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abstract: 'The minimality of the penalty function associated with a convex risk measure is analyzed in this paper. First, in a general static framework, we provide necessary and sufficient conditions for a penalty function defined in a convex and closed subset of the absolutely continuous measures with respect to s... | 1 | member_0 |
the positive homogeneity, and in further works it was removed, defining the concept of convex risk measure introduced by Föllmer and Schied [@FoellSch; @2002; @a], [@FoellSch; @2002; @b], Frittelli and Rosazza Gianin [@FritRsza; @2002], [@FritRsza; @2004] and Heath [@Heath; @2000].
This is a rich area that has receive... | 1 | member_0 |
semicontinuity of the penalty function, and the complications to prove this property depend on the structure of the probability space.
We first provide a general framework, within a measurable space with a reference probability measure $\mathbb{P}$, and show necessary and sufficient conditions for a penalty function d... | 1 | member_0 |
minimal for the risk measures generated by duality.
The paper is organized as follows. Section 2 contains the description of the minimal penalty functions for a general probability space, providing necessary and sufficient conditions, the last one rectricted to a subset of equivalent probability measures. Section 3 re... | 1 | member_0 |
results from the theory of static risk measures, and then a characterization for minimal penalties is presented.
Preliminaries from static measures of risk [Subsect:\_Preliminaries\_SCRM]{}
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Let $X:\Omega \rightarrow \mathbb{R}$ be a mapping ... | 1 | member_0 |
respect to a measure $\mathbb{P}$, when for each $X\in L_{+}^{\infty }\left( \mathbb{P}\right) $ with $\mathbb{P}\left[ X>0\right] >0$ we have that $\rho \left( -X\right) >\rho \left(
0\right) .$
We say that a set function $\mathbb{Q}:\mathcal{F}\rightarrow \left[ 0,1\right] $ is a *probability content* if it is f... | 1 | member_0 |
}\left( \mathbb{Q}\right) :=\sup\limits_{X\in \mathcal{A}\rho }\mathbb{E}_{\mathbb{Q}}\left[ -X\right] , \label{Def._minimal_penalty}$$and $\mathcal{A}_{\rho }:=\left\{ X\in \mathfrak{M}_{b}:\rho (X)\leq
0\right\} $ is the *acceptance set* of $\rho .$
The penalty $\psi _{\rho }^{\ast }$ is called the *minimal penalty... | 1 | member_0 |
risk is concentrated on the set $\mathcal{Q}$. Later, Krätschmer [@Kraetschmer; @2005 Prop. 3 p. 601] established that the sequential continuity from below is not only a sufficient but also a necessary condition in order to have a representation, by means of the minimal penalty function in terms of probability measures... | 1 | member_0 |
for example, Schied [@Schd; @2007] and Hernández-Hernández and Pérez-Hernández [@PerHer]), using techniques of duality, the minimality property is a necessary condition in order to have a well posed dual problem. More recently, the dual representations of dynamic risk measures were analyzed by Barrieu and El Karoui [@B... | 1 | member_0 |
is closed in the original topology [@FoellSch; @2004 Thm A.59].
\[static minimal penalty funct. in Q(<<) <=>\] Let $\psi :\mathcal{K}\subset \mathcal{Q}_{\ll }(\mathbb{P})\rightarrow \mathbb{R}\cup \{+\infty
\} $ be a function with $\inf\nolimits_{\mathbb{Q}\in \mathcal{K}}\psi (\mathbb{Q})\in \mathbb{R},$... | 1 | member_0 |
$ is the coarsest topology on $L^{1}\left( \mathbb{P}\right) $ under which every linear operator is continuous, and hence $\Psi _{0}^{X}\left( Z\right) :=\mathbb{E}_{\mathbb{P}}\left[ Z\left(
-X\right) \right] $, with $Z\in L^1$, is a continuous function for each $X\in
\mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right) ... | 1 | member_0 |
}^{\ast }\left( \mathbb{Q}\right) =\sup_{X\in \mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right)
}\left\{ \int Z\left( -X\right) d\mathbb{P}-\rho \left( X\right) \right\},$$ where $Z:=d\mathbb{Q}/d\mathbb{P},$ we have that $\Psi \left( Z\right)
=\sup_{X\in \mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right) }\left\{ \Psi
... | 1 | member_0 |
_{\rho
}^{\ast }\left( \mathbb{Q}\right) =+\infty =\psi \left( \mathbb{Q}\right).$$ Therefore, under the conditions of Lemma \[static minimal penalty funct. in Q(<<) <=>\] $\left( b\right) $ the penalty function $\psi $ might differ from $\psi _{\rho
}^{\ast }$ on $\mathcal{Q}_{cont}^{\ll }\setminus... | 1 | member_0 |
