Christelle Al Hasrouty
email: alhasrouty@labri.fr

Mohamed Lamine Lamali
email: mohamed-lamine.lamali@labri.fr

Damien Magoni
email: magoni@labri.fr

John Murphy
email: j.murphy@ucd.ie

SDN-enabled Adaptation of Videoconference Streams to Network Dynamics

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Videoconferencing has become a mainstream communication method but often suffers from poor Quality of Experience (QoE). During a videoconferencing call, QoE is mainly affected by the heterogeneous participants' characteristics and their channel capacity. The wireless channel is a shared-access medium, and the available bandwidth varies with the number of hosts contending for the channel. In wireless networks, the available access bandwidth experiences quick modifications due to physical obstacles, mobility from one access network to another and channel blurring. An access link with a low bandwidth capacity can prevent the reception of a high-quality stream regardless of the participant characteristics. Traditional networks find it difficult to detect and adapt to the bandwidth variations in real time. Software Defined Networking (SDN) provides new ways to manage network bandwidth. SDN has a centralized view that allows the management of the network variations in a more efficient way.

In [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF], our team proposes algorithms that use SDN to create multicast adaptive trees with Scalable Video Coding (SVC) [START_REF] Schwarz | Overview of the scalable video coding extension of the H.264/AVC standard[END_REF] layers inside the network. The main goal of this work is to save the network bandwidth consumption while adapting to the participants' heterogeneity. However, the proposed algorithms do not consider the dynamic adaptation to time-evolving user access bandwidth. The only solution to use them in a dynamic context is to run them again when a change occurs. This may be costly in processing time and resources. In this paper, we complete the work in [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF] and propose new algorithms with a lower complexity that require less processing time and resources while minimizing the bandwidth consumption in the core network. We evaluate these algorithms through simulations and compare them to the algorithms of [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF]. The simulation results show that the processing time of the new solution is much lower than that of the other algorithms. Moreover, it achieves the same bandwidth saving. This suggests that the new solution is more adapted to a dynamic context and provides high reactivity and QoS to the users, while minimizing the controller resource consumption and the used bandwidth in the core network.

The paper is organized as follows: Section II describes the problem, presents a brief state of the art on bandwidth and calls variations in a dynamic environment, and places our solution in this context. Section III details the model and assumptions of our proposal and describes our algorithms together with their complexity. Section IV details our simulation methodology, the used parameters and the results of the evaluation of the different algorithms. Finally, Section V concludes the paper.

II. MULTICAST FOR VIDEOCONFERENCE MANAGEMENT

A. Problem Statement

We consider multi-party video calls where all the participants send their video stream and receive those of the other participants. Thus, for each video stream, there are one sender and several receivers. The unicast method consists in building a path between each pair of participants while the multicast one consists in computing a tree from each sender to the receivers. A layering technique (SVC) for adapting video stream is available on each SDN switch in the core network. An adaptation of a video stream is a dropping of high-quality video layers while allowing lower quality ones. In [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF], our team aims to minimize the bandwidth consumption in the core network by accurately computing the multicast trees and opportunely placing the SVC adaptation rules inside the network. The problem we address is double: How these methods behave in a dynamic context (bandwidth variation)? And how to design less costly and more reactive algorithms to adapt video streams when variations occur.

B. Related Work

Using multicast to deal with multi-party (video) calls is a natural approach and is investigated since the development of multimedia telecommunication. Together with video layering techniques such as SVC, it provides mechanisms to manage video calls while providing different quality streams to the participants. For example, some works [START_REF] De Amorim | Single-loop packet merging for receiver-oriented multicast multi-layered video[END_REF]- [START_REF] Chandrasekar | Performance of video conferencing in unicast and multicast communication using protocol independent multicast routing[END_REF] use these mechanisms to improve the participants' QoE. However the video layering adaptation in these works can only be performed at the application layer, and thus the core network bandwidth consumption cannot benefit from it. With the emergence of SDN, several works used the new paradigm to improve the participants' QoE and simplify the network management [START_REF] Zhao | Software defined networkenabled multicast for multi-party video conferencing systems[END_REF]- [START_REF] Yang | A video conferencing system based on sdn-enabled svc multicast[END_REF]. These works propose that the SDN controller directly manages the multicast tree construction and the video layering inside the network. However, while some of them minimize the bandwidth consumption but degrade the video quality stream of the participants, the others provide high quality video stream but at the price of higher bandwidth consumption. For example, the latter constructs a different multicast tree for each video layer, which is not optimal in term of bandwidth saving. In [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF], our team proposes a solution to provide the best possible video quality to the participants while minimizing the bandwidth consumption in the core network. The proposed solution consists in constructing a tree from each sender to the receivers (the same tree for all the video quality layers) and then placing SVC adaptation rules (video quality degradation) on the SDN switches in the network. Two algorithms are proposed for the tree construction: Shortest Path Tree (SPT), which computes the shortest path tree between each sender and the other participants (the receivers), and Minimizing Spanning Tree (MST) which proceeds iteratively. MST first computes the shortest path between the sender and the first receiver (according to a specific ranking rule), then the shortest path between any node of the previous path and the second receiver, etc. At each step, it computes the shortest path between the current multicast tree and the next receiver. Both algorithms perform a very good bandwidth saving while providing to each participant the maximum video quality that it can receive. However, this method was not evaluated in a dynamic context, i.e., under access channel bandwidth variations, which is a realistic assumption in current networks.

C. Dynamic Aspects

There is an extensive literature on video calls under bandwidth variations. For example, [START_REF] Wu | Streaming video over the internet: approaches and directions[END_REF] provides an architecture and several mechanisms to perform video streaming that adapts to congestion and bandwidth variations. It investigates both unicast and multicast methods. But the congestion control is performed at the application layer, and thus at the endpoints of the communication. The authors of [START_REF] De Cicco | Skype video responsiveness to bandwidth variations[END_REF] investigate the effect of congestion and bandwidth variation on a Skype video call. However, the case study includes only two hosts and, again, the adaptation mechanism is performed within the application layer. Some other works focus on the reactivity to bandwidth variation. For example, [START_REF] Vutukuru | Cross-layer wireless bit rate adaptation[END_REF] proposes a cross-layer approach where the bitrate is directly estimated on the physical layer, allowing a quick adaptation to the new bandwidth. More recently, the new approach of Pseudo analog video transmission, which allows a direct adaptation of the video stream quality according to the channel noise, is investigated in [START_REF] Ding | Pseudo analog video transmission based on LTE physical layer[END_REF], [START_REF] He | Progressive pseudo-analog transmission for mobile video streaming[END_REF]. The drawback of these cited works on video call under bandwidth variation is that the video quality adaptation is at the endpoints. In the case of multicast approach, this has two consequences: i) If the video quality adaptation is performed at the sender, it will adapt to the video quality to the receiver with the lowest bandwidth. This means that the receivers with higher bandwidth cannot benefit from a quality corresponding to their access channel. ii) If the adaptation is performed at the receivers, the sender has to emit the highest quality stream through the multicast tree, which implies higher bandwidth consumption in the core network.

D. Proposed Solution

In the perspective of providing the highest possible video quality to each receiver while minimizing the bandwidth consumption, we first propose to evaluate the algorithm of [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF] under random access bandwidth variations. It consists in running the algorithm that computes the multicast trees and replaces the adaptation rules at every bandwidth variation, which may be costly. We propose algorithms that do not rebuild the multicast trees but replace optimally the adaptation rules. This does not require any path computation but just requires moving upward or downward the rules in the trees. These methods are especially fast when the height of the multicast trees is small, thus improving considerably the complexity, the processing time and the reactivity. Our algorithms complete those of [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF] which can still be used to construct the initial multicast trees.

III. SYSTEM MODEL AND ALGORITHMS

A. Technical Assumptions

We assume that the network is SDN-enabled and that the controller manages the calls and has a global view of the network. All the switches can communicate with the controller and are supposed to be able to adapt SVC streams by dropping unused layers. We consider that there are at most 4 SVC stream layers as mentioned in Table I and a given layer can be used only if all the lower layers are also received. In addition, each participant accepts the highest number of layers allowable by its access downlink capacity. If a participant does not have enough bandwidth to receive any layer, the video call is rejected. Many participants can be connected wirelessly to the same node, which makes the access channel subject to variable communication conditions. This randomness in bandwidth availability can lead to a large heterogeneity of bandwidth between the participants.

B. Formal Model

The network is a graph G = (V, E), where |V | = n is the number of nodes and |E| = m is the number of links. The set of participants is denoted by P = {1, . . . , p}. There is p multicast trees, one for each participant as a sender. Thus, a multicast tree T i is associated to a sender s i and a set of receivers R i = {r i,j | i ∈ P and j = i}. For simplicity, we consider that the participants are also nodes of the trees (the receivers as leaves and the sender as root).

P i (r)
Parent of a node r in the multicast tree T i R i

Set of receivers in the tree T i r i,j

Receiver j in the tree T i Rule(b, b )

The adaptation of bitrate b to b s i Sender i T i

Multicast tree associated to the sender s i for any link specified by a pair of nodes (x, y), the bandwidth1 (or capacity) is denoted by B(x, y) and the bitrate (the actual used capacity in a specified tree T i ) is denoted by b i (x, y). In a tree T i , the parent of a node x is denoted by P i (x) and the set of its children (if any) by C i (x). Note that each receiver has the same parent in all the trees, as it has only one access link. For sake of simplicity, B(r) (resp b i (r)) can denote the download bandwidth (resp. the bitrate in T i ) of the receiver r. Likewise, B(s i ) (resp. b(s i )) can denote the upload bandwidth (resp. bitrate) of sender s i . At each node in the network and for each tree, there is a set of adaptation rules. This corresponds to SVC layer adaptation in the network. In a specified tree and on a node x, an adaptation rule (b, b ) for a child c of x means that x receives a bitrate b but sends a bitrate b < b to c. SVC layers above b are dropped before transmitting the video stream to c.

