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# Explanations can be manipulated and geometry is to blame

Explanation methods aim to make neural networks more trustworthy and interpretable. In this paper, we demonstrate a property of explanation methods which is disconcerting for both of these purposes. Namely, we show that explanations can be manipulated arbitrarily by applying visually hardly perceptible perturbations to the input that keep the network's output approximately constant. We establish theoretically that this phenomenon can be related to certain geometrical properties of neural networks. This allows us to derive an upper bound on the susceptibility of explanations to manipulations. Based on this result, we propose effective mechanisms to enhance the robustness of explanations.

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03/04/2022

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## 1 Introduction

Explanation methods have attracted significant attention over the last years due to their promise to open the black box of deep neural networks. Interpretability is crucial for scientific understanding and safety critical applications. Explanations can be provided in terms of explanation maps[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] that visualize the relevance attributed to each input feature for the overall classification result. In this work, we establish that these explanation maps can be changed to an arbitrary target map. This is done by applying a visually hardly perceptible perturbation to the input. We refer to Figure 1

for an example. This perturbation does not change the output of the neural network, i.e. in addition to the classification result also the vector of all class probabilities is (approximately) the same. This finding is clearly problematic if a user, say a medical doctor, is expecting a robustly interpretable explanation map to rely on in the clinical decision making process. Motivated by this unexpected observation, we provide a theoretical analysis that establishes a relation of this phenomenon to the geometry of the neural network’s output manifold. This novel understanding allows us to derive a bound on the degree of possible manipulation of the explanation map. This bound is proportional to two differential geometric quantities: the principle curvatures and the geodesic distance between the original input and its manipulated counterpart. Given this theoretical insight, we propose efficient ways to limit possible manipulations and thus enhance resilience of explanation methods. In summary, this work provides the following key contributions:

• We propose an algorithm which allows to manipulate an image with a hardly perceptible perturbation such that the explanation matches an arbitrary target map. We demonstrate its effectiveness for six different explanation methods and on four network architectures as well as two datasets.

• We provide a theoretical understanding of this phenomenon for gradient-based methods in terms of differential geometry. We derive a bound on the principle curvatures of the hypersurface of equal network output. This implies a constraint on the maximal change of the explanation map due to small perturbations.

• Using these insights, we propose methods to undo the manipulations and increase the robustness of explanation maps by smoothing the explanation method. We demonstrate experimentally that smoothing leads to increased robustness not only for gradient but also for propagation-based methods.

### 1.1 Related work

In [20], it was demonstrated that explanation maps can be sensitive to small perturbations in the image. Their results may be thought of as untargeted manipulations, i.e. perturbations to the image which lead to an unstructured change in the explanation map. Our work focuses on targeted manipulations instead, i.e. to reproduce a given target map. Another approach [21] adds a constant shift to the input image, which is then eliminated by changing the bias of the first layer. For some methods, this leads to a change in the explanation map. Contrary to our approach, this requires to change the network’s biases. In [22], explanation maps are changed by randomization of (some of) the network weights. This is different from our method as it does not aim to change the explanation in a targeted manner and modifies the weights of the network.

## 2 Manipulating explanations

We consider a neural network

which classifies an image

in categories with the predicted class given by . The explanation map is denoted by and associates an image with a vector of the same dimension whose components encode the relevance score of each pixel for the neural network’s prediction. For a given explanation method and specified target , a manipulated image has the following properties:

1. The output of the network stays approximately constant, i.e. .

2. The explanation is close to the target map, i.e. .

3. The norm of the perturbation added to the input image is small, i.e. and therefore not perceptible.

Throughout this paper, we will use the following explanation methods:

• Gradient: The map is used and quantifies how infinitesimal perturbations in each pixel change the prediction [2, 1].

• Gradient Input: This method uses the map [14]. For linear models, this measure gives the exact contribution of each pixel to the prediction.

• Integrated Gradients: This method defines where is a suitable baseline. See the original reference [13] for more details.

• Guided Backpropagation (GBP)

: This method is a variation of the gradient explanation for which negative components of the gradient are set to zero while backpropagating through the non-linearities

[4].

• Layer-wise Relevance Propagation (LRP): This method [5, 16] propagates relevance backwards through the network. For the output layer, relevance is defined by111Here we use the Kronecker symbol .

 RLi=δi,k, (1)

which is then propagated backwards through all layers but the first using the rule

 Rli=∑jxli(Wl)+ji∑ixli(Wl)+jiRl+1j, (2)

where denotes the positive weights of the -th layer and is the activation vector of the -th layer. For the first layer, we use the rule to account for the bounded input domain

 R0i=∑jx0jW0ji−lj(W0)+ji−hj(W0)−ji∑i(x0jW0ji−lj(W0)+ji−hj(W0)−ji)R1j, (3)

where and are the lower and upper bounds of the input domain respectively.

• Pattern Attribution (PA): This method is equivalent to standard backpropagation upon element-wise multiplication of the weights with learned patterns . We refer to the original publication for more details [17].

These methods cover two classes of attribution methods, namely gradient-based and propagation-based explanations, and are frequently used in practice [23, 24].

### 2.1 Manipulation Method

Let be a given target explanation map and an input image. As explained previously, we want to construct a manipulated image such that it has an explanation very similar to the target but the output of the network stays approximately constant, i.e.

. We obtain such manipulations by optimizing the loss function

with respect to using gradient descent. We clamp after each iteration so that it is a valid image. The first term in the loss function (4

) ensures that the manipulated explanation map is close to the target while the second term encourages the network to have the same output. The relative weighting of these two summands is controlled by the hyperparameter

.

The gradient with respect to the input of the explanation often depends on the vanishing second derivative of the relu non-linearities. This causes problems during optimization of the loss (4). As an example, the gradient method leads to

We therefore replace the relu by softplus non-linearities

 softplusβ(x)=1βlog(1+eβx). (5)

For large values, the softplus approximates the relu closely but has a well-defined second derivative. After optimization is complete, we test the manipulated image with the original relu network. Similarity metrics: In our analysis, we assess the similarity between both images and explanation maps. To this end, we use three metrics following [22]: the structural similarity index (SSIM), the Pearson correlation coefficient (PCC) and the mean squared error (MSE). SSIM and PCC are relative similarity measures with values in , where larger values indicate high similarity. The MSE is an absolute error measure for which values close to zero indicate high similarity. We normalize the sum of the explanation maps to be one and the images to have values between 0 and 1.

### 2.2 Experiments

To evaluate our approach, we apply our algorithm to 100 randomly selected images for each explanation method. We use a pre-trained VGG-16 network [25]

and the ImageNet dataset

[26]. For each run, we randomly select two images from the test set. One of the two images is used to generate a target explanation map . The other image is perturbed by our algorithm with the goal of replicating the target using a few thousand iterations of gradient descent. We sum over the absolute values of the channels of the explanation map to get the relevance per pixel. Further details about the experiments are summarized in Supplement A. Qualitative analysis: Our method is illustrated in Figure 2 in which a dog image is manipulated in order to have an explanation of a cat. For all explanation methods, the target is closely emulated and the perturbation of the dog image is small. More examples can be found in the supplement. Quantitative analysis: Figure 3 shows similarity measures between the target and the manipulated explanation map as well as between the original image and perturbed image .222Throughout this paper, boxes denote 25th and 75th percentiles, whiskers denote 10th and 90th

percentiles, solid lines show the medians and outliers are depicted by circles.

All considered metrics show that the perturbed images have an explanation closely resembling the targets. At the same time, the perturbed images are very similar to the corresponding original images. We also verified by visual inspection that the results look very similar. We have uploaded the results of all runs so that interested readers can assess their similarity themselves and will provide code to reproduce them. In addition, the output of the neural network is approximately unchanged by the perturbations, i.e. the classification of all examples is unchanged and the median of is of the order of magnitude for all methods. See Supplement B for further details. Other architectures and datasets: We checked that comparable results are obtained for ResNet-18 [27], AlexNet [28] and Densenet-121 [29]. Moreover, we also successfully tested our algorithm on the CIFAR-10 dataset [30]. We refer to the Supplement C for further details.

## 3 Theoretical considerations

In this section, we analyze the vulnerability of explanations theoretically. We argue that this phenomenon can be related to the large curvature of the output manifold of the neural network. We focus on the gradient method starting with an intuitive discussion before developing mathematically precise statements. We have demonstrated that one can drastically change the explanation map while keeping the output of the neural network constant

 g(x+δx)=g(x)=c (6)

using only a small perturbation in the input . The perturbed image therefore lies on the hypersurface of constant network output .444It is sufficient to consider the hypersurface in a neighbourhood of the unperturbed input . We can exclusively consider the winning class output, i.e. with

because the gradient method only depends on this component of the output. Therefore, the hyperplane

is of co-dimension one. The gradient for every is normal to this hypersurface. The fact that the normal vector can be drastically changed by slightly perturbing the input along the hypersurface suggests that the curvature of is large. While the latter statement may seem intuitive, it requires non-trivial concepts of differential geometry to make it precise, in particular the notion of the second fundamental form. We will briefly summarize these concepts in the following (see e.g. [31] for a standard textbook). To this end, it is advantageous to consider a normalized version of the gradient method

 n(x)=∇g(x)∥∇g(x)∥. (7)

This normalization is merely conventional as it does not change the relative importance of any pixel with respect to the others. For any point , we define the tangent space as the vector space spanned by the tangent vectors of all possible curves with . For , we denote their inner product by . For any , the directional derivative is uniquely defined for any choice of by

 Duf(p)=ddtf(γ(t))∣∣∣t=0 with γ(0)=pand˙γ(0)=u. (8)

We then define the Weingarten map as555The fact that follows by taking the directional derivative with respect to on both sides of .

 L:{TpS→TpSu↦−Dun(p),

where the unit normal can be written as (7). This map quantifies how much the unit normal changes as we infinitesimally move away from in the direction . The second fundamental form is then given by

 L:{TpS×TpS→Ru,v↦−⟨v,L(u)⟩=−⟨v,Dun(p)⟩.