represent every local martingale as a combination of two components, namely as an stochastic integral with respect to the continuous part of the semimartingale and an integral with respect to its compensated jump measure. This is of course the case for local martingales, and with more reason this observation about the ... | 1 | member_0 |
by $\mu :\Omega \times \left( \mathcal{B}\left(
\mathbb{R}_{+}\right) \otimes \mathcal{B}\left( \mathbb{R}_{0}\right)
\right) \rightarrow \mathbb{N}$ where $\mathbb{R}_{0}:=\mathbb{R}\setminus
\left\{ 0\right\} $. The dual predictable projection of this measure, also known as its Lévy system, satisfies the relation $\m... | 1 | member_0 |
the variation of $A$ over the finite interval $\left[ 0,t\right] $. The subset $\mathcal{A}^{+}=\mathcal{A\cap V}^{+}$ represents those processes which are also increasing i.e. with non-negative right-continuous increasing trajectories. Furthermore, $\mathcal{A}_{loc}$ (resp. $\mathcal{A}_{loc}^{+}$) is the collection ... | 1 | member_0 |
n\in \mathbb{N}\right\} &
\end{array}
\label{Def._L(U)}$$be the class of predictable processes $\theta \in \mathcal{P}$ integrable with respect to $U^{c}$ in the sense of local martingale, and by $$\Lambda \left( U^{c}\right) :=\left\{ \int \theta _{0}dU^{c}:\theta _{0}\in
\mathcal{L}\left( U^{c}\right) \right\}$$the l... | 1 | member_0 |
_{1}\left( t,x\right) \left( \mu ^{\prime }\right) ^{\mathcal{P}}\left( \left\{ t\right\} ,dx\right) .$$
We use the notation $\int
\theta _{1}d\left( \mu ^{\prime }-\left( \mu ^{\prime }\right) ^{\mathcal{P}}\right) $ to write the value of this functional in $\theta _{1}$. It is important to point out that this functi... | 1 | member_0 |
càdlàg uniformly integrable martingale respectively. The following lemma is interesting by itself to understand the continuity properties of the quadratic variation for a given convergent sequence of uniformly integrable martingale . It will play a central role in the proof of the lower semicontinuity of the penalizati... | 1 | member_0 |
}\right\vert \right) \right] \vee
\mathbb{E}\left[ G\left( \left\vert M_{\infty }\right\vert \right) \right]
<\infty .$$Now, define the stopping times $$\tau _{k}^{n}:=\inf \left\{ u>0:\sup_{t\leq u}\left\vert M_{t}^{\left(
n\right) }-M_{t}\right\vert \geq k\right\} .$$Observe that the estimation $\sup_{n\in \mathbb{N}... | 1 | member_0 |
k\in \mathbb{N},$$and hence $\left[ M^{\left( n\right) }-M\right] _{\tau _{k}^{n}}\underset{n\rightarrow \infty }{\overset{\mathbb{P}}{\longrightarrow }}0$ for all $k\in \mathbb{N}.$
Finally, to prove that $\left[ M^{\left( n\right) }-M\right] _{\infty }\overset{\mathbb{P}}{\rightarrow }0$ we assume that it is not tru... | 1 | member_0 |
n_{k}\right) }-M_{t}\right\vert \geq k\right\} \right] ,$$it follows that $$d\left( \left[ M^{\left( n_{k}\right) }-M\right] _{\tau _{k\left(
n_{k}\right) }^{n_{k}}},\left[ M^{\left( n_{k}\right) }-M\right] _{\tau
_{k}^{n_{k}}}\right) \underset{k\rightarrow \infty }{\longrightarrow }0,$$which yields a contradiction wit... | 1 | member_0 |
Wang and Yan [@HeWanYan] or Theorem III.5.19 in Jacod and Shiryaev .
Denote by $D_{t}:=\mathbb{E}\left[ \left. \frac{d\mathbb{Q}}{d\mathbb{P}}\right\vert \mathcal{F}_{t}\right] $ the càdlàg version of the density process. For the increasing sequence of stopping times $\tau
_{n}:=\inf \left\{ t\geq 0:D_{t}<\frac{1}{n}\... | 1 | member_0 |
measure $\mathbb{Q}\ll \mathbb{P}$, there exist coefficients $\theta _{0}\in \mathcal{L}\left(
W\right) \ $and $\theta _{1}\in \mathcal{G}\left( \mu \right) $ such that $$\frac{d\mathbb{Q}_{t}}{d\mathbb{P}_{t}}=\frac{d\mathbb{Q}_{t}}{d\mathbb{P}_{t}}\mathbf{1}_{[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[}=\mathcal{... | 1 | member_0 |
s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)
\},\end{array}$$and thus$$\begin{aligned}
\triangle D_{t} &=&D_{t-}\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu
\left( dx\right) \right) \right) \\
&=&D_{t-}\tria... | 1 | member_0 |
**New Penrose Limits and AdS/CFT**
1.8cm
0.5cm
*$^1$ Dipartimento di Fisica, Università di Perugia,\
I.N.F.N. Sezione di Perugia,\
Via Pascoli, I-06123 Perugia, Italy\
0.4cm *$^2$ NORDITA\
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 0.4cm *$^3$ The Niels Bohr Institute\
*Blegdamsvej 17, DK-2100 Copenhagen Ø, ... | 1 | member_1 |
the strong coupling regime of either theory using the weak coupling limit of the other, on the other hand this makes it hard to test directly since it is not easy to find situations where approximate computations in both theories have an overlapping domain of validity.
