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# Joint Altitude, Beamwidth, Location and Bandwidth Optimization for UAV-Enabled Communications

This letter investigates an uplink power control problem for unmanned aerial vehicles (UAVs) assisted wireless communications. We jointly optimize the UAV's flying altitude, antenna beamwidth, UAV's location and ground terminals' allocated bandwidth and transmit power to minimize the sum uplink power subject to the minimal rate demand. An iterative algorithm is proposed with low complexity to obtain a suboptimal solution. Numerical results show that the proposed algorithm can achieve good performance in terms of uplink sum power saving.

## Authors

• 39 publications
• 37 publications
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• 146 publications
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• 45 publications
• ### Joint Location and Power Optimization for THz-enabled UAV Communications

In this paper, the problem of unmanned aerial vehicle (UAV) deployment a...
11/02/2020 ∙ by Luyao Xu, et al. ∙ 0

• ### Cellular UAV-to-UAV Communications

Reliable and direct communication between unmanned aerial vehicles (UAVs...
04/10/2019 ∙ by M. Mahdi Azari, et al. ∙ 0

• ### UAV Positioning and Power Control for Two-Way Wireless Relaying

This paper considers an unmanned-aerial-vehicle-enabled (UAV-enabled) wi...
04/17/2019 ∙ by Lei Li, et al. ∙ 0

• ### UAV Data Collection over NOMA Backscatter Networks: UAV Altitude and Trajectory Optimization

The recent evolution of ambient backscattering technology has the potent...
02/08/2019 ∙ by Amin Farajzadeh, et al. ∙ 0

• ### Power Control for a URLLC-enabled UAV system incorporated with DNN-Based Channel Estimation

This letter is concerned with power control for a ultra-reliable and low...
11/14/2020 ∙ by Peng Yang, et al. ∙ 0

• ### Location-aware Predictive Beamforming for UAV Communications: A Deep Learning Approach

Unmanned aerial vehicle (UAV)-assisted communication becomes a promising...
09/16/2020 ∙ by Chang Liu, et al. ∙ 0

09/17/2020 ∙ by Shu Sun, et al. ∙ 0

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## I Introduction

Unmanned aerial vehicles (UAVs) assisted wireless communications have attracted considerable attention recently due to its maneuverability and increasing affordability [1]. Compared to conventional wireless communications, UAV-enabled wireless communications can provide higher wireless connectivity in areas without infrastructure coverage and achieve higher capacity for line-of-sight (LoS) communication links with the ground terminals (GTs).

To fully exploit the design degrees of freedom for UAV-enabled communications, it is crucial to investigate the UAV mobility in the three-dimensional space.

In [2], the altitude of UAV was optimized to provide maximum radio coverage on the ground. For an underlaid device-to-device (D2D) communication network with one UAV, the optimal values for the UAV altitude were analyzed in [3]

for the maximum system sum rate and coverage probability. Considering the adjustable UAVs’ locations over time, the UAV number and trajectory optimization problems were respectively considered in

[4] and [5]. Further optimizing user-UAV association, [6] investigated the sum power minimization problem of the UAV. Different from [2, 3, 4, 5, 6] with fixed-beamwidth antenna, the beamwidth of the directional antenna and the altitude of the UAV were jointly optimized in [7] to improve the system throughput. However, the optimal beamwidth was only examined numerically and simple equal bandwidth allocation was assumed in [7], even though proper bandwidth allocation can further enhance the system performance.

In this letter, we aim to minimize the sum power for an uplink UAV-enabled wireless communication. There are two main contributions. One contribution is that we consider joint altitude, beamwidth, location and bandwidth allocation, and an algorithm is proposed by solving three subproblems iteratively, where each subproblem can be solved optimally. We also provide the complexity analysis of the proposed algorithm. Numerical results verify that the proposed algorithm outperforms the existing algorithms with fixed beamwidth or bandwidth allocation in terms of sum power, especially when the minimal rate demand is high. The other contribution is to effectively obtain the optimal beamwidth with the bisection method when the pathloss exponent is two, and to obtain the optimal solution in closed form for bandwidth allocation subproblem.

## Ii System Model and Problem Formulation

Consider an uplink UAV-enabled wireless communication system with one flying UAV and GTs. The UAV is deployed as a flying BS with horizontal and vertical location at hight . The horizontal and vertical location of GT is denoted by , and the hight of each GT is assumed to be zero compared with the hight of the UAV.

Assume that the UAV is equipped with a directional antenna with adjustable beamwidth, while each GT is equipped with an omnidirectional antenna with unit gain. The azimuth and elevation half-power beamwidths of the UAV antenna are equal, which are both denoted by . According to [8, Eq. (2-51)], the antenna gain in the direction with azimuth angle and elevation angle can be modeled as

 G={G0Θ2if 0≤θ≤Θ and 0≤ϕ≤Θg≈0otherwise, (1)

where , and means the channel gain outside the beamwidth of the antenna. For simplify, we set . We consider the case that the GTs are located outdoors, and the channel between each GT and the UAV is mainly a LoS path. The uplink channel gain between GT and the UAV is

 (2)

where denotes the Euclidian norm, is the channel power gain at the reference distance 1 m, is the hight of the UAV, is the distance between GT and the UAV, and is the pathloss exponent. Based on (1) and (2), the uplink achievable rate of GT in the coverage area of the UAV is

 rk=wklog2⎛⎝1+pkg0G0wkσ2Θ2(∥y−xk∥2+H2)α2⎞⎠, (3)

where is the allocated bandwidth for GT , is the transmit power of GT , is the noise power density and is the noise power for decoding the information of GT at the UAV side. For GT , the minimal rate constraint should be satisfied. Since we aim at minimizing uplink sum power of all GTs, it is always energy saving to transmit with minimal rate. Setting in (3) , we have

 pk=awk(2Rkwk−1)Θ2(∥y−xk∥2+H2)α2, (4)

where .

We aim at minimizing the uplink sum power of all GTs whilst satisfying the minimal rate constraints. Mathematically, the sum power minimization problem is formulated as

 minH,Θ,y,w K∑k=1awk(2Rkwk−1)Θ2(∥y−xk∥2+H2)α2 (5a) s.t. awk(2Rkwk−1)Θ2(∥y−xk∥2+H2)α2≤Pk, ∀k=1,⋯,K (5b) ∥y−xk∥2≤H2tan2Θ,∀k=1,⋯,K (5c) K∑k=1wk≤B (5d) Hmin≤H≤Hmax,Θmin≤Θ≤Θmax (5e) wk≥0,∀k=1,⋯,K. (5f)

where , is the maximal bandwidth of the system, is the maximum transmit power of GT , is the feasible region of height determined by obstacle heights and authority regulations, and is the feasible region of half-beamwidth determined by practical antenna beamwidth tuning technique. Constraints in (5c) ensure that all GTs are in the coverage area of the UAV.

## Iii Proposed Algorithm

Due to nonconvex objective function (5a) and constraints (5b)-(5c), Problem (5) is a nonconvex problem. It is generally hard to obtain the globally optimal solution to Problem (5). To solve this problem, we propose an iterative algorithm with low complexity through sequently optimizing , and . It is fortunate that we can globally optimize each variable with other variables fixed in each step.

### Iii-a Optimal Altitude and Beamwidth

With fixed and , Problem (5) is formulated as

 minH,Θ K∑k=1AkΘ2(Dk+H2)α2 (6a) s.t. AkΘ2(Dk+H2)α2≤Pk,∀k=1,⋯,K (6b) H2tan2Θ≥Dmax (6c) Hmin≤H≤Hmax,Θmin≤Θ≤Θmax, (6d)

where , , and . Denoting as optimal value of and observing that the objective function (6a) is an increasing function in with given , we can claim that

 H∗=max{√DmaxtanΘ,Hmin}, (7)

for the optimal solution. This claim can be proved by the contradiction method. If is the optimal solution of Problem (6) with , we find that solution is also a feasible solution of Problem (6) with , which contradicts the hypothesis that is the optimal solution. Based on (7), we consider the value of in the following two cases.

1) Case 1: With , Problem (6) is equivalent to

 minΘ Θ (8a) s.t. AkΘ2(Dk+H2min)α2≤Pk,∀k=1,⋯,K (8b) H2mintan2Θ≥Dmax (8c) Θmin≤Θ≤Θmax. (8d)

Since Problem (8) is a minimization of , the optimal solution is thus

 Θ∗=max{arctan√DmaxHmin,Θmin}, (9)

which is the minimal value of satisfying (8b), (8c) and (8d). Note that Problem (8) is feasible if and only if

 Θ∗≤min⎧⎨⎩mink=1,⋯,K ⎷PkAk(Dk+H2min)α2,Θmax⎫⎬⎭. (10)

2) Case 2: With , Problem (6) becomes

 minΘ K∑k=1AkΘ2(Dk+Dmaxtan2Θ)α2 (11a) s.t. AkΘ2(Dk+Dmaxtan2Θ)α2≤Pk,∀k=1,⋯,K (11b) H2mintan2Θ≤Dmax (11c) Θmin≤Θ≤Θmax. (11d)

Due to the complicated objective function (11a), it is generally difficult to obtain the optimal of Problem (11) in closed form. can be obtained via a one-dimension exhaustive search over .

Specifically, for the special case where pathloss exponent , we can fortunately obtain the optimal through a simple bisection method. When the GTs are located outdoors in rural areas, and the communication channel between the UAV and each GT is dominated by the LoS path, i.e., [7, Eq. (2)]. For , we define function

 fk(x)=x2(Dk+Dmaxtan2x),x∈[0,π/2), (12)
 h1(x)=(cotx−x−xcot2x)cotx, (13)
 h2(x)=x+2xcos2(x)−3/2sin(2x), (14)

and then we have

 f′k(x)=2x(Dk+Dmaxh1(x)), (15)
 h′1(x)=csc4(x)h2(x), (16)
 h′2(x)=−4(xcos(x)−sin(x))sin(x), (17)

for . Since for , we have , i.e., . As a result, , , i.e., is an increasing function, and . Due to that , is equivalent to . To show the monotonicity of , we consider the following two situations:

• If , then for all , i.e., is monotonically increasing.

• If , there must exist one solution such that due to the fact that and is a continuous function. In this situation, first decreases for and then increases with .

According to the above analysis, is equivalent to , where and can be obtained by using the bisection method. As a result, constraints (11b)-(11d) can be equivalently transformed to

 ¯Θmin≤Θ≤¯Θmax, (18)

where , .

Based on the definition of , the objective function (11a) can be expressed as . We have . To show the monotonicity of in , we also consider the following three situations:

• If , then for , i.e., is monotonically increasing. The optimal beamwidth is .

• If , is monotonically decreasing and .

• If and , there must exist one solution such that . In this situation, first decreases for and then increases with , i.e., .

Note that Problem (11) is feasible if and only if . By comparing the objective values of the solutions obtained in the above two cases, the one with lower objective value is chosen as the optimal solution to Problem (6).

### Iii-B Optimal Location Planning

For Problem (5) with fixed and , the location planning problem can be formulated as

 miny K∑k=1Ck(∥y−xk∥2+H2)α2 (19a) s.t. ∥y−xk∥2+H2≤¯Ek,∀k=1,⋯,K (19b) ∥y−xk∥2≤H2tan2Θ,∀k=1,⋯,K, (19c)

where , and . Since is a convex function and is convex and nondecreasing, is convex based on the scalar composition property of convex functions [9]. As a result, Problem (19) is a convex problem, which can be effectively solved via the standard interior point method.