C. Algorithms

As said in section II, SPT and MST methods compute the multicast trees and place the adaptation rules at each node. We will use them in our algorithms in the initialization phase, i.e., when a call is established. However, when a bandwidth variation occurs, as recomputing the trees is costly, we opt for keeping them unchanged. But in order to guarantee that each participant receives the highest video quality allowed by its bandwidth, our algorithm recomputes the bitrates according to the following formulas:

For each sender s i b i (r i,j ) ← min B(r i,j ) p -1 , B(s i ) b(s i ) ← min B(s i ), max ri,j ∈Ri b i (r i,j ) (1)
These formulas are used in [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF] in order to set the definitive values of the bitrates. In a dynamic context, they should be used at each bandwidth variation. The first one ensures that a receiver r i,j gets the maximum among what is allowed by its bandwidth ( B(r i,j ) ) divided by the number of streams it receives (p -1) and the access uplink bandwidth of the sender ( B(s i ) ). The second one ensures that the sender emits the maximum needed video quality ( b(r i,j ) for each receiver) and allowed by its uplink bandwidth. The basic principle of our algorithms is to always satisfy these constraints but with minimum computation. Thus, when a bandwidth variation occurs, the bitrates are recomputed. If there is a bitrate change at a receiver, this change is propagated and the adaptation rules are pushed upward the tree. In the same way, if a bitrate change occurs at the sender, the propagation is done downward.

1) Change at a receiver: Figure 1 shows the propagation mechanism. Figure 1(a) depicts a multicast tree where node a is the sender and nodes e, f , and g are the receivers. The other nodes are SDN switches. The bitrate of each link is beside it. In Figure 1(b), because of a bandwidth change, the bitrate of (c, e) is no longer 250Kbps but 90Kbps. While neither e nor f need a stream of 250Kbps, the layer corresponding to this rate is dropped at b, the parent of c. In Figure 1(c), the bitrate of (d, g) changes from 1Mbps to 0.5Mbps. This change is propagated until b, but neither c nor d require 1Mbps, thus the bitrate of a is also adapted. This method adapts the bitrate upward from a receiver, but it is infrequent that it reaches the sender. It happens only in the case where the bitrate of the receiver and the sender are the same, and a change of the first one affects the second one. The goal of this algorithm is to minimize the computation task while minimizing the consumed bandwidth. To do so, it places the adaptation rules that drop higher quality streams as close as possible to the sender, thus saving the bandwidth in the core network. Algorithm 1 formalizes this method. It takes as input Algorithm 1: Relocate Up 1 Input: A tree T i and a node r 2 bmax ← max 1≤j≤k (b i (P i (r), c i,j ) with c i,j ∈ C i (P i (r)) 3 b min ← min(bmax, b(s i )) 4 if (b min = b i (P i (P i (r)), P i (r)) then 5 b i (P i (P i (r)), P i (r)) ← b min 6 AdaptRule(P i (r))

7 if P i (P i (r)) = s i ) then 8 RelocateUp(T i , P i (r)) 9 AdaptRule(P i (r))
the receiver where a change occurs, then propagates the change recursively. At each step, it checks if all the children of a node P i (r) receive less than their parent (line 3). If it is the case, P i (r) does not need to receive its current bitrate from P i (P i (r)). The latter bitrate is adapted (line 5) and the adaptation rules at P i (r) are updated by Algorithm 3 (line 6) further described in this paper. The algorithm operates recursively (line 8) until it reaches the sender (line 7) or there is no need to go further, i.e., the bitrate received by P i (r) has not to be changed. Complexity: In Algorithm 1, the number of recursive calls is bounded by the height of the multicast tree, which is itself bounded by the diameter of the network. The diameter is in O(n) in the worst case, however, it is much smaller in random and realistic networks. It is well known that the diameter of an Erdős-Rényi 2 graph G(n, φ) is "almost constant" when φ is fixed, and the diameter of a scalefree graph is ∼ log n/ log log n [START_REF] Bollobás | The diameter of a scale-free random graph[END_REF]. At each recursive call, Algorithm 1 checks the bandwidth between P (r) and its children. This suggests that this operation is in O(p) (the number of children of any node in the tree is smaller than the number of participants). The same argument applies to the complexity of Algorithm 3 (AdaptRule). Assuming that the diameter of the network is in O(log n), the complexity of Algorithm 1 is in O(p log n).

2) Change at the sender: When the bitrates are recomputed and the value of b(s i ) is modified. The method used is almost the same, except that the propagation is performed from the sender to the receivers. Again, the goal is to keep the adaptation rules as close as possible to the sender. The main difference between this case and the previous one is that the propagation is not only done on a branch of the tree but can occur on several branches down to the leaves (receivers). Algorithm 2 is performed when the sender's bitrate decreases. It starts at each sender's child. From each one of these nodes, if their own children receive more than the sender's bitrate (line 3), the children bitrates should be changed (line 4) and 2 Erdős-Rényi graphs are usually denoted by G(n, p) where p is the probability for each link to exist. We replace it by φ in order to avoid confusion with the number of participants.

Algorithm 2: Relocate Down

1 Input: A tree T i and node r 2 foreach c i,j ∈ C i (r) do 3 if b i (r, c i,j ) = b i (P i (r), r) then 4 b i (r, c i,j ) ← b i (P i (r), r) 5 AdaptRule(P i (r)) 6 RelocateDown(c i,j )
the adaptation rules updated (line 5). The algorithm performs recursively (line 6) until reaching a leaf. Unlike Algorithm 1 that starts from a leaf but infrequently reaches the sender, Algorithm 2 always reaches at least one receiver. Because there is always a receiver that receives the same bitrate as the sender (otherwise the sender could send a lower bitrate), and this receiver's bitrate must be updated. Complexity: Algorithm 2 browses all the tree in the worst case. Assuming that the diameter is in O(log n), the size of the tree is at most O(p log n), which is also the complexity of Algorithm 2.

Algorithm 3: Adapt rules

1 Input: A tree T i and node r 2 foreach c i,j ∈ C i (r) do 3 if b i (r, c i,j ) = b i (P i (r), r) then 4 Delete rule if exists 5 else 6 Rule(r, c i,j ) ← (b i (P i (r), r), b i (r, c i,j ))
3) Adapting the SVC Downsizing Rules: When incoming and outgoing bitrates are changed at a node r, the adaptation rules should be updated. Algorithm 3 takes as input a node r and, for each one of its children, deletes the old rule if it exists and replaces it by the correct one, i.e., the pair (incoming bitrate into r, outgoing bitrate from r to its child). On any node, there is at worst one rule per child. The number of children of any node is smaller than the number of participants, this gives a complexity of O(p).

4) The General Algorithm: Algorithm 4 is performed in the SDN controller. The call is first established by using algorithms MST or SPT of [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF]. Then at each event the controller reacts and adapts the multicast trees. We consider four events:

• B(r) ↓: The downlink bandwidth of a node r decreases.

This can lead to a change of b i (r) and b(s i ) for each tree T i because of formulas [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF]. In this case, the bitrates are recomputed and the new bitrate of r is propagated in all the trees (except the one where it is the sender) using Algorithm 1. Thus the complexity of processing this event is in O(p 2 log n) . • B(s i ) ↓: The Uplink bandwidth of a node s i decreases.

This can affect the bitrate of s i and those of the receivers, but only in the tree T i where s i is the sender. The change is propagated using Algorithm 2 on the access node of s i , which is its only child. The complexity is that of Algorithm 2, i.e., O(p log n).

• B(s i ) ↑: The uplink bandwidth of a node s i increases.

Again, this can impact the bitrates of s i and the receivers r i,j . After recomputation, if the receivers' bitrates do not change, there is no need to propagate the new bitrate of s i , because no receiver can get a higher bitrate. Otherwise, the new bitrates are propagated in T i using Algorithm 1. Likewise the case B(r) ↓, the complexity is in O(p 2 log n). • B(r) ↑: The downlink bandwidth of a node r increases.

This case is more complex since, according to formulas 1, it can impact all the bitrates in all the trees (except the one where r is the sender). For each tree, there are two possible cases. i) The sender's bitrate is not affected, in this case, the new downlink bitrate of r is propagated upward. ii) The sender's bitrate is affected, in this case it can, in turn, affect the other receivers' bitrates. This case is similar B(s i ) ↑. The complexity of processing this event is in O(p 3 log n) .

Algorithm 4: The general algorithm Note that the number of participants is much smaller than the network size, i.e., p n. If we consider the network size as a parameter, Algorithm 4 processes each event in O(n) if the diameter is linear and in O(log n) in the more realistic case where the diameter is logarithmic. This complexity is much lower than the complexity of MST/SPT that is in O(n 3 ) if the network is dense and O(n 2 log n) if the network is sparse [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF]. A lower complexity when processing an event implies less consumed resources in the controller and better reactivity for the participants.

IV. EVALUATION

A. Parameters

In order to evaluate the efficiency of our solution, we performed simulations that compared it with three other approaches:

• Unicast method with recomputation at each event, called Uni-recomp. At each change of access bandwidth, the shortest paths between each pair of participants are recomputed.

• MST method with recomputation at each event, called MST-recomp. • SPT method with recomputation at each event, called SPT-recomp. • Our solution, called MST-adapt. MST method is performed to establish the video call then the multicast trees are adapted at each event according to Algorithm 4. The simulations are performed on two types of random topologies with varying size: Erdős-Rényi (ER) G(n, φ) with φ = 0.5 and Magoni-Pansiot (MP) [START_REF] Magoni | Internet topology modeler based on map sampling[END_REF] which is a scalefree topology derived from real networks. There are 6 participants for each call and 10 events chosen randomly in {B(r) ↓, B(r) ↑, B(s i ) ↓, B(s i ) ↑}. Each result is obtained by averaging 100 runs.