It can be shown that the second fundamental form is bilinear and symmetric

. It is therefore diagonalizable with real eigenvalues

which are called principle curvatures. We have therefore established the remarkable fact that the sensitivity of the gradient map (7) is described by the principle curvatures, a key concept of differential geometry. In particular, this allows us to derive an upper bound on the maximal change of the gradient map as we move slightly on . To this end, we define the geodesic distance of two points as the length of the shortest curve on connecting and . In the supplement, we show that:

###### Theorem 1

Let be a network with non-linearities and an environment of a point such that is fully connected. Let have bounded derivatives for all . It then follows for all that

 ∥h(p)−h(p0)∥≤|λmax|dg(p,p0)≤βCdg(p,p0), (9)

where is the principle curvature with the largest absolute value for any point in and the constant depends on the weights of the neural network.

This theorem can intuitively be motivated as follows: for relu non-linearities, the lines of equal network output are piece-wise linear and therefore have kinks, i.e. points of divergent curvature. These relu non-linearities are well approximated by softplus non-linearities (5) with large . Reducing smoothes out the kinks and therefore leads to reduced maximal curvature, i.e. . For each point on the geodesic curve connecting and , the normal can at worst be affected by the maximal curvature, i.e. the change in explanation is bounded by .

There are two important lessons to be learned from this theorem: the geodesic distance can be substantially greater than the Euclidean distance for curved manifolds. In this case, inputs which are very similar to each other, i.e. the Euclidean distance is small, can have explanations that are drastically different. Secondly, the upper bound is proportional to the parameter of the softplus non-linearity. Therefore, smaller values of provably result in increased robustness with respect to manipulations.

## 4 Robust explanations

Using the fact that the upper bound of the last section is proportional to the parameter of the softplus non-linearities, we propose -smoothing of explanations. This method calculates an explanation using a network for which the relu non-linearities are replaced by softplus with a small parameter to smooth the principle curvatures. The precise value of is a hyperparameter of the method, but we find that a value around one works well in practice. As shown in the supplement, a relation between SmoothGrad [12] and -smoothing can be proven for a one-layer neural network:

###### Theorem 2

For a one-layer neural network and its -smoothed counterpart , it holds that

 Eϵ∼pβ[∇g(x−ϵ)]=∇gβ∥w∥(x),

where .

Since

closely resembles a normal distribution with variance

, -smoothing can be understood as limit of SmoothGrad where . We emphasize that the theorem only holds for a one-layer neural network, but for deeper networks we empirically observe that both lead to visually similar maps as they are considerably less noisy than the gradient map. The theorem therefore suggests that SmoothGrad can similarly be used to smooth the curvatures and can thereby make explanations more robust.666For explanation methods other than gradient, SmoothGrad needs to be used in a slightly generalized form, i.e. . Experiments: Figure 4 demonstrates that -smoothing allows us to recover the orginal explanation map by lowering the value of the

parameter. We stress that this works for all considered methods. We also note that the same effect can be observed using SmoothGrad by successively increasing the standard deviation

of the noise distribution. This further underlines the similarity between the two smoothing methods.

If an attacker knew that smoothing was used to undo the manipulation, they could try to attack the smoothed method directly. However, both -smoothing and SmoothGrad are substantially more robust than their non-smoothed counterparts, see Figure 5. It is important to note that -smoothing achieves this at considerably lower computational cost: -smoothing only requires a single forward and backward pass, while SmoothGrad requires as many as the number of noise samples (typically between 10 to 50). We refer to Supplement D for more details on these experiments.

## 5 Conclusion

Explanation methods have recently become increasingly popular among practitioners. In this contribution we show that dedicated imperceptible manipulations of the input data can yield arbitrary and drastic changes of the explanation map. We demonstrate both qualitatively and quantitatively that explanation maps of many popular explanation methods can be arbitrarily manipulated. Crucially, this can be achieved while keeping the model’s output constant. A novel theoretical analysis reveals that in fact the large curvature of the network’s decision function is one important culprit for this unexpected vulnerability. Using this theoretical insight, we can profoundly increase the resilience to manipulations by smoothing only the explanation process while leaving the model itself unchanged. Future work will investigate possibilities to modify the training process of neural networks itself such that they can become less vulnerable to manipulations of explanations. Another interesting future direction is to generalize our theoretical analysis from gradient to propagation-based methods. This seems particularly promising because our experiments strongly suggest that similar theoretical findings should also hold for these explanation methods.

## Appendix A Details on experiments

We provide a run_attack.py file in our reference implementation which allows one to produce manipulated images. The hyperparameter choices used in our experiments are summarized in Table 1. We set and for beta growth (see section below for a description). The column ’factors’ summarizes the weighting of the mean squared error of the heatmaps and the images respectively.

The patterns for explanation method PA are trained on a subset of the ImageNet training set. The baseline for explanation method IG was set to zero. To approximate the integral, we use steps for which we verified that the attributions approximately adds up to the score at the input.

### a.1 Beta growth

In practise, we observe that we get slightly better results by increasing the value of of the softplus during training a start value to a final value using

 β(t)=β0(βeβ0)t/T, (10)

where is the current optimization step and denotes the total number of steps. Figure 6 shows the MSE for images and explanation maps during training with and without -growth. This strategy is however not essential for our results.

We use beta growth for all methods except LRP for which we do not find any speed-up in the optimization as the LRP rules do not explicitly depend on the second derivative of the relu activations. Figure 7 demonstrates that for large beta values the softplus networks approximate the relu network well. Figure 8 and Figure 9 show this for an example for the gradient and the LRP explanation method. We also note that for small beta the gradient explanation maps become more similar to LRP/GPB/PA explanation maps.

## Appendix B Difference in network output

Figure 10 summarizes the change in the output of the network due to the manipulation. We note that all images have the same classification result as the orginals. Furthermore, we note that the change in confidence is small. Last but not least, norm of the vector of all class probabilities is also very small.

## Appendix C Generalization over architectures and data sets

Manipulable explanations are not only a property of the VGG-16 network. In this section, we show that our algorithm to manipulate explanations can also be applied to other architectures and data sets. For the experiments, we optimize the loss function given in the main text. We keep the pre-activation for all network architectures approximately constant, which also leads to approximately constant activation.

In addition to the VGG architecture we also analyzed the explanation’s susceptibility to manipulations for the AlexNet, Densenet and ResNet architectures. The hyperparameter choices used in our experiments are summarized in Table 2. We set and for beta growth. Only for Densenet we set and as for smaller beta values the explanation map produced with softplus does not resemble the explanation map produced with relu. Figure 12 and 11 show that the similarity measures are comparable for all network architectures for the gradient method. Figure 13, 15, 16 and 14 show one example image for each architecture.

We trained the VGG-16 architecture on the CIFAR-10 dataset777code for training VGG on CIFAR-10 from https://github.com/chengyangfu/pytorch-vgg-cifar10. The test accuracy is approximately . We then used our algorithm to manipulate the explanations for the LRP method. The hyperparameters are summarized in Table 3. Two example images can be seen in Figure 17.

## Appendix D Smoothing explanation methods

One can achieve a smoothing effect when substituting the relu activations for softplus activations and then applying the usual rules for the different explanation methods. A smoothing effect can also be achieved by applying the smoothgrad explanation method, see Figure 18. That is adding random perturbation to the image and then averaging over the resulting explanation maps. We average over 10 perturbed images with different values for the standard deviation of the Gaussian noise. The noise level is related to as , where and are the maximum and minimum values the input image can have.

The -smoothing or SmoothGrad explanation maps are more robust with respect to manipulations. Figure 1920 and 21 show results (MSE, SSIM and PCC) for 100 targeted attacks on the original explanation, the SmoothGrad explanation and the -smoothed explanation for explanation methods Gradient and LRP.

For manipulation of SmoothGrad we use beta growth with and . For manipulation of -Smoothing we set for all runs. The hyperparameters for SmoothGrad and -Smoothing are summarized in Table 4 and Table 5.

In Figure 22 and Figure 23, we directly compare the original explanation methods with the -smoothed explanation methods. An increase in robustness can be seen for all methods: explanation maps for -smoothed explanations have higher MSE and lower SSIM and PCC than explanation maps for the original methods. The similarity measures for the manipulated images are of comparable magnitude.

## Appendix E Proofs

In this section, we collect the proofs of the theorems stated in the main text.

### e.1 Theorem 1

###### Theorem 3

Let be a network with non-linearities and an environment of a point such that is fully connected. Let have bounded derivatives for all . It then follows for all that

 ∥h(p)−h(p0)∥≤|λmax|dg(p,p0)≤βCdg(p,p0), (11)

where is the principle curvatures with the largest absolute value for any point in and the constant depends on the weights of the neural network.