In [@Berenstein:2002jq] a way out of this difficu... | 1 | member_1 |
were found in the case of the planar limit using the idea of integrability and the connection to spin chains [@Minahan:2002ve; @Beisert:2003tq; @Beisert:2003yb][^1] particularly by considering a near plane wave limit with curvature corrections to the pp-wave background [@Callan:2003xr; @Callan:2004uv]. A high point of ... | 1 | member_1 |
limit is that in comparing the spectrum of operators one assumes that most of the operators of the gauge theory receive an infinitely large correction to the bare dimension in the large ’t Hooft coupling limit $\lambda \rightarrow \infty$. This is of course a built in feature of the Asymptotic Bethe Ansatz for ${\mathc... | 1 | member_1 |
as well, and these sectors are crucial for approaching non-perturbative physics of type IIB string theory in $\mbox{AdS}_5\times S^5$, such as D-branes and black holes. This means that there should be additional Penrose limits of $\mbox{AdS}_5\times S^5$ in addition to the ones of [@Blau:2002dy; @Berenstein:2002jq; @Be... | 1 | member_1 |
to the new Penrose limit of $\mbox{AdS}_5\times S^5$ we also explore Penrose limits of $\mbox{AdS}_4 \times {\mathbb{C}}P^3$. Here two different classes of Penrose limits have been found, one in which there are no explicit space-like isometries [@Nishioka:2008gz; @Gaiotto:2008cg] and another in which there are two expl... | 1 | member_1 |
generally the behavior of string theory above the Hagedorn temperature and to study the connection between gauge theory and black holes in $\mbox{AdS}_5 \times S^5$ [@Grignani:2009ua][^4].
Interesting related work in other less supersymmetric gauge theories can be found in Refs. [@Grignani:2007xz; @Larsen:2007bm; @Ham... | 1 | member_1 |
{\mathbb{C}}P^3$ background of type IIA supergravity with one explicit space-like isometry.
Penrose limits and pp-waves with explicit isometries {#sec:stringtheory}
====================================================
In this section we derive a Penrose limit of $\mbox{AdS}_5 \times S^5$ which results in a new pp-wav... | 1 | member_1 |
to emphasize is that one can choose any light-like geodesic of $\mbox{AdS}_5 \times S^5$ for implementing the procedure. While the pp-wave background always corresponds to the maximally supersymmetric pp-wave background of type IIB supergravity [@Blau:2001ne], different choices of light-like geodesics can give this bac... | 1 | member_1 |
the quadratic potential terms for the transverse directions are massive for all eight transverse directions. Another coordinate system was introduced in [@Michelson:2002wa; @Bertolini:2002nr] and we will refer to it as the [*one flat direction pp-wave background*]{} due to the presence of a space-like isometry in the p... | 1 | member_1 |
string theory in this new background, we can complete the matching between the spectrum of anomalous dimensions of gauge theory operators in certain sectors of $\neqf$ SYM theory and the spectrum of the dual string theory states.
We show below in Section \[sec:stringrotspectra\] that all the pp-wave s achievable throu... | 1 | member_1 |
= {g_{\rm YM}}^2 N/(4\pi^2)$ is the ’t Hooft coupling of $SU(N)$ ${\mathcal{N}}=4$ SYM.[^5] The energy $E$ of type IIB string states on $\mbox{AdS}_5\times S^5$ is identified with the energy $E$ of the dual ${\mathcal{N}}=4$ SYM states on ${\mathbb{R}}\times S^3$, or equivalently, with the scaling dimension of the dual... | 1 | member_1 |
d\theta d\alpha d\Omega_3 )\, .$$ We parameterize the two three-spheres as $$\begin{aligned}
\label{3sph}
d\Omega_3^2 &= d\psi^2 + \sin^2 \psi d\phi^2 + \cos^2 \psi d\chi^2\, , \\
\label{3sphAdS}
d\Omega_3'^2 &= d\beta^2 + \sin^2 \beta d\gamma^2 + \cos^2 \beta d\xi^2\, .\end{aligned}$$ The three angular momenta on the ... | 1 | member_1 |
\quad
\alpha = \eta_2 \varphi_0 + \varphi_2\, , \quad
\gamma = \eta_3 \varphi_0 + \varphi_3\, , \quad
\xi = \eta_4 \varphi_0 + \varphi_4\,,\end{aligned}$$ and we define the light-cone coordinates as $$\begin{aligned}
z^- = \frac{1}{2} \mu R^2 (t-\varphi_0)\, , \quad
z^+ = \frac{1}{2\mu} (t+\varphi_0)\, .
