and constitutive description of several alloys by BP-ANN model. Phaniraj et al. [10], Mandal et al. [11] and Lin et al. [12] conducted BP-ANN models to predict the flow behaviors of carbon steels, type 304 L stainless steel, 42CrMo steel, and so on. These reports about BP-ANN revealed that BP-ANN is an effective tool to predict the complex nonlinear hot flow behaviors by self-training to be adaptable to the material characteristics.

As for ultra-high-strength steel, there are few reports of the constitutive modeling of the hot flow behaviors by BP-ANN. As is known, the stress–strain data amount and accuracy input into the FEM has a significant influence on the simulation accuracy of the hot stamping process of ultra-high-strength steel sheet. The BP-ANN meets this request just right since it has not only a high accuracy of characterization but also a high predictability. In this work, the stress–strain data for ultra-high-strength steel sheet, here BR1500HS, beyond experimental conditions were predicted by BP-ANN; then the predicted data were interpolated densely. The most important is that a three-dimensional (3D) continuous interaction space representing the continuous response of stress to strain, strain rate and temperature has been constructed by the surface fitting of limited dense data. In the 3D continuous interaction space, all the stress–strain points are digital and can be determined, which means the stress under any temperatures, strain rates and strains is known as long as the deformation conditions are within the scope of such a 3D continuous interaction space. Thus, it provides continuous stress–strain data for a series of studies of ultra-high-strength steel sheet, such as processing maps, dynamic recrystallization (DRX) kinetics, ductile damage evolution, even more important, FEM simulations of the hot stamping processes. The accuracy of such 3D continuous mapping relationships among temperature, strain rate and strain is strongly guaranteed by the high accuracy of BP-ANN model, which undoubtedly induces the high accuracy of relative studies needing stress–strain data. This 3D continuous interaction space was put forward for the first time in the characterization field of material flow behaviors.

In this work, the stress–strain data of BR1500HS sheet were collected from a series of isothermal hot tensile tests carried out in a wide temperature range of 973–1,123 K and a strain rate range of 0.01–10 s−1 on a Gleeble 3800 thermo-mechanical simulator. A BP-ANN model that takes temperature (T), strain rate ($\stackrel{˙}{\mathrm{\epsilon }}$) and strain ($\mathrm{\epsilon }$) as the input variables and true stress ($\mathrm{\sigma }$) as the output variable was established by determining proper network structure and parameters to predict the nonlinear complex flow behaviors of BR1500HS sheet. The predictability and adaptability of this BP-ANN model were evaluated, having admirable performance by a series of evaluators such as correlation coefficient (R), average absolute relative error (AARE) and relative error (δ). Meanwhile, a comparison between BP-ANN model and a phenomenological constitutive model, i. e. improved Arrhenius-type constitutive model of BR1500HS sheet, was implemented, which predictably indicates that the former has higher prediction accuracy. In the following, as described previously, a 3D continuous interaction space within the temperature range of 973–1,223 K, strain rate range of 0.01–10 s−1 and strain range of 0–0.16 was constructed.

## Experimental procedures

The detailed chemical compositions of as-rolled ultra-high-strength steel, BR1500HS, are as follows (wt. %): C 0.23, Si 0.25, S 0.006, P 0.015, Cr 0.19, Ni 0.028, Mo 0.04, B 0.003, Al 0.04, Ti 0.03, Cu 0.016, V 0.004 and balance Fe. Just as shown in Figure 1, the initial material was made up of ferrite and pearlite. From the as-received material with initial thickness of 1.8 mm, totally 16 specimens corresponding to the tensile tests schedule of four strain rates (0.01, 0.1, 1 and 10 s−1) as well as four temperatures (973, 1,023, 1,073 and 1,123 K) were machined by wire-electrode cutting. The shape and size of the specimen as shown in Figure 2 were designed according to the ASTM Standard: E8M. In the isothermal hot tensile tests, a computer-controlled, servo-hydraulic Gleeble 3800 testing machine produced by DSI was applied. The detailed experimental procedures of isothermal hot tensile tests are shown in Figure 3. In order to keep the temperature accurate and invariable, two K-type thermocouple wires were welded on the specimen before the test to record the real-time temperature of the specimen. Then the specimen was fixed at the exact center of the anvils, heated to 1,223 K with a fixed heating rate of 5 K/s and kept in this temperature for 180 s to reduce anisotropy and eliminate internal temperature gradient. Subsequently, a series of specimens were stretched one by one till the complete break separation according to the previous tensile test schedule of four strain rates and four temperatures. After the tensile tests, the specimens were water quenched immediately to keep the deformed microstructures at elevated temperature. In these tensile tests, a computer-controlled automatic data acquisition system was used to monitor the nominal stress and nominal strain continuously, and afterward these data were converted into homologous true stress and true strain according to eq. (1) and (2): ${\mathrm{\sigma }}_{\mathrm{T}}={\mathrm{\sigma }}_{\mathrm{N}}\left(1+{\mathrm{\epsilon }}_{\mathrm{N}}\right)$(1) ${\mathrm{\epsilon }}_{\mathrm{T}}=\mathrm{l}\mathrm{n}\left(1+{\mathrm{\epsilon }}_{\mathrm{N}}\right)$(2)

Figure 1:

The original metallographs of BR1500HS.

Figure 2:

Specimen for hot tensile tests (unit: mm).

Figure 3:

Experimental procedures for the isothermal hot tensile tests.

where ${\mathrm{\epsilon }}_{\mathrm{N}}$ is the nominal strain, ${\mathrm{\sigma }}_{\mathrm{N}}$ is the nominal strain, ${\mathrm{\sigma }}_{\mathrm{T}}$ is the true stress and ${\mathrm{\epsilon }}_{\mathrm{T}}$ is the true strain [13].

