text:
Robust Directional Modulation Design for Secrecy Rate Maximization in Multi-User Networks

In this paper, based on directional modulation (DM), robust beamforming matrix design for sum secrecy rate maximization is investigated in multi-user systems. The base station (BS) is assumed to have the imperfect knowledge of the direction angle toward each eavesdropper, with the estimation error following the Von Mises distribution. To this end, a Von Mises distribution-Sum Secrecy Rate Maximization (VMD-SSRM) method is proposed to maximize the sum secrecy rate by employing semi-definite relaxation and first-order approximation based on Taylor expansion to solve the optimization problem. Then in order to optimize the sum secrecy rate in the case of the worst estimation error of direction angle toward each eavesdropper, we propose a maximum angle estimation error-SSRM (MAEE-SSRM) method. The optimization problem is constructed based on the upper and lower bounds of the estimated eavesdropping channel related coefficient and then solved by the change of the variable method. Simulation results show that our two proposed methods have better sum secrecy rate than zero-forcing (ZF) method and signal-to-leakage-and-noise ratio (SLNR) method. Furthermore, the sum secrecy rate performance of our VMD-SSRM method is better than that of our MAEE-SSRM method.

Authors

• 4 publications
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• 10 publications
• Robust Beamforming Design for Sum Secrecy Rate Optimization in MU-MISO Networks

This paper studies the beamforming design problem of a multi-user downli...
03/24/2020 ∙ by Pu Zhao, et al. ∙ 0

• Secure SWIPT for Directional Modulation Aided AF Relaying Networks

Secure wireless information and power transfer based on directional modu...
03/14/2018 ∙ by Xiaobo Zhou, et al. ∙ 0

• Secrecy Rate Maximization in Multi-IRS Millimeter Wave Networks

In this paper, the problem of physical layer security enhancement in a m...
10/02/2020 ∙ by Anahid Rafieifar, et al. ∙ 0

• Sum Secrecy Rate Maximization in a Multi-Carrier MIMO Wiretap Channel with Full-Duplex Jamming

In this paper we address a worst-case weighted sum rate maximization pro...
02/14/2018 ∙ by Tianyu Yang, et al. ∙ 0

• Adaptive Robust Null-Space Projection Beamforming Scheme for Secure Wireless Transmission with AN-aided Directional Modulation

In this paper, an adaptive robust secure null-space projection (NSP) syn...
12/06/2017 ∙ by Feng Shu, et al. ∙ 0

• Single cell and multi-cell performance analysis of OFDM index modulation

This paper addresses the achievable rate of single cell and sum rate of ...
03/07/2018 ∙ by Shangbin Wu, et al. ∙ 0

• Comments on "Precoding and Artificial Noise Design for Cognitive MIMOME Wiretap Channels"

Several gaps and errors in [1] are identified and corrected. While accom...
10/24/2020 ∙ by Mahdi Khojastehnia, et al. ∙ 0

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I Introduction

With the explosive growth of the mobile Internet, wireless communications have played an increasingly important role in daily life [1, 2]. However, due to the inherent broadcasting nature of the wireless communications, the transmit signals are easily wiretapped by the unauthorized receivers. Therefore, wireless communications are facing the serious privacy and security problems. Traditional communication methods based on the cryptographic techniques need an additional secure channel for private key exchanging, which may be insufficient with the development of the mobile Internet [3, 4, 5]. Recently, physical layer security exploiting the characteristics of the wireless channels can realize the secure wireless communications without the upper layer data encryption, being identified as a significant complement to cryptographic techniques [6, 7, 8, 9].

Directional modulation (DM), which is regarded as one of the promising physical layer wireless security techniques, has attracted wide attentions in recent years [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The main ideal of DM is to project the confidential information signals into the desired spatial directions while simultaneously distorting the constellation of the signals in the other directions [10]. In [11], the authors proposed a DM method which employed the near-field direct antenna modulation technique to overcome the security challenges. A similar simplified method of DM synthesis was proposed in [12] for a far-field scenario. With the development of the artificial noise (AN) aided technique, DM has been further developed. The AN is added to the transmit signals expecting that the AN would interfere with the eavesdroppers and not affect the legitimate receivers. In [13]

, a novel baseband signal synthesis method was proposed to construct the AN signal based on the null space of the channel vector in the desired direction. An orthogonal projection (OP) method in

[10] was designed for the multi-beam DM systems. And in [14], the authors proposed a multiple-direction DM transmission scheme which employed the space domain to transmit the multiple data streams independently and increased the capacity of the system.

All the methods mentioned above assumed that the transmitter knows the precise direction angles toward all users (including the eavesdroppers). However, in practical scenarios, the perfect direction angles are hard to obtain because of the estimation errors induced by the widely-used estimation algorithms such as multiple signal classification (MUSIC) and Capon. So in [21]

the authors considered a single-user DM system where the estimation error of the direction angle toward the desired receiver was assumed to follow the uniform distribution.

[22]

considered the imperfect desired direction angles in a multi-beam DM scenario, and assumed the desired direction angle estimation errors follow the truncated Gaussian distribution.

Compared to the direction angles of the desired receivers, the precise direction angles toward the eavesdroppers are more difficult to be obtained as they rarely expose their accurate locations. In this paper, we consider the imperfect direction angles toward eavesdroppers. Moreover, since the Von Mises distribution is regarded as the best statistical model for the direction angles, we assume that the angle estimation error follows the Von Mises distribution [23]. Meanwhile, the transmit beamforming technique and the AN-aided technique are also employed to realize the secure wireless communications. The main contributions of this paper are summarized as follows:

1) We propose a Von Mises distribution-Sum Secrecy Rate Maximization (VMD-SSRM) robust DM scheme which designs the robust signal beamforming matrix and AN beamforming matrix on the assumption that the estimation errors of direction angles toward the eavesdroppers follow the Von Mises distribution. First, the expectation of the estimated eavesdropping channel related coefficient is derived. Then the system sum secrecy rate maximization problem subject to the total transmit power of the base station (BS) is formulated. Comprising the logarithms of the product of fractional quadratic functions, the sum secrecy rate expression is non-convex and difficult to tackle. In order to solve this problem, we employ semi-definite relaxation and first-order approximation based on Taylor expansion to transform the original problem into a convex problem.

2) In order to optimize the system sum secrecy rate in the case of the worst estimation error of direction angle toward each eavesdropper, we propose a maximum angle estimation error-SSRM (MAEE-SSRM) method. To design the robust signal beamforming matrix and AN beamforming matrix, first the upper and lower bounds of the estimated eavesdropping channel related coefficient are derived, then the system sum secrecy rate maximization problem based on the derived upper and lower bounds is constructed. Since the optimization problem is still non-convex, we use the change of the variable method to convert it into a convex problem and then solve it by convex optimization tools.

3) Through simulation and evaluation, we investigate the performance of our two proposed methods. We compare our methods with some typical DM methods such as ZF method [24] and SLNR method [25]. Simulation results show that our two proposed robust methods have much better sum secrecy rate than ZF method and SLNR method. Meanwhile the performance of our VMD-SSRM method is better than that of MAEE-SSRM method.

The rest of this paper is organized as follows. Section II introduces the channel model and the system model. In Section III, we design the information beamforming matrix and AN beamforming matrix to maximize the worst-case system sum secrecy rate with the VMD-SSRM method. In Section IV, we consider the maximum angle estimation error of the eavesdropper and design the information beamforming matrix and AN beamforming matrix to maximize the worst-case system sum secrecy rate with the MAEE-SSRM method. Section V provides the simulation results to validate the effectiveness of the two proposed methods as well as the complexity of our two proposed methods, and followed by our conclusions in Section VI.

Notations: In this paper, we use boldface lowercase and uppercase letters to denote the vectors and matrices, respectively. , , , and denote the transpose, the conjugate transpose, the trace, the rank and the real part of the matrix A, respectively. denotes that A is a positive semi-definite matrix. and denote the module of the scalars and the Frobenius norm of the matrices. , , and denote the statistical expectation, the logarithm to base 2, the logarithm to base 10 and the natural logarithm, respectively. denotes the inner product of the vector x and y. denotes the set of all complex matrices. denotes an identity matrix.

denotes a circularly symmetric complex Gaussian random variable

with zero mean and variance

.

Ii system model and problem formulation

Ii-a Channel Model

In this paper, the DM transmitter is assumed to be equipped with uniform isotropic array. The array is composed of elements with an adjacent distance of , while the direction angle between the transmit antenna array and the receiver is denoted by . The normalized steering vector for the transmit antenna array is expressed as

 ^h(θTR)=1√N[ej2πΨθTR(1),ej2πΨθTR(2),⋯,ej2πΨθTR(N)]T, (1)

where is

 ΨθTR(n)=−(n−(N+1)/2)lcos(θTR)λ,n=1,2,⋯,N, (2)

where is the wavelength of the transmit signal radio.

In the line of sight (LOS) communication scenarios, e.g., unmanned aerial vehicles (UAV) communication and suburban mobile communication, it is widely accepted to use free-space path loss model [26], given by

 PL(d)=10log10(4πdfc)2, (3)

where is the path loss in dB at a given distance , is the frequency of the transmit signal radio and is the radio velocity. Then according to (1) and (3), the channel model between the transmit antenna array and the receiver can be expressed as

 h(θTR)=√(c4πdTRf)2×^h(θTR) (4) =√gTR1√N[ej2πΨθTR(1),ej2πΨθTR(2),⋯,ej2πΨθTR(N)]T,

where is the distance between the transmit antenna array and the receiver, and denotes the path loss between the transmit antenna array and the receiver.