### Iii-C Optimal Bandwidth Allocation

It remains to investigate the bandwidth allocation with fixed location, altitude and beamwidth. Define function for , and we have

 u′k(x)=2Rkx−(ln2)Rk2Rkxx−1,u′′k(x)=(ln2)2R2k2Rkxx3>0. (20)

From (20), we observe that function is a convex function, which indicates that Problem (5) is a convex problem with fixed and . Based on (20),we have , i.e., is a monotonically decreasing function, which is helpful in transforming constraints (5b). With optimized and , Problem (5) is equivalent to

 minw K∑k=1Fkwk(2Rkwk−1) (21a) s.t. K∑k=1wk≤B (21b) wk≥Wk∀k=1,⋯,K, (21c)

where , , and is the inverse function of . The lagrangian of convex Problem (21) is

 L(w,λ)=K∑k=1Fkwk(2Rkwk−1)+λ(K∑k=1wk−B), (22)

where is the non-negative Lagrange multiplier associated with constraint (21b). According to [9] and [10, Appendix A], the KKT conditions of (21) are

 ∂L∂wk=Fk(2Rkwk−(ln2)Rkwk2Rkwk−1)+λ=0 (23)

From (23), we have

 λ=Fk(−e(ln2)Rkwk+(ln2)Rkwke(ln2)Rkwk+1). (24)

Define function , . We have , . Thus, function is strictly increasing and , . Based on (24) and (21c), we have

 wk=max⎧⎪ ⎪⎨⎪ ⎪⎩(ln2)Rku−1(λFk),Wk⎫⎪ ⎪⎬⎪ ⎪⎭,∀k=1,⋯,K, (25)

where is the inverse function of . According to (24), , which implies that (21b) holds with equality. Plugging (25) into (21b) yields

 B=K∑k=1max⎧⎪ ⎪⎨⎪ ⎪⎩(ln2)Rku−1(λFk),Wk⎫⎪ ⎪⎬⎪ ⎪⎭≜^u(λ). (26)

Equation (26) has a unique solution . Since is strictly increasing, inverse function is also strictly increasing in . Thus, is a strictly decreasing function in . Owing to the fact that and , there exists one unique satisfying , and the solution can be obtained by using the bisection method. Having obtained the value of , the optimal can be obtained from (25).

### Iii-D Iterative Algorithm and Complexity Analysis

The iterative procedure for solving Problem (5) is given in Algorithm 1. The main complexity of Algorithm 1 lies in Problem (6) and Problem (21). For Problem (6), the major computation comes from case 2, which needs to solve Problem (11) via a one-dimension exhaustive search method with complexity for minimal step . To solve Problem (21), the major complexity lies in solving (26) with complexity for the bisection method with accuracy . As a result, the total complexity of Algorithm 1 is , where is the number of iterations of the iterative Algorithm 1.

## Iv Numerical Results

We consider that there are

GTs uniformly distributed in a circular area with radius

m. We set , MHz, dBm, dBm/Hz, m, m, , and rad. We consider equal minimal rate demand, i.e., .

In Fig. 1, we consider the sum power (11a), which equals to (5a) with fixed and , versus for various minimal rate demands with equal bandwidth allocation and the UAV located at the center of the circle. With given minimal rate demand, it is observed that the sum power first decreases and then increases with the increase of , which verifies the theoretical analysis in Section III.A.

We compare the proposed algorithm with the following four methods: fixed location method with optimized altitude, beamwidth and bandwidth (labeled as ‘FL’), fixed altitude and beamwidth method with optimized location and bandwidth (labeled as ‘FAB’), fixed bandwidth method with optimized location altitude and beamwidth (labeled as ‘FB’), and exhaustive method via running Algorithm 1 with 1000 initial points (labeled as ‘Exhaustive’). We investigate the sum power versus the minimal rate demand in Fig. 2. It can be seen that the proposed algorithm outperforms FL, FAB and FB, especially when the minimal rate demand is large. This is because the proposed algorithm jointly optimizes altitude, beamwidth, location and bandwidth. It can be seen that the sum power of the exhaustive method is slightly lower than that of the proposed algorithm, which indicates that the proposed algorithm approaches the globally optimal solution.

## V Conclusion

In this letter, we investigated the sum power minimization problem in uplink UAV-enabled communications. We showed that the sum power first decreases and then decreases with the beamwidth. Numerical results showed that the uplink sum power performance can be improved by the proposed algorithm.

## References

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## Results (1-50 of 453 matches)

Next
Label $\alpha$ $A$ $d$ $N$ $\chi$ $\mu$ $\nu$ $w$ prim arith $\mathbb{Q}$ self-dual $\operatorname{Arg}(\epsilon)$ $r$ First zero Origin
2-1441-1441.130-c0-0-0 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.130 $$0.0 0 -0.217 0 0.325713 Modular form 1441.1.u.a.130.1 2-1441-1441.130-c0-0-1 0.848 0.719 2 11 \cdot 131 1441.130$$ $0.0$ $0$ $-0.417$ $0$ $0.515793$ Modular form 1441.1.u.a.130.2
2-1441-1441.130-c0-0-2 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.130 $$0.0 0 -0.0172 0 1.06009 Modular form 1441.1.u.a.130.3 2-1441-1441.130-c0-0-3 0.848 0.719 2 11 \cdot 131 1441.130$$ $0.0$ $0$ $0.182$ $0$ $1.86122$ Modular form 1441.1.u.a.130.5
2-1441-1441.130-c0-0-4 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.130 $$0.0 0 0.382 0 2.41893 Modular form 1441.1.u.a.130.4 2-1441-1441.654-c0-0-0 0.848 0.719 2 11 \cdot 131 1441.654$$ $0.0$ $0$ $0.217$ $0$ $0.826779$ Modular form 1441.1.u.a.654.1
2-1441-1441.654-c0-0-1 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.654 $$0.0 0 -0.382 0 0.922909 Modular form 1441.1.u.a.654.4 2-1441-1441.654-c0-0-2 0.848 0.719 2 11 \cdot 131 1441.654$$ $0.0$ $0$ $-0.182$ $0$ $1.25191$ Modular form 1441.1.u.a.654.5
2-1441-1441.654-c0-0-3 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.654 $$0.0 0 0.0172 0 1.33723 Modular form 1441.1.u.a.654.3 2-1441-1441.654-c0-0-4 0.848 0.719 2 11 \cdot 131 1441.654$$ $0.0$ $0$ $0.417$ $0$ $1.93637$ Modular form 1441.1.u.a.654.2
2-1441-1441.785-c0-0-0 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.785 $$0.0 0 0.376 0 0.0556337 Modular form 1441.1.u.a.785.3 2-1441-1441.785-c0-0-1 0.848 0.719 2 11 \cdot 131 1441.785$$ $0.0$ $0$ $-0.423$ $0$ $1.03024$ Modular form 1441.1.u.a.785.1
2-1441-1441.785-c0-0-2 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.785 $$0.0 0 -0.223 0 1.11117 Modular form 1441.1.u.a.785.2 2-1441-1441.916-c0-0-0 0.848 0.719 2 11 \cdot 131 1441.916$$ $0.0$ $0$ $-0.176$ $0$ $1.59040$ Modular form 1441.1.u.a.916.5
2-1441-1441.916-c0-0-1 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.916 $$0.0 0 0.0237 0 1.59425 Modular form 1441.1.u.a.916.4 2-1441-1441.916-c0-0-2 0.848 0.719 2 11 \cdot 131 1441.916$$ $0.0$ $0$ $0.223$ $0$ $1.60724$ Modular form 1441.1.u.a.916.2
2-1441-1441.916-c0-0-3 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.916 $$0.0 0 0.423 0 2.29300 Modular form 1441.1.u.a.916.1 2-1441-1441.916-c0-0-4 0.848 0.719 2 11 \cdot 131 1441.916$$ $0.0$ $0$ $-0.376$ $0$ $2.73235$ Modular form 1441.1.u.a.916.3
2-1441-1441.785-c0-0-3 $0.848$ $0.719$ $2$ $11 \cdot 131$ 1441.785 $$0.0 0 -0.0237 0 1.11721 Modular form 1441.1.u.a.785.4 2-1441-1441.785-c0-0-4 0.848 0.719 2 11 \cdot 131 1441.785$$ $0.0$ $0$ $0.176$ $0$ $1.96735$ Modular form 1441.1.u.a.785.5
2-1441-1.1-c1-0-28 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0 0 0.891551 Modular form 1441.2.a.e.1.15 2-1441-1.1-c1-0-29 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $0.905368$ Modular form 1441.2.a.f.1.2
2-1441-1.1-c1-0-3 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0 0 0.294989 Modular form 1441.2.a.e.1.6 2-1441-1.1-c1-0-30 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $0.926359$ Modular form 1441.2.a.e.1.14
2-1441-1.1-c1-0-31 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0 0 0.938457 Modular form 1441.2.a.f.1.5 2-1441-1.1-c1-0-32 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $0.952588$ Modular form 1441.2.a.f.1.8
2-1441-1.1-c1-0-33 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0 0 0.986048 Modular form 1441.2.a.e.1.9 2-1441-1.1-c1-0-34 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0.5$ $1$ $1.00558$ Modular form 1441.2.a.d.1.6
2-1441-1.1-c1-0-35 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0.5 1 1.03896 Modular form 1441.2.a.c.1.11 2-1441-1.1-c1-0-36 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0.5$ $1$ $1.05117$ Modular form 1441.2.a.c.1.8
2-1441-1.1-c1-0-37 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0 0 1.06721 Modular form 1441.2.a.f.1.30 2-1441-1.1-c1-0-38 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $1.06867$ Modular form 1441.2.a.e.1.19
2-1441-1.1-c1-0-39 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0 0 1.09382 Modular form 1441.2.a.e.1.3 2-1441-1.1-c1-0-4 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $0.297343$ Modular form 1441.2.a.f.1.1
2-1441-1.1-c1-0-40 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0.5 1 1.10166 Modular form 1441.2.a.c.1.13 2-1441-1.1-c1-0-41 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $1.15294$ Modular form 1441.2.a.e.1.23
2-1441-1.1-c1-0-42 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0 0 1.16448 Modular form 1441.2.a.e.1.4 2-1441-1.1-c1-0-43 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0.5$ $1$ $1.22131$ Modular form 1441.2.a.d.1.10
2-1441-1.1-c1-0-44 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0.5 1 1.23550 Modular form 1441.2.a.c.1.16 2-1441-1.1-c1-0-45 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $1.24631$ Modular form 1441.2.a.f.1.19
2-1441-1.1-c1-0-46 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0.5 1 1.25121 Modular form 1441.2.a.d.1.12 2-1441-1.1-c1-0-47 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $1.25729$ Modular form 1441.2.a.f.1.14
2-1441-1.1-c1-0-48 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0.5 1 1.26672 Modular form 1441.2.a.d.1.1 2-1441-1.1-c1-0-49 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $1.28357$ Modular form 1441.2.a.e.1.7
2-1441-1.1-c1-0-5 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0 0 0.323626 Modular form 1441.2.a.f.1.18 2-1441-1.1-c1-0-50 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $1.29398$ Modular form 1441.2.a.e.1.10
2-1441-1.1-c1-0-51 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0.5 1 1.30723 Modular form 1441.2.a.d.1.4 2-1441-1.1-c1-0-52 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0.5$ $1$ $1.30857$ Modular form 1441.2.a.c.1.4
2-1441-1.1-c1-0-53 $3.39$ $11.5$ $2$ $11 \cdot 131$ 1.1 $$1.0 1 0.5 1 1.34459 Modular form 1441.2.a.d.1.16 2-1441-1.1-c1-0-54 3.39 11.5 2 11 \cdot 131 1.1$$ $1.0$ $1$ $0$ $0$ $1.37482$ Modular form 1441.2.a.e.1.27

format: plain


text:
## The Annals of Probability

### The local quantization behavior of absolutely continuous probabilities

#### Abstract

For a large class of absolutely continuous probabilities $P$ it is shown that, for $r>0$, for $n$-optimal $L^{r}(P)$-codebooks $\alpha_{n}$, and any Voronoi partition $V_{n,a}$ with respect to $\alpha_{n}$ the local probabilities $P(V_{n,a})$ satisfy $P(V_{a,n})\approx n^{-1}$ while the local $L^{r}$-quantization errors satisfy $\int_{V_{n,a}}\|x-a\|^{r}\,dP(x)\approx n^{-(1+r/d)}$ as long as the partition sets $V_{n,a}$ intersect a fixed compact set $K$ in the interior of the support of $P$.