We aim to estimate the trade-off performed by our solution between bandwidth saving and processing time. It is clear that MST and SPT methods should perform equal or more bandwidth saving since they recompute the multicast trees and replace the SVC adaptation rules at each event. However, our solution has a lower complexity, which leads to less processing time and used resources.

B. Results

Figures 2(a) and 2(b) show the results obtained on Erdős-Rényi topologies for networks from 500 to 4000 nodes. On Figure 2(a), one can observe the average processing time per event of the different algorithms. Note the logarithmic scale. Our solution is much faster than the other approaches. For instance, on a network of 4000 nodes, our solution takes around 4.10 -4 s while all the others take around 4s. Thus achieving a ratio of 10 -4 . The same results are obtained using SPT or unicast instead of MST to establish the video call. Figure 2(b) shows the average used bandwidth in the core network per call. Surprisingly, it is the same for our solution and MST-recomputing, and better than the other algorithms. It appears that the bandwidth variations at an access link are very small compared to the bandwidth scale in the core links. They are too small to induce a change in the multicast tree topologies. Consequently, our solution is thousands of times faster than a total recomputing while providing exactly the same bandwidth saving.

The same results can be observed on MP networks on figures 3(a) and 3(b). All the algorithms are faster because MP networks are sparse while ER ones are dense. However, the ratio between processing time of our solution and the other ones is still around 10 -4 . The results on both topologies with different sizes show that our approach of replacing the SVC adaptation rules in the multicast trees is well adapted to a dynamic context. While the algorithms proposed in [START_REF] Hasrouty | SVC Videoconferencing Call Adaptation and Bandwidth Usage in SDN Networks[END_REF] are very efficient to establish a video call, it appears that our adaptation algorithms are more suited for processing the different variations occurring during the call.

V. CONCLUSION

Videoconferencing is a critical tool in today's connected world. Regrettably, the users still suffer from poor QoS because of management difficulties of the video calls. The emergence of SDN represents a chance to simplify this management and to provide more flexibility. Some works propose to use SDN together with video layering techniques with the aim to adapt the video quality to the users' heterogeneity. In order to minimize the bandwidth consumption in the core network while providing the best possible video quality to the users, multicast trees are computed and video layer adaptations are performed in the nodes of the core network. To be serviceable, these methods should be evaluated in a dynamic context where access channel bandwidth variations occur. In this paper, we proposed to evaluate these methods in such a context and to show their reactivity. We proposed much faster algorithms, based on tree traversal ideas, that adapt the video layering without recomputing the multicast trees. The low complexity of these algorithms allows high reactivity to network changes and low resource consumption at the SDN controller. The simulation results confirm the efficiency of our algorithms in terms or processing time. Moreover, they show that our solution performs as good bandwidth saving as more costly and elaborate methods.

Fig. 1 .

 1 Fig. 1. Bitrate change propagation

Fig. 2 .

 2 Fig. 2. Average processing time/bandwidth usage vs ER network size.

Fig. 3 .

 3 Fig. 3. Average processing time/bandwidth usage vs MP network size.

Note that it is possible that B(x, y) = B(y, x) if the uplink and downlink bandwidths are not the same.

ACKNOWLEDGMENT

This work was supported, in part, by SFI grant 13/RC/2094 to Lero -the Irish Software Research Centre, and in part by ANR grant 10IDEX-03-02 to the University of Bordeaux.


Hassan S Bakouch
email: hnbakouch@yahoo.com

Marcelo Bourguignon
email: m.p.bourguignon@gmail.com

Christophe Chesneau
email: christophe.chesneau@unicaen.fr

On some properties of the hazard rate function for compound distributions

Keywords: Compound distributions, Hazard rate function, Power series distributions. 2000 MSC: 60E05, 62E15

The hazard rate function is an important quantity characterizing several random phenomena. In this paper, we investigate some crucial properties satisfied by the hazard rate function for a general power series class of distributions, as monotonicity properties, sharp bounds and convexity properties. We highlight how these properties are related to those of the baseline distribution and the corresponding probability-generating function of discrete power series distributions.

Introduction

Several aspects of an absolutely continuous distribution can be seen more clearly from the hazard rate function (hrf) than from either the distribution or density functions. The hrf is an important quantity characterizing life phenomena. Let X be a random variable with probability density function (pdf) f (x) and cumulative distribution function (cdf) F (x). The hrf of X is defined by f (x)/F (x), where F (x) = 1 -F (x) is the survival function. The hrf may be increase, decrease, constant, upside-down bathtub (unimodal), bathtub-shaped or indicate a more complicated process. In many applications there is a qualitative information about the hrf shape, which can help in selecting a specified model.

Several classes of distributions were proposed by compounding some useful lifetime and discrete distributions in the last few years. The compounding procedure follows the pioneering work of [START_REF] Marshall | A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families[END_REF]. Chakandi and Ganjali (2009) defined the exponential power series (EPS) class of distributions, which contains as special cases involving the exponential geometric [START_REF] Adamidis | A lifetime distribution with decreasing failure rate[END_REF], exponential Poisson [START_REF] Kus | A new lifetime distribution[END_REF] and exponential logarithmic (Tamasbi and Rezaei , 2008) distributions. [START_REF] Morais | A Compound Class of Weibull and Power Series Distributions[END_REF] defined the Weibull power series (WPS) class which includes the EPS distribution as a sub-model. The WPS distributions can have increasing, decreasing and upside down bathtub hrf. The generalized exponential power series distributions were proposed by [START_REF] Mahmoudi | Generalized exponential?power series distributions[END_REF]. [START_REF] Silva | The compound family of extended Weibull power series distributions[END_REF] studied the extended Weibull power series family of distributions, which includes as special models the EPS and WPS classes of distributions. [START_REF] Bourguignon | A new class of fatigue life distributions[END_REF] and [START_REF] Silva | The Burr XII power series distributions: A new compounding family[END_REF] proposed the Birnbaum-Saunders power series (BSPS) and Burr XII power series (BXIIPS) classes of distributions, respectively. Recently, [START_REF] Silva | A new compounding family of distributions: The generalized gamma power series distributions[END_REF] proposed the generalized gamma power series (GGPS) distributions and defines 76 sub-models as special cases. The GGPS hrf can be constant, decreasing, increasing, upside-down bathtub or bathtub-shaped.

In this paper, we consider a general power series class of distributions characterized by the distributions of the random variables X = min(X 1 , . . . , X N ), where X 1 , X 2 , . . . , X N are independent and identically distributed (i.i.d.) continuous random variables and N is a discrete random variable having a power series distributions (including translated Poisson distribution, truncated Poisson distribution, geometric distribution, binomial distribution, and many more). For example, X can represent the lifetime of a system with N components where the lifetime of the i-th component is represented by X i .

Further details on this general power series class can be found in [START_REF] Tahir | Compounding of distributions: a survey and new generalized classes[END_REF], and the reference therein. In this study, we focus our attention on the nature of the hrf of X. We investigate some of its properties according to the properties of the distributions of X 1 and N . In particular, we prove that if the hrf of X 1 is decreasing, the hrf of X is decreasing too, without extra assumptions on the distribution of N ; the power series family of distributions preserve the decreasing hrf property of the baseline distribution. We determine sharp and simple lower and upper bounds for the hrf of X depending only of the hrf of X 1 and the expectation of N . An immediate consequence is the preservation of the discontinuities for these hrfs. We also provide a differential equation where the hrf of X is the solution, with some consequences n on its monotonicity properties. Additional properties are given for three specific distributions on N : the translated Poisson distribution, the truncated Poisson distribution and the geometric distribution.

The rest of the paper is organized as follows. In Section 2, we introduce the power series family of distributions and present our main results. Section 3 provided more contributions for three specific class of distributions.

The power series class of distributions

Let N be a discrete random variable having a power series distribution (truncated at zero) with probability mass function

p n = P (N = n) = a n θ n C(θ) , n = 1, 2, . . . , (1)
where the support S of N is a subset of the non-negative integers, a n ≥ 0 depends only on n and there

is r > 0 such that C(θ) = ∞ n=1
a n θ n is finite for all θ ∈ (0, r) (r can be ∞). Although we will always consider θ as a value in (0, r). Note that C(θ) has derivatives of all orders in (0, r) and those derivatives can be obtained by differentiating the power series term to term. Also, because a n ≥ 0 for all n, C(θ) and all its derivatives will be positive in (0, r).

The moments of the random variable N can be easily obtained from the probability-generating function K(s) = E(s N ). By noticing that K(s) = C(s θ)/C(θ), they also can be determined using C(θ). By setting

G(θ) = log[C(θ)], G (θ) = d G(θ)/dθ and G (θ) = d 2 G(θ)/dθ 2 ,
the expectation of N is given by E(N ) = θ G (θ) and the variance is given by Var

(N ) = θ 2 G (θ) + θ G (θ).
Table 1 presents well-known power series distributions and some useful quantities (to determine moments, quantile . . . ): C(θ), C (θ), C (θ) and C -1 (θ).

Distribution an C(θ) C (θ) C (θ) C -1 (θ) Θ Trunc. Poisson n! -1 e θ -1 e θ e θ log(θ + 1) θ ∈ (0, +∞) Logarithmic n -1 -log(1 -θ) (1 -θ) -1 (1 -θ) -2 1 -e -θ θ ∈ (0, 1) Geometric 1 θ(1 -θ) -1 (1 -θ) -2 2(1 -θ) -3 θ(θ + 1) -1 θ ∈ (0, 1) Binomial m n (θ + 1) m -1 m(θ + 1) m-1 m(m-1) (θ+1) 2-m (θ -1) 1/m -1 θ ∈ (0, 1)
Table 1: Useful quantities for some power series distributions.