Ii-B Multi-User System Model with Multiple Eavesdroppers

In this paper, we consider a flat fading multiuser multi-input single-output (MU-MISO) downlink system as shown in Fig. 1. The system comprises a BS with transmit antennas, () single-antenna destination users (), and single-antenna eavesdroppers (), wishing to wiretap the confidential information sent from the BS to the destination users. The total transmit power of the BS is , and the direction angle between the BS and each destination user is (). The transmit beamforming technique is employed by the BS to steer the information signal beams toward the destination users. The direction angle between the BS and each eavesdropper is (). In order to decode the confidential messages sent from the BS to all destination users, the eavesdroppers are assumed to locate closer to the BS than the destination users. And in order to interfere with the eavesdroppers, the BS employs the beamforming technique to steer () AN beams toward the eavesdroppers. These AN beams are transmitted along with the information signal beams, thus the received information signal qualities of the eavesdroppers are degraded. Let denote the signal vector formed by the independent data streams sent from the BS to the destination users with . Let and denote the information beamforming matrix and the AN beamforming matrix, respectively. The total transmit signals of the BS can be expressed as

 x=M∑m=1wmxm+L∑l=1qlzl, (5)

where is the AN signal sent by the BS. Then the received signal at the -th destination user is given by

 yDi =hH(θBDi)x+nDi (6) =hH(θBDi)wixi+M∑m=1m≠ihH(θBDi)wmxm +L∑l=1hH(θBDi)qlzl+nDi,

where the first term of the right hand side of (6) is the useful signal received by , the second one is the multiuser interference from other destination users, the third one is the AN, and the last one is the additive white Gaussian noise (AWGN) with .

Similarly, the signal received at the -th eavesdropper is

 yEk =hH(θBEk)x+nEk (7) =hH(θBEk)wixi+M∑m=1m≠ihH(θBEk)wmxm +L∑l=1hH(θBEk)qlzl+nEk,

where the right hand side of (7) is composed of the -th confidential message intercepted by , the messages sent to the other (-1) destination users that are intercepted by , the AN to disturb and the AWGN satisfying .

It should be noted that we consider the non-colluding eavesdropping scenario where each eavesdropper relies on itself to overhear and decode the messages and there is no information exchange between the eavesdroppers.

In this paper, the worst-case system sum secrecy rate is used to evaluate the system performance in the multiuser scenario, which can be expressed as

 Rs =M∑i=1mink Rki (8) =M∑i=1mink (log(1+SINRDi)−log(1+SINREk))

where denotes the system sum secrecy rate, and denote the signal-to-interference-plus-noise ratio (SINR) at the destination user and the eavesdropper , respectively.

Then, according to (6), the SINR at the -th destination user is

 SINRDi=|hH(θBDi)wi|2M∑m=1m≠i|hH(θBDi)wm|2+L∑l=1|hH(θBDi)ql|2+σ2D. (9)

And according to (7), the SINR at the -th eavesdropper intending to intercept the -th destination user is

 SINREk=|hH(θBEk)wi|2M∑m=1m≠i|hH(θBEk)wm|2+L∑l=1|hH(θBEk)ql|2+σ2E. (10)

Now, according to (8), (9) and (10), the optimization problem that maximizes the worst-case system sum secrecy rate under the constraint of the total transmit power of the BS is constructed as shown in (II-B) on the following page.

Ii-C Direction Angle Estimation Error Following Von Mises Distribution

Since the BS is assumed to have the imperfect knowledge of the direction angles toward the eavesdroppers, the actual direction angle between the BS and the -th eavesdropper is expressed as

 θBEk=^θBEk+ΔθBEk,k=1,⋯,K, (12)

where is the estimated direction angle between the BS and the -th eavesdropper, denotes the estimation error which is assumed to follow the Von Mises distribution [27] over the interval . Note that represents the maximum angle error and is a positive value up to the beamwidth between the first nulls [21].

The Von Mises distribution, also known as the circular normal distribution, is a continuous probability distribution on the circle and mainly describes the information related to the directional angles. In

[23], the Von Mises distribution was considered to be the best statistical model for circular parameters such as the direction angle. Subsequently, according to (12), the estimation error of the direction angle should also follow the Von Mises distribution.

The probability density function (PDF) of the Von Mises distribution is

 f(θ|μ,κ)=eκcos(θ−μ)2πI0(κ),θ∈[−π,π), (13)

where represents the estimation error of the direction angle, denotes the mean of , denotes the concentration parameter and controls the width of , and with is the modified Bessel function of the first kind with order computed by

 In(x)=1π∫π0excos(t)cos(nt)dt,x∈(−∞,∞). (14)

Iii Robust Beamforming Matrix Design With Von-Mises Distributed Direction Angle Error at Eavesdroppers

In the previous section, the estimation error of the direction angle toward each eavesdropper has been modeled to follow the Von Mises distribution. Then in order to maximize the worst-case system sum secrecy rate, in this section, we propose a Von Mises distribution-SSRM (VMD-SSRM) robust DM scheme. The procedure of the proposed scheme, i.e., the design of the optimal beamforming matrices, will be illustrated as follows.

In (II-B), the item can be expressed as

 |hH(θBEk)wm|2= (15) Tr(h(θBEk)hH(θBEk)wmwHm) =Tr(h(^θBEk+ΔθBEk)hH(^θBEk+ΔθBEk)wmwHm).

To simplify the expression, we define

 HBEk=E{h(^θBEk+ΔθBEk)hH(^θBEk+ΔθBEk)},k∈K, (16)

where . The necessity of the expectation in (16) is explained as follows. If we remove the expectation, since is a random variable, then would be random as well as the objective function of (II-B), and it would be very difficult to find a solution to (II-B).

Let denote the -th row and the -th column entry of , then can be written as

 HBEk(u,v)=Υ1k(u,v)−jΥ2k(u,v). (17)

The detailed procedure for deriving and based on the Von Mises distribution is shown in Appendix A. And and are expressed as (Appendix A) and (Appendix A), respectively.

In order to make (II-B) more tractable, we substitute the new matrix as well as the positive semi-definite matrix variables , into (II-B). Then the optimization problem (II-B) can be rewritten in a simple form as shown in (III) at the top of the next page.

By solving the problem (III), we can obtain the optimal information beamforming matrices and the AN beamforming matrix Q. However, the objective function in (III) is non-convex and difficult to tackle, as it comprises the logarithms of the product of fractional quadratic functions. In order to solve this problem, we employ the semi-definite relaxation and the first-order approximation technique based on the Taylor expansion to transform the original problem into a convex problem [28]. Then this convex problem can be solved by using the convex optimization toolbox (CVX).

The specific process for converting (III) into a convex one is illustrated as follows. First, the exponential variables are used to substitute the numerators and denominators of the fractions in the objective function in (III), i.e.,

 epi=M∑m=1Tr(H(θBDi)Wm)+Tr(H(θBDi)Q% )+σ2D, (19)
 eqi=M∑m=1m≠iTr(H(θBDi)Wm)+Tr(H(θBDi)Q% )+σ2D, (20)
 eck=M∑m=1Tr(HBEk% Wm)+Tr(HBEkQ)+σ2E, (21)
 edi,k=M∑m=1m≠iTr(HBEkWm)+Tr(HBEkQ)+σ2E. (22)

Then according to the properties of exponential and logarithmic functions, the objective function in (III) can be rewritten as

 M∑i=1((pi−qi)−maxk=1,⋯,K(ck−di,k)). (23)

Meanwhile , , , are constrained by the right hand sides of each equation in (19), (20), (21) and (22). So the problem (III) can be transformed to a semi-definite programming (SDP) problem as

 maximizeS M∑i=1((pi−qi)−maxk=1,⋯,K(ck−di,k)) (24a) s.t. M∑m=1Tr(H(θBDi)Wm)+Tr(H(θBDi)Q)+σ2D≥epi,∀i, (24b) M∑m=1m≠iTr(H(θBDi)Wm)+Tr(H(θBDi)Q% )+σ2D≤eqi,∀i, (24c) M∑m=1Tr(HBEkWm)+Tr(HBEkQ)+σ2E≤eck,∀k, (24d) M∑m=1m≠iTr(HBEkWm)+Tr(HBEkQ)+σ2E≥edi,k,∀i,∀k, (24e) Tr(Q)+M∑i=1Tr(% Wi)≤Pt, (24f) Wi,Q⪰0,∀i. (24g)

The objective function (24a) is a concave function because it comprises a sum of the affine functions minus a sum of the maxima of the affine functions. However, the constraints (24c) and (24d) are non-convex. In order to transform these two constraints into convex ones, and can be linearized based on the first-order Taylor approximation as

 eqi=e¯qi(qi−¯qi+1), (25)
 eck=e¯ck(ck−¯ck+1), (26)

where

 ¯qi=ln(M∑m=1m≠iTr(H(θBDi)Wm)+Tr(H(θBDi)Q)+σ2D), (27)
 ¯ck=ln(M∑m=1Tr(H% BEkWm)+Tr(HBEkQ)+σ2E), (28)

are the points around which the approximation are made. So the problem (II-B) eventually becomes

 maximizeS M∑i=1((pi−qi)−maxk=1,⋯,K(ck−di,k)) (29a) s.t. M∑m=1Tr(H(θBDi)Wm)+Tr(H(θBDi)Q)+σ2D≥epi,∀i, (29b) M∑m=1m≠iTr(H(θBDi)Wm)+Tr(H(θBDi)Q)+σ2D ≤e¯qi(qi−¯qi+1),∀i, (29c) M∑m=1Tr(HBEkWm)+Tr(HBEkQ)+σ2E ≤e¯ck(ck−¯ck+1),∀k, (29d) M∑m=1m≠iTr(HBEkWm)+Tr(HBEkQ)+σ2E≥edi,k,∀i,∀k, (29e) Tr(Q)+M∑i=1Tr(% Wi)≤Pt, (29f) Wi,Q⪰0,∀i. (29g)

As a convex problem, (29) can be solved iteratively by the Algorithm 1 using the CVX optimization software. According to the definition, has to satisfy , so the optimal of (29) should satisfy this rank-one constraint. Otherwise the randomization technology [29] would be utilized to get a rank-one approximation.

Iv Robust Beamforming Matrix Design With Maximum Direction Angle Error at Eavesdroppers

In the previous section, the system was designed based on the Von Mises distribution of the eavesdropper direction angle error. Thus the system sum secrecy rate was maximized for all possible values of the direction angle error. Nevertheless, it is also important to exclusively investigate the case of the worst direction angle error. Thus in this section, the system will be designed when the eavesdropper direction angle error reaches its maximum value and the robust beamforming matrices are obtained by the maximum angle estimation error-SSRM (MAEE-SSRM) method.