#### Article information

Source
Ann. Probab., Volume 40, Number 4 (2012), 1795-1828.

Dates
First available in Project Euclid: 4 July 2012

https://projecteuclid.org/euclid.aop/1341401149

Digital Object Identifier
doi:10.1214/11-AOP663

Mathematical Reviews number (MathSciNet)
MR2978138

Zentralblatt MATH identifier
1260.60032

#### Citation

Graf, Siegfried; Luschgy, Harald; Pagès, Gilles. The local quantization behavior of absolutely continuous probabilities. Ann. Probab. 40 (2012), no. 4, 1795--1828. doi:10.1214/11-AOP663. https://projecteuclid.org/euclid.aop/1341401149

#### References

• [1] Bucklew, J. A. and Wise, G. L. (1982). Multidimensional asymptotic quantization theory with $r$th power distortion measures. IEEE Trans. Inform. Theory 28 239–247.
• [2] Chernaya, E. V. (1995). An asymptotically sharp estimate for the remainder of weighted cubature formulas that are optimal on certain classes of continuous functions. Ukraïn. Mat. Zh. 47 1405–1415.
• [3] Chornaya, E. V. (1995). On the optimization of weighted cubature formulae on certain classes of continuous functions. East J. Approx. 1 47–60.
• [4] Cohn, D. L. (1980). Measure Theory. Birkhäuser, Boston, MA.
• [5] Delattre, S., Graf, S., Luschgy, H. and Pagès, G. (2004). Quantization of probability distributions under norm-based distortion measures. Statist. Decisions 22 261–282.
• [6] Fort, J.-C. and Pagès, G. (2002). Asymptotics of optimal quantizers for some scalar distributions. J. Comput. Appl. Math. 146 253–275.
• [7] Gersho, A. (1979). Asymptotically optimal block quantization. IEEE Trans. Inform. Theory 25 373–380.
• [8] Graf, S. and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions. Lecture Notes in Math. 1730. Springer, Berlin.
• [9] Graf, S. and Luschgy, H. (2002). Rates of convergence for the empirical quantization error. Ann. Probab. 30 874–897.
• [10] Graf, S., Luschgy, H. and Pagès, G. (2008). Distortion mismatch in the quantization of probability measures. ESAIM Probab. Stat. 12 127–153.
• [11] Gray, R. M. and Neuhoff, D. L. (1998). Quantization. IEEE Trans. Inform. Theory 44 2325–2383.
• [12] Gruber, P. M. (2004). Optimum quantization and its applications. Adv. Math. 186 456–497.
• [13] Pagès, G. (1993). Voronoi tessellation, space quantization algorithms and numerical integration. In Proc. ESANN’93 (M. Verleysen, ed.) 221–228. Quorum Editions, Bruxelles.
• [14] Pagès, G. (1998). A space quantization method for numerical integration. J. Comput. Appl. Math. 89 1–38.
• [15] Pagès, G. and Printems, J. (2009). Optimal quantization for finance: From random vectors to stochastic processes. In Mathematical Modeling and Numerical Methods in Finance (Special Volume) (A. Bensoussan and Q. Zhang, guest eds.), Coll. Handbook of Numerical Analysis (P. G. Ciarlet, ed.) 595–649. North-Holland, Amsterdam.
• [16] Pagès, G. and Sagna, A. (2012). Asymptotics of the maximal radius of an $L^{r}$-optimal sequence of quantizers. Bernoulli 18 360–389.
• [17] Zador, P. L. (1963). Development and evaluation of procedures for quantizing multivariate distributions. Ph.D. thesis, Stanford Univ.
• [18] Zador, P. L. (1982). Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28 139–149.


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# Posts Tagged proton

## Recent Postings from proton

### The recoil correction to the proton-finite-size contribution to the Lamb shift in muonic hydrogen

The Lamb shift in muonic hydrogen was measured some time ago to a high accuracy. The theoretical prediction of this value is very sensitive to the proton-finite-size effects. The proton radius extracted from muonic hydrogen is in contradiction with the results extracted from elastic electron-proton scattering. That creates a certain problem for the interpretation of the results from the muonic hydrogen Lamb shift. For the latter we need also to take into account the two-photon-exchange contribution with the proton finite size involved. The only way to describe it relies on the data from the scattering, which may produce an internal inconsistency of theory. Recently the leading proton-finite-size contribution to the two-photon exchange was found within the external field approximation. The recoil part of the two-photon-exchange has not been considered. Here we revisit calculation of the external-field part and take the recoil correction to the finite-size effects into account.

### Calculation of the Electric and Magnetic Root Mean Squared Radiuses of Proton Based on MIT Bag Model [Cross-Listing]

The electric and magnetic bag radiuses of the proton can be determined by MIT bag model based on electric and magnetic form factors of the proton. Also we determined electric and magnetic root mean squared radiuses of the proton, using of bag radius and compared with other results suggests a suitable compatibility.

### Parity violation in neutron capture on the proton: determining the weak pion-nucleon coupling [Cross-Listing]

We investigate the parity-violating analyzing power in neutron capture on the proton at thermal energies in the framework of chiral effective field theory. By combining this analysis with a previous analysis of parity violation in proton-proton scattering, we are able to extract the size of the weak pion-nucleon coupling constant. The uncertainty is significant and dominated by the experimental error which is expected to be reduced soon.

### Parity violation in neutron capture on the proton: determining the weak pion-nucleon coupling [Cross-Listing]

We investigate the parity-violating analyzing power in neutron capture on the proton at thermal energies in the framework of chiral effective field theory. By combining this analysis with a previous analysis of parity violation in proton-proton scattering, we are able to extract the size of the weak pion-nucleon coupling constant. The uncertainty is significant and dominated by the experimental error which is expected to be reduced soon.

### Non-Perturbative Lepton Sea Fermions in the Nucleon and the Proton Radius Puzzle [Replacement]

A potential explanation [U.~D.~Jentschura, Phys.\ Rev.\ A {\bf 88}, 062514 (2013)] of the proton radius puzzle originating from the non-perturbative lepton-pair content of the proton is studied. Well-defined quantities that depend on this lepton-pair content are evaluated. Each is found to be of the order of $10({\alpha\over \pi})^2$, so that we find such a lepton-pair content exists in the proton. However, we argue that this relatively large result and general features of loop diagrams rule out the possibility of lepton-pair content as an explanation of the proton radius puzzle. In addition, the contributions of a class of explanations of the proton radius puzzle (for which the dependence on the $\mu p$ relative distance is as a contact interaction) is determined.

### Non-Perturbative Lepton Sea Fermions in the Nucleon and the Proton Radius Puzzle [Cross-Listing]

A potential explanation [U.~D.~Jentschura, Phys.\ Rev.\ A {\bf 88}, 062514 (2013)] of the proton radius puzzle originating from the non-perturbative lepton-pair content of the proton is studied. Observable quantities that depend on this lepton-pair content are evaluated. Each is found to be of the order of $10({\alpha\over \pi})^2$, so that we find such a lepton-pair content exists in the proton. However, we argue that this relatively large result and general features of loop diagrams rule out the possibility of lepton-pair content as an explanation of the proton radius puzzle. In addition, the nuclear dependence of a class of explanations of the proton radius puzzle is determined.

### Single and double polarization observables in timelike Compton scattering off proton

We study the exclusive photoproduction of a lepton pair off the proton with the aim of studying the proton quark structure via the Generalized Parton Distributions (GPD) formalism. After deriving the amplitudes of the processes contributing to the $\gamma P\to P’ e^+e^-$, the Timelike Compton Scattering and the Bethe-Heitler process, we calculate all unpolarized, single- and double- beam-target spin observables in the valence region in terms of GPDs.

### On the Proton charge extensions

It is shown that the recent determination of the various proton charge extensions is compatible with Standard Model expectations.

### Directed Flow Indicates a Crossover Deconfinement Transition in Relativistic Nuclear Collisions [Cross-Listing]

Analysis of directed flow ($v_1$) of protons, antiprotons and pions in heavy-ion collisions is performed in the range of incident energies $\sqrt{s_{NN}}$ = 2.7–27 GeV. Simulations have been done within a three-fluid model employing a purely hadronic equation of state (EoS) and two versions of the EoS involving deconfinement transitions: a first-order phase transition and a smooth crossover transition. High sensitivity of the directed flow, especially the proton one, to the EoS is found. The crossover EoS is favored by the most part of considered experimental data. A strong wiggle in the excitation function of the proton $v_1$ slope at the midrapidity obtained with the first-order-phase-transition EoS and a smooth proton $v_1$ with positive midrapidity slope, within the hadronic EoS unambiguously disagree with the data. The pion and antiproton $v_1$ also definitely testify in favor of the crossover EoS. The results obtained with deconfinement EoS’s apparently indicate that these EoS’s in the quark-gluon sector should be stiffer at high baryon densities than those used in the calculation.

### Nonperturbative parton distributions and the proton spin problem

The Lorentz contracted form of the static wave functions is used to calculate the scaled parton distributions for mesons and baryons, boosting the rest frame solutions of the path integral Hamiltonian. Both unpolarized and polarized parton distributions are derived, and the proton spin problem is solved in the new setting. It is argued, that the DIS parton densities are not related to the ground state nucleon.

### Nonperturbative parton distributions and the proton spin problem [Replacement]

The Lorentz contracted form of the static wave functions is used to calculate the scaled parton distributions for mesons and baryons, boosting the rest frame solutions of the path integral Hamiltonian. Both unpolarized and polarized parton distributions are derived, and the proton spin problem is solved in the new setting. It is argued, that the DIS parton densities are not related to the ground state nucleon.

### Strong-Field Quantum Electrodynamics and Muonic Hydrogen

We explore the possibility of a breakdown of perturbative quantum electrodynamics in light muonic bound systems, notably, muonic hydrogen. The average electric field seen by a muon orbiting a proton is shown to be comparable to hydrogenlike Uranium and, notably, larger than the electric field achievable using even the most advanced strong-laser facilities. Following Maltman and Isgur who have shown that fundamental forces such as the meson exchange force may undergo a qualitative change in the strong-coupling regime, we investigate a concomitant possible existence of muon-proton and electron-proton contact interactions, of nonperturbative origin, and their influence on transition frequencies in light one-muon ions.