Proof: This proof will proceed in four steps. We will first bound the Frobenius norm of the Hessian of the network . From this, we will deduce an upper bound on the Frobenius norm of the second fundamental form. This in turn will allow us to bound the largest principle curvature . Finally, we will bound the maximal and minimal change in explanation. Step 1: Let where are the weights of layer .888We do not make the dependence of softplus on its parameter explicit to ease notation. We note that

 ∂ksoftplus(∑jWijxj)=Wikσ(∑jWijxj) (12) ∂lσ(∑jWijxj)=βWilg(∑jWijxj)) (13)

where

 σ(x)=1(1+e−βx), g(x)=1(eβx/2+e−βx/2)2. (14)

The activation at layer is then given by

 a(L)(x)=(softplus(L)∘⋯∘softplus(1))(x) (15)

Its derivative is given by

 ∂ka(L)i =∑s2…sLW(L)isLσ(∑jW(L)ija(L−1)j)W(L−1)sLsL−1σ(∑jW(L−1)sLja(L−2)j) …W(1)s2kσ(∑jW(1)s2jxj)

We therefore obtain

 ∥∥∇a(L)∥∥≤L∏l=1∥∥W(l)∥∥F (16)

Deriving the expression for again, we obtain

 ∂l∂ka(L)i= ∑m∑s2…sL{ W(L)isLσ(∑jW(L)ija(L−1)j)W(L−1)sLsL−1σ(∑jW(L−1)ija(L−2)j) …β∑^smW(m)sm+1^smW(m)sm+1smg(∑jW(m)sm+1ja(m−1)j(x))∂la(m−1)^sm(x) …W(1)s2kσ(∑jW(1)s2jxj)}

We now restrict to the case for which the index only takes a single value and use . The Hessian is then bounded by

 ∥H∥F≤β~C (17)

where the constant is given by

 ~C=∑m∥∥W(L)∥∥F∥∥W(L−1)∥∥F…∥∥W(m)∥∥2F…∥∥W(1)∥∥2F. (18)

Step 2: Let be a basis of the tangent space . Then the second fundamental form for the hypersurface at point is given by

 L(ei,ej) =−⟨Dein(p),ej⟩ (19) =−⟨Dei∇f(p)∥∇f(p)∥,ej⟩ =−1∥∇f(p)∥⟨H[f]ei,ej⟩+(…)⟨∇f(p),ej⟩ (20)

We now use the fact that , i.e. the gradient of is normal to the tangent space. This property was explained in the main text. This allows us to deduce that

 L(ei,ej)=−1∥∇f(p)∥H[f]ij. (21)

Step 3: The Frobenius norm of the second fundamental form (considered as a matrix in the sense of step 2) can be written as

 ∥L∥F=√λ21+⋯+λ2d−1, (22)

where

are the principle curvatures. This property follows from the fact that the second fundamental form is symmetric and can therefore be diagonalized with real eigenvectors, e.g. the principle curvatures. Using the fact that the derivative of the network is bounded from below,

, we obtain

 |λmax|≤β~Cc. (23)

Step 4: For , we choose a curve with and . Furthermore, we use the notation . It then follows that

 n(p)−n(p0)=∫tt0ddt(n(γ(t)))dt=∫tt0Du(t)n(γ(t))dt (24)

Using the fact that and choosing an orthonormal basis for the tangent spaces, we obtain

 ∫tt0Du(t)n(γ(t))dt =∫tt0∑j⟨ej(t),Du(t)n(γ(t))⟩ej(t)dt (25) =∫tt0∑jL(ej(t),u(t))ej(t)dt (26)

The second fundamental form is bilinear and therefore

 ∫tt0∑iL(ej(t),u(t))ej(t)dt=∫tt0∑i,jL(ej(t),ei(t))ui(t)ej(t)dt (27)

We now use the notation and choose its eigenbasis for . We then obtain for the difference in the unit normals:

 n(p)−n(p0)=∫tt0∑iλi(t)ui(t)ei(t)dt, (28)

where denote the principle curvatures at . By orthonormality of the eigenbasis, it can be easily checked that

 ⟨∑iλi(t)ui(t)ei(t),∑jλj(t)uj(t)ej(t)⟩≤|λmax|2∑iui(t)2 ⇒∥∥ ∥∥∑iλi(t)ui(t)ei(t)∥∥ ∥∥≤|λmax|∥u(t)∥

Using this relation and the triangle inequality, we then obtain by taking the norm on both sides of (28):

 ∥n(p)−n(p0)∥≤|λmax|∫tt0∥˙γ(t)∥dt. (29)

This inequality holds for any curve connecting and but the tightest bound follows by choosing to be the shortest possible path in with length , i.e. the geodesic distance on . The second inequality of the theorem is obtained by the upper bound on the largest principle curvature derived above, i.e. (23).

### e.2 Theorem 2

###### Theorem 4

For one layer neural networks and , it holds that

 Eϵ∼pβ[∇g(x−ϵ)]=∇gβ∥w∥(x), (30)

where .

Proof: We first show that

 softplusβ(x)=Eϵ∼pβ[relu(x))], (31)

for a scalar input . This follows by defining implicitly as

 softplusβ(x)=∫+∞−∞p(ϵ)relu(x−ϵ)dϵ. (32)

Differentiating both sides of this equation with respect to results in

 σβ(x)=∫+∞−∞p(ϵ)Θ(x−ϵ)dϵ=∫x−∞p(ϵ)dϵ, (33)

where is the Heaviside step function and . Differentiating both sides with respect to again results in

 pβ(x)=p(x). (34)

Therefore, (31) holds. For a vector input , we define the distribution of its perturbation by

 pβ(→ϵ)=∏ipβ(ϵi), (35)

where denotes the components of . We will suppress any arrows denoting vector-valued variables in the following in order to ease notation. We choose an orthogonal basis such that

 ϵ=ϵp^w+∑iϵ(i)o^w(i)o with ^w⋅^w(i)o=0 and w=∥w∥^w. (36)

This allows us to rewrite

 Eϵ∼pβ[relu(wT(x−ϵ))] =Eϵ∼pβ[relu(wTx−∥w∥ϵp))] =∫pβ(ϵp)(relu(wTx−∥w∥ϵp))dϵp

By changing the integration variable to and using (31), we obtain

 softplusβ∥w∥(wTx)=Eϵ∼pβ[relu(wT(x−ϵ))], (37)

The theorem then follows by deriving both sides of the equation with respect to .


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Determination of ionization constants by chromatography

Darawan Palalikit

# Determination of ionization constants by chromatography

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### Determination of ionization constants by chromatography by Darawan Palalikit Download PDF EPUB FB2

Simultaneous determination of the lipophilicity and dissociation constants of dialkyl phosphinic acids by negligible depletion hollow fiber membrane-protected liquid-phase microextraction. Journal of Chromatography A, Cited by: This practical manual is devised for organic chemists and biochemists who, in the course of their researches and without previous experience, need to determine an ionization constant.

We are gratified that earlier editions were much used for this purpose and that they also proved adequate for the.

Buy The Determination of Ionization Constants: A Laboratory Manual by Adrien Albert online at Alibris. We have new and used copies available, in 3 editions - starting at \$ Shop now. The other describes the use of more recently introduced techniques for the determination of ionization constants, such as Raman and nuclear magnetic resonance spectroscopy, thermometric titrations, and paper electro phoresis.

Chapter 1 gives enhanced help in choosing between alternative methods for determining ionization constants. About this book This practical manual is devised for organic chemists and biochemists who, in the course of their researches and without previous experience, need to determine an ionization constant.

Prediction of solubility from ionization constants Determination of ionization constants from solubilities Determination of ionization constants from vapour pressure, by partitioning between a pair of solvents, or by other phase equilibria 6 Determination of Ionization Constants by Conductimetry Try the new Google Books.

Check out the new look and enjoy easier access to your favorite features. Try it now. No thanks. Try the new Google Books. View eBook. Get this book in print. ; Barnes& The Determination of Ionization Constants: A Laboratory Manual.

An illustration of an open book. Books. An illustration of two cells of a film strip. Video. An illustration of an audio speaker. Audio An illustration of a " floppy disk. The determination of ionization constants: a laboratory manual Item Preview remove-circle Share or Embed This Item.

Determination of the values of ionization constants of ethylenebisdithiocarbamic acid Š. GERGELY Department of Analytical Chemistry\ Slovak Technical University, CS 37 Bratislava Received 14 January Accepted for publication 11 September The values of ionization constants К тл and K T, 2 of dibasic ethylenebis.

Ionization constants of some flavanols (catechin and epicatechin) and flavonols (kaempherol, fisetin, morin, and quercetin) are determined by capillary zone electrophoresis (CZE).

The other describes the use of more recently introduced techniques for the determination of ionization constants, such as Raman and nuclear magnetic resonance spectroscopy, thermometric titrations, and paper electro phoresis. Chapter 1 gives enhanced help in choosing between alternative methods for determining ionization : Adrien Albert.

The determination of ionization constants by ultraviolet or visible spectrophotometry is more time-consuming than by potentiometry.

However, spectrometry is an ideal method when a substance is too insoluble for potentiometry or when its p Ka value is particularly low or.

6. Conclusions In the determination of the ionization constant of p-nitrophenol from electrophoretic mobility, spectroscopic analysis based on peak areas was used as corroborating data.

Peak areas divided by migration time versus pH for a constant solute concentration gave an ionic distribution plot which was fit using a non-linear regression.

We have developed a methodology that enables for the rapid measurement of ionization constants (pKa) of series of compounds by UV spectrophotometry.

This protocol, which is straightforward to set up, takes advantage of the sensitivity of UV spectroscopy and the throughput enabled by the well microplate (as opposed to the use of 1 cm quartz cuvette). The compounds, in stock solutions in.

In order to select the most appropriate method for the determination of an unknown ionization constant, the experimenter should be able to forecast an approximate value for the expected pK a.

Title:Determination of The Ionization Constant of Local Anaesthetics in Organic Solvent-Water Media by The Spectrofluorimetric Method VOLUME: 9 ISSUE: 4 Author(s):S. Beniz Gunduz, Eda Ediz and Zeynep Altınordu Affiliation:Selcuk University, Faculty of Science, Department of Chemistry,Campus-Konya/Turkiye Keywords:Ionization constant, local anaesthetics, pKa, spectrofluorimetric.

We have learned that an ionization constant quantifies the degree a substance will ionize in a solution (typically water).

K a, K b, and K w are constants for acids, bases, and finally water, respectively and are related by $K_a \times K_b = K_w$ However, these constants are also used to find concentrations as well as pH. Abstract. The stoichiometric ionization constants of N,N-dialkyl--(4-substituted benzoyl) thiourea (Substitutes: H, Cl, and Br; alkyl groups: ethyl, n-propyl, n-butyl, and phenyl) derivatives have been determined potentiometrically in dioxane-water (v:v, ) mixture at ionic strength of M and °C.

The ionization constants were calculated with the BEST computer program and the formation. High-performance liquid chromatography as a technique to determine pKa values of drug candidates that show poor solubility in water. [22] Determination of dissociation constants of weak acids by feedback-based flow ratiometry.