## Flow behavior characteristics

The true stress–strain data of ultra-high-strength steel sheet, BR1500HS, measured by a series of hot tensile tests were shown in Figure 4(a)–(d). It is valuable to note that the true stress–strain data in fracture stage have been removed. As expected, stress–strain relationships are highly nonlinear, and highly susceptible to three parameters, including temperature, strain and strain rate. By comparison of one stress–strain curve with another, it can be summarized that the stress level decreases with increase in temperature or decrease in strain rate. This is due to the following facts. When the strain rate is accelerated, the dislocations participating in deformation in unit time increase; meanwhile, there is no enough time to coordinate deformation by means of dislocation and grain movements. The multiplication of dislocation leads to the improvement of stress level. When raising the deformation temperature level, the heat activation increases the average kinetic energy in atoms and makes the atomic diffusion and dislocation movement easier. And this induces the decrease of stress level. As for each stress–strain curve, it shows an initial rapid and the following more and more slow work hardening (WH), following which two types of curve evolution exist. In the first case, the stress decreases gradually after a single peak value, an evidence of DRX softening, which corresponds to the conditions of 0.01 s−1 and 973–1,023 K, 1 s−1 and 1,073–1,123 K, and 10 s−1 and 973–1,073 K. Taking the situation at 1 s−1 and 973 K as example, equiaxed dynamically recrystallized grains nucleate and grow along the grain boundaries, and the final microstructure can be seen in Figure 5. In the second case, the stress approximately keeps a steady state, an evidence of dynamic recovery (DRV) softening, under the conditions of 0.01 s−1 and 1,073–1,123 K, 0.1 s−1 and 973–1,123 K, 1 s−1 and 973–1,023 K, and 10 s−1 and 1,123 K. From the previous descriptions, it can be concluded that the hot tensile flow behaviors of ultra-high-strength steel sheet, BR1500HS, is extremely complex and highly nonlinear, not only due to the effects of temperature and strain rate to stress level, but also due to the evolution characteristics involving WH, DRX and DRV indicated in each stress–strain curve. Consequently, as for this case, it is a difficult issue to characterize such flow behaviors accurately.

Figure 4:

The stress–strain curves of BR1500HS under different temperatures with the strain rates of (a) 0.01 s−1, (b) 0.1 s−1, (c) 1 s−1 and (d) 10 s−1.

Figure 5:

Microstructure of BR1500HS under the condition of 1 s−1 and 1,073 K.

## Development of BP-ANN model

A BP-ANN model for the hot tensile flow behaviors of BR1500HS sheet in respect of dependency of strain, strain rate and temperature was developed by taking deformation temperature (T), strain rate ($\stackrel{˙}{\mathrm{\epsilon }}$) and strain ($\mathrm{\epsilon }$) as the input variables and true stress ($\mathrm{\sigma }$)as the only output variable [14]. The schematic representation of the BP-ANN architecture was shown in Figure 6. In this network, all the continuous stress–strain curves measured from a series of hot tensile tests were discrete with a strain interval of 0.001. The total 1,971 discrete data points from the 14 stress–strain curves except the two curves corresponding to 1,023 K and 0.1 s−1 and 1,073 K and 1 s−1 were defined as the training data of the BP-ANN work. As the BP-ANN work had been trained, the stress values of 322 points picked out from the two test curves with a strain interval of 0.001, and 218 points from the other 14 training curves in a strain range of 0.004–0.154 with a strain interval of 0.01 were predicted, and then these total 540 predicted data points were considered as the test data for the BP-ANN work performance. Among them, the 218 predicted data points were used for the training performance evaluation, while the left 322 predicted data points were used for the evaluation of the generalization performance, i. e. the adaptability of BP-ANN work to fresh forming conditions beyond training. Subsequently, the corresponding total 540 experimental data points were picked out and considered as the reference points for the test work. The test work would be conducted between the reference data and the test data.

Figure 6:

The schematic of the BP-ANN architecture.

It is common that the determination of the appropriate number of hidden layers and neurons in each hidden layer is one of the most critical tasks in obtaining an accurate BP-ANN model. It was assumed that one and two hidden layers were adopted to test, respectively, and then the appropriate number needed to be finally determined through the appropriate tolerance evaluation between the predicted data from training and test work and the experimental data. Meanwhile, an empirical formula as eq. (3) was utilized to determine e-value range, i. e. the neuron number range of each hidden layer, and accordingly the range was calculated as 3–12: $e=\sqrt{n+m}+a$(3)where e is the neuron number of each hidden layer; n and m are the neuron numbers of input and output layers, respectively, in the network shown in Figure 6, and here n=3 and m=1; a is a constant ranging from 1 to 10.

An indicator of mean square error (MSE) as in eq. (4) [15] was introduced to evaluate the training performance and the generalization performance of the BP-ANN work, and accordingly the optimal hidden layer number and neuron number would be determined by comparing the performance of different networks deriving from different network structure parameters. It is noted that smaller MSE value indicates better network performance. As for the two different network structures, i. e. one or two hidden layers, the relative MSE plots along with the number of neurons in each hidden layer were calculated, respectively, as shown in Figure 7. Figure 7(a) shows the training performance, while Figure 7(b) shows the generalization performance of the developed BP-ANN work. The comparison of the MSE plots between the different network structures in Figure 6 shows that the structure of two hidden layers induces lower MSE value level, and thus it corresponds to higher training and generalization performance. Moreover, in Figure 7, MSE value declines with neuron number increasing, while it appears an inverse trend when the neuron number is above 10, which indicates that increasing neuron number up to 10 elevates the network performance significantly. In Figure 7(a) and (b), MSE values reach the minimum at the same time when the neuron number is 10, which indicates that this neuron number is corresponding to the highest training and generalization performance. Above all, two hidden layers and 10 neurons in each hidden layer are determined for the final network architecture: $\mathrm{M}\mathrm{S}\mathrm{E}=\frac{1}{N}\sum _{i=1}^{N}{\left({E}_{i}-{P}_{i}\right)}^{2}$(4)

Figure 7:

The influence of the hidden layer number and neuron number for each hidden layer on (a) the training performance of the neural network and (b) the generalization performance of the neural network.

where Ei is an experimental stress value, Pi is a predicted stress value and N is the number of data samples.

It is noteworthy that the numerical values of the input and output variables distribute in distinct ranges and even dimensions, which induces the poor convergence speed and prediction accuracy of a BP-ANN model. Hence, a normalization process for initial true stress–strain data is essential to ensure the input and output variables being dimensionless and being in an approximately same magnitude. In this research, the normalization processing was realized in eq. (5). The coefficients of 0.05 and 0.25 in eq. (5) are regulating parameters for the sake of narrowing the magnitude of the normalized data within 0–0.3. It had been demonstrated by a trial-and-error method that such a magnitude could bring a promotion in convergence speed and prediction accuracy. Besides, it should be noted that the initial numerical values of true stain rates exhibit great magnitude distinction, thereby logarithm was taken for transforming the true stain rate data before normalization processing: ${x}_{\mathrm{n}}=0.05+0.25×\frac{x-0.95{x}_{min}}{1.05{x}_{max}-0.95{x}_{min}}$(5)where x is the initial data of input or output variables, xmin is the minimum value of x and xmax is the maximum and xn is the value of x after normalization processing.