### Hard Excluseive $\Phi$ and $J/\Psi$ Photoproduction off a Proton

We study the photoproduction of the vector mesons $\Phi$ and $J/\Psi$ off a proton in the kinematical regime of large energies and scattering angles within the framework of perturbative QCD. Our investigations are based on the hard scattering approach. This means that the hadrons are replaced by their valence Fock states and scattering on the partonic level is described by tree graphs in which the large transferred momentum is redistributed between the valence partons via the exchange of hard gluons. We find that the unpolarized photoproduction cross sections are dominated by Compton-scattering-like graphs in which the photon couples to the proton, whereas vector-meson-dominance-like graphs, in which the photon fluctuates into the heavy quark-antiquark pair which then exchanges two gluons with the proton, play a minor role. We give explicit predictions for unpolarized scattering cross sections and compare them with experimental data where possible.

### On the role of quark orbital angular momentum in the proton spin

We study the covariant version of the quark-parton model, in which the general rules of the angular momentum composition are accurately taken into account. We demonstrate how these rules affect the relativistic interplay between the quark spins and orbital angular momenta, which collectively contribute to the proton spin. The spin structure functions $g_{1}$ and $g_{2}$ corresponding to the many-quark state $J=1/2$ are studied and it is shown they satisfy constraints and relations well compatible with the available experimental data including proton spin content $\Delta\Sigma\lesssim1/3$. The suggested Lorentz invariant 3D approach for calculation of the structure functions is compared with the approach based on the conventional collinear parton model.

### HERAFitter, Open Source QCD Fit Project

HERAFitter is an open-source package that provides a framework for the determination of the parton distribution functions (PDFs) of the proton and for many different kinds of analyses in Quantum Chromodynamics (QCD). It encodes results from a wide range of experimental measurements in lepton-proton deep inelastic scattering and proton-proton (proton-antiproton) collisions at hadron colliders. These are complemented with a variety of theoretical options for calculating PDF-dependent cross section predictions corresponding to the measurements. The framework covers a large number of the existing methods and schemes used for PDF determination. The data and theoretical predictions are brought together through numerous methodological options for carrying out PDF fits and plotting tools to help visualise the results. While primarily based on the approach of collinear factorisation, HERAFitter also provides facilities for fits of dipole models and transverse-momentum dependent PDFs. The package can be used to study the impact of new precise measurements from hadron colliders. This paper describes the general structure of HERAFitter and its wide choice of options.

### HERAFitter, Open Source QCD Fit Project [Replacement]

HERAFitter is an open-source package that provides a framework for the determination of the parton distribution functions (PDFs) of the proton and for many different kinds of analyses in Quantum Chromodynamics (QCD). It encodes results from a wide range of experimental measurements in lepton-proton deep inelastic scattering and proton-proton (proton-antiproton) collisions at hadron colliders. These are complemented with a variety of theoretical options for calculating PDF-dependent cross section predictions corresponding to the measurements. The framework covers a large number of the existing methods and schemes used for PDF determination. The data and theoretical predictions are brought together through numerous methodological options for carrying out PDF fits and plotting tools to help visualise the results. While primarily based on the approach of collinear factorisation, HERAFitter also provides facilities for fits of dipole models and transverse-momentum dependent PDFs. The package can be used to study the impact of new precise measurements from hadron colliders. This paper describes the general structure of HERAFitter and its wide choice of options.

### Relativistic corrections to static properties of the proton

A new method to relate the proton electromagnetic form factors in momentum space to the corresponding charge and magnetization densities with the inclusion of relativistic corrections is presented by extending the standard Breit equation to higher orders in its $1/c^2$ expansion. Applying a Lorentz boost to the relativistically corrected form factors $\tilde{G}_{E,M}({\bf q}^{2})$ in the Breit frame, moments of the charge and magnetization distributions are evaluated. The proton charge radius thus determined is found to be smaller and hence in better agreement with recent spectroscopy results.

### Transverse single-spin asymmetries in $p^\uparrow p \to \gamma X$ from quark-gluon-quark correlations in the proton [Replacement]

We analyze the transverse single-spin asymmetry in direct photon production from proton-proton collisions, denoted $A_N^\gamma$, within collinear twist-3 factorization. We provide a calculation of the contribution due to quark-gluon-quark correlations in the unpolarized proton as well as summarize previous studies on those effects in the polarized proton. Both soft-gluon poles and soft-fermion poles are considered. From this complete result we then estimate $A_N^\gamma$, including error bands due to uncertainties in the non-perturbative inputs, at kinematics relevant for planned measurements of this observable at the Relativistic Heavy Ion Collider. We find $A_N^\gamma$ can allow for a "clean" extraction of the Qiu-Sterman function, which could lead to a definitive solution to the so-called "sign mismatch" crisis. Since we use the Sivers function extracted from semi-inclusive deep-inelastic scattering to develop our input for the Qiu-Sterman function, this reaction can also make a statement about the process dependence of the Sivers function.

### Transverse single-spin asymmetries in $p^\uparrow p \to \gamma X$ from quark-gluon-quark correlations in the proton [Replacement]

We analyze the transverse single-spin asymmetry in direct photon production from proton-proton collisions, denoted $A_N^\gamma$, within collinear twist-3 factorization. We provide a calculation of the contribution due to quark-gluon-quark correlations in the unpolarized proton as well as summarize previous studies on those effects in the polarized proton. Both soft-gluon poles and soft-fermion poles are considered. From this complete result we then estimate $A_N^\gamma$, including error bands due to uncertainties in the non-perturbative inputs, at kinematics relevant for planned measurements of this observable at the Relativistic Heavy Ion Collider. We find $A_N^\gamma$ can allow for a "clean" extraction of the Qiu-Sterman function, which could lead to a definitive solution to the so-called "sign mismatch" crisis. Since we use the Sivers function extracted from semi-inclusive deep-inelastic scattering to develop our input for the Qiu-Sterman function, this reaction can also make a statement about the process dependence of the Sivers function.

### Transverse single-spin asymmetries in $p^\uparrow p \to \gamma X$ from quark-gluon-quark correlations in the proton

We analyze the transverse single-spin asymmetry in direct photon production from proton-proton collisions, denoted $A_N^\gamma$, within collinear twist-3 factorization. We provide a calculation of the contribution due to quark-gluon-quark correlations in the unpolarized proton as well as summarize previous studies on those effects in the polarized proton. Both soft-gluon poles and soft-fermion poles are considered. From this complete result we then estimate $A_N^\gamma$, including error bands due to uncertainties in the non-perturbative inputs, at kinematics relevant for planned measurements of this observable at the Relativistic Heavy Ion Collider. We find $A_N^\gamma$ can allow for a "clean" extraction of the Qiu-Sterman function, which could lead to a definitive solution to the so-called "sign mismatch" crisis. Since we use the Sivers function extracted from semi-inclusive deep-inelastic scattering to develop our input for the Qiu-Sterman function, this reaction can also make a statement about the process dependence of the Sivers function.

### Interpretation of an "edge" in proton-proton scattering [Replacement]

A study of proton-proton collisions at very high energy has revealed a "black disk," whose radius grows with the logarithm of the center-of-mass energy, surrounded by an edge of approximately constant width 1 fm. We interpret this behavior as the maximum length of a QCD string connecting the color triplet and antitriplet components of the proton, and propose further tests of this explanation.

### Polarizability relations across real and virtual Compton scattering processes [Replacement]

We derive two relations involving spin polarizabilities of a spin-1/2 particle and consider their empirical implications for the proton. Using the empirical values of the proton anomalous magnetic moment, electric and magnetic charge radii, moments of the spin structure functions $g_1$, $g_2$, and of two spin polarizabilities, the present relations constrain the low-momentum behavior of generalized polarizabilities appearing in virtual Compton scattering. In the case of the proton, the dispersive model evaluations of the spin and generalized polarizabilities appear to be consistent with these relations. The ongoing measurements of different electromagnetic observables at the MAMI, Jefferson Lab, and HI$\gamma$S facilities may be able to put these relations to a test, or use them to unravel the low-energy spin structure of the nucleon.

### Polarizability sum rule across real and virtual Compton scattering processes [Cross-Listing]

We derive a sum rule relating various electromagnetic properties of a spin-1/2 particle and consider its empirical implications for the proton. Given the measured values of the proton anomalous magnetic moment, electric and magnetic charge radii, the slope of the first moment of the spin structure function $g_1$, and the recently determined proton spin polarizability $\gamma_{E1M2}$, the sum rule yields a constraint on the low-momentum behavior of a generalized polarizability appearing in virtual Compton scattering. With the help of the presently ongoing measurements of different electromagnetic observables at the MAMI, Jefferson Lab, and HI$\gamma$S facilities, the sum rule will provide a model-independent test of the low-energy spin structure of the nucleon.

### Polarizability sum rule across real and virtual Compton scattering processes [Cross-Listing]

We derive a sum rule relating various electromagnetic properties of a spin-1/2 particle and consider its empirical implications for the proton. Given the measured values of the proton anomalous magnetic moment, electric and magnetic charge radii, the slope of the first moment of the spin structure function $g_1$, and the recently determined proton spin polarizability $\gamma_{E1M2}$, the sum rule yields a constraint on the low-momentum behavior of a generalized polarizability appearing in virtual Compton scattering. With the help of the presently ongoing measurements of different electromagnetic observables at the MAMI, Jefferson Lab, and HI$\gamma$S facilities, the sum rule will provide a model-independent test of the low-energy spin structure of the nucleon.

### Quark spin-orbit correlations

The proton spin puzzle issue focused the attention on the parton spin and orbital angular momentum contributions to the proton spin. However, a complete characterization of the proton spin structure requires also the knowledge of the parton spin-orbit correlation. We showed that this quantity can be expressed in terms of moments of measurable parton distributions. Using the available phenomenological information about the valence quarks, we concluded that this correlation is negative, meaning that the valence quark spin and kinetic orbital angular momentum are, in average, opposite. The quark spin-orbit correlation can also be expressed more intuitively in terms of relativistic phase-space distributions, which can be seen as the mother distributions of the standard generalized and transverse-momentum dependent parton distributions. We present here for the first time some examples of the general multipole decomposition of these phase-space distributions.

### Nuclear Waffles [Replacement]

The dense neutron-rich matter found in supernovae and neutron stars is expected to form complex nonuniform phases referred to as nuclear pasta. The pasta shapes depend on density, temperature and proton fraction and determine many transport properties in supernovae and neutron star crusts. We use two recently developed hybrid CPU/GPU codes to perform large scale molecular dynamics (MD) simulations with $51200$ and $409600$ nucleons of nuclear pasta. From the output of the MD simulations we characterize the topology and compute two observables, the radial distribution function $g(r)$ and the structure factor $S(q)$, for systems with proton fractions $Y_p=0.10, 0.20, 0.30$ and $0.40$ at about one third of nuclear saturation density and temperatures near $1.0$ MeV. We observe that the two lowest proton fraction systems simulated, $Y_p=0.10$ and $0.20$, equilibrate quickly and form liquid-like structures. Meanwhile, the two higher proton fraction systems, $Y_p=0.30$ and $0.40$, take a longer time to equilibrate and organize themselves in solid-like periodic structures. Furthermore, the $Y_p=0.40$ system is made up of slabs, lasagna phase, interconnected by defects while the $Y_p=0.30$ systems consist of a stack of perforated plates, the nuclear waffle phase. The periodic configurations observed in our MD simulations for proton fractions $Y_p\ge0.30$ have important consequences for the structure factors $S(q)$ of protons and neutrons, which relate to many transport properties of supernovae and neutron star crust. A detailed study of the waffle phase and how its structure depends on temperature, size of the simulation and the screening length showed that finite-size effects appear to be under control and, also, that the plates in the waffle phase merge at temperatures slightly above $1.0$ MeV and the holes in the plates form an hexagonal lattice at temperatures slightly lower than $1.0$ MeV.