Let us now present the considered general power series (GPS) class of distributions. Let ξ be a parameter vector. Let X 1 , X 2 , . . . be i.i.d. random variables with cdf G(x; ξ) (baseline distribution), pdf g(x; ξ) and hrf r(x; ξ) = g(x; ξ)/(1 -G(x; ξ)). We set X = min(X 1 , . . . , X N ). The conditional cdf of X|N = n is given by P

(X ≤ x | N = n) = 1 -[1 -G(x; ξ)] n .
The GPS class of distributions is characterized by the cdf:

F X (x; θ, ξ) = +∞ n=1 P (X ≤ x | N = n)P (N = n) = 1 - C(θ G(x; ξ)) C(θ) , (2)
where G(x; ξ) = 1 -G(x; ξ). Using the derivative of C(θ) according to θ, i.e. C (θ) = +∞ n=1 a n nθ n-1 , the pdf is given by

f X (x; θ, ξ) = θ g(x; ξ) C (θ G(x; ξ)) C(θ) .
The hrf of GPS class of distributions is given by

h X (x; θ, ξ) = θ g(x; ξ) C (θ G(x; ξ)) C(θ G(x; ξ)) .
The following result shows a sufficient condition to ensure that h X (x; θ, ξ) decreases.

Proposition 1. If r(x; ξ) is a decreasing function of x, then h X (x; θ, ξ) is a decreasing function of x.

Proof of Proposition 1. Let us introduce the function M (s) defined by

M (s) = E(N s N ) E(s N ) , s ∈ (0, 1].
It is of interest because it allows us to write h X (x; θ, ξ) in the following form:

h X (x; θ, ξ) = g(x; ξ) G(x; ξ) C (θ G(x; ξ)) θ G(x; ξ) C(θ G(x; ξ)) = r(x; ξ)M (G(x; ξ)) (3)
and, since r(x; ξ) is positive and is supposed to be a decreasing function of x, and G(x; ξ) is a decreasing function of x, it is enough to prove that M (s) is an increasing function of s with s ∈ (0, 1]. The standard derivative ratio formula with [E(N s N )] = E(N 2 s N -1 ) and [E(s N )] = E(N s N -1 ) gives

M (s) = E(N 2 s N -1 )E(s N ) -E(N s N )E(N s N -1 ) [E(s N )] 2 = E(N 2 s N -1 )E(s N ) -s E(N s N -1 ) 2 [E(s N )] 2 . ( 4
)
Clearly, [E(s N )] 2 > 0. The Cauchy-Schwarz inequality gives

s E(N s N -1 ) 2 = s E(N s N 2 -1 s N 2 ) 2 ≤ sE(N 2 s N -2 )E(s N ) = E(N 2 s N -1 )E(s N ),
implying that E(N 2 s N -1 )E(s N ) -s E(N s N -1 ) 2 ≥ 0. Thus M (s) ≥ 0, so M (s) is an increasing function of s, implying that h X (x; θ, ξ) is a decreasing function of x. This ends the proof.

Thus, Proposition 1 ensures that the power series family of distributions preserve the decreasing hrf property of the baseline distribution.

The following result presents lower and upper bounds for h X (x; θ, ξ) depending on r(x; ξ) and E(N ) (without assumptions on the monotony of r(x; ξ)).

Proposition 2. For any x ∈ R, we have

r(x; ξ) ≤ h X (x; θ, ξ) ≤ E(N )r(x; ξ).
Proof of Proposition 2. We use the expression (3). Owing to N ≥ 1 and s N > 0 for s ∈ (0, 1], we have

M (s) = E(N s N ) E(s N ) ≥ E(s N ) E(s N ) = 1. Since r(x; ξ) is positive, we obtain r(x; ξ) ≤ h X (x; θ, ξ). Now using the fact that M (s) is increasing in s, G(x; ξ) ∈ [0, 1] and M (1) = E(N ), we have h X (x; θ, ξ) = r(x; ξ)M (G(x; ξ)) ≤ r(x; ξ)M (1) = E(N )r(x; ξ).
The proof of Proposition 2 is complete.

Owing to Proposition 2, we have lim Proposition 3. The hrf h X (x; θ, ξ) satisfies the non-linear equation:

h X (x; θ, ξ) h X (x; θ, ξ) = g (x; ξ) g(x; ξ) - M (G(x; ξ)) M (G(x; ξ)) g(x; ξ) + r(x; ξ),
where M (s) is given by (4).

Proof of Proposition 3. Using the expression (3), we have

h X (x; θ, ξ) = r(x; ξ)M (G(x; ξ)) = r (x; ξ)M (G(x; ξ)) -r(x; ξ)M (G(x; ξ))g(x; ξ).
Hence, using again (3),

h X (x; θ, ξ) h X (x; θ, ξ) = r (x; ξ) r(x; ξ) - M (G(x; ξ)) M (G(x; ξ)) g(x; ξ).
It is well-known that a hrf as r(x; ξ) satisfies the non-linear differential equation:

r (x; ξ) r(x; ξ) = g (x; ξ) g(x; ξ) + r(x; ξ). Therefore h X (x; θ, ξ) h X (x; θ, ξ) = g (x; ξ) g(x; ξ) - M (G(x; ξ)) M (G(x; ξ)) g(x; ξ) + r(x; ξ).
Proposition 3 is proved.

An important consequence of Proposition 3 is that, since h X (x; θ, ξ) > 0, the sign of h X (x; θ, ξ) is the same as the sign of g (x;ξ)

g(x;ξ) -M (G(x;ξ)) M (G(x;ξ)) g(x; ξ) + r(x; ξ) (or r (x;ξ) r(x;ξ) -M (G(x;ξ)) M (G(x;ξ)) g(x; ξ)).
This can provide important informations on the monotonicity properties of h X (x; θ, ξ) according to the complexity of the functions considered (discontinuities, . . . ).

Also, remark that we can rediscover the result of Proposition 1: in the proof of Proposition 1, it is shown that M (s) ≥ 0. Since h X (x; θ, ξ) > 0, g(x; ξ) > 0, M (G(x; ξ)) > 0 and r(x; ξ) > 0, if r (x; ξ) is decreasing, so r (x; ξ) < 0, then r (x;ξ) r(x;ξ) -M (G(x;ξ)) M (G(x;ξ)) g(x; ξ) < 0 implying that h X (x; θ, ξ) < 0, thus h X (x; θ, ξ) is a decreasing function of x.

The next section presents more properties of h X (x; θ, ξ) by considering specific distributions for N , namely the translated Poisson distribution, the truncated Poisson distribution and the geometric distribution.

Study of three examples

Translated Poisson distribution case

Here we supposed that N follows the translated Poisson distribution of parameter θ > 0, i.e. P (N = n) = e -θ θ n-1

(n-1)! , n = 1, 2, . . . . Then one can apply Propositions 1, 2 (with E(N ) = 1+θ) and 3. However, since we have the analytic expression of h X (x; θ, ξ), some properties can be refined. The result below provides some equalities in this case.

Proposition 4. If N follows the translated Poisson distribution with parameter θ > 0, then we have h X (x; θ, ξ) = r(x; ξ) 1 + θG(x; ξ) , h X (x; θ, ξ) = r (x; ξ) + θg (x; ξ).

Proof of Proposition 4. We have E(s N ) = s e θ(s-1) and E(N s N ) = [E(s N )] s = (1 + θs)s e θ(s-1) . We now use the expression (3). Since M (s) = E(N s N ) E(s N ) = 1 + θ s, we have h X (x; θ, ξ) = r(x; ξ)M (G(x; ξ)) = r(x; ξ) 1 + θG(x; ξ) . Moreover h X (x; θ, ξ) = r (x; ξ) 1 + θG(x; ξ) -θg(x; ξ)r(x; ξ) = r (x; ξ) + θ r (x; ξ)G(x; ξ) -g(x; ξ)r(x; ξ) .

We end the proof by observing that r(x; ξ)G(x; ξ) = g(x; ξ) and r (x; ξ)G(x; ξ) -g(x; ξ)r(x; ξ) = (r(x; ξ)G(x; ξ)) = g (x; ξ).

Proposition 4 is proved.

Some additional consequences of Proposition 4 are described below. Note that the extrema of h X (x; θ, ξ) are given by the solution x of the equation h X (x; θ, ξ) = 0, i.e. r (x; ξ) + θg (x; ξ) = 0.

As ever shown in Proposition 1, if r(x; ξ) is a decreasing function of x, we have r (x; ξ) ≤ 0, implying that g (x; ξ) ≤ 0, so h X (x; θ, ξ) = r (x; ξ) + θg (x; ξ) ≤ 0, hence h X (x; θ, ξ) is a decreasing function of x.

On the other hand, if r(x; ξ) and g(x; ξ) are increasing functions of x, then h X (x; θ, ξ) = r (x; ξ) + θg (x; ξ) ≥ 0, so h X (x; θ, ξ) is an increasing functions of x.

Also note that if r(x; ξ) and g(x; ξ) are convex (resp concave), then h X (x; θ, ξ) = r (x; ξ)+θg (x; ξ) ≥ 0 (resp h X (x; θ, ξ) ≤ 0), then h X (x; θ, ξ) is convex (resp concave).

Truncated Poisson distribution case

Here we supposed that N follows the truncated Poisson distribution of parameter θ > 0, i.e. P (N = n) = 1 e θ -1 θ n n! , n = 1, 2, . . . . This case is standard in the literature. See, for instance, the truncatedexponential skew-symmetric distributions [START_REF] Nadarajah | Truncated-exponential skew-symmetric distributions[END_REF], and the references therein. In this case, one can apply Propositions 1, 2 (with E(N ) = θe θ e θ -1 ) and 3. The analytic expression of h X (x; θ, ξ) allows us to prove some new properties. The result below provides some equalities in this case.