[23] Study of the micro-volume flow titration and screening the dissociation constants (pKa) of weak acids. [24]. The effect of the mobile phase composition on the ionization constant was studied by measuring the pK a at different methanol–water mixtures, ranging between 50 and 65% (v/v), using LC–DAD method.

Two simple, accurate, precise and fully validated analytical methods for the simultaneous determination of enalapril and lercanidipine in.

Stanford Libraries' official online search tool for books, media, journals, databases, government documents and more. The determination of ionization constants: a. Background. In this study, a RP - HPLC method was applied for determination of pK a value by using the dependence of the capacity factor (k) on the pH of the mobile phase for oxysophocarpine (OSP).

Findings. The effect of the mobile phase composition on the ionization constant was studied by measuring the pK a value at different MeOH concentrations, ranging from 10.

Determination of ionization constants by paper chromatography. (PMID) Abstract Citations; Related Articles; Data; BioEntities; External Links ' ' ROUFFIAC R, ' ' LATTES A, ' ' VALDIGUIE P Annales Pharmaceutiques Francaises [01 May] Type.

determination of aqueous ionization constant (pK a) of drugs or other organic chemical substances. It is not applicable for the determination of ionization constant of poorly-water soluble drugs because reliable titration curves are not obtained. However, the use of cosolvent.

Electrogravimetric Determination of Copper Polarographic Determination of Cadmium Polarographic Determination of Riboflavin Appendix I Mathematical Operations Appendix II Solubility Product Constants Appendix III Ionization Constants of Weak Acids and Bases Appendix IV Reduction Potentials Appendix V Log Formation Constants for Metal.

chromatography, the mobile phase is a gas that carries the analytes through the column and is therefore referred to as the carrier gas. The stationary phase for this experiment is a silicon phase. In this experiment, the carrier gas is inert helium.

Just like with a HPLC instrument, a GC is composed of several components. The Equilibrium Constant The Acid Ionization Constant Calculating the Hydrogen Ion Concentration The Base Ionization Constant Ion Product for Water The Solubility Product Constant (Ksp) The pH of a Solution Measuring the pH Buffered Solutions—Description and.

The acid ionization represents the fraction of the original acid that has been ionized in solution. Therefore, the numerical value of K a is a reflection of the strength of the acid.

Weak acids with relatively higher K a values are stronger than acids with relatively lower K a values. Because strong acids are essentially % ionized, the concentration of the acid in the denominator is nearly. Two fluoroalcohols—1,1,1,3,3,3-hexafluoropropanol (HFIP) and 1,1,1,3,3,3-hexafluoromethylpropanol (HFTB)—were evaluated for the first time as volatile buffer acids in the basic mobile phase for reversed-phase chromatography with electrospray ionization-mass spectrometric (LC.

The effect of the mobile phase composition on the ionization constant was studied by measuring the pK (a) at different ACN concentrations, ranging from 30 to 40%.

The adequate condition for the chromatographic determination of these compounds in pharmaceutical dosage forms was established based on the different retention behaviors of the species. determination of trace elements in hazardous wastes by ion chromatography project report Posted By C. Lewis Public Library TEXT ID d72 Online PDF Ebook Epub Library protection agency isbn kostenloser versand fur alle bucher mit versand und aug determination of trace elements in hazardous wastes by ion determination.

determination of trace elements in hazardous wastes by ion chromatography project report Posted By Hermann Hesse Media TEXT ID d72 Online PDF Ebook Epub Library report u s environmental protection agency isbn kostenloser versand fur alle bucher mit versand und verkauf duch amazon wastes by ion chromatography.

A gas chromatography (Agilent N) with mass spectrometer (Waters, Boston, USA) and an auto-sampler (Agilent ) with electron ionization (EI+) mode were used. Separation of analytes was carried out using HP-5MS column (30 m. PKA Determination Determination of the dissociation constant (pKa) for a drug capable of ionization with in a pH range of 1 to 10 is important, since solubility and absorption, can be altered by orders of magnitude with changing pH.

o Determination of pKa value by Ultraviolet spectroscopy o By Potentiometric Titrations 17 D Test Method for Determination of the Boiling Range Distribution of Gasoline by Wide-Bore Capillary Gas Chromatography. D Test Method for Boiling Point Distribution of Samples with Residues Such as Crude Oils and Atmospheric and Vacuum Residues by High Temperature Gas Chromatography.

E Practice for Packed Column Gas Chromatography. The acid dissociation constant (pK a) is among the most frequently used physicochemical parameters, and its determination is of interest to a wide range of research present a brief introduction on the conceptual development of pK a as a physical parameter and its relationship to the concept of the pH of a solution.

This is followed by a general summary of the historical development. ceptable for determination of OSP. The effect of different mobile phase composition on the ionization constant was studied by measuring the pK a values of OSP at different MeOH concentrations, ranging from 10% to 20% (v/v).

The results showed that the mobile phase at 10% (v/v)MeOHwastoo weak to elute the compound when the pH > On the. GET LIPIDAT “THE BOOK” ISBN ISBN MEMBRANE BIOPHYSICS LABORATORY.

Structural parameters of aqueous phospholipid mixtures. Currently, an increasing demand of cannabis-derived products for recreational and medical use is observed. Therefore, the reliable and fast quantification of cannabinoids in hemp samples is essential for the control of product from Cannabis sativa, L.

strains. In general, gas chromatography (GC) is the method of choice for the quantification of cannabinoids whereas this method is time.

The effect of the mobile phase composition on the ionization constant was studied by measuring the pKa at different acetonitrile-water mixtures, ranging between 35 and 50% (v/v) using the reversed-phase liquid chromatography (RP-LC) method with UV detector.

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text:
Lemma 12.14.4. Let $\mathcal{A}$ be an abelian category. Let

$0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$

be a short exact sequence of complexes. Suppose that $\{ s_ n : C_ n \to B_ n\}$ is a family of morphisms which split the short exact sequences $0 \to A_ n \to B_ n \to C_ n \to 0$. Let $\pi _ n : B_ n \to A_ n$ be the associated projections, see Lemma 12.5.10. Then the family of morphisms

$\pi _{n - 1} \circ d_{B, n} \circ s_ n : C_ n \to A_{n - 1}$

define a morphism of complexes $\delta (s) : C_\bullet \to A[-1]_\bullet$.

Proof. Denote $i : A_\bullet \to B_\bullet$ and $q : B_\bullet \to C_\bullet$ the maps of complexes in the short exact sequence. Then $i_{n - 1} \circ \pi _{n - 1} \circ d_{B, n} \circ s_ n = d_{B, n} \circ s_ n - s_{n - 1} \circ d_{C, n}$. Hence $i_{n - 2} \circ d_{A, n - 1} \circ \pi _{n - 1} \circ d_{B, n} \circ s_ n = d_{B, n - 1} \circ (d_{B, n} \circ s_ n - s_{n - 1} \circ d_{C, n}) = - d_{B, n - 1} \circ s_{n - 1} \circ d_{C, n}$ as desired. $\square$

Comment #2655 by Ko Aoki on

Typo in the second sentence: "... be a sort exact sequence of complexes." should be replaced by "... be a short exact sequence of complexes."

There are also:

• 2 comment(s) on Section 12.14: Homotopy and the shift functor

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
# Assignment 6: Speech Recognition

Out: Thu, Apr 14      Due: 11:59 pm EST on Thu, Apr 28

### Objective

Build a template-based isolated-word speech recognizer.

### Setup

You may work with someone you have collaborated with before. Use the Google Sheet to find a partner and set up your joint repository as instructed.

Install the scikits.talkbox using pip. This relies on scipy being installed on your computer.

pip install scikits.talkbox
Unless you already have an audio program that can resample files, also install the sox command-line program.

If these installations give you trouble, use tempest as instructed in the Tools doc.

We will use a portion of the CSLU ISOLET corpus, consisting of recordings of isolated letters by different speakers. Each filename is of the form speakername-let1-t.wav (a recording of speakername saying the letter "let").

Download the corpus from this link, and place the unzipped isolet folder in your repository clone (but don't add it when you push your code). If you are working on tempest, you should use the dataset stored in /home/sravana/nlp/isolet directly, without copying it to your home directory. Do this by creating a symbolic link to the wav directories.

cd your_repo_clone/
mkdir isolet
mkdir isolet/train
mkdir isolet/test
mkdir isolet/train/mfc
mkdir isolet/test/mfc
ln -s /home/sravana/nlp/isolet/test/wav isolet/test/wav
ln -s /home/sravana/nlp/isolet/train/wav isolet/train/wav

Important: Before you start, you must run the provided feature extraction script to compute MFCCs from your WAV files. Execute featurize.py appropriately on the train and test folders, which will write .mfc files corresponding to every .wav file into train/mfc and test/mfc. Each row in a .mfc file represents the cepstral coefficients from one time frame.

The default Python on tempest has changed recently. Please explicitly call /opt/bin/python2.7 since this is the one that has scipy, or set your PYTHONPATH.

### Submitting

Before the deadline (11:59 pm EST on Thu, Apr 28), push the files below to your repository following these instructions.

• the completed dtwrecognize.py
• the .csv output file created on your dataset for section (b)
• README.md and reflection.md

If you are working with a partner, make sure you have set things up as described so you are both contributing to a single repository.

## Section (a): Word Recognition with Dynamic Time Warping [35 pts]

utils.py contains a helper function for kNN classification (like the one for A5). This function takes an integer k, a 2-d array distances, and a list trainlabels. distances[i, j] contains the DTW distance between the ith test audio file and the jth training audio file trainlabels[j] contains the label of the jth training file. (In this case, the labels are the transcription of the file -- "a", "b", "q", etc.) The function returns a list testlabels where testlabels[i] is the predicted transcription of the ith audio file.