### Nuclear Waffles [Cross-Listing]

The dense neutron-rich matter found in supernovae and neutron stars is expected to form complex nonuniform phases referred to as nuclear pasta. The pasta shapes depend on density, temperature and proton fraction and determine many transport properties in supernovae and neutron star crusts. We use two recently developed hybrid CPU/GPU codes to perform large scale molecular dynamics (MD) simulations with $51200$ and $409600$ nucleons of nuclear pasta. From the output of the MD simulations we characterize the topology and compute two observables, the radial distribution function $g(r)$ and the structure factor $S(q)$, for systems with proton fractions $Y_p=0.10, 0.20, 0.30$ and $0.40$ at about one third of nuclear saturation density and temperatures near $1.0$ MeV. We observe that the two lowest proton fraction systems simulated, $Y_p=0.10$ and $0.20$, equilibrate quickly and form liquid-like structures. Meanwhile, the two higher proton fraction systems, $Y_p=0.30$ and $0.40$, take a longer time to equilibrate and organize themselves in solid-like periodic structures. Furthermore, the $Y_p=0.40$ system is made up of slabs, lasagna phase, interconnected by defects while the $Y_p=0.30$ systems consist of a stack of perforated plates, the nuclear waffle phase. The periodic configurations observed in our MD simulations for proton fractions $Y_p\ge0.30$ have important consequences for the structure factors $S(q)$ of protons and neutrons, which relate to many transport properties of supernovae and neutron star crust. A detailed study of the waffle phase and how its structure depends on temperature, size of the simulation and the screening length showed that finite-size effects appear to be under control and, also, that the plates in the waffle phase merge at temperatures slightly above $1.0$ MeV and the holes in the plates form an hexagonal lattice at temperatures slightly lower than $1.0$ MeV.

### TMD parton densities in associated real and virtual photon and jet production at LHC [Replacement]

We study the associated production of real (isolated) or virtual photons (with their subsequent leptonic decay) and hadronic jets in proton-proton collisions at the LHC using the $k_T$-factorization approach of QCD. The consideration is based on the off-shell quark-gluon QCD Compton scattering subprocesses. In the case of virtual photon production, the contributions from Z boson exchange as well as gamma*-Z interference with the full spin correlations are included. The transverse momentum dependent (TMD) quark and gluon densities in a proton are determined from the Kimber-Martin-Ryskin prescription or Catani-Ciafoloni-Fiorani-Marchesini (CCFM) equation. In the latter, we restricted to the case where the gluon-to-quark splitting occurs at the last evolution step and calculate the sea quark density as a convolution of the CCFM-evolved gluon distribution and the TMD gluon-to-quark splitting function. Our numerical predictions are compared with the recent experimental data taken by the ATLAS Collaboration. We discuss the theoretical uncertainties of our calculations and argue that further studies are capable of constraining the TMD parton densities in a proton.

### Statistical description of the proton spin with a large gluon helicity distribution

The quantum statistical parton distributions approach proposed more than one decade ago is revisited by considering a larger set of recent and accurate Deep Inelastic Scattering (DIS) experimental results. It enables us to improve the description of the data by means of a new determination of the parton distributions. We will see that a large gluon polarization emerges, giving a significant contribution to the proton spin.

### Particle rapidity distribution in proton-nucleus collisions using the proton-contributor reference frame

I define the proton-contributor reference frame in proton nucleus (p–A) collisions as the center of mass of the system formed by the proton and the participant nucleons of the nucleus. Assuming that the rapidity distribution of produced particles is symmetric in the proton-contributor reference frame, several measurements in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02 {\rm TeV}$ can be described qualitatively. These include rapidity distributions of charged particles, $J/\psi$ and Z bosons.

### Nuclear spin-dependent interactions: Searches for WIMP, Axion and Topological Defect Dark Matter, and Tests of Fundamental Symmetries [Replacement]

We calculate the proton and neutron spin contributions for nuclei using semi-empirical methods, as well as a novel hybrid \emph{ab initio}/semi-empirical method, for interpretation of experimental data. We demonstrate that core-polarisation corrections to \emph{ab initio} nuclear shell model calculations generally reduce discrepancies in proton and neutron spin expectation values from different calculations. We derive constraints on the spin-dependent P,T-violating interaction of a bound proton with nucleons, which for certain ranges of exchanged pseudoscalar boson masses improve on the most stringent laboratory limits by several orders of magnitude. We derive a limit on the CPT and Lorentz-invariance-violating parameter $|\tilde{b}_{\perp}^p| < 7.6 \times 10^{-33}$ GeV, which improves on the most stringent existing limit by a factor of 8, and demonstrate sensitivities to the parameters $\tilde{d}_{\perp}^p$ and $\tilde{g}_{ D\perp}^p$ at the level $\sim 10^{-29} – 10^{-28}$ GeV, which is a one order of magnitude improvement compared to the corresponding existing sensitivities. We extend previous analysis of nuclear anapole moment data for Cs to obtain new limits on several other CPT and Lorentz-invariance-violating parameters: $\left|b_0^p \right| < 7 \times 10^{-8}$ GeV, $\left|d_{00}^p \right| < 8 \times 10^{-8}$, $\left|b_0^n \right| < 3 \times 10^{-7}$ GeV and $\left|d_{00}^n \right| < 3 \times 10^{-7}$.

### Nuclear spin-dependent interactions: Searches for WIMP, Axion and Topological Defect Dark Matter, and Tests of Fundamental Symmetries [Replacement]

We calculate the proton and neutron spin contributions for nuclei using semi-empirical methods, as well as a novel hybrid \emph{ab initio}/semi-empirical method, for interpretation of experimental data. We demonstrate that core-polarisation corrections to \emph{ab initio} nuclear shell model calculations generally reduce discrepancies in proton and neutron spin expectation values from different calculations. We derive constraints on the spin-dependent P,T-violating interaction of a bound proton with nucleons, which for certain ranges of exchanged pseudoscalar boson masses improve on the most stringent laboratory limits by several orders of magnitude. We derive a limit on the CPT and Lorentz-invariance-violating parameter $|\tilde{b}_{\perp}^p| < 7.6 \times 10^{-33}$ GeV, which improves on the most stringent existing limit by a factor of 8, and demonstrate sensitivities to the parameters $\tilde{d}_{\perp}^p$ and $\tilde{g}_{ D\perp}^p$ at the level $\sim 10^{-29} – 10^{-28}$ GeV, which is a one order of magnitude improvement compared to the corresponding existing sensitivities. We extend previous analysis of nuclear anapole moment data for Cs to obtain new limits on several other CPT and Lorentz-invariance-violating parameters: $\left|b_0^p \right| < 7 \times 10^{-8}$ GeV, $\left|d_{00}^p \right| < 8 \times 10^{-8}$, $\left|b_0^n \right| < 3 \times 10^{-7}$ GeV and $\left|d_{00}^n \right| < 3 \times 10^{-7}$.

### Baseline for the cumulants of net-proton distributions at STAR [Replacement]

We present a systematic comparison between the recently measured cumulants of the net-proton distributions by STAR for 0-5% central Au+Au collisions at $\sqrt{s_{NN}}$=7.7-200 GeV and two kinds of possible baseline measures, the Poisson and Binomial baselines. These baseline measures are assuming that the proton and anti-proton distributions independently follow Poisson statistics or Binomial statistics. The higher order cumulant net-proton data are observed to deviate from all the baseline measures studied at 19.6 and 27 GeV. We also compare the net-proton with net-baryon fluctuations in UrQMD and AMPT model, and convert the net-proton fluctuations to net-baryon fluctuations in AMPT model by using a set of formula.

### Baseline for the cumulants of net-proton distributions at STAR [Cross-Listing]

We present a systematic comparison between the recently measured cumulants of the net-proton distributions by STAR for 0-5$\%$ central Au+Au collisions at $\sqrt{s_{NN}}$=7.7-200 GeV and two kinds of possible baseline measures, the Poisson and Binomial baselines. These baseline measures are assuming that the proton and anti-proton distributions independently follow Poisson statistics or Binomial statistics. The higher order cumulant net-proton data are observed to deviate from all the baseline measures studied at 19.6 and 27 GeV. We also compare the net-proton with net-baryon fluctuations in UrQMD and AMPT model, and convert the net-proton fluctuations to net-baryon fluctuations in AMPT model by using a set of formula.

### The shape of the proton at high energies

We present first calculations of the fluctuating gluon distribution in a proton as a function of impact parameter and rapidity employing the functional Langevin form of the JIMWLK renormalization group equation. We demonstrate that when including effects of confinement by screening the long range Coulomb field of the color charges, the evolution is unitary. The large-x structure of the proton, characterized by the position of three valence quarks, retains an effect on the proton shape down to very small values of x. We determine the dipole scattering amplitude as a function of impact parameter and dipole size and extract the rapidity evolution of the saturation scale and the proton radius.

### Model independent extraction of the proton magnetic radius from electron scattering [Replacement]

We combine constraints from analyticity with experimental electron-proton scattering data to determine the proton magnetic radius without model-dependent assumptions on the shape of the form factor. We also study the impact of including electron-neutron scattering data, and $\pi\pi\to N\bar{N}$ data. Using representative datasets we find for a cut of $Q^2\leq0.5$ GeV$^2$, $r_M^p=0.91^{+0.03}_{-0.06}\pm0.02$ fm using just proton scattering data; $r_M^p=0.87^{+0.04}_{-0.05}\pm0.01$ fm adding neutron data; and $r_M^p=0.87^{+0.02}_{-0.02}$ fm adding $\pi\pi$ data. We also extract the neutron magnetic radius from these data sets obtaining $r_M^n=0.89^{+0.03}_{-0.03}$ fm from the combined proton, neutron, and $\pi\pi$ data.

### Model independent extraction of the proton magnetic radius from electron scattering

We combine constraints from analyticity with experimental electron-proton scattering data to determine the proton magnetic radius without model-dependent assumptions on the shape of the form factor. We also study the impact of including electron-neutron scattering data, and $\pi\pi\to N\bar{N}$ data. Using representative datasets we find for a cut of $Q^2\leq0.5$ GeV$^2$, $r_M^p=0.91^{+0.03}_{-0.06}\pm0.02$ fm using just proton scattering data; $r_M^p=0.87^{+0.04}_{-0.05}\pm0.01$ fm adding neutron data; and $r_M^p=0.87^{+0.02}_{-0.02}$ fm adding $\pi\pi$ data. We also extract the neutron magnetic radius from these data sets obtaining $r_M^n=0.89^{+0.03}_{-0.03}$ fm from the combined proton, neutron, and $\pi\pi$ data.