If the direction angle estimation error toward each eavesdropper is bounded, we can prove that the channel estimation error between the BS and each eavesdropper is norm-bounded (the detailed procedure is illustrated in Appendix B). Then the channel between the BS and the -th eavesdropper can be expressed as

 h(θBEk)=h(^θBEk)+ΔhBEk,∥ΔhBEk∥≤εk,k=1,⋯,K, (30)

where is the estimated channel between the BS and the -th eavesdropper, is the channel estimation error, and is the bound of the norm of the channel estimation error of the -th eavesdropper.

Then we derive the lower bound of the objective function in (II-B). For that purpose, we first derive the upper and lower bounds of and in (II-B). The upper bound of is

 |hH(θBEk)wm|2 (31) =hH(θBEk)wmwHmh(θBEk) =(hH(^θBEk)+ΔhHBEk)wmwHm(h(^θBEk)+ΔhBEk) =hH(^θBEk)wmwHmh(^θBEk)+2Re{ΔhHBEkwmwHmh(^θBEk)} (a)≤hH(^θBEk)%wmwHmh(^θBEk)+2εk∥wmwHmh(^θBEk)∥.

Similarly, the lower bound of is

 |hH(θBEk)wm|2 (32) =hH(^θBEk)wmwHmh(^θBEk)+2Re{ΔhHBEkwmwHmh(^θBEk)} ≥hH(^θBEk)wm% wHmh(^θBEk)−2εk∥wmwHmh(^θBEk)∥.

It should be noted that in (31) and (32) the second-order error term is omitted because it is quite small compared to the other two terms. And note that the step () in (31) results from the following derivation process. According to the Cauchy-Schwarz inequality theorem, we have

 |xHy|≤∥x∥∥y∥. (33)

Since the inequality

 −|xHy|≤Re{xHy}≤|xHy|, (34)

is obviously true, then we can obtain the following inequality

 −∥x∥∥y∥≤Re{xHy}≤∥x∥∥y∥. (35)

The inequality in (35) holds with equality when and only when x and y are linearly dependent. The derivation of the step () in (31) is completed.

Similarly, the upper bound of is

 |hH(θBEk)ql|2 (36) ≤hH(^θBEk)ql% qHlh(^θBEk)+2εk∥qlqHlh(^θBEk)∥,

and the lower bounds of is

 |hH(θBEk)ql|2 (37) ≥hH(^θBEk)ql% qHlh(^θBEk)−2εk∥ql


text:
# The number of solutions of x

Question:

The number of solutions of $x^{2}+|x-1|=1$ is

(a) 0

(b) 1

(c) 2

(d) 3

Solution:

(c) 2

$x^{2}+|x-1|=x^{2}+x-1, x \geq 1$

$=x^{2}-x+1 \quad, x<1$

(i) $x^{2}+x-1=1$

$\Rightarrow x^{2}+x-2=0$

$\Rightarrow x^{2}+2 x-x-2=0$

$\Rightarrow x(x+2)-1(x+2)=0$

$\Rightarrow(x+2)(x-1)=0$

$\Rightarrow x+2=0$ or, $x-1=0$

$\Rightarrow x=-2$ or $x=1$

Since $-2$ does not satisfy the condition $x \geq 1$

(ii) $x^{2}-x+1=1$

$\Rightarrow x^{2}-x=0$

$\Rightarrow x^{2}-x=0$

$\Rightarrow x(x-1)=0$

$\Rightarrow x=0$ or, $(x-1)=0$

$\Rightarrow x=0, \quad x=1$

$x=1$ does not satisfy the condition $x<1$

So, there are two solutions.

format: plain


text:
# Plzzzzzzzzzzzzzzzzzz

2 answers
###### Question:

plzzzzzzzzzzzzzzzzzz

## Answers

1 answer

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1 answer

### ANYONE HELP THIS IS HARD PLS

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### Help me pls!!! asap please answer correctly

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### Define neuron and nervecell

Define neuron and nervecell...
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### Find the circumference of a circle with diameter, d = 3cm. Give your answer in terms of pi

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The oldest recorded effort to organize a fire fighting service dates back to the:...
1 answer

### 3An aeroplane flies 4200 km at a speed of 800 km/ how long is the flight

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### Determine weather the following table represents an exponential function. EASY!!!!!!!!!!!!!!

Determine weather the following table represents an exponential function. EASY!!!!!!!!!!!!!!...
1 answer

### Explain how a change in thermal energy causes matter to change from one state to another. Give two examples

Explain how a change in thermal energy causes matter to change from one state to another. Give two examples...

-- 0.049444--


text:
# [Dot Product] Vector Proection

[Dot Product] Vector Projection

## The Attempt at a Solution

I am not sure what to do here -- I know that the projection of u onto a "dotted" with w = 0 by definition, but I don't know how to show this.

added this second part after plugging in for the definition of the projection we derived in class, then simplified. . .

Last edited:

anyone?

I like Serena
Homework Helper
Hi Highway!

You have an expression for the projection.
Can you substitute that (and only that) in the formula you have for w?

To show that 2 vectors are orthogonal, you need to show that their dot product is zero. That is, that $\vec a \cdot \vec w = 0$.

What you need to know, is that there are calculation rules for dot products.
For instance $\vec a \cdot (\vec b+\vec c) = \vec a \cdot \vec b + \vec a \cdot \vec c$.

Can you simplify the expression for $\vec a \cdot \vec w = 0$?

Hi Highway!

You have an expression for the projection.
Can you substitute that (and only that) in the formula you have for w?

To show that 2 vectors are orthogonal, you need to show that their dot product is zero. That is, that $\vec a \cdot \vec w = 0$.

What you need to know, is that there are calculation rules for dot products.
For instance $\vec a \cdot (\vec b+\vec c) = \vec a \cdot \vec b + \vec a \cdot \vec c$.

Can you simplify the expression for $\vec a \cdot \vec w = 0$?

Thanks!! I got it figured out :P

I like Serena
Homework Helper
Congrats!


text:
# Whether to apply structural equation modelling separately to each of a set of heterogeneous correlation matrices in a meta-analysis context?

I'm in the process of performing an Structural Equation Modelling (SEM) meta-analysis on some psychological data. Ultimately I want to examine a mediational model based on a set of correlation matrices. I've spent a little time reviewing the literature regarding what to do when you have heterogenous correlation matrices. I've written up a brief review of what I've found on Cognitive Sciences SE.

To summarise what I've found:

• Most researchers use a two-step approach. Step 1 involves extracting estimates of true correlations between a set of variables using a variety of meta-analytic techniques. Step 2 involves inputting the correlation into your SEM software and in general doing analyses as per normal non-meta-analytic SEM.
• Despite an awareness of the problems, many researchers still use the two-step approach even though most datasets have variation in true correlations that can not merely be attributable to random sampling (i.e., a random effects model is generally more applicable).
• Some researchers mention the possibility of clustering the dataset or performing SEM Meta-analysis on values of moderators in the hope that the true score variability will be removed once you divide the analysis up into groups. However, in most applications such moderators are insufficient to explain all the true score variation.

So instead of adopting a two-step approach, I thought about maybe doing the following, at least for cases where you are able to compute a full correlation matrix for each study:

"perform SEM on each sample and treat the parameters and fit statistics as values that vary between samples. You could then summarise the distribution (e.g.,mean and SD) of these SEM parameters and fit statistics. This would be similar to how correlations and other effect sizes are typically modelled as random effects. So, for example you could look at the variation in the indirect effect across samples. The challenging bit would be to tease out what is true score variation and what is due to random sampling."

Excluding the issue of teasing out true score from random sampling, it would be straight forward to do such an analysis in R with any of the SEM packages.

### Questions

So, I was wondering:

• Has this approach of fitting SEM to each study separately ever been applied?
• What are the pros and cons of the approach?
• How might you tease out what variation in SEM parameters is due to random sampling and what is due to true score variance?
• Or is there some better approach to doing SEM meta-analysis on heterogenous correlation matrices?

Cheung and Cheung (2010) gave a presentation discussing two appraoches to meta-analytic Structural Equation Modelling (SEM). They label the two approaches:

1. Correlation matrices Meta-Analysis: An average correlation matrix is extracted from the primary studies and SEM is applied to this single correlation matrix.
2. Model Parameters Meta-Analysis: Model fitting is applied separately to the correlation matrix of each primary study and the variation in parameters is modelled using random effects multivariate meta-analysis (Cheung and Cheung, 2010 cite Cheung 2009, and Kalaian & Raudenbush, 1996)

Cheung and Cheung (2010) note that this is an area of research needing further development. They recommend trying both approaches especially where the full correlation matrix is available at least in a subset of studies.

Discussions of how to perform meta-analysis SEM are also closely related with general discussions of multivariate meta-analysis (e.g., see the tutorial by Van Houwelingen et al 2002).

I think that the standIndirect function in the metaSEM package in R (Cheung, 2011) implements a three variable test of mediation using the model parameters meta-analysis approach.

### References

• Becker, B.J. & Wu, M.J. (2007). The synthesis of regression slopes in meta-analysis. Statistical Science, 22, 414-429. PDF on arxiv
• Cheung, S. F., & Cheung, M. W.-L. (2010). Random effects models for meta-analytic structural equation modeling. Presented at The 7th Conference of the International Test Commission, July 19-21, 2010, Hong Kong.
• Cheung, M.W.L. (2011). metaSEM: Meta-Analysis using Structural Equation Modeling.
• Cheung, M.W.L. (2009). Comparison of methods for constructing confidence intervals of standardized indirect effects. Behavior Research Methods, 41, 425-438.
• Cheung, M.W.L. (2009). Constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16, 267-294.
• Kalaian, H.A. & Raudenbush, S.W. (1996). A multivariate mixed linear model for meta-analysis.. Psychological methods, 1, 227. PDF
• Van Houwelingen, H.C., Arends, L.R. & Stijnen, T. (2002). Advanced methods in meta-analysis: multivariate approach and meta-regression. Statistics in medicine, 21, 589-624. PDF


text:
5 and 7

1. Sep 12, 2007

mathslover

guys,

From the solutions of the Pell's equation x*x-2*y*y=-1,
how can we prove that whenever y ends in digit 5, then 7 | x ?

-Mathslover

Perhaps I should clarify a bit,x*x-2*y*y=-1 has solution
x=1, 7, 41, 239, 1393, 8119, 47321, 275807,.....
y=1, 5, 29, 169, 985, 5741, 33461, 167305,...
and the general solution is (xn+yn*sqrt(2))=(1+sqrt(2))^(2*n+1)
Apart from induction,how can we prove whenever 5|y then 7|x ?