Since you're mature Python programmers by now, no other starter code is provided for this section. Write your entire program in a file named dtwrecognize.py. This program should take these arguments in order:

1. path to the training directory with MFCC files
2. integer value of k for kNN classification
3. a string specifying whether to run in "batch" or "single"-file mode. Batch mode computes transcription predictions for several test files using pre-processed MFCCs, while single mode reads a WAV file and predicts its transcription.
4. the test directory name with MFCC files in case of "batch" mode, or a filename of a .wav file in case of "single" mode.

### Batch Mode

Calling

python dtwrecognize.py isolet/train/mfc 1 batch isolet/test/mfc
1. Computes the DTW distances (with Euclidean distance for each frame) between every MFCC training file in isolet/train/mfc and every MFCC test file in isolet/test/mfc.

Debug by checking that the DTW distance between isolet/test/mfc/mvcw0-m1-t.mfc and isolet/train/mfc/fgw0-x1-t.mfc is 271.04.
2. Runs kNN with k=1 on this distance matrix to predict the transcriptions of the test files.
3. Produces a hypothesis file named isolet-1.csv identical (except for ordering) to isolet-1.csv where each line contains the test filename,actual letter,predicted letter.
4. Finally, prints out the accuracy score to standard output, which is 78.84% for this example.

The program takes about 1 minute on the isolet data. You can run it on a subset of the test data for faster debugging.

### Single-File Mode

Testing on a set of MFCC files and computing accuracy is great for development and evaluation, but it's not a particularly fun tool to show your friends.

Add an option for the program to read a single wav file of someone saying a letter, compute its MFCC representation on the fly, and predict its transcription using your code from section (a). Use the function from featurize.py. For example, executing

python dtwrecognize.py isolet/train/mfc 3 single isolet/test/wav/fmf0-x1-t.wav
should print out
Predicted label: x

## Section (b): Personalized Speech Recognition [10 pts]

Partly relies on section (a).

Your program can be used out-of-the-box to build a recognizer for any small vocabulary task. Come up with your own vocabulary: names, commands for a device ["open", "read", "turnon"], anything that floats your boat. The vocabulary does not even have to be in English. Each "word" must be relatively short, but may contain multiple syllables.

Make about 5 recordings of each word. Partners should both contribute recordings -- data from multiple speakers makes a better system. You can also clip speech segments containing these words from other sound files (from YouTube, poscasts, etc) if you prefer. Save them as 16000 Hz mono-channel files. Use SoX or any audio program to convert the recordings.

sox infile.wav -r 16000 -c 1 outfile.wav

Store these files in a directory named myrecordings having the same directory and filename structure as isolet, with about 4 recordings per word for training and 1 for testing.

Test the performance of the recognizer on this dataset in batch mode. You can vary k if you think it will help. What's the accuracy? Is it better or worse than you expected given the results on the isolet data? Push the resulting myrecordings-k.csv file with the best accuracy result.

Finally, record yourself saying a few isolated letters, and run the recognizer trained on the isolet data in single-file mode on those recordings. (Remember to resample the wav files.) Is it comparable to the performance on the isolet test files?

## Misc [5 pts]

• Answer the remaining questions in reflection.md.
• Fill out README.md and submit your work correctly.
• Check that your code is reasonably documented, elegant, and efficient.


text:
The square root of 0.007 is 0.08366600265340755. See below the step-by-step solution:

In this case we are going to use the Babylonian Method to get the square root of any positive number.

Step 1:

Divide the number (0.007) by 2 to get the first guess for the square root .
First guess = 0.007/2 = 0.0035.

Step 2:

Divide 0.007 by the previous result. d = 0.007/0.0035 = 2.
Average this value (d) with that of step1: (2 + 0.0035)/2 = 1.00175 (New guess).
Error = current value - previous value = 0.0035 - 1.00175 = 0.99825.
0.99825 > 0.0001. So, error > precision. Let's repeat this step again.

Step 3:

Divide 0.007 by the previous result. d = 0.007/1.00175 = 0.006988.
Average this value (d) with that of step2: (0.006988 + 1.00175)/2 = 0.504369 (New guess).
Error = current value - previous value = 1.00175 - 0.504369 = 0.49738099999999996.
0.49738099999999996 > 0.0001. So, error > precision. Let's repeat this step again.

Step 4:

Divide 0.007 by the previous result. d = 0.007/0.504369 = 0.013879.
Average this value (d) with that of step3: (0.013879 + 0.504369)/2 = 0.259124 (New guess).
Error = current value - previous value = 0.504369 - 0.259124 = 0.24524499999999994.
0.24524499999999994 > 0.0001. So, error > precision. Let's repeat this step again.

Step 5:

Divide 0.007 by the previous result. d = 0.007/0.259124 = 0.027014.
Average this value (d) with that of step4: (0.027014 + 0.259124)/2 = 0.143069 (New guess).
Error = current value - previous value = 0.259124 - 0.143069 = 0.11605500000000002.
0.11605500000000002 > 0.0001. So, error > precision. Let's repeat this step again.

Step 6:

Divide 0.007 by the previous result. d = 0.007/0.143069 = 0.048927.
Average this value (d) with that of step5: (0.048927 + 0.143069)/2 = 0.095998 (New guess).
Error = current value - previous value = 0.143069 - 0.095998 = 0.047071.
0.047071 > 0.0001. So, error > precision. Let's repeat this step again.

Step 7:

Divide 0.007 by the previous result. d = 0.007/0.095998 = 0.072918.
Average this value (d) with that of step6: (0.072918 + 0.095998)/2 = 0.084458 (New guess).
Error = current value - previous value = 0.095998 - 0.084458 = 0.011539999999999995.
0.011539999999999995 > 0.0001. So, error > precision. Let's repeat this step again.

Step 8:

Divide 0.007 by the previous result. d = 0.007/0.084458 = 0.082881.
Average this value (d) with that of step7: (0.082881 + 0.084458)/2 = 0.08367 (New guess).
Error = current value - previous value = 0.084458 - 0.08367 = 0.0007880000000000109.
0.0007880000000000109 > 0.0001. So, error > precision. Let's repeat this step again.

Step 9:

Divide 0.007 by the previous result. d = 0.007/0.08367 = 0.083662.
Average this value (d) with that of step8: (0.083662 + 0.08367)/2 = 0.083666 (New guess).
Error = current value - previous value = 0.08367 - 0.083666 = 0.0000039999999999901226.
0.0000039999999999901226 <= 0.0001. So, error <= precision. We can now stop and use 0.083666 as the square root.

Finally, the square root of 0.007 is 0.083666 with an error smaller than 0.0001.


text:
Spearman's Rank Correlation Coefficient Calculator » payaenable.com

Spearman Rank-Order Correlation Coefficient. The logic and computational details of rank-order correlation are described in Subchapter 3b of Concepts and Applications. this calculator is not recommended for cases where the number of XY pairs is greater than about N=40. Spearman’s Rank correlation coefficient. A correlation can easily be drawn as a scatter graph, but the most precise way to compare several pairs of data is to use a statistical test - this establishes whether the correlation is really significant or if it could have been the result of chance alone.

Spearman's Rho Calculator. Spearman's Rho is a non-parametric test used to measure the strength of association between two variables, where the value r = 1 means a perfect positive correlation and the value r = -1 means a perfect negataive correlation. So, for example, you could use this test to find out whether people's height and shoe size are correlated they will be - the taller people are. More About this Spearman's Critical Correlation Calculator. Critical Values are used to be compared with a test statistic to assess whether or not the null hypothesis is rejected. In this case, Spearman's sample correlation \\rho\ will be compared with the critical correlation values \\rho_c\ found by this calculator. The calculation of Spearman’s correlation coefficient and subsequent significance testing of it requires the following data assumptions to hold: interval or ratio level or ordinal; monotonically related. Note, unlike Pearson’s correlation, there is no requirement of normality and hence it.

Spearman's Rank-Order Correlation. This guide will tell you when you should use Spearman's rank-order correlation to analyse your data, what assumptions you have to satisfy, how to calculate it, and how to report it. If you want to know how to run a Spearman correlation in SPSS Statistics, go to our Spearman's correlation in SPSS Statistics guide. 19.12.2014 · Spearman's Rank Correlation: Case of Tied Ranks Correlation Coefficient Quantitative Techniques in Management: How to find the Coefficient of Correlation also called the Correlation Coefficient. Spearman correlation coefficient: Definition. The Spearman’s rank coefficient of correlation is a nonparametric measure of rank correlation statistical dependence of ranking between two variables. Named after Charles Spearman, it is often denoted by the Greek letter ‘ρ’. The method of Spearman's rank correlation coefficient calculation is actually pretty simple. It's like the Pearson correlation coefficient, but designed not for measurements of random variables only but for their ranking values. That is. We have only to understand what is the rank.

Correlation Coefficients > Spearman Rank Correlation / Spearman’s Rho. What is Spearman Rank Correlation / Spearman’s Rho? The Spearman rank correlation coefficient, r s, is the nonparametric version of the Pearson correlation coefficient.Your data must be ordinal, interval or ratio.Spearman’s returns a value from -1 to 1, where. 24.04.2019 · Use the Spearman Rank Correlation Coefficient R to measure the relationship between two variables where one or both is not normally distributed. This.

Note. R-squared is always a positive number, hence the deduced Spearman rank correlation coefficient will also be always positive. To add an appropriate sign, just look at the line in your correlation graph - an upward slope indicates a positive correlation plus sign and a downward slope indicates a negative correlation minus sign. Spearmans Rank Correlation Coefficient Calculation. Spearmans Rank-Order Correlation Coeff. An online spearmans rank-order RHO calculation. Number of Datas. Spearman Coefficient Formula1 Spearmans Coefficient Formula2. Related.

This free online software calculator computes the Spearman Rank Correlation and the two-sided p-value H0: rho = 0. The ordinary scatterplot and the scatterplot between ranks of X & Y is also shown. Spearman’s Rank Order Correlation Coefficient • In this lesson, we will learn how to measure the coefficient of correlation for two sets of ranking. • The coefficient of correlation, r, measures the strength of association or correlation between two sets of data that can be. Hey, anyone here know if you can do spearman's rank coefficient on the calculator casio fx 9750gii I have tried everywhere online and the manual and I ca.