Proposition 5. If N follows the truncated Poisson distribution with parameter θ > 0, then we have h X (x; θ, ξ) = r(x; ξ) θG(x; ξ)

1 -e -θG(x;ξ) .

Let us now present two expressions of h X (x; θ, ξ).

First expression:

h X (x; θ, ξ) = r (x; ξ) θG(x; ξ)

1 -e -θG(x;ξ) -θg(x; ξ)r(x; ξ)e -θG(x;ξ) w(x), where w(x) = e θG(x;ξ) -1 -θG(x; ξ)

(1 -e -θG(x;ξ) ) 2 .

Second expression:

h X (x; θ, ξ) = θ
1 -e -θG(x;ξ) g (x; ξ) + θ 2 [g(x; ξ)] 2 e -θG(x;ξ)

(1 -e -θG(x;ξ) ) 2 .

Proof of Proposition 5. We have E(s N ) = 1 e θ -1 (e θs -1) and E(N s N ) = [E(s N )] s = 1 e θ -1 θs e θs . We now use the expression (3). Since M (s) = E(N s N ) E(s N ) = θs 1-e -θs , we have h X (x; θ, ξ) = r(x; ξ)M (G(x; ξ)) = r(x; ξ) θG(x; ξ)

1 -e -θG(x;ξ) .

Now remark that M (s) = θe -θs e θs -1-θs (1-e -θs ) 2 . So h X (x; θ, ξ) = r (x; ξ)M (G(x; ξ)) -r(x; ξ)M (G(x; ξ))g(x; ξ) = r (x; ξ) θG(x; ξ)

1 -e -θG(x;ξ) -θg(x; ξ)r(x; ξ)e -θG(x;ξ) w(x).

Proposition 6 is proved.

From the second equality, we see that the sign of h X (x; θ, ξ) is the same as the sign of w(x) since h X (x; θ, ξ) ≥ 0. As ever shown in Proposition 1, if r(x; ξ) is a decreasing function of x, we have r (x; ξ) ≤ 0, so w(x) ≤ 0, hence h X (x; θ, ξ) is a decreasing function of x.

  ξ) = +∞ if and only if lim x→x * h X (x; θ, ξ) = +∞. The discontinuities of r(x; ξ) are preserved in h X (x; θ, ξ).Proposition 3 below show a differential equation where h X (x; θ, ξ) is the solution.

This first expression is proved.

On the other hand, one can also express M (s) as M (s) = θ 1 1-e -θs -θs e -θs (1-e -θs ) 2 . So h X (x; θ, ξ) = r (x; ξ)M (G(x; ξ)) -r(x; ξ)M (G(x; ξ))g(x; ξ) = r (x; ξ) θG(x; ξ)

1 -e -θG(x;ξ) -θg(x; ξ)r(x; ξ) 1

1 -e -θG(x;ξ) -θG(x; ξ) e -θG(x;ξ)

(1 -e -θG(x;ξ) ) 2

= θ

1 -e -θG(x;ξ) [r (x; ξ)G(x; ξ) -g(x; ξ)r(x; ξ)] θG(x;ξ) (1 -e -θG(x;ξ) ) 2 .

Let us now observe that r(x; ξ)G(x; ξ) = g(x; ξ) and θG(x;ξ) (1 -e -θG(x;ξ) ) 2 .

The second expression is proved; Proposition 5 is proved.

Some additional consequences of Proposition 5 are described below. Note that the extreme of h X (x; θ, ξ) are given by the solution x of the equation h X (x; θ, ξ) = 0. One can consider the most simple expression for h X (x; θ, ξ) between the first one and the second one.

Let us now show a fact ever proved in Proposition 1 by considering the first expression of h X (x; θ, ξ). If r(x; ξ) is a decreasing function of x, we have r (x; ξ) ≤ 0. Since e x -1 -x ≥ 0, we have w(x) ≥ 0, implying that h X (x; θ, ξ) ≤ 0, so h X (x; θ, ξ) is a decreasing function of x.

On the other hand, if g(x; ξ) are increasing functions of x, then g (x; ξ) ≥ 0 and it follows from the second expression of h X (x; θ, ξ) that h X (x; θ, ξ) ≥ 0, so h X (x; θ, ξ) is an increasing functions of x.

Geometric distribution case

Here we supposed that N follows the geometric distribution of parameter θ ∈ (0, 1), i. e. P (N = n) = (1 -θ)θ n-1 , n = 1, 2, . . . . For an application, we refer the reader to [START_REF] Marshall | A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families[END_REF]. Naturally, we can apply Propositions 1, 2 (with E(N ) = 1 1-θ ) and 3. However, since we have access to the expression of h X (x; θ, ξ), some properties can be refined. The result below provides some equalities in this case. Proposition 6. If N follows the geometric distribution with parameter θ ∈ (0, 1), then we have

where

Proof of Proposition 6. We have = h X (x; θ, ξ) r (x; ξ) r(x; ξ) -θ g(x; ξ) 1 -θ G(x; ξ) = h X (x; θ, ξ)w(x).


Alexander G Ramm
email: ramm@math.ksu.edu

Semion Gutman
email: sgutman@ou.edu

MODIFIED RAYLEIGH CONJECTURE FOR SCATTERING BY PERIODIC STRUCTURES

This paper contains a self-contained brief presentation of the scattering theory for periodic structures. Its main result is a theorem (the Modified Rayleigh Conjecture, or MRC), which gives a rigorous foundation for a numerical method for solving the direct scattering problem for periodic structures. A numerical example illustrating the procedure is presented.

Introduction

For simplicity we consider a 2-D setting, but our arguments can be as easily applied to n-dimensional problems, n ≥ 2. Let f : R → R, f (x + L) = f (x) be an L-periodic Lipschitz continuous function, and let D be the domain

D = {(x, y) : y ≥ f (x), x ∈ R}.
Without loss of generality we assume that f ≥ 0. If it is not, one can choose the origin so that this assumption is satisfied, because M := sup 0≤x≤L |f (x)| < ∞.

Let x = (x, y) and u(x) be the total field satisfying where the unit vector α = (cos θ, -sin θ), 0 < θ < π/2, and v(x) is the scattered field, whose asymptotic behavior as y → ∞ will be specified below, and

(1.4) u(x + L, y) = νu(x, y), u x (x + L, y) = νu x (x, y) in D, ν : = e ikL cos θ .

Conditions (1.4) are the qp (quasiperiodicity) conditions. To find the proper radiation condition for the scattered field v(x) consider the spectral problem (1.5) ϕ + λ 2 ϕ = 0, 0 < x < L,

(1.6) ϕ(L) = νϕ(0), ϕ (L) = νϕ (0)
arising from the separation of variables in (1.1)- (1.4). This problem has a discrete spectrum, and its eigenfunctions form a basis in L 2 (0, L). One has ϕ = Ae iλx + Be -iλx , A, B = const, Ae iλL + Be -iλL = ν(A + B), iλAe iλL -iλBe -iλL = iλν(A -B). Thus e iλL -ν e -iλL -ν iλ(e iλL -ν) -iλ(e -iλL -ν) = 0.

So, iλ(e iλL -ν)(e -iλL -ν) = 0. If λ = 0, then ϕ = A + Bx, A + BL = νA, B = νB. Since ν = e ikL cos θ , one has no eigenvalue λ = 0 unless kL sin θ = 2πm, m > 0 is an integer. Let us assume that kL cos θ = 2πm. Then e iλL = e ikL cos θ or e -iλL = e ikL cos θ , that is λ + j = k cos θ + 2πj L , or λ - j = -k cos θ + 2πj L , j = 0, ±1, ±2, . . .

The corresponding eigenfunctions are e iλ + j x and e -iλ - j x . We will use the system e iλ + j x , which forms an orthogonal bais in L 2 (0, L). One has:

L 0 e iλ + j x e -iλ + m x dx = L 0 e 2πi L (j-m) dx = 0, j = m.
The normalized eigenfunctions are

ϕ j (x) = e iλ + j x
√ L , j = 0, ±1, ±2, . . . These functions form an orthonormal basis of L 2 (0, L). Let us look for v(x) = v(x, y) of the form

(1.7) v(x, y) = ∞ j=-∞ c j v j (y)ϕ j (x), y > M, c j = const.
For y > M , equation (

) v j + (k 2 -λ 2 j )v j = 0. Let us assume that λ 2 j = k 2 for all j. Then (1.9) v j (y) = e iµj y , 1.1) implies (1.8
where, for finitely many j, the set of which is denoted by J, one has:

(1.10)

µ j = (k 2 -λ 2 j ) 1/2 > 0, if λ 2 j < k 2 , j ∈ J, and
(1.11) µ j = i(λ 2 j -k 2 ) 1/2 , if λ 2 j > k 2 , j / ∈ J.
The radiation condition at infinity requires that the scattered field v(x, y) be representable in the form (1.7) with v j (y) defined by (1.9)- (1.11).

The Periodic Scattering Problem consists of finding the solution to (1.1)-(1.4) satisfying the radiation condition (1.7), (1.9)- (1.11).