\label{lcc}... | 1 | member_1 |
– is a generalization of it. Type IIB string theory can be quantized on this background and the light-cone Hamiltonian that one obtains is $$\begin{aligned}
H_\textrm{lc} \sim E-J_1, \qquad p^+ \sim \frac{E+J_1}{R^2}\, .\end{aligned}$$ From the condition that $H_\textrm{lc}$ and $p^+$ should stay finite in the limit, ... | 1 | member_1 |
way $$\begin{aligned}
\label{eq:oneflatphi}
\chi = \varphi_0 -\varphi_1\, , \quad
\phi = \varphi_0 + \varphi_1\, , \quad
\alpha = \eta_2 \varphi_0 + \varphi_2\, , \quad
\gamma = \eta_3\varphi_0 + \varphi_3\, , \quad
\xi = \eta_4\varphi_0 + \varphi_4\, ,\end{aligned}$$ with the light-cone variables still given by eq.n... | 1 | member_1 |
means that $J_3$, $S_1$ and $S_2$ are fixed. But now the condition that $H_\textrm{lc}$, $p^+$ and $p^1$ have to remain finite in the limit tells us that the quantities $$E-J_1-J_2 , \quad \frac{E+J_1+J_2}{R^2}, \quad \frac{J_1+J_2}{R^2} , \quad \frac{J_1-J_2}{R} , \quad g_s,l_s$$ are all fixed when $R \to \infty$. Thi... | 1 | member_1 |
are defined as $$\begin{array}{lcl}
z^1 = R \varphi_1 \, , & \phantom{qquad} & z^2 =
\displaystyle{ \frac{R}{\sqrt{2}}} \left(\displaystyle{\frac{ \pi}{4}-\psi}\right) \, , \\[4mm]
z^3 = R \varphi_2 \, , & & z^4 = R \left(\displaystyle{\frac{ \pi}{4}}-\theta \right) \, .
\end{array}$... | 1 | member_1 |
compute $H_\textrm{lc}$, $p^+$, $p^1$ and $p^3$ and request that they should stay finite in the Penrose limit we get that the quantities $$E-J_1-J_2-J_3 , \quad \frac{E+J_1+J_2+J_3}{R^2},
\quad
\frac{J_1+J_2+J_3}{R^2}, \quad \frac{J_1 - J_2}{R} ,\quad \frac{J_3 -J_1 - J_2}{R}, \quad g_s,l_s$$ are fixed as $R$ goes to ... | 1 | member_1 |
set of pp-wave s (one for each point of the parameter space). We refer to them as to *rotated pp-wave backgrounds*.
Note that the backgrounds obtained in this way do not necessarily have any specific meaning in an AdS/CFT context. They will only have a meaning in the AdS/CFT context if we derive them from a Penrose li... | 1 | member_1 |
- \mu^2 x^ix^i\left(dx^+\right)^2+dx^idx^i\, ,$$ where $i=1,2,\dots,8$ and five-form field strength $$\label{fff}
F_{(5)}=2\mu dx^{+}\left(dx^{1}dx^{2}dx^{3}dx^{4}+dx^{5}dx^{6}dx^{7}dx^{8}\right)\, .$$ We consider the following coordinate transformation $$\label{transfrot}
\begin{split}
x^- =z^- &+\frac{\mu}{2}\lef... | 1 | member_1 |
background and we now proceed in finding the superstring spectrum.
Bosonic sector
--------------
We work in the light-cone gauge $z^+ = p^+ \tau$ with $l_s=1$. The light-cone Lagrangian density of the bosonic $\sigma$-model is given by $$\label{boslagr}
\begin{split}
\mathscr{L}_{lc}^{B}= &- \frac{1}{4\pi p^+}\... | 1 | member_1 |
mode expansions
\[bosmodeex\] $$\begin{aligned}
Y^k&=i \sum_{n=-\infty}^{+\infty} \frac{1}{\sqrt{\omega_n}}\left(a_{n}^{k}e^{-i (\omega_n \tau -n\sigma)}- \left(\tilde{a}_{n}^{k}\right)^\dagger e^{i (\omega_n \tau -n\sigma)}\right)\, , \\
\bar{Y}^k&=i \sum_{n=-\infty}^{+\infty} \frac{1}{\sqrt{\omega_n}}\left(\tilde{... | 1 | member_1 |
are the Pauli matrices and $\omega_{a,b,c}$ are the spin connections. The non-vanishing components of the five-form field strength are $F_{+1234}=F_{+5678}=2\mu$.