The result of Spearman's Rank-Order Correlation Coefficient Calculator. Now you can understand how the answer was arrived at. Spearman’s Rank Correlation Coefficient Definition: The Spearman’s Rank Correlation Coefficient is the non-parametric statistical measure used to study the strength of association between the two ranked variables. This method is applied to the ordinal set of numbers, which can be arranged in order, i.e. one after the other so that ranks can be given to each. The Formula for Spearman Rank Correlation $$r_R = 1 – \frac6\Sigma_i d_i^2nn^2 – 1$$ where n is the number of data points of the two variables and d i is the difference in the ranks of the i th element of each random variable considered. The Spearman correlation coefficient, ρ,.


text:
20 views
is grammar CFG =    S->AS/epsilon

A->aA/a

in Chomsky Nomal form ?
0
i think its not in chomsky normal form.

second prodn should be A->AA /a

A grammar is in Chomsky normal form if the productions are of the form:

$$A \rightarrow BC \;\text{or} \\ A \rightarrow a \; \text{or} \\ S \rightarrow \epsilon$$

where $A,B,C$ are non-terminal symbols and $S$ is the starting symbol and $a$ is a terminal.

In your grammar, due to the presence of $A \rightarrow aA$, it is not in CNF.

It can be converted into CNF by replacing $A \rightarrow aA$ as $A \rightarrow X, X \rightarrow a$.

+1 vote
1
2
+1 vote


text:
# Dan Sitaru's Cyclic Inequality in Three Variables VIII

### Solution

Define $O=(0,0),$ $A=(x,0),$ $\displaystyle B=\left(-\frac{y}{2},\frac{y\sqrt{3}}{2}\right),$ $\displaystyle C=\left(-\frac{z}{2},-\frac{z\sqrt{3}}{2}\right).$ Then in $\Delta ABC,$ $a=BC=\sqrt{y^2+yz+z^2},$ $b=AC=\sqrt{z^2+zx+x^2},$ $c=AB=\sqrt{x^2+xy+y^2},$ as $O$ appears to be Fermat's point of $\Delta ABC.$ Further, if $S=[\Delta ABC],$

$4S=4(\Delta OAB]+[\Delta OBC]+[\Delta OCA])=\sqrt{3}(xy+yz+zx).$

Hence, the required inequality reduces to $\displaystyle \frac{8Ss}{\sqrt{3}}\le 3abc$ which is equivalent to $\displaystyle s\le\frac{3R\sqrt{3}}{2},$ which is well known. $(s$ here is the semiperimeter $\displaystyle s=\frac{a+b+c}{2}$ and $R$ the circumradius of $\Delta ABC.)$

### Acknowledgment

Leo Giugiuc has kindly posted problem by Dan Sitaru at the CutTheKnotMath facebook page, with a solution of his. The problem appeared earlier at the Romanian Mathematical Magazine.


text:
# 8 (3 pts): The components of a vector are Az 10 m and 46u What angle does this vector make

###### Question:

8 (3 pts): The components of a vector are Az 10 m and 46u What angle does this vector make with the positive T axis? (6 pts):Threc vectors A, B. and ( have the following ' and " com ponents: 4 WB Lc component 6 53 4 2 W componem 33 44 5 Culenlate 4 ! B ! â‚¬. magnitndle a] direction (angle Abore Lhe Hi