The existence and uniqueness for such a scattering problem is established in Section 2. Our presentation is essentially self-contained. In [START_REF] Alber | A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent[END_REF] the scattering by a periodic structure was considered earlier, and was based on a uniqueness theorem from [START_REF] Eidus | Some boundary-value problems in infinite regions[END_REF]. Our proofs differ from the proofs in [START_REF] Alber | A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent[END_REF]. There are many papers on scattering by periodic structures, of which we mention a few [START_REF] Alber | A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent[END_REF], [START_REF] Albertsen | Comparison of four software packages applied to a scattering problem[END_REF], [START_REF] Bonnet-Bendhia | Guided waves by eletromagnetic gratings and non-uniqueness examples for the diffraction problem[END_REF], [START_REF] Bonnet-Bendhia | Diffraction by an acoustic grating perturbed by a bounded obstacle[END_REF], [START_REF] Christiansen | On a misconception involving point collocation and the Rayleigh hypothesis[END_REF], [START_REF] Kazandjian | Rayleigh-Fourier and extinction theorem methods applied to scattering and transmission at a rough solid-solid interface[END_REF], [START_REF] Kazandjian | Reflection from a corrugated surface revisited[END_REF], [START_REF] Millar | The Rayleigh hypothesis and a related least-squares solution to the scattering problems for periodic surfaces and other scatterers[END_REF], [START_REF] Millar | On the Rayleigh assumption in scattering by a periodic surface[END_REF], [START_REF]Electromagnetic theory of gratings[END_REF], [START_REF] Rayleigh | On the dynamical theory of gratings[END_REF]. The Rayleigh conjecture is discussed in several of the above papers. It was shown (e.g. [START_REF]Electromagnetic theory of gratings[END_REF], [START_REF] Barantsev | Concerning the Rayleigh hypothesis in the problem of scattering from finite bodies of arbitrary shapes[END_REF]) that this conjecture is incorrect, in general. The modified Rayleigh conjecture is a theorem proved in [START_REF] Ramm | Modified Rayleigh Conjecture and Applications[END_REF] for scattering by bounded obstacles. A numerical method for solving obstacle scattering problems, based on the modified Rayleigh conjecture is developed in [START_REF] Gutman | Numerical implementation of the MRC method for obstacle scattering problems[END_REF]. The main results of our paper are: the modified Rayleigh conjecture for periodic structures (Theorem 4.4) and a rigorous numerical method for solving scattering problems by periodic structures, based on the modified Rayleigh conjecture (Section 4). The proof of the limiting absorption principle (LAP) and the rigorous and self-contained development of the plane wave scattering theory by periodic structures is also of interest for broad audience. This theory is based partly on the ideas developed in [START_REF] Ramm | Scattering by Obstacles[END_REF], [START_REF] Ramm | Investigation of the scattering problem in some domains with infinite boundaries I, II[END_REF], [START_REF] Ramm | Existence and uniqueness of the scattering solutions in the exterior of rough domains, in the book "Operator Theory and Its Applications[END_REF], [START_REF] Ramm | Scattering by obstacles in acoustic waveguides, Spectral and scattering theory[END_REF]. The proof of the key lemma 2.2 is based on a version of Ramm's identity (2.16). Numerical implementation of the method for solving scattering problems by periodic structures, based on the modified Rayleigh conjecture, is constructed using the approach developed in [START_REF] Gutman | Numerical implementation of the MRC method for obstacle scattering problems[END_REF] and in [START_REF] Ramm | Modified Rayleigh Conjecture method for multidimensional obstacle scattering problems[END_REF]. Applications to inverse problems are discussed in [START_REF] Ramm | Modified Rayleigh Conjecture and Applications[END_REF] and [START_REF] Ramm | Inverse Problems[END_REF].

Periodic Scattering Problem

Existence and uniqueness of solutions of the Periodic Scattering Problem can be proved easily, if one establishes first the existence and uniqueness of the resolvent kernel G(x, y, ξ, η, k) of the Dirichlet Laplacian in D:

(2.1) (∆ + k 2 )G(x, y, ξ, η, k) = -δ(x -ξ)δ(y -η), G = 0 on S, (2.2) G(x + L, y, ξ, η, k) = νG(x, y, ξ, η, k), G(x, y, ξ + L, η, k) = νG(x, y, ξ, η, k), (2.3) G x (x + L, y, ξ, η, k) = νG x (x, y, ξ, η, k), G x (x, y, ξ + L, η, k) = νG x (x, y, ξ, η, k),
and G satisfies the LAP, see (2.5) below. The overbar here and below stands for the complex conjugation.

Indeed, if such a function G exists, then v can be found by the Green's formula

(2.4) v(x, y) = - S L u 0 (ξ, η)G N (x, y, ξ, η, k) ds,
where N is the unit normal vector to S pointing into D.

To prove the existence and uniqueness of G(x, y, ξ, η, k) define 0 = -∆ to be the Laplacian on the set of C 2 (D) quasiperiodic functions vanishing on the boundary S, and vanishing near infinity. Let

D L : = {(x, y) : 0 ≤ x ≤ L, (x, y) ∈ D}.
Then D L is a section of D, and 0 is a symmetric operator in L 2 (D L ). This operator is nonnegative , and therefore [START_REF] Kato | Perturbation theory for linear operators[END_REF] there exists its unique selfadjoint Friedrichs' extension, which will be denoted by .

Let Im(k 2 ) > 0. Then there exists a unique resolvent operator ( -k 2 ) -1 . Thus its kernel G(x, y, ξ, η, k) also exists and it is unique. To establish the existence and uniqueness of the kernel for k > 0 we are going to prove the following Limiting Absorption Principle (LAP). Let k > 0, > 0 and assume that k 2 is not equal to λ 2 j . Then the limit (2.5) lim

→0+ G(x, y, ξ, η, k + i ) = G(x, y, ξ, η, k),
exists for all (x, y) ∈ D, x = y. The proof is based on the following two lemmas.

Lemma 2.1. Let 0 < < 1, and a > 2. Then

(2.6) D L |G(x, y, ξ, η, k + i )| 2 (1 + ξ 2 + η 2 ) a/2 dξdη ≤ c,
where c = const > 0 does not depend on , and (x, y) is running on compact sets.

Proof of Lemma 2.1. It is sufficient to prove that the solution to the problem

(2.7) (∆ + k 2 + i )w = F, in D L , w ∈ L 2 (D L ), w = 0 on S L (2.8) w (x + L, y) = νw (x, y), w x (x + L, y) = νw x (x, y),
satisfies the estimate (2.9)

N 2 : = sup 0< <1 D L |w (x, y)| 2 (1 + x 2 + y 2 ) a/2 dxdy : = N 2 (w ) ≤ c, where F ∈ C ∞ 0 (D L
) is arbitrary, and c = const > 0 is independent of > 0. If (2.9) fails, then N n → ∞, n → 0. Define ψ : = w /N , where : = n . Then N (ψ ) = 1, ψ solves (2.7) (with F replaced by F : = F/N ), and satisfies (2.8). From N (ψ ) = 1 it follows that ψ ψ as → 0, where denotes the weak convergence in

L 2 (D L , 1/(1 + x 2 + y 2 ) a/2 ) : = L 2
a . By elliptic estimates, ψ ψ in H 2 loc (D L ), and therefore ψ → ψ in L 2 loc (D L ). This and (2.7)-(2.8) imply ψ → ψ in H 2 loc (D L ). Thus ψ solves the homogeneous (F = 0) problem (2.7)-(2.8). If we prove that ψ = 0, then we get a contradiction, which shows that (2.9) holds. The contradiction comes from the relationship 0

= N (ψ) = lim →0 N (ψ ) = 1. One proves that (2.10) lim →0 N (ψ ) = N (ψ) as follows. If (x, y) ∈ D R : = {(x, y) : f (x) ≤ y ≤ R, 0 ≤ x ≤ L},
where R > M is an arbitrary large fixed number, then lim →0 N (ψ η R ) = N (ψη R ), where

η R : = 1, f (x) < y < R, 0, y > R.
In the region

D R = {(x, y) : y > R, 0 ≤ x ≤ L}, one has |ψ (x, y)| ≤ c, (x, y) ∈ D R . Thus sup 0< <1 N (ψ (χ L -η R )) ≤ O 1 R γ , 0 < γ < a -2.
The desired result (2.10) follows.

To complete the proof let us show that the problem (2.7)-(2.8), with F = 0, and = 0, has only the trivial solution w, provided that w is "outgoing" in the sense w jy -iµ j w j = o(1), as y → ∞, w j : = L 0 wϕ j dx.

One has (2.11) lim

R→∞ S R (ww y -w y w) ds = 0,
where S R : = {(x, y) : y = R, 0 ≤ x ≤ L}, ds = dx is the element of the arclength of S R , and the overbar stands for the complex conjugate. Let us outline the steps of the further argument.

Step 1: we prove that (2.11) implies

(2.12) w ∈ L 2 (D L ), |w| + |∇w| ≤ ce -γ|y| , γ = const > 0,
if w is outgoing.

Step 2: we prove that if w ∈ L 2 (D L ) solves (2.7)-(2.8), with F = = 0, then w = 0. Then we conclude that (2.9) (and (2.6)) holds, and, therefore, (2.5) holds.

Let us prove (2.12). One has

0 = D LR [ w(∆ + k 2 )w -w(∆ + k 2 ) w] dxdy = - S L ( ww N -w wN ) ds + S R ( ww N -w wN ) ds = S R ( ww N -w wN ) ds, (2.13)
where the Dirichlet condition (2.7) was used, and the integrals over the lines x = 0 and x = L are cancelled due to the qp conditions (2.8): If w is outgoing, then (2.14) implies w j (y) = 0 for j ∈ J, and |w j (y)| ≤ e -γ|y| , γ = const > 0, so (2.12) holds.

Lemma 2.2. Assume that w ∈ L 2 (D L ), w solves (2.7) with = 0 and F = 0, and w satisfies (2.8). Then w = 0.