We can write the action as $$\label{feract}
\begin{split}
S_{lc}^{F}=& \frac{i}{2\pi p^+ }\int d\tau d\sigma \Bigg\{{\left(S^1\right)^T} \left[\partia... | 1 | member_1 |
the form $$H_{lc}^{F}= \frac{i}{2\pi p^+ }\int^{2\pi}_{0}d\sigma
\left({\left(S^1\right)^T}\dot{S^1}+{\left(S^2\right)^T}\dot{S^2}\right)\,$$ where we used the equations of motion . Now we quantize the theory imposing the canonical equal time anticommutation relations $$\left\{S_{n}^{a},\left(S_{m}^{b}\right)^{\dag... | 1 | member_1 |
---
abstract: 'In this paper, we consider femtocell CR networks, where femto base stations (FBS) are deployed to greatly improve network coverage and capacity. We investigate the problem of generic data multicast in femtocell networks. We reformulate the resulting MINLP problem into a simpler form, and derive upper and... | 1 | member_2 |
the base station, a considerably larger transmit power is needed to overcome attenuation, while causing interference to other users and deteriorating network capacity. To this end, femtocells provide an effective solution that brings network infrastructure closer to mobile users. A femtocell is a small (e.g., residenti... | 1 | member_2 |
often observed in wireline networks. By allowing multiple users to share the same downlink multicast transmission, significant spectrum and power savings can be achieved.
In particular, we adopt [*superposition coding*]{} (SC) and [*successive interference cancellation*]{} (SIC), two well-known PHY techniques, for dat... | 1 | member_2 |
general. The objective is to minimize the total BS power consumption. Then we reformulate the MINLP problem into a simpler form, and derive upper and lower performance bounds. We also derive a simple heuristic scheme that assigns users to the BS’s with a greedy approach. Finally, we consider three typical connection sc... | 1 | member_2 |
scalability is very useful for video streaming systems, especially in CR networks, to accommodate heterogeneous channel availabilities and dynamic network conditions. We consider H.264/SVC medium grain scalable (MGS) videos, since MGS can achieve better rate-distortion performance over Fine-Granularity-Scalability (FGS... | 1 | member_2 |
remainder of this paper is organized as follows. The related work is discussed in Section \[sec:femto\_work\]. We investigate the problem of data multicast over fenmtocell networks in Section \[sec:femto\_mcast\_sic\]. The problem of streaming multiple MGS videos in a femtocell CR network is discussed in Section \[sec:... | 1 | member_2 |
macrocell was improved by subband partitioning and modest gains were achieved by interference cancellation. In [@Bharucha09], the interference was controlled by denying the access of femtocell base stations to protect the transmission of nearby macro base station. A novel algorithmic framework was presented in [@Madan1... | 1 | member_2 |
for a multicode MIMO system and showed the proposed scheme significantly outperformed the equal power allocation scheme. A scheme on joint power control and receiver optimization of CDMA transceivers was presented in [@Benvenuto07]. In [@Agrawal05; @Andrews03], the impact of imperfect channel estimation and imperfect i... | 1 | member_2 |
proposed algorithms have low computational complexity, and are shown to outperform the heuristic scheme with considerable gains.
System Model and Problem Statement \[sec:mod3\]
-----------------------------------------------
### System Model
Consider a femtocell network with an MBS (indexed $0$) and $M$ FBS’s (index... | 1 | member_2 |
focus on a multicast scenario, where the MBS and FBS’s multicast a data file to the $K$ users. The data file is divided into multiple packets with equal length and transmitted in sequence with the same modulation scheme. Once packet $l$ is successfully received and decoded at the user, it requests packet $(l+1)$ in the... | 1 | member_2 |
frequency reuse. The annual power bill is a large part of a mobile operator’s costs [@Ulf10]. Minimizing BS power consumption is important to reduce not only the operator’s OPEX, but also the global CO$_2$ emission; an important step towards “green” communications.
![Superposition coding and successive interference ca... | 1 | member_2 |
\hspace{-0.2in} P_l^m \ge 0, \mbox{ for all } l, m,\end{aligned}$$ where $P_l^m$ is the power of BS $m$ for transmitting the level $l$ packet; $\gamma_m^k$ is the SNR at user $k$ if it connects to BS $m$. Constraint (\[eq:cntrate\]) guarantees the minimum rate at each user. Constraint (\[eq:cnttransceiver\]) is due to ... | 1 | member_2 |
is an MINLP problem, which is NP-hard in general. In Section \[sec:alg\], we first reformulate the problem to a obtain a simpler form, and then develop effective algorithms for optimal and suboptimal solutions.