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4. Recently the value of the U.S. dollar has been falling. This has made products produced in the United States less costly for foreigners. Is it possible for American suppliers to gain a comparative advantage in the production of certain goods simply because there has been a devaluation of the doll...
##### Question 16 Which of the following are determinants of the long-run supply of goods and services?...
Question 16 Which of the following are determinants of the long-run supply of goods and services? 1. Natural resources II. Technology III. The price level IV. The quantity of labor V. The money supply O a. I, II, III, IV, and v O b. I, II, and IV only OC. I, II, IV, and V only d. I and I only...
##### (8 = 2)(82 Mc 1 28 +10) 1;d 178 2s + 17)
(8 = 2)(82 Mc 1 28 +10) 1 ;d 178 2s + 17)...
##### Find the general solution of the differential equation y" _ 2y +y= % 14tUse C1, Cz, for the constants of integration_Enclose arguments of functions in parentheses For example, sin (2z)
Find the general solution of the differential equation y" _ 2y +y= % 14t Use C1, Cz, for the constants of integration_ Enclose arguments of functions in parentheses For example, sin (2z)...
##### 6F only please nas ne sam nection as - (+.-2,-1) (0) ns oppositely directed to v...
6F only please nas ne sam nection as - (+.-2,-1) (0) ns oppositely directed to v = (4,-2,-1) 6. Let u (-3,1.2), v (4. 0,-8), and w (6,-1. -4). Find the components of (a) v- (b) 6u + 2v (c) -vu (d) 5(v-4u) (e) -3( -8w) (f) ( 7. Lct u, v, and w be the vectors in Exercise 6 Find the comnanente oFtha...
##### THIS ASSIGNMENT CONTRIBUTES TO YOUR COURSE MARK. Given the forces P = 2.4 kN and Q = 4.6 kN and the height L = 12 m: THIS ASSIGNMENT CONTRIBUTES TO YOUR COURSE MARK Q KN L m P kN P kN Given the forces...
THIS ASSIGNMENT CONTRIBUTES TO YOUR COURSE MARK. Given the forces P = 2.4 kN and Q = 4.6 kN and the height L = 12 m: THIS ASSIGNMENT CONTRIBUTES TO YOUR COURSE MARK Q KN L m P kN P kN Given the forces P = 2.4 kN and Q = 4.6 kN and the height L = 12 m: 4 m 8 m 8 m 4 m How many zero force members are ...
##### Suppose the correlation coefficient between the rates of return on ABC Mutual Fund and the market...
Suppose the correlation coefficient between the rates of return on ABC Mutual Fund and the market portfolio is 0.6. The standard deviations of the rates return are 0.30 for ABC and 0.20 for the market portfolio. How would you combine the fund and the riskless asset to obtain a portfolio with a relat...
##### Just some questions i need answers to Part 1: Multiple choice Highlight the correct answer in...
Just some questions i need answers to Part 1: Multiple choice Highlight the correct answer in colour. Max: 5 points 1. 'Runoff ratio' is the proportion of precipitation that appears as runoff a. True b. False s by which water 'soaks' into the soil and moves vertically downwards?...
##### Examples of civility and incivility
examples of civility and incivility...
##### Volatility of a liquid increases as temperature increasesbecauseGroup of answer choicesLondon forces increase as temperature increasesthe molecules of the liquid decompose into lightermolecules.the number of molecules with enough kinetic energy to escapefrom the liquid increases.the kinetic energy of the molecules decreases.an equilibrium between liquid and vapor is established
Volatility of a liquid increases as temperature increases because Group of answer choices London forces increase as temperature increases the molecules of the liquid decompose into lighter molecules. the number of molecules with enough kinetic energy to escape from the liquid increases. the kinetic ...
##### Find the area between the curves_X= -2,X=3,y=2x2+9,y=0The area between the curves is (Round to the nearest whole number as needed:)
Find the area between the curves_ X= -2,X=3,y=2x2+9,y=0 The area between the curves is (Round to the nearest whole number as needed:)...
##### The half-life for the $\alpha$ decay of uranium ${ }_{92}^{238} \mathrm{U}$ is $4.47 \times 10^{9} \mathrm{yr}$. Determine the age (in years) of a rock specimen that contains $60.0 \%$ of its original number of ${ }_{92}^{238} \mathrm{U}$ atoms.
The half-life for the $\alpha$ decay of uranium ${ }_{92}^{238} \mathrm{U}$ is $4.47 \times 10^{9} \mathrm{yr}$. Determine the age (in years) of a rock specimen that contains $60.0 \%$ of its original number of ${ }_{92}^{238} \mathrm{U}$ atoms....
##### 10. Aniline (MW 93 g/mol} Co-distils with water at 98.2FC The vapour pressure of water at 98.28â‚¬ 712 mmllg: You need t0 isolate 10G ol_aniline: What the minimmum volume onWater required t0 isolate LQ gofaniline in steam distlllation a5760 mmHg? Rotae 76ommty Qcrehtt 4876t Py = n rt Ratr 712 m Ha manilw nanclill= lmel (Haalint - 933 /al Manlutc Peyu Vuakr nuttr Pbstr Rater nanalnL Rakr Potal Penalunt nsettr Parun~ Denb- Rater Ranelte 0, Il mol 7/2mm nisakr 76ommld Tlmlla - PJim 48 mm' ra
10. Aniline (MW 93 g/mol} Co-distils with water at 98.2FC The vapour pressure of water at 98.28â‚¬ 712 mmllg: You need t0 isolate 10G ol_aniline: What the minimmum volume onWater required t0 isolate LQ gofaniline in steam distlllation a5760 mmHg? Rotae 76ommty Qcrehtt 4876t Py = n rt Ratr 712 m ...
##### The graph of two functions andillustrated below: Use the graph to answer parts (a) through (0).(e) (f+913)- D (Simplify your answer:)
The graph of two functions and illustrated below: Use the graph to answer parts (a) through (0). (e) (f+913)- D (Simplify your answer:)...
##### Pls make reference to the graph in your explanation. Alpha is the only company that produces...
pls make reference to the graph in your explanation. Alpha is the only company that produces smartphones in a city. The diagram below depicts the demand and the cost function of producing smartphones by Alpha. Price $1000$350 $180$100 ATC MR. t o Quantity 40 90 160 180 How would Alpha sustain the ...
##### Explain the roles of phosphatidylinositol, small monomeric GTPases, phospholipase C, and PI3K in the determination of...
Explain the roles of phosphatidylinositol, small monomeric GTPases, phospholipase C, and PI3K in the determination of polar movement of a mammalian cell. Specifically, note how phospholipid gradients are established, where you might find Cdc42, RhoA, and Rac1 during polarized cell movement (e.g. in ...
##### In a lecture demonstration, a professor pulls apart two hemispherical steel shells (diameter $D )$ with ease using their attached handles. She then places them together, pumps out the air to an absolute pressure of $p,$ and hands them to a bodybuilder in the back row to pull apart. (a) If atmospheric pressure is $p_{0},$ how much force must the bodybuilder exert on each shell? (b) Evaluate your answer for the case $p=0.025$ atm, $D=10.0 \mathrm{cm} .$
In a lecture demonstration, a professor pulls apart two hemispherical steel shells (diameter $D )$ with ease using their attached handles. She then places them together, pumps out the air to an absolute pressure of $p,$ and hands them to a bodybuilder in the back row to pull apart. (a) If atmospheri...
##### Question 48 1. Trace the path of sound from the outside of the ear to stimulating...
Question 48 1. Trace the path of sound from the outside of the ear to stimulating cranial nerve VIL 2. Describe the auditory pathway from stimulation of cranial nerve Vill to the brain HTML Editor -LE35 BIVA % + 2 y = x W 12pt o here to search...
##### Problem 6-29 Simple Interest versus Compound Interest (L04) First Simple Bank pays 8.3 percent simple interest...
Problem 6-29 Simple Interest versus Compound Interest (L04) First Simple Bank pays 8.3 percent simple interest on its investment accounts. If First Complex Bank pays interest on its accounts compounded annually, what rate should the bank set if it wants to match First Simple Bank over an investment ...
##### How are they finding the total change in the oxidation number?
I can not understand the part in the box. How can one H_{2}0_{2} give 2H_{2}O and 2H^{+} . Is this a standard equation and do I have to memorise this equation?...
##### 0.009lcragularly shaped solid density of the object with mass 0l 75 4 metal? 40ml AonL J5m 35mlgraduated30 ml30 ml20 ml45 ml15 MLIoml10 mL2.91 gcm' 2.906 glem'782 g cm' 0.344 glcm"Which of the following is derived SI unit?the highest point in Arizona is 12,600 The summit of Mt: Humphreys 1.0936 yd) 34,565 m 4593 m 3841 m 1096 m Dalton atomic theory of matter did not include which of the following Each element is made of a number of isotopes_ some of which are Each element is
0.009 lcragularly shaped solid density of the object with mass 0l 75 4 metal? 40ml AonL J5m 35ml graduated 30 ml 30 ml 20 ml 45 ml 15 ML Ioml 10 mL 2.91 gcm' 2.906 glem' 782 g cm' 0.344 glcm" Which of the following is derived SI unit? the highest point in Arizona is 12,600 The su...
##### Find the average rate of change of the function in the graph shown below between_land x = 132 3 7 *
Find the average rate of change of the function in the graph shown below between _land x = 13 2 3 7 *...
##### Current Attempt in ProgressAcircular curveof highway is designed for traffic movingat 66 km/h Assume the traffic consists of cars without negative lift (a) If the radius of the curve is 180m whatis the correct angle of banking of the road? (b) If the curve were not banked; what would be the. minimum coefficient of friction between tires and road that would keep traffic trom skidding out of the turn when traveling at 66 kmh?NumberUnits(b) NumberUnits
Current Attempt in Progress Acircular curveof highway is designed for traffic movingat 66 km/h Assume the traffic consists of cars without negative lift (a) If the radius of the curve is 180m whatis the correct angle of banking of the road? (b) If the curve were not banked; what would be the. minimu...
##### How do you use the product rule to find the derivative of y=x*ln(x) ?
How do you use the product rule to find the derivative of y=x*ln(x) ?...
##### Calculate the mass/volume percent (m/v) of 16.0 g NaCl in 70.5 mL of solution.
Calculate the mass/volume percent (m/v) of 16.0 g NaCl in 70.5 mL of solution....
##### Determine whether the (ollowing integral converges or diverges: ifit converges, tind thee value: a. [%-"'b L dx G+I)-
Determine whether the (ollowing integral converges or diverges: ifit converges, tind thee value: a. [%-"' b L dx G+I)-...
##### 1 Sea solucioflus) usando Ia una V Hi orden. diferencial3Sy=Encuentre
1 Sea solucioflus) usando Ia una V Hi orden. diferencial 3 Sy= Encuentre...
##### [10 Points] How much work is done by Goku as he launches & 114kg refrlgerator initially at rest to a speed of 100 mph? 5,70 x 105] 222x 105 ] 9001 k LAx 105]
[10 Points] How much work is done by Goku as he launches & 114kg refrlgerator initially at rest to a speed of 100 mph? 5,70 x 105] 222x 105 ] 9001 k LAx 105]...
##### Explain the mistake and work the problem correctly. $\begin{array}{l}[12-2 \cdot 3]+4(3)^{2} \\ \quad=[12-6]+4(3)^{2} \\ \quad=6+4(3)^{2} \\ \quad=6+144 \\ \quad=150\end{array}$
Explain the mistake and work the problem correctly. $\begin{array}{l}[12-2 \cdot 3]+4(3)^{2} \\ \quad=[12-6]+4(3)^{2} \\ \quad=6+4(3)^{2} \\ \quad=6+144 \\ \quad=150\end{array}$...
##### Suppose that X and Y are independent and X Exp(A) and Y Geom(p) , where A > 0 and p â‚¬ (0,1). (Recall that X has has an exponential distribution if it's cumulative distribution function is Fx(c) = 0 if x < 0 and Fx(c) = 1 - e Az jf â‚¬ 20). Let [x] be the floor function (i.e. "round down ) Find P(LX] = Y)
Suppose that X and Y are independent and X Exp(A) and Y Geom(p) , where A > 0 and p â‚¬ (0,1). (Recall that X has has an exponential distribution if it's cumulative distribution function is Fx(c) = 0 if x < 0 and Fx(c) = 1 - e Az jf â‚¬ 20). Let [x] be the floor function (i.e. &qu...
##### At one instant, the electric and magnetic fields at one point of an electromagnetic wave are...
At one instant, the electric and magnetic fields at one point of an electromagnetic wave are E = (200i + 310 j – 50k)V/m and B = (7.2î – 7.0j + ak)Bo. Part A What is the value of a? Express your answer using two significant figures. IVO AXO o 2 ? Submit Request Answer Part B What i...
##### Pacticle Wa nlong elliptic helix -space andthe porition function the particle time given by 7 () Ecoat Jsint ] 3k Khere I-, and =-coordinate meterd Ad in Reconds. (20 pte) Is tlue prticle moing downward? If+o hen? Whet dogs the particle rench 12 metera above the pround? Find the veloclty; "1 Wid acekration o t partkk te tme Then I 42 meters nbove the Kollle| 12 meteTE abne the ground_ kave - the helux n When the purtick Mott Nonz Fiud tho pitiou (1,0 ) o the purtic ke Atok altet Lthc (l
pacticle Wa nlong elliptic helix -space andthe porition function the particle time given by 7 () Ecoat Jsint ] 3k Khere I-, and =-coordinate meterd Ad in Reconds. (20 pte) Is tlue prticle moing downward? If+o hen? Whet dogs the particle rench 12 metera above the pround? Find the veloclty; "1 ...


text:
# A number theory problem by dipanjan mitra

Number Theory Level pending

3^2002+7^2002+2002 is divided by 29 .The remainder is

×


text:
# Electronics Letters

## Filter Results

Displaying Results 1 - 25 of 44
• ### In brief

Publication Year: 2010, Page(s): 666
| | PDF (156 KB)

First Page of the Article
View full abstract»

• ### Interview

Publication Year: 2010, Page(s): 666
| | PDF (118 KB)

Dr Xiaobin Yuan from the Microelectronics Division of IBM and author of the Letter 'Transistor mismatch in 32 nm high-k metal-gate process' on page 708 tells us more about his research into advanced process technologies. View full abstract»

• ### Strength in numbers

Publication Year: 2010, Page(s): 667
| | PDF (174 KB)

The first coherent vertical cavity surface-emitting laser (VCSEL) array with the in-phase mode using a simple fabrication process has been demonstrated at the University of Illinois at Urbana-Champaign in the USA. The 2 Ã 2 VCSEL array was made using conventional fabrication processes that are already in use to manufacture millions of VCSELs, and photonic crystal coupling was used to control the ... View full abstract»

• ### A fluid approach to microwaves

Publication Year: 2010, Page(s): 668
| | PDF (492 KB)

Collaborative researchers from the University of Cambridge in the UK and Technische UniversitÃ¿t Darmstadt (TUD) in Germany have signifi cantly increased the dielectric anisotropy of liquid crystals in the microwave range by adding minute amounts of carbon nanotubes. This result is a signifi cant step for the use of liquid crystals as a tunable material in a wide range of microwave devices at freq... View full abstract»

• ### Planar multi-band monopole antenna with L-shaped parasitic strip for WiMAX application

Publication Year: 2010, Page(s):671 - 672
Cited by:  Papers (15)
| | PDF (291 KB)

A multi-band design of a planar inverted L-shaped monopole antenna with an L-shaped parasitic strip for the WiMAX system is proposed. With the use of L-shaped monopole and parasitic strips, multi-resonant modes close to the 2.6/3.5/5.5 GHz bands are excited to meet the specifications of the WiMAX system (IEEE 802.16e). Also, obtained impedance bandwidth across the operating bands can reach about 2... View full abstract»

• ### Small square slot antenna with circular polarisation characteristics for WLAN/WiMAX applications

Publication Year: 2010, Page(s):672 - 673
Cited by:  Papers (8)
| | PDF (338 KB)

A small square slot antenna with circular polarisation characteristics for WLAN/WiMAX applications is presented. The proposed antenna consists of a microstrip feed line and a ground plane with a novel slot configuration which provides usable circular polarisation characteristics on the WLAN/WiMAX frequency bands. By optimising the length of the feed line, the total bandwidth of the antenna is grea... View full abstract»