Proof of Lemma 2.2. If w solves equation (2.7) with = 0 and F = 0, then w = j w j (y)ϕ j (x). Since {ϕ j (x)} is an orthonormal basis and w ∈ L 2 (D L ), it follows that w j (y) = 0 for all j ∈ J, and (2.12) holds. Let us use a version of Ramm's identity ( [START_REF] Ramm | Scattering by obstacles in acoustic waveguides, Spectral and scattering theory[END_REF], p. 92), which is valid for any solution w of equation (1.1) which is outgoing in the sense that

(2.15) w = j c j v j (y)ϕ j (x), c j = const, j ∈ J.
Note, that v j (y) = v j (y) for j ∈ J. The identity is:

(2.16) 0 = (x 2 w,2 w ,j ) ,j + (k 2 |w| 2 x 2 -|∇w| 2 x 2 ) ,2 2 + |∇w| 2 -k 2 |w| 2 2 -|w ,2 | 2 ,
where w ,j : = ∂w/∂x j , j = 1, 2, x 1 = x, x 2 = y, over the repeated indices one sums up, |w| 2 : = w w. The right-hand side of (2.16) equals to 1 2 [x 2 ( w,2j w ,j -w ,2j w,j ) + k 2 x 2 (w ,2 w -w,2 w)] = 0, because w ,2 w = w,2 w for outgoing w. One has (2.17)

|w| + |∇w| ≤ ce -γ|y| , γ = const > 0, c = const > 0.
Let R > max f (x). Integrate (2.16) over D LR : = {(x, y) : (x, y) ∈ D L , y ≤ R} and use Green's formula to get:

0 = -lim R→∞ S L ∪S R [x 2 w,2 w ,j N j + (k 2 |w| 2 x 2 -|∇w| 2 x 2 )N 2 2 ] ds -lim R→∞ D LR |w ,2 | 2 dx 1 dx 2 , (2.18)
where N is the normal pointing into D LR , and we have used the relation

(2.19) lim R→∞ D LR |∇w| 2 dx 1 dx 2 = k 2 lim R→∞ D LR |w| 2 dx 1 dx 2 ,
which follows from the equation ∆w + k 2 w = 0, boundary condition w = 0 on S, quasiperiodicity of w, and from (2.17). We have also used the relation w,2 w ,j N j = x 2 |∇w| 2 N 2 , which follows from the condition u = 0 on S. From (2.18) one gets:

(2.20) lim

R→∞ D LR |w ,2 | 2 dx 1 dx 2 = - 1 2 S L x 2 N 2 |∇w| 2 ds.
Since f (x) is a graph, one has N 2 x 2 ≥ 0, and it follows from (2.20) that w ,2 = 0, so w = const, and const = 0 because w| S = 0. Lemma 2.2 is proved.

Remark 2.3. Condition of the type

(2.21) N 2 x 2 ≥ 0 on S L
was also used in [START_REF] Ramm | Scattering by obstacles in acoustic waveguides, Spectral and scattering theory[END_REF]. The proof of Lemma 2.2 is not valid if the Neumann boundary condition is imposed on S.

Integral equations method

In this Section we present another proof of the existence and uniqueness of the resolvent kernel G. We want to construct a scattering theory quite similar to the one for the exterior of a bounded obstacle [START_REF] Ramm | Scattering by Obstacles[END_REF]. The first step is to construct an analog to the halfspace Dirichlet Green's function. The function g = g(x, ξ, k) can be constructed analytically

(x = (x 1 , x 2 ), ξ = (ξ 1 , ξ 2 )): (3.1) g(x, ξ) = j ϕ j (x 1 )ϕ j (ξ 1 )g j (x 2 , ξ 2 , k), g j := g j (x 2 , ξ 2 , k) = v j (x 2 )ψ j (ξ 2 ), x 2 > ξ 2 v j (ξ 2 )ψ j (x 2 ), x 2 < ξ 2 ψ j = (µ j ) -1 e iµj b sin[µ j (ξ 2 + b)], µ j = [k 2 -λ 2 j ] 1/2 , v j (x 2 ) = e iµj x2
, where

ψ j + (k 2 -λ 2 j )ψ j = 0, ψ j (-b) = 0, W [v j , ψ j ] = 1, λ j = k cos(θ) + 2πj L ,
and W [v, ψ] is the Wronskian. The function g is analytic with respect to k on the complex plain with cuts along the rays λ j -iτ, 0 ≤ τ < ∞, j = 0, ±1, ±2, ..., in particular, in the region k > 0, up to the real positive half-axis except for the set {λ j } j=0,±1,±2,.. . Choose b > 0 such that k 2 > 0 is not an eigenvalue of the problem:

(3.2) (∆ + k 2 )ψ = 0, in D -b : = {(x, y) : -b ≤ y ≤ f (x), 0 ≤ x ≤ L}.
ψ| y=-b = 0, ψ N = 0 on S, ψ(x + L, y) = νψ(x, y), ψ x (x + L, y) = νψ x (x, y). Thus, we have proved the existence and uniqueness of µ, and, therefore, of G, and got a representation formula

(3.3) One has (∆ + k 2 )g = -δ(x -ξ), x = (x 1 , x 2 ), ξ = (ξ 1 , ξ 2 ), x ∈ {(x, y) : -b < y < ∞, 0 ≤ x ≤ L}, (3.4)
(3.9) G = g - S L
gµ ds.

This representation shows that the rate of decay of G as y → ∞ is essentially the same as that of g. The G is analytic with respect to k on the complex plain with cuts along the rays λ jiτ, 0 ≤ τ < ∞, j = 0, ±1, ±2, ..., in particular, in the region k > 0, up to the real positive half-axis except for the set {λ j } j=0,±1,±2,.. . This follows from (3.8), (3.9), and the general result [START_REF] Ramm | Scattering by Obstacles[END_REF], p. 57, [START_REF] Ramm | Singularities of the inverses of Fredholm operators[END_REF], concerning analyticity of the solution to a Fredholm equation with respect to a parameter. Suppose a bounded obstacle D 0 is placed inside D L , u = 0 on S 0 = ∂D 0 , S 0 is a Lipschitz boundary. If qp condition is imposed, then Green's function G 0 in the presence of the obstacle satisfies equations similar to (3.9) and (3.8):

(3.10) G 0 (x, y) = G(x, y) - S0 G(x, s)µ 0 (s, y) ds, µ 0 = G 0N ,
where N is the unit normal to S 0 pointing into D L , and Rayleigh conjectured [START_REF] Rayleigh | On the dynamical theory of gratings[END_REF] hypothesis") that the series (1.7) converges up to the boundary S L . This conjecture is wrong ( [START_REF]Electromagnetic theory of gratings[END_REF]) for some f (x). Since the Rayleigh hypothesis has been widely used for numerical solution of the scattering problem by physicists and engineers, and because these practitioners reported high instability of the numerical solution, and there are no error estimates, we propose a modification of the Rayleigh conjecture, which is a Theorem. This MRC (Modified Rayleigh Conjecture) can be used for a numerical solution of the scattering problem, and it gives an error estimate for this solution. Our arguments are very similar to the ones in [START_REF] Ramm | Modified Rayleigh Conjecture and Applications[END_REF].

(3.11) µ 0 = -A 0 µ 0 + 2 ∂G ∂N on S 0 , A 0 µ 0 : = 2 S0 ∂G(s, σ) ∂N s µ 0 (σ) dσ.
Rewrite the scattering problem (1.1)-(1.4) as

(4.1) (∆ + k 2 )v = 0 in D, v = -u 0 on S L ,
where v satisfies (1.4), and v has representation (1.7), that is, v is "outgoing", it satisfies the radiation condition. Fix an arbitrarily small > 0, and assume that (4.2)

u 0 + |j|≤j( ) c j ( )v j (y)ϕ j (x) ≤ , 0 ≤ x ≤ L, y = f (x),
where

• = • L 2 (S L ) .
Lemma 4.1. For any > 0, however small, and for any u 0 ∈ L 2 (S L ), there exists j( ) and c j ( ) such that (4.2) holds.

Proof. Lemma 4.1 follows from the completeness of the system {ϕ j (x)v j (f (x))} j=0,±1,±2,.... in L 2 (S L ). Let us prove this completeness. Assume that there is an h ∈ L 2 (S L ), h ≡ 0 such that ( Thus ψ = 0 in D L , and

h = ψ + N -ψ - N = 0. Lemma 4.1 is proved. Lemma 4.2. If (4.2) holds, then |v(x) - |j|≤j( ) c j ( )v j (y)ϕ j (x) | ≤ c , ∀x, y ∈ D L , 0 ≤ x ≤ L, y ≥ f (x),
where c = const > 0 does not depend on , x, y, and R; R > M is an arbitrary fixed number, and

|w | = sup x∈D\D LR |w(x)| + ||w|| H 1/2 (D LR ) .
Proof. Let w : = v -|j|≤j( ) c j ( )v j (y)ϕ j (x). Then w solves equation (1.1), w satisfies (1.4), w is outgoing, and w L 2 (S L ) ≤ . One has (cf. (2.4))

(4.5) w(x) = - S L
wG N (x, ξ) ds.

Thus (4.2), i.e. w : = w L 2 (S L ) ≤ , implies (4.6) |w(x)| y=R ≤ w L 2 (S L ) G N (x, ξ) L 2 (S L ) ≤ c , c = const > 0,
where c is independent of , and R > max f (x) is arbitrary. Now let us use the elliptic inequality

(4.7) w H m (D LR ) ≤ c w H m-0.5 (S L ) + w H m-0.5 (S R ) ,
where we have used the equation ∆w + k 2 w = 0, and assumed that k

Numerical solution of the scattering problem

According to the MRC method (Theorem 4.4), if the restriction of the incident field -u 0 (x, y) to S L is approximated as in (4.9), then the series (4.10) approximates the scattered field in the entire region above the profile y = f (x). However, a numerical method that uses (4.9) does not produce satisfactory results as reported in [START_REF]Electromagnetic theory of gratings[END_REF] and elsewhere. Our own numerical experiments confirm this observation. A way to overcome this difficulty is to realize that the numerical approximation of the field -u 0 | S L can be carried out by using outgoing solutions described below.