Reformulation and Power Allocation \[sec:alg\]
----------------------------------------------
In this secti... | 1 | member_2 |
\Gamma_m I_m^k,
\nonumber \\
&& \hspace{0.7in}
\mbox{ for all } k \in \mathcal{U}_l, l=1, \cdots, L \label{eq:cntrate2} \\
&& \hspace{-0.2in} Q_l^m \ge Q_{l+1}^m, l=1, \cdots, L \\
&& \hspace{-0.2in} \sum_{m=0}^M I_m^k=1, \mbox{ for all } k. \end{aligned}$$
For $l\le L$, constraint (\[eq:cntrate2\]) can be rew... | 1 | member_2 |
none of the subsets $\mathcal{U}_l^m$ ($l=1,\cdots,L$) is empty, we can expand the above recursive term using (\[eq:FmDef\]). It follows that $$\label{eq:FoldTerm}
Q_1^m = N_0 \Gamma_m \sum_{l=1}^L (1 + \Gamma_m)^{c_l^m} \max_{k \in \mathcal{U}_l^m} \left\{1 / H_m^k \right\},$$ where the exponent $c_l^m$ is defined as ... | 1 | member_2 |
It follows that: $$\begin{aligned}
\sum_{m=0}^M Q_1^m &\hspace{-0.1in} =& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m}\max_{k\in\mathcal{U}_l^m}\left\{\Gamma_m / H_m^k\right\} \nonumber \\
&\hspace{-0.1in} \le& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m} \overline{G}_m \no... | 1 | member_2 |
(1+\underline{\Gamma})^{c_l^m} \nonumber \\
&\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 (M+1) \sum_{l=1}^L \underline{G}^l (1+\underline{\Gamma})^{\frac{\sum_{m=0}^Mc_l^m}{M+1}}\nonumber \\
&\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 (M+1) \sum_{l=1}^L \underline{G}^l(1+\underline{\Gamma})^{\frac{l-1}{M+1}}. \label{eq:L... | 1 | member_2 |
with $Q_{L+1}^m=0$, we can apply (\[eq:QmlEqu\]) iteratively to find the $Q_l^m$’s. Finally, the transmit powers $P_l^m$ can be computed using (\[eq:QrepP\]).
With this approach, among the users requesting the level $l$ packet, it is more likely that some of them connect to the MBS and the rest connect to some FBS’s, ... | 1 | member_2 |
Q_l^{m\ast} &=& (1+\Gamma_m) Q_{l+1}^{m\ast} + \max_{k \in \mathcal{U}_l^m} \left\{ N_0\Gamma_m / H_m^k \right\}, \nonumber \\
&=& N_0 \Gamma_m \sum_{i=l}^L (1+\Gamma_m)^{i-l} \max_{k \in \mathcal{U}_l^m} \left\{1/H_m^k\right\},
%\nonumber \\
%&& \hspace{1.3in}
\;\; l=1,2, \cdots,L.
%\\
%Q_{L+1}^{m\ast} &=& 0,\e... | 1 | member_2 |
will yield larger power savings, since more exponents (i.e., $c_l^m, c_{l+1}^m, \cdots, c_{l'-1}^m$) will assume smaller values. Therefore, we should let these two subsets be empty whenever possible, i.e., either $\mathcal{U}_l^0=\emptyset$ or $\mathcal{U}_l^1=\emptyset$. According to this policy, all the users request... | 1 | member_2 |
determined subsets $\mathcal{U}_l^0$ and $\mathcal{U}_l^1$.
The computational complexity of this algorithm is $\mathcal{O}(L)$.
----- ---------------------------------------------------------------------------------------
1: Initialize all $c_l^0$, $c_l^1$, $Q_{L+1}^0$ and $Q_{L+1}^1$ to zero;
2: FOR $l=1... | 1 | member_2 |
to the MBS or an FBS. The second property is because each user can connect to only one BS. The user subsets connecting to different BS’s do not overlap. Therefore, $\mathcal{U}_l^m$’s is a [*partition*]{} of $\mathcal{U}_l$ with respect to $m$.
In addition, we define $\mathcal{S}_l^m$ as the set of possible users that... | 1 | member_2 |
$W_m(\emptyset)=0$. For example, consider Case II where $M=1$. We have $\mathcal{S}_l^0 = \mathcal{S}_l^1 = \mathcal{U}_l$, $W_0(\mathcal{U}_l)=G_l^0$, and $W_1(\mathcal{U}_l)=G_l^1$.