Last edited: Sep 13, 2007
2. Sep 12, 2007

bel

3. Sep 12, 2007

CRGreathouse

For y = 15 there are no solutions, since sqrt(449) isn't an integer... right?

4. Sep 12, 2007

D H

Staff Emeritus
The first solution pair (x,y) is (1,1). Develop a recursive relationship for the nth such pair (xn, yn) in terms of previous pair(s). The relation you want to prove will fall right out.

5. Sep 13, 2007

Kummer

Do you the general solution to this Pellian equation? I think that is the way to approach this problem. Followed by induction.

6. Sep 20, 2007

ramsey2879

$$(1 +\sqrt{2})^n$$ gives $$X_{n}\sqrt{2} + Y_{n}$$ where $$X_{n}$$ and $$Y_{n}$$ for odd $$n$$ are solutions.

thus the first 3 solutions pairs X,Y are
1,1
5,7
29,41

from this one might notice that 5 = 3*1+2*1 and 29 = 3*5+2*7 and also notice that
7 = 4*1 + 3*1 and that 41=4*5+3*7.
Or simply multiply $$(X\sqrt{2} + Y)*(1+\sqrt{2})^2$$ and put back into the same X,Y form to find the new X and Y.

So prove that if X and Y are a solution pair then (3X+2Y) and (4X+3Y) are the next solution pair then consider X mod 5 and Y mod 7 and you can follow the rest by induction.

7. Sep 21, 2007

ramsey2879

Since $$(1+\sqrt{2})^2 = 3+2\sqrt{2}$$ and since $$(1+\sqrt{2})^3 = 7+5\sqrt{2}$$ then
$$X_{n} = 0 \mod 5$$ implies that $$n = 0 \mod 3$$
but $$X_{n}$$ is only a solution for odd n so we calculate $$(1 + \sqrt{2})^6 = 99 + 70\sqrt{2}$$ so the next solution after n = 3 (x=5, y = 7) is
$$X_{n+6} = 70*Y_{n} + 99*X_{n} \| Y_{n+6} = 70*2*X_{n} + 99*Y_{n}$$
From inspection of the above formula it can be seen that
$$X_{n} = 0 \mod 5$$ if and only if $$X_{n+6} = 0 \mod 5$$ and
$$Y_{n}=0 \mod 7$$ if and only if $$Y_{n+6} = 0 \mod 7$$
So our proof is complete


text:
# Projective module (The lifting property)

I'm looking for an example for a left R-module that doesn't have the lifting property, From theorems I read, $Z/2Z$ as a $Z$ module should be an example since it' not a direct sum (there isn't a sub-module $K$ of $Z$ such that the direct sum of $Z/2Z$ and $K$ is $Z$).

But I don't understand why $Z/2Z$ doesn't have the lifting property, can someone explain why?

-

Suppose $Z/2Z$ had the lifting property. Then in particular the identity map would have a lift, i.e. there would exist a module homomorphism $\varphi: Z/2Z\to Z$ with $\pi\circ\varphi=id$. Show that this yields a contradiction.
The identity map $Z/2Z$ to $Z/2Z$ would have to have a lift. The epimorphism is $\pi: Z\to Z/2Z$. –  Julian Kuelshammer Mar 16 '12 at 23:08


text:
# Just started learning Java...

## Recommended Posts

Kambiz    758
I have just started learning Java but there are at least 3 thing I'm missing from c/c++/c#: 1)unsigned types 2)function pointers/delegates 3)something like .Net Framework's CodeDomProvider If there are no Java equivalents for 2 & 3 I would give it up! So, dear Java fans: can you motivate me to learn Java? [Edited by - Kambiz on May 17, 2006 12:15:46 PM]

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Guest Anonymous Poster
Hahaha, what a lame message! Did you try to buy a book? Like "Head first Java" or "Core Java". First study a little, later start bitching and whining.

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Kambiz    758
There are some Java books available I can use and I have read some chapters from them, I thought about using Java for my next Project to make it cross platform but if the language is so much weaker than c#(at least from my point of view... (I really need dynamic code...)) I will go with Microsoft. Why should I spend so much time learning a new language when it is useless for me?

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Fred304    382
Quote:
 Original post by Kambiz 1)unsigned types

There aren't any. Why exactly do you need them?

Quote:
 2)function pointers/delegates

Simply declare an interface and have different classes implement it.
public interface FunctionPointer{	public int function(String a, int b, Object x);}public class Foo implements FunctionPointer{	public int function(String a, int b, Object x)	{		return a.length() + 5*b + x.hashCode();	}}public class Bar implements FunctionPointer{	public int function(String a, int b, Object x)	{		return a.indexOf('n') * b - x.toString().length() & 0x1234;	}}

Quote:
 3)something like .Net Framework's CodeDomProvider

Since I have no idea what this is I cannot comment on that.

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Kambiz    758
Thanks...
I think the reflection library could help me with number 3...

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Guest Anonymous Poster
Quote:
 Original post by KambizThere are some Java books available I can use and I have read some chapters from them, I thought about using Java for my next Project to make it cross platform but if the language is so much weaker than c#(at least from my point of view... (I really need dynamic code...)) I will go with Microsoft. Why should I spend so much time learning a new language when it is useless for me?

I don't think you understand what you are talking about at all. Each language has its own way of doing stuff, trying to program C++ in Java only leads to poor code and headaches.

All your so much "needed features" are done in one way or the other:

- Why in the h*** are you going to need unsigned types? Unless you are programming for embedded devices, and even so this is not a problem all the times since the data is there anyways, you would care about it?

Java developers don't use, don't need and don't want unsigned types. The only people I saw complaining about it are C++ developers trying to find a pretext for not to use it.

- What you do with "function pointers / delegates" is done with anonymous classes / listeners in Java in a much clearer way.

- I have no idea what CodeDomProvider does but have everything you could ever need for XML processing.

Java is Java, so either you make up your mind about learning it or not. The choice is yours, we don't need to prove anything to you.

Quote:
 language is so much weaker than c#

Millions of people disagree with you, all over IT industry where Java is #1. In the opensource too, take a look at sourceforge.

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Zahlman    1682
OP: Your post (a) sure looks like a troll to me, and (b) probably belongs in General Programming anyway. But I'll bite:

Quote:
Original post by Fred304
Quote:
 Original post by Kambiz 1)unsigned types

There aren't any. Why exactly do you need them?

Actually, there is one: 'char' is effectively 'unsigned short'.

But yeah, why do you think you need them? I can think of two reasons:

- "unsigned correctness", similar to "const correctness", which also isn't provided in Java. Well, you can choose a language according to how much type-checking you want it to do, but there are lots of people who prefer that the language check basically nothing at all at compile-time. So, to each their own.

- Optimizations where you need a value to range 0 to 255 and you want it to fit in a byte. Well, this can certainly be dealt with when you do need to: As long as you can keep track of which bytes you want to be "unsigned", just convert the signed value whenever you use it in an expression. Java promotes numeric subexpressions at least to 'int' anyway, so this is the trick:

byte x, y;// laterx = (byte)200; // it stores -56, but we can recover an unsigned valuey = (byte)150; // similarlyint z = (x & 0xff) * (y & 0xff); // 30000

Each bitwise-and causes the x and y to be promoted to int, and sign-extended. The 0xff literals already are ints, so nothing changes there. We use the bitwise-and to mask off the sign bits, and then multiply the new unsigned values.

Quote:

Quote:
 2)function pointers/delegates

Simply declare an interface and have different classes implement it.

Right. The main reason function pointers exist in C++ in the first place, and that C# has delegates to refine the concept, is C backwards-compatibility. You know, that non-OO language where you had to create those abstractions yourself.

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Guest Anonymous Poster
Quote:
 Original post by KambizIf there are no Java equivalents for 2 & 3 I would give it up! So, dear Java fans: can you motivate me to learn Java?

news flash: no one (with any luck) cares what language you use. if you have a question try asking it.

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Quote:
 Original post by Kambiz3)something like .Net Framework's CodeDomProvider

If all you want to do is compile Java to native binaries, then you might want to look into the GNU Java Compiler. If you actually want to compile code at run-time, try poking around for information on JIT (just-in-time) compiling.

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Guest Anonymous Poster
Quote:
 Original post by programwizardIf you actually want to compile code at run-time, try poking around for information on JIT (just-in-time) compiling.

What!?

Java already does that out of the box, no one needs to worry about "getting information on JIT".

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jfclavette    1058
Java doesn't have anything standardized akin to CodeDom and I'm somewhat unsure as to wether it is technically possible to implement something similar in a JVM. You could certainly load another class in another thread, but reloading code in the same thread is going to be harder, I believe.

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Guest Anonymous Poster
Quote:
 Original post by jfclavetteJava doesn't have anything standardized akin to CodeDom ...

You know, it would be really helpful if you all said what you plan to do instead of using some "MS patented terminology (tm)". WTF is CodeDom!? Will I need a dictionary to translate what you are saying?

Sorry, Java developers can't help you if they can't understand what you want.

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jfclavette    1058
Quote:
Original post by Anonymous Poster
Quote:
 Original post by jfclavetteJava doesn't have anything standardized akin to CodeDom ...

You know, it would be really helpful if you all said what you plan to do instead of using some "MS patented terminology (tm)". WTF is CodeDom!? Will I need a dictionary to translate what you are saying?

Sorry, Java developers can't help you if they can't understand what you want.

Compiling a .java file at runtime, loading the class and instantiating it. This is only a microscopic subset of what CodeDom can do, but I assume it's what the OP wants to do. It's also a significant enough building block to start building CodeDom-like functionnality in Java.

I tought 'Java developers' (somehow tying a language to the word programmers seems so pathetic to me) could use Google.

[Edited by - jfclavette on May 17, 2006 4:13:21 PM]

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Guest Anonymous Poster
Quote:
 Original post by jfclavetteCompiling a .java file at runtime, loading the class and instantiating it. This is only a microscopic subset of what CodeDom can do, but I assume it's what the OP wants to do. I tought 'Java developers' (somehow tying a language to the word programmers seems so pathetic to me) could use Google.