• ### Spiraphase-type element with optimal transformation of switch impedances

Publication Year: 2010, Page(s):673 - 675
Cited by:  Papers (4)
| | PDF (264 KB)

An electronically-reconfigurable spiraphase-type element based on a ring slot resonator is analysed. It is demonstrated with numerical simulation and experimental measurements that optimal transformation of the switch impedances is possible in this element. As a result, low loss of 0.38 dB is obtained for the one-bit element at the frequency of 11.55 GHz. View full abstract»

• ### Wideband cavity backed slot antenna for automotive monopulse radars

Publication Year: 2010, Page(s):675 - 677
| | PDF (195 KB)

A wideband cavity backed slot antenna is designed for automotive-borne monopulse radar applications. A rectangular slot is used to realise broad bandwidth; the slot is backed by a cavity, and the vertical walls of the cavity are implemented by discrete vias; and the slot is fed by a microstrip line. The proposed antenna is compact, robust, fully planar and easy to integrate. A two-element array is... View full abstract»

• ### Cognitive memory network

Publication Year: 2010, Page(s):677 - 678
Cited by:  Papers (8)
| | PDF (237 KB)

A resistive memory network that has no crossover wiring is proposed to overcome the hardware limitations to size and functional complexity that is associated with conventional analogue neural networks. The proposed memory network is based on simple network cells that are arranged in a hierarchical modular architecture. Cognitive functionality of this network is demonstrated by an example of charac... View full abstract»

• ### Development of capacitance measurement system for human serum albumin detection

Publication Year: 2010, Page(s):678 - 680
| | PDF (257 KB)

The use of interdigitated electrodes and binding with Cibacron blue F3GA by self-assembly technology, which has high affinity for human serum albumin (HSA), and serves as a receptor for HSA detection, is reported. In preliminary tests, the interdigitated chip measured the impedance of HSA by electrochemical impedance spectroscopy and this result indicated the characteristic frequency of impedance ... View full abstract»

• ### Multiscale principal contour direction for varying lighting face recognition

Publication Year: 2010, Page(s):680 - 682
| | PDF (236 KB)

Proposed is a new type of image feature for varying lighting face recognition, namely, multiscale principal contour direction (MPCD). It can capture true neighbourhood contour direction information and is robust to different illumination conditions. In addition, MPCD is constructed for extracting illumination invariant by applying the non-subsampled contourlet transform (NSCT) to multiscale cont... View full abstract»

• ### Regularisation of eigenfeatures by extrapolation of scatter-matrix in face-recognition problem

Publication Year: 2010, Page(s):682 - 683
| | PDF (91 KB)

The proposed face recognition algorithm regularises the within-class scatter matrix by extrapolating its eigenvalues. This way both the range space and the null space information of the within-class scatter matrix is utilised. The results are promising when the algorithm is experimented on face recognition datasets and compared with several other methods. View full abstract»

• ### Detector for signals in generalised normal-Laplace distributed noise

Publication Year: 2010, Page(s):683 - 684
| | PDF (211 KB)

A novel near-optimal receiver is introduced for the detection of signals in noise modelled with a new probability distribution called a generalised normal-Laplace (GNL) distribution. The near-optimal detector is based on the approximation of GNL probability density function (PDF) while the optimum detector uses the PDF obtained by the Fourier inversion of the characteristic function. The performan... View full abstract»

• ### Load balancing for two-tier Chord system

Publication Year: 2010, Page(s):685 - 686
| | PDF (301 KB)

Compared to flat distributed hash table (DHT) systems, hierarchical DHT systems can use some powerful and stable peers (called superpeers) to achieve efficient look-up under churn. Several hierarchical DHT systems have been proposed so far. However, a crucial problem faced by all these systems is the load imbalance among superpeers. Proposed is a novel load balancing algorithm to efficiently solve... View full abstract»

• ### Frequency stability measurements of ultrastable BVA resonators and oscillators

Publication Year: 2010, Page(s):686 - 688
Cited by:  Papers (5)
| | PDF (239 KB)

The performances of instruments developed for the measurement of 5 MHz ultra-stable resonators, before and after being integrated in an oscillator, are presented. The chosen resonator has an estimated stability of several 10-14. The intrinsic noise floor of these benches are about 7 to 9.5 10-15. View full abstract»

• ### Compact dual-band power divider with single allpass coupled lines sections

Publication Year: 2010, Page(s):688 - 689
Cited by:  Papers (12)
| | PDF (271 KB)

A compact dual-band power divider is presented. An allpass coupled lines section is used to realise the necessary impedance transformation at two selected frequencies. The design procedure, limitations and examples are provided first. To implement the design in microstrip, stepped-impedance coupled lines are used without changing the circuit size to compensate for the different even- and odd-mode ... View full abstract»

• ### Compact microstrip lowpass filter with ultra-wide stopband

Publication Year: 2010, Page(s):689 - 691
Cited by:  Papers (3)
| | PDF (244 KB)

A compact microstrip lowpass filter with an ultra-wide stopband is proposed. To achieve wide stopband suppression and compact size, both symmetrically loaded resonant patches and meander transmission line are adopted. A demonstration filter with 3 dB cutoff frequency at 1.3 GHz has been designed, fabricated and measured. The proposed filter also exhibits a very wide stopband of 152 , and is able t... View full abstract»

• ### Quad-band bandpass filter based on dual-plane microstrip/DGS slot structure

Publication Year: 2010, Page(s):691 - 692
Cited by:  Papers (17)
| | PDF (416 KB)

An approach for designing a quad-band bandpass filter is presented. The dual-plane structure composed of both a microstrip and a defected ground structure (DGS) slot is employed to obtain quad-frequency responses. The slot, with interdigital shape in the DGS, is designed to obtain dual-frequency responses and transmission zeros are introduced to split both passbands into two. The equivalent-circui... View full abstract»

• ### UWB bandpass filter with notched band using DSRR

Publication Year: 2010, Page(s):692 - 693
Cited by:  Papers (5)
| | PDF (358 KB)

A topology of a compact planar microstrip filter with a notched band is introduced, which employs a split-ring resonator defected ground structure (DSRR). Folded short-circuited stubs are used to realise the bandpass filter with good selectivity and out-of-band characteristics. By employing a split-ring resonator defected ground structure, a narrow notched band was introduced to reject the undesir... View full abstract»

• ### Dielectric anisotropy of nematic liquid crystals loaded with carbon nanotubes in microwave range

Publication Year: 2010, Page(s):693 - 695
| | PDF (163 KB)

Liquid crystals are attractive materials for microwave applications as tunable dielectrics owing to low losses and high anisotropy of dielectric properties. The possibility of further enhancing their dielectric anisotropy is studied by loading with highly polarisable and anisotropic rods-carbon nanotubes at various concentrations. The studies are performed using two different methods, one in the r... View full abstract»

• ### Improved thermal model for optical fibre coating owing to small bending diameter and high power signals

Publication Year: 2010, Page(s):695 - 697
Cited by:  Papers (4)
| | PDF (147 KB)

Long-term optical network reliability is highly dependent on the fibre quality. However, poor handling or installation can impose bendings in the optical fibres, leading to a significant increase in link losses, which results in fibre coating degradation owing to heating occurring in the presence of high power signals propagation. Presented, for the first time, is an accurate thermal model for opt... View full abstract»

• ### InGaAsP/InP optical digital-to-analogue converter for 40 Gbit/s, 3-bit operation

Publication Year: 2010, Page(s):697 - 698
Cited by:  Papers (1)
| | PDF (433 KB)

A 40 Gbit/s, 3-bit operation of a semiconductor optical digital-to-analogue converter, consisting of inputand output-side 1 ?? 3 MMI couplers and 1and 2-bit optical delay lines based on InGaAsP/InP materials, is presented. Splitting the ratio of an input-side 1 ?? 3 MMI coupler is designed to have almost equal signal quality in each output level under the extinction ratio of the input signal of 23... View full abstract»

• ### Laser phase noise induced error-rate floors in differential n-level phase-shift-keying coherent receivers

Publication Year: 2010, Page(s):698 - 700
| | PDF (111 KB)

A simple and illustrative analytical model for the phase noise influence in differential n-level phase shift keying (nPSK) systems is presented. Results for the phase noise sensitivity are given for a dual polarisation system configuration with basic baud-rate of 25 GS/s. Results emphasise the need for highly phase stable optical signal sources in higher constellation configurations as needed for ... View full abstract»

• ### Phase noise characteristics of all-optical single sideband frequency upconverter for radio-over-fibre applications

Publication Year: 2010, Page(s):700 - 701
Cited by:  Papers (2)
| | PDF (239 KB)

The phase noise characteristics of an all-optical single sideband frequency upconverter that uses an optical interleaver and a semiconductor optical amplifier are experimentally investigated. While the phase noise characteristic of the upconverter is limited by that of the LO signal source, it becomes degraded owing to the path imbalance of the two optical signal paths. Compensation for the path i... View full abstract»

• ### Running-code optical CDMA at 2 x 10 Gbit/s and 40 Gbit/s

Publication Year: 2010, Page(s):701 - 703
Cited by:  Papers (1)
| | PDF (342 KB)

A continuously running-code optical CDMA system useful for secure communications is reported. The system utilises acousto-optic modulator-based pulse shapers as spectral phase en-/decoding devices, which are capable of microsecond-scale code sequence refresh rates. The codes are generated on-the-fly by a stream cipher. Transmit/ receive code synchronisation is independent of the data stream, allow... View full abstract»

## Aims & Scope

Electronics Letters is internationally renowned for its rapid communication of new developments and emerging topics across the broad and interdisciplinary field of modern electronics and electrical engineering.

Full Aims & Scope

## Meet Our Editors

Editors-in-Chief

Professor Chris Toumazou
Director of the Biomedical Institute
Imperial College, London, UK

Professor Ian White