The power allocation algorithm for Case III is presented in Table \[tab:Case3Algo\]. The algorithm iteratively picks users from the [*e... | 1 | member_2 |
FOR $m=0$ TO $M$
4: $\;\;\;\;\;\;$ $\Delta_l^m=\Gamma_m(1+\Gamma_m)^{c_l^m}W_m(\mathcal{S}_l^m)$;
5: $\;\;\;$ END FOR
6: $\;\;\;$ Set $\Omega=\{1,\cdots,M\}$ and $\Psi=\emptyset$;
7: $\;\;\;$ WHILE ($\Omega \neq \emptyset$)
8: $\;\;\;\;\;\;$ $m'=\arg\min_{m\in\Omega} \Delta_l^m$;
9: $\;\;\... | 1 | member_2 |
MBS and one FBS; and (iii) Case III: one MBS and three FBS’s.
Since we do not find any similar schemes in the literature, we made the following comparisons. First, we compare Cases I and II with respect to BS power consumption and interference footprint. In both cases, there are $K=8$ users and $L=4$ levels. In Case I... | 1 | member_2 |
---
abstract: 'We have analyzed available optical data for Au in the mid-infrared range which is important for a precise prediction of the Casimir force. Significant variation of the data demonstrates genuine sample dependence of the dielectric function. We demonstrate that the Casimir force is largely determined by th... | 1 | member_3 |
France'
- '$^2$MESA+ Research Institute, University of Twente, P.O. 217, 7500 AE Enschede, The Netherlands'
author:
- 'I. Pirozhenko$^1$, A. Lambrecht$^1$, and V. B. Svetovoy$^2$'
title: Sample dependence of the Casimir force
---
Introduction\[Sec1\]
====================
The Casimir force [@Cas48] between uncharged m... | 1 | member_3 |
material, surface, geometry and temperature dependent. Here we will only focus on the material dependence, which is easy to treat on a level of some percent precision but which will turn out difficult to tackle on a high level of precision since different uncontrolled factors are involved.
In its original form, the Ca... | 1 | member_3 |
} \label{rThick}\end{aligned}$$
The force between dielectric materials had first been derived by Lifshitz [@Lif56; @LP9]. The material properties enter these formulas via the dielectric function $\varepsilon \left( i\zeta
\right) $ at angular imaginary frequencies $\omega=i\zeta $, which is related to the physical qua... | 1 | member_3 |
a characteristic imaginary frequency $\zeta_{\rm ch}=c/2L$ of electromagnetic field fluctuations in the gap. Fluctuations of frequency $\zeta \sim
\zeta _{\rm ch}$ give the dominant contribution to the Casimir force. For example, for a plate separation of $L=100$ nm the characteristic imaginary frequency is $\zeta _{\r... | 1 | member_3 |
extracted form the optical data at the lowest accessible frequencies. The exact values of the Drude parameters are very important for the precise evaluation of the force. Lambrecht and Reynaud [@Lam00] fixed the plasma frequency using the relation
$$\omega _{\rm p}^{2}=\frac{Ne^{2}}{\varepsilon _{0}m_{e}^{\ast }},
\la... | 1 | member_3 |
[@Bos00b] and Svetovoy and Lokhanin [@Sve00b] extracted the low-frequency optical data by fitting them with Eq. (\[Drude\]). For one set of data from Ref. [@HB2] the result [@Sve00b] was close to that found by the first approach, but using different sources for the optical data collected in Ref. [@HB2] an appreciable d... | 1 | member_3 |
performed with high experimental accuracy. On the other hand, sophisticated theoretical calculations predict the Casimir force at the level of 1% or better. These results illustrate the considerable progress achieved in the field in only one decade. In order to assure a comparison between theory and experiment at the s... | 1 | member_3 |
Drude parameters are extracted from the data by fitting both real and imaginary parts of the dielectric function at low frequencies in Sec. \[Sec4\]. In Section \[Sec5\] the Drude parameters are estimated by a different method using Kramers-Kroning analysis. The uncertainty in the Casimir force due to the sample depend... | 1 | member_3 |
Fig. \[fig1\] (left panel). The region marked as [1]{} corresponds to the frequencies smaller than $\omega _{\rm c}$. The region [2]{} defining the Drude parameters extends from the cutoff frequency to the edge of the interband absorption $\omega _{0}$. The high energy domain $\omega>\omega _{0}$ is denoted by [3]{}.
... | 1 | member_3 |
imaginary frequencies up to $\zeta \approx
5$ eV. $\varepsilon _{2}\left( i\zeta \right) $ gives a perceptible contribution to $\varepsilon \left( i\zeta \right)$, while $\varepsilon_{3}\left( i\zeta \right)$ produces minor contribution negligible for $\zeta<0.5$ eV.
As mentioned in the Introduction, we may introduce ... | 1 | member_3 |
inappropriate for our purposes. Among recent experiments let us mention the measurement of normal reflectance for evaporated gold films [@Sot03], which was performed in the wide wavelength range $0.3-50$ $\mu$m, but unfortunately does not permit to evaluate independently both real and imaginary parts of the dielectric ... | 1 | member_3 |
right graph of Fig. \[fig2\]. These data sets demonstrate considerable variations of the dielectric function from one sample to another.