Google returned nothing but a bunch of gibberish. I thought '.Net developers' were could use complete meaningful sentences, is that difficult?

BTW, it's possible to mess with classloaders in Java, dynamically instantiate classes, use external scripting languages (from the 6 it will be in the standard JDK), etc, but I will leave that for you to look up on Google, ok?

Good search.

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jfclavette    1058
Quote:
Original post by Anonymous Poster
Quote:
 Original post by jfclavetteCompiling a .java file at runtime, loading the class and instantiating it. This is only a microscopic subset of what CodeDom can do, but I assume it's what the OP wants to do. I tought 'Java developers' (somehow tying a language to the word programmers seems so pathetic to me) could use Google.

Google returned nothing but a bunch of gibberish. I thought '.Net developers' were could use complete meaningful sentences, is that difficult?

BTW, it's possible to mess with classloaders in Java, dynamically instantiate classes, use external scripting languages (from the 6 it will be in the standard JDK), etc, but I will leave that for you to look up on Google, ok?

Good search.

I know it's possible to load from a JAR. What I'm interested in is compiling them without resorting to external tools and in a way where I can assume anyone with any JVM can do the same. Does Java maintain a reliable accessible path to the compiler (is the compiler included in the JVM ?) that is guaranteed to work in all case on any machine ? I'd be fine with that, even if it wouldn't offer the synctactic sugar .NET offers.

I would also like to load binaries from languages other than Java. I don't want to use an arbitrary interpreted scripting language, but say, C++ or C#. (Altough .NET requires a third-party library to load Java classes, the underlying used functionnality is CodeDOM, so loading C# classes in Java wouldn't be as easy, and as such, no popular projects doing so exist.)

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Kambiz    758
I know *exactly* what I want:
My application will have a small kernel, the kernel will load many PlugIns, compile them and use them when needed. In Managed C++ or C# I would use C# like a scripting language and compile all the PlugIns using the CodeDomProvider, now I want to know if it is possible to do the same thing in Java.
Please do not tell me I need to use a scripting language! If that is the only way in Java I will use c# for this Project at the cost of portability.

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rip-off    10979
Quote:
 Original post by KambizI know *exactly* what I want:My application will have a small kernel, the kernel will load many PlugIns, compile them and use them when needed. In Managed C++ or C# I would use C# like a scripting language and compile all the PlugIns using the CodeDomProvider, now I want to know if it is possible to do the same thing in Java.

AFAIK the java compiler is only available to those with the sdk, not the java runtime. Java can load and use arbitrary class files during execution (not only those included during compilation ), and can load DLLs ( but you need some special interfacing to talk to the JVM, namely the JNI ).

If I read the MSDN page here correctly:
Quote:
 . Several languages are supported by CodeDomProvider implementations that ship with the .NET Framework SDK.

Does this mean that an implementation of CodeDom only exists if you have a .Net SDK installed? Or idoes the SDK come with the runtime?

However, anyone who is going to be writing plugins for your program will surely have access to a compiler, so why would it be neccesary to have your program compile the .javas as opposed to just using the .class / .jar files?

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jfclavette    1058
Quote:
Original post by rip-off
Quote:
 Original post by KambizI know *exactly* what I want:My application will have a small kernel, the kernel will load many PlugIns, compile them and use them when needed. In Managed C++ or C# I would use C# like a scripting language and compile all the PlugIns using the CodeDomProvider, now I want to know if it is possible to do the same thing in Java.

AFAIK the java compiler is only available to those with the sdk, not the java runtime. Java can load and use arbitrary class files during execution (not only those included during compilation ), and can load DLLs ( but you need some special interfacing to talk to the JVM, namely the JNI ).

If I read the MSDN page here correctly:
Quote:
 . Several languages are supported by CodeDomProvider implementations that ship with the .NET Framework SDK.

Does this mean that an implementation of CodeDom only exists if you have a .Net SDK installed? Or idoes the SDK come with the runtime?

However, anyone whoe is going to be writing plugins for your program will surely have access to a compiler, so why would it be neccesary to have your program compile the .javas as opposed to just using the .class / .jar files?

The compilers, not the SDK, come with the Framework redistributable. As to why, say I want the users to be able to script my game trough a level editor. The scripting language I want is a subset of C#. They script the game trough my level editor, save, load the game, and their modifications take effect automatically.

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Guest Anonymous Poster
Quote:
 Original post by jfclavetteI know it's possible to load from a JAR. What I'm interested in is compiling them without resorting to external tools and in a way where I can assume anyone with any JVM can do the same. Does Java maintain a reliable accessible path to the compiler (is the compiler included in the JVM ?) that is guaranteed to work in all case on any machine ? I'd be fine with that, even if it wouldn't offer the synctactic sugar .NET offers.

See Java 6 Mustang releases for the Compiler API. Prior to that version you will to look up for libraries on the net.

Quote:
 I would also like to load binaries from languages other than Java. I don't want to use an arbitrary interpreted scripting language, but say, C++ or C#. (Altough .NET requires a third-party library to load Java classes, the underlying used functionnality is CodeDOM, so loading C# classes in Java wouldn't be as easy, and as such, no popular projects doing so exist.)

[/quote]

Anything else? Maybe a cup of coffee? Listen, you don't expect to produce decent production code in C++ for running in the JVM, do you?

So basically you are asking for some idiotic stuff that shouldn't be done nor attempted? If you need to use C++ then use C++, if you need C# then use C#.

Don't even come with J# as an example of Java for .Net, that's based in a very old version of Java that Microsoft had but was sued by Sun for having "incompatibilities".

There's the Java 6 Mustang scripting language support. Prior to that there was a framework for adding such languages to Java at Apache website that's not locked to a specific language.

Have a nice day.

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jfclavette    1058
Quote:
 Original post by Anonymous PosterAnything else? Maybe a cup of coffee? Listen, you don't expect to produce decent production code in C++ for running in the JVM, do you?

Yes. It works in .NET.

Quote:
 So basically you are asking for some idiotic stuff that shouldn't be done nor attempted? If you need to use C++ then use C++, if you need C# then use C#.

I've written my whole game server in C++. I want to write the client in Java while reusing the common classes I need. Isn't that valid ?

Quote:
 Don't even come with J# as an example of Java for .Net, that's based in a very old version of Java that Microsoft had but was sued by Sun for having "incompatibilities".

I wasn't talking about J# but about the flurry of Open Source or proprietary hooks to compile and run Java code for .NET.

Quote:
 There's the Java 6 Mustang scripting language support. Prior to that there was a framework for adding such languages to Java at Apache website that's not locked to a specific language.

I wasn't aware of that. Why is it so underused compared to similar .NET functionnality ? I see tons of .NET hooks for languages such as FORTRAN, COBOL and friends, yet nothing for Java.

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Guest Anonymous Poster
Quote:
 Original post by KambizI know *exactly* what I want:My application will have a small kernel, the kernel will load many PlugIns, compile them and use them when needed. In Managed C++ or C# I would use C# like a scripting language and compile all the PlugIns using the CodeDomProvider, now I want to know if it is possible to do the same thing in Java.Please do not tell me I need to use a scripting language! If that is the only way in Java I will use c# for this Project at the cost of portability.

Wow! Microkernels!

OSGI has been doing this for the last six years, and some big projects use it. http://www.osgi.org/ , http://download.eclipse.org/eclipse/equinox/

For the scripting stuff use either the framework at Apache or Java 6 that has the scripting engine bundled.

For compiler stuff use the compiler API in Java 6.

Good luck.

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Kambiz    758
Quote:
 Does this mean that an implementation of CodeDom only exists if you have a .Net SDK installed? Or idoes the SDK come with the runtime?

You can use the c# and jscript.net compilers directly in your application. They are integrate in the .Net framework and the user does not need to install the SDK. You can even compile the code to an assembly in memory and then use it.
I have used c# and jscript.net as scripting languages in some of my c# programs, having such a compatible scripting language makes everything easy and such compiled scripts run as fast as your code.

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Guest Anonymous Poster
Quote:
 Original post by jfclavetteI've written my whole game server in C++. I want to write the client in Java while reusing the common classes I need. Isn't that valid ?

You would need to use JNI for interfacing with C++, which is not good because the VM don't have a clue of what you are doing with the memory.

JNI should be used only for legacy stuff.

Not sure what would be the best way in this case.

Quote:
 Original post by jfclavette

There's the Java 6 Mustang scripting language support. Prior to that there was a framework for adding such languages to Java at Apache website that's not locked to a specific language.
[/quote]

I wasn't aware of that. Why is it so underused compared to similar .NET functionnality ?[/quote]

The Java 6 feature is fairly new. There are tens of languages ported to the JVM but only now it's going to be official part of the JDK to support them.

I think the reason for people not to talk about it too loud is that they don't trust such languages too much and prefer to use Java.


text:
normalization to "housekeeping" genes of RNAseq data vs qPCR result
1
0
Entering edit mode
10 months ago
cwwong13 ▴ 10

It is relatively common to use "housekeeping" genes when performing qPCR experiments. I would like to know how common (or uncommon) that a gene expression found in RNAseq can be replicated with qPCR.

Indeed, I usually get fairly comparable results between these two experimental techniques. However, I shift to study metabolism-related pathways recently and found that many of the transcriptional changes are not consistent between RNAseq and qPCR. One possibility is the metabolism shift will induce a large array of gene changes, that might also include the "housekeeping" gene I chose (I usually use actin or 36b4).

an example is: we found that from RNAseq, the actin level is downregulated after the treatment, thus, indeed make using actin as a "housekeeping" gene invalid. On the other hand, I wonder if that down-regulation of actin could result from generally up-regulation of all other genes? Given that many RNAseq will still be normalized to the total library size, thus, I am not sure the analysis pipeline, such as DESeq2, can deal with this.

I wonder are there any means to correct this, or at least diagnosis on the data structure/ quality so that I can know I should interpret the RNAseq/ qPCR result with cautions?