Let ξ = (ξ 1 , ξ 2 ) ∈ D -b , where b > 0,

D -b : = {(ξ 1 , ξ 2 ) : -b ≤ ξ 2 ≤ f (x), 0 ≤ ξ 1 ≤ L},
and g(x, ξ) be defined as in Section 3. Then g(x, ξ) is an outgoing solution satisfying ∆g + k 2 g = 0 in D L , according to (3.4).

To implement the MRC method numerically one proceeds as follows:

(1) Choose the nodes x i , i = 1, 2, ..., N on the profile S L . These points are used to approximate L 2 norms on S L . (2) Choose points ξ (1) , ξ (2) , ..., ξ (M ) in D -b , M < N .

(3) Form the vectors b = (u 0 (x i )), and a (m) = (g(x i , ξ (m) )), i = 1, 2, ..., N, m = 1, 2, ..., M . Let A be the N × M matrix containing vectors a (m) as its columns. (4) Find the Singular Value Decomposition of A . Use a predetermined w min > 0 to eliminate its small singular values. Use the decomposition to compute

r min = min{ b + Ac , c ∈ C M },
where

a 2 = 1 N N i=1 |a i | 2 .
( (b) If r min > , then increase N, M by the order of 2, readjust the location of points ξ (m) ∈ D -b as needed, and repeat the procedure. We have conducted numerical experiments for four different profiles. In each case we used L = π, k = 1.0 and three values for the angle θ. Table 1 shows the resulting residuals r min . Note that b = 1. Thus, in all the considered cases, the MRC method achieved 0.04% to 2% accuracy of the approximation. Other parameters used in the experiments were chosen as follows: N = 256, M = 64, w min = 10 -8 , b = 1.2. The value of b > 0, used in the definition of g, was chosen experimentally, but the dependency of r min on b was slight. The Singular Value Decomposition (SVD) is used in Step 4 since the vectors a (m) , m = 1, 2, ..., M may be nearly linearly dependent, which leads to an instability in the determination of the minimizer c. According to the SVD method this instability is eliminated by cutting off small singular values of the matrix A, see e.g. [START_REF] Press | Numerical Recepies in FORTRAN[END_REF] for details. The cut-off value w min > 0 was chosen experimentally. We used the truncated series (3.1) with |j| ≤ 120 to compute functions g(x, y, ξ). A typical run time on a 333 MHz PC was about 40s for each experiment.

The following is a description of the profiles y = f (x), the nodes x i ∈ S L , and the poles ξ (m) ∈ D -b used in the computation of g(x i , ξ (m) ) in Step 3. For example, in profile I the x-coordinates of the N nodes x i ∈ S L are uniformly distributed on the interval 0 ≤ x ≤ L. The poles ξ (m) ∈ D -b were chosen as follows: every fourth node x i was moved by a fixed amount -0.1 parallel to the y axis, so it would be within the region D -b . The location of the poles was chosen experimentally to give the smallest value of the residual r min .

Profile I. f (x) = sin(2x) for 0 ≤ x ≤ L, t i = iL/N, x i = (t i , f (t i )), i = 1, 2, ..., N, ξ (m) = (x 4m , y 4m -0.1), m = 1, 2, ..., M .

Profile II. f (x) = sin(0.2x) for 0 ≤ x ≤ L, t i = iL/N, x i = (t i , f (t i )), i = 1, 2, ..., N, ξ (m) = (x 4m , y 4m -0.1), m = 1, 2, ..., M .

Profile III. f (x) = x for 0 ≤ x ≤ L/2, f (x) = L -x for L/2 ≤ x ≤ L, t i = iL/N, x i = (t i , f (t i )), i = 1, 2, ..., N, ξ (m) = (x 4m , y 4m -0.1), m = 1, 2, ..., M .

Profile IV. f (x) = x for 0 ≤ x ≤ L, t i = 2iL/N, x i = (t i , f (t i ), i = 1, ..., N/2, x i = (L, f (2(i -N/2)L/N )), i = N/2 + 1, ..., N, ξ (m) = (x 4m -0.03, y 4m -0.05), m = 1, 2, ..., M . In this profile N/2 nodes x i are uniformly distributed on its slant part, and N/2 nodes are uniformly distributed on its vertical portion x = L.

The experiments show that the MRC method provides a competitive alternative to other methods for the computation of fields scattered from periodic structures. It is fast and inexpensive. The results depend on the number of the internal points ξ (m) and on their location. A similar MRC method for the computation of fields scattered by a bounded obstacle was presented in [START_REF] Gutman | Numerical implementation of the MRC method for obstacle scattering problems[END_REF].

  k 2 )u = 0, x ∈ D, k = const > 0 (1.2) u = 0 on S : = ∂D, (1.3) u = u 0 + v, u 0 : = e ikα•x ,

x=0(

  -ww x + w wx ) dy + x=L ( ww x -w wx ) dy = x=0 (w wx -ww x ) dy -x=0 ν ν(w wx -ww x ) dy = 0. Here we have used the relation ν ν = 1. Thus (2.13) implies (2.14) 0 = S R ( ww y -w wy ) dx, ∀R > M.

  k 2 )G = -δ(x -ξ), G = 0 on S, G satisfies the qp condition and the radiation condition ( it is outgoing at infinity).Multiply(3.4) by G, (3.6) by g, subtract from the second equation the first one, integrate over D LR , and take R → ∞, to get(3.7) G = g + S L (Gg N -G N g)ds = g -S L gµ ds, µ : = G N | S L .The qp condition allows one to cancel the integrals over the lateral boundary (x 0 and x = L), and the radiation condition allows one to have limR→∞ S R (Gg N -G N g)ds = 0This is a Fredholm equation for µ in L 2 (S L ), if S L is C 1,m , m > 0.The homogeneous equation (3.8) has only the trivial solution: if µ + Aµ = 0, then the function ψ : = S L gµ ds satisfies ψ + N | S L = 0, where + N (ψ - N ) is the normal derivative of ψ from D -b (D L ), and we use the known formula for the normal derivative of the single layer potential at the boundary. The ψ satisfies also (3.2) and (3.3), and, by the choice of b, one has ψ = 0 in D -b . Also ψ = 0 in D L , because (∆ + k 2 )ψ = 0 in D L , ψ| S L = 0 (by the continuity of the single layer potential), ψ satisfies the qp condition (because g satisfies it), and ψ is outgoing (because g is). Since ψ = 0 in D -b and in D L , one concludes that µ = ψ + N -ψ - N , where ψ + N (ψ - N ) is the normal derivative of ψ from D -b (D L ), and we use the jump relation for the normal derivative of the single layer potential.

S

  L hϕ j (x)v j (f (x)) ds = 0 for any j. From (4.3) one derives (cf. [17], p.162-163) (4.4) ψ(x) : = S L hg(x, ξ)dξ = 0, x ∈ D -b .

Remark 4 . 3 .

 43 In (4.7) there are no terms with boundary norms over the lateral boundary (lines x = 0 and x = L) because of the quasiperiodicity condition. From Lemma 4.2 the basic result, Theorem 4.4, follows immediately: Theorem 4.4. MRC-Modified Rayleigh Conjecture. Fix > 0, however small, and choose a positive integer p. Find (4.9) min cj u 0 + |j|≤p c j ϕ j (x)v j (y) : = m(p).Let {c j (p)} be the minimizer of (4.9). If m(p) ≤ , then(4.10) v(p) = |j|≤p c j (p)ϕ j (x)v j (y)satisfies the inequality (4.11) |v -v(p) | ≤ c , where c = const > 0 does not depend on . If m(p) > , then there exists j = j( ) > p such that m(j( )) < . Denote c j (j( )) : = c j ( ) and v(j( )) : = v . Then(4.12) |v -v | ≤ c .

)

  Stopping criterion. Let > 0. (a) r min ≤ , then stop. Use the coefficients c = {c 1 , c 2 , ..., c M } obtained in the above minimization step to compute the scattered field by v(x, y) = M m=1 c m g(x, y, ξ (m) ).

  This is a Fredholm equation (with index zero). Ifk 2 is not an eigenvalue of the Neumann Laplacian in D 0 (=not exceptional), then equation (3.11) is uniqueley solvable and, by (3.10), G 0 exists and is unique for this k > 0. It is not known what are nontrivial sufficient

conditions for k > 0 to be not exceptional. The exceptional k form a discrete countable set on the positive semi-axis k > 0. If the Neumann boundary condition is imposed on S L , then, even in the absence of the obstacle D 0 , it is not known if LAP holds, because the proof of Lemma 2.2 is not valid for the Neumann boundary condition on S L .

4. Modified Rayleigh Conjecture (MRC)

  2 is not a Dirichlet eigenvalue of the Laplacian in D LR , which can be done without loss of generality, because one can vary R. The integer m 0 is arbitrary if S L is sufficiently smooth, and m ≤ 1 if S L is Lipschitz. Taking m = 0.5 and using (4.2) and (4.6) one gets(4.8) wH 1/2 (D LR ) ≤ c .Thus, in a neighborhood of S L , we have proved estimate (4.8), and in a complement of this neighborhood in D L we have proved estimate (4.6). Lemma 4.2 is proved.

Table 1 .

 1 Residuals attained in the numerical

	Profile	θ	r min
	I	π/4 0.000424
		π/3 0.000407
		π/2 0.000371
	II	π/4 0.001491
		π/3 0.001815
		π/2 0.002089
	III	π/4 0.009623
		π/3 0.011903
		π/2 0.013828
	IV	π/4 0.014398
		π/3 0.017648
		π/2 0.020451