### Photoproduction of the omega meson off the proton near threshold [Cross-Listing]

An experimental study of omega photoproduction off the proton has been conducted by using the Crystal Ball and TAPS multiphoton spectrometers together with the photon tagging facility at the Mainz Microtron MAMI. The gp–>omega p differential cross sections are measured from threshold to the incident-photon energy Eg=1.4 GeV with ~15 MeV binning and full production-angle coverage. The quality of the present data near threshold gives access to a variety of interesting physics, including an estimation of the omegaN scattering length alpha_{omega p}.

### Hadronic supercriticality as a trigger for GRB emission

We explore a one-zone hadronic model that may be able to reproduce $\gamma$-ray burst (GRB) prompt emission with a minimum of free parameters. Assuming only that GRBs are efficient high-energy proton accelerators and without the presence of an ab initio photon field, we investigate the conditions under which the system becomes supercritical, i.e. there is a fast, non-linear transfer of energy from protons to secondary particles initiated by the spontaneous quenching of proton-produced $\gamma$-rays. We first show analytically that the transition to supercriticality occurs whenever the proton injection compactness exceeds a critical value, which favours high proton injection luminosities and a wide range of bulk Lorentz factors. The properties of supercriticality are then studied with a time-dependent numerical code that solves concurrently the coupled equations of proton, photon, electron, neutron and neutrino distributions. For conditions that drive the system deep into the supercriticality we find that the photon spectra obtain a Band-like shape due to Comptonization by cooled pairs and that the energy transfer efficiency from protons to $\gamma$-rays and neutrinos is high reaching $\sim 0.3$. Although some questions concerning its full adaptability to the GRB prompt emission remain open, supercriticality is found to be a promising process in that regard.

### Reduction of the proton radius discrepancy by 3 sigma

We show that in previous analyses of electron-proton scattering, the uncertainties in the proton charge radius extraction are underestimated. Using a fit function based on a conformal mapping, we can describe the scattering data with high precision and extract a radius value in agreement with the one obtained from muonic hydrogen.

### Reduction of the proton radius discrepancy by 3 sigma [Replacement]

We show that in previous analyses of electron-proton scattering, the uncertainties in the statistical procedure to extract the proton charge radius are underestimated. Using a fit function based on a conformal mapping, we can describe the scattering data with high precision and extract a radius value in agreement with the one obtained from muonic hydrogen.

### The photon PDF of the proton

We show how the photon input parton distribution function (PDF) may be calculated with good accuracy, and used in an extended DGLAP global parton analysis in which the photon is treated as an additional point-like parton. The uncertainty of the input photon PDF is relatively small, since the major part of the distribution (which is produced by the coherent emission of the photon from a proton that remains intact) is well known. We present the expected photon PDFs, and compare the predictions with ZEUS data for isolated photon electroproduction at negative rapidities.

### Ensemble Simulations of Proton Heating in the Solar Wind via Turbulence and Ion Cyclotron Resonance

Protons in the solar corona and heliosphere exhibit anisotropic velocity distributions, violation of magnetic moment conservation, and a general lack of thermal equilibrium with the other particle species. There is no agreement about the identity of the physical processes that energize non-Maxwellian protons in the solar wind, but a traditional favorite has been the dissipation of ion cyclotron resonant Alfven waves. This paper presents kinetic models of how ion cyclotron waves heat protons on their journey from the corona to interplanetary space. It also derives a wide range of new solutions for the relevant dispersion relations, marginal stability boundaries, and nonresonant velocity-space diffusion rates. A phenomenological model containing both cyclotron damping and turbulent cascade is constructed to explain the suppression of proton heating at low alpha-proton differential flow speeds. These effects are implemented in a large-scale model of proton thermal evolution from the corona to 1 AU. A Monte Carlo ensemble of realistic wind speeds, densities, magnetic field strengths, and heating rates produces a filled region of parameter space (in a plane described by the parallel plasma beta and the proton temperature anisotropy ratio) similar to what is measured. The high-beta edges of this filled region are governed by plasma instabilities and strong heating rates. The low-beta edges correspond to weaker proton heating and a range of relative contributions from cyclotron resonance. On balance, the models are consistent with other studies that find only a small fraction of the turbulent power spectrum needs to consist of ion cyclotron waves.

### Probing the transverse dynamics and polarization of gluons inside the proton at the LHC

Transverse momentum dependent gluon distributions encode fundamental information on the structure of the proton. Here we show how they can be accessed in heavy quarkonium production in proton-proton collisions at the LHC. In particular, their first determination could come from the study of an isolated J/psi or Upsilon particle, produced back to back with a photon.

### The role of hadronic cascades in GRB models of efficient neutrino production

We investigate the effects of hadronic cascades on the gamma-ray burst (GRB) prompt emission spectra in scenarios of efficient neutrino production. By assuming a fiducial GRB spectrum and a power-law proton distribution extending to ultra-high energies, we calculate the proton cooling rate and the neutrino emission produced through photopion processes. For this, we employ a numerical code that follows the formation of the hadronic cascade by taking into account non-linear feedback effects, such as the evolution of the target photon field itself due to the contribution of secondary particles. We show that in cases of efficient proton cooling and subsequently efficient high-energy neutrino production, the emission from the hadronic cascade distorts and may even dominate the GRB spectrum. Taking this into account, we constrain the allowable values of the ratio $\eta_p=L_p/L_{\gamma}$, where $L_{p}$ and $L_{\gamma}$ are the isotropic equivalent proton and prompt gamma-ray luminosities. For the highest value of $\eta_{p}$ that does not lead to the dominance of the cascading emission, we then calculate the maximum neutrino luminosity from a single burst and show that it ranges between $(0.01- 0.6)L_{p}$ and $(0.5-1.4)L_{\gamma}$ for various parameter sets. We discuss possible implications of other parameters, such as the magnetic field strength and the shape of the initial gamma-ray spectrum, on our results. Finally, we compare the upper limit on $\eta_p$ derived here with various studies in the field, and we point out the necessity of a self-consistent treatment of the hadronic emission in order to avoid erroneously high neutrino fluxes from GRB models.

### Photoproduction of $J/\psi$ and $\psi(2S)$ in proton-proton ultraperipheral collisions at the LHC

Using the framework of leading order perturbative QCD along with the CTEQ6L1 gluon distribution of the proton and high-energy HERA data on the $\gamma p \to J/\psi p$ and $\gamma p \to \psi(2S) p$ cross sections along with the corresponding H1 fit, we obtain a good description of the rapidity dependence of the cross sections of photoproduction of $J/\psi$ or $\psi(2S)$ mesons in proton–proton ultraperipheral collisions (UPCs) measured by the LHCb collaboration at the LHC. Within the same framework we also make predictions for the $pp \to pp J/\psi$ and $pp \to pp \psi(2S)$ UPC cross sections at $\sqrt{s_{NN}}=8$ and 14 TeV. We show that the possible contribution of the $p \to \Delta \gamma$ transition to the photon flux discernibly increases the $pp \to ppV$ UPC cross section and thus can affect the theoretical interpretation of results.

### Investigation of diffractive photoproduction of $J/\Psi$ in hadronic collisions [Replacement]

In this work we study the diffractive photoproduction of $J/\Psi$ in proton-proton, proton – nucleus and nucleus – nucleus collisions at LHC energies using the color dipole formalism and different models for the forward dipole – target scattering amplitude. Our goal is to estimate the theoretical uncertainty present in the current predictions in the literature. Our results are compared with the experimental data and predictions for higher energies are presented.

### Diffractive photoproduction of $J/\Psi$ in hadronic collisions: A closer look

In this work we study the diffractive photoproduction of $J/\Psi$ in proton-proton, proton – nucleus and nucleus – nucleus collisions at LHC energies using the color dipole formalism and different models for the forward dipole – target scattering amplitude. Our goal is to estimate the theoretical uncertainty present in the current predictions in the literature. Our results are compared with the experimental data and predictions for higher energies are presented.

### Model-independent determination of the magnetic radius of the proton from spectroscopy of ordinary and muonic hydrogen

To date the magnetic radius of the proton has been determined only by means of electron-proton scattering, which is not free of controversies. Any existing atomic determinations are irrelevant because they are strongly model-dependent. We consider a so-called Zemach contribution to the hyperfine interval in ordinary and muonic hydrogen and derive a self-consistent model-independent value of the magnetic radius of the proton. More accurately, we constrain not a value of the magnetic radius by itself, but its certain combination with the electric-charge radius of the proton, namely, R_E^2+R_M^2. The result from the ordinary hydrogen is found to be R_E^2+R_M^2=1.35(12) fm^2, while the derived muonic value is 1.49(18) fm^2. That allows us to constrain the value of the magnetic radius of proton R_M=0.78(8) fm at the 10% level.

### A self-consistent value of the electric radius of the proton from the Lamb shift in muonic hydrogen

Recently a high-precision measurement of the Lamb shift in muonic hydrogen has been performed. An accurate value of the proton charge radius can be extracted from this datum with a high accuracy. To do that a sufficient accuracy should be achieved also on the theoretical side, including an appropriate treatment of higher-order proton-structure effects. Here we consider a higher-order contribution of the finite size of the proton to the Lamb shift in muonic hydrogen. Only model-dependent results for this correction have been known up to date. Meantime, the involved models are not consistent either with the existing experimental data on the electron-proton scattering or with the value for the electric charge radius of the proton extracted from the Lamb shift in muonic hydrogen. We consider the higher-order contribution of the proton finite size in a model-independent way and eventually derive a self-consistent value of the electric radius of the proton. The re-evaluated value of the proton charge radius is found to be R_E=0.84022(56) fm.


text:
# [NTG-context] prevent gobbling of spaces when omitting optional argument

Florian Wobbe Florian.Wobbe at awi.de
Sat Jan 29 15:57:12 CET 2011

>> Hi,
>>
>> please consider the following minimal example:
>>
>> \def\before{\dodoubleempty\dobefore}%
>> \def\dobefore[#1][#2]{before}
>>
>> \starttext
>> \before[] after (should be: \before[]\ after without explicit \type{\ })\par
>> \before[]after\par
>> \before[][] after\par % this works
>> \before[][]after\par
>> \stoptext
>>
>> How can I prevent gobbling the following space, when the second optional argument is not given (i.e. first line). The macro itself should not insert space if the command is followed by any other character (2nd and 4th line).
>
>
> What do you want to achieve, maybe there is already something available.

Wolfgang, thank you very much! Recursion is really an elegant way. I wanted a wrapper around \cite with the same functionality like natbib's \citep[][]{} and \citet[][]{}, which saves a lot of typing. Especially if you start writing a text and at a later point decide to have a citation in parenthesis instead of textual. And of cause, if you switch between both styles in the same text.