Thanks!

qPCR RNAseq • 1.1k views
2
Entering edit mode

I do not generally buy that qPCR is a good verification for RNA-seq. For a really meaningful and solid qPCR comparison, assuming that the RNA-seq was conducted and analysed properly in terms of replication and statistics, should imho probably address these points:

• definition of housekeeping genes: While there are a couple of established house keeping genes one is essentially setting the baseline of an entire experiment to (often) a single gene. One should rather use a panel of housekeeping genes and then use the something like the median of the housekeeping panel for normalization. RNA-seq commonly uses approaches that make use of all genes while trying to find a baseline that centers most genes at a fold change of somewhat zero (RLE DESeq2, TMM edgeR for example) or at least tries to find a median ratio that properly captures the size relationship between samples.

• primer efficiency and uniqueness: primers in PCR are usually between 18-25bp. That makes them far less unique than e.g. 2x75 or 2x150bp reads. One must therefore (for a good qPCR) ensure that primers have no detectable off-targets and have decent primer efficiency as one gets only one readout per gene while in RNA-seq the coverage is highly redundant and therefore (probably) more robust. Also, one would probably need multiple primer sets for the same gene to exclude amplicon-specific biases. Primer validation is tedious, and something like melting curves (at least in my head) are rather pointless as after 40 PCR cycles any tiny contamination will produce some kind of band that may introduce shoulders in these profiles. RNA-seq is highly redundant per gene in terms of coverage, so I think these kinds of biases should be better compensated here, especially with paired-end reads.

• non-standardized statistics: while RNA-seq has a couple of very robust statistical frameworks such as DESeq2 and edgeR people often do custom (and sometimes inappropriate) statistics for qPCR in combination with few housekeeping controls and primers per gene/transcript. Tests such as the commonly used t-test require normality of data which is not necessarily the case. Non-parametric statistics such as the Wilcox-test also require more replicates than RNA-seq to even have the chance to reach significance. A 3 vs 3 comparison with a Wilcox-test will at best yield a p-value of 0.1 even if all ranks of group1 are lower than of group2, even without multiple testing correction, e.g.:

> wilcox.test(c(1,2,3), c(10,11,12))

Wilcoxon rank sum exact test

data:  c(1, 2, 3) and c(10, 11, 12)
W = 0, p-value = 0.1
alternative hypothesis: true location shift is not equal to 0

So you would need several housekeeping genes, several sets of primers, many replicates to make a fair qPCR experiment if challenging RNA-seq. But this is obviously lots of work, anyone who got cramps in the thumb after pipetting several 96-well plates in a row knows that.

RNA-seq is often used to build a hypothesis rather than checking individual genes. I would therefore prefer to validate the generated hypothesis, e.g. via a knockdown experiment of a pathway that was hypothesized to be altered after analysing the expression data, rather than qPCR-ing some individual pathway genes. If you are interested in individual genes then do qPCR, for the global picture I would always prefer RNA-seq followed by experimental functional validation of results.

Not sure whether this contributed anything to your question, I just felt like ranting about qPCR :p

0
Entering edit mode

I totally agree with you on doing the qPCR validation thing, but sadly, it seems still a common practice among experimental biologists.

I am also curious that what should be a proper normalization. Do you think we should normalize to total cell number in bulk RNAseq? Do you have an idea (or suggestion on a published article that will be even better, so I can just cite it or show it to my supervisor) of how library size is correlated to the cell number?

Sorry for asking this fundamental question here. I just realized I really missed all the basics of statistics and bioinformatics even I can run/ modify the code from online tutorials.

0
Entering edit mode

Do you think we should normalize to total cell number in bulk RNAseq?

No, I do not see how this a) would be possible and b) would make sense. DESeq2 and edgeR work very well for this purpose. I in fact never heard of normalization to a cell number.

Do you have an idea (or suggestion on a published article that will be even better, so I can just cite it or show it to my supervisor) of how library size is correlated to the cell number?

If with library size you mean sequencing depth then there is no correlation. The more reads, the more library size, it is a technical thing.

0
Entering edit mode
10 months ago
JC 13k

There is the concept of any gene can be variable at some point, so there are no "housekeeping" genes, for the experiments you should include a few typically non-variable genes and select those that are not variable for normalization.

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text:
# Reproduce neat Algorithm Style

I've seen this nice algorithm sytle in Stachurski (2009), "Economic Dynamics", p.74.

• Does someone know what package, environment or settings he used?

• Use package algorith2e – Yiannis Lazarides Jun 26 '15 at 12:55
• Do you mean algorithm2e? – Paul Stiverson Jun 26 '15 at 13:28
• @PaulStiverson Yes sure, that was a typo. Just watch out for hyperref problems with it. – Yiannis Lazarides Jun 26 '15 at 17:06

Algorithm2e is the package to use:

\documentclass{amsart}

\usepackage[ruled]{algorithm2e}
\usepackage{mathpazo}
\SetAlgoInsideSkip{medskip}

\begin{document}

\begin{algorithm}
%\BlankLine
\DontPrintSemicolon
\For{i in 1 to n}{
draw $X\sim\psi$\;
\For{j in 1 to t}{
draw $X\sim p(X,dy)$
}
set $X^i_t=X$\;
}
\Return{$(1/n)\sum^n_{i=1}\mathbb{1}\{X^i_t=y\}$}
%\BlankLine
\caption{Approximate marginal distribution}
\end{algorithm}
\end{document}

Later: I noticed that the original sample had some additional space at the beginning and end of the algorithm. At first, as you can see, I tried \BlankLine, but that seemed like a kluge, so I used \SetAlgoInsideSkip{medskip} so as to be the default.

You can use the algorithm2e package. The documentation is available from CTAN. Here is a quick workup of the algorithm, but note that there are many options you can tweak to get the appearance just as you want it.

\documentclass{article}
\usepackage{algorithm2e, bbold}

\begin{document}
\begin{algorithm}
\For{i in 1 to n} {
draw $X \sim \psi$\;
\For{j in 1 to t}{
draw $X \sim p(X,dy)$\;
}
set $X_t^i = X$\;
}
\Return  $(1/n) \sum_{i=1}^n \mathbb{1} \left\{X_t^i = y\right\}$
\end{algorithm}
\end{document}

The above LaTeX gives the following output:


text:
# Missing Matter Problem and Galactic Kinematics

This is a re-wording of an earlier problem I posted.

The standard example of the Missing Mass Problem comes from the rotational profiles of galaxies. By counting up the visible matter, we extrapolate a mass profile for a galaxy. We then apply Kepler's laws (the enclosed mass of a stable orbit can be modeled as a point mass) to calculate the expected velocity: $$f(v) = \sqrt{\frac{GM}{r}}$$ Where G is the gravitational constant, M is the enclosed mass of an elliptical orbit, r is the radius of the orbit. But this formula assumes a closed, elliptical orbit. I'm sure the data exists, but I haven't been able to find it. How do we know that the Earth, for instance, is not falling towards or away from the center of the Milky Way. That is, when we apply the rules of Keplar's orbits, what information do we have that the orbits of the observed galactic bodies describe a closed ellipse and not a spiral in or out (which would change the amount of missing mass considerably)?

• Interesting point. Here something to think about. It is mass missing and the movement would be outward then. So there is not enough gravitational force to hold the galaxy together. Then the question is: Why did it form in the first place. – mikuszefski Mar 5 '15 at 13:20
• Anyway, there should be data that shows that stars are basically moving tangential on average. – mikuszefski Mar 5 '15 at 13:21
• they take into account the type of galaxy en.wikipedia.org/wiki/Galaxy#Modern_research – anna v Mar 5 '15 at 13:25
• @mikuszefski I'm not an expert on galactic evolution, but I could suggest that two elliptical galaxies collided and the disk is the ejecta of the collision, but I don't want to take away from the root question. Is the orbit of galactic material in the disks an assumption or are there kinematic studies that tell us, within a reasonable margin of error, that the orbits are closed ellipses? – Quarkly Mar 5 '15 at 13:29
• My cosmology classes are also long long time ago, but I think you get the same problem of missing mass in elliptic galaxies, only that orbits are less well defined. In disc galaxies, I think the error in radial movement does not make up for the missing mass. – mikuszefski Mar 5 '15 at 13:41

The local standard of rest is the restframe circular orbit of a star at the position of the Sun in the azimuthally averaged Galactic potential. The Sun moves with 3d Galactic velocity coordinates of (11, 12, 7) km/s with respect to the LSR (Schonrich et al. 2009. i.e The Sun has a velocity component (wrt the LSR) of 11 km/s towards the Galactic centre (it of course has a tangential velocity of $>200$ km/s).

Sjouwerman et al. (1998) measure the line of sight velocities of 229 Maser sources towards the Galactic centre finding a mean velocity wrt the LST of $4 \pm 5$ km/s.

Reid et al. (2007) find a mean line of sight velocity, with respect to the LSR, for masers even closer to the Galactic centre as $-22 \pm 28$ km/s.

Li et al. (2010) measure velocities for 20 masers source within 2pc projected distance of the Galactic centre finding a mean velocity wrt to the LSR of $5 \pm 11$ km/s.

i.e. The Sun travels at no more than $\sim 10$ km/s radially with respect to the Galactic centre.

The tangential velocity of the Sun can be fixed with respect to the Galactic centre by observing the proper motion of Sag A* (Reid & Brunthaler 2004), which they find is almost entirely along the Galactic plane (i.e almost no vertical motion of the Sun wrt the Galactic plane). Assuming a distance of $8.0\pm 0.5$ kpc to the centre (for which there is a variety of evidence), the tangential velocity of the Sun is $241 \pm 15$ km/s, translating to a LSR tangential velocity of $236\pm 15$ km/s wrt the Galactic centre.

The Sun's motion wrt the Galactic centre is therefore almost entirely tangential and in the absence of anything but a nearly axially symmetric Galactic potential, the Sun executes a nearly circular orbit, with epicycles in the radial and vertical directions. The speed of the Sun's orbit and the amplitude and period of its epicycles depend on the size and shape of the Galactic potential.

I suppose you could hypothesise that the Sun was at the apogee of a highly elliptical, or otherwise non-circular orbit, but then the fact that it has very similar kinematics to 99 per cent of the nearby stars means they too would have to be on highly elliptical orbits with similar apogees (as they are moving in the same potential). But why would this be? Why should stars born over billions of years in different parts of the Galaxy have organised themselves to align their semi-major axes? By far the simplest explanation is that the orbits are close to circular and that is why the solar peculiar motion is small wrt to most stars - but large wrt halo (Population II) stars, which do have highly elliptical orbits and little circular motion.