To construct an excellent BP-ANN model, appropriate preferences are quite vital. Here, trainbr function and learngd function were empirically chosen as the training function and learning function, respectively. In the meantime, the transfer function of the hidden layers was assumed as tansig function, whereas the output layer adopted purelin function. It is well accepted that learning rate directly determines the revised value of weight in each training cycle, which simultaneously plays an important role in the convergence property of networks. Commonly, adopting a high learning rate may cause instability in training, while conversely may get low convergence speed but could keep off the local least values and approach the true least error. Consequently, a lower learning rate is usually selected for better stability and convergence property, and 0.02 was applied in the present network. In addition, the goal of training error before anti-normalization processing was set as 0.0001.

## Evaluation of BP-ANN model

To synthetically estimate the prediction capability of BP-ANN model, two commonly used statistical indicators of R and AARE [15], which were expressed as eqs (6) and (7), were introduced. A high R value close to 1 illustrates that the predicted values conform to the experimental ones well; meanwhile, a low AARE value close to 0 indicates that the sum of the errors between the predicted and experimental values tends to be 0. Thereby, such R and AARE are expected: $R=\frac{{\sum }_{i=1}^{N}\left({E}_{i}-\stackrel{‾}{E}\right)\left({P}_{i}-\stackrel{‾}{P}\right)}{\sqrt{{\sum }_{i=1}^{N}\left({E}_{i}-\stackrel{‾}{E}{\right)}^{2}{\sum }_{i=1}^{N}{\left({P}_{i}-\stackrel{‾}{P}\right)}^{2}}}$(6) $\mathrm{A}\mathrm{A}\mathrm{R}\mathrm{E}\left(\mathrm{%}\right)=\frac{1}{N}\sum _{i=1}^{N}\left|\frac{{P}_{i}-{E}_{i}}{{E}_{i}}\right|×100\mathrm{%}$(7)in which E and P are, respectively, experimental value and predicted value of true stress; $\stackrel{‾}{E}$ and $\stackrel{‾}{P}$ are the mean values of E and P, respectively; N is the number of predicted points.

Based on the well-trained BP-ANN model, the true stress values under experimental conditions that include the deformation conditions corresponding to the previous training points and test points were predicted. After that, the correlation relationships between the experimental and predicted true stresses were expressed in Figure 8. It is discovered that the points in Figure 8, which take experimental true stress as horizontal axis and predicted true stress as vertical axis, lie fairly close to the best linear fitted line, suggesting that the predicted stress–strain values conform very well to the homologous experimental ones. Besides, the R values for the predicted true stress of training points and test points are 0.99989 and 0.99876, respectively, from another quantitative perspective proving the strong linear relationships between the predicted and experimental true stresses. Additionally, the AAREs were calculated as well. The AARE value deriving from the test part is 1.2621 %, while that from the training part is merely 0.2675 %. Such minor errors bode the high accuracy exhibited in the training and test work by BP-ANN.

Figure 8:

The correlation relationships between the predicted and experimental true stress for the (a) training part and (b) test part.

The relative error (δ) [11] in eq. (8) represents the percentage error of each predicted stress–strain value relative to the homologous experimental one; it was introduced in order to further and more detailedly evaluate the BP-ANN model. By calculating and making a count on the δ-values from all predicted points involving training and test ones, some information on δ-values was provided. In the training part, the relative errors range from –5.09 % to 5.68 %, and for the test part their distribution ranges from –3.89 % to 4.75 %. Figure 9 expresses the columnar distribution maps of the relative errors in the training and test parts. In terms of Figure 9, it is not so difficult to find that the relative errors, no matter in training or test part, are within a narrow range of 0 ± 6 %. However, it is more noteworthy that most of the δ-values are miraculously concentrated in the vicinity of the ideal value 0: in the training part, the δ-values of 93.97 % points are within the interval of [–1 %, 1 %], and for the test part, 60.22 % points are concentrated in [–1 %, 1 %], at the same time, 91.82 % points are concentrated in [–1 %, 4 %]. These results arisen from unbiased statistical data provide direct evidence that high precision in both the training and test stages is acquired by the BP-ANN model: $\mathrm{\delta }\left(\mathrm{%}\right)=\frac{{P}_{i}-{E}_{i}}{{E}_{i}}×100\mathrm{%}$(8)where Pi is a predicted stress value and Ei is the homologous experimental stress value.

Figure 9:

The relative errors distribution on the predicted true stress corresponding to (a) the training points and (b) the test points.

Figure 10 exhibits the comparisons between the true stresses predicted by BP-ANN model and the corresponding experimental true stresses. Apparently, just as the phenomenon in the isothermal hot tensile tests, the predicted true stress decreases with increase in temperature or decrease in strain rate, predictably indicating that the BP-ANN model is able to effectively grasp the stress–strain evolution rules, that is, it possesses excellent capability to track the dynamic softening and WH regions of BR1500HS. Moreover, as shown in Figure 10, the predicted points on the 14 training curves almost coincide with the experimental stress–strain curves. Even under 1,023 K and 10 s−1, where discontinuous DRX causes nonsmooth in stress–strain curves thus enhancing prediction difficulties, the predicted true stress points still present minor deviations to the experimental ones that the maximum relative error is only 2.84 %. Compared with the predicted points on the training curves, the predicted points on the test curves of 1,023 K and 0.1 s−1 and 1,073 K and 1 s−1 have relatively larger deviations, especially at the beginning or end of each curve where the true stress shows large fluctuation. Nevertheless, the maximum relative error is merely 5.68 %, a quite acceptable value, which appears at the temperature of 1,073 K, strain rate of 1 s−1 and strain of 0.16. By comparing the predicted and experimental true stresses corresponding to training points and test points, the learning and generalization capabilities of BP-ANN model were validated, respectively. Conclusively, the present BP-ANN model has admirable performance in describing and predicting the true stress of BR1500HS.

Figure 10:

The comparisons between the experimental and predicted true stresses by the BP-ANN model at different temperatures and the strain rates of (a) 0.01 s−1, (b) 0.1 s−1, (c) 1 s−1 and (d) 10 s−1.