Following your example, I came up with this now:

\def\citep[#1]%
{\strictdoifnextoptionalelse{\docitep[#1]}{\nocitep[#1]}}

\def\docitep[#1][#2]%
{\strictdoifnextoptionalelse{\dodocitep[#1][#2]}{\nodocitep[#1][#2]}}

\def\dodocitep[#1][#2][#3]%
{\cite[alternative=authoryears, left={(#1 }, extras={#2}][#3]}

\def\nodocitep[#1][#2]%
{\cite[alternative=authoryears, extras={#1}][#2]}

\def\nocitep[#1]%
{\cite[alternative=authoryears][#1]}

\setuppublications[state=start,refcommand=authoryear]
\setupbibtex[database=sample]

\starttext
\startlines
%\cite[hh2010a] % commenting this out does not work
\stoplines
\citep[, p.\ 1][hh2010a].\par
\citep[, p.\ 1][hh2010a] more text.\par
\citep[see][, p.\ 1][hh2010a].\par
\citep[see][][hh2010a] more text.\par
\startlines
\cite[hh2010a] % this is ok at the end
\stoplines
\stoptext

I noticed there is this error: using the first cite in \startlines...\stoplines does not work.

Florian


text:
# Math Help - Implicit differentiation

1. ## Implicit differentiation

Find dy/dx by implicit differentiation.

(3xy^2 + 1)^4 = 2x - 3y

4(3xy^2 + 1)^3 (3xy^2 + 1) = 2 -3

(3xy^2 + 1) +3 = 2-4(3xy^2 +1)^3

dy/dx = 2 - 4(3xy^2 +1)^3/(3xy^2 +1) +3

Am I getting any closer to understanding this?

2. You started out ok, but your second line is incorrect.

This part is right 4(3xy^2+1)^3. Now you must multiply by the derivative of the inside function.

So d/dx (3xy^2+1)= 6xy(dy/dx)+3y^2.

Putting the LHS together it's 4(3xy^2+1)^3 * (6xy(dy/dx)+3y^2)

The RHS of your work is correct.

3. Hello, Becky!

I don't see dy/dx or y' in your work.
Then it suddenly appears in your last step.

Find dy/dx by implicit differentiation: .(3xy² + 1)^4 .= .2x - 3y

. . . . . . 4(3xy² + 1)³(6xyy' + 3y²) . = . 2 - 3y'

. . . . . . 24xy(3xy² + 1)3y' + 12y²(3xy² + 1)³ . = . 2 - 3y'

. . . . . . 24xy(3xy² + 1)3y' + 3y' . = . 2 - 12y²(3xy² + 1)³

Factor: .3[8xy(3xy(3xy² + 1)³ + 1]y' . = . 2[1 - 6y²(3xy² + 1)³]

. . . . . . . . . . . . . . .2[1 - 6y²(3xy² + 1)³]
Therefore: . y' . = . ---------------------------
. . . . . . . . . . . . . . 3[8xy(3xy² + 1)³ + 1]

4. Originally Posted by becky
Find dy/dx by implicit differentiation.

(3xy^2 + 1)^4 = 2x - 3y...
Hi,

I've attached an image to show you what I've calculated:


text:
## FANDOM

36 Pages

Springer UTX, 2000

"The aim of this book is to describe the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today."

# Chapter 3 : Projective VarietiesEdit

## Section 3.3 : The Projective Closure of an Algebraic VarietyEdit

• The closure $\overline{V}$ of the twisted cubic: $[1:t:t^2:t^3] = \left[\frac{1}{t^3}:\frac{1}{t^2}:\frac{1}{t}:1\right] \rightarrow [0:0:0:1]$ as $t \rightarrow \infty$

## Section 3.4 : Morphisms of Projective VarietiesEdit

• no single choice of polynomials will work for all points of C: is this supposed to be immediately clear?
• For example, the point at infinity is [0:0:1] when we consider the chart $U_x$: At first it seemed like there should be lots of points at infinity, but then I realized that the points on the "line at infinity" don't count: $\mathbb{P}^2 = U_x \sqcup (\text{the line at infinity}) \sqcup (\text{the point at infinity})$
• A "form" is a homogeneous polynomial (in some number of variables).
• Exercise 3.4.4: $\mathbb{V}(x)$ can be parametrized as [p:0:q:r], and $\mathbb{V}(x-y^4-z^4)$ can be parametrized as $\left[s^4:t^4+u^4:ts^3:us^3\right]$

## Section 3.5 : Automorphisms of Projective SpaceEdit

• Exercise 3.5.1: The 3 is very important here! This does not hold in other dimensions.

# Chapter 5 : Classical ConstructionsEdit

## Section 5.4 : GrassmanniansEdit

• as the action of g on the determinants is just multiplication by the nonzero constant deg g: clearly k x k subsets must be special somehow... ah - just delete the irrelevant rows of $(a_{ij})$ and work in ${\rm GL}_k(\mathbb{C})$. (The rows are linearly independent by construction.)

# Chapter 8 : Maps to Projective SpaceEdit

## Section 8.1 : Embedding a Smooth Curve in Three-SpaceEdit

• Apparently the intention is that $\phi_x(\lambda) = \varphi_x(\lambda)$ here (both are versions of the lowercase Greek letter phi), which confused me briefly since I assumed one or the other must refer to the parametrization of X.

## Section 8.2 : Vector Bundles and Line BundlesEdit

• Definition of vector bundle: the definition says "the diagram commutes," without indicating a diagram. What's going on would be obvious except for the badly-placed page break -- the diagram it's talking about is at the top of the next page.
• They start talking about sections here before they're defined in 8.3.
• I guess the "atlas" functions of a vector bundle are called "local trivializations" because $U_i \times \mathbb{C}^n$ is trivially a vector bundle, which would make the functions $(\varphi_i)|_{\pi^{-1}(U)}$ "local isomorphisms to trivial bundles."

## Section 8.4 : Examples of Vector BundlesEdit

• The tautological line bundle: We apparently need to assume that X has positive dimension -- if our projective variety is a single point then we do seem to have a nontrivial section of the tautological bundle.
• The tautological line bundle: This example is defined for a projective variety X (rather than for a quasi-projective variety). This resolves the apparent ambiguity between "tautological line bundles have no nonzero global sections" and the fact that the sheaf of sections isn't just the zero sheaf. The hyperplane bundle on the next page, on the other hand, is defined for a quasi-projective variety X.


text:
The phase_diagram using a uniform initial velocity $vx = vy = vz = 160 km/s$ and no gravity ( cooling is on ) and $\eta=0$ (always using total internal energy):

Row 1: Normal pressure on the flux of E and momentum

Row 2: No pressure on the flux of E and momentum

No spurious heating of the gas!

To track the differences on the hydro conserved variables I plot projections ( 5 Mpc deep );

I used PPCM for the reconstruction and HLLC for the Reimann solver.

First using $\eta = 0.001$ same as enzo:

Now using $\eta =0.02$, here we are using more of the advected internal energy

Now using $\eta=1$ always using advected internal energy and not synchronizing the total internal energy to see the independent evolution of the total internal energy.

Here is the phase diagram for the simulation using $\eta=1$ (only advected internal energy )

Here I show the fraction of the 90 percentile of the internal energy to the 90 percentile of the kinetic energy to show how small errors in the kinetic energy result in large errors on the internal energy.


text:
# How do you write the following situation in function notation: Sadie has a bank account with a balance of $650.00 and plans to spend$55 per week?

Let $A \left(t\right)$ be the amount of Sadie's bank account after $t$ weeks.
$A \left(t\right) = 650 - 55 t$


text:
# In what sense is the Inductive effect a permanent effect?

Almost everywhere I checked, the Inductive effect is described as a permanent effect with almost no mention of what that actually means.

• What exactly is permanent about the effect?

• If the effect is permanent, does that mean that even if the attached electron withdrawing/donating group is removed, the induced effect still continues to exist?

The effect is permanent because the group that does the 'induction' is always there in the molecule. For example, if the induction is caused by a t-butyl group, that group is held by a strong carbon to carbon ($\ce{C-C}$) bond that will not easily be broken. One could also consider the inductive effect of the Fluorine held by a very strong $\ce{C-F}$ bond.

A temporary effect might be caused by protontation of an oxygen atom where the protonation can catalyse a reaction. The protonation is reversible. The bond between the proton and oxygen would be much weaker, as $\ce{C=O^+-H}$ will be in equilibrium with $\ce{C=O + H^+}$.

Removal of the group that does the induction will remove the inductive effect.

• Doesn't the same happen in Electromeric Effect? I mean, for the time that the reagent is present, the pi-bond is polarized. On its removal, the molecule returns to its original configuration. But then we say that Electromeric effect is temporary. What am I missing? May 30 '15 at 9:36
• @Technotoes - might be worth asking another question about that... wikipedia I think summarizes it neatly by saying "This effect is shown by those compounds containing multiple bonds. When a double bond or triple bond is exposed to an attack by a reagent, a pair of bonding electrons involved in the pi bond is transferred completely from one atom to another. This effect will remain as long as the attacking reagent is present. As soon as the reagent is removed, the polarized molecule will come back to the original state." - so because the attacking reagent causes it the effect is 'temporary'
– tom
May 30 '15 at 20:52


text:
# Thread: A number theory puzzle

1. ## A number theory puzzle

I give you 7x - 17y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).

2. Originally Posted by wonderboy1953
I give you 7x - 7y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).
Typo? 7*5 - 7*2 most definitely does not equal 1.

In fact it is easily shown that 7x - 7y = 1 has no integer solutions; the LHS is a multiple of 7 while the RHS is not.

3. Originally Posted by undefined
Typo? 7*5 - 7*2 most definitely does not equal 1.

In fact it is easily shown that 7x - 7y = 1 has no integer solutions; the LHS is a multiple of 7 while the RHS is not.

4. Originally Posted by wonderboy1953
I give you 7x - 17y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).
We have Bezout's identity with a sign change. There are infinite solutions, and a solution exists iff x > 4 and $x\equiv 5\pmod{17}$.

For $k \ge 0$ we solve for y

7*(17k + 5) - 17y = 1

7*17k + 35 - 17y = 1

17y = 7*17k + 34

y = 7k + 2

So the solution set is given by (x,y) = (5 + 17k, 2 + 7k) where k ranges over the non-negative integers.

5. That's the solution using parametric equations.

6. Methinks Sir Diophantine just turned over in his grave.

7. Originally Posted by Wilmer
Methinks Sir Diophantine just turned over in his grave.
I realize my solving for y could be shortened realizing that

7(x + 17) - 17(y + 7) = 7x - 17y

but what exactly are you referring to?

8. Originally Posted by Wilmer
Methinks Sir Diophantine just turned over in his grave.
Certainly not English.

9. Hello, wonderboy1953!

I have a very elementary solution . . .

$\text{Given: }\;7x - 17y \:=\: 1$

$\text{How many solutions in natural numbers are there for the equation?}$

Answer: there are a brizillian solutions.

Solve for $x\!:\;\;x \:=\:\dfrac{17y+1}{7} \;=\;2y + \dfrac{3y+1}{7}$

Since $\,x$ is a natural number, $(3y+1)$ must be divisible by 7.

We find that: . $y \;=\;2,\,9,\,16,\,\hdots,\,7t-5\;\text{ for }t \in N$

Then: . $x \;=\;2(7t-5) + \dfrac{3(7t-5)+1}{7} \;=\;17t - 12$

Solutions: . $\begin{Bmatrix}x &=& 17t - 12 \\ y &=& 7t-5 \end{Bmatrix}\;\;\text{ for }t \in N$

Of course, this is basicaly undefined's solution.

10. Originally Posted by undefined
I realize my solving for y could be shortened realizing that
7(x + 17) - 17(y + 7) = 7x - 17y
but what exactly are you referring to?
Nothing in particular UnD; I had just skimmed over:
Diophantine Equations
and saw WB's equation...


text:
## TextFileToTable Version 1.1

NEVER MIND. THE GNU PROJECT IMPRESSES AGAIN.