EDIT: Part of the premise of this question is incorrect, since stars do not execute Keplerian orbits in the potential of a disk galaxy (see Why don't stars have Keplerian orbits? ). Keplerian orbits apply either to cases where objects orbit a much larger, point-like mass, or are orbiting in a spherically symmetric mass distribution where Newton's shell theorem can be applied. Neither of these are true in detail for the Milky Way and serious research work does not make these assumptions unless it is shown to be reasonable (e.g. for objects at large distances from the centre of the Galaxy).

As it says in the Introduction of the classic "Galactic Dynamics" by Binney & Tremaine 2nd ed. (2008) - " "The simplest approximate dynamical description of the Galaxy is obtained by assuming that its mass distribution is spherical. Let the mass interior to radius r be M(r). From Newton’s theorems the gravitational acceleration at radius r is equal to that of a point whose mass is the same as the total mass interior to r; thus the inward acceleration is $GM(r)/r^2$, where the gravitational constant $G = 6.674\times 10^{-11}\ m^{3} kg^{-1} s^{-2}$. The central or centripetal acceleration required to hold a body in a circular orbit with speed $v_0$ is $v_0^{2}/r$. Thus the mass interior to the solar radius $R_0$ in this crude model is $M(R_0)=v_0^{2}R_0/G$. The approximation that the mass distribution is spherical is reasonable for the dark halo, but not for the flat stellar disk."

The closed/not-closed issue I don't understand. Galactic orbits are not closed at all in the sense that orbital paths do not repeat, they undergo epicycles because the potential is not that of a point mass. A "spiral" orbit would imply that energy was being dissipated somehow (or added if the spiral were outward) - but the motion of stars in the galactic potential is essentially collisionless, there is no reason that this should happen.

• The ~10 km/s could be part of a spiral in or part of an elliptical orbit (the assumption being that we'd measure a ~10 km/s outward component on the other side of the Milky Way). Our assumptions about the missing mass (Dark Matter) in the Milky Way assumes that the material in the disk follows Keplar's rules (specifically, the 3rd law). I've read through the documents you've provided (thank you, btw), but I can't see any conclusions about the total orbit. Do you know if anyone has drawn a conclusion based in this kinematic data? – Quarkly Mar 10 '15 at 15:15
• Take a look at this simulation. astronomynow.com/news/n1109/21milkyway Mass/energy is often added to spiral galaxies as they merge. I think it's a mistake to think about them as a closed system like a solar system. – Quarkly Mar 14 '15 at 19:24

A rotation curve is an attempt to measure the mass enclosed as a function of radius based on the fact that:

$$v_{\rm circular} = \sqrt{\frac{GM(<r)}{r}}$$

As you correctly point out this depends critically on the assumption that the orbits are, in fact, circular. The motion of individual stars or bits of gas aren't that interesting in the context of the "missing mass" problem, but rather the collective behaviour of all the matter at a given radius. In a rotationally dominated system (e.g. classic spiral galaxy), you would typically measure some bulk rotation that is circular at every radius (think of a differentially rotating disk), and some velocity dispersion at each radius, which would account for motions like inward or outward radial migrations of some stars/gas, or motion that is not in the plane of the disk. As long as the system is rotationally dominated, as in the rotational component of the motion dominates the dispersion component, then it is straightforward to model the total mass profile. You just pay attention to the bulk rotation and ignore the other components when doing your modelling.

Things get more complicated if the system is dispersion dominated. Elliptical/spheroidal systems are still not too difficult to model, but things like dwarf irregulars that are not rotationally dominated can be a real mess. The usual term is "non-circular motions". It is still usually possible to get a total mass profile that is reliable by fitting a "tilted ring model". It is in these small systems that the presence of dark matter is most obvious since the dark matter mass exceeds the mass of luminous matter within any radius (for larger systems the luminous matter typically dominates the dark matter within the extent of the luminous galaxy).

Incidentally, the current version of the "missing mass" problem is that there seems to be some dark matter missing from the central regions of some galaxies. This is usually referred to as the "cusp-core problem" these days. This is distinct from the old problem, which was "things seem to rotate too fast given the amount of mass we think we can see". Rather, this is "the standard model for the dark matter distribution (based on standard LCDM cosmology) seems to overpredict the amount of dark matter in the central regions of some galaxies". I've just written an entire journal article on the topic, which I will try to remember to link as soon as is reasonable (e.g. a couple of months once it's refereed and so on).

• This answer has some glaring lack of referencing, which I will correct as soon as I have time, hopefully tomorrow. – Kyle Oman Mar 11 '15 at 0:12


text:
VGAM (version 1.0-4)

# zanegbinomial: Zero-Altered Negative Binomial Distribution

## Description

Fits a zero-altered negative binomial distribution based on a conditional model involving a binomial distribution and a positive-negative binomial distribution.

## Usage

zanegbinomial(zero = "size", type.fitted = c("mean", "munb", "pobs0"),
mds.min = 1e-3, nsimEIM = 500, cutoff.prob = 0.999,
eps.trig = 1e-7, max.support = 4000, max.chunk.MB = 30,
lpobs0 = "logit", lmunb = "loge", lsize = "loge",
imethod = 1, ipobs0 = NULL,
imunb = NULL, iprobs.y = NULL, gprobs.y = (0:9)/10,
isize = NULL, gsize.mux = exp(c(-30, -20, -15, -10, -6:3)))
zanegbinomialff(lmunb = "loge", lsize = "loge", lonempobs0 = "logit",
type.fitted = c("mean", "munb", "pobs0", "onempobs0"),
isize = NULL, ionempobs0 = NULL, zero = c("size",
"onempobs0"), mds.min = 1e-3, iprobs.y = NULL, gprobs.y = (0:9)/10,
cutoff.prob = 0.999, eps.trig = 1e-7, max.support = 4000,
max.chunk.MB = 30, gsize.mux = exp(c(-30, -20, -15, -10, -6:3)),
imethod = 1, imunb = NULL,
nsimEIM = 500)

## Arguments

lpobs0

Link function for the parameter $$p_0$$, called pobs0 here. See Links for more choices.

lmunb

Link function applied to the munb parameter, which is the mean $$\mu_{nb}$$ of an ordinary negative binomial distribution. See Links for more choices.

lsize

Parameter link function applied to the reciprocal of the dispersion parameter, called k. That is, as k increases, the variance of the response decreases. See Links for more choices.

type.fitted

See CommonVGAMffArguments and fittedvlm for information.

lonempobs0, ionempobs0

Corresponding argument for the other parameterization. See details below.

ipobs0, imunb, isize

Optional initial values for $$p_0$$ and munb and k. If given then it is okay to give one value for each response/species by inputting a vector whose length is the number of columns of the response matrix.

zero

Specifies which of the three linear predictors are modelled as intercept-only. All parameters can be modelled as a function of the explanatory variables by setting zero = NULL (not recommended). A negative value means that the value is recycled, e.g., setting $$-3$$ means all k are intercept-only for zanegbinomial. See CommonVGAMffArguments for more information.

nsimEIM, imethod

See CommonVGAMffArguments.

iprobs.y, gsize.mux, gprobs.y

See negbinomial.

cutoff.prob, eps.trig

See negbinomial.

mds.min, max.support, max.chunk.MB

See negbinomial.

## Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

The fitted.values slot of the fitted object, which should be extracted by the generic function fitted, returns the mean $$\mu$$ (default) which is given by $$\mu = (1-p_0) \mu_{nb} / [1 - (k/(k+\mu_{nb}))^k].$$ If type.fitted = "pobs0" then $$p_0$$ is returned.

## Warning

This family function is fragile; it inherits the same difficulties as posnegbinomial. Convergence for this VGAM family function seems to depend quite strongly on providing good initial values.

This VGAM family function is computationally expensive and usually runs slowly; setting trace = TRUE is useful for monitoring convergence.

Inference obtained from summary.vglm and summary.vgam may or may not be correct. In particular, the p-values, standard errors and degrees of freedom may need adjustment. Use simulation on artificial data to check that these are reasonable.

## Details

The response $$Y$$ is zero with probability $$p_0$$, or $$Y$$ has a positive-negative binomial distribution with probability $$1-p_0$$. Thus $$0 < p_0 < 1$$, which is modelled as a function of the covariates. The zero-altered negative binomial distribution differs from the zero-inflated negative binomial distribution in that the former has zeros coming from one source, whereas the latter has zeros coming from the negative binomial distribution too. The zero-inflated negative binomial distribution is implemented in the VGAM package. Some people call the zero-altered negative binomial a hurdle model.

For one response/species, by default, the three linear/additive predictors for zanegbinomial() are $$(logit(p_0), \log(\mu_{nb}), \log(k))^T$$. This vector is recycled for multiple species.

The VGAM family function zanegbinomialff() has a few changes compared to zanegbinomial(). These are: (i) the order of the linear/additive predictors is switched so the negative binomial mean comes first; (ii) argument onempobs0 is now 1 minus the probability of an observed 0, i.e., the probability of the positive negative binomial distribution, i.e., onempobs0 is 1-pobs0; (iii) argument zero has a new default so that the pobs0 is intercept-only by default. Now zanegbinomialff() is generally recommended over zanegbinomial(). Both functions implement Fisher scoring and can handle multiple responses.

## References

Welsh, A. H., Cunningham, R. B., Donnelly, C. F. and Lindenmayer, D. B. (1996) Modelling the abundances of rare species: statistical models for counts with extra zeros. Ecological Modelling, 88, 297--308.

Yee, T. W. (2014) Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics and Data Analysis, 71, 889--902.

dzanegbin, posnegbinomial, negbinomial, binomialff, rposnegbin, zinegbinomial, zipoisson, dnbinom, CommonVGAMffArguments, simulate.vlm.

## Examples

Run this code
# NOT RUN {
zdata <- data.frame(x2 = runif(nn <- 2000))
zdata <- transform(zdata, pobs0 = logit(-1 + 2*x2, inverse = TRUE))
zdata <- transform(zdata,
y1 = rzanegbin(nn, munb = exp(0+2*x2), size = exp(1), pobs0 = pobs0),
y2 = rzanegbin(nn, munb = exp(1+2*x2), size = exp(1), pobs0 = pobs0))
with(zdata, table(y1))
with(zdata, table(y2))

fit <- vglm(cbind(y1, y2) ~ x2, zanegbinomial, data = zdata, trace = TRUE)
coef(fit, matrix = TRUE)


text:
MMM LXV

Image Details

Once again, this image is a result of the nebulabrot rendering technique. It is the result of applying that technique to a the standard Mandelbrot function of degree seven, i.e.

$$z_{k+1} = {z_k}^7 + z_0$$.