## Introduction of the improved Arrhenius-type constitutive equation

Over the past decades, the Arrhenius-type constitutive equation, a type of phenomenological constitutive model, was widely used to describe the constitutive relationship completely omitting the effect of strain, while the strain is a vital parameter in hot working processes and has a critical influence on the flow behaviors, especially for those strain-sensing materials. Therefore, the previous Arrhenius-type constitutive equations often received poor results, until a revised model taking the influence of strain into account was put forward and successfully utilized to accurately describe the deformation behaviors of 42CrMo steel under high temperature by Lin et al. [16]. Subsequently, this kind of improved Arrhenius-type constitutive equation was successfully applied in constitutive relationship descriptions of various materials [1719] due to such superiority.

Quan et al. [20] calculated and reported the improved Arrhenius-type constitutive equation of BR1500HS, and it was expressed in eq. (9): $\sigma =8.55676\mathrm{ln}\left\{{\left(\frac{\stackrel{˙}{\epsilon }\mathrm{exp}\left[j\left(\epsilon \right)/8.314T\right]}{f\left(\epsilon \right)}\right)}^{1/h\left(\epsilon \right)}+{\left[{\left(\frac{\stackrel{˙}{\epsilon }\mathrm{exp}\left(j\left(\epsilon \right)/8.314T\right)}{f\left(\epsilon \right)}\right)}^{2/h\left(\epsilon \right)}+1\right]}^{0.5}\right\}$(9)where $\mathrm{\sigma }$ is flow stress, $f\left(\mathrm{\epsilon }\right)$, $g\left(\mathrm{\epsilon }\right)$, $h\left(\mathrm{\epsilon }\right)$ and $j\left(\mathrm{\epsilon }\right)$ are the polynomial functions of $lnA$, $\mathrm{\alpha }$, $n$ and $Q$ at different strains, and their expressions were as shown in eq. (10). The coefficients in eq. (10) were listed in Table 1: $\left\{\begin{array}{l}Q={B}_{0}+{B}_{1}\epsilon +{B}_{2}{\epsilon }^{2}+{B}_{3}{\epsilon }^{3}+{B}_{4}{\epsilon }^{4}+{B}_{5}{\epsilon }^{5}\hfill \\ n={C}_{0}+{C}_{1}\epsilon +{C}_{2}{\epsilon }^{2}+{C}_{3}{\epsilon }^{3}+{C}_{4}{\epsilon }^{4}+{C}_{5}{\epsilon }^{5}\hfill \\ \mathrm{ln}A={D}_{0}+{D}_{1}\epsilon +{D}_{2}{\epsilon }^{2}+{D}_{3}{\epsilon }^{3}+{D}_{4}{\epsilon }^{4}+{D}_{5}{\epsilon }^{5}\hfill \\ \alpha =a+b{\epsilon }^{0.5}+c\epsilon +d{\epsilon }^{1.5}+e{\epsilon }^{2}+f{\epsilon }^{2.5}+g{\epsilon }^{3}+h{\epsilon }^{3.5}+i{\epsilon }^{4}+j{\epsilon }^{4.5}+k{\epsilon }^{5}\hfill \end{array}$(10)

Table 1:

The polynomial fit results of Q, n, ln A and α.

## Prediction capability comparison between the BP-ANN model and Arrhenius-type constitutive equation

Depending on the improved Arrhenius-type constitutive equation, in this paper, the true stress of 32 points under the condition of 1,023 K and 0.1 s−1 and 1,073 K and 1 s−1 with a strain interval of 0.01 was calculated to compare with the true stress predicted by BP-ANN model and obtained from the tensile tests. For the sake of the contrast of prediction accuracy between these two models, the AARE values and δ-values relative to the experimental true stress were calculated in eqs (7) and (8) and listed in Table 2. According to the calculation results, it is manifest that the AARE value for the BP-ANN model is 2.12 %, but for the constitutive equation, it reaches a higher level, 4.18 %. Lower AARE value means a smaller deviation on the whole; therefore, to make a prejudgment, the BP-ANN model has higher accuracy in predicting the true stress of BR1500HS than the constitutive equation. To prove this view, on the basis of the data in Table 2, Figures 11 and 12 were illustrated for more profound comparison of two models.

Table 2:

Relative errors of the predicted results by the BP-ANN model and constitutive equation to experimental results under the condition of 1,023 K and 0.1 s−1 and 1,073 K and 1 s−1.

Figure 11:

The relative errors distribution on the true stress points predicted by (a) the BP-ANN model and (b) the Arrhenius-type constitutive equation relative to the experimental ones.

What can be learned from Table 2 is that the δ-values from the BP-ANN model range from –3.89 % to 5.68 %; however, the δ-values obtained by the constitutive equation vary in [–6.49 %, 19.8451 %]. Sometimes narrower range of δ-values does not always indicate high prediction accuracy because of the uncertainty of the errors distribution inside the given interval. Hence, in Figure 11, which shows the δ-values distribution of true stress gained by the BP-ANN model and constitutive equation, the values of relative frequencies were fitted by a typical Gaussian distribution function in eq. (11). In the function, the two parameters of μ and w represent the mean value and standard deviation, respectively, which are the two of the most important indexes in statistical work. The mean value and standard deviation calculated in eqs (12) and (13) [9], respectively, reflect the central tendency and discrete degree of a set of data, and smaller w-value and μ-value close to 0 hint that better errors distribution is achieved. As shown in Figure 11, the μ-value and w-value of the BP-ANN model are 1.2790 % and 1.6173 %, respectively; in another aspect, the ones of the constitutive equation are, respectively, –1.6751 % and 1.9150 %. It is obvious that the μ-values and w-values corresponding to the BP-ANN model and constitutive equation are both fairly close to 0. Yet, in spite of this, compared with the constitutive equation, not only the μ-value corresponding to the BP-ANN model is still more close to 0, but also the w-value is smaller. That means the relative errors deriving from the BP-ANN model are not only more centralized but also more approaching to 0, providing further evidence that the BP-ANN model is more accurate than the constitutive equation: $y={y}_{\text{0}}+A{e}^{-\left({\left({\delta }_{i}-\mu \right)}^{2}/2{w}^{2}\right)}$(11) $\mathrm{\mu }=\frac{1}{N}\sum _{i=1}^{N}{\mathrm{\delta }}_{i}$(12) $w=\sqrt{\frac{1}{N-1}\sum _{i=1}^{N}\left({\mathrm{\delta }}_{i}-\mathrm{\mu }\right){}^{2}}$(13)where δi is a value of the relative error; μ, w and y are the mean value, standard deviation and probability density of δ, respectively; y0 and A are constants; and N is the number of relative errors, here N=32.