Please check out the org-mode to emacs for a program that did all this years ago.

http://orgmode.org/

Made a small change to the TextFileToTable script.  Now the user is able to exploit more of the Text::ASCIITable work with the addition of horizontal lines.

By inserting a ‘<<hr>>’ on a line by itself, the program will put another horizontal rule there.

```# COMMAND

perl TextFileToTable.pl [delimiter] [File to be read]
perl TextFileToTable.pl \& data.txt

# INPUT

ID&Fruit&Color
1&apple&red
2&banana&yellow
<<hr>>
3&orange&orange

# OUTPUT

.----------------------.
| ID | Fruit  | Color  |
+----+--------+--------+
|  1 | apple  | red    |
|  2 | banana | yellow |
+----+--------+--------+
|  3 | orange | orange |
'----+--------+--------'```

Grab the new script here:

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## Choosing an Acetophenetidin Solvent

When synthesizing Acetophenetidin (Phenacetin) in an Organic Laboratory, you might end up with a cake of crude product filled with impurities from the synthesis process.  A giveaway sign of this is a calculated percent yield in excess of one hundred percent.  This is not high-quality product.  It must be recrystallized in order to refine the product for larger crystals and purer yield.

In order to recrystallize a product for better purity, a solvent must be found that will solvate everything at high temperatures, but will precipitate the pure product at cooler temperatures while keeping the impurities solvated and out of the crystal structure of the reforming product.

Perhaps you are given three options: deionized water, ethanol, and hexane.  To test which would be the better solvent, add a bit of the crude product to small amounts of the potential solvents in test tubes.  Observe solubility.  Then heat the solvents in a boiling hot-water bath and observe solubility at that state.  Finish the mock recrystallization by removing the test tubes from the bath and letting them cool to room temperature before sticking them in an ice bath.  Observe solubility once again.

The better solvent will preferably not solvate the acetophenetidin at room temperature.  This means that the recrystallization will begin earlier in the cooling phase.  Everything must be dissolved at boiling.  A solvent with a low boiling point would not help here, as it will evaporate away leaving everything as an impure solid stuck to the sides of the test tube.  When cooled in ice, crystals must form for the solvent to be worth anything.

Here is some sample data:

Water

• Room Temperature – Crude product appeared to be insoluble
• Boiling – Completely solvated the crude product
• Freezing – Crystals reappeared

Ethanol

• Room Temperature – Completely solvated
• Boiling – Still solvated
• Freezing – Remained solvated

Hexane

• Room Temperature – Insoluble
• Boiling – Solvent evaporated
• Freezing – The solvent evaporated away in the previous step

The hexane is out as a solvent; its boiling point is too low.  Ethanol does not work because the acetophenetidin never precipitates out of it.  This leaves only water, and the observations associated with it prove it to be a useful solvent.

It is important to know what to look for when observing the Freezing solubility.  Even though the impure cake may enter the test tube as a lump, it will not precipitate that way.  Once the test tubes are sitting in the ice bath, let them sit undisturbed for five minutes.  Then remove and observe.  Wipe the condensation off the tubes and look very closely at them.  From farther away, the water and ethanol tubes look much the same.  Upon closer inspection and perhaps a swirl, one can see very tiny white particles floating in the water that are absent in the ethanol.  That is the pure acetophenetidin and a good sign.  These particles may be even smaller than any dust in the tube; close observation is important here.

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## Inkscape Bug #485269 – Strange Black Boxes

Was working with Inkscape and upon viewing the SVG file in an image viewer, I noticed conspicuous black boxes that were not present in the Inkscape view of the SVG or the exported PNG version of the SVG. A quick Google query turned up a standards conflict.

The Inkscape FAQ reports:

When flowed text support was added to Inkscape, it was conformant to the then-current unfinished draft of SVG 1.2 specification (and was always described as an experimental feature). Unfortunately, in further SVG 1.2 drafts, the W3C decided to change the way this feature is specified. Currently SVG 1.2 is still not finished, and as a result, very few SVG renderers currently implement either the old or the new syntax of SVG 1.2 flowed text. So, technically, Inkscape SVG files that use flowed text are not valid SVG 1.1, and usually cause problems (errors or just black boxes with no text).

I found out that it was the flowed text. Now to remove the flowed text. It did not seem to exist, though, making deleting the flowed text difficult. One could remove the visible text, but the black box would remain.

Another Google query turned up this Inkscape bug to which this simple workaround was suggested:

Another workaround to find empty ‘Flowed text’ objects:
1) ‘Edit > Deselect’
2) activate the text tool
3) use <TAB> to cycle through all text objects in the drawing and watch the status line for the message “Type or edit flowed text (0 characters); Enter to start new paragraph”
4) use <DEL> or <Backspace> to delete the selected empty ‘Flowed Text’ object
5) continue with until the first text object is selected again

I was unable to delete the boxes with Delete or Backspace.  I settled with right-clicking on the box and deleting it that way.  Save, then view the image again.  The black boxes should be gone.  To prevent them from returning, use the “Text > Convert to Text” tool to turn all flowed text into static text.

Filed under Computing, How-To

## Debian 5.0 DHCP Hostname

Was setting up a seedbox on old hardware recently. Was going to run Screen and rTorrent on top of Debian. The problem that arose was the router did not report the computer’s hostname. It assigned it an IP address via DHCP, but the lack of a hostname prevented it from port forwarding correctly. If the IP address to the machine changed, the forwarded ports did not follow as they were assigned to a hostname-less static IP.

After some research, I discovered it was not a problem but a feature. I needed to set what the DHCP program sent to the router as a hostname. So the computer could have one hostname, and send a different one to the router.

A minimal Debian 5.0 install (no desktop environment or pre-packaged server setup) has a program by the name of “dhcp3-client” to take care of this function.

Read through the documentation for “man dhclient.conf” to find the sample configuration. The line with “send host-name” is what we are interested in.

Now to edit the configuration file. Fish on down to “/etc/dhcp3/” and open up “dhclient.conf” if it exists. Edit the “send host-name” option to whatever you want the router to call the machine. Uncomment the line if it is commented.

If “dhclient.conf” does not exist, check to make sure “dhcp3-client” is installed:

aptitude search dhcp3-client

The package will have an “i” to the left if it is installed.

If dhcp3-client is installed, drop this line in a file by the name of “dhclient.conf”.

send host-name “Seedbox”;

Save and restart the machine.

Filed under Computing, How-To

## Perl Script – Text File to a Table

I could not find a utility to turn a delimited text file into an ASCII table. So I built a simple one in Perl. It builds on this work by preparing the data for it.

The user creates a file with some delimiter, such as a Comma-Separated Values file, and then runs it through the script. The table is sent to standard output, and the user can do what they want with it.

It is run as such:

perl TextFileToTable.pl [delimiter] [file to be read]

For instance:

perl TextFileToTable.pl \& data.txt

The order is important.

The code, in an archive.

Filed under Computing, Content

## Getting Tripped By Markovnikov

Markovnikov’s rule is a decent tool to predict some simple organic reactions. It states that in simple electrophilic additions of alkenes, such as Hydrogen Chloride to n-Pentene, the electrophile will attach itself to the carbon of the double bond that has more Hydrogen atoms already attached; the less substituted carbon. It can trip a user up, however, if they do not understand the actual process taking place. Example:

1-phenyl-2-methylpropene has two “sp2” hybridized carbons in its Propene backbone. In a reaction with Hydrogen Chloride, we will only consider these two points and will leave out attachments to the Benzene ring itself for simplicity.

When the PI bond reattaches itself to a passing Hydrogen (HCl), it creates an intermediate carbocation and a Chloride ion. Which of the two carbons the intermediate forms on is determined by stability considerations. Markovnikov’s rule would predict the hydrogen attaching to the Phenyl side, as there is only one substituent there. The reverse actually happens, creating a “non-Markovnikov” scenario where the Chloride ion attaches to the molecule close to the Phenyl substituent.

Instead of relying on Markovnikov’s rule for simple additions which create carbocations, use the idea of hyperconjugation. This boils down to the idea that carbocations (positively charged) can be stabilized by the sigma bonds of nearby substituents (usually carbon to hydrogen bonds) because the electrons involved in sigma bonding are negatively charged. The more substituents, the more stabilized a present carbocation becomes. The stability of carbons with positive charges becomes:

$1^{\circ} < 2^{\circ} < 3^{\circ}$

Primary carbocations are less stable than tertiary because tertiary carbocations have two more sets of hyperconjugation sigma bonds to stabilize the positive charge.

However, sigma hyperconjugation is nowhere near as stabilizing as the effects of lone pairs of nearby electrons such as those found in “sp2” hybridized carbons. The negative charge is less diluted in the lone pair. This makes Allyl and Benzene groups even more effective than tertiary carbons. The order stands at:

$1^{\circ} < 2^{\circ} < 3^{\circ} < Allyl < Benzene$

The point of this is that when a carbocation intermediate forms, it will form in the most probable place most of the time. The most probable is the most stable. Carbocations will form on carbons attached to Benzene before they attach to carbons with only two other carbons (methyl groups, for instance) attached to themselves.

Filed under Science

Feh can Stretch, Center, and Tile a background image, among other functions, but cannot Scale an image. That is fine. Feh follows the Unix Philosophy. This is where Image Magick and some bash scripting come in.

The plan is to create a directory of images that a bash script will peruse and feh will plaster to the background.

Start by compiling a directory of images to use. Copy these images to a new directory as the next step could irrevocably change the size of the used images. This is unlikely, however, as Image Magick tends to leave the originals alone.

Enter this into a terminal emulator:

convert * -resize 1024×768 background

It runs “convert” on all files. Resizes them to a best fit within a 1024 by 768 pixel box. Then names the new copy after the last argument. Adjust the numbers to your screen’s resolution. You can find it using:

xdpyinfo | grep dimensions

After the images are properly sized, it is time to set up the random selector. This is the bash code used. It assumes you have feh installed. Stick it in a file named “Roller” (or whatever you want; change the \$NAME variable so that it can prevent itself from being selected) and make it executable.

``` #!/bin/bash # Roller # # Version 1 # # Does not sterilize input and such # Assumes feh is installed `````` # Change to match the name of the file the script sits in # Only need to change the script's name once this way NAME="Roller" `````` # Change to the directory that contains the images # /home/user/Pictures/Wallpaper/* dir="/home/user/Pictures/Wallpaper/*" `````` # Counts the number of files in the directory count=`ls \$dir | wc -l` `````` # Sets a variable for counting the number of files skipped to that point j=0 `````` # Generates a psuedo-random number ran_num=\$RANDOM `````` # Sets ran_num to an integer in range: # 0 to \$count - 1 let "ran_num %= \$count" `````` # For all files in the directory for file in \$dir do # If the file to be used has been found if [ "\$j" -eq "\$ran_num" ] then # If the image to be used is not this script # (Originally, this script sat in the same directory as the images) if [ "\$file" != "\$NAME" ] then # Set the image using feh feh --bg-center \$file # Break out of the loop: it is done break else # Skip past Roller, will use the next file found let "j++" let "ran_num++" fi `````` # Appropriate file not found yet: increment and try again else let "j++" fi done `````` exit 0 ```

Edit the \$NAME and \$dir variables to your needs.

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Filed under Computing, How-To