The coloring was done a little differently than usual. In the past, individual layers were colored, then the layers were “added” together, pixel by pixel. This image, on the other hand, started with a layer of white, then layers of nebulabrot renderings were subtracted. Because the base image is white, rather than black, the final image doesn’t have the space-like appearance of previous renderings.

This entry was posted in MMM and tagged , , . Bookmark the permalink.


text:
I need to highlight specific entries of my beamer navigation according to highlighted entries within an overview figure. For this purpose, I need to redraw the navigation from within the frame environment. The problem: The current solution requires a manual adjustment of the alignment. What can be considered as hacky. A cleaner solution for the exact alignment at the original position would be highly appreciated. Big thx in advance for your ideas and suggestions!

\documentclass[t]{beamer}

\usetheme{Singapore}
\useoutertheme{miniframes}

\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\section{Section}
\subsection{Subsection}

\begin{frame}[c]
\frametitle{Title}
\framesubtitle{Subtitle}
%
\begin{beamerboxesrounded}[]{}
The intention is to redraw the navigation from \emph{within} the frame environment to e.\,g. allow highlights of specific sections. The problem: The current solution needs a manual adjustment of the alignment. What can be considered as hacky. A cleaner solution for the \emph{exact} alignment at the original position would be highly appreciated.
\end{beamerboxesrounded}
%
\begin{tikzpicture}[remember picture,overlay]%
{\node[anchor=north west,text width=\paperwidth] at ($(current page.north west) + (0ex,0ex)$) {%
\end{beamercolorbox}};}%
\end{tikzpicture}
%
\end{frame}

\end{document}

• Even after adding the obvious things like \documentclass{beamer}, \usepackage{tikz} and \begin{document}...\end{document} your code fragment does not compile nor would nor would it show any navigation bar. – user36296 Jul 30 '17 at 11:28
• @samcarter: Added MWE as requested ;) – Albert Jul 30 '17 at 19:43
• What exactly do you want to highlight? I am pretty sure redrawing the whole mini frames navigation is not the best way to do that, there might be other ways to achieve the highlighting using beamer templates, but it's hard to say without knowing your actual requirements. – diabonas Sep 29 '17 at 8:12
• @diabonas: My aim is to highlight specific sections (\alert<1>{Section 1}, \alert<2>{Section 2}, ...) according to an overview figure that is presented step by step. – Albert Oct 1 '17 at 14:45
• Ok, to make sure I've understood you correctly: you want to have one slide (\begin{frame}...\end{frame}) with multiple overlays where on each overlay you highlight a different (only one?) section in the navigation bar? How should the highlight look like, is it the same as if you were actually in this section (so simply in a darker blue font)? Which of the mini frames (the little circles below the section name) should be filled out? – diabonas Oct 1 '17 at 15:03

Partial solution

The following code reduces the manual spacing to only one dimension, but unless you use some technique like \tikzmark to actually record the position of the navigation bar, adjusting the spacing in vertical direction seems necessary, as the original navigation bar is not on the very top of the frame.

The original definition of the headline in the miniframe theme is

\defbeamertemplate*{headline}{miniframes theme}
{%
\end{beamercolorbox}
\end{beamercolorbox}%
\ifbeamer@theme@subsection%
\end{beamercolorbox}
\begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,%
\end{beamercolorbox}%
\fi%
\end{beamercolorbox}
}

As you can see, this definition includes

• manual spacing, e.g. the \vskip2pt
• a separation line at the top of the frame and
• the margins of the colorboxes

This space has to be somehow compensated.

\documentclass[t]{beamer}

\usetheme{Singapore}
\useoutertheme{miniframes}

\usepackage{tikz}
\usetikzlibrary{calc,positioning}

\begin{document}

\section{Section}
\subsection{Subsection}

\section{Section}
\subsection{Subsection}

\begin{frame}[c]
\frametitle{Title}
\framesubtitle{Subtitle}
%
\begin{beamerboxesrounded}[]{}
The intention is to redraw the navigation from \emph{within} the frame environment to e.\,g. allow highlights of specific sections. The problem: The current solution needs a manual adjustment of the alignment. What can be considered as hacky. A cleaner solution for the \emph{exact} alignment at the original position would be highly appreciated.
\end{beamerboxesrounded}
\begin{tikzpicture}[remember picture,overlay]%
{\node[text width=\paperwidth] at (current page.north) {%
\vskip13.25pt%


text:
+0

0
42
1

Let $$\mathbf{A}$$ be a matrix with columns $$\mathbf{col}_1, \mathbf{col}_2, and \mathbf{col}_3.$$ Consider points T, U, V, W in the output grid below: Then what is the list$$\mathbf{A}^{-1} \overrightarrow{T}, \mathbf{A}^{-1} \overrightarrow{U}, \mathbf{A}^{-1} \overrightarrow{V}, \mathbf{A}^{-1} \overrightarrow{W}?$$

Mar 31, 2020


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# Article

 Title: The periodic problem for semilinear differential inclusions in Banach spaces (English) Author: Bader, Ralf Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 39 Issue: 4 Year: 1998 Pages: 671-684 . Category: math . Summary: Sufficient conditions on the existence of periodic solutions for semilinear differential inclusions are given in general Banach space. In our approach we apply the technique of the translation operator along trajectories. Due to recent results it is possible to show that this operator is a so-called decomposable map and thus admissible for certain fixed point index theories for set-valued maps. Compactness conditions are formulated in terms of the Hausdorff measure of noncompactness. (English) Keyword: periodic solutions Keyword: translation operator along trajectories Keyword: set-valued maps Keyword: $C_0$-semigroup Keyword: $R_\delta$-sets MSC: 34A60 MSC: 34C25 MSC: 34G25 MSC: 47H11 MSC: 47N20 idZBL: Zbl 1060.34508 idMR: MR1715457 . Date available: 2009-01-08T18:47:24Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119043 . Reference: [1] Akhmerov R.R., Kamenskii M.I., Potapov A.S., Rodkina A.E., Sadovskii B.N.: Measures of Noncompactness and Condensing Operators.Birkhäuser Verlag, Basel-Boston-Berlin, 1992. MR 1153247 Reference: [2] Bader R.: Fixpunktindextheorie mengenwertiger Abbildungen und einige Anwendungen.Ph.D. Dissertation, Universität München, 1995. Zbl 0865.47049 Reference: [3] Bader R.: Fixed point theorems for compositions of set-valued maps with single-valued maps.submitted. Zbl 1012.47043 Reference: [4] Bader R., Kryszewski W.: Fixed-point index for compositions of set-valued maps with proximally $\infty$-connected values on arbitrary ANR's.Set-Valued Analysis 2 (1994), 459-480. Zbl 0846.55001, MR 1304049 Reference: [5] Bothe D.: Multivalued perturbations of $m$-accretive differential inclusions.to appear in Israel J. Math. Zbl 0922.47048, MR 1669396 Reference: [6] Conti G., Obukhovskii V., Zecca P.: On the topological structure of the solution set for a semilinear functional-differential inclusion in a Banach space.in: Topology in Nonlinear Analysis, K. Geba and L. Górniewicz (eds.), Polish Academy of Sciences, Institute of Mathematics, Banach Center Publications 35, Warszawa, 1996, pp.159-169. MR 1448435 Reference: [7] Deimling K.: Multivalued differential equations.de Gruyter, Berlin-New York, 1992. Zbl 0820.34009, MR 1189795 Reference: [8] Diestel J.: Geometry of Banach Spaces - Selected Topics.LNM 485, Springer-Verlag, Berlin-Heidelberg-New York, 1975. Zbl 0466.46021, MR 0461094 Reference: [9] Górniewicz L.: Topological approach to differential inclusions in: Topological methods in differential equations and inclusions, A. Granas and M. Frigon (eds.), NATO ASI Series C 472, Kluwer Academic Publishers, 1995, pp.129-190.. MR 1368672 Reference: [10] Hyman D.M.: On decreasing sequences of compact absolute retracts.Fund. Math. 64 (1969), 91-97. Zbl 0174.25804, MR 0253303 Reference: [11] Kamenskii M.I., Obukhovskii V.V.: Condensing multioperators and periodic solutions of parabolic functional-differential inclusions in Banach spaces.Nonlinear Anal. 20 (1993), 781-792. MR 1214743 Reference: [12] Kamenskii M., Obukhovskii V., Zecca P.: Condensing multivalued maps and semilinear differential inclusions in Banach spaces.book in preparation. Zbl 0988.34001 Reference: [13] Kamenskii M., Obukhovskii V., Zecca P.: On the translation multioperator along the solutions of semilinear differential inclusions in Banach spaces.to appear in Rocky Mountain J. Math. MR 1661823 Reference: [14] Krasnoselskii M.A.: The operator of translation along trajectories of differential equations.American Math. Soc., Translation of Math. Monographs, vol. 19, Providence, 1968. MR 0223640 Reference: [15] Lasota A., Opial Z.: Fixed-point theorems for multi-valued mappings and optimal control problems.Bull. Polish Acad. Sci. Math. 16 (1968), 645-649. Zbl 0165.43304, MR 0248580 Reference: [16] Mönch H., von Harten G.-F.: On the Cauchy problem for ordinary differential equations in Banach spaces.Archiv Math. 39 (1982), 153-160. MR 0675655 Reference: [17] Muresan M.: On a boundary value problem for quasi-linear differential inclusions of evolution.Collect. Math. 45 2 (1994), 165-175. Zbl 0824.34017, MR 1316934 Reference: [18] Papageorgiou N.S.: Boundary value problems for evolution inclusions.Comment. Math. Univ. Carolinae 29 (1988), 355-363. Zbl 0696.35074, MR 0957404 Reference: [19] Pazy A.: Semigroups of linear operators and applications to partial differential equations.Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. Zbl 0516.47023, MR 0710486 .

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