The experimental stress–strain curves, the stress–strain points predicted by the BP-ANN model and constitutive equation under the conditions of 1,023 K and 0.1 s−1, and 1,073 K and 1 s−1 were plotted in Figure 12(a) and (b), respectively. It is easy to find that, relative to the experimental stress–strain curves, both the stress–strain points predicted by the BP-ANN model and constitutive equation appear certain deviations although they are in qualitative agreement with the stress–strain evolution rules. But the predicted stress–strain points by the BP-ANN model lie more close to the experimental curves from the global aspect. It is valuable to note that, in the training stage of the BP-ANN model, the experimental stress–strain data of two test curves under the conditions of 1,023 K and 0.1 s−1, and 1,073 K and 1 s−1 did not participate in. However, when establishing the constitutive equation, they were involved. But even on this premise, the BP-ANN model still shows smaller errors, giving the full proof that the present BP-ANN model has better prediction capability than the constitutive equation in the flow characteristics of ultra-high-strength steel, BR1500HS. The reason why the constitutive equation cannot follow the tracks of varied true stress so effectively as the BP-ANN model is that the constitutive equation needs to take the physical interpretation into account. Yet, on the contrary, the BP-ANN model can just predict the true stress under different deformation conditions regardless of the physical interpretation, and the essential tasks are preparing proper training and test data, afterward developing the optimal network structure and parameters for the BP-ANN model.

Figure 12:

The comparisons among the experimental and predicted true stresses by the BP-ANN model and Arrhenius-type constitutive equation at (a) 1,023 K, 0.1 s−1 and (b) 1,073 K, 1 s−1.

## Construction of 3D continuous interaction space

Sections “BP-ANN model for BR1500HS” and “Comparison between BP-ANN model and improved Arrhenius-type constitutive equation” make it known that the BP-ANN model is highly reliable and more accurate than the improved Arrhenius-type constitutive equation. Therefore, due to outstanding generalization capability of the BP-ANN model, it can undoubtedly be employed to predict the true stress of BR1500HS outside of the experimental conditions. In this investigation, the true stress data under the temperature of 973, 1,023, 1,073, 1,123, 1,173 and 1,223 K, the strain rate of 0.01, 0.0316, 0.1, 0.316, 1, 3.16 and 10 s−1, as well as the strain of 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, 0.11, 0.12, 0.13, 0.14,0.15 and 0.16 were predicted, whereafter, the results were illustrated in Figure 13(a)–(g). It is clearly observed that the temperature, strain rate, strain and stress in Figure 13 articulate similar response relationships with the experimental stress–strain data. On the foundation of these predicted stress–strain data, an interpolation method was employed to densely insert stress–strain data into predicted ones; going a step further, a 3D continuous interaction space (illustrated in Figure 14) was eventually constructed by a surface fitting process for these limited dense stress–strain data. In Figure 14, the X-axis, Y-axis, Z-axis and V-axis, respectively, represent deformation temperature, logarithm of strain rate, strain and true stress. The values on V-axis are indicated by different colors. Figure 14(a) shows the 3D continuous interaction space representing the continuous response of stress to strain, strain rate and temperature of BR1500HS. Figure 14(b)–(d) shows the cutting slices of the 3D continuous interaction space along with different variables, including temperature, strain rate and strain. In the 3D continuous interaction space, all the stress–strain points are digital and can be determined, since the surface fitting step has transformed the discrete stress–strain points into continuous stress–strain surface and space. The accuracy of such a 3D continuous interaction space is strongly guaranteed by the excellent prediction performance of optimally constructed and well-trained BP-ANN model.

Figure 13:

The true stress predicted by the BP-ANN model at the temperature range of 973–1,223 K, the strain range of 0–0.16 and the strain rate of (a) 0.01 s−1, (b) 0.0316 s−1, (c) 0.1 s−1, (d) 0.316 s−1, (e) 1 s−1, (f) 3.16 s−1 and (g) 10 s−1.

Figure 14:

The 3D relationships among temperature, strain rate, strain and stress: (a) 3D continuous interaction space; 3D continuous mapping relationships under different (b) temperatures, (c) strain rates and (d) strains.

It is well known that the stress–strain data play a critical role in many studies, such as the DRX kinetics model [21], processing maps [22], ductile fracture criteria [23] and so on. In these studies, the amount and quality of stress–strain data have a significant influence on the research results. Exactly, the 3D continuous interaction space constructed in this paper can provide abundant and accurate stress–strain data for the relevant studies on ultra-high-strength steel sheet, BR1500HS, to guarantee the creditability of research findings. Beyond that, the 3D continuous interaction space plays an essential part in FEM. As is known, the stress–strain data are the most fundamental data to predict the deformation behaviors of the ultra-high-strength steels in hot stamping processes with FEM. It is realizable to pick out dense stress–strain data from the 3D continuous interaction space and insert such continuous mapping relationships into commercial software such as Marc, and so on by program codes. In this way, the accurate simulation of one certain forming process is able to perform. On the other hand, the 3D continuous interaction space makes it possible to automatically and accurately solve DRX kinetics, DRV kinetics, processing maps, and so on. For example, in processing maps, the flow instability domains with higher accuracy can be identified at different time–space domains automatically [24], and then more reasonable optimization of the process parameters such as the deforming temperature, the shape of die cavity, deformation velocity, and so on, can be conducted automatically.

## Conclusions

• (1)

The true stress level of BR1500HS decreases with increasing temperature or decreasing strain rate. The true stress varies along with strain highly nonlinearly, which represents the nonlinear variation of the comprehensive effects of different action mechanisms, including DRX, DRV and WH.

• (2)

A BP-ANN model taking the deformation temperature (T), strain rate ($\stackrel{˙}{\mathrm{\epsilon }}$) and strain ($\mathrm{\epsilon }$) as input variables and the true stress ($\mathrm{\sigma }$) as output variable was constructed for the tensile flow behaviors of ultra-high-strength steel sheet, BR1500HS. The evaluation via the indicators of R, AARE and relative error (δ) revealed that the present BP-ANN model has admirable performance in describing and predicting the flow behaviors.

• (3)

A comparison between the improved Arrhenius-type constitutive equation and BP-ANN model shows that the latter has higher prediction accuracy in predicting the flow behaviors of BR1500HS.

• (4)

The true stress data within the temperature range of 973–1,223 K, the strain rate range of 0.01–10 s−1 and the strain range of 0–0.16 were predicted densely. According to these abundant data, a 3D continuous interaction space was constructed by interpolation and surface fitting methods. It significantly contributes to all the researches requesting abundant and accurate stress–strain data of BR1500HS.

## Acknowledgments

The corresponding author was also a part of Chongqing Higher School Youth-Backbone Teacher Support Program.

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Accepted: 2015-11-24

Published Online: 2016-03-12

Published in Print: 2017-01-01

Funding: This work was supported by Fundamental Research Funds for the Central Universities (CDJZR14135503).

Citation Information: High Temperature Materials and Processes, Volume 36, Issue 1, Pages 29–43, ISSN (Online) 2191-0324, ISSN (Print) 0334-6455,

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## Citing Articles

[1]
Guo-zheng Quan, Zong-yang Zhan, Le Zhang, Dong-sen Wu, Gui-chang Luo, and Yu-feng Xia
Materials Science and Engineering: A, 2016, Volume 673, Page 24


text:
# (molecular spectroscopy)what is the meaning of the symbol

1. Apr 4, 2013

### einstein1921

Hi, everyone. In the picture attached, what is the meaning of 0oo? what should I read to understand the meaning?Thank you!!

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2. Apr 5, 2013

### DrDu

I suppose it is some symbol for vibrational modes excited. Maybe you could write something about what is measured? Is this a mass spectrum after two photon excitation?

3. Apr 10, 2013

### einstein1921

thank your for your answer! I also think it should be vibraitonal modes exceted. However, what confused me is that 0 to 0 transition is non-meaningful!! attached is the paper!!

#### Attached Files:

• ###### JChemPhys_79_571.pdf
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1.2 MB
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139
4. Apr 10, 2013

### DrDu

From the paper and ref. 6 therein I understand that the big number is the $\nu_1$ number of the excited state while the small superscript is the $\nu_2$ quantum number of the excited state.
The sub-script is the $\nu_2$ vibrational number of the ground state.
I suppose that due to the higher frequency, the $\nu_1$ mode is never excited in the ground state.
$\nu_1$ is the symmetric stretch and $\nu_2$ the out of plane bending mode.

Why is a 0 to 0 transition not meaningful? Take in mind that in electronic transitions, in contrast to IR or Raman purely vibrational transitions, vibrational levels don't have to change (with the exception of symmetry forbidden transitions). See e.g.:
http://en.wikipedia.org/wiki/Franck–Condon_principle

Last edited: Apr 10, 2013

format: plain


text:
# Adaptive model selection using empirical complexities

## Author Info

• Gábor Lugosi

()

• Andrew B. Nobel

## Abstract

Given $n$ independent replicates of a jointly distributed pair $(X,Y) \in {\cal R}^d \times {\cal R}$, we wish to select from a fixed sequence of model classes ${\cal F}_1, {\cal F}_2, \ldots$ a deterministic prediction rule $f: {\cal R}^d \to {\cal R}$ whose risk is small. We investigate the possibility of empirically assessing the {\em complexity} of each model class, that is, the actual difficulty of the estimation problem within each class. The estimated complexities are in turn used to define an adaptive model selection procedure, which is based on complexity penalized empirical risk. The available data are divided into two parts. The first is used to form an empirical cover of each model class, and the second is used to select a candidate rule from each cover based on empirical risk. The covering radii are determined empirically to optimize a tight upper bound on the estimation error. An estimate is chosen from the list of candidates in order to minimize the sum of class complexity and empirical risk. A distinguishing feature of the approach is that the complexity of each model class is assessed empirically, based on the size of its empirical cover. Finite sample performance bounds are established for the estimates, and these bounds are applied to several non-parametric estimation problems. The estimates are shown to achieve a favorable tradeoff between approximation and estimation error, and to perform as well as if the distribution-dependent complexities of the model classes were known beforehand. In addition, it is shown that the estimate can be consistent, and even possess near optimal rates of convergence, when each model class has an infinite VC or pseudo dimension. For regression estimation with squared loss we modify our estimate to achieve a faster rate of convergence.

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## Bibliographic Info

Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 323.

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## Related research

Keywords: Complexity regularization; classification; pattern recognition; regression estimation; curve fitting; minimum description length;

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Cited by:
1. Peter Bickel & Bo Li & Alexandre Tsybakov & Sara Geer & Bin Yu & Teófilo Valdés & Carlos Rivero & Jianqing Fan & Aad Vaart, 2006. "Regularization in statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 15(2), pages 271-344, September.
2. Stéphane Boucheron & Gábor Lugosi & Pascal Massart, 1999. "A sharp concentration inequality with applications," Economics Working Papers 376, Department of Economics and Business, Universitat Pompeu Fabra.
3. Marta Horvath & Gábor Lugosi, 1996. "A data-dependent skeleton estimate and a scale-sensitive dimension for classification," Economics Working Papers 199, Department of Economics and Business, Universitat Pompeu Fabra.

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# American Institute of Mathematical Sciences

July  2009, 23(3): 847-865. doi: 10.3934/dcds.2009.23.847

## Topology of some tiling spaces without finite local complexity

 1 Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, United States 2 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, United States

Received  October 2007 Revised  June 2008 Published  November 2008

A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which tiles can slide past one another. We investigate the topology of a certain class of tiling spaces of this type. We show that they can be written as inverse limits of CW complexes, and their Čech cohomology is related to properties of the fault lines.
Citation: Natalie Priebe Frank, Lorenzo Sadun. Topology of some tiling spaces without finite local complexity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 847-865. doi: 10.3934/dcds.2009.23.847
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Build a city of skyscrapers—one synonym at a time. Common knowledge need not concern one specific subject, e.g., science or history. Common knowledge is distinct from general knowledge. Enrich your vocabulary with the English Definition dictionary For example, an engineer who counters the idea that solar energy could never completely serve a nation's energy needs by calculating the total amount of solar energy that reaches the territory of that nation each year. As an example, this type of information may include the temperature in which water freezes or boils. to determine what constitutes common knowledge, particularly as the way this idea is interpreted varies among disciplines and practices. Please tell us where you read or heard it (including the quote, if possible). Common knowledge is often the first source for scientific investigating. The extension consists of the introduction of a group G of agents, and of n modal operators Ki (with i = 1, ..., n) with the intended meaning that "agent i knows." Definition of be common knowledge in the Idioms Dictionary. Some experts say that a fact is common knowledge if it can be found in three independent sources. "Conventional wisdom" is a similar term also referring to ostensibly pervasive knowledge or analysis. Define Common Knowledge. All Free. The “common” way to talk about common knowledge is to say that it is knowledge that most educated people know or can find out easily in an encyclopedia or dictionary. Learn a new word every day. Delivered to your inbox! Common Knowledge 255 I. That candy is sweet. Common knowledge can be given a logical definition in multi-modal logic systems in which the modal operators are interpreted epistemically. Readers may want to ask themselves what, ifany, distinctive aspects of mutual and common knowledge reasoning eachexample illustrates. Common knowledge need not concern one specific subject, e.g., science or history. "The Moon orbits the Earth." be common knowledge phrase. means a well known fact, quote, figure or information that is known to most of the people; INTRODUCTION People, no matter how rational they are, usually act on the basis of incomplete informa- tion. Common-Knowledge Exception Law and Legal Definition Common-knowledge exception refers to a legal principle exempting testimony concerning simple medical procedures in a medical mal practice action. Of course not! Definitions by the largest Idiom Dictionary. Rather, common knowledge can be about a broad range of subjects, such as science, literature, history, and entertainment. Which word describes a musical performance marked by the absence of instrumental accompaniment. phrase. Common knowledge definition: something widely or generally known | Meaning, pronunciation, translations and examples 'All Intensive Purposes' or 'All Intents and Purposes'? In addition. Common knowledge need not concern one specific subject, e.g., science or history. Common knowledge is knowing water is wet and when the sun sets it gets dark. How to use common in a sentence. Definition of common knowledge. anything generally known to everyone ; Thanks for visiting The Crossword Solver. Another word for common knowledge. What Google Ngram Viewer shows is that "it is common knowledge" is more common than "it is a common knowledge," but that doesn't tell us anything about the syntax of these constructions. Purdue’s Online Writing Lab recommends finding five independent sources before considering a fact common knowledge. Post the Definition of common knowledge to Facebook, Share the Definition of common knowledge on Twitter. Common-knowledge exception is considered a narrow exception. If they are rational they recognize their own ignorance and reflect carefully on what they know and what they do not know, before choosing how to act. Common knowledge is a condition usually required in game theory, so the model is completely specified and its analysis is coherently undertaken. If you have common sense you will know this. Look it up now! Learn more. This includes: Information that most people know , such as that water freezes at 32 degrees Fahrenheit or that Barack Obama was the first American of mixed race to be elected president. Translate common knowledge into Spanish. Learn how and when to remove this template message, Fifth Amendment to the United States Constitution, https://en.wikipedia.org/w/index.php?title=Common_knowledge&oldid=991105360, Articles lacking in-text citations from August 2019, Creative Commons Attribution-ShareAlike License. If you don't you'll believe them to be the same thing. Many philosophers even suggest that science is common knowledge made more exact, clear and reliable. To determine if information should be considered common knowledge, you can ask yourself who your audience is, are you able to assume they already have some familiarity with the topic, or will the information's credibility come into question. Synonyms for common knowledge include open secret, no secret, public knowledge, current, contemporary, modern, present, present-day, ongoing and topical. Often, common knowledge does not need to be cited. Common knowledge definition at Dictionary.com, a free online dictionary with pronunciation, synonyms and translation. When deciding whether or not to cite what might appear to be common knowledge, you can make use of these three general guidelines: 1. … Observation of the moon shows us that this happens. Many techniques have been developed in response to the question of distinguishing truth from fact in matters that have become "common knowledge". First Principles Knowledge can be deduced from first principles that are known or assumed to be valid. If you are familiar with the notion of “common knowledge” from earlier writing experiences, you may have noticed that its definition is easy to state, but can be hard to apply in a particular case. In legal settings, rules of evidence generally exclude hearsay (which may draw on "facts" someone believes to be "common knowledge"). Common knowledge is knowledge that is known by everyone or nearly everyone, usually with reference to the community in which the term is used. Scientific Knowledge VS. Common Knowledge Similarities What is Knowledge? Definition and synonyms of common knowledge from the online English dictionary from Macmillan Education.. 1. common knowledge - anything generally known to everyone. Learn more. Definition of common knowledge in the Definitions.net dictionary. It completes the notion of complete information, which requires all players to know the rules of the game and each other’s utility function. : something that many or most people know It's common knowledge that she plans to run for mayor. We can define an operator EG with the intended meaning of "everyone in group G knows" by defining it with the axiom You don’t need to cite a source when referencing common knowledge in your papers, as everyone knows it so it doesn’t usually need to be evidenced. Often, common knowledge does not need to be cited. Meaning of common knowledge. See synonyms for common knowledge. This is the British English definition of common knowledge.View American English definition of common knowledge.. Change your default dictionary to American English. R. Fagin, J. Y. Halpern, Y. Moses, and M. Y. Vardi. The point is that common knowledge can be found in a variety of sources. 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# Minimum MSE estimation derivation

1. Aug 4, 2011

### EmmaSaunders1

Hello,

Would anyone be-able to recommend a good, easy to read article which outlines MMSE and its derivation. Specifically I am having trouble finding this term

$$+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2$$

from

$$E({\left | \left | X-z \right | \right |}^2|Y=y) =\int (x-z)'(x-z)P(x|y)dx\\ =[z'-\int x'P(x|y)dx][z-\int xP(x|y)dx] + \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2$$

Thank you

2. Aug 4, 2011

### EmmaSaunders1

Shouldnt the term just be zero - I cant understand it's presence - are there any conditions in which it is not zero??

3. Aug 5, 2011

### EmmaSaunders1

For anyone who is interested - the last term

$$+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2$$

is necessary to account for the difference between E(x^2) and [E(x)]^2. When Z = E[x|Y=y] the term

$$E({\left | \left | X-z \right | \right |}^2|Y=y)$$

Is a minimum and reduces to

$$+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2 =$$

Then

$$E({\left | \left | X \right | \right |}^2|Y=y)-E(X|Y=y)^2\\ =E({\left | \left | X \right | \right |}^2|Y=y)-{\left | \left | \hat{X} \right | \right |}^2$$

which is the average mean square error

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• (1968)

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• ([Urbana, Ill.] : College of Commerce and Business Administration, University of Illinois at Urbana-Champaign,, Aug 14 198)

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• (Urbana : Cooperative Extension Service, Dept. of Agricultural Economics, College of Agriculture, University of Illinois at Urbana-Champaign., 2004-05-10)

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This thesis adopts an informational perspective to explain why managers with private information engage in income smoothing and to provide new insights into income smoothing. In particular, this study extends the existing ...

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SFAS 133 requires most types of hedge ineffectiveness to be measured on a fair value basis and reported in earnings. This earnings recognition requirement was the focal point of controversy surrounding the adoption of SFAS ...

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• ([Urbana] : College of Commerce and Business Administration, Bureau of Economic and Business Research, University of Illinois, Urbana-Champaign,, Apr 1981)

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This thesis analyzes the effect of the personal income tax on the choice of labor supply and non-monetary job characteristics. Previous studies of income taxation and labor supply have implicitly assumed that the wage rate ...

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• (American Institute of Physics, 2006)
In a nematic elastomer the deformation of the polymer network chains is coupled to the orientational order of the mesogenic groups. Statistical arguments have derived the so-called neoclassical free energy that models this ...

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• (Department of Theoretical and Applied Mechanics (UIUC), 2005-08)
In a nematic elastomer the deformation of the polymer network chains is coupled to the orientational order of the mesogenic groups. Here we use the neo-classical model for this coupling supplemented by the usual Frank ...

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A new family of preconditioners for conjugate gradient-like iterative methods applied to large sparse linear least squares problems, $min\Vert Ax-b\Vert\sb2$, is proposed. The family is based on incomplete Gram-Schmidt ...

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My thesis consists of three essays that investigate strategic interactions between individuals engaging in risky collective action in uncertain environments. The first essay analyzes a broad class of incomplete information ...

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We discuss an object-based, multi-paradigm approach to the development of large-scale, high performance parallel applications. Our approach is characterized by three essential ingredients: (i) Plurality, i.e. a programmer ...

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The objective of my dissertation is to create a general approach to evaluating IS/IT projects using Real Option Analysis (ROA). This is an important problem because an IT Project Portfolio (ITPP) can represent hundreds of ...

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Existing network simulators perform reasonably well in evaluating the performance of network protocols, but lack the capability of verifying the correctness of network protocols. In this paper, we present our ongoing ...

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静岡大学解析セミナー 日　時： 2013年11月20日（水）16:30--18:00 場　所： 静岡大学理学部 C 棟 309 号室 講演者： Prof. Dr. Reinhard Farwig (Technische Universität Darmstadt, Germany) 講演タイトル： Optimal initial values and regularity conditions of Besov space type for weak solutions to the Navier-Stokes system

(+ := @ipc.shizuoka.ac.jp)

その場合，1コマ目に遅刻する可能性があれば，事前に連絡しておいてもらった方が無難です．
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### プログラム

#### 12月13日（金）

13:45 - 14:45 三内顕義（東京大学，数理科学研究科）
A numerical invariant and singularities in positive characteristic, I

15:00 - 16:00 Luo Xueyu（名古屋大学，多元数理科学研究科）
Ice Quivers with Potential associated with Triangulations and Cohen-Macaulay Modules over Orders, I

16:15 - 17:15 相原琢磨（名古屋大学，多元数理科学研究科）
Flipping Brauer graphs, I

18:30 - 懇親会

#### 12月14日（土）

09:30 - 10:30 Luo Xueyu（名古屋大学，多元数理科学研究科）
Ice Quivers with Potential associated with Triangulations and Cohen-Macaulay Modules over Orders, II

10:45 - 11:45 三内顕義（東京大学，数理科学研究科）
A numerical invariant and singularities in positive characteristic, II

13:15 - 14:15 相原琢磨（名古屋大学，多元数理科学研究科）
Flipping Brauer graphs, II

14:30 - 15:30 浅芝 秀人（静岡大学，理学研究科）
Gluing of derived equivalences along bimodules

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### 講演概要

• #### 三内 顕義

A numerical invariant and singularities in positive characteristic
Huneke, Leuschkeによって定義された不変量であるF-signatureを加群の不変量として再定義し、F-rational, Gorenstein, regularなどの特徴付けを行う。またこの不変量はMCMの圏の構造と密接な関係にあり、それについても解説する。
• #### Xueyu Luo

Ice Quivers with Potential associated with Triangulations and Cohen-Macaulay Modules over Orders
This is a joint work with Laurent Demonet. A quiver together with a linear combination of cyclic paths in the quiver is called a quiver with potential (QP for short). For each triangulation of a bordered surface with marked points, a QP was introduced by Labardini-Fragoso. In the first talk, we extend Labardini's construction and associate an ice QP to each triangulation of a polygon with $n$ vertices by adding a set of $n$ frozen vertices corresponding to the edges of the polygon and certain arrows. We study the associated frozen Jacobian algebra and show that it has the structure of a special kind of $K[x]$-order $\Lambda$ (i.e. a $K[x]$-algebra which is finitely generated free as $K[x]$-module). In the second talk, we will construct a bijection between the set of the isomorphism classes of all indecomposable Cohen-Macaulay $\Lambda$-modules (i.e. all the $\Lambda$-modules which are also finitely generated free as $K[x]$-module) and the set of all sides and diagonals of the polygon. We show that the stable category of the category of Cohen-Macaulay $\Lambda$-modules is $2$-Calabi-Yau and is triangle-equivalent to the cluster category of type $A_{n-3}$.
• #### 相原 琢磨

Flipping Brauer graphs
The class of tilting complexes is one of the most important classes in representation theory of algebras, since tilting complexes give derived equivalences. By tilting mutation we can get a new tilting complex from a given one. A problem is to explicitly describe the endomorphism algebra of a tilting complex given by tilting mutation. In this talk, we give a complete answer to this problem for Brauer graph algebras.
• #### 浅芝 秀人

Gluing of derived equivalences along bimodules
Let k be a commutative ring, and consider the bicategory k-Cat^b of small k-categories C, bimodules over them and bimodule morphisms (the 1-morphisms C --> C' are the C'-C-bimodules and the composite C --> C' --> C'' is given by the tensor over C'). We give a definition of the Grothendieck construction Gr(X) of a lax functor X from a small category I to the bicategory k-Cat^b, which extends the corresponding one of a colax functor from I to the 2-category k-Cat of small k-categories, k-functors and natural transformations, and which enables us to construct new k-categories by gluing k-categories together along bimodules, in particular, triangular matrix algebras and the tensor algebras of k-species. When k is a field, we can construct a derived equivalence between Grothendieck constructions Gr(X) and Gr(X') of lax functors X and X': I --> k-Cat^b by gluing derived equivalences between X(i) and X'(i) (i is an object of I) together along bimodules X(a) and X'(a) (a is a morphism of I) if X and X' are "derived equivalent".

http://www.ipc.shizuoka.ac.jp/~shasash/

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