Let us briefly discuss the sets of data [@HB1; @Wea81; @Mot64; @Pad61] used in our analysis and the corresponding samples. The commonly used Handbook of Optical Constants of Solids [... | 1 | member_3 |
and transmission-electron-microscopy methods. The dc resistivity of the films was found to be very sensitive to the preparation conditions. The errors in the optical characteristics of the films were estimated on the level of a few percents.
The handbook [@Wea81] embraces the optical data from $0.1$ eV to $28.6$ eV (m... | 1 | member_3 |
in the spectral range $1-12\ \mu$m. The errors in $n$ and $k$ were estimated as 2-3% and 0.5-1%, respectively.
Finally, the triangles represent Padalka and Shklarevskii data [@Pad61] for unannealed Au films evaporated onto glass.
The variation of the data points from different sources cannot be explained by experimen... | 1 | member_3 |
}}{\omega \left( \omega ^{2}+\omega _{\tau }^{2}\right). }
\label{ImDrude}$$
For $\omega \gg \omega _{\tau }$, the data on the log-log plot should fit straight lines with the slopes $-2$ and $-3$ for $\varepsilon ^{\prime }$ and $\varepsilon ^{\prime\prime }$, respectively, shifted along the ordinate due to variation ... | 1 | member_3 |
follow from Eq. (\[ImDrude\]) with an additional polarization term ${\cal P}$ in $\varepsilon
^{\prime }$:
$$\varepsilon ^{\prime }\left( \omega \right) ={\cal P}-\frac{\omega
_{p}^{2}}{\omega ^{2}+\omega _{\tau }^{2}},\ \ \varepsilon ^{\prime
\prime }\left( \omega \right) =\frac{\omega _{p}^{2}\omega _{\tau
}}{\omega... | 1 | member_3 |
this integral does not depend on $\omega $, leading to a constant $\varepsilon _{\left( i\right)
}^{\prime }\left( \omega \right) $. In reality the situation is more complicated because the transition is not sharp and many factors can influence the transition region. We will assume here that ${\cal P}$ is a constant bu... | 1 | member_3 |
4 $6.85\pm 0.02$ $3.6\pm 0.05$ $-12.33\pm 9.13$ Padalka 11 points, $\blacktriangledown,\triangledown$
: The Drude parameters found by fitting the available infrared data for $\varepsilon ^{\prime }\left( \omega \right)$ and $\varepsilon ^{\prime \prime }\left( \omega \right) $ w... | 1 | member_3 |
we interpret them as systematic errors, we can propagate the errors in $\varepsilon$ or $n,k$ to the values of $\omega_{\rm p}$ and $\omega_{\tau}$, leading to an additional error in $\omega_{\rm p}$ of about 5% for the Weaver data and 1% for the Motulevich data and twice as large in $\omega_{\tau}$.
Significant varia... | 1 | member_3 |
of a wide transition between inter- and intraband absorption. Coincidence of the solid and dashed lines for $\varepsilon ^{\prime
\prime }$ is accidental. The fits for $\varepsilon ^{\prime }$ are nearly perfect for both data sets.
It is interesting to see on the same figure how well the parameters $\omega _{\rm p}=9.... | 1 | member_3 |
Ref. [@Ben66], where the authors checked the validity of the Drude theory by measuring reflectivity of carefully prepared gold films in ultrahigh vacuum in the spectral range $0.04<\omega<0.6$ eV. Therefore, this value is good if one disposes of well prepared samples.
The Drude parameters from Kramers-Kronig analysis\... | 1 | member_3 |
frequencies the experimental data from different sources [@HB1; @Wea81; @Mot64; @Pad61] are used. The data in Refs. [@Mot64; @Pad61] must be extended to high frequencies starting from $\omega=1.25$ eV. We do this using the handbook data [@HB1].
Let us start from the data for bulk Au(110) [@Wea81]. This data set is giv... | 1 | member_3 |
$\varepsilon^{\prime}(\omega)$, leading to $\omega_{\rm p}=8.40$ eV and $\omega_{\tau}=0.020$ eV. These parameters are in reasonable agreement with the ones indicated in Tab. \[tab1\]. In Fig. \[fig6\] the experimental data (dots) and $|\varepsilon^{\prime}(\omega)|$ found from Eq. (\[KKrel\]) (solid line) are plotted,... | 1 | member_3 |
Motulevich and Shubin data [@Mot64], we find the Drude parameters $\omega_{\rm p}=8.81$ eV, $\omega_{\tau}=0.044$ eV which are close to the values in Tab. \[tab1\]. The experimental data and calculated function $|\varepsilon^{\prime}(\omega)|$ are shown in Fig. \[fig8\]. There is good agreement for frequencies $\omega<... | 1 | member_